**Section 2**

**The Performance of Metamaterials** 

168 Metamaterial

Zhang, S.; Yin, L. & Fang, N. (2009). Focusing Ultrasound with an Acoustic Metamaterial

Zhou, X. & Hu, G. (2009). Analytic Model of Elastic Metamaterials with Local Resonances, *Physical Review B*, Vol.75, No.19, (May 2009), pp.195109, ISSN 1098-0121

9007

Network, *Physical Review Letters,* Vol.102, No.19, (May 2009), pp. 194301, ISSN 0031-

**0**

**7**

**Resonant Negative Refractive**

Negative refractive index (NRI) media are extensively studied nowadays. The interest in these materials keeps on increasing since the year 2000 when a team at the university of California in San Diego (UCSD) published an experimental demonstration of the existence of a material presenting both a negative permittivity and negative permeability (Shelby et al. (2001); Smith, Padilla, Vier, Nemat-Nasser & Schultz (2000)). They also showed that it is necessary to attribute a negative refractive index to such media (Smith & Kroll (2000)). Novel physical phenomena such as the inversion of Doppler's effect, Cerenkov effect and focusing using flat slabs are then predicted based on the theoretical publication V. G. Veselago dating back to

Though different terminologies are used for these media (the actual terms are "left handed media", "double negative metamaterial", "negative refractive index metamaterial"), the concept of backward wave propagation (wave with a phase velocity propagating in the opposite direction with respect to the propagation of energy) dates back to at least 1904 (Moroz (n.d.); Tretyakov (2005)). Indeed, H. Lamb has studied this concept for mechanical systems and A. Schuster in the field of electromagnetism. Independently, H. C. Pocklington (Pocklington (1905)) demonstrated theoretically that in a media supporting backward wave propagation, the phase velocity can be directed in the direction of the source, in the inverse direction of the group velocity. Forty years later (in 1944), L. I. Mandelshtam studied the properties of NRI media (Mandel'shtam (1944)) and more than twenty years later, V. G. Veselago published an exhaustive study on NRI media. The interest in these media then

The actual revival of interest for these media can certainly be explained published (Smith, Padilla, Vier, Nemat-Nasser & Schultz (2000)). This demonstration has been performed at microwaves by assembling a medium of periodic metallic wires (for negative permittivity) (Pendry et al. (1996)) and a medium of split ring resonators presenting a negative permeability (Pendry et al. (1999)). These media can be assimilated to a crystalline structure

Different technological solutions have been proposed to synthesize negative refractive index media, such as the use of backward wave transmission lines (Eleftheriades et al. (2003); Lai et al. (2004)) and photonic crystals in negative phase velocity regime (Gadot et al. (2003); Gralak et al. (2000); Qiu et al. (2003)). The approach to be chosen depends mainly on the

**1. Introduction**

1967 (Veselago (1968)).

decreased up to the year 2000.

of artificial molecules hence the term metamaterial.

**Index Metamaterials**

Divitha Seetharamdoo

*IFSTTAR, LEOST*

*France*

### **Resonant Negative Refractive Index Metamaterials**

Divitha Seetharamdoo *IFSTTAR, LEOST France*

#### **1. Introduction**

Negative refractive index (NRI) media are extensively studied nowadays. The interest in these materials keeps on increasing since the year 2000 when a team at the university of California in San Diego (UCSD) published an experimental demonstration of the existence of a material presenting both a negative permittivity and negative permeability (Shelby et al. (2001); Smith, Padilla, Vier, Nemat-Nasser & Schultz (2000)). They also showed that it is necessary to attribute a negative refractive index to such media (Smith & Kroll (2000)). Novel physical phenomena such as the inversion of Doppler's effect, Cerenkov effect and focusing using flat slabs are then predicted based on the theoretical publication V. G. Veselago dating back to 1967 (Veselago (1968)).

Though different terminologies are used for these media (the actual terms are "left handed media", "double negative metamaterial", "negative refractive index metamaterial"), the concept of backward wave propagation (wave with a phase velocity propagating in the opposite direction with respect to the propagation of energy) dates back to at least 1904 (Moroz (n.d.); Tretyakov (2005)). Indeed, H. Lamb has studied this concept for mechanical systems and A. Schuster in the field of electromagnetism. Independently, H. C. Pocklington (Pocklington (1905)) demonstrated theoretically that in a media supporting backward wave propagation, the phase velocity can be directed in the direction of the source, in the inverse direction of the group velocity. Forty years later (in 1944), L. I. Mandelshtam studied the properties of NRI media (Mandel'shtam (1944)) and more than twenty years later, V. G. Veselago published an exhaustive study on NRI media. The interest in these media then decreased up to the year 2000.

The actual revival of interest for these media can certainly be explained published (Smith, Padilla, Vier, Nemat-Nasser & Schultz (2000)). This demonstration has been performed at microwaves by assembling a medium of periodic metallic wires (for negative permittivity) (Pendry et al. (1996)) and a medium of split ring resonators presenting a negative permeability (Pendry et al. (1999)). These media can be assimilated to a crystalline structure of artificial molecules hence the term metamaterial.

Different technological solutions have been proposed to synthesize negative refractive index media, such as the use of backward wave transmission lines (Eleftheriades et al. (2003); Lai et al. (2004)) and photonic crystals in negative phase velocity regime (Gadot et al. (2003); Gralak et al. (2000); Qiu et al. (2003)). The approach to be chosen depends mainly on the

Fig. 1. Surface current *J*<sup>0</sup> in *x* = *x*<sup>0</sup> radiating in the medium (*ε*(*ω*) < 0, *μ*(*ω*) < 0). The

Resonant Negative Refractive Index Metamaterials 173

<sup>2</sup>*k*0*<sup>n</sup>* <sup>=</sup> <sup>−</sup>*i*0*η*<sup>0</sup>

However, if the power *P* delivered by the current �*J*<sup>0</sup> to the volume *V* (Balanis (1989)) is

This equation represents the work done by the source and it must be positive, which implies that *P* > 0 (Balanis (1989)). The ratio *μr*/*n* must also be positive. *If μ<sup>r</sup> is negative, then n must also be negative*. An equivalent demonstration can be done for *εr*. For a propagative medium,

To determine the constraints with respect to the sign choice of the imaginary part of the

2 *μr*

> 2 <sup>0</sup>*η*<sup>0</sup> 2 *μr*

�<sup>0</sup> ·�*rn*��) exp[*j*(*ω<sup>t</sup>* <sup>−</sup> *<sup>k</sup>*

<sup>0</sup> is the free space wave vector and �*uE* is the unit vector along the direction of the

*E*. If a stable propagation is to be ensured, the magnitude of Re[�

�

*<sup>n</sup>* , (5)

*<sup>n</sup>* . (7)

*<sup>n</sup>* exp(*jk* <sup>|</sup>*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0|). (6)

*E*(�*r*, *ω*) in a medium with *n* = *n*� − *jn*�� for a

)]�*uE* 

, (8)

*E*(�*r*, *t*)] must

� <sup>0</sup> ·�*rn*�

<sup>0</sup> ·�*rn*�� must be positive and:

and substitute the expressions (4) and (3) in equation (2). *α* is then given by:

*<sup>E</sup>*(*x*) = <sup>−</sup>*i*0*η*<sup>0</sup>

*<sup>P</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> 2 *V* � *<sup>E</sup>* ·�*<sup>J</sup>* ∗ <sup>0</sup> *dV* <sup>=</sup> *<sup>i</sup>*

*<sup>α</sup>* <sup>=</sup> <sup>−</sup> *μωi*<sup>0</sup>

2 *μr*

the solution retained for the wave equation verifies backward wave propagation.

current distribution is considered uniform and infinite in *y*ˆ et *z*ˆ.

and the wave equation becomes:

calculated, the fol. equation is obtained:

refractive index, let us consider the electric field �

*<sup>E</sup>*(�*r*, *<sup>t</sup>*) = Re

decrease with time. This implies that the term *k*

 � *E*(�*r*) exp(−*<sup>k</sup>*

time dependence in exp(*jωt*):

where *k* �

E-field vector �

�

applications and the frequency band of interest. All the approaches have the same aim i.e to synthesize NRI metamaterials for the potential applications and considering the industrial and economic potential of such materials.

The synthesis and study of NRI metamaterials is however difficult because of its heterogeneous characteristics. For an easier study of applications of metamaterials in microwave frequency range, homogenization and macroscopic description of these metamaterials can prove to be very helpful. It can specially allow a higher degree of freedom to overcome the fundamental limitations imposed by natural materials on the performances of microwave devices.

In this chapter, the description of NRI resonant metamaterials in terms of a continuous medium will be analyzed. The electrodynamics of NRI materials will first be described. The effective medium theory as applied to NRI resonant metamaterials as well as the calculation methods will then be detailed. Finally numerical results will be presented together with a thorough analysis and interpretation of the effective parameters calculated with respect to the electrodynamics of negative refractive index materials presented for continuous media.

#### **2. Electrodynamics of negative refractive index materials**

#### **2.1 Adequate choice of the sign of the refractive index and wave impedance**

For backward wave propagation, an adequate choice of the sign of the refractive index *n*(*ω*) given by (1) is necessary.

$$
\pi(\omega) = \pm \sqrt{\varepsilon(\omega)\mu(\omega)}\tag{1}
$$

where *ε*(*ω*) and *μ*(*ω*) represent the effective permittivity and permeability respectively.

#### **2.1.1 Refractive index**

The determination of the sign in front of the square root of (1) is done thanks to causal properties which the solutions of wave propagation should respect and energy conservation principles. The choice of this sign allows to define, among other parameters, the direction of the outgoing wave with respect to an interface between a NRI and a conventional material.

To demonstrate that for a material with (*ε*(*ω*) < 0, *μ*(*ω*) < 0), the sign of the refractive index should be negative, let us consider a current surface in *x* = *x*<sup>0</sup> (Smith (2000)). The radiation of this surface current in the medium (*ε*(*ω*) < 0, *μ*(*ω*) < 0) is then studied as shown on figure 1.

The wave equation in the medium can be written as fol.:

$$\frac{\partial^2}{\partial \mathbf{x}^2} E(\mathbf{x}) + k^2 E(\mathbf{x}) = -j\omega\mu I\_0(z),\tag{2}$$

where *<sup>E</sup>*(*x*) is the electric fied component along *<sup>x</sup>*ˆ, �*J*<sup>0</sup> <sup>=</sup> *<sup>i</sup>*0*δ*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0) *<sup>z</sup>*<sup>ˆ</sup> et *<sup>μ</sup>* <sup>=</sup> *<sup>μ</sup>*0*μr*. The solution of this equation is given by:

$$E(\mathbf{x}) = \mathfrak{a} \exp(jk \, |\mathbf{x} - \mathbf{x}\_0|). \tag{3}$$

To determine *α*, let us first calculate :

$$\frac{\partial^2 E(\mathbf{x})}{\partial \mathbf{x}^2} = -ak^2 \exp(jk(|\mathbf{x} - \mathbf{x}\_0|) + 2jak\delta(\mathbf{x} - \mathbf{x}\_0))\tag{4}$$

2 Will-be-set-by-IN-TECH

applications and the frequency band of interest. All the approaches have the same aim i.e to synthesize NRI metamaterials for the potential applications and considering the industrial

The synthesis and study of NRI metamaterials is however difficult because of its heterogeneous characteristics. For an easier study of applications of metamaterials in microwave frequency range, homogenization and macroscopic description of these metamaterials can prove to be very helpful. It can specially allow a higher degree of freedom to overcome the fundamental limitations imposed by natural materials on the performances

In this chapter, the description of NRI resonant metamaterials in terms of a continuous medium will be analyzed. The electrodynamics of NRI materials will first be described. The effective medium theory as applied to NRI resonant metamaterials as well as the calculation methods will then be detailed. Finally numerical results will be presented together with a thorough analysis and interpretation of the effective parameters calculated with respect to the electrodynamics of negative refractive index materials presented for continuous media.

For backward wave propagation, an adequate choice of the sign of the refractive index *n*(*ω*)

The determination of the sign in front of the square root of (1) is done thanks to causal properties which the solutions of wave propagation should respect and energy conservation principles. The choice of this sign allows to define, among other parameters, the direction of the outgoing wave with respect to an interface between a NRI and a conventional material. To demonstrate that for a material with (*ε*(*ω*) < 0, *μ*(*ω*) < 0), the sign of the refractive index should be negative, let us consider a current surface in *x* = *x*<sup>0</sup> (Smith (2000)). The radiation of this surface current in the medium (*ε*(*ω*) < 0, *μ*(*ω*) < 0) is then studied as shown on figure 1.

where *<sup>E</sup>*(*x*) is the electric fied component along *<sup>x</sup>*ˆ, �*J*<sup>0</sup> <sup>=</sup> *<sup>i</sup>*0*δ*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0) *<sup>z</sup>*<sup>ˆ</sup> et *<sup>μ</sup>* <sup>=</sup> *<sup>μ</sup>*0*μr*. The solution

where *ε*(*ω*) and *μ*(*ω*) represent the effective permittivity and permeability respectively.

*ε*(*ω*)*μ*(*ω*), (1)

*<sup>∂</sup>x*<sup>2</sup> *<sup>E</sup>*(*x*) + *<sup>k</sup>*2*E*(*x*) = <sup>−</sup>*jωμJ*0(*z*), (2)

*E*(*x*) = *α* exp(*jk* |*x* − *x*0|). (3)

*<sup>∂</sup>x*<sup>2</sup> <sup>=</sup> <sup>−</sup>*αk*<sup>2</sup> exp(*jk* <sup>|</sup>*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0|) + <sup>2</sup>*jαkδ*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*0), (4)

**2. Electrodynamics of negative refractive index materials**

The wave equation in the medium can be written as fol.:

*∂*2*E*(*x*)

*∂*2

**2.1 Adequate choice of the sign of the refractive index and wave impedance**

*n*(*ω*) = ±

and economic potential of such materials.

of microwave devices.

given by (1) is necessary.

**2.1.1 Refractive index**

of this equation is given by:

To determine *α*, let us first calculate :

Fig. 1. Surface current *J*<sup>0</sup> in *x* = *x*<sup>0</sup> radiating in the medium (*ε*(*ω*) < 0, *μ*(*ω*) < 0). The current distribution is considered uniform and infinite in *y*ˆ et *z*ˆ.

and substitute the expressions (4) and (3) in equation (2). *α* is then given by:

$$\alpha = -\frac{\mu \omega i\_0}{2k\_0 n} = -\frac{i\_0 \eta\_0}{2} \frac{\mu\_r}{n} \,\prime \tag{5}$$

and the wave equation becomes:

$$E(\mathbf{x}) = -\frac{i\_0 \eta\_0}{2} \frac{\mu\_r}{n} \exp(jk \left| \mathbf{x} - \mathbf{x}\_0 \right|). \tag{6}$$

However, if the power *P* delivered by the current �*J*<sup>0</sup> to the volume *V* (Balanis (1989)) is calculated, the fol. equation is obtained:

$$P = -\frac{1}{2} \int\_{V} \vec{E} \cdot \vec{J}\_0^\* dV = \frac{i\_0^2 \eta\_0}{2} \frac{\mu\_r}{n}. \tag{7}$$

This equation represents the work done by the source and it must be positive, which implies that *P* > 0 (Balanis (1989)). The ratio *μr*/*n* must also be positive. *If μ<sup>r</sup> is negative, then n must also be negative*. An equivalent demonstration can be done for *εr*. For a propagative medium, the solution retained for the wave equation verifies backward wave propagation.

To determine the constraints with respect to the sign choice of the imaginary part of the refractive index, let us consider the electric field � *E*(�*r*, *ω*) in a medium with *n* = *n*� − *jn*�� for a time dependence in exp(*jωt*):

$$\vec{E}(\vec{r},t) = \text{Re}\left[ \left| \vec{E}(\vec{r}) \right| \exp(-\vec{k\_0} \cdot \vec{r} n'') \exp[j(\omega t - \vec{k\_0} \cdot \vec{r} n')] \vec{u}\_E \right],\tag{8}$$

where *k* � <sup>0</sup> is the free space wave vector and �*uE* is the unit vector along the direction of the E-field vector � *E*. If a stable propagation is to be ensured, the magnitude of Re[� *E*(�*r*, *t*)] must decrease with time. This implies that the term *k* � <sup>0</sup> ·�*rn*�� must be positive and:

The mean of Poynting's vector �

2 Re[�

for both conventional and NRI materials3.

 � *E*(�*r*, *ω*) 

*Sav*(�*r*, *<sup>ω</sup>*) = <sup>1</sup>

*E*(�*r*, *ω*) =

*Sav*(�*r*).

medium implies that �

�

where �

vector of �

component (�

permittivity *ε*(*ω*) = *ε*�

vector �

*Sav* is calculated, :

2 � *E*(�*r*, *ω*) *H*� (�*r*, *ω*) 

Resonant Negative Refractive Index Metamaterials 175

Knowing that passivity of a medium implies that the energy flux must be directed inside the

medium irrespective of the sign of their refractive index(as it can be verified on figure 2). If

There is no particular sign restriction on the imaginary part of the wave impedance. The complex wave impedance provides information non only on wave propagation (as described above) but it also allows physical understanding when there is no wave propagation in a medium (*i.e.* when its imaginary part is much higher that its real part)as to which field

of artificial magnetic medium such as those based on split-ring resonators. Indeed, if the imaginary part of *Z* is negative, the medium can be said to be capacitive and there is no wave propagation because of H-field filtering. The response of the medium to an applied magnetic field is thus non-negligible and it can be considered as an artificial magnetic medium.

There are fundamental restrictions limiting the signs that the imaginary part of *ε*(*ω*) et *μ*(*ω*) can admit for linear, passive, isotropic homogeneous medium.For conventional material, these restrictions are derived from fundamental theorems of macroscopic electrodynamics (Depine & Lakhtakia (2004); Efros (2004)) and it has been demonstrated in various ways, namely by Callen *et al.* thanks to the fluctuation-dissipation theorems (Callen & Welton (1951)) for arbitrary linear and dissipative systems, and by Landau *et al*. (Landau et al. (1984)) for electromagnetic waves. Based on this last demonstration, we propose to

Let us consider a passive, linear, homogeneous, isotropic and dispersive medium of

*S*(�*r*, *t*) provides the definition of the power flux density in a medium with variable

*∂t* �

(*ω*) − *jε*��(*ω*) and permeability *μ*(*ω*) = *μ*�

Using Maxwell-Faraday et Maxwell-Ampere equations in the absence of sources,

*<sup>E</sup>*(�*r*, *<sup>t</sup>*) = <sup>−</sup> *<sup>∂</sup>*

**2.2 Adequate choice of the sign of the effective permittivity and permeability**

demonstrate the extension of these limitations for NRI materials.

fields. It can be written in time-domain for dispersive medium as:

� *S*(�*r*, *t*) = �

∇ × �

<sup>3</sup> This restriction is identical in both conventions exp(−*jωt*) and exp(*jωt*)

*E* or *H*� fields) is canceled. This information is indeed interesting for the design

 *H*� (�*r*, *ω*) 

*Sav*(�*r*, *ω*) > 0. The term cos(*ϕ<sup>H</sup>* − *ϕE*) is thus always positive for all

Re[*Z*(*ω*)] > 0, (12)

cos(*ϕ<sup>H</sup>* − *ϕE*)�*uS*, (11)

exp(−*jϕH*) and �*uS* is the unit

(*ω*) − *jμ*��(*ω*). The Poynting

*<sup>E</sup>*(�*r*, *<sup>t</sup>*) <sup>×</sup> *<sup>H</sup>*� (�*r*, *<sup>t</sup>*) (13)

*B*(�*r*, *t*), (14)

*<sup>E</sup>*(�*r*, *<sup>ω</sup>*) <sup>×</sup> *<sup>H</sup>*�∗(�*r*, *<sup>ω</sup>*)] = <sup>1</sup>

we apply this restricion to equation (10), the fol. condition is obtained:

exp(−*jϕE*), *<sup>H</sup>*� (�*r*, *<sup>ω</sup>*) =

$$n'' > 0,\tag{9}$$

irrespective of the sign of *n*� <sup>1</sup>

#### **2.1.2 Wave impedance**

Impedance is a concept generally applied to circuits but is also extended to electromagnetic wave propagation. This extension was developped by Schelkunoff and the analogy between the impedance of a medium for wave propagation and the impedance of a transmission line is fully described in (Stratton (1941)). The physical interpretation of the wave impedance *Z* given here is based on this analogy.

The complex wave impedance of a medium is strongly related to the flux of energy of the wave propagating in the medium. This is why there are fundamental limitations to the values that *Z* can admit; one of the limitations is directly linked to the passivity of the medium. These limitations apply to both positive and negative refractive index medium.

The passivity or absence of activity within a medium implies that for a plane progressive electromagnetic wave, the mean energy flux must be directed inside the medium in which the wave propagates (Wohlers (1971)). The directions of the vectors (� *E*, *H*� , �*k*) and the energy flux � *S* for a plane progressive wave at the interface between a conventional material and an NRI is shown in figure 2.

Fig. 2. Direction of the field vectors (� *E*, *H*� ,�*k*) et � *S* for the interaction of a plane wave with at the interface of (a) two conventional material with positive refractive index, and (b) a conventional material and a negative refractive index material.

The wave impedance<sup>2</sup> is defined as the ration of the electric field to the magnetic field component in the propagation plane; the real part is thus given by:

$$\operatorname{Re}[Z(\omega)] = \operatorname{Re}\left[\frac{\bar{E}(\omega)}{\bar{H}(\omega)}\right] = \frac{|\bar{E}(\omega)|}{|\bar{H}(\omega)|}\cos(\varphi\_H - \varphi\_E),\tag{10}$$

where *<sup>E</sup>*¯(*ω*) = <sup>|</sup>*E*¯(*ω*)<sup>|</sup> exp(−*jϕE*) et *<sup>H</sup>*¯(*ω*) = <sup>|</sup>*H*¯ (*ω*)<sup>|</sup> exp(−*jϕH*). Equation (10) is verified for both positive and negative refractive indexes. The sign of Re[*Z*(*ω*)] depends only on the term cos(*ϕ<sup>H</sup>* − *ϕE*).

<sup>1</sup> It can be shown that for the convention exp(−*jωt*), *<sup>n</sup>*�� is positive also but in this case *<sup>n</sup>* should be written as *n* = *n*� + *jn*��. This is quite similar for the wave impedance, the permittivity and permeability.

<sup>2</sup> The wave impedance is generally defined for a *single* plane wave and in the case of a guided or periodic structure, monomodal wave propagation is assumed and an impedance is assigned to each mode.

4 Will-be-set-by-IN-TECH

Impedance is a concept generally applied to circuits but is also extended to electromagnetic wave propagation. This extension was developped by Schelkunoff and the analogy between the impedance of a medium for wave propagation and the impedance of a transmission line is fully described in (Stratton (1941)). The physical interpretation of the wave impedance *Z*

The complex wave impedance of a medium is strongly related to the flux of energy of the wave propagating in the medium. This is why there are fundamental limitations to the values that *Z* can admit; one of the limitations is directly linked to the passivity of the medium. These

The passivity or absence of activity within a medium implies that for a plane progressive electromagnetic wave, the mean energy flux must be directed inside the medium in which the

*S* for a plane progressive wave at the interface between a conventional material and an NRI is

(a) (b)

The wave impedance<sup>2</sup> is defined as the ration of the electric field to the magnetic field

where *<sup>E</sup>*¯(*ω*) = <sup>|</sup>*E*¯(*ω*)<sup>|</sup> exp(−*jϕE*) et *<sup>H</sup>*¯(*ω*) = <sup>|</sup>*H*¯ (*ω*)<sup>|</sup> exp(−*jϕH*). Equation (10) is verified for both positive and negative refractive indexes. The sign of Re[*Z*(*ω*)] depends only on the term

<sup>1</sup> It can be shown that for the convention exp(−*jωt*), *<sup>n</sup>*�� is positive also but in this case *<sup>n</sup>* should be written as *n* = *n*� + *jn*��. This is quite similar for the wave impedance, the permittivity and permeability. <sup>2</sup> The wave impedance is generally defined for a *single* plane wave and in the case of a guided or periodic structure, monomodal wave propagation is assumed and an impedance is assigned to each mode.

<sup>=</sup> <sup>|</sup>*E*¯(*ω*)<sup>|</sup> <sup>|</sup>*H*¯ (*ω*)<sup>|</sup>

*E*, *H*� ,�*k*) et �

the interface of (a) two conventional material with positive refractive index, and (b) a

 *E*¯(*ω*) *H*¯ (*ω*)

limitations apply to both positive and negative refractive index medium.

wave propagates (Wohlers (1971)). The directions of the vectors (�

conventional material and a negative refractive index material.

Re[*Z*(*ω*)] = Re

component in the propagation plane; the real part is thus given by:

irrespective of the sign of *n*� <sup>1</sup>

given here is based on this analogy.

Fig. 2. Direction of the field vectors (�

**2.1.2 Wave impedance**

�

shown in figure 2.

cos(*ϕ<sup>H</sup>* − *ϕE*).

*n*�� > 0, (9)

*E*, *H*� ,

*S* for the interaction of a plane wave with at

cos(*ϕ<sup>H</sup>* − *ϕE*), (10)

�*k*) and the energy flux

The mean of Poynting's vector � *Sav* is calculated, :

$$\vec{S}\_{\text{av}}(\vec{r},\omega) = \frac{1}{2} \text{Re}[\vec{E}(\vec{r},\omega) \times \vec{H}^\*(\vec{r},\omega)] = \frac{1}{2} \left|\vec{E}(\vec{r},\omega)\right| \left|\vec{H}(\vec{r},\omega)\right| \cos(\varphi\_H - \varphi\_E) \vec{u}\_{\text{S}\prime} \tag{11}$$

where � *E*(�*r*, *ω*) = � *E*(�*r*, *ω*) exp(−*jϕE*), *<sup>H</sup>*� (�*r*, *<sup>ω</sup>*) = *H*� (�*r*, *ω*) exp(−*jϕH*) and �*uS* is the unit vector of � *Sav*(�*r*).

Knowing that passivity of a medium implies that the energy flux must be directed inside the medium implies that � *Sav*(�*r*, *ω*) > 0. The term cos(*ϕ<sup>H</sup>* − *ϕE*) is thus always positive for all medium irrespective of the sign of their refractive index(as it can be verified on figure 2). If we apply this restricion to equation (10), the fol. condition is obtained:

$$\operatorname{Re}[Z(\omega)] > 0,\tag{12}$$

for both conventional and NRI materials3.

There is no particular sign restriction on the imaginary part of the wave impedance. The complex wave impedance provides information non only on wave propagation (as described above) but it also allows physical understanding when there is no wave propagation in a medium (*i.e.* when its imaginary part is much higher that its real part)as to which field component (� *E* or *H*� fields) is canceled. This information is indeed interesting for the design of artificial magnetic medium such as those based on split-ring resonators. Indeed, if the imaginary part of *Z* is negative, the medium can be said to be capacitive and there is no wave propagation because of H-field filtering. The response of the medium to an applied magnetic field is thus non-negligible and it can be considered as an artificial magnetic medium.

#### **2.2 Adequate choice of the sign of the effective permittivity and permeability**

There are fundamental restrictions limiting the signs that the imaginary part of *ε*(*ω*) et *μ*(*ω*) can admit for linear, passive, isotropic homogeneous medium.For conventional material, these restrictions are derived from fundamental theorems of macroscopic electrodynamics (Depine & Lakhtakia (2004); Efros (2004)) and it has been demonstrated in various ways, namely by Callen *et al.* thanks to the fluctuation-dissipation theorems (Callen & Welton (1951)) for arbitrary linear and dissipative systems, and by Landau *et al*. (Landau et al. (1984)) for electromagnetic waves. Based on this last demonstration, we propose to demonstrate the extension of these limitations for NRI materials.

Let us consider a passive, linear, homogeneous, isotropic and dispersive medium of permittivity *ε*(*ω*) = *ε*� (*ω*) − *jε*��(*ω*) and permeability *μ*(*ω*) = *μ*� (*ω*) − *jμ*��(*ω*). The Poynting vector � *S*(�*r*, *t*) provides the definition of the power flux density in a medium with variable fields. It can be written in time-domain for dispersive medium as:

$$
\vec{S}(\vec{r},t) = \vec{E}(\vec{r},t) \times \vec{H}(\vec{r},t) \tag{13}
$$

Using Maxwell-Faraday et Maxwell-Ampere equations in the absence of sources,

$$\nabla \times \vec{E}(\vec{r}, t) = -\frac{\partial}{\partial t} \vec{B}(\vec{r}, t), \tag{14}$$

<sup>3</sup> This restriction is identical in both conventions exp(−*jωt*) and exp(*jωt*)

canceled because the integrand of equation (20) is an odd function of *ω* 4. The two terms of the right-hand side of Eq. (22) represent respectively the dielectric and magnetic losses. The second law of thermodynamics, stating that the entropy of a isolated macroscopic system never decreases, imposes that *Q* > 0. It is thus necessary according to equation (22) that:

Resonant Negative Refractive Index Metamaterials 177

+ *μ*��(*ω*)

for positive frequencies (*ω* > 0). The laws of Thermodynamics also express the irreversible nature of physical processes and the fundamental difference between two types of energy: work and heat (Yavorski & Detlaf (1975)). The energy dissipated by fields into heat is irreversible. In other terms, there can be no exchange between the work done by either the

This demonstration can be very easily extended to NRI materials. Indeed, the starting point of the demonstration is energy conservation through the expression of the divergence of Poynting vector [Eq. (16)] written thanks to Maxwell-Ampère [Eq. (15)] and Maxwell-Faraday [Eq. (14)] equations as well Poynting theorem [Eq. (13)]. For a NRI material, these equations and theorems are valid (Veselago (1968)). Indeed, only the direction of the vector �*k* changes

for conventional materials). This is easily verified for a monochromatic wave but can actually

Let us consider for instance, the mean of Poynting vector for a dispersive material excited by the superposition of two monochromatic waves of angular frequency *ω*<sup>1</sup> et *ω*<sup>2</sup> such that

�*k*(*ω*1)

and for any propagative medium, *i.e.* when �*k* is real, the ratio *k*/*μ* is positive. As shown

The concept of effective medium for the description of heterogeneous systems by a homogeneous one is very attractive in different field of physics. Homogenization procedures allowing the definition of an effective macroscopic response from physical parameters characterizing the heterogeneous system are generally developed. In our case, from the microscopic parameters (geometrical and physical definitions) of the metamaterial, a macroscopic electromagnetic response can be obtained. If this macroscopic definition

<sup>4</sup> Principle of causality imposes that (Good & Nelson (1971)): *<sup>ε</sup>*(−*ω*) = *<sup>ε</sup>*(*ω*)<sup>∗</sup> and *<sup>μ</sup>*(−*ω*) = *<sup>μ</sup>*(*ω*)∗.

*<sup>ω</sup>*1*μ*(*ω*1) <sup>+</sup> �*k*(*ω*2)

*ω*2*μ*(*ω*2)

*S* is independent of the sign of the refractive index

*S* and that of the wave vector�*k* is clearly

2 2

 *H*� (�*r*, *ω*) 2

*E*(�*r*, *ω*) or magnetic field *H*� (�*r*, *ω*)] and the heat dissipated by the other, implying:

*ε*(*ω*)�� > 0 and *μ*(*ω*)�� > 0 (24)

> 0 (23)

*<sup>S</sup>* ·�*<sup>k</sup>* (This product is positive

(25)

*ε* ��(*ω*) � *E*(�*r*, *ω*) 2

in a NRI material thus giving a negative value for the product �

*<sup>S</sup>*(�*r*, *<sup>t</sup>*) <sup>&</sup>gt;<sup>=</sup> <sup>|</sup>*E*<sup>|</sup>

also be verified for non-monochromatic wave.

< �

The relation between the direction of Poynting vector �

before, if *k* takes negative values, then *μ* will be negative too.

**3. Effective parameters of resonant NRI metamaterials**

**3.1 Effective medium theory as applied to metamaterials**

shown by this equation. The direction of �

*ω*<sup>1</sup> �= *ω*<sup>2</sup> (Pacheco-Jr et al. (2002)):

electric field [�

and

$$
\nabla \times \vec{H}(\vec{r}, t) = \frac{\partial}{\partial t} \vec{D}(\vec{r}, t),
\tag{15}
$$

the divergence of Poynting vector is given as:

$$-\nabla \cdot \vec{S}(\vec{r},t) = \vec{E}(\vec{r},t) \cdot \frac{\partial}{\partial t} \vec{D}(\vec{r},t) + \vec{H}(\vec{r},t) \cdot \frac{\partial}{\partial t} \vec{B}(\vec{r},t). \tag{16}$$

This equation provides an expression of the energy conservation in a dispersive material in time-domain (Balanis (1989)). In frequency domain, the electric and magnetic fields are given, after Fourier transform, by:

$$\vec{E}(\vec{r},t) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} \vec{E}(\vec{r},\omega) \exp(j\omega t) \mathrm{d}\omega \,\tag{17}$$

$$\frac{\partial}{\partial t}\vec{D}(\vec{r},t) = \frac{j}{2\pi} \int\_{-\infty}^{\infty} \omega \varepsilon(\omega) \vec{E}(\vec{r},\omega) \exp(j\omega t) \mathrm{d}\omega. \tag{18}$$

For equation (18), we assume an isotropic material with *D* (*r*, *ω*) = *ε*(*ω*) *E*(*r*, *ω*). Integration of the product of Eq. (17) an Eq. (18) with respect to time gives:

$$\int\_{-\infty}^{\infty} \vec{E}(\vec{r},t) \cdot \frac{\partial}{\partial t} \vec{D}(\vec{r},t) \mathrm{d}t = \frac{\mathrm{j}}{4\pi^2} \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \omega \varepsilon(\omega) \vec{E}(\vec{r},\omega) \vec{E}(\vec{r},\omega') \exp\left(\mathrm{j}(\omega + \omega')t\right) \mathrm{d}\omega \mathrm{d}\omega' \mathrm{d}t. \tag{19}$$

A first integration with respect to *t* of the right-hand side of Eq. (19) is done using

$$\int\_{-\infty}^{\infty} \exp\left(j(\omega + \omega')t\right) \mathrm{d}t = 2\pi\delta(\omega + \omega').$$

The Dirac distribution is then eliminated by the second integration with respect to *ω*� . The principle of causality and reality of fields impose (Good & Nelson (1971)):

$$
\vec{E}(\vec{r}\_\prime - \omega) = \vec{E}(\vec{r}\_\prime \omega)^\* \, , \,\, \vec{r}\_\prime
$$

The right-hand side of the equation (19)can finally be written as:

$$\frac{j}{2\pi} \int\_{-\infty}^{\infty} \omega \epsilon(\omega) \left| \vec{E}(\vec{r}, \omega) \right|^2 \mathbf{d}\omega. \tag{20}$$

After application of the same procedure for magnetic fields *H* , we obtain:

$$\int\_{-\infty}^{\infty} \vec{H}(\vec{r}, t) \cdot \frac{\partial}{\partial t} \vec{B}(\vec{r}, t) \mathrm{d}t = \frac{j}{2\pi} \int\_{-\infty}^{\infty} \omega \mu(\omega) \left| \vec{H}(\vec{r}, \omega) \right|^2 \mathrm{d}\omega. \tag{21}$$

Then substituting *ε*(*ω*) et *μ*(*ω*) by their complex expression, the energy dissipated (in the period of field variations) in frequency domain is:

$$\int\_{-\infty}^{\infty} \mathbf{Q} \mathbf{d}t = \frac{1}{2\pi} \int\_{-\infty}^{\infty} \omega \left( \boldsymbol{\varepsilon}^{\prime\prime}(\omega) \left| \vec{\mathbf{E}}(\vec{r}, \omega) \right|^{2} + \mu^{\prime\prime}(\omega) \left| \vec{H}(\vec{r}, \omega) \right|^{2} \right) \mathbf{d}\omega. \tag{22}$$

The divergence of Poynting vector is expressed as the rate of energy transformation to heat: this dissipated energy depends on *ε*��(*ω*) et *μ*��(*ω*). The dependence on *ε*� (*ω*) et *μ*� (*ω*) is 6 Will-be-set-by-IN-TECH

This equation provides an expression of the energy conservation in a dispersive material in time-domain (Balanis (1989)). In frequency domain, the electric and magnetic fields are given,

*ωε*(*ω*)

<sup>−</sup><sup>∞</sup> *ωε*(*ω*)

)*t* 

The Dirac distribution is then eliminated by the second integration with respect to *ω*�

*<sup>E</sup>*(*<sup>r</sup>*, <sup>−</sup>*ω*) =

*ωε*(*ω*) *E*(*r*, *ω*) 2

2*π*

Then substituting *ε*(*ω*) et *μ*(*ω*) by their complex expression, the energy dissipated (in the

The divergence of Poynting vector is expressed as the rate of energy transformation to heat:

 ∞ −∞ *ωμ*(*ω*) *H* (*r*, *ω*) 2

+ *μ*��(*ω*)

 *H* (*r*, *ω*) 2 

*E*(*r*, *ω*)

d*t* = 2*πδ*(*ω* + *ω*�

*E*(*r*, *ω*)∗,

*E*(*r*, *ω*�

).

*∂t*

*<sup>D</sup>* (*<sup>r</sup>*, *<sup>t</sup>*) + *<sup>H</sup>* (*<sup>r</sup>*, *<sup>t</sup>*) · *<sup>∂</sup>*

*∂t* 

*E*(*r*, *ω*) exp(*jωt*)d*ω*, (17)

) exp

*E*(*r*, *ω*) exp(*jωt*)d*ω*. (18)

*D* (*r*, *t*), (15)

*B*(*r*, *t*). (16)

*E*(*r*, *ω*). Integration of

)*t* d*ω*d*ω*�

d*t*. (19)

. The

*j*(*ω* + *ω*�

d*ω*. (20)

d*ω*. (21)

d*ω*. (22)

(*ω*) is

(*ω*) et *μ*�

∇ × *<sup>H</sup>* (*<sup>r</sup>*, *<sup>t</sup>*) = *<sup>∂</sup>*

*<sup>E</sup>*(*<sup>r</sup>*, *<sup>t</sup>*) · *<sup>∂</sup> ∂t*

the divergence of Poynting vector is given as:

after Fourier transform, by:

−∇·

*∂ ∂t*

*<sup>D</sup>* (*<sup>r</sup>*, *<sup>t</sup>*)d*<sup>t</sup>* <sup>=</sup> *<sup>j</sup>*

 ∞ −∞

 ∞ −∞ *S*(*r*, *t*) =

*<sup>E</sup>*(*<sup>r</sup>*, *<sup>t</sup>*) = <sup>1</sup>

*<sup>D</sup>* (*<sup>r</sup>*, *<sup>t</sup>*) = *<sup>j</sup>*

the product of Eq. (17) an Eq. (18) with respect to time gives:

4*π*<sup>2</sup>

 ∞ <sup>−</sup><sup>∞</sup> exp 2*π*

 ∞ −∞

2*π*

For equation (18), we assume an isotropic material with *D* (*r*, *ω*) = *ε*(*ω*)

 ∞ −∞ ∞

A first integration with respect to *t* of the right-hand side of Eq. (19) is done using

*j*(*ω* + *ω*�

principle of causality and reality of fields impose (Good & Nelson (1971)):

 ∞ −∞

After application of the same procedure for magnetic fields *H* , we obtain:

*<sup>B</sup>*(*<sup>r</sup>*, *<sup>t</sup>*)d*<sup>t</sup>* <sup>=</sup> *<sup>j</sup>*

this dissipated energy depends on *ε*��(*ω*) et *μ*��(*ω*). The dependence on *ε*�

The right-hand side of the equation (19)can finally be written as:

*∂t* 

 ∞ −∞ *ω ε* ��(*ω*) *E*(*r*, *ω*) 2

*<sup>H</sup>* (*<sup>r</sup>*, *<sup>t</sup>*) · *<sup>∂</sup>*

period of field variations) in frequency domain is:

*<sup>Q</sup>*d*<sup>t</sup>* <sup>=</sup> <sup>1</sup> 2*π*

*j* 2*π*

 ∞ −∞  ∞ −∞ 

and

 ∞ −∞ *<sup>E</sup>*(*<sup>r</sup>*, *<sup>t</sup>*)· *<sup>∂</sup> ∂t* canceled because the integrand of equation (20) is an odd function of *ω* 4. The two terms of the right-hand side of Eq. (22) represent respectively the dielectric and magnetic losses.

The second law of thermodynamics, stating that the entropy of a isolated macroscopic system never decreases, imposes that *Q* > 0. It is thus necessary according to equation (22) that:

$$
\varepsilon''(\omega) \left| \vec{E}(\vec{r}, \omega) \right|^2 + \mu''(\omega) \left| \vec{H}(\vec{r}, \omega) \right|^2 > 0 \tag{23}
$$

for positive frequencies (*ω* > 0). The laws of Thermodynamics also express the irreversible nature of physical processes and the fundamental difference between two types of energy: work and heat (Yavorski & Detlaf (1975)). The energy dissipated by fields into heat is irreversible. In other terms, there can be no exchange between the work done by either the electric field [� *E*(�*r*, *ω*) or magnetic field *H*� (�*r*, *ω*)] and the heat dissipated by the other, implying:

$$
\varepsilon(\omega)^{\prime\prime} > 0 \quad \text{and} \quad \mu(\omega)^{\prime\prime} > 0 \tag{24}
$$

This demonstration can be very easily extended to NRI materials. Indeed, the starting point of the demonstration is energy conservation through the expression of the divergence of Poynting vector [Eq. (16)] written thanks to Maxwell-Ampère [Eq. (15)] and Maxwell-Faraday [Eq. (14)] equations as well Poynting theorem [Eq. (13)]. For a NRI material, these equations and theorems are valid (Veselago (1968)). Indeed, only the direction of the vector �*k* changes in a NRI material thus giving a negative value for the product � *<sup>S</sup>* ·�*<sup>k</sup>* (This product is positive for conventional materials). This is easily verified for a monochromatic wave but can actually also be verified for non-monochromatic wave.

Let us consider for instance, the mean of Poynting vector for a dispersive material excited by the superposition of two monochromatic waves of angular frequency *ω*<sup>1</sup> et *ω*<sup>2</sup> such that *ω*<sup>1</sup> �= *ω*<sup>2</sup> (Pacheco-Jr et al. (2002)):

$$<\vec{S}(\vec{r},t)> = \frac{|E|^2}{2} \left( \frac{\vec{k}(\omega\_1)}{\omega\_1 \mu(\omega\_1)} + \frac{\vec{k}(\omega\_2)}{\omega\_2 \mu(\omega\_2)} \right) \tag{25}$$

The relation between the direction of Poynting vector � *S* and that of the wave vector�*k* is clearly shown by this equation. The direction of � *S* is independent of the sign of the refractive index and for any propagative medium, *i.e.* when �*k* is real, the ratio *k*/*μ* is positive. As shown before, if *k* takes negative values, then *μ* will be negative too.

#### **3. Effective parameters of resonant NRI metamaterials**

#### **3.1 Effective medium theory as applied to metamaterials**

The concept of effective medium for the description of heterogeneous systems by a homogeneous one is very attractive in different field of physics. Homogenization procedures allowing the definition of an effective macroscopic response from physical parameters characterizing the heterogeneous system are generally developed. In our case, from the microscopic parameters (geometrical and physical definitions) of the metamaterial, a macroscopic electromagnetic response can be obtained. If this macroscopic definition

<sup>4</sup> Principle of causality imposes that (Good & Nelson (1971)): *<sup>ε</sup>*(−*ω*) = *<sup>ε</sup>*(*ω*)<sup>∗</sup> and *<sup>μ</sup>*(−*ω*) = *<sup>μ</sup>*(*ω*)∗.

polarize the neighboring inclusions. The group of inclusions then react by creating a modified local field. The presence of inclusions (or perturbations) in a given environment can make an initially evanescent field propagative (de Fornel (1997)). A common example is the insertion of inclusions in a guide under the cut-off frequency. All these complex interactions are visible at the microscopic or local scale. However, the field in a material as expressed in Maxwell's

Resonant Negative Refractive Index Metamaterials 179

The definition of constitutive parameters require the determination of a relationship between the local field, the applied field and the macroscopic field. the theory of local field of Lorentz (Berthier (1993); Tretyakov (2003)) can be used but it is not always adequate (Baker-Jarvis, Janezic, Riddle, Johnk, Kabos, Holloway, Geyer & Grosvenor (2004); Baker-Jarvis, Kabos & Holloway (2004)). It has been however applied to certain types of composites. The polarizabilities are calculated analytically and the theory of Lorentz then provides the macroscopic parameters. Numerous examples of such calculations are given in ( (Tretyakov (2003)) and the papers there cited there. Other methods have also been used in the literature such as those introduced by O. Keller et J. Baker-Jarvis (Baker-Jarvis, Kabos & Holloway (2004); Keller (1996)) but they rely on statistical and quantum approaches. The

The definition of effective medium can be mainly performed in two distinct categories of approaches. The first category can be termed *locale* ( ˘g 3.3) and the second one *global* ( ˘g 3.4). In the first case, the effective parameters are defined directly from local fields while the second one allows a definition based on global propagation characteristics of the periodic system, for

When they are not based on analytical approaches, the input data are the fields or electric and magnetic induction calculated using full-wave numerical methods. The definition of effective parameters from local fields is not straightforward. Three methodologies can be distinguished. The first one consists in defining an equivalence between the local field calculated and the effective parameters of a corresponding homogeneous medium (Pincemin (1995); Silveirinha & Fernandes (2004a;b; 2005a;b)). The second methodology consists in the calculation of the propagation constant from the phase velocity locally determined using time-domain numerical modeling methods (Moss et al. (2002)). Finally, the third methodology consists in the definition of effective parameters by calculation a linear, surface-based or volume-based mean field values on adequately chosen geometries. Several methods are available in the literature (Acher et al. (2000); Bardi et al. (2002); Lerat et al. (2005); Lubkowski et al. (2005); Pendry et al. (1999); Smith (2005); Smith, Vier, Kroll & Schultz (2000); Weiland et al. (2001)). In the method given by Acher *et al.* (Acher et al. (2000)), it is worth noting that a convergence is demonstrated between effective parameters calculated using this type of numerical approach and those obtained by the Bruggeman extended theory of effective medium (taking into account magnetic polarizability) for the asymptotic case of

Global approaches provide effective parameters starting from global responses of the periodic system. These responses such as the scattering matrix, the reflection and transmissions

equations is the macroscopic field, usually defined by constitutive equations.

discussion will be restricted to the context of *classical* electrodynamics.

instance from the model of scattering parameters.

**3.3 Local approaches**

metal-dielectric slab.

**3.4 Global approaches**

is performed in accordance with the electrodynamics of continuous medium, they can afterwards be used in Maxwell's equations to predict propagation phenomena and provide physical insight into the design of metamaterial-based microwave and optical devices.

In this chapter, the NRI metamaterials considered are assumed periodic and based on resonant inclusions such as the combination of Split Ring Resonator and wire medium. The general definition of the relevant dimensions for the definition of the effective parameters of such periodic medium is depicted on figure 3 (Baker-Jarvis, Janezic, Riddle, Johnk, Kabos, Holloway, Geyer & Grosvenor (2004)).

Fig. 3. Dimensions for effective medium of resonant periodic metamaterial.

The left-hand side region represents the quasi-static region where the wavelength is much bigger than the periodicity of the inclusions. The effective parameters of the composite in this zone can be easily calculated using quasi-static solutions or classical mixing rules (Berthier (1993)).

In the right-hand side region, the composite is heterogeneous and the resonances of the medium can be directly linked to the periodicity. Such a composite cannot be considered homogeneous. To study the propagation characteristics of these media, full-wave numerical methods are generally required. The volume under study has to be discretized : a unit-cell is defined and Floquet-Bloch boundary conditions are used. This case is typically the working regime of photonic crystals.

The intermediate region is a region where the inclusions are resonant. The electrical dimensions of the inclusions as well as the periodicity are small compared to the wavelength. Resonant NRI metamaterials belong to this intermediate region. Such a medium is generally considered homogeneous. However, the question which remains to be answered is: how should one study the characteristics of such a medium and how should the associated effective medium be defined?

There is indeed no simple or unique definition to the effective medium concept. The possible approach and definition which will be used in this chapter for NRI metamaterials will be described hereafter.

#### **3.2 Definition of the effective medium concept for composites of the intermediate region**

When an EM field is applied to a composite, the fields in the composite results from the interaction between the applied field and the reaction of the inclusions constituting the composite (Baker-Jarvis, Janezic, Riddle, Johnk, Kabos, Holloway, Geyer & Grosvenor (2004); Baker-Jarvis, Kabos & Holloway (2004)). The local field in the composite can be freely propagative, propagative with attenuation or evanescent. The resulting local field is a complicated physical process whereby the applied field polarizes the inclusions which in turn 8 Will-be-set-by-IN-TECH

is performed in accordance with the electrodynamics of continuous medium, they can afterwards be used in Maxwell's equations to predict propagation phenomena and provide physical insight into the design of metamaterial-based microwave and optical devices.

In this chapter, the NRI metamaterials considered are assumed periodic and based on resonant inclusions such as the combination of Split Ring Resonator and wire medium. The general definition of the relevant dimensions for the definition of the effective parameters of such periodic medium is depicted on figure 3 (Baker-Jarvis, Janezic, Riddle, Johnk, Kabos,

The left-hand side region represents the quasi-static region where the wavelength is much bigger than the periodicity of the inclusions. The effective parameters of the composite in this zone can be easily calculated using quasi-static solutions or classical mixing rules (Berthier

In the right-hand side region, the composite is heterogeneous and the resonances of the medium can be directly linked to the periodicity. Such a composite cannot be considered homogeneous. To study the propagation characteristics of these media, full-wave numerical methods are generally required. The volume under study has to be discretized : a unit-cell is defined and Floquet-Bloch boundary conditions are used. This case is typically the working

The intermediate region is a region where the inclusions are resonant. The electrical dimensions of the inclusions as well as the periodicity are small compared to the wavelength. Resonant NRI metamaterials belong to this intermediate region. Such a medium is generally considered homogeneous. However, the question which remains to be answered is: how should one study the characteristics of such a medium and how should the associated effective

There is indeed no simple or unique definition to the effective medium concept. The possible approach and definition which will be used in this chapter for NRI metamaterials will be

**3.2 Definition of the effective medium concept for composites of the intermediate region** When an EM field is applied to a composite, the fields in the composite results from the interaction between the applied field and the reaction of the inclusions constituting the composite (Baker-Jarvis, Janezic, Riddle, Johnk, Kabos, Holloway, Geyer & Grosvenor (2004); Baker-Jarvis, Kabos & Holloway (2004)). The local field in the composite can be freely propagative, propagative with attenuation or evanescent. The resulting local field is a complicated physical process whereby the applied field polarizes the inclusions which in turn

Fig. 3. Dimensions for effective medium of resonant periodic metamaterial.

Holloway, Geyer & Grosvenor (2004)).

(1993)).

regime of photonic crystals.

medium be defined?

described hereafter.

polarize the neighboring inclusions. The group of inclusions then react by creating a modified local field. The presence of inclusions (or perturbations) in a given environment can make an initially evanescent field propagative (de Fornel (1997)). A common example is the insertion of inclusions in a guide under the cut-off frequency. All these complex interactions are visible at the microscopic or local scale. However, the field in a material as expressed in Maxwell's equations is the macroscopic field, usually defined by constitutive equations.

The definition of constitutive parameters require the determination of a relationship between the local field, the applied field and the macroscopic field. the theory of local field of Lorentz (Berthier (1993); Tretyakov (2003)) can be used but it is not always adequate (Baker-Jarvis, Janezic, Riddle, Johnk, Kabos, Holloway, Geyer & Grosvenor (2004); Baker-Jarvis, Kabos & Holloway (2004)). It has been however applied to certain types of composites. The polarizabilities are calculated analytically and the theory of Lorentz then provides the macroscopic parameters. Numerous examples of such calculations are given in ( (Tretyakov (2003)) and the papers there cited there. Other methods have also been used in the literature such as those introduced by O. Keller et J. Baker-Jarvis (Baker-Jarvis, Kabos & Holloway (2004); Keller (1996)) but they rely on statistical and quantum approaches. The discussion will be restricted to the context of *classical* electrodynamics.

The definition of effective medium can be mainly performed in two distinct categories of approaches. The first category can be termed *locale* ( ˘g 3.3) and the second one *global* ( ˘g 3.4). In the first case, the effective parameters are defined directly from local fields while the second one allows a definition based on global propagation characteristics of the periodic system, for instance from the model of scattering parameters.

#### **3.3 Local approaches**

When they are not based on analytical approaches, the input data are the fields or electric and magnetic induction calculated using full-wave numerical methods. The definition of effective parameters from local fields is not straightforward. Three methodologies can be distinguished. The first one consists in defining an equivalence between the local field calculated and the effective parameters of a corresponding homogeneous medium (Pincemin (1995); Silveirinha & Fernandes (2004a;b; 2005a;b)). The second methodology consists in the calculation of the propagation constant from the phase velocity locally determined using time-domain numerical modeling methods (Moss et al. (2002)). Finally, the third methodology consists in the definition of effective parameters by calculation a linear, surface-based or volume-based mean field values on adequately chosen geometries. Several methods are available in the literature (Acher et al. (2000); Bardi et al. (2002); Lerat et al. (2005); Lubkowski et al. (2005); Pendry et al. (1999); Smith (2005); Smith, Vier, Kroll & Schultz (2000); Weiland et al. (2001)). In the method given by Acher *et al.* (Acher et al. (2000)), it is worth noting that a convergence is demonstrated between effective parameters calculated using this type of numerical approach and those obtained by the Bruggeman extended theory of effective medium (taking into account magnetic polarizability) for the asymptotic case of metal-dielectric slab.

#### **3.4 Global approaches**

Global approaches provide effective parameters starting from global responses of the periodic system. These responses such as the scattering matrix, the reflection and transmissions

**3.5.1 NRI metamaterials finite in the propagation direction**

in figure 5.

valid.

For NRI metamaterials finite in the direction of wave propagation, the problem to be solved is the substitution of this periodic structure by a homogeneous slab of same thickness as shown

Resonant Negative Refractive Index Metamaterials 181

Fig. 5. Equivalence between a periodic composite of transverse periodicity *PT* by a homogeneous slab of same thickness *d*, effective permittivity *ε*(*ω*) and permeability *μ*(*ω*). This substitution or equivalence is only valid under a few assumptions (Lalanne & Hutley (2003); Lalanne & Lalanne (1996)): (i) Only the first mode propagates in the incident medium, transmitted medium and the periodic structure. This condition is given by the fol. equation:

> <sup>|</sup>*β*| ≤ *<sup>π</sup> PT*

*β* is the propagation constant in each medium. (ii) Evanescent modes should not be present in the *x*0*y* plane. If only the first mode can propagate in the periodic structure with a speed of *c*0/*n*(*ω*) where *n*(*ω*) the interference phenomenon is identical to the one which occurs in a homogeneous slab. However, the existence of higher order modes give a much more complex interference phenomenon such that the equivalence with a homogeneous slab is no longer

In the case of a metal-dielectric composite, these equivalence conditions are not automatically satisfied. Indeed, the existing propagation modes and their associated propagation constants

The calculation of effective parameters of NRI metamaterials under these assumptions is done in two steps. The first one consists in the determination of the complex reflection and transmission coefficients which can be numerically calculated according to the generalized scattering parameters described in section 3.4. They can also be obtained by experimentally. The second step is then the calculation of the effective permittivity and permeability (*ε*(*ω*), *μ*(*ω*)) from these reflection and transmission using inversion methods. These methods can either be direct using analytical inversion of Fresnel equations, (*r*, *t*) = *f*(*ε*(*ω*), *μ*(*ω*)) or

In the NRW method, the wave impedance and refractive index are first calculated. The effective permittivity and permeability are then deduced. The normalized wave impedance of a slab can be described by analogy as the input impedance of a transmission line thus containing information not only on *E*¯/*H*¯ at the interface of two lines or medium but also of

(<sup>1</sup> <sup>+</sup> *<sup>r</sup>*)<sup>2</sup> <sup>−</sup> *<sup>t</sup>*2*e*−2*jk*0*<sup>d</sup>*

(<sup>1</sup> <sup>−</sup> *<sup>r</sup>*)<sup>2</sup> <sup>−</sup> *<sup>t</sup>*2*e*−2*jk*0*<sup>d</sup>* , (27)

depend highly on the nature of the inclusions: their geometry, size and distribution.

performed by an iterative approach. Both approaches are described here.

the propagation constant inside the propagating medium. *Z* is given by:

*Z* = ±

3.5.1.1 Direct method - Nicholson Ross Weir (NRW) approach

. (26)

coefficients, resonant frequencies are observable or measurable quantities with can be either experimentally determined or numerically calculated from fields defined locally in the unit cell of the periodic NRI metamaterial.

The transformation of the local field to the scattering matrix relies on an analogy between propagation in a periodic structure and in waveguides and circuits. It consists in assimilating the periodic structure in a multiple-access system and to study transmission and reflection between the different accesses. The development of such an analogy requires a few assumptions (Hélier (2001); Richalot (1998); Rivier & Sardos (1982)), namely:


Figure 4 depicts an example of a system with three physical accesses modeled using the generalized scattering matrix method described before. Each physical access is artificially decomposed in N virtual accesses, where N represent the number of modes to be taken into account at each physical access.

Fig. 4. Example of a system with three physical accesses modeled by a system of N virtual accesses to take inbto account the number of modes present or excited at each physical access.

Such a scattering matrix allows the complete characterization of the structure both in emitting and receiving modes both in near and far fields. The reflection and transmission matrices can thus be directly determined from this matrix followed by the effective parameters; this procedure is further detailed hereafter.

#### **3.5 Calculation of effective parameters of resonant metamaterials**

The calculation method described here is belongs to the category of global approaches as defined in section (3.4) and can be divided in two parts, namely for NRI metamaterials structures finite and infinite in the direction of propagation.

10 Will-be-set-by-IN-TECH

coefficients, resonant frequencies are observable or measurable quantities with can be either experimentally determined or numerically calculated from fields defined locally in the unit

The transformation of the local field to the scattering matrix relies on an analogy between propagation in a periodic structure and in waveguides and circuits. It consists in assimilating the periodic structure in a multiple-access system and to study transmission and reflection between the different accesses. The development of such an analogy requires a few

• absence of radiation: the system is closed and energy exchange can only exist between the

• existence of pure mode: each access of the system supports a pure mode, i.e. a unique propagation mode characterized by a given propagation constant. If this assumption is not verified, then sufficient supplementary virtual accesses have to be defined to account

Figure 4 depicts an example of a system with three physical accesses modeled using the generalized scattering matrix method described before. Each physical access is artificially decomposed in N virtual accesses, where N represent the number of modes to be taken into

Fig. 4. Example of a system with three physical accesses modeled by a system of N virtual accesses to take inbto account the number of modes present or excited at each physical access. Such a scattering matrix allows the complete characterization of the structure both in emitting and receiving modes both in near and far fields. The reflection and transmission matrices can thus be directly determined from this matrix followed by the effective parameters; this

The calculation method described here is belongs to the category of global approaches as defined in section (3.4) and can be divided in two parts, namely for NRI metamaterials

*B* et *H* are linked to one another by linear relationships,

assumptions (Hélier (2001); Richalot (1998); Rivier & Sardos (1982)), namely:

• stationarity: the properties of the system are invariant with respect to time,

*E* et *D* ,

cell of the periodic NRI metamaterial.

for higher propagation modes.

procedure is further detailed hereafter.

**3.5 Calculation of effective parameters of resonant metamaterials**

structures finite and infinite in the direction of propagation.

account at each physical access.

• linearity: the vectors

system accesses,

#### **3.5.1 NRI metamaterials finite in the propagation direction**

For NRI metamaterials finite in the direction of wave propagation, the problem to be solved is the substitution of this periodic structure by a homogeneous slab of same thickness as shown in figure 5.

Fig. 5. Equivalence between a periodic composite of transverse periodicity *PT* by a homogeneous slab of same thickness *d*, effective permittivity *ε*(*ω*) and permeability *μ*(*ω*).

This substitution or equivalence is only valid under a few assumptions (Lalanne & Hutley (2003); Lalanne & Lalanne (1996)): (i) Only the first mode propagates in the incident medium, transmitted medium and the periodic structure. This condition is given by the fol. equation:

$$|\beta| \le \frac{\pi}{P\_T}.\tag{26}$$

*β* is the propagation constant in each medium. (ii) Evanescent modes should not be present in the *x*0*y* plane. If only the first mode can propagate in the periodic structure with a speed of *c*0/*n*(*ω*) where *n*(*ω*) the interference phenomenon is identical to the one which occurs in a homogeneous slab. However, the existence of higher order modes give a much more complex interference phenomenon such that the equivalence with a homogeneous slab is no longer valid.

In the case of a metal-dielectric composite, these equivalence conditions are not automatically satisfied. Indeed, the existing propagation modes and their associated propagation constants depend highly on the nature of the inclusions: their geometry, size and distribution.

The calculation of effective parameters of NRI metamaterials under these assumptions is done in two steps. The first one consists in the determination of the complex reflection and transmission coefficients which can be numerically calculated according to the generalized scattering parameters described in section 3.4. They can also be obtained by experimentally. The second step is then the calculation of the effective permittivity and permeability (*ε*(*ω*), *μ*(*ω*)) from these reflection and transmission using inversion methods. These methods can either be direct using analytical inversion of Fresnel equations, (*r*, *t*) = *f*(*ε*(*ω*), *μ*(*ω*)) or performed by an iterative approach. Both approaches are described here.

#### 3.5.1.1 Direct method - Nicholson Ross Weir (NRW) approach

In the NRW method, the wave impedance and refractive index are first calculated. The effective permittivity and permeability are then deduced. The normalized wave impedance of a slab can be described by analogy as the input impedance of a transmission line thus containing information not only on *E*¯/*H*¯ at the interface of two lines or medium but also of the propagation constant inside the propagating medium. *Z* is given by:

$$Z = \pm \sqrt{\frac{(1+r)^2 - t^2 e^{-2jk\_0d}}{(1-r)^2 - t^2 e^{-2jk\_0d}}},\tag{27}$$

transverse dimensions. For a plane wave having an incident angle of *θ<sup>i</sup>* and a polarization *TE*

Resonant Negative Refractive Index Metamaterials 183

*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>* et *<sup>t</sup>* <sup>=</sup> <sup>2</sup>*<sup>A</sup>*

, *B* = *Z*<sup>0</sup>

and *<sup>Z</sup>*<sup>0</sup> <sup>=</sup> cos *<sup>θ</sup><sup>i</sup>*

and *<sup>Z</sup>*<sup>0</sup> <sup>=</sup> *<sup>μ</sup>*0*c*<sup>0</sup>

 *j* sin *qd <sup>Z</sup>* <sup>+</sup>

*ε*0*c*<sup>0</sup>

cos *θ<sup>i</sup>*

*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*, (35)

. (36)

. (37)

cos *qd Z*0

or *TM*, the reflection coefficient *r* and transmission *t* are defined by:

avec *<sup>A</sup>* <sup>=</sup> cos *qd* <sup>+</sup> *<sup>j</sup>* sin *qd*

For the TE polarization,

For TM polarization,

the previous frequency point.

index must be close to zero.

to zero,

medium.

*<sup>r</sup>* <sup>=</sup> *<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*

*Z*0

 1 − 1 *n* sin *θ<sup>i</sup>* <sup>2</sup> .

In our case, the functions *F*1(*x*) and *F*2(*x*) of the equation (34) are replaced by *r* et *t* equation (35). The cost function (34) is minimized using the non-linear mean square algorithm of Levenberg-Marquardt. To ensure good results, this algorithm needs to start from a feasible point. This is why a large choice of values for the couple (*ε*(*ω*), *μ*(*ω*)) is done for the starting point in frequency. Then if the frequency sampling from the numerical simulations is fine enough and the functions considered to be continuous, the starting point chosen is the one for

For a few composites, the algorithm can converge to many solutions for large values of the thickness of the slab. These solutions are not local minima but are solutions to Fresnel

• If the scattering parameter *S*<sup>21</sup> is close to 1, the structure is propagative. The refractive index, being a parameter representative of propagation 5, its imaginary part must be close

• If *S*<sup>11</sup> is close to one, there is no propagation in the structure ; the real part of the refractive

These criteria are particularly appropriate for NRI resonant metamaterials and may not be adequate for all type of composites. The principal limitation is that the starting point must be

If the EM wave propagation is considered in a periodic medium such as the one presented on figure 6(a), the solution of the wave equation provides solution for the propagation constant

<sup>5</sup> The imaginary part of the refractive index does not allow to calculate losses by dissipation of a medium (Landau et al. (1984)). It can only represent the presence or absence of propagation in a

equations. To choose the right solutions, two physical criteria have been defined:

far from resonance and the composite should not present high dissipative losses.

**3.5.2 Periodic structure infinite in the propagation direction**

which are given by *kn* = *k* + 2*mπ*/*P* where *m* ∈ *Z* and *P* = *PL* or *PT*.

et *q* = *k*<sup>0</sup>

*<sup>Z</sup>* <sup>=</sup> *ωμeff q*

*<sup>Z</sup>* <sup>=</sup> *<sup>q</sup> ωεeff*

where *d* is the slab thickness, *k*<sup>0</sup> = 2*π*/*λ*<sup>0</sup> is the wave number *λ*<sup>0</sup> the free-space wavelength. The choice of the sign in front of the square root of *Z* is done according to the definitions given in section 2.1.

The real part of the refractive index *n* is given by equation (28):

$$m' = \frac{\arctan\left(\operatorname{Im}(Y)/\operatorname{Re}(Y)\right) \pm m\pi}{k\_0 d},\tag{28}$$

where *m* ∈ *Z*. The variable *Y* is defined as:

$$Y = e^{-j\text{inkd}} = X \pm \sqrt{X^2 - 1},\tag{29}$$

where

$$X = \frac{e^{jk\_0d}}{2t} \left(1 - r^2 + t^2 e^{-2jk\_0d}\right). \tag{30}$$

The choice of the value of *m* in equation (28) constitute one of the ambiguities of this method which can be solved in different ways, namely (i) by considering various thicknesses and assuming that there is no coupling between different layers of metamaterials in the direction of propagation (Markos & Soukoulis (2001)), (ii) by comparing the measured group arrival time to the calculated one (Baker-Jarvis, Janezic, Riddle, Johnk, Kabos, Holloway, Geyer & Grosvenor (2004)).

The imaginary part of the refractive index *n*�� is given by:

$$m'' = \frac{\ln|Y|}{k\_0 d} \tag{31}$$

*n*�� is calculated using the fundamental limitation described in 2.1, i.e. *n*�� > 0 for both positive or negative refractive index materials.

Using the two independent equations (32 and 33) the effective permittivity and permeability can be deduced.

$$m = \sqrt{\varepsilon(\omega)}\sqrt{\mu(\omega)} \quad \text{and} \quad Z = \sqrt{\mu(\omega)/\varepsilon(\omega)}\tag{32}$$

$$
\epsilon(\omega) = \frac{n}{Z} \quad \text{and} \quad \mu(\omega) = nZ \tag{33}
$$

It should be noted that the refractive index is defined as: *<sup>n</sup>* = √*εeff* √*μeff* such that when *ε*(*ω*) and *μ*(*ω*) are simultaneously negative, the real part of *n* is also negative. The common formula *<sup>n</sup>* = √*εeff <sup>μ</sup>eff* should not be used.

#### 3.5.1.2 Iterative method - Optimization approach

This method consists in the minimization of the difference between the functions *F*1(*x*), *F*2(*x*) and the scattering parameters *S*<sup>11</sup> et *S*<sup>21</sup> according to the fol. cost function:

$$E(\mathbf{x}) = |F\_1(\mathbf{x}) - \mathbf{S}\_{11}|^2 + |F\_2(\mathbf{x}) - \mathbf{S}\_{21}|^2 \qquad \text{with} \quad \mathbf{x} = \{\varepsilon(\omega), \mu(\omega)\} \tag{34}$$

The functions *F*1(*x*) et *F*2(*x*) are complex and represent respectively the Fresnel reflection and transmission coefficients defined for a magneto-dielectric slab of thickness *d* and of infinite transverse dimensions. For a plane wave having an incident angle of *θ<sup>i</sup>* and a polarization *TE* or *TM*, the reflection coefficient *r* and transmission *t* are defined by:

$$r = \frac{A - B}{A + B} \qquad \text{et} \qquad t = \frac{2A}{A + B}, \tag{35}$$

$$\begin{array}{ll} \text{avec} & A = \cos qd + \frac{j \sin qd}{Z\_0}, & B = Z\_0 \left(\frac{j \sin qd}{Z} + \frac{\cos qd}{Z\_0}\right) \\\\ & \text{et} & q = k\_0 \sqrt{1 - \left(\frac{1}{n} \sin \theta\_i\right)^2}. \end{array}$$

For the TE polarization,

12 Will-be-set-by-IN-TECH

where *d* is the slab thickness, *k*<sup>0</sup> = 2*π*/*λ*<sup>0</sup> is the wave number *λ*<sup>0</sup> the free-space wavelength. The choice of the sign in front of the square root of *Z* is done according to the definitions given

*<sup>n</sup>*� <sup>=</sup> arctan (*Im*(*Y*)/*Re*(*Y*)) <sup>±</sup> *<sup>m</sup><sup>π</sup>*

The choice of the value of *m* in equation (28) constitute one of the ambiguities of this method which can be solved in different ways, namely (i) by considering various thicknesses and assuming that there is no coupling between different layers of metamaterials in the direction of propagation (Markos & Soukoulis (2001)), (ii) by comparing the measured group arrival time to the calculated one (Baker-Jarvis, Janezic, Riddle, Johnk, Kabos, Holloway, Geyer &

*<sup>n</sup>*�� <sup>=</sup> *ln* <sup>|</sup>*Y*<sup>|</sup>

*n*�� is calculated using the fundamental limitation described in 2.1, i.e. *n*�� > 0 for both positive

Using the two independent equations (32 and 33) the effective permittivity and permeability

*μ*(*ω*) and *Z* =

It should be noted that the refractive index is defined as: *<sup>n</sup>* = √*εeff* √*μeff* such that when *ε*(*ω*) and *μ*(*ω*) are simultaneously negative, the real part of *n* is also negative. The common

This method consists in the minimization of the difference between the functions *F*1(*x*), *F*2(*x*)

The functions *F*1(*x*) et *F*2(*x*) are complex and represent respectively the Fresnel reflection and transmission coefficients defined for a magneto-dielectric slab of thickness *d* and of infinite

*<sup>ε</sup>*(*ω*) = *<sup>n</sup>*

and the scattering parameters *S*<sup>11</sup> et *S*<sup>21</sup> according to the fol. cost function:

<sup>2</sup> <sup>+</sup> <sup>|</sup>*F*2(*x*) <sup>−</sup> *<sup>S</sup>*21<sup>|</sup>

*<sup>Y</sup>* <sup>=</sup> *<sup>e</sup>*−*jnkd* <sup>=</sup> *<sup>X</sup>* <sup>±</sup>

*<sup>X</sup>* <sup>=</sup> *<sup>e</sup>jk*0*<sup>d</sup>* 2*t* 1 − *r* <sup>2</sup> + *t* 2*e* −2*jk*0*d* 

The imaginary part of the refractive index *n*�� is given by:

*n* = *ε*(*ω*) 

formula *<sup>n</sup>* = √*εeff <sup>μ</sup>eff* should not be used.

3.5.1.2 Iterative method - Optimization approach

*E*(*x*) = |*F*1(*x*) − *S*11|

or negative refractive index materials.

*<sup>k</sup>*0*<sup>d</sup>* , (28)

*<sup>k</sup>*0*<sup>d</sup>* (31)

*<sup>Z</sup>* and *<sup>μ</sup>*(*ω*) = *nZ* (33)

<sup>2</sup> with *<sup>x</sup>* <sup>=</sup> {*ε*(*ω*), *<sup>μ</sup>*(*ω*)} (34)

*μ*(*ω*)/*ε*(*ω*) (32)

*<sup>X</sup>*<sup>2</sup> − 1, (29)

. (30)

The real part of the refractive index *n* is given by equation (28):

where *m* ∈ *Z*. The variable *Y* is defined as:

in section 2.1.

where

Grosvenor (2004)).

can be deduced.

$$Z = \frac{\omega \mu\_{eff}}{q} \qquad \text{and} \qquad Z\_0 = \frac{\cos \theta\_i}{\varepsilon\_0 \varepsilon\_0}. \tag{36}$$

For TM polarization,

$$Z = \frac{q}{\omega \varepsilon\_{eff}} \qquad \text{and} \qquad Z\_0 = \frac{\mu\_0 \varepsilon\_0}{\cos \theta\_i}. \tag{37}$$

In our case, the functions *F*1(*x*) and *F*2(*x*) of the equation (34) are replaced by *r* et *t* equation (35). The cost function (34) is minimized using the non-linear mean square algorithm of Levenberg-Marquardt. To ensure good results, this algorithm needs to start from a feasible point. This is why a large choice of values for the couple (*ε*(*ω*), *μ*(*ω*)) is done for the starting point in frequency. Then if the frequency sampling from the numerical simulations is fine enough and the functions considered to be continuous, the starting point chosen is the one for the previous frequency point.

For a few composites, the algorithm can converge to many solutions for large values of the thickness of the slab. These solutions are not local minima but are solutions to Fresnel equations. To choose the right solutions, two physical criteria have been defined:


These criteria are particularly appropriate for NRI resonant metamaterials and may not be adequate for all type of composites. The principal limitation is that the starting point must be far from resonance and the composite should not present high dissipative losses.

#### **3.5.2 Periodic structure infinite in the propagation direction**

If the EM wave propagation is considered in a periodic medium such as the one presented on figure 6(a), the solution of the wave equation provides solution for the propagation constant which are given by *kn* = *k* + 2*mπ*/*P* where *m* ∈ *Z* and *P* = *PL* or *PT*.

<sup>5</sup> The imaginary part of the refractive index does not allow to calculate losses by dissipation of a medium (Landau et al. (1984)). It can only represent the presence or absence of propagation in a medium.

NRI metamaterials (figure 7b) proposed in (Seetharamdoo et al. (2004)) for the reduced

Resonant Negative Refractive Index Metamaterials 185

(a) EC-SRR unit cell (b) BC-SRR unit cell

The unit cell are simulated using Ansoft HFSS and the reflection and transmission coefficients are shown figure 8. A resonance can be observed where the metamaterials are transparent to

0

0

1

2

Phase (rad)

3

1

2

Phase (rad)

3

10 12 14 16

*r t*

*r t*

Frequency (GHz)

6 7 8 9 10

Frequency (GHz)

(d) Phase of *r* and *t* for BC-SRR

(b) Phase of *r* and *t* for EC-SRR

Fig. 7. Unit cell of NRI resonant metamaterials constituted of BC-SRR and EC-SRR and metallic lines. These inclusions are printed on the dielectric teflon substrates (*ε* = 2.2,

tan *<sup>δ</sup>* <sup>=</sup> <sup>9</sup> <sup>×</sup> <sup>10</sup>−4). The periodicities *PH*=4.5 mm, *PT*=3.3 mm, and *<sup>d</sup>*=3.3 mm.

*r t*

*r t*

Fig. 8. Reflection and Transmission coefficients of metamaterials constituted of BC-SRR et

.

10 12 14 16

Frequency (GHz)

6 7 8 9 10

Frequency (GHz)

(c) Magnitude of *r* and *t* for BC-SRR

(a) Magnitude of *r* and *t* for EC-SRR

−40

−40

−30

−20

Magnitude (dB)

EC-SRR

−10

0

−30

−20

Magnitude (dB)

−10

0

bianisotropic properties they present.

(a) Réseau périodique bi-dimensionnel (b) Contour de Brillouin associé

Because the structure is periodic, the analysis of the propagation in only a unit cell of the constant *k* is enough : the fundamental analysis domain is defined for −*π* < *kP* < *π*. This fundamental domain consists sufficient information for defining propagation inside the whole structure. Figure 6(b) shows the irreducible Brillouin zone associated to the periodic structure for *PT* = *PL*. This Brillouin zone can be briefly described in terms of the propagation constants (*kx*, *ky*) in the following way:


To calculate the two dimensional dispersion diagram of an arbitrary periodic NRI resonant metamaterial, a source-free eigenmode solver of a numerical modeling tool such as Ansoft HFSS (HFSS (2004)) can be used. The calculation volume is sampled by finite elements in the case of the software HFSS and for specific periodic boundary conditions, the eigenvalues of the "periodic cavity" are searched. A couple of propagation constants (*kx*, *ky*) belonging to the Brillouin zone is imposed as boundary condition and the eigenfrequency is calculated such that the source-free Maxwell equations with the boundary conditions are satisfied. The calculation of each eigenfrequency is performed in an iterative manner (Chang (2005)). To ensure reasonable calculation time, it is thus necessary to impose two parameters which are the lowest eigenfrequency to be calculated and a limited number of eigen frequencies.

#### **4. Numerical results and interpretation of effective parameters of resonant NRI metamaterials**

The first unit cell (figure 7a) considered here as resonant NRI metamaterials is based on the metamaterial edge-side coupled split-ring resonators (EC-SRR) proposed in (Greegor et al. (2003)) because the dissipative losses presented by these metamaterials are relatively low. The second NRI metamaterial unit cell considered are based on broad-side coupled 14 Will-be-set-by-IN-TECH

(a) Réseau périodique bi-dimensionnel (b) Contour de Brillouin associé

Because the structure is periodic, the analysis of the propagation in only a unit cell of the constant *k* is enough : the fundamental analysis domain is defined for −*π* < *kP* < *π*. This fundamental domain consists sufficient information for defining propagation inside the whole structure. Figure 6(b) shows the irreducible Brillouin zone associated to the periodic structure for *PT* = *PL*. This Brillouin zone can be briefly described in terms of the propagation constants

1. the Γ*X* contour corresponds to propagation constants *kyP* = 0 et *kxP* ∈ [0; *π*]. for this

2. the *XM* contour corresponds to propagation constants *kxP* = *π* et *kyP* ∈ [0; *π*]. The

3. the *M*Γ contour corresponds to propagation constants *kxP* et *kyP* ∈ [0; *π*]. The only

To calculate the two dimensional dispersion diagram of an arbitrary periodic NRI resonant metamaterial, a source-free eigenmode solver of a numerical modeling tool such as Ansoft HFSS (HFSS (2004)) can be used. The calculation volume is sampled by finite elements in the case of the software HFSS and for specific periodic boundary conditions, the eigenvalues of the "periodic cavity" are searched. A couple of propagation constants (*kx*, *ky*) belonging to the Brillouin zone is imposed as boundary condition and the eigenfrequency is calculated such that the source-free Maxwell equations with the boundary conditions are satisfied. The calculation of each eigenfrequency is performed in an iterative manner (Chang (2005)). To ensure reasonable calculation time, it is thus necessary to impose two parameters which are the lowest eigenfrequency to be calculated and a limited number of eigen frequencies.

**4. Numerical results and interpretation of effective parameters of resonant NRI**

The first unit cell (figure 7a) considered here as resonant NRI metamaterials is based on the metamaterial edge-side coupled split-ring resonators (EC-SRR) proposed in (Greegor et al. (2003)) because the dissipative losses presented by these metamaterials are relatively low. The second NRI metamaterial unit cell considered are based on broad-side coupled

Fig. 6. Réseau périodique bi-dimensionnel et zone irréductible de Brillouin associée.

(*kx*, *ky*) in the following way:

**metamaterials**

contour, only normal incidence is considered.

incidence angles vary from 0°to 45°.

incidence angle considered is 45°.

NRI metamaterials (figure 7b) proposed in (Seetharamdoo et al. (2004)) for the reduced bianisotropic properties they present.

(a) EC-SRR unit cell (b) BC-SRR unit cell

Fig. 7. Unit cell of NRI resonant metamaterials constituted of BC-SRR and EC-SRR and metallic lines. These inclusions are printed on the dielectric teflon substrates (*ε* = 2.2, tan *<sup>δ</sup>* <sup>=</sup> <sup>9</sup> <sup>×</sup> <sup>10</sup>−4). The periodicities *PH*=4.5 mm, *PT*=3.3 mm, and *<sup>d</sup>*=3.3 mm.

The unit cell are simulated using Ansoft HFSS and the reflection and transmission coefficients are shown figure 8. A resonance can be observed where the metamaterials are transparent to

Fig. 8. Reflection and Transmission coefficients of metamaterials constituted of BC-SRR et EC-SRR

.

also positive values for the imaginary parts. The main frequency bands of interest of these

Resonant Negative Refractive Index Metamaterials 187

Negative refractive index 7.7 -8.7 GHz 11.5-13.3 GHz Negative permeability 7.9 -8.7 GHz 12.3-13.3 GHz Negative permittivity 6-10 GHz 9-16 GHz Saturation of the real part of refractive index 7.7-7.9 GHz 11.5-12.3 GHz *m*(*ε*) > 0 7.7-7.9 GHz 11.5-12.3 GHz *m*(*μ*) > 0 7.7-7.9 GHz 11.5-12.3 GHz Table 1. Frequency bands of interest for NRI metamaterials based on BC-SRR and EC-SRR.

A similar behavior can be observed for both metamaterials but with a shift in frequency. This shift as explained in (Seetharamdoo et al. (2004)) is due to higher capacitive coupling in the BC-SRR compared to the EC-SRR. There is indeed a frequency band for which the real part of the refractive index, effective permittivity and permeability are negative. However, in a part of this frequency band the imaginary parts of *ε*(*ω*) and *μ*(*ω*) are positive which not a physically correct as described in section 2.2. This frequency band deserves further analysis and in the next sections for better understanding of these results, a dispersion diagram as well as a multimodal analysis will be proposed for the BC-SRR NRI metamaterial. The choice of this metamaterial for further analysis is justified by the fact that it has also been shown to be

The dispersion diagram is calculated using the method described in section **??**. This diagram shown on figure 10(a) gives information on the modes that can propagate in the periodic medium in two dimensions in the irreducible Brillouin zone. The dispersion diagram of the

(a) Dispersion diagram (b) Real part of Refractive index

Fig. 10. (a) Two dimensional dispersion diagram of the medium with BC-SRR only and the NRI metamaterial in the irreducible Brillouin zone. (b) Superposition of the refractive index calculated from the dispersion diagram and the one calculated by the inversion methods. The shaded frequency band represents the frequency band where the refractive index is

BC-SRR EC-SRR

NRI metamaterials are given in table 1.

2D-isotropic (Seetharamdoo (2006)).

**4.2 Dispersion diagram of NRI metamaterials**

negative and where there is backward propagation.

the incident wave at the frequencies of 12.3 GHz for the EC-SRR and 8.1 GHz for the BC-SRR respectively.

#### **4.1 Effective parameters calculated by inversion methods**

The effective parameters are then calculated by inversion methods presented in the previous section and the refractive index, wave impedance and permittivity and permeability are shown on figure 9.

Fig. 9. Effective paramaters of NRI metamaterials. The upper frequency scale correspond to EC-SRR structures and lower one to BC-SRR.

The NRI metamaterials constituted of EC-SRR and BC-SRR present respectively a negative refractive index from 11.5 GHz - 13.3 GHz and from 7.7 GHz - 8.7 GHz [figure 9(a)]. It should be noted that the refractive index saturates in both cases (11.5 GHz < *f* < 12.3 GHz for the EC-SRR and 7.7 GHz < *f* < 7.9 GHz for the BC-SRR). This maximum value can be predicted by equation (26). The effective permeability shown on figure 9(d) is resonant and the imaginary part is positive. The effective permittivity shown on figure 9(c) is anti-resonant and presents 16 Will-be-set-by-IN-TECH

the incident wave at the frequencies of 12.3 GHz for the EC-SRR and 8.1 GHz for the BC-SRR

The effective parameters are then calculated by inversion methods presented in the previous section and the refractive index, wave impedance and permittivity and permeability are

> −8 −6 −4 −2 0

−5

Real part - EC-SRR Imaginary part - EC-SRR Real part- BC-SRR Imaginary part - BC-SRR

Fig. 9. Effective paramaters of NRI metamaterials. The upper frequency scale correspond to

The NRI metamaterials constituted of EC-SRR and BC-SRR present respectively a negative refractive index from 11.5 GHz - 13.3 GHz and from 7.7 GHz - 8.7 GHz [figure 9(a)]. It should be noted that the refractive index saturates in both cases (11.5 GHz < *f* < 12.3 GHz for the EC-SRR and 7.7 GHz < *f* < 7.9 GHz for the BC-SRR). This maximum value can be predicted by equation (26). The effective permeability shown on figure 9(d) is resonant and the imaginary part is positive. The effective permittivity shown on figure 9(c) is anti-resonant and presents

0

5

−20

−10

μ ( *f* ) 0

10

*Z* ( *f* )

6 7 8 9 10

−20

−10

0

10

9 10.75 12.5 14.25 16

Frequency (GHz)

9 10.75 12.5 14.25 16

6 7 8 9 10

Frequency (GHz)

(d) Complex permeability

(b) Normalized wave impedance

**4.1 Effective parameters calculated by inversion methods**

6 7 8 9 10

9 10.75 12.5 14.25 16

Frequency (GHz)

9 10.75 12.5 14.25 16

6 7 8 9 10

Frequency (GHz)

(c) Complex permittivity

EC-SRR structures and lower one to BC-SRR.

(a) Refractive index

respectively.

shown on figure 9.

−8 −6 −4 −2 0

−5

ε ( *f* ) 0

5

*n* ( *f* ) also positive values for the imaginary parts. The main frequency bands of interest of these NRI metamaterials are given in table 1.


Table 1. Frequency bands of interest for NRI metamaterials based on BC-SRR and EC-SRR.

A similar behavior can be observed for both metamaterials but with a shift in frequency. This shift as explained in (Seetharamdoo et al. (2004)) is due to higher capacitive coupling in the BC-SRR compared to the EC-SRR. There is indeed a frequency band for which the real part of the refractive index, effective permittivity and permeability are negative. However, in a part of this frequency band the imaginary parts of *ε*(*ω*) and *μ*(*ω*) are positive which not a physically correct as described in section 2.2. This frequency band deserves further analysis and in the next sections for better understanding of these results, a dispersion diagram as well as a multimodal analysis will be proposed for the BC-SRR NRI metamaterial. The choice of this metamaterial for further analysis is justified by the fact that it has also been shown to be 2D-isotropic (Seetharamdoo (2006)).

#### **4.2 Dispersion diagram of NRI metamaterials**

The dispersion diagram is calculated using the method described in section **??**. This diagram shown on figure 10(a) gives information on the modes that can propagate in the periodic medium in two dimensions in the irreducible Brillouin zone. The dispersion diagram of the

Fig. 10. (a) Two dimensional dispersion diagram of the medium with BC-SRR only and the NRI metamaterial in the irreducible Brillouin zone. (b) Superposition of the refractive index calculated from the dispersion diagram and the one calculated by the inversion methods. The shaded frequency band represents the frequency band where the refractive index is negative and where there is backward propagation.

observed whose magnitude is higher that that of the fundamental mode. In figure 11(b), this second mode can be seen to be evanescent while the first one is propagative. The value of the propagation constant of the evanescent mode is low enough to shown that it can propagate through a few layers of the structure. This implies that the evanescent modes do participate to the interference phenomena is this frequency band and this effect will be more visible with

Resonant Negative Refractive Index Metamaterials 189

This analysis can prove to be very useful to verify if the assumptions made while using the inversion methods are violated. In this case, one can conclude that the effective parameters calculated in this frequency band is incorrect and non-physical and should thus not be

The electrodynamics of NRI materials and the fundamental limitations related to the signs of refractive index, wave impedance, effective permittivity and permeability, both in real and imaginary parts have been fully described. The effective medium theory as it is applied to NRI resonant materials have been detailed with a description of the assumptions linked to this theory for cases of finite thickness in the direction of propagation and infinite dimensions. The methods used for the calculation of effective parameters have been given and applied to numerical models of NRI resonant metamaterials. Unphysical results have been obtained: the imaginary part of the effective permittivity and permeability takes positive values. It has been shown that this is mainly due to the finite size of the structure and that there is a frequency band where the results obtained by the classical inversion methods for the calculation of effective parameters are not correct and this frequency band can be defined thanks to complementary analysis like the calculation of a dispersion diagram and

Acher, O., Adenot, A. L. & Duverger, F. (2000). Fresnel coefficients at an interface with a

Baker-Jarvis, J., Janezic, M. D., Riddle, B. F., Johnk, R. T., Kabos, P., Holloway, C. L., Geyer, R. G.

division, Natl. Inst. Stand. Technol. (NIST) : Tech. Note 1536, Boulder, USA. Baker-Jarvis, J., Kabos, P. & Holloway, C. L. (2004). Nonequilibrium electromagnetics: Local and macroscopic fields and constitutive relationships, *Phys. Rev. E* 70: 036615–1–13.

Callen, H. B. & Welton, T. A. (1951). Irreversibility and generalized noise, *Phys. Rev.* 83: 34–40. Chang, K. (2005). Encyclopedia of rf and microwave engineering, John Wiley and Sons Inc

de Fornel, F. (1997). *Les Ondes évanescentes en optique et en optoélectronique*, Eyrolles, Paris. Depine, R. A. & Lakhtakia, A. (2004). Comment i on: Resonant and anti-resonant frequency

Balanis, C. A. (1989). *Advanced engineering electromagnetics*, John Wiley and Sons, Canada. Bardi, I., Remski, R., Perry, D. & Cendes, Z. (2002). Plane wave scattering from

& Grosvenor, C. A. (2004). *Measuring the permittivity and permeability of lossy materials : solids, liquids, metals, building materials and negative-index materials*, Electromagnetics

frequency-selective surfaces by the finite element method, *IEEE Trans. on Mag.*

dependence of the effective parameters of metamaterials, *Phys. Rev. E* pp. 048601–1.

lamellar composite material, *Phys. Rev. B* 62: 13748.

Berthier, S. (1993). *Optique des milieux composites*, Polytechnica, Paris.

lower dissipative losses in the resonant NRI metamaterial (Seetharamdoo (2006)).

presented or interpreted (Seetharamdoo et al. (2005)).

**5. Conclusion**

a multimodal analysis.

38: 641.

(USA).

**6. References**

metamaterial constituted of only BC-SRR (without the metallic line medium) is also shown. In the shaded frequency band, the metamaterial with BC-SRR only presents a forbidden frequency band while in association with the metallic lines, a propagated frequency band is observed. The phase velocity given by the slope of the curve is negative; the propagation is hence a backward wave propagation. The refractive index can be calculated from this phase velocity and it is compared to the one calculated using inversion methods. As it can be observed, there is indeed a frequency band (7.9 - 8.7 GHz) where both results are in good agreement.

However, in the frequency band (7.7 - 7.9 GHz) where the calculation of effective parameters by inversion methods yield unphysical results, the dispersion diagram shows no propagation. This strongly suggests that the results obtained by the inversion method in this frequency band is not correct and is caused by the finite thickness of the structure. If the structure were large enough in the direction of propagation to represent a periodic or a continuous medium, these unphysical results would not have been obtained. Unfortunately, either in measurements or in the design of NRI metamaterials using numerical modeling, it is not always possible to analyze large structures due to the cost or resources required for the calculation.

#### **4.3 Solution proposed: multi-modal analysis**

A simple solution to verify the validity of the results given by inversion methods is to make a multimodal analysis of the periodic NRI resonant metamaterial to detect the existence of higher order modes which would definitely result in incorrect effective parameters calculation by inversion methods using a finite-size structure in the direction of propagation. Figure 11 depicts the modal *S*<sup>21</sup> parameters and the associated propagation constants for the first two modes of the periodic structure 6.

Fig. 11. (a) Modal Scattering parameter *S*<sup>21</sup> for the first two modes. (b) Propagation constants of these first two modes. Only Im(*γ*) is shown for the first mode Re(*γ*) for the second mode because the corresponding imaginary and real parts are close to zero. The shaded frequency band represents the frequency band where the unphysical results have been observed.

The scattering parameter *S*<sup>21</sup> of the fundamental mode presents a resonance at frequency close to 7.8 GHz. Around this frequency and in the shaded frequency band, a second mode can be

<sup>6</sup> *S*<sup>21</sup> Mode2:Mode1 represents for instance what is observed from the profile of the second mode on access 2 when only the first mode is excited on access 1.

observed whose magnitude is higher that that of the fundamental mode. In figure 11(b), this second mode can be seen to be evanescent while the first one is propagative. The value of the propagation constant of the evanescent mode is low enough to shown that it can propagate through a few layers of the structure. This implies that the evanescent modes do participate to the interference phenomena is this frequency band and this effect will be more visible with lower dissipative losses in the resonant NRI metamaterial (Seetharamdoo (2006)).

This analysis can prove to be very useful to verify if the assumptions made while using the inversion methods are violated. In this case, one can conclude that the effective parameters calculated in this frequency band is incorrect and non-physical and should thus not be presented or interpreted (Seetharamdoo et al. (2005)).

#### **5. Conclusion**

18 Will-be-set-by-IN-TECH

metamaterial constituted of only BC-SRR (without the metallic line medium) is also shown. In the shaded frequency band, the metamaterial with BC-SRR only presents a forbidden frequency band while in association with the metallic lines, a propagated frequency band is observed. The phase velocity given by the slope of the curve is negative; the propagation is hence a backward wave propagation. The refractive index can be calculated from this phase velocity and it is compared to the one calculated using inversion methods. As it can be observed, there is indeed a frequency band (7.9 - 8.7 GHz) where both results are in good

However, in the frequency band (7.7 - 7.9 GHz) where the calculation of effective parameters by inversion methods yield unphysical results, the dispersion diagram shows no propagation. This strongly suggests that the results obtained by the inversion method in this frequency band is not correct and is caused by the finite thickness of the structure. If the structure were large enough in the direction of propagation to represent a periodic or a continuous medium, these unphysical results would not have been obtained. Unfortunately, either in measurements or in the design of NRI metamaterials using numerical modeling, it is not always possible to analyze large structures due to the cost or resources required for the

A simple solution to verify the validity of the results given by inversion methods is to make a multimodal analysis of the periodic NRI resonant metamaterial to detect the existence of higher order modes which would definitely result in incorrect effective parameters calculation by inversion methods using a finite-size structure in the direction of propagation. Figure 11 depicts the modal *S*<sup>21</sup> parameters and the associated propagation constants for the first two

Fig. 11. (a) Modal Scattering parameter *S*<sup>21</sup> for the first two modes. (b) Propagation constants of these first two modes. Only Im(*γ*) is shown for the first mode Re(*γ*) for the second mode because the corresponding imaginary and real parts are close to zero. The shaded frequency band represents the frequency band where the unphysical results have been observed.

The scattering parameter *S*<sup>21</sup> of the fundamental mode presents a resonance at frequency close to 7.8 GHz. Around this frequency and in the shaded frequency band, a second mode can be <sup>6</sup> *S*<sup>21</sup> Mode2:Mode1 represents for instance what is observed from the profile of the second mode on

6 7 8 9 10

Im(γ) mode 1 Re(γ) mode 2

Fréquence (GHz)

(b)

γ

agreement.

calculation.

**4.3 Solution proposed: multi-modal analysis**

6 7 8 9 10

S21 mode 1:mode 1 S21 mode 2:mode 1

Fréquence (GHz)

access 2 when only the first mode is excited on access 1.

(a)

modes of the periodic structure 6.

−40

−30

−20

Magnitude (dB)

−10

0

The electrodynamics of NRI materials and the fundamental limitations related to the signs of refractive index, wave impedance, effective permittivity and permeability, both in real and imaginary parts have been fully described. The effective medium theory as it is applied to NRI resonant materials have been detailed with a description of the assumptions linked to this theory for cases of finite thickness in the direction of propagation and infinite dimensions. The methods used for the calculation of effective parameters have been given and applied to numerical models of NRI resonant metamaterials. Unphysical results have been obtained: the imaginary part of the effective permittivity and permeability takes positive values. It has been shown that this is mainly due to the finite size of the structure and that there is a frequency band where the results obtained by the classical inversion methods for the calculation of effective parameters are not correct and this frequency band can be defined thanks to complementary analysis like the calculation of a dispersion diagram and a multimodal analysis.

#### **6. References**


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**1. Introduction** 

conventional real-world materials).

which is impossible with conventional lenses.

**8** 

**Nonlinear Left-Handed Metamaterials** 

Metamaterials are artificial structures that are designed to exhibit specific electromagnetic properties required for different applications but not commonly found in nature. The methodology of synthesizing materials composed of micro- and nano-structured components that mimic the electromagnetic response of individual atoms and molecules (meta-atoms and meta-molecules) has proven to be very productive and resulted in the development of metamaterials exhibiting strong magnetic response at microwave and optical frequencies and so-called left-handed metamaterials (LHMs) (both impossible in

LHMs are designed to exhibit simultaneously negative permittivity and permeability (Veselago, 1968; Engheta & Ziolkowski, 2006). In 2000, Smith et al. developed the first experimental left-handed structure, which was composed of metallic split-ring resonators and thin metal wires (Smith et. al., 2000; Shelby et. al., 2001). An alternative transmission line approach for left-handed materials was proposed, almost simultaneously, by several different groups (Belyantsev & Kozyrev, 2002; Caloz & Itoh, 2002; Iyer & Eleftheriades, 2002). This approach, based on nonresonant components, allows for low-loss left-handed structures with broad bandwidth. The unique electrodynamic properties of these materials, first postulated by Veselago in 1968, include the reversal of Snell's law, the Doppler effect, Vavilov-Cherenkov radiation, and negative refractive index, making theses materials attractive for new types of RF and microwave components. The range of applications for LHMs is extensive, and opportunities abound for development of new and powerful imaging and communication techniques. The most tantalizing of these potential applications is the possibility of realizing "perfect" (diffraction-free) lenses based on their inherent negative index of refraction (Pendry, 2000). The slab of LHM can act as an ideal (diffractionfree) lens and thus capable of producing images of objects without any loss of information

Most studies of LHMs have been concerned with linear wave propagation, and have inspired many applications that were unthinkable in the past (Engheta & Ziolkowski, 2006; Lai et. al., 2004) such as LH phase shifters (Anioniades & Eleftheriades, 2003), LH directional couplers (Caloz et. al., 2004a; Liu et. al., 2002a), and leaky-wave antennas (Lim et. al., 2005;

Alexander B. Kozyrev and Daniel W. van der Weide

*Department of Electrical and Computer Engineering* 

*University of Wisconsin,* 

*Madison USA* 

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## **Nonlinear Left-Handed Metamaterials**

Alexander B. Kozyrev and Daniel W. van der Weide

*Department of Electrical and Computer Engineering University of Wisconsin, Madison USA* 

#### **1. Introduction**

22 Will-be-set-by-IN-TECH

192 Metamaterial

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Veselago, V. G. (1968). The electrodynamics of substances with simultaneously negative values

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Stratton, J. A. (1941). *Electromagnetic theory*, McGraw-Hill book company, New-York. Tretyakov, S. (2003). *Analytical modeling in applied electromagnetics*, Artech House, USA. Tretyakov, S. A. (2005). Research on negative refraction and backward-wave media: A

*to optics - EPFL Latsis symposium*, Lausanne, Suisse, pp. 30–35.

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of *ε* and *μ*, *Sov. Phys. Usp.* 10: 509.

*Trans. on circuits and systems* 18: 332–336.

Metamaterials are artificial structures that are designed to exhibit specific electromagnetic properties required for different applications but not commonly found in nature. The methodology of synthesizing materials composed of micro- and nano-structured components that mimic the electromagnetic response of individual atoms and molecules (meta-atoms and meta-molecules) has proven to be very productive and resulted in the development of metamaterials exhibiting strong magnetic response at microwave and optical frequencies and so-called left-handed metamaterials (LHMs) (both impossible in conventional real-world materials).

LHMs are designed to exhibit simultaneously negative permittivity and permeability (Veselago, 1968; Engheta & Ziolkowski, 2006). In 2000, Smith et al. developed the first experimental left-handed structure, which was composed of metallic split-ring resonators and thin metal wires (Smith et. al., 2000; Shelby et. al., 2001). An alternative transmission line approach for left-handed materials was proposed, almost simultaneously, by several different groups (Belyantsev & Kozyrev, 2002; Caloz & Itoh, 2002; Iyer & Eleftheriades, 2002). This approach, based on nonresonant components, allows for low-loss left-handed structures with broad bandwidth. The unique electrodynamic properties of these materials, first postulated by Veselago in 1968, include the reversal of Snell's law, the Doppler effect, Vavilov-Cherenkov radiation, and negative refractive index, making theses materials attractive for new types of RF and microwave components. The range of applications for LHMs is extensive, and opportunities abound for development of new and powerful imaging and communication techniques. The most tantalizing of these potential applications is the possibility of realizing "perfect" (diffraction-free) lenses based on their inherent negative index of refraction (Pendry, 2000). The slab of LHM can act as an ideal (diffractionfree) lens and thus capable of producing images of objects without any loss of information which is impossible with conventional lenses.

Most studies of LHMs have been concerned with linear wave propagation, and have inspired many applications that were unthinkable in the past (Engheta & Ziolkowski, 2006; Lai et. al., 2004) such as LH phase shifters (Anioniades & Eleftheriades, 2003), LH directional couplers (Caloz et. al., 2004a; Liu et. al., 2002a), and leaky-wave antennas (Lim et. al., 2005;

Nonlinear Left-Handed Metamaterials 195

ω

ω*B* 

4ω*B* 

> π

2 <sup>0</sup> <sup>0</sup> 1

**(c)**

*L C*

*B*

ω=

**LH**

*j*

**RH** 

β

Fig. 1. (a) Equivalent circuit of a LH NLTL; (b) Equivalent circuit of a dual RH NLTL; (c) Typical dispersion curves of LH NLTL (solid line) and RH NLTL (dashed line). Here

Nonlinear transmission lines first drew attention in connection with the idea of distributed parametric amplification. It had been predicted that a distributed parametric amplifier or oscillator circuit could exhibit superior stability of operation and efficiency over lumped parametric circuits (Cullen, 1958; Tien, 1958). Lumped parametric amplifiers were popular as very low-noise alternatives to vacuum tubes prior to the widespread use of semiconductor amplifiers (Louisell, 1960). (Parametric resonance responsible for amplification in lumped circuits is similar to the physical mechanism playing on a swing which allows large amplitudes by alternately raising and lowering the center of mass at a certain relation between the frequency of the swing and the frequency of external force.) Their complexity (they require external resonators and matching circuits) and low efficiencies however made them less attractive for widespread use. Conventional NLTLs were thought to be very promising candidates for use in distributed amplifiers because they do not require external resonant circuits and conversion efficiency was claimed to be very high due to a cumulative effect of parametrically interacting waves propagating along

It turned out that parametric interactions (such as three- and four-wave mixing of phase matched waves) in RH NLTLs typically compete with shock wave formation. For instance, parametric generation and amplification in dispersionless RH transmission lines is entirely suppressed by shock wave formation (Landauer, 1960a; Landauer, 1960b). In contrast to conventional NLTLs, both nonlinearity and dispersion present in LH NLTLs (see Fig. 1) lead to waveform spreading (Caloz et. al., 2004b), consequently making shock wave and electronic soliton formation impossible. Anomalous dispersion makes sharp field transients in left-handed NLTL unstable. Once created, they decompose very quickly during propagation of the waveform due to substantial difference in the phase velocities of the propagating waves. This inability to form shock waves enables a variety of parametric processes to occur instead (Kozyrev et. al., 2005; Kozyrev & van der Weide, 2005a). Furthermore, since the parametric interactions no longer compete with shock wave

( )1/2 0 0

*L*<sup>0</sup> *L*0*L*0*L*<sup>0</sup>

*L*<sup>0</sup> *L*<sup>0</sup> *L*0*L*0*L*<sup>0</sup>

**(a)** 

*C(V) C(V) C(V)*

**(b)**

*C(V) C(V) C(V) C(V) C(V)*

*<sup>B</sup>* = 1 / *L C* .

ω

NLTLs.

Liu et. al., 2002b; Grbic & Eleftheriades, 2002). Materials that combine *nonlinearity* with the anomalous dispersion exhibited by LH media (Lapine & Gorkunov, 2004; Lapine et. al., 2003; Powell et. al., 2007; Shadrivov et.al., 2006), however, give rise to a new class of phenomena and promising applications (Zharov et.al., 2003; Shadrivov & Kivshar, 2005; Shalaev, 2007). Here we present a review of the basic *nonlinear* wave propagation phenomena in LH media. We consider left-handed nonlinear transmission lines (LH NLTL) as the simplest systems that would allow us to combine anomalous dispersion with nonlinearity in a controlled fashion. Understanding the nonlinear phenomena in LH NLTL media is important for both the development of new devices and improvement of the performance of recent tunable devices based on LH NLTLs like phase shifters (Kim et. al., 2005a), tunable leaky-wave antennas (Lai et. al., 2002; Sievenpiper, 2005) and notch filters (Gil et. al., 2004).

#### **2. Comparison of conventional RH and LH nonlinear transmission lines**

The transmission line approach proves to be a useful description of LH media. It provides insight into the physical phenomena of LH media and is an efficient design tool for LH applications (Lai et. al., 2004). A LH NLTL is the dual of a conventional nonlinear transmission line shown in Fig. 1*b* where inductors are replaced with capacitors and capacitors with inductors. The effective permeability and permittivity of one-dimensional transmission line metamaterials in the lossless case are expressed as follows:

$$\mu\_{\rm eff} = -\frac{2\,d}{\,\rho^2 \mathcal{C}\_L} \; ; \; \mathcal{E}\_{\rm eff} = -\frac{d}{\,\rho^2 \mathcal{L}\_L} \;$$

where *d* is the period of the LH NLTL and ω is the radian frequency. In contrast with RH NLTL where capacitance gives rise to electric nonlinearity, nonlinear capacitances *CL* introduce magnetic-type nonlinearity into the LH NLTL (i.e. effective magnetic permeability becomes nonlinear).

Although both the RH and LH NLTLs use the same components arranged in a similar way, the performance of these two circuits is dramatically different. This difference primarily comes from the difference in their dispersion characteristics (see Fig. 1*c*).

A conventional (right-handed) nonlinear transmission line has normal dispersion and frequency increases with the wavenumber. The fundamental wave can travel synchronously with its higher harmonics. In contrast to the RH NLTL, the LH transmission line exhibits anomalous dispersion and frequency decreases with the wave number (see Fig. 1*c*). The waves propagating in such media are also known as backward waves because the direction of group velocity *vg* is opposite to phase velocity ( 0 *p g v v*⋅ < ). The fundamental wave can travel synchronously with its higher harmonics. The nonlinearity in RH NLTLs provides energy flow to higher frequencies, which results in waveform sharpening and shock wave formation (Gaponov et. al., 1967; Kataev, 1966). Dispersion, however, results in waveform spreading. If a transmission line exhibits both nonlinearity and dispersion, the latter may compensate the nonlinearity, thus resulting in the formation of temporal solitons (Hirota & Suzuki, 1973).

Liu et. al., 2002b; Grbic & Eleftheriades, 2002). Materials that combine *nonlinearity* with the anomalous dispersion exhibited by LH media (Lapine & Gorkunov, 2004; Lapine et. al., 2003; Powell et. al., 2007; Shadrivov et.al., 2006), however, give rise to a new class of phenomena and promising applications (Zharov et.al., 2003; Shadrivov & Kivshar, 2005; Shalaev, 2007). Here we present a review of the basic *nonlinear* wave propagation phenomena in LH media. We consider left-handed nonlinear transmission lines (LH NLTL) as the simplest systems that would allow us to combine anomalous dispersion with nonlinearity in a controlled fashion. Understanding the nonlinear phenomena in LH NLTL media is important for both the development of new devices and improvement of the performance of recent tunable devices based on LH NLTLs like phase shifters (Kim et. al., 2005a), tunable leaky-wave antennas (Lai et. al., 2002; Sievenpiper, 2005) and notch filters

**2. Comparison of conventional RH and LH nonlinear transmission lines** 

transmission line metamaterials in the lossless case are expressed as follows:

*eff*

comes from the difference in their dispersion characteristics (see Fig. 1*c*).

μ

where *d* is the period of the LH NLTL and

2 2

ω

*L d C*

ω

NLTL where capacitance gives rise to electric nonlinearity, nonlinear capacitances *CL* introduce magnetic-type nonlinearity into the LH NLTL (i.e. effective magnetic permeability

Although both the RH and LH NLTLs use the same components arranged in a similar way, the performance of these two circuits is dramatically different. This difference primarily

A conventional (right-handed) nonlinear transmission line has normal dispersion and frequency increases with the wavenumber. The fundamental wave can travel synchronously with its higher harmonics. In contrast to the RH NLTL, the LH transmission line exhibits anomalous dispersion and frequency decreases with the wave number (see Fig. 1*c*). The waves propagating in such media are also known as backward waves because the direction of group velocity *vg* is opposite to phase velocity ( 0 *p g v v*⋅ < ). The fundamental wave can travel synchronously with its higher harmonics. The nonlinearity in RH NLTLs provides energy flow to higher frequencies, which results in waveform sharpening and shock wave formation (Gaponov et. al., 1967; Kataev, 1966). Dispersion, however, results in waveform spreading. If a transmission line exhibits both nonlinearity and dispersion, the latter may compensate the nonlinearity, thus resulting in the formation of temporal solitons (Hirota &

= − ; *eff* <sup>2</sup>

ε

*L d L*

is the radian frequency. In contrast with RH

ω= −

The transmission line approach proves to be a useful description of LH media. It provides insight into the physical phenomena of LH media and is an efficient design tool for LH applications (Lai et. al., 2004). A LH NLTL is the dual of a conventional nonlinear transmission line shown in Fig. 1*b* where inductors are replaced with capacitors and capacitors with inductors. The effective permeability and permittivity of one-dimensional

(Gil et. al., 2004).

becomes nonlinear).

Suzuki, 1973).

Fig. 1. (a) Equivalent circuit of a LH NLTL; (b) Equivalent circuit of a dual RH NLTL; (c) Typical dispersion curves of LH NLTL (solid line) and RH NLTL (dashed line). Here ( )1/2 0 0 ω*<sup>B</sup>* = 1 / *L C* .

Nonlinear transmission lines first drew attention in connection with the idea of distributed parametric amplification. It had been predicted that a distributed parametric amplifier or oscillator circuit could exhibit superior stability of operation and efficiency over lumped parametric circuits (Cullen, 1958; Tien, 1958). Lumped parametric amplifiers were popular as very low-noise alternatives to vacuum tubes prior to the widespread use of semiconductor amplifiers (Louisell, 1960). (Parametric resonance responsible for amplification in lumped circuits is similar to the physical mechanism playing on a swing which allows large amplitudes by alternately raising and lowering the center of mass at a certain relation between the frequency of the swing and the frequency of external force.) Their complexity (they require external resonators and matching circuits) and low efficiencies however made them less attractive for widespread use. Conventional NLTLs were thought to be very promising candidates for use in distributed amplifiers because they do not require external resonant circuits and conversion efficiency was claimed to be very high due to a cumulative effect of parametrically interacting waves propagating along NLTLs.

It turned out that parametric interactions (such as three- and four-wave mixing of phase matched waves) in RH NLTLs typically compete with shock wave formation. For instance, parametric generation and amplification in dispersionless RH transmission lines is entirely suppressed by shock wave formation (Landauer, 1960a; Landauer, 1960b). In contrast to conventional NLTLs, both nonlinearity and dispersion present in LH NLTLs (see Fig. 1) lead to waveform spreading (Caloz et. al., 2004b), consequently making shock wave and electronic soliton formation impossible. Anomalous dispersion makes sharp field transients in left-handed NLTL unstable. Once created, they decompose very quickly during propagation of the waveform due to substantial difference in the phase velocities of the propagating waves. This inability to form shock waves enables a variety of parametric processes to occur instead (Kozyrev et. al., 2005; Kozyrev & van der Weide, 2005a). Furthermore, since the parametric interactions no longer compete with shock wave

Nonlinear Left-Handed Metamaterials 197

other parametric instabilities which require long distances of propagation for energy exchange to occur because of long coherence distance (due to phase matching). A somewhat similar singular behavior of the second harmonic amplitude was predicted for the wave

We fabricated a 4-section LH NLTL having identical sections (shown in Fig. 2*a*) (Kozyrev

thickness *h* = 1.27 mm. The nonlinear capacitance in each section is formed by two back-toback M/A-COM hyperabrupt junction GaAs flip-chip varactor diodes (MA46H120) with DC bias applied between them. Shunt inductances were implemented with 0.12 mm diameter copper wires connecting the pads to the ground plane on the back side of the board. The pads on the board surface, together with inherent parasitics introduce unavoidable series inductance and shunt capacitance, making the whole circuit a composite right/left-handed transmission line having the equivalent circuit shown in Fig. 2*b*. The dispersion characteristic of a composite right/left-handed transmission line has two passbands divided by the stop band. The low frequency passband exhibits anomalous dispersion (left-handed passband) while the high-frequency one is right-handed. Fig. 3 shows the magnitude of the linear wave transmission (*S*21) of the LH NLTLs. We measured a –6 dB cut-off frequency at 2.7 GHz for 0 V-bias. The frequency region from 2.7 GHz to 8 GHz for 0 V bias corresponds to the left-handed passband. Parameters of the circuit model in Fig. 2*b* were extracted from the *S*-parameters measured at 0 V bias. They are *CL* = 0.99 pF,

> **Wire Inductors**

> > *LR*/2

**Varactors**

**Copper pad**

*<sup>R</sup> CL <sup>d</sup>*

*LL*

*CR*/4 *CR*/4

(b)

*CR*/2

Fig. 2. (a) Fabricated 4-section LH NLTL and (b) equivalent circuit of one stage.

*C<sup>L</sup> LR*/2

*Rd*

(a)

*V*dc 1.5 kΩ 1.5 kΩ **Bias supply** ε

= 10.2 and

et.al, 2005). The circuit was realized on a Rogers RT/Duroid 3010 board with *<sup>r</sup>*

reflected from a slab of nonlinear LH medium (Agranovich et. al., 2004).

**3.2 Experiment** 

*LL* = 1.695 nH, *CR* = 0.05 pF, *LR* = 0.966 nH.

formation, it is possible to use stronger nonlinearities, consequently achieving considerable gain in shorter transmission lines (Kozyrev et. al., 2006).

Both theoretical (Kozyrev & van der Weide, 2005a, 2004) and experimental (Kozyrev et. al., 2005, 2006) investigation demonstrate that nonlinear wave form evolution in a LH NLTL can be understood in terms of competition between harmonic generation, subharmonic generation, frequency down conversion and parametric instabilities.

#### **3. Higher harmonic generation**

#### **3.1 Theoretical consideration**

In short LH NLTLs, harmonic generation dominates over parametric instabilities (Kozyrev et. al., 2005). The amplitude of the second harmonic in the *n*-th section of a LH NLTL *V n* <sup>2</sup> ( ) can be obtained using a small signal approach described in (Kozyrev & van der Weide, 2005):

$$\left| V\_{\perp} \left( n \right) \right| = \frac{4K\_{N}V\_{1}^{2} \left( 0 \right) \sin^{2} \left( \frac{\beta\_{1}}{2} \right) \cdot \sin^{2} \left( \beta\_{1} \right)}{\sin \left( \frac{\beta\_{2}}{2} \right) \cdot \left[ \sin^{2} \left( \frac{\beta\_{2}}{2} \right) - \sin^{2} \left( \beta\_{1} \right) \right]} \cdot e^{-m} \cdot \left| \sin \left[ \left( \beta\_{2} - 2\beta\_{1} \right) n / 2 \right] \right| \tag{1}$$

where *KN* is a "nonlinearity factor" dependent only on diode parameters, 1 *V* (0) is the voltage at the input of the LH NLTL, α is the attenuation constant, *n* is the section number and β1 and β2 are the propagation constants (phase shift per section) for the fundamental wave and its second harmonic, respectively.

The fundamental wave propagating in the LH NLTL is always badly mismatched with its higher harmonics due to inherent anomalous dispersion, yet the generation of higher harmonics can still be very effective. This is possible because of "amplitude singularities". The denominator in (1) has zeros when

$$
\sin^2\left(\beta\_2/2\right) - \sin^2\left(\beta\_1\right) \to 0 \,\,.\tag{2}
$$

Due to phase mismatch, the amplitude of the second harmonic varies rapidly with distance. This gives rise to a highly localized energy exchange between the fundamental wave and its second harmonic. It is apparent from (1) that the maximum amplitude of the second harmonic at the end of the *N*-section line is achieved when

$$(\beta\_2 - 2\beta\_1)N = (2k+1)\pi \; , \; k = 0, 1, 2, 3\dots \tag{3}$$

The same approach applied to RH NLTL predicts linear growth of the second harmonic amplitude (in the lossless case) due to its phase-matching with fundamental wave (Kozyrev & van der Weide, 2005a; Champlin & Singh, 1986). Thus, the theoretical analysis of 2nd harmonic generation in LH NLTLs shows that, despite the large phase mismatch, inherent anomalous dispersion enables the possibility of faster-than-linear growth of the second harmonic amplitude as predicted by (1) in a narrow frequency range where condition (2) is satisfied. This fact explains the dominance of harmonic generation in short LH NLTLs over other parametric instabilities which require long distances of propagation for energy exchange to occur because of long coherence distance (due to phase matching). A somewhat similar singular behavior of the second harmonic amplitude was predicted for the wave reflected from a slab of nonlinear LH medium (Agranovich et. al., 2004).

#### **3.2 Experiment**

196 Metamaterial

formation, it is possible to use stronger nonlinearities, consequently achieving considerable

Both theoretical (Kozyrev & van der Weide, 2005a, 2004) and experimental (Kozyrev et. al., 2005, 2006) investigation demonstrate that nonlinear wave form evolution in a LH NLTL can be understood in terms of competition between harmonic generation, subharmonic

In short LH NLTLs, harmonic generation dominates over parametric instabilities (Kozyrev et. al., 2005). The amplitude of the second harmonic in the *n*-th section of a LH NLTL *V n* <sup>2</sup> ( ) can be obtained using a small signal approach described in (Kozyrev & van der Weide,

( )

β

<sup>≈</sup> ⋅ ⋅ <sup>−</sup> ⋅ −

β

1

2 are the propagation constants (phase shift per section) for the fundamental

<sup>2</sup> sin 2 2

α

−

2 1

 β (1)

β

is the attenuation constant, *n* is the section number

− → . (2)

*N k* , *k* = 0,1,2,3... (3)

( ) ( )

<sup>⋅</sup>

β

*<sup>N</sup> <sup>n</sup>*

22 2 1 1 1

4 0 sin sin

*K V*

β

harmonic at the end of the *N*-section line is achieved when

β

sin sin sin 2 2

2 2 2 2

 β

α

β

( )( ) 2 1

 − =+ 2 21 β

( ) <sup>2</sup>

where *KN* is a "nonlinearity factor" dependent only on diode parameters, 1 *V* (0) is the

The fundamental wave propagating in the LH NLTL is always badly mismatched with its higher harmonics due to inherent anomalous dispersion, yet the generation of higher harmonics can still be very effective. This is possible because of "amplitude singularities".

> ( ) ( ) 2 2 2 1 sin 2 sin 0

Due to phase mismatch, the amplitude of the second harmonic varies rapidly with distance. This gives rise to a highly localized energy exchange between the fundamental wave and its second harmonic. It is apparent from (1) that the maximum amplitude of the second

The same approach applied to RH NLTL predicts linear growth of the second harmonic amplitude (in the lossless case) due to its phase-matching with fundamental wave (Kozyrev & van der Weide, 2005a; Champlin & Singh, 1986). Thus, the theoretical analysis of 2nd harmonic generation in LH NLTLs shows that, despite the large phase mismatch, inherent anomalous dispersion enables the possibility of faster-than-linear growth of the second harmonic amplitude as predicted by (1) in a narrow frequency range where condition (2) is satisfied. This fact explains the dominance of harmonic generation in short LH NLTLs over

 β

 π

*V n e n*

gain in shorter transmission lines (Kozyrev et. al., 2006).

**3. Higher harmonic generation** 

( )

voltage at the input of the LH NLTL,

The denominator in (1) has zeros when

wave and its second harmonic, respectively.

**3.1 Theoretical consideration** 

2005):

and β1 and β

generation, frequency down conversion and parametric instabilities.

We fabricated a 4-section LH NLTL having identical sections (shown in Fig. 2*a*) (Kozyrev et.al, 2005). The circuit was realized on a Rogers RT/Duroid 3010 board with *<sup>r</sup>* ε = 10.2 and thickness *h* = 1.27 mm. The nonlinear capacitance in each section is formed by two back-toback M/A-COM hyperabrupt junction GaAs flip-chip varactor diodes (MA46H120) with DC bias applied between them. Shunt inductances were implemented with 0.12 mm diameter copper wires connecting the pads to the ground plane on the back side of the board. The pads on the board surface, together with inherent parasitics introduce unavoidable series inductance and shunt capacitance, making the whole circuit a composite right/left-handed transmission line having the equivalent circuit shown in Fig. 2*b*. The dispersion characteristic of a composite right/left-handed transmission line has two passbands divided by the stop band. The low frequency passband exhibits anomalous dispersion (left-handed passband) while the high-frequency one is right-handed. Fig. 3 shows the magnitude of the linear wave transmission (*S*21) of the LH NLTLs. We measured a –6 dB cut-off frequency at 2.7 GHz for 0 V-bias. The frequency region from 2.7 GHz to 8 GHz for 0 V bias corresponds to the left-handed passband. Parameters of the circuit model in Fig. 2*b* were extracted from the *S*-parameters measured at 0 V bias. They are *CL* = 0.99 pF, *LL* = 1.695 nH, *CR* = 0.05 pF, *LR* = 0.966 nH.

*<sup>R</sup> CL <sup>d</sup> LL LR*/2 *CR*/2 *C<sup>L</sup> LR*/2 *CR*/4 *CR*/4 *Rd* (b)

Fig. 2. (a) Fabricated 4-section LH NLTL and (b) equivalent circuit of one stage.

Nonlinear Left-Handed Metamaterials 199

is close to the transmission maximum, which is located in the middle of the left-handed passband. A fundamental of 2.875 GHz generates numerous higher harmonics, with the second harmonic dominating over the fundamental and the other harmonics. Thus, the LH NLTL combines the properties of both a harmonic generator and a bandpass filter, and

The conversion efficiency observed in the LH NLTL is comparable with the per-stage efficiency of a hybrid Schottky-diode RH NLTL operated in a lower frequency range

Under certain circumstances, harmonic generation may compete with different parametric processes, resulting in unstable harmonic generation. Effective parametric interaction in medium exhibiting a second-order nonlinearity generally requires phase matching of three waves. The anomalous dispersion of a LH NLTL system enables effective parametric

In the "parametric oscillator configuration", a high-frequency backward pump wave having

port of a LH NLTL. It generates two other waves having frequencies 1*f* and 2*f* , such that 1 2 *f* < *f* and 12 3 *f* + = *f f* . The wave having frequency 2*f* propagates in the opposite direction relative to the pump wave and the wave having frequency 1*f* (this is emphasized in (4) with the minus sign). We therefore have a similar situation to backward wave parametric generation (Gorshkov et. al., 1998; Harris, 1966). The backward-propagating parametrically generated wave 2*f* enables internal feedback that results in a considerable

If the amplitude of a high-frequency pump wave exceeds a certain threshold value, it may parametrically generate two other waves. This threshold value depends on the loss present in the LH NLTL, its length and the boundary conditions (matching) at the input and output. No parametric generation occurs when the amplitude of the voltage source is below this value. However, when a weak signal wave is fed into the LH NLTL together with a pump wave having an amplitude below the threshold value, a parametric amplification is observed. In this case, we have two input waves: an intense pump wave and a weak signal wave (Yariv, 1988). The power from the pump wave is transferred to the signal wave, thus amplifying it. A third parasitic idler wave is generated which provides phase matching. From a previous analysis (Gorshkov, et. al. 1998), for the lossless case, the frequencies and

Though generation of higher harmonics dominates in short LH NLTLs, in longer transmission lines parametric interactions predominate. We observed efficient parametric

ββ

 β

12 3 − = (4)

3 is excited by the voltage source connected at the input

12 3 *fff* + = ,

β

energy transfer from the pump wave to the parametrically excited waves.

powers of these waves also obey the nonlinear Manley-Rowe relations.

under certain conditions may provide an almost pure higher harmonic at its output.

(Duchamp et. al., 2003).

interactions of the type:

**4.2 Experiment** 

a frequency *f*3 and wavenumber

**4.1 Theory** 

**4. Parametric generation and amplification** 

The measured results qualitatively confirm our predictions using small-signal analysis. Figure 4 shows the spectrum of the waveform from the output of 4-section LH NLTL as measured with an Agilent E4448A Spectrum Analyzer, and corresponds to the maximum of the second harmonic conversion efficiency.

Fig. 3. Measured magnitude of *S*21 parameter for four-section LH NLTL.

Fig. 4. Spectrum of the output waveform generated by a four-section LH NLTL fed by 2.875 GHz, +17.9 dBm input signal at reverse bias voltage of 6.4 V.

The measured value for the second harmonic conversion efficiency in this 4-section LH NLTL was 19% at 2.875 GHz, using a +17.9 dBm input signal and a reverse bias voltage of 6.4 V. The second harmonic power delivered into a 50 Ω load was +10.72 dBm. The fundamental wave is close to the Bragg cutoff frequency (note the magnitude of *S*21 for the 4 section LH NLTL at bias voltage 6 V as shown in Fig. 2*a*), and thus falls into frequency range for which small-signal analysis predicts amplitude singularity. The second harmonic wave is close to the transmission maximum, which is located in the middle of the left-handed passband. A fundamental of 2.875 GHz generates numerous higher harmonics, with the second harmonic dominating over the fundamental and the other harmonics. Thus, the LH NLTL combines the properties of both a harmonic generator and a bandpass filter, and under certain conditions may provide an almost pure higher harmonic at its output.

The conversion efficiency observed in the LH NLTL is comparable with the per-stage efficiency of a hybrid Schottky-diode RH NLTL operated in a lower frequency range (Duchamp et. al., 2003).

#### **4. Parametric generation and amplification**

#### **4.1 Theory**

198 Metamaterial

The measured results qualitatively confirm our predictions using small-signal analysis. Figure 4 shows the spectrum of the waveform from the output of 4-section LH NLTL as measured with an Agilent E4448A Spectrum Analyzer, and corresponds to the maximum of

2468

Fig. 3. Measured magnitude of *S*21 parameter for four-section LH NLTL.

frequency (GHz)

2 4 6 8 10 12 14 16 18

frequency (GHz)

Fig. 4. Spectrum of the output waveform generated by a four-section LH NLTL fed by 2.875

The measured value for the second harmonic conversion efficiency in this 4-section LH NLTL was 19% at 2.875 GHz, using a +17.9 dBm input signal and a reverse bias voltage of 6.4 V. The second harmonic power delivered into a 50 Ω load was +10.72 dBm. The fundamental wave is close to the Bragg cutoff frequency (note the magnitude of *S*21 for the 4 section LH NLTL at bias voltage 6 V as shown in Fig. 2*a*), and thus falls into frequency range for which small-signal analysis predicts amplitude singularity. The second harmonic wave

 4-sec (*V*bias=0 V) 4-sec (*V*bias

=6 V)

the second harmonic conversion efficiency.



GHz, +17.9 dBm input signal at reverse bias voltage of 6.4 V.



Spectrum at the output (dBm)

0

10

20


Magnitude S

21 (dB)


0

Under certain circumstances, harmonic generation may compete with different parametric processes, resulting in unstable harmonic generation. Effective parametric interaction in medium exhibiting a second-order nonlinearity generally requires phase matching of three waves. The anomalous dispersion of a LH NLTL system enables effective parametric interactions of the type:

$$f\_1 + f\_2 = f\_3 \ \ \ \ \ \beta\_1 - \beta\_2 = \beta\_3 \tag{4}$$

In the "parametric oscillator configuration", a high-frequency backward pump wave having a frequency *f*3 and wavenumber β3 is excited by the voltage source connected at the input port of a LH NLTL. It generates two other waves having frequencies 1*f* and 2*f* , such that 1 2 *f* < *f* and 12 3 *f* + = *f f* . The wave having frequency 2*f* propagates in the opposite direction relative to the pump wave and the wave having frequency 1*f* (this is emphasized in (4) with the minus sign). We therefore have a similar situation to backward wave parametric generation (Gorshkov et. al., 1998; Harris, 1966). The backward-propagating parametrically generated wave 2*f* enables internal feedback that results in a considerable energy transfer from the pump wave to the parametrically excited waves.

If the amplitude of a high-frequency pump wave exceeds a certain threshold value, it may parametrically generate two other waves. This threshold value depends on the loss present in the LH NLTL, its length and the boundary conditions (matching) at the input and output. No parametric generation occurs when the amplitude of the voltage source is below this value. However, when a weak signal wave is fed into the LH NLTL together with a pump wave having an amplitude below the threshold value, a parametric amplification is observed. In this case, we have two input waves: an intense pump wave and a weak signal wave (Yariv, 1988). The power from the pump wave is transferred to the signal wave, thus amplifying it. A third parasitic idler wave is generated which provides phase matching. From a previous analysis (Gorshkov, et. al. 1998), for the lossless case, the frequencies and powers of these waves also obey the nonlinear Manley-Rowe relations.

#### **4.2 Experiment**

Though generation of higher harmonics dominates in short LH NLTLs, in longer transmission lines parametric interactions predominate. We observed efficient parametric

Nonlinear Left-Handed Metamaterials 201

Figure 7 demonstrates the effect of the intensive pump wave, having frequency *f*p = *f*3, on a weak signal wave (*f*s = *f*2). Figure 7*b* shows the spectrum at the output of the 7-section LH NLTL when only a 1.7279 GHz, 13.96 dBm intensive pump wave is applied at the input. The magnitude of the pump wave was chosen so as to be 0.1 dB below the threshold value required for the occurrence of parametric generation, which manifests itself in distinct,

> -80 -60 -40 -20 <sup>0</sup> **(b)**

Output power (dBm)

Fig. 7. Spectra of the output waveforms generated by a 7-section LH NLTL fed by: (a) only weak signal source 864.252 MHz, -28 dBm; (b) only pump source 1.7279 GHz, 13.96 dBm; (c) simultaneously signal and pump sources specified in (a) and (b). Reverse bias voltage is 3.87 V.

Figure 7a shows the spectrum at the output when only the 864.252 MHz, -28 dBm signal wave is applied at the LH NLTL input (no pump wave). The graph shows 11.7 dB attenuation of the weak signal wave at the output due loss in the NLTL and power conversion to higher harmonics. And finally, Fig. 7c shows the spectrum at the output when the signal and the pump wave are both applied concurrently at the input of the 7-section LH NLTL. In this spectrum, the components corresponding to the signal wave (*f*s = *f*2), idler wave (*f*1), as well as many difference frequencies generated due to the strong nonlinearity in LH NLTL, are evident. Thus, the application of the intensive pump wave results in

It should be mentioned that parametric amplification of the signal wave propagating in the backward direction with respect to the pump wave has been observed in a round-trip configuration (when both the signal and the pump waves are still applied at the TL input). The parametric interaction of the counter-propagating signal and pump waves becomes possible since both the signal and the pump waves are subject to a strong reflection while propagating along the LH NLTL. This reflection originates from the mismatch at the input and output ports (which is unavoidable due to strong nonlinear variation of capacitance) and results in a standing wave formation. This enables the phase matching of the incident signal wave and the reflected from the output end of the LH NLTL pump wave and hence

862 863 864 865 866

Frequency (MHz)

narrow peaks corresponding to the parametrically generated frequencies.

*f signal*

862 863 864 865 866

the parametric interaction of the counterpropagating backward waves.

*f signal*

862 863 864 865 866

Frequency (MHz)

Frequency (MHz)


amplification of the weak signal by 9 dB.


Output power (dBm)

Output power (dBm)

<sup>0</sup> **(c)** *<sup>f</sup>*

**(a)**

*idler*

amplification in 7-section LH NLTL (Kozyrev et. al., 2006) shown in Fig. 5. The design of the 7-section LH NLTL is similar to the design of four-section LH NLTL described in the previous section.

However, this time, the series nonlinear capacitance has been implemented with Skyworks Inc. SMV 1233 silicon hyperabrupt varactors and shunt inductances with high-Q 10 nH chip inductors (Murata LQW18A\_00).

Fig. 6 shows the magnitude and phase of the linear wave transmission (*S*21) of this 7-section LH NLTL. Parameters of the circuit model in Fig. 2*b* were extracted from the measured *S*-parameters using Agilent ADS software. They are *CL* (3.823 V) = 1.34 pF, *LL* = 11.43 nH, *CR* = 0.62 pF, *LR* = 3.18 nH. The dashed line in Fig. 6 shows the magnitude of *S*21 calculated for the circuit model shown in Fig. 2*b* with component values specified above, and it is in a good agreement with measured data. The circuit model of Fig. 2*b* has also been used to calculate the dispersion curve of the LH transmission line as shown in the inset in Fig. 6. As is evident from *S*-parameters and dispersion curve presented in Fig. 6, the transmission line has a left-handed passband (phase velocity is anti-parallel with the group velocity) from 800 MHz to 1.9 GHz at -3.823 V bias.

Fig. 5. Fabricated seven-section LH NLTL.

Fig. 6. Measured and simulated (dotted lines) magnitudes of *S*21 parameter for seven-section LH NLTLs for the reverse bias voltage *VB* = 3.823 V. Inset shows dispersion curve of the LH NLTL (dependence of the frequency vs relative wave number β).

amplification in 7-section LH NLTL (Kozyrev et. al., 2006) shown in Fig. 5. The design of the 7-section LH NLTL is similar to the design of four-section LH NLTL described in the

However, this time, the series nonlinear capacitance has been implemented with Skyworks Inc. SMV 1233 silicon hyperabrupt varactors and shunt inductances with high-Q 10 nH chip

Fig. 6 shows the magnitude and phase of the linear wave transmission (*S*21) of this 7-section LH NLTL. Parameters of the circuit model in Fig. 2*b* were extracted from the measured *S*-parameters using Agilent ADS software. They are *CL* (3.823 V) = 1.34 pF, *LL* = 11.43 nH, *CR* = 0.62 pF, *LR* = 3.18 nH. The dashed line in Fig. 6 shows the magnitude of *S*21 calculated for the circuit model shown in Fig. 2*b* with component values specified above, and it is in a good agreement with measured data. The circuit model of Fig. 2*b* has also been used to calculate the dispersion curve of the LH transmission line as shown in the inset in Fig. 6. As is evident from *S*-parameters and dispersion curve presented in Fig. 6, the transmission line has a left-handed passband (phase velocity is anti-parallel with the group velocity) from 800

012345

Frequency (GHz)

Fig. 6. Measured and simulated (dotted lines) magnitudes of *S*21 parameter for seven-section LH NLTLs for the reverse bias voltage *VB* = 3.823 V. Inset shows dispersion curve of the LH

0 2 4 Frequency (GHz)

0123

β

β).

previous section.

inductors (Murata LQW18A\_00).

MHz to 1.9 GHz at -3.823 V bias.

Fig. 5. Fabricated seven-section LH NLTL.


NLTL (dependence of the frequency vs relative wave number



mag(S21)


0

Figure 7 demonstrates the effect of the intensive pump wave, having frequency *f*p = *f*3, on a weak signal wave (*f*s = *f*2). Figure 7*b* shows the spectrum at the output of the 7-section LH NLTL when only a 1.7279 GHz, 13.96 dBm intensive pump wave is applied at the input. The magnitude of the pump wave was chosen so as to be 0.1 dB below the threshold value required for the occurrence of parametric generation, which manifests itself in distinct, narrow peaks corresponding to the parametrically generated frequencies.

Fig. 7. Spectra of the output waveforms generated by a 7-section LH NLTL fed by: (a) only weak signal source 864.252 MHz, -28 dBm; (b) only pump source 1.7279 GHz, 13.96 dBm; (c) simultaneously signal and pump sources specified in (a) and (b). Reverse bias voltage is 3.87 V.

Figure 7a shows the spectrum at the output when only the 864.252 MHz, -28 dBm signal wave is applied at the LH NLTL input (no pump wave). The graph shows 11.7 dB attenuation of the weak signal wave at the output due loss in the NLTL and power conversion to higher harmonics. And finally, Fig. 7c shows the spectrum at the output when the signal and the pump wave are both applied concurrently at the input of the 7-section LH NLTL. In this spectrum, the components corresponding to the signal wave (*f*s = *f*2), idler wave (*f*1), as well as many difference frequencies generated due to the strong nonlinearity in LH NLTL, are evident. Thus, the application of the intensive pump wave results in amplification of the weak signal by 9 dB.

It should be mentioned that parametric amplification of the signal wave propagating in the backward direction with respect to the pump wave has been observed in a round-trip configuration (when both the signal and the pump waves are still applied at the TL input). The parametric interaction of the counter-propagating signal and pump waves becomes possible since both the signal and the pump waves are subject to a strong reflection while propagating along the LH NLTL. This reflection originates from the mismatch at the input and output ports (which is unavoidable due to strong nonlinear variation of capacitance) and results in a standing wave formation. This enables the phase matching of the incident signal wave and the reflected from the output end of the LH NLTL pump wave and hence the parametric interaction of the counterpropagating backward waves.

Nonlinear Left-Handed Metamaterials 203





*f* 2 *f* 1


Harmonic magnitude (dBm)

Harmonic magnitude (dBm)

frequency (GHz) 0 5 10 15 20 -60

Fig. 9. Spectra of the output waveform generated by a 4-section LH NLTL fed by 2.875 GHz, +19 dBm input signal for different values of the bias voltage (*a* – 4 V, *b* – 4.95 V, *c* – 5 V, *d* –

Parametric amplification can be of interest for building "active" or "amplifying" metamaterials and for providing a means to compensate for inherent LH media loss, a challenge for currently existing metamaterials (Kozyrev et. al., 2006). The primary drawback of current negative-index metamaterials (NIMs) (for example those composed of the arays of metallic wires and split-ring resonators) is their considerable loss, which renders the results ambiguous and the materials all but useless for practical applications. These losses have been overcome to some extent by careful fabrication and assembly techniques (Houck et. al. 2003), but still remain the primary obstacle to using NIMs in imaging applications. It was shown (Tretyakov, 2001) that due to causality requirements, the use of conventional composite NIMs (based on arrays of metallic wires and arrays of split-ring resonators) does not allow for the realization of low-loss NIMs without the incorporation of some active components (transistor amplifiers, etc.) in a composite NIM. The idea of using parametric amplification to compensate for inherent loss in optical left-handed systems has been also

Besides the nonlinear evolution of a waveform itself, another class of phenomena involving evolution of amplitude and phase of continuous waves is also possible. This type of nonlinear wave propagation phenomena arise in NLTLs having strong frequency dispersion with respect to the average amplitude for amplitude-modulated wave containing a carrier of relatively high frequency and slow (optical-type) nonlinearity. This dispersion may lead to amplitude instability as well as to formation of envelope solitons and periodic modulation of a carrier wave propagating in a stationary manner. The observation of amplitude


**4.4 Higher-order parametric processes** 

discussed in (Popov & Shalaev, 2006).

**5. Envelope solitons in LH NLTLs** 




Harmonic magnitude (dBm)

Harmonic magnitude (dBm)

0 (c)

0 5 10 15 20 -60

frequency (GHz)

0 (a)

frequency (GHz)

6.3 V). The results here are decreased by 6 dB due to a protection attenuator.

0 5 10 15 20 -60

0 (d)

0 5 10 15 20

frequency (GHz)

0 (b)

Figure 8 represents the measured gain of a weak 864.252 MHz signal stimulated by an intense 1.7279 GHz pump wave versus the power of the signal at the input, for fixed values of the pump power. The gain was calculated as the difference between the power of the signal at the output and the power at the input when both are expressed in dBm. Thus, we measured a greater than 10 dB amplification of the signal with power of −32 dBm and below for the power of the pump wave at the input of 13.96 dBm. The measured dependencies of gain verses input signal power becomes flatter with decreasing pump power, thus revealing the potential for amplification in a broad band of the signal power. The results of our measurements in Fig. 8 are in a good agreement with the results of simulations reported in (Kozyrev & van der Weide, 2005b).

Fig. 8. Measured gain vs power of the signal at the input of LH NLTL for different values of the power of the pump wave at the input *Pp,in*.: Green squares - *Pp,in* = 13.96 dBm; Blue circles - *Pp,in* = 13.86 dBm; Red up triangles - *Pp,in* = 13.76 dBm; Black down triangles - *Pp,in* = 13.66 dBm; Magenta left triangles - *Pp,in* = 13.56.

#### **4.3 Higher-order parametric processes**

Efficiently generated higher harmonics may also initiate the parametric process. A wave at 2.875 GHz cannot parametrically generate any other waves in the 4-section LH NLTL, shown in Fig. 2, since they would exist below the line's cutoff frequency. The second harmonic at 5.75 GHz excites waves with frequencies of 2.2 GHz and 3.55 GHz depicted in Fig. 9*b* as 1*f* and 2*f* . This basic parametric process then initiates multiple higher-order parametric interactions, resulting in multiple peaks in the spectrum of the output waveform. The progression of this process is shown in Fig. 9*c*, which illustrates conversion of a monochromatic input signal into a wideband output. Further increase in the reverse bias voltage leads to the stabilization of the harmonic generation and suppression of parametric instability (Fig. 9*d*). The variation of bias voltage results in a corresponding change in the dispersion characteristics of the LH NLTL (propagation constants of the interacting waves). This change allows for enabling or disabling of certain nonlinear interactions (phase matching). In our particular case it enables/disables higher-order parametric interactions.

Figure 8 represents the measured gain of a weak 864.252 MHz signal stimulated by an intense 1.7279 GHz pump wave versus the power of the signal at the input, for fixed values of the pump power. The gain was calculated as the difference between the power of the signal at the output and the power at the input when both are expressed in dBm. Thus, we measured a greater than 10 dB amplification of the signal with power of −32 dBm and below for the power of the pump wave at the input of 13.96 dBm. The measured dependencies of gain verses input signal power becomes flatter with decreasing pump power, thus revealing the potential for amplification in a broad band of the signal power. The results of our measurements in Fig. 8 are in a good agreement with the results of simulations reported in


*Ps* (dBm) (at input)

Fig. 8. Measured gain vs power of the signal at the input of LH NLTL for different values of the power of the pump wave at the input *Pp,in*.: Green squares - *Pp,in* = 13.96 dBm; Blue circles - *Pp,in* = 13.86 dBm; Red up triangles - *Pp,in* = 13.76 dBm; Black down triangles -

Efficiently generated higher harmonics may also initiate the parametric process. A wave at 2.875 GHz cannot parametrically generate any other waves in the 4-section LH NLTL, shown in Fig. 2, since they would exist below the line's cutoff frequency. The second harmonic at 5.75 GHz excites waves with frequencies of 2.2 GHz and 3.55 GHz depicted in Fig. 9*b* as 1*f* and 2*f* . This basic parametric process then initiates multiple higher-order parametric interactions, resulting in multiple peaks in the spectrum of the output waveform. The progression of this process is shown in Fig. 9*c*, which illustrates conversion of a monochromatic input signal into a wideband output. Further increase in the reverse bias voltage leads to the stabilization of the harmonic generation and suppression of parametric instability (Fig. 9*d*). The variation of bias voltage results in a corresponding change in the dispersion characteristics of the LH NLTL (propagation constants of the interacting waves). This change allows for enabling or disabling of certain nonlinear interactions (phase matching). In our particular case it enables/disables higher-order parametric interactions.

(Kozyrev & van der Weide, 2005b).

0

*Pp,in* = 13.66 dBm; Magenta left triangles - *Pp,in* = 13.56.

**4.3 Higher-order parametric processes** 

2

4

6

Gain (dB)

8

10

12

Fig. 9. Spectra of the output waveform generated by a 4-section LH NLTL fed by 2.875 GHz, +19 dBm input signal for different values of the bias voltage (*a* – 4 V, *b* – 4.95 V, *c* – 5 V, *d* – 6.3 V). The results here are decreased by 6 dB due to a protection attenuator.

#### **4.4 Higher-order parametric processes**

Parametric amplification can be of interest for building "active" or "amplifying" metamaterials and for providing a means to compensate for inherent LH media loss, a challenge for currently existing metamaterials (Kozyrev et. al., 2006). The primary drawback of current negative-index metamaterials (NIMs) (for example those composed of the arays of metallic wires and split-ring resonators) is their considerable loss, which renders the results ambiguous and the materials all but useless for practical applications. These losses have been overcome to some extent by careful fabrication and assembly techniques (Houck et. al. 2003), but still remain the primary obstacle to using NIMs in imaging applications. It was shown (Tretyakov, 2001) that due to causality requirements, the use of conventional composite NIMs (based on arrays of metallic wires and arrays of split-ring resonators) does not allow for the realization of low-loss NIMs without the incorporation of some active components (transistor amplifiers, etc.) in a composite NIM. The idea of using parametric amplification to compensate for inherent loss in optical left-handed systems has been also discussed in (Popov & Shalaev, 2006).

#### **5. Envelope solitons in LH NLTLs**

Besides the nonlinear evolution of a waveform itself, another class of phenomena involving evolution of amplitude and phase of continuous waves is also possible. This type of nonlinear wave propagation phenomena arise in NLTLs having strong frequency dispersion with respect to the average amplitude for amplitude-modulated wave containing a carrier of relatively high frequency and slow (optical-type) nonlinearity. This dispersion may lead to amplitude instability as well as to formation of envelope solitons and periodic modulation of a carrier wave propagating in a stationary manner. The observation of amplitude

Nonlinear Left-Handed Metamaterials 205

numerous closely spaced spectral harmonics. The interval between adjacent spectral

is the period of the train of solitons.

20



Spectrum (dBm)

Fig. 10. Voltage waveform (a) and its spectrum (b) measured at the output of 7-section LH


Fig. 11. Measured trains of envelope solitons for different power *P*inp and the frequency *f*inp of the input signal. a) *f*inp = 1.3723 GHz, *P*inp = 24.66 dBm; b) *f*inp = 1.3125 GHz, *P*inp = 21.60 dBm; c) *f*inp = 1.321596 GHz, *P*inp = 19.34 dBm; d) *f*inp = 1.2974 GHz, *P*inp = 24.64 dBm; e) *f*inp =

A small variation of the parameters of the input signal leads to switching between the generation of bright and dark solitons [compare traces (a) and (b)] in contrast to the scenario described by the NSE. As is known, systems described by the NSE can be characterized by

number). According to the Lighthill criterion (Lighthill, 1965), either dark or bright solitons are observed depending on the sign of these two parameters. Bright solitons exist when *DN*<0 and dark solitons exits when *DN*>0. The observed switching is enabled by the

time (nsec)

<sup>2</sup> ∂ ∂ ω*A* (

ω

ω

and *A* are the carrier

*k* (*k* is the wave

0

012345

Frequency (GHz)

components is Δ = *f* 1

τ, where


Voltage waveform (Volts)

τ


Time (nsec) (a) (b)

NLTL fed by a 1.3125 GHz, +21.6 dBm input signal.

0.4 0.6 0.8

1.102 GHz, *P*inp = 23.62 dBm

*e*

two main parameters: the nonlinearity parameter *N* =

frequency and the amplitude) and the dispersion parameter *D* = 2 2 ∂ ∂

*d*

*c*

*b*

*a*

0.4 0.6

0.4 0.6

Envelope Amplitude (V)

 0.4 0.6 0.4 0.6

instability and envelope soliton generation in conventional (RH) NLTLs has already been the subject of many publications (Lonngren & Scott, 1978; Ostrovskii & Soustov, 1972; Yagi & Noguchi, 1976). The experimental observation of the generation of the trains of envelope solitons in LH NLTLs arising from the self-modulational instability under certain conditions of the amplitude and frequency of the pump wave was first reported in (Kozyrev & van der Weide, 2007).

The analysis of LH NLTLs is straightforward when the equations governing envelope evolution can be reduced to the one-dimensional cubic nonlinear Schrodinger equation (NSE), which provides a canonical description for the envelope dynamics of a quasimonochromatic plane wave (the carrier) propagating in a weakly nonlinear dispersive medium when dissipative processes (including nonlinear damping due to higher harmonic generation and nonlinear wave mixing) are negligible (Gupta & Caloz, 2007; Narahara et. al., 2007). However, in most of the practical situations the parametric decay instabilities and higher harmonic generation can be very significant (Kozyrev et. al, 2005, 2006; Gorshkov et. al., 1998; Lighthill, 1965). The threshold for parametric generation is known to be very low (lower then in conventional RH NLTLs). In order to realize the scenario described by the NSE, the LH NLTL should be operated below this threshold so that the nonlinearity should be very weak and the NLTL impractically long. In contrast, we performed an experimental study of nonlinear envelope evolution and envelope soliton generation in relatively short LH NLTLs and when nonlinear damping is very strong. We are also taking advantage of a fast nonlinearity introduced by Schottky diodes when nonlinear capacitance is a function of the instantaneous value of voltage along the line rather then its amplitude, a type of nonlinearity not described in the framework of the NSE and its modifications developed for slow (retarding) nonlinearity.

Figure 10 shows a typical voltage waveform and its spectrum measured at the output of 7 section LH NLTL in the envelope soliton generation regime. This voltage waveform is a cw signal with carrier at fundamental (pump) frequency and with an envelope representing itself a train of bright solitons appearing as periodic pulses above a cw background. The scenario of the development of modulational instability and/or generation of envelope solitons is very sensitive to the parameters of the signal applied at the input of NLTL. Depending on the amplitude and frequency of the input signal, trains of envelope solitons of different shape and types can be generated. Figure 11 shows envelopes of the measured waveforms at the output of 7-section LH NLTL. These envelopes' functions have been obtained by applying the Hilbert transform to the original voltage waveforms.

Traces (a), (b) and (c) in Fig. 11 show trains of bright envelope solitons of different shapes while traces (d*)* and (e) show periodic trains of so-called dark solitons (dips in the cw background). The trains of envelope solitons that we observed are also known as cnoidal waves. The interval between individual solitons depends on the amplitude and frequency of the input signal but does not depend on the length of LH NLTL. Comparison of voltage waveforms at the output of 7-, 10- and 17-section LH NLTL for the same input signal parameters shows that the distance between solitons and their shape are preserved during propagation along transmission line and that we deal with the generation of stationary train of solitons. The envelope shape is not smooth since strong nonlinearity gives rise to numerous higher harmonics and subharmonics of carrier frequency. In the spectral domain, generation of envelope solitons manifests itself in the appearance of spectral regions with

instability and envelope soliton generation in conventional (RH) NLTLs has already been the subject of many publications (Lonngren & Scott, 1978; Ostrovskii & Soustov, 1972; Yagi & Noguchi, 1976). The experimental observation of the generation of the trains of envelope solitons in LH NLTLs arising from the self-modulational instability under certain conditions of the amplitude and frequency of the pump wave was first reported in (Kozyrev & van der

The analysis of LH NLTLs is straightforward when the equations governing envelope evolution can be reduced to the one-dimensional cubic nonlinear Schrodinger equation (NSE), which provides a canonical description for the envelope dynamics of a quasimonochromatic plane wave (the carrier) propagating in a weakly nonlinear dispersive medium when dissipative processes (including nonlinear damping due to higher harmonic generation and nonlinear wave mixing) are negligible (Gupta & Caloz, 2007; Narahara et. al., 2007). However, in most of the practical situations the parametric decay instabilities and higher harmonic generation can be very significant (Kozyrev et. al, 2005, 2006; Gorshkov et. al., 1998; Lighthill, 1965). The threshold for parametric generation is known to be very low (lower then in conventional RH NLTLs). In order to realize the scenario described by the NSE, the LH NLTL should be operated below this threshold so that the nonlinearity should be very weak and the NLTL impractically long. In contrast, we performed an experimental study of nonlinear envelope evolution and envelope soliton generation in relatively short LH NLTLs and when nonlinear damping is very strong. We are also taking advantage of a fast nonlinearity introduced by Schottky diodes when nonlinear capacitance is a function of the instantaneous value of voltage along the line rather then its amplitude, a type of nonlinearity not described in the framework of the NSE and its modifications developed for

Figure 10 shows a typical voltage waveform and its spectrum measured at the output of 7 section LH NLTL in the envelope soliton generation regime. This voltage waveform is a cw signal with carrier at fundamental (pump) frequency and with an envelope representing itself a train of bright solitons appearing as periodic pulses above a cw background. The scenario of the development of modulational instability and/or generation of envelope solitons is very sensitive to the parameters of the signal applied at the input of NLTL. Depending on the amplitude and frequency of the input signal, trains of envelope solitons of different shape and types can be generated. Figure 11 shows envelopes of the measured waveforms at the output of 7-section LH NLTL. These envelopes' functions have been

Traces (a), (b) and (c) in Fig. 11 show trains of bright envelope solitons of different shapes while traces (d*)* and (e) show periodic trains of so-called dark solitons (dips in the cw background). The trains of envelope solitons that we observed are also known as cnoidal waves. The interval between individual solitons depends on the amplitude and frequency of the input signal but does not depend on the length of LH NLTL. Comparison of voltage waveforms at the output of 7-, 10- and 17-section LH NLTL for the same input signal parameters shows that the distance between solitons and their shape are preserved during propagation along transmission line and that we deal with the generation of stationary train of solitons. The envelope shape is not smooth since strong nonlinearity gives rise to numerous higher harmonics and subharmonics of carrier frequency. In the spectral domain, generation of envelope solitons manifests itself in the appearance of spectral regions with

obtained by applying the Hilbert transform to the original voltage waveforms.

Weide, 2007).

slow (retarding) nonlinearity.

numerous closely spaced spectral harmonics. The interval between adjacent spectral components is Δ = *f* 1 τ , where τis the period of the train of solitons.

Fig. 10. Voltage waveform (a) and its spectrum (b) measured at the output of 7-section LH NLTL fed by a 1.3125 GHz, +21.6 dBm input signal.

Fig. 11. Measured trains of envelope solitons for different power *P*inp and the frequency *f*inp of the input signal. a) *f*inp = 1.3723 GHz, *P*inp = 24.66 dBm; b) *f*inp = 1.3125 GHz, *P*inp = 21.60 dBm; c) *f*inp = 1.321596 GHz, *P*inp = 19.34 dBm; d) *f*inp = 1.2974 GHz, *P*inp = 24.64 dBm; e) *f*inp = 1.102 GHz, *P*inp = 23.62 dBm

A small variation of the parameters of the input signal leads to switching between the generation of bright and dark solitons [compare traces (a) and (b)] in contrast to the scenario described by the NSE. As is known, systems described by the NSE can be characterized by two main parameters: the nonlinearity parameter *N* = <sup>2</sup> ∂ ∂ ω *A* (ω and *A* are the carrier frequency and the amplitude) and the dispersion parameter *D* = 2 2 ∂ ∂ ω *k* (*k* is the wave number). According to the Lighthill criterion (Lighthill, 1965), either dark or bright solitons are observed depending on the sign of these two parameters. Bright solitons exist when *DN*<0 and dark solitons exits when *DN*>0. The observed switching is enabled by the

Nonlinear Left-Handed Metamaterials 207

16 18 20 22 24

Power of fundamental wave at input (dBm)

The step-like dependence of the second harmonic power on the power of the fundamental signal may impact significantly the output waveform if the amplitude in the fundamental wave is modulated around the threshold value. To verify this assumption the LH NLTL was fed by a 783 MHz, +20.5 dBm sinusoidal signal modulated at 100 kHz with the depth of modulation of 50 %. Power of the input signal corresponds to the threshold value in Fig. 12. Figure 13 shows voltage waveforms at the input and output port and spectrum at the output of 7-section LH NLTL. As expected, the voltage waveform at the input is a sinusoidal wave modulated by another sinusoidal signal at 100 MHz. The envelope of the output waveform is dramatically different from the one of the input wave. It represents itself a series of pulses with the shape approaching a rectangular. Furthermore, the carrier frequency of the output signal is the second harmonic of the fundamental signal as revealed by the spectrum presented in Fig. 13*c*. Modulated signal switches second harmonic generation on and off thus enabling generation of a train of RF pulses at the output. Since the fundamental frequency is chosen below the cut-off frequency, it is heavily attenuated in transmission line and only second harmonic is present at the output. Some asymmetry of the shape of the RF pulses at the output is related to the existence of hysteresis and narrow multistability region. The experimental results presented in Fig. 13 clearly demonstrate that a small modulation signal can be used to control the shape, duration and repetition rate of the RF pulses at the

Our experimental results correlate very well with speculations in (Agranovich et. al., 2004) where authors predicted that the shape of pulses at the output of LH media can be

Potential applications may include pulse forming circuits, amplifiers of digital signals as well as very efficient modulators at power levels or in frequency ranges not attainable by

drastically different from those expected from an ordinary nonlinear medium.

Fig. 12. Dependence of the power of the 2nd harmonic at the output on the power of the fundamental signal at the input in 7-section LH NLTL shown in Fig. 5 measured at 783 MHz


output which is very promising to numerous applications.

conventional semiconductor devices.

Power of secon

d

at output (dBm)

and the reverse bias voltage *VB* = - 4.1 V.

harmonic

counterplay of the significant nonlinear damping (due to strong and fast nonlinearity) and strong spatial dispersion exhibited by the periodic LH NLTLs. Neither is taken into account by standard NSE yet both are known to lead to co-existence of bright and dark solitons in other physical systems (Kivshar et. al., 1994; Scott et.al., 2005). For example, somewhat similar processes have recently been observed in the system of an in-plane magnetized single crystal yttrium-iron-garnet (YIG) film in the magnetostatic backward volume wave configuration. However, there is a fundamental issue that distinguishes our work from (Scott et.al., 2005) where the soliton trains have been generated through the nonlinear mode beating of two copropagating magnetostatic backward volume wave excitations in thin YIG film. Thus, a pre-modulated signal was used to achieve soliton generation. In contrast to this work, we applied non-modulated sine wave at the input.

#### **6***.* **Pulse formation in LH NLTL media**

As it has already been mentioned in Section 2, both the nonlinearity and dispersion present in LH NLTLs lead to waveform spreading, consequently making shock wave and electronic soliton formation impossible, making them at first blush useless for pulse forming applications. However, this inability to form shock waves enables a variety of parametric processes leading to amplitude istabilitity as well as formation of envelope solitons and periodic modulation of a carrier wave (Kozyrev & van der Weide, 2007) as discussed in the previous section. Here we describe another type of envelope evolution resulting in generation of RF pulses of limited duration with stable amplitude and very short rise/fall times (sharp transients). This type of envelope evolution is primarily enabled by the amplitude-dependent higher harmonic generation rather than self-modulation instability leading to generation of the envelope solitons (Kozyrev & van der Weide, 2010).

Fig. 12 shows a typical dependence of the magnitude of the second harmonics at the output of 7-section LH NLTL shown in Fig. 5 vs magnitude of the input sinusoidal signal. This dependence has three distinct regions. In the first region the power of the generated second harmonic follows a square law as predicted by the small signal analysis (1). When the power of the fundamental wave reaches certain threshold level the second harmonic power jumps by almost 5 dB indicating a bifurcation (multistability region) followed by the saturation region where second harmonic amplitude changes insignificantly with the input power. Step-like dependence of the second harmonic power indicates a bifurcation-type change in the field distribution along the line and formation of field patterns that change dispersion properties of the line resulting in significant increase of the generation efficiency. These field patterns (nonlinear mode build-up) result from the nonlinear interactions and reflection of both fundamental and second harmonic signals from input and output interfaces. An example of such patterns has been investigated in (Kozyrev & van der Weide, 2005a; Kozyrev et. al., 2005) where a significant increase of the 3rd harmonic generation efficiency correlated with self-induced periodicity of the voltage oscillations across nonlinear capacitances on LH NLTL. This self-induced periodicity of the voltage amplitude cross the nonlinear capacitors leads to periodic variation of the capacitance along the line. Due to strong nonlinearity (large capacitance ratio), this periodicity results in a considerable change of the dispersion characteristics and enables quasi-phase matching of the fundamental wave and its higher harmonics.

counterplay of the significant nonlinear damping (due to strong and fast nonlinearity) and strong spatial dispersion exhibited by the periodic LH NLTLs. Neither is taken into account by standard NSE yet both are known to lead to co-existence of bright and dark solitons in other physical systems (Kivshar et. al., 1994; Scott et.al., 2005). For example, somewhat similar processes have recently been observed in the system of an in-plane magnetized single crystal yttrium-iron-garnet (YIG) film in the magnetostatic backward volume wave configuration. However, there is a fundamental issue that distinguishes our work from (Scott et.al., 2005) where the soliton trains have been generated through the nonlinear mode beating of two copropagating magnetostatic backward volume wave excitations in thin YIG film. Thus, a pre-modulated signal was used to achieve soliton generation. In contrast to this

As it has already been mentioned in Section 2, both the nonlinearity and dispersion present in LH NLTLs lead to waveform spreading, consequently making shock wave and electronic soliton formation impossible, making them at first blush useless for pulse forming applications. However, this inability to form shock waves enables a variety of parametric processes leading to amplitude istabilitity as well as formation of envelope solitons and periodic modulation of a carrier wave (Kozyrev & van der Weide, 2007) as discussed in the previous section. Here we describe another type of envelope evolution resulting in generation of RF pulses of limited duration with stable amplitude and very short rise/fall times (sharp transients). This type of envelope evolution is primarily enabled by the amplitude-dependent higher harmonic generation rather than self-modulation instability

Fig. 12 shows a typical dependence of the magnitude of the second harmonics at the output of 7-section LH NLTL shown in Fig. 5 vs magnitude of the input sinusoidal signal. This dependence has three distinct regions. In the first region the power of the generated second harmonic follows a square law as predicted by the small signal analysis (1). When the power of the fundamental wave reaches certain threshold level the second harmonic power jumps by almost 5 dB indicating a bifurcation (multistability region) followed by the saturation region where second harmonic amplitude changes insignificantly with the input power. Step-like dependence of the second harmonic power indicates a bifurcation-type change in the field distribution along the line and formation of field patterns that change dispersion properties of the line resulting in significant increase of the generation efficiency. These field patterns (nonlinear mode build-up) result from the nonlinear interactions and reflection of both fundamental and second harmonic signals from input and output interfaces. An example of such patterns has been investigated in (Kozyrev & van der Weide, 2005a; Kozyrev et. al., 2005) where a significant increase of the 3rd harmonic generation efficiency correlated with self-induced periodicity of the voltage oscillations across nonlinear capacitances on LH NLTL. This self-induced periodicity of the voltage amplitude cross the nonlinear capacitors leads to periodic variation of the capacitance along the line. Due to strong nonlinearity (large capacitance ratio), this periodicity results in a considerable change of the dispersion characteristics and enables quasi-phase matching of the fundamental wave

leading to generation of the envelope solitons (Kozyrev & van der Weide, 2010).

work, we applied non-modulated sine wave at the input.

**6***.* **Pulse formation in LH NLTL media** 

and its higher harmonics.

Fig. 12. Dependence of the power of the 2nd harmonic at the output on the power of the fundamental signal at the input in 7-section LH NLTL shown in Fig. 5 measured at 783 MHz and the reverse bias voltage *VB* = - 4.1 V.

The step-like dependence of the second harmonic power on the power of the fundamental signal may impact significantly the output waveform if the amplitude in the fundamental wave is modulated around the threshold value. To verify this assumption the LH NLTL was fed by a 783 MHz, +20.5 dBm sinusoidal signal modulated at 100 kHz with the depth of modulation of 50 %. Power of the input signal corresponds to the threshold value in Fig. 12. Figure 13 shows voltage waveforms at the input and output port and spectrum at the output of 7-section LH NLTL. As expected, the voltage waveform at the input is a sinusoidal wave modulated by another sinusoidal signal at 100 MHz. The envelope of the output waveform is dramatically different from the one of the input wave. It represents itself a series of pulses with the shape approaching a rectangular. Furthermore, the carrier frequency of the output signal is the second harmonic of the fundamental signal as revealed by the spectrum presented in Fig. 13*c*. Modulated signal switches second harmonic generation on and off thus enabling generation of a train of RF pulses at the output. Since the fundamental frequency is chosen below the cut-off frequency, it is heavily attenuated in transmission line and only second harmonic is present at the output. Some asymmetry of the shape of the RF pulses at the output is related to the existence of hysteresis and narrow multistability region. The experimental results presented in Fig. 13 clearly demonstrate that a small modulation signal can be used to control the shape, duration and repetition rate of the RF pulses at the output which is very promising to numerous applications.

Our experimental results correlate very well with speculations in (Agranovich et. al., 2004) where authors predicted that the shape of pulses at the output of LH media can be drastically different from those expected from an ordinary nonlinear medium.

Potential applications may include pulse forming circuits, amplifiers of digital signals as well as very efficient modulators at power levels or in frequency ranges not attainable by conventional semiconductor devices.

Nonlinear Left-Handed Metamaterials 209

The first experimental observation of an inverse Doppler effect, in which the frequency of a wave is increased upon reflection from a receding boundary, was reported in (Seddon & Bearpark, 2003). They used an experimental scheme based on a magnetic nonlinear transmission line which was suggested recently in (Belyantsev & Kozyrev, 2002, 1999). This scheme falls into a general class of systems that involve the emission of phase matched highfrequency waves by an electromagnetic shock wave propagating along a NLTL with dispersion (Belyantsev et. al., 1995; Belyantsev & Kozyrev, 1998, 2000). The moving boundary that is used to produce a Doppler shift is the discontinuity that is formed between regions of unsaturated and saturated nonlinearity in the transmission line at the leading edge of the pump pulse. Under appropriate conditions, this shock wave (moving discontinuity) generates a Bloch wave propagating in the opposite direction to the moving discontinuity. It occurs when this shock wave is phase matched with a backward spatial harmonic of the exited Bloch wave. Following its reflection from the NLTL input interface, the excited Bloch wave catches up with the moving discontinuity and produces an anomalous Doppler shift (Belyantsev & Kozyrev, 2002). The detailed theory of this

Nonlinear phenomena similar to ones described in Sections 3-7 have also been observed in volumetric metamaterials (Shadrivov et. al., 2008a, 2008b). For instance, the selective generation of higher harmonics has been observed in metamaterials consisting of split-ring resonantors (SRR) and metal wires shown in Fig. 14. Each SRR contains a variable capacity diode (model Skyworks SMV-1405) which introduces nonlinear current-voltage dependence and resulting nonlinear magnetic dipole moment to each SRR (Shadrivov et. al., 2008a). In terms of effective medium parameters, the manufactured structure has nonlinear

Fig. 14. Photograph of the nonlinear tunable metamaterial created by square arrays of wires and nonlinear SRRs. Each SRR contains a varactor (see the inset) which provides power-

phenomenon is presented in (Kozyrev & van der Weide, 2005, 2006).

magnetization and non- linear effective magnetic permittivity.

**8. Nonlinear volumetric metamaterials** 

dependent nonlinear response.

Fig. 13. Voltage waveforms at the input (a) and output (b) port and spectrum at the output (c) of 7-section LH NLTL. Voltage was measured at the coupled output of directional couplers connected at the input and output ports of NLTL.

#### **7. Inverse Doppler effect in nonlinear transmission lines**

Some interesting phenomena arise in nonlinear periodic systems (periodically loaded nonlinear transmission lines) supporting propagation of backward spatial harmonics. It is well known that the periodic systems support propagation of the Bloch waves which can be expanded into an infinite set of spatial harmonics (Collin, 1992) so that the field in a periodic structure can be represented as

$$V\_B = \sum\_{n=-\infty}^{n=\infty} V\_{p,n} e^{-j\beta\_n z}$$

Each term in this expansion is a spatial harmonic, is periodic in space amplitude

$$V\_{p,n}(z+d) = V\_{p,n}(z)$$

and has propagation phase constant

$$
\beta\_n = \beta + 2n\pi \,/\, d\,\,.
$$

All harmonics propagate with the same group velocity; however, some of the spatial harmonics have phase and group velocities that are oppositely directed (backward spatial harmonics) since βn can be both positive and negative and thus exhibit anomalous dispersion:

$$
\upsilon\_{\mathcal{S}}^{(n)} = d \frac{\partial}{\partial\_n} \quad \upsilon\_p^{(n)} = d \frac{\partial \, \rho \sigma}{\partial \beta\_n} = d \frac{\partial \, \rho \sigma}{\partial \beta} \, \, \, \, \,
$$

Fig. 13. Voltage waveforms at the input (a) and output (b) port and spectrum at the output (c) of 7-section LH NLTL. Voltage was measured at the coupled output of directional

Some interesting phenomena arise in nonlinear periodic systems (periodically loaded nonlinear transmission lines) supporting propagation of backward spatial harmonics. It is well known that the periodic systems support propagation of the Bloch waves which can be expanded into an infinite set of spatial harmonics (Collin, 1992) so that the field in a periodic

> *B pn n V Ve*

=−∞ <sup>=</sup>

( ) () *V zd V z p n*, , + = *p n*

 π*<sup>n</sup>* = + 2 / *n d* .

n can be both positive and negative and thus exhibit anomalous

 ω

 β

*n vd d* ω

∂ ∂ = = ∂ ∂ .

β

All harmonics propagate with the same group velocity; however, some of the spatial harmonics have phase and group velocities that are oppositely directed (backward spatial

Each term in this expansion is a spatial harmonic, is periodic in space amplitude

ββ

*n*

β <sup>=</sup> ( ) *<sup>n</sup> p*

ω

( ) *n g*

*v d*

, *<sup>n</sup> <sup>n</sup> <sup>j</sup> <sup>z</sup>*

=∞ <sup>−</sup>

β

couplers connected at the input and output ports of NLTL.

structure can be represented as

and has propagation phase constant

β

harmonics) since

dispersion:

**7. Inverse Doppler effect in nonlinear transmission lines** 

The first experimental observation of an inverse Doppler effect, in which the frequency of a wave is increased upon reflection from a receding boundary, was reported in (Seddon & Bearpark, 2003). They used an experimental scheme based on a magnetic nonlinear

transmission line which was suggested recently in (Belyantsev & Kozyrev, 2002, 1999). This scheme falls into a general class of systems that involve the emission of phase matched highfrequency waves by an electromagnetic shock wave propagating along a NLTL with dispersion (Belyantsev et. al., 1995; Belyantsev & Kozyrev, 1998, 2000). The moving boundary that is used to produce a Doppler shift is the discontinuity that is formed between regions of unsaturated and saturated nonlinearity in the transmission line at the leading edge of the pump pulse. Under appropriate conditions, this shock wave (moving discontinuity) generates a Bloch wave propagating in the opposite direction to the moving discontinuity. It occurs when this shock wave is phase matched with a backward spatial harmonic of the exited Bloch wave. Following its reflection from the NLTL input interface, the excited Bloch wave catches up with the moving discontinuity and produces an anomalous Doppler shift (Belyantsev & Kozyrev, 2002). The detailed theory of this phenomenon is presented in (Kozyrev & van der Weide, 2005, 2006).

### **8. Nonlinear volumetric metamaterials**

Nonlinear phenomena similar to ones described in Sections 3-7 have also been observed in volumetric metamaterials (Shadrivov et. al., 2008a, 2008b). For instance, the selective generation of higher harmonics has been observed in metamaterials consisting of split-ring resonantors (SRR) and metal wires shown in Fig. 14. Each SRR contains a variable capacity diode (model Skyworks SMV-1405) which introduces nonlinear current-voltage dependence and resulting nonlinear magnetic dipole moment to each SRR (Shadrivov et. al., 2008a). In terms of effective medium parameters, the manufactured structure has nonlinear magnetization and non- linear effective magnetic permittivity.

Fig. 14. Photograph of the nonlinear tunable metamaterial created by square arrays of wires and nonlinear SRRs. Each SRR contains a varactor (see the inset) which provides powerdependent nonlinear response.

Nonlinear Left-Handed Metamaterials 211

Selective generation of higher harmonics observed in our experiments is related to the transmission properties of the metamaterial. A particular harmonics dominates over fundamental harmonic and the other higher harmonics when its frequency corresponds to the transparency band. Results of the transmission coefficient measurements performed on our nonlinear LHMs indicate a right-handed transparency band with a maximum transparency at around 7 GHz. This value agrees well with the values of the higher harmonics dominating in our measurements. Furthermore, the presence of very high order harmonics in the spectrum of the trans- mitted signal manifests strong nonlinearity inside the metamaterial which potentially may lead to significant enhancement in nonlinear

We have reviewed several nonlinear wave phenomena in LH media, including harmonic generation, parametric amplification and generation of traveling waves, generation of the train of envelope solitons and their competition. Furthermore, LH NLTLs which were considered as a model system in this paper, can be also of interest from the design perspective for development of various compact and robust applications for wireless communications and imaging. LH NLTLs have already been used as the key counterparts of recently designed and implemented tunable phase-shifters, tunable band-pass filters, and the arbitrary waveform generator based on fourier decomposition (by combining broadband power divider, LPF, BPF, HPF, harmonic generator, vector modulator and broadband LNAs on copper board) (Kim et. al., 2005a, 2005b, 2006, 2007 ). Moreover, extending the results for 1-D LH NLTL to higher dimensions would enable combining harmonic generation in LH NLTL media with focusing (Grbic & Eleftheriades, 2003, 2004), due to the negative refractive index of 2-D or 3-D LH transmission line media. This may lead to the development of highly efficient and powerful frequency multipliers, as well as to building "active" or "amplifying" super lenses. Furthermore, our approach can be also scaled from its current microwave form into terahertz, infrared, and, ultimately, visible

This work was supported under the Air Force Office of Scientific Research, MURI Grant No

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F49620-03-1-0420, 'Nanoprobe Tools for Molecular Spectroscopy and Control'.

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processes in artificial metamaterials as compared to conventional materials.

form (Goussetis et. al., 2005; Qin et. al., 2007, 2008).

**10. Acknowledgment** 

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**11. References** 

**9. Conclusion** 

Arrays of SRRs and wires form a square lattice with 29 x 4 x 1 unit cells of the size of 10.5 mm.

To measure the em field scattering on our samples, the metamaterial slab is placed in a parallel plate waveguide. The planes of SRRs are aligned perpendicular to the parallel plate surfaces. We have measured the spectrum of the transmitted signal for different frequencies of the incident em wave. For this purpose, the input antenna (placed at the midpoint of the lower plate, 2 mm from the metamaterial slab, in front of the central unit cell) was fed by the signal generated by an Agilent E4428C ESG vector signal generator and amplified by a 38 dB amplifier. The signal detected by the receiving antenna placed 2 cm behind the metamaterial slab was analyzed using an Agilent E4448A PSA series spectrum analyzer.

Figure 15 shows spectra of the signal detected by the receiving antenna behind the nonlinear LHM slab. Varying the input frequency, we observed efficient selective harmonic generation. Namely, second (Fig. 15*a*), third (Fig. 15*b*), and fourth (Fig. 15*c*) harmonics were selectively generated. Moreover, the generation of a comblike signal was also observed (Fig. 15*d*).

Fig. 15. Spectra of the signals detected by the receiving antenna located behind the nonlinear LHM slab for different source frequencies: (a) 3.415 GHz, (b) 2.29 GHz, (c) 1.733 GHz, and (d) 1.668 GHz. The frequency on each graph is normalized to the corresponding source frequency. Power at the input antenna is +30 dBm.

Selective generation of higher harmonics observed in our experiments is related to the transmission properties of the metamaterial. A particular harmonics dominates over fundamental harmonic and the other higher harmonics when its frequency corresponds to the transparency band. Results of the transmission coefficient measurements performed on our nonlinear LHMs indicate a right-handed transparency band with a maximum transparency at around 7 GHz. This value agrees well with the values of the higher harmonics dominating in our measurements. Furthermore, the presence of very high order harmonics in the spectrum of the trans- mitted signal manifests strong nonlinearity inside the metamaterial which potentially may lead to significant enhancement in nonlinear processes in artificial metamaterials as compared to conventional materials.

#### **9. Conclusion**

210 Metamaterial

Arrays of SRRs and wires form a square lattice with 29 x 4 x 1 unit cells of the size of 10.5

To measure the em field scattering on our samples, the metamaterial slab is placed in a parallel plate waveguide. The planes of SRRs are aligned perpendicular to the parallel plate surfaces. We have measured the spectrum of the transmitted signal for different frequencies of the incident em wave. For this purpose, the input antenna (placed at the midpoint of the lower plate, 2 mm from the metamaterial slab, in front of the central unit cell) was fed by the signal generated by an Agilent E4428C ESG vector signal generator and amplified by a 38 dB amplifier. The signal detected by the receiving antenna placed 2 cm behind the metamaterial slab was analyzed using an Agilent E4448A PSA series spectrum analyzer.

Figure 15 shows spectra of the signal detected by the receiving antenna behind the nonlinear LHM slab. Varying the input frequency, we observed efficient selective harmonic generation. Namely, second (Fig. 15*a*), third (Fig. 15*b*), and fourth (Fig. 15*c*) harmonics were selectively generated. Moreover, the generation of a comblike signal was also observed (Fig.

Fig. 15. Spectra of the signals detected by the receiving antenna located behind the nonlinear LHM slab for different source frequencies: (a) 3.415 GHz, (b) 2.29 GHz, (c) 1.733 GHz, and (d) 1.668 GHz. The frequency on each graph is normalized to the corresponding source fre-

quency. Power at the input antenna is +30 dBm.

mm.

15*d*).

We have reviewed several nonlinear wave phenomena in LH media, including harmonic generation, parametric amplification and generation of traveling waves, generation of the train of envelope solitons and their competition. Furthermore, LH NLTLs which were considered as a model system in this paper, can be also of interest from the design perspective for development of various compact and robust applications for wireless communications and imaging. LH NLTLs have already been used as the key counterparts of recently designed and implemented tunable phase-shifters, tunable band-pass filters, and the arbitrary waveform generator based on fourier decomposition (by combining broadband power divider, LPF, BPF, HPF, harmonic generator, vector modulator and broadband LNAs on copper board) (Kim et. al., 2005a, 2005b, 2006, 2007 ). Moreover, extending the results for 1-D LH NLTL to higher dimensions would enable combining harmonic generation in LH NLTL media with focusing (Grbic & Eleftheriades, 2003, 2004), due to the negative refractive index of 2-D or 3-D LH transmission line media. This may lead to the development of highly efficient and powerful frequency multipliers, as well as to building "active" or "amplifying" super lenses. Furthermore, our approach can be also scaled from its current microwave form into terahertz, infrared, and, ultimately, visible form (Goussetis et. al., 2005; Qin et. al., 2007, 2008).

#### **10. Acknowledgment**

This work was supported under the Air Force Office of Scientific Research, MURI Grant No F49620-03-1-0420, 'Nanoprobe Tools for Molecular Spectroscopy and Control'.

#### **11. References**


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**9** 

*Japan* 

**Non-Unity Permeability in InP-Based Photonic** 

The relative permeability of every natural material is 1 at optical frequencies because the magnetization of natural materials does not follow the alternating magnetic field of light (see Fig. 1). If we can overcome this restriction and control both the permeability and the permittivity at optical frequencies, we will be able to establish a new field involving optical/photonic devices for future communication technologies. In this paper, we move one step closer to this goal—we demonstrate that in photonic devices, the relative

permeability can be controlled by adopting metamaterials.

Fig. 1. Constitutive parameters at optical frequencies.

**1. Introduction** 

**Device Combined with Metamaterial** 

*1Quantum Nanoelectronics Research Center, Tokyo Institute of Technology, 2Dept. of Electrical and Electronic Engineering, Tokyo Institute of Technology,* 

T. Amemiya1, T. Shindo2, S. Myoga2, E. Murai2,

N. Nishiyama2 and S. Arai1,2


### **Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial**

T. Amemiya1, T. Shindo2, S. Myoga2, E. Murai2, N. Nishiyama2 and S. Arai1,2 *1Quantum Nanoelectronics Research Center, Tokyo Institute of Technology, 2Dept. of Electrical and Electronic Engineering, Tokyo Institute of Technology, Japan* 

#### **1. Introduction**

214 Metamaterial

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Yagi, T. & Noguchi, A. (1976). *Electronics and Communications in Japan,* Vol.59, p. 1.

Zharov, A. A.; Shadrivov, I. V. & Kivshar, Y. S. (2003). *Physical Review Letters,* Vol.91, p.

Shalaev, V. M. (2007). *Nature Photonics,* Vol.1, p. 41.

*Review Letters,* Vol.84, p. 4184.

Yariv, A. (1988). *Quantum electronics*: Wiley.

037401.

Tien, P. K. (1958). *Journal of Applied Physics,* Vol.29, p.1347.

Veselago, V.G. (1968) *Soviet Physics Uspekhi, Vol.*10, p. 509.

The relative permeability of every natural material is 1 at optical frequencies because the magnetization of natural materials does not follow the alternating magnetic field of light (see Fig. 1). If we can overcome this restriction and control both the permeability and the permittivity at optical frequencies, we will be able to establish a new field involving optical/photonic devices for future communication technologies. In this paper, we move one step closer to this goal—we demonstrate that in photonic devices, the relative permeability can be controlled by adopting metamaterials.

Fig. 1. Constitutive parameters at optical frequencies.

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 217

The operating characteristics of the device, i.e., the transmission dependences on the polarization and wavelength of incident light, are also given in this Section. In Sections 4, we report a measurement method for retrieving accurate constitutive parameters (relative permittivity and permeability) from experimental data of the devices. We hope that this paper will be helpful to readers who are aiming to combine photonic integrated devices and

**2. Recent progress in photonic devices using the concept metamaterials** 

**2.2 Fiber-based metamaterial device which functions nanoscale light source** 

simultaneously for imaging and targeted electron beam excitation of samples.

Figure 3 shows a schematic of the fiber-based photonic metamaterial devices which act as nanoscale light sources. The device consists of fiber-coupled gold asymmetrically-split ring (ASR) array excited by an electron beam with a trajectory parallel to the surface. Light emission from nanoscale planar photonic metamaterials is induced by beams of free electrons, at wavelengths determined by both the dimensions of metamaterials and the electron beam energy. The ASR resonators are manufactured by focused ion-beam milling through ~70 nm gold films evaporated onto the end faces of standard or tapered optical fibers. Experiments are performed in a scanning electron microscope, which provided

In this device, energy is coupled from incident electrons to the plasmonic modes of the metamaterial structure for which propagating light modes then constitute a decay channel. Low energy beams of free-electrons can act as a broadband excitation sources for the collective plasmonic modes of photonic metamaterials, thereby driving resonant light emission at wavelengths determined by the structural design parameters of the metamaterial, which may be adjusted for operation across the visible to infrared range.

There are several strategies to develop advanced photonic applications which combine with metamaterial. The strategies can be classified into two types. One is based on the optical fiber as in conventional photonic communication system. Transferring the principle of metamaterials to a optical fiber system raises a number of inherent difficulties such as the discoherence of polarization rotation induced by structural birefringence. Therefore new idea is needed to use metamaterials in optical fiber structure. Sophisticated example is the nanoscale light sources consisting of fiber-coupled gold asymmetrically-split ring [19]. This device have attracted attention in recent years because of its compact techniques for producing the device. The other strategy is to combine semiconductor devices with metamaterials. Leading examples are the Si-based modulator which enables active tuning of metamaterial [20-22] and the III-V semiconductor-based waveguide in which relative permeability is not unity [23, 24]. The latter in particular is now the focus of attention because it is compatible with other standard waveguide-based optical devices such as lasers. In the following sections, we give the outline of the fiber-based nanoscale light sources and the Si-based modulator. The III-V semiconductor-based waveguide combined with metamaterials, which has been developed in our laboratory, is explained in detail in

**2.1 Photonic devices combined with metamaterials** 

metamaterials.

Section 3.

Metamaterials are artificial materials designed to have permittivity and permeability values that are not possible in nature [1]–[4]. They have recently attracted considerable interest because they exhibit unusual properties such as negative refractive indexes and have potential for unique applications such as high-resolution superlenses and invisibility cloaking devices [5, 6].

It is a challenging task to introduce the concept of metamaterials to actual photonic devices (see [7]–[12] for ordinary photonic devices). We hope to apply metamaterials to realize novel optical functionalities that can potentially establish a new field, *meta-photonics*. For this reason, many efforts have been expended in developing advanced optical applications using the concept of metamaterials. Some novel optical functionalities have been realized previously; for example, it has been shown that in theory, it is possible to achieve sophisticated manipulation of light such as slowing, trapping, and storing of light signals [13]–[18].

This paper provides an overview of the present state of research on novel photonic devices with the concept of metamaterials. In Section 2, we outline two promising approaches of making photonic devices combined with metamaterials. One of them is a fiberbased metamaterial device which functions nanoscale light source; another is a Si-based modulator which enables active tuning of metamaterials; the third has the form of III-V semiconductor-based waveguide combined with metamaterials which is compatible with other conventional photonic devices such as lasers and optical amplifier. Although these researches on *meta-photonics* are still in the experimental stage, they will probably reach a level of producing practical devices in the near future. In the succeeding sections, we focus on the III-V semiconductor-based waveguide combined with metamaterials shown in Fig. 2 and make a detailed explanation of the device. In Section 3, we take up the multimodeinterferometer (MMI) as an example of photonic devices. First, theoretical investigations of the device are given. Actual devices based on this phenomenon are then developed.

Fig. 2. III-V semiconductor-based waveguide optical device combined with metamaterials

The operating characteristics of the device, i.e., the transmission dependences on the polarization and wavelength of incident light, are also given in this Section. In Sections 4, we report a measurement method for retrieving accurate constitutive parameters (relative permittivity and permeability) from experimental data of the devices. We hope that this paper will be helpful to readers who are aiming to combine photonic integrated devices and metamaterials.

#### **2. Recent progress in photonic devices using the concept metamaterials**

#### **2.1 Photonic devices combined with metamaterials**

216 Metamaterial

Metamaterials are artificial materials designed to have permittivity and permeability values that are not possible in nature [1]–[4]. They have recently attracted considerable interest because they exhibit unusual properties such as negative refractive indexes and have potential for unique applications such as high-resolution superlenses and invisibility

It is a challenging task to introduce the concept of metamaterials to actual photonic devices (see [7]–[12] for ordinary photonic devices). We hope to apply metamaterials to realize novel optical functionalities that can potentially establish a new field, *meta-photonics*. For this reason, many efforts have been expended in developing advanced optical applications using the concept of metamaterials. Some novel optical functionalities have been realized previously; for example, it has been shown that in theory, it is possible to achieve sophisticated manipulation

This paper provides an overview of the present state of research on novel photonic devices with the concept of metamaterials. In Section 2, we outline two promising approaches of making photonic devices combined with metamaterials. One of them is a fiberbased metamaterial device which functions nanoscale light source; another is a Si-based modulator which enables active tuning of metamaterials; the third has the form of III-V semiconductor-based waveguide combined with metamaterials which is compatible with other conventional photonic devices such as lasers and optical amplifier. Although these researches on *meta-photonics* are still in the experimental stage, they will probably reach a level of producing practical devices in the near future. In the succeeding sections, we focus on the III-V semiconductor-based waveguide combined with metamaterials shown in Fig. 2 and make a detailed explanation of the device. In Section 3, we take up the multimodeinterferometer (MMI) as an example of photonic devices. First, theoretical investigations of the device are given. Actual devices based on this phenomenon are then developed.

Fig. 2. III-V semiconductor-based waveguide optical device combined with metamaterials

of light such as slowing, trapping, and storing of light signals [13]–[18].

cloaking devices [5, 6].

There are several strategies to develop advanced photonic applications which combine with metamaterial. The strategies can be classified into two types. One is based on the optical fiber as in conventional photonic communication system. Transferring the principle of metamaterials to a optical fiber system raises a number of inherent difficulties such as the discoherence of polarization rotation induced by structural birefringence. Therefore new idea is needed to use metamaterials in optical fiber structure. Sophisticated example is the nanoscale light sources consisting of fiber-coupled gold asymmetrically-split ring [19]. This device have attracted attention in recent years because of its compact techniques for producing the device. The other strategy is to combine semiconductor devices with metamaterials. Leading examples are the Si-based modulator which enables active tuning of metamaterial [20-22] and the III-V semiconductor-based waveguide in which relative permeability is not unity [23, 24]. The latter in particular is now the focus of attention because it is compatible with other standard waveguide-based optical devices such as lasers. In the following sections, we give the outline of the fiber-based nanoscale light sources and the Si-based modulator. The III-V semiconductor-based waveguide combined with metamaterials, which has been developed in our laboratory, is explained in detail in Section 3.

#### **2.2 Fiber-based metamaterial device which functions nanoscale light source**

Figure 3 shows a schematic of the fiber-based photonic metamaterial devices which act as nanoscale light sources. The device consists of fiber-coupled gold asymmetrically-split ring (ASR) array excited by an electron beam with a trajectory parallel to the surface. Light emission from nanoscale planar photonic metamaterials is induced by beams of free electrons, at wavelengths determined by both the dimensions of metamaterials and the electron beam energy. The ASR resonators are manufactured by focused ion-beam milling through ~70 nm gold films evaporated onto the end faces of standard or tapered optical fibers. Experiments are performed in a scanning electron microscope, which provided simultaneously for imaging and targeted electron beam excitation of samples.

In this device, energy is coupled from incident electrons to the plasmonic modes of the metamaterial structure for which propagating light modes then constitute a decay channel. Low energy beams of free-electrons can act as a broadband excitation sources for the collective plasmonic modes of photonic metamaterials, thereby driving resonant light emission at wavelengths determined by the structural design parameters of the metamaterial, which may be adjusted for operation across the visible to infrared range.

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 219

Fig. 4. Si-based modulator which enables active tuning of metamaterials [22].

**3.1 Waveguide-based photonic devices combined with metamaterials** 

value under this condition (this causes the absorption loss of light).

Section.

**3. III-V semiconductor-based waveguide optical device with metamaterials** 

Encouraged by the results stated in Section 2, we consider introducing metamaterials into conventional photonic devices such as lasers, optical amplifiers, and modulators which have the form of III-V semiconductor-based waveguide. In this Section, the feasibility of employing semiconductor-based photonic devices combined with split-ring resonator (SRR)-based metamaterials is examined both theoretically and experimentally. We used a MMI as the stage of interaction between SRRs and light because the input light dispersed over the whole MMI and therefore gives no saturation to each SRR even if the light was very strong. First, theoretical investigations of the device structures of the SRRs and a MMI for use in the 1.5-μm-wavelength region are given in the former of this Section. Fabrication processes and operating characteristics of the device are then explained in the latter of this

Our metamaterial MMI device is shown in Fig. 5. It consists of a waveguide-based GaInAsP/InP 1 × 1 MMI on which a gold SRR array is attached. If transverse electric (TE) mode input light for the MMI has a frequency close to the SRR-resonant frequency, magnetic interactions occur between the TE light and SRR array. Therefore, the real part of the macroscopic permeability becomes large positive and negative at frequencies below and above SRR resonance, respectively. Consequently, the SRR array operates as a metamaterial layer to control permeability. The imaginary part of the permeability is not 0 but a finite

Fig. 3. Fiber-based photonic metamaterial devices which act as nanoscale light sources [19].

#### **2.3 Si-based modulator which enables active tuning of metamaterials**

When applying the concept of metamaterials to actual applications, it is indispensable to control properties of metamaterials for tuning electromagnetic responses (i.e., tunable metamaterials). A typical way to create tunable metamaterials is to integrate a reconfigurable material into a metamaterial structure; thereby the active tuning is achieved by applying an external stimulus. GaAs-based modulators with split ring resonators (SRRs) [25] and metamaterial memories on a VO2 film [26] are such device prototypes operating at 1-10 THz.

Recent researches have shown that one can modulate the optical properties of a metamaterial on a sub-picosecond timescale enabling ultrafast photonic devices [20-22]. In these demonstrations, the authors used a fishnet structure metamaterial [27, 28] with two negative index resonances corresponding to two different periodic wavevectors of the internal gap-mode surface plasmon polaritons. Figure 4 shows one leading example of such device. The device reported here is composed of a BK7 glass substrate and a single metaldielectric-metal (Ag/α-Si/Ag) functional layer with an inter-penetrating two-dimensional square array of elliptical apertures. In this device, the metamaterial is photoexcited with a visible pump pulse and then the pump-induced, time-resolved change in transmission (ΔT/T) is measured around both the resonances. The longer wavelength resonance has a significantly stronger nonlinear response (ΔT/T~70%) corresponding to its larger absolute value of the negative index and the stronger Drude response of photocarriers at longer wavelengths. These results provide insight into engineering various aspects of the nonlinear response of fishnet structure metamaterials.

Fig. 3. Fiber-based photonic metamaterial devices which act as nanoscale light sources [19].

When applying the concept of metamaterials to actual applications, it is indispensable to control properties of metamaterials for tuning electromagnetic responses (i.e., tunable metamaterials). A typical way to create tunable metamaterials is to integrate a reconfigurable material into a metamaterial structure; thereby the active tuning is achieved by applying an external stimulus. GaAs-based modulators with split ring resonators (SRRs) [25] and metamaterial memories on a VO2 film [26] are such device prototypes operating at

Recent researches have shown that one can modulate the optical properties of a metamaterial on a sub-picosecond timescale enabling ultrafast photonic devices [20-22]. In these demonstrations, the authors used a fishnet structure metamaterial [27, 28] with two negative index resonances corresponding to two different periodic wavevectors of the internal gap-mode surface plasmon polaritons. Figure 4 shows one leading example of such device. The device reported here is composed of a BK7 glass substrate and a single metaldielectric-metal (Ag/α-Si/Ag) functional layer with an inter-penetrating two-dimensional square array of elliptical apertures. In this device, the metamaterial is photoexcited with a visible pump pulse and then the pump-induced, time-resolved change in transmission (ΔT/T) is measured around both the resonances. The longer wavelength resonance has a significantly stronger nonlinear response (ΔT/T~70%) corresponding to its larger absolute value of the negative index and the stronger Drude response of photocarriers at longer wavelengths. These results provide insight into engineering various aspects of the nonlinear

**2.3 Si-based modulator which enables active tuning of metamaterials** 

1-10 THz.

response of fishnet structure metamaterials.

Fig. 4. Si-based modulator which enables active tuning of metamaterials [22].

### **3. III-V semiconductor-based waveguide optical device with metamaterials**

#### **3.1 Waveguide-based photonic devices combined with metamaterials**

Encouraged by the results stated in Section 2, we consider introducing metamaterials into conventional photonic devices such as lasers, optical amplifiers, and modulators which have the form of III-V semiconductor-based waveguide. In this Section, the feasibility of employing semiconductor-based photonic devices combined with split-ring resonator (SRR)-based metamaterials is examined both theoretically and experimentally. We used a MMI as the stage of interaction between SRRs and light because the input light dispersed over the whole MMI and therefore gives no saturation to each SRR even if the light was very strong. First, theoretical investigations of the device structures of the SRRs and a MMI for use in the 1.5-μm-wavelength region are given in the former of this Section. Fabrication processes and operating characteristics of the device are then explained in the latter of this Section.

Our metamaterial MMI device is shown in Fig. 5. It consists of a waveguide-based GaInAsP/InP 1 × 1 MMI on which a gold SRR array is attached. If transverse electric (TE) mode input light for the MMI has a frequency close to the SRR-resonant frequency, magnetic interactions occur between the TE light and SRR array. Therefore, the real part of the macroscopic permeability becomes large positive and negative at frequencies below and above SRR resonance, respectively. Consequently, the SRR array operates as a metamaterial layer to control permeability. The imaginary part of the permeability is not 0 but a finite value under this condition (this causes the absorption loss of light).

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 221

0 0

where *ω* is the angular frequency of light, *ε*0 and *μ*0 are the permittivity and permeability of vacuum, respectively, *τ* is the thickness of the plane conductor, and *σ*(*ω*) is the conductivity

Figure 6 shows the internal impedance as a function of frequency for a gold layer whose thickness is larger than the penetration depth. As the frequency increases, the real part of the internal impedance first increases sharply and then saturates at around 100 THz; at frequencies higher than 100 THz, it gradually decreases. This dispersion property corresponds to the dielectric behavior of gold. In contrast, the imaginary part changes monotonously with frequency and has large negative values at optical frequencies; this

Fig. 6. Internal impedance of gold as a function of frequency (real part is surface resistivity,

0 100 200 300 400 500 Frequency (THz)

Imaginary part

Using these dispersion curves, we calculated the magnetic response of a gold SRR at optical frequencies. A four-cut SRR was considered because it has high resonant frequency due to its small gap capacitance [36, 37]. The SRR was placed in a homogeneous host material (air), as shown in Fig. 5. If an incident AC magnetic field is applied to the SRR, an induced circular current flows in the ring through the gap capacitance. The circular current produces an internal magnetic field, and this produces a magnetic interaction between the SRR and

= +

( ) *k i* () 1

ω ω εμ

0

, (2)





Impedance (Imaginary part)


0

ωε

Real part

σ ω

Here, *k*(*ω*) is given by

of the metal as defined by the Drude model.

corresponds to ohmic losses in gold.

1.2

1.0

0.8

0.6

Impedance (Real part)

0.4

0.2

0.0

and imaginary part is internal reactance).

light.

Fig. 5. InP-based 1 × 1 MMI with on which a gold SRR array is attached.

#### **3.2 Theory of waveguide optical devices combined with metamaterials**

The key is to create optical metamaterials that can be used to control permeability and obtain *non-unity* values at optical frequencies (note that, on the other hand, permittivity can be controlled more easily than permeability). A promising method for controlling permeability involves the use of a split-ring resonator (SRR). An SRR produces a circular current in response to an incident magnetic flux, thereby producing its own flux to enhance or oppose the incident field. Consequently, an array of extremely small SRRs operates as a metamaterial layer with non-unity permeability [29-33]. We now investigate the optimal structure of a four-cut SRR device for use at an optical frequency of 193 THz (corresponding to 1.55 μm wavelength for low-loss optical fiber communications). The transmission characteristics are obtained by considering the magnetic interactions between the SRRs and light traveling in the MMI.

#### *A. Design of SRR structure for optical frequency*

We must first determine the optimal dimensions of the SRR for the 1.5-μm-band frequency. Because the magnetic response of an SRR strongly depends on the conduction characteristics of the metal that forms the SRR, the dispersion of the internal impedance *Z* of the gold used in our SRRs was calculated. The internal impedance is the ratio of the surface electric field to the total current [34, 35]. *Z* for a unit length and unit width of a metal plane conductor is given by

$$Z(\tau) = \left( \sigma(o) \right)\_0^{\tau} \frac{\exp\left[ik(o)z\right] + \exp\left[ik(o)(\tau - z)\right]}{1 + \exp\left[ik(o)z\right]} dz \tag{1}$$

Here, *k*(*ω*) is given by

220 Metamaterial

Fig. 5. InP-based 1 × 1 MMI with on which a gold SRR array is attached.

light traveling in the MMI.

conductor is given by

*A. Design of SRR structure for optical frequency* 

τ

 σω

**3.2 Theory of waveguide optical devices combined with metamaterials** 

The key is to create optical metamaterials that can be used to control permeability and obtain *non-unity* values at optical frequencies (note that, on the other hand, permittivity can be controlled more easily than permeability). A promising method for controlling permeability involves the use of a split-ring resonator (SRR). An SRR produces a circular current in response to an incident magnetic flux, thereby producing its own flux to enhance or oppose the incident field. Consequently, an array of extremely small SRRs operates as a metamaterial layer with non-unity permeability [29-33]. We now investigate the optimal structure of a four-cut SRR device for use at an optical frequency of 193 THz (corresponding to 1.55 μm wavelength for low-loss optical fiber communications). The transmission characteristics are obtained by considering the magnetic interactions between the SRRs and

We must first determine the optimal dimensions of the SRR for the 1.5-μm-band frequency. Because the magnetic response of an SRR strongly depends on the conduction characteristics of the metal that forms the SRR, the dispersion of the internal impedance *Z* of the gold used in our SRRs was calculated. The internal impedance is the ratio of the surface electric field to the total current [34, 35]. *Z* for a unit length and unit width of a metal plane

( ) [ ][ ]

ω

*Z dz*

exp ( ) exp ( )( ) ( ) 1 exp ( )

<sup>−</sup> + − <sup>=</sup>

0

τ

[ ]

*ik z*

ω

 ωτ

<sup>+</sup> (1)

*ik z ik z*

1

$$k(\alpha) = \alpha \sqrt{\varepsilon\_0 \mu\_0 \left[1 + i \frac{\sigma(\alpha)}{\alpha \varepsilon\_0}\right]},\tag{2}$$

where *ω* is the angular frequency of light, *ε*0 and *μ*0 are the permittivity and permeability of vacuum, respectively, *τ* is the thickness of the plane conductor, and *σ*(*ω*) is the conductivity of the metal as defined by the Drude model.

Figure 6 shows the internal impedance as a function of frequency for a gold layer whose thickness is larger than the penetration depth. As the frequency increases, the real part of the internal impedance first increases sharply and then saturates at around 100 THz; at frequencies higher than 100 THz, it gradually decreases. This dispersion property corresponds to the dielectric behavior of gold. In contrast, the imaginary part changes monotonously with frequency and has large negative values at optical frequencies; this corresponds to ohmic losses in gold.

Fig. 6. Internal impedance of gold as a function of frequency (real part is surface resistivity, and imaginary part is internal reactance).

Using these dispersion curves, we calculated the magnetic response of a gold SRR at optical frequencies. A four-cut SRR was considered because it has high resonant frequency due to its small gap capacitance [36, 37]. The SRR was placed in a homogeneous host material (air), as shown in Fig. 5. If an incident AC magnetic field is applied to the SRR, an induced circular current flows in the ring through the gap capacitance. The circular current produces an internal magnetic field, and this produces a magnetic interaction between the SRR and light.

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 223

where *By* and *Hy* with over lines represent the average values of magnetic flux density and magnetic field, respectively, in the SRR array layer. To get high accuracy in this homogenization approximation, the size of the integration region should be larger than the wavelength of light. In our calculation, we therefore integrated magnetic field over a large cubic tregion that cantained a 3 × 3 cell array (this means that we replaced d with 3d in Eq. 4 ). Figure 8 shows the real and imaginary parts of permeability as a function of frequency, and the SRR size *L* is a parameter; here, *W*1 and *W*2 were set as 100 nm and *d* was 1.8 μm. The thickness *τ* of the SRR was set at twice the penetration depth of gold at each frequency. As *L* decreases, the magnetic resonant frequency increases. Magnetic response could be obtained at the 1.55-μm-band frequency (approximately 193 THz) for the SRR size *L* of 750 nm (red curves). As *L* decreases, the magnetic response becomes weaker because the inductance of each SRR decreases; this increases the effective resistance (or decreases the Q

Real part

Imaginary part

*w1* = 100 nm *L* = 650 nm

*L* = 750 nm

25

20

15

Imaginary part of permeability

10

5

0

*L* = 850 nm

Fig. 8. Real and imaginary parts of the effective permeability of gold SRR array in air (*ε* = 1) as a function of SRR size; here, *W1* = *W2* = 100 nm, d = 1.8 µm, and frequencies = 100–300

100 150 200 250 300 Frequency (THz)

Further, the magnetic response of the SRR array layer depends on the array pitch *d*. If *d* is large, the response is weak because the area density of SRRs is small; on the other hand, if *d* is small, the magnetic field in each SRR is canceled by the fields of the neighboring SRRs,

Using the abovementioned results, the transmission characteristics of an MMI with a metamaterial, which is shown in Fig. 5, were estimated using computer simulations based

and this weakens the total response of the SRR array layer.

*w2* = 100 nm *d* = 1.8 μm

on the transfer-matrix method.

*C. Transmission characteristics of waveguide device with metamaterial* 

factor) of the SRR.

THz.






Real part of permeability

0

2

4

6

The total electromotive force (emf) induced around the SRR is given by the magnetic flux density *B*. Therefore, we can equate the potential drop to the emf as follows:

$$\begin{split} \partial\_t \int\_{SRR} B d\sigma &= i\alpha \mu\_0 \int\_{SRR} \left( H\_{ext} + \frac{1}{4\pi} \oint \frac{j d\mathbf{s} \times \mathbf{r}}{r^3} \right) d\sigma \\ &= V\_{ring} + 4V\_{gap} = \left[ Z(\mathbf{r}) \cdot \frac{4L}{\mathcal{W}\_1} - \frac{4\mathcal{W}\_2}{i\alpha \varepsilon\_0 \varepsilon\_m \mathcal{W}\_1 \pi} \right] j \end{split} \tag{3}$$

Here, we have used Biot-Savart's law. In this equation, *Hext* is the magnetic field of light, *ε<sup>m</sup>* and *Z*(*τ*) are the relative permittivity and internal impedance of gold, respectively, *τ* is the thickness of the SRR, *j* is the induced circular current in the SRR; and *L*, *W*1, and *W*2 are the dimensions of the SRR (see Fig. 7(a)). The distribution of the magnetic field around the SRR can be calculated using Eq. 3, as illustrated in Fig. 7(b).

Fig. 7. Four-cut single SRR placed in glass: (a) plane pattern and (b) magnetic field distribution around the SRR at resonant frequency (L = 300 nm). The distribution of field intensity is visualized by a rainbow color map.

#### *B. Macroscopic permeability of SRR array*

The results for a single SRR were used to calculate the effective permeability of an SRR array taking into consideration a two-dimensional array layer comprising cubic unit cells, each with an SRR at its center; both the side lengths of each cell and the array pitch of the cells were *d*. For simplicity, the SRRs were assumed to be placed in air. The macroscopic permeability *μyy* of the SRR array layer can be calculated using the field averaging equation [38], and it is given by

$$\overline{\mu}\_{yy} = \frac{\overline{B}\_y(0, d/2, 0)}{\overline{H}\_y(0, d/2, 0)} = \mu\_0 \frac{(d)^{-2} \int\_{-d/2}^{d/2} dx \int\_{-d/2}^{d/2} H\_z(x, d/2, z) dz}{(d)^{-1} \int\_0^d H\_z(0, y, 0) dy},\tag{4}$$

The total electromotive force (emf) induced around the SRR is given by the magnetic flux

*jd Bd i H <sup>d</sup>*

( )

= + = ⋅−

τ

Here, we have used Biot-Savart's law. In this equation, *Hext* is the magnetic field of light, *ε<sup>m</sup>* and *Z*(*τ*) are the relative permittivity and internal impedance of gold, respectively, *τ* is the thickness of the SRR, *j* is the induced circular current in the SRR; and *L*, *W*1, and *W*2 are the dimensions of the SRR (see Fig. 7(a)). The distribution of the magnetic field around the SRR

a b

The results for a single SRR were used to calculate the effective permeability of an SRR array taking into consideration a two-dimensional array layer comprising cubic unit cells, each with an SRR at its center; both the side lengths of each cell and the array pitch of the cells were *d*. For simplicity, the SRRs were assumed to be placed in air. The macroscopic permeability *μyy* of the SRR array layer can be calculated using the field averaging equation

<sup>0</sup> <sup>1</sup>

(0 /2 0) ( ) (0 0)

*H , d , d H , y, dy*

−

*<sup>z</sup> <sup>y</sup> d d*

*<sup>y</sup> <sup>z</sup>*

*yy d*

<sup>−</sup> = =

 μ

( ) ( /2 ) (0 /2 0)

*B , d , d dx H x, d , z dz*

/2 /2 2 /2 /2

− −

*d d*

0

, (4)

Fig. 7. Four-cut single SRR placed in glass: (a) plane pattern and (b) magnetic field distribution around the SRR at resonant frequency (L = 300 nm). The distribution of field

*L W V VZ <sup>j</sup> Wi W*

0 3

4

π

 

2

σ

(3)

*m*

1 01

ωε ε τ

1 s r

*r*

density *B*. Therefore, we can equate the potential drop to the emf as follows:

*<sup>t</sup> ext SRR SRR*

σ ωμ

can be calculated using Eq. 3, as illustrated in Fig. 7(b).

intensity is visualized by a rainbow color map.

*B. Macroscopic permeability of SRR array* 

μ

[38], and it is given by

*ring gap*

4 4 4

<sup>×</sup> ∂= +

where *By* and *Hy* with over lines represent the average values of magnetic flux density and magnetic field, respectively, in the SRR array layer. To get high accuracy in this homogenization approximation, the size of the integration region should be larger than the wavelength of light. In our calculation, we therefore integrated magnetic field over a large cubic tregion that cantained a 3 × 3 cell array (this means that we replaced d with 3d in Eq. 4 ). Figure 8 shows the real and imaginary parts of permeability as a function of frequency, and the SRR size *L* is a parameter; here, *W*1 and *W*2 were set as 100 nm and *d* was 1.8 μm. The thickness *τ* of the SRR was set at twice the penetration depth of gold at each frequency. As *L* decreases, the magnetic resonant frequency increases. Magnetic response could be obtained at the 1.55-μm-band frequency (approximately 193 THz) for the SRR size *L* of 750 nm (red curves). As *L* decreases, the magnetic response becomes weaker because the inductance of each SRR decreases; this increases the effective resistance (or decreases the Q factor) of the SRR.

Fig. 8. Real and imaginary parts of the effective permeability of gold SRR array in air (*ε* = 1) as a function of SRR size; here, *W1* = *W2* = 100 nm, d = 1.8 µm, and frequencies = 100–300 THz.

Further, the magnetic response of the SRR array layer depends on the array pitch *d*. If *d* is large, the response is weak because the area density of SRRs is small; on the other hand, if *d* is small, the magnetic field in each SRR is canceled by the fields of the neighboring SRRs, and this weakens the total response of the SRR array layer.

#### *C. Transmission characteristics of waveguide device with metamaterial*

Using the abovementioned results, the transmission characteristics of an MMI with a metamaterial, which is shown in Fig. 5, were estimated using computer simulations based on the transfer-matrix method.

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 225

51 1 11 12 21 22 00 0

β

<sup>=</sup>

( ) 2 2 *n z y* 0 *x z*

Here *dp* is the thickness of the *p*-th layer. We solved these eigenvalue equations and obtained the effective refractive index *βn*/*k*0 of each layer. With these results, we calculated the transmission characteristics of the metamaterial MMI using the Fourier expansion method,

Figure 9 illustrates the example of the distribution profile of light traveling in the metamaterial MMI device. The intensity of the TE electric field at the cross-section (*x*-*z* plane) of the device is shown. The MMI is 15-μm wide, 650-μm long, and has a 450-nm-thick InP cladding layer. The wavelength of light is 1565 nm. Figure 9(a) shows our device with a *4-cut* SRR array. The light traveling in the device suffers large propagation losses because of the magnetic interactions between the SRRs and light. Figure 9(b) shows the results for a control device with *no-cut* gold square rings having the same size as a *4-cut* SRR. The *no-cut* ring has no resonant frequency and shows no magnetic interactions with 1.5-μm light; hence, the propagation loss is approximately 10 dB smaller than that of the *4-cut* SRR. These results show that the *4-cut* SRR array can successfully operate as a metamaterial layer at

Fig. 9. Distribution profile of light traveling in metamaterial MMI device, calculated for devices with (a) *4-cut* SRR array and (b) *no-cut* gold square rings. Electric field intensity at the cross-section (*x*-*z* plane) of the device is visualized by shading. Wavelength of light is

0

7.5


Width (μm)

*jj j mm mm*

ωμ μ

0

 μμβ= − *k*

β

ωμ μ

*j*

β

which is commonly used for MMI propagation analysis [34].

Propagation direction (μm) 0 325 650

*z*

*m m d d*

ββ

ωμ μ

11 12 21 22 All

*m m*

where *m*11–*m*22 are given by

optical frequencies.

1565 nm.

Width (μm)

0

7.5


*zz z*

 − − +− =

0

 β

(11)

Propagation direction (μm) 0 325 650

 β , (9)

 β

ωμ μ

( ) ( )

β

ωμ μ

cosh sinh

sinh cosh

β

*<sup>n</sup> n p n p*

ε μ 0

∏ (10)

*<sup>z</sup> n p n p n*

*<sup>j</sup> d d*

() ()

We first designed a device for use at 1.55-μm wavelength. The resultant structure is as follows. The substrate is an InP (refractive index *n* = 3.16). The constituent layers of the MMI are (i) a core guiding layer: 200-nm-thick Ga0.25In0.75As0.54P0.46 (bandgap wavelength *λg* = 1.25 μm; *n* = 3.38), (ii) InP upper cladding layer (*n* = 3.16), and (iii) SRR metamaterial layer: 50 nm-thick gold SRR array (*L* = 750 nm; *W*1 = *W*2 = 100 nm; *d* = 1.6 μm).

The thickness of the InP cladding layer affects the strength of interaction between the light traveling in the MMI and the SRR array attached to the surface of the cladding layer. Therefore, we determined the optimal thickness of the cladding layer from the following calculations.

After having designed the device structure, we calculated the transmission characteristics of the device as follows. The permeability tensor of the *p*-th layer, i.e., the InP substrate (*p* = 1), GaInAsP core layer (*p* = 2), InP upper cladding layer (*p* = 3), SRR array layer (*p* = 4), and air (*p* = 5), is given by

$$
\tilde{\mu}\_p = \begin{pmatrix}
\mu\_{xx}\,^p & 0 & 0 \\
0 & \mu\_{yy}\,^p & 0 \\
0 & 0 & \mu\_{zz}\,^p
\end{pmatrix}\,'\tag{5}
$$

where the diagonal elements *μ*xx, *μ*yy, and *μ*zz are 1 at optical frequencies except in the SRR array layer. Using this tensor and the permittivity tensor *εp* for the *p*-th layer, Maxwell's equations are written as follows:

$$
\nabla \times \mathbf{H} = j\alpha \mathbf{e}\_0 \bar{\mathbf{e}}\_p \mathbf{E}
$$

$$
\nabla \times \mathbf{E} = -j\alpha \mu\_0 \tilde{\mu}\_p \mathbf{H} \tag{6}
$$

We solved Eq. 6 under the condition that the electric and magnetic fields are invariant in the x-direction, that is, ∂*x* = 0, and their tangential components are continuous at the boundary between the layers. For TE-mode light, the electric field *Ex* parallel to the *z*-axis is given by the following differential equation:

$$\left(\partial\_y \, \prescript{2}{}{E}\_x + \left(k\_0 \, \prescript{2}{}{\mathcal{E}}\_x \mu\_z - \frac{\mu\_z}{\mu\_y} \mathcal{J}^2\right) E\_x = 0\tag{7}$$

where *εx* is the diagonal element of the permittivity tensor, and 0 00 *k* = = ω μ ε πλ 2 is the free-space propagation constant. The magnetic field *Hz* parallel to the *z*-axis (propagation direction) can be calculated using *Ex* as follows:

$$
\Delta H\_z = -\frac{\dot{j}}{\alpha \mu\_0 \mu\_z} \partial\_y E\_x \tag{8}
$$

An eigenvalue equation can be obtained using the boundary conditions with continuous *Ex* and *Hz*. In the calculations, we assumed that *Ex* and *Hz* decrease exponentially outside the GaInAsP guiding layer (i.e., in the air and the InP layers). For simplicity, we also assumed that all the layers except the SRR array layer are birefringent. The eigenvalue equation is given by

$$-\frac{j\beta\_5}{o\mu\_0\mu\_z}\left(m\_{11} - m\_{12}\frac{j\beta\_1}{o\mu\_0\mu\_z}\right) + \left(m\_{21} - m\_{22}\frac{j\beta\_1}{o\mu\_0\mu\_z}\right) = 0,\tag{9}$$

where *m*11–*m*22 are given by

224 Metamaterial

We first designed a device for use at 1.55-μm wavelength. The resultant structure is as follows. The substrate is an InP (refractive index *n* = 3.16). The constituent layers of the MMI are (i) a core guiding layer: 200-nm-thick Ga0.25In0.75As0.54P0.46 (bandgap wavelength *λg* = 1.25 μm; *n* = 3.38), (ii) InP upper cladding layer (*n* = 3.16), and (iii) SRR metamaterial layer: 50-

The thickness of the InP cladding layer affects the strength of interaction between the light traveling in the MMI and the SRR array attached to the surface of the cladding layer. Therefore, we determined the optimal thickness of the cladding layer from the following

After having designed the device structure, we calculated the transmission characteristics of the device as follows. The permeability tensor of the *p*-th layer, i.e., the InP substrate (*p* = 1), GaInAsP core layer (*p* = 2), InP upper cladding layer (*p* = 3), SRR array layer (*p* = 4), and air

0 0

*p zz*

, (5)

(6)

ω μ ε

=− ∂ (8)

 πλ2 is the

(7)

μ

0 0

 

> μ

where the diagonal elements *μ*xx, *μ*yy, and *μ*zz are 1 at optical frequencies except in the SRR array layer. Using this tensor and the permittivity tensor *εp* for the *p*-th layer, Maxwell's

> H E <sup>0</sup> *<sup>p</sup>* ∇× = *j*ωε ε

E H <sup>0</sup> *<sup>p</sup>* ∇× =− *j*ωμ μ

We solved Eq. 6 under the condition that the electric and magnetic fields are invariant in the x-direction, that is, ∂*x* = 0, and their tangential components are continuous at the boundary between the layers. For TE-mode light, the electric field *Ex* parallel to the *z*-axis is given by

> 22 2 <sup>0</sup> 0 *<sup>z</sup> y x xz <sup>x</sup> y*

where *εx* is the diagonal element of the permittivity tensor, and 0 00 *k* = =

εμ

*Ek E* μ

 ∂+ − = 

free-space propagation constant. The magnetic field *Hz* parallel to the *z*-axis (propagation

0 *z y x z*

*<sup>j</sup> H E* ωμ μ

An eigenvalue equation can be obtained using the boundary conditions with continuous *Ex* and *Hz*. In the calculations, we assumed that *Ex* and *Hz* decrease exponentially outside the GaInAsP guiding layer (i.e., in the air and the InP layers). For simplicity, we also assumed that all the layers except the SRR array layer are birefringent. The eigenvalue equation is given by

 β

μ

*p*

0 0

*p xx*

*p yy*

μ

μ

=

nm-thick gold SRR array (*L* = 750 nm; *W*1 = *W*2 = 100 nm; *d* = 1.6 μm).

calculations.

(*p* = 5), is given by

equations are written as follows:

the following differential equation:

direction) can be calculated using *Ex* as follows:

$$
\begin{pmatrix} m\_{11} & m\_{12} \\ m\_{21} & m\_{22} \end{pmatrix} = \prod\_{\text{All}} \begin{pmatrix} \cosh\left(\mathcal{\beta}\_{n} d\_{p}\right) & \frac{j o \mu\_{0} \mu\_{z}}{\mathcal{\beta}\_{n}} \sinh\left(\mathcal{\beta}\_{n} d\_{p}\right) \\ \mathcal{\beta}\_{n} & \cosh\left(\mathcal{\beta}\_{n} d\_{p}\right) \end{pmatrix} \tag{10}
$$

$$
\beta\_n = \sqrt{\left(\mu\_z / \mu\_y\right) \beta^2 - k\_0^{-2} \varepsilon\_x \mu\_z} \tag{11}
$$

Here *dp* is the thickness of the *p*-th layer. We solved these eigenvalue equations and obtained the effective refractive index *βn*/*k*0 of each layer. With these results, we calculated the transmission characteristics of the metamaterial MMI using the Fourier expansion method, which is commonly used for MMI propagation analysis [34].

Figure 9 illustrates the example of the distribution profile of light traveling in the metamaterial MMI device. The intensity of the TE electric field at the cross-section (*x*-*z* plane) of the device is shown. The MMI is 15-μm wide, 650-μm long, and has a 450-nm-thick InP cladding layer. The wavelength of light is 1565 nm. Figure 9(a) shows our device with a *4-cut* SRR array. The light traveling in the device suffers large propagation losses because of the magnetic interactions between the SRRs and light. Figure 9(b) shows the results for a control device with *no-cut* gold square rings having the same size as a *4-cut* SRR. The *no-cut* ring has no resonant frequency and shows no magnetic interactions with 1.5-μm light; hence, the propagation loss is approximately 10 dB smaller than that of the *4-cut* SRR. These results show that the *4-cut* SRR array can successfully operate as a metamaterial layer at optical frequencies.

Fig. 9. Distribution profile of light traveling in metamaterial MMI device, calculated for devices with (a) *4-cut* SRR array and (b) *no-cut* gold square rings. Electric field intensity at the cross-section (*x*-*z* plane) of the device is visualized by shading. Wavelength of light is 1565 nm.

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 227

device. Since it is possible to know the occurrence of the interaction by measuring the propagation loss of light in the device, we primarily measured the transmission and

(a) (b) (c)

InP sub.

GaInAsP core InP sub. InP sub.

InP clad GaInAsP core

PMMA

Au

InP clad GaInAsP core

InP sub.

(f)

InP sub.

InP clad GaInAsP core

SiO2

Fig. 10. Fabrication process for metamaterial MMI devices. SRR array was prepared using

PMMA

InP clad GaInAsP core SiO2

InP sub.

In the measurement, light was sent from a tunable laser to the device through a polarization controller. The wavelength was changed in a range of 1420-1575 nm. To clarify the effect of the magnetic interaction, we took the difference between the transmission intensity for the experimental samples (with *4-cut* SRRs) and that for the control samples (with *2-cut* SRRs). This difference shows an intrinsic change in transmission intensity induced by the SRR resonance without including parasitic factors such as wavelength-dependent ohmic loss in SRR metal, lensed-fiber coupling loss, and wavelength-dependent propagation

electron-beam lithography and lift-off process.

(d) (e)

InP sub. InP sub.

characteristics in the MMI.

absorption of light in the device.

InP sub.

InP clad

InP sub.

Au

InP sub. InP clad GaInAsP core

#### **3.3 Device fabrication and measurement**

To move one step closer to the development of actual advanced optical-communication devices using the concept of metamaterials, we fabricated a trial device to confirm the magnetic response of a metamaterial comprising SRRs arrayed on a GaInAsP/InP 1 × 1 MMI coupler. The trial device was fabricated as follows. An undoped Ga0.25In0.75As0.54P0.46 core layer (λg = 1.22 μm, 200-nm thick) and an undoped InP cladding layer (420-nm thick) were grown on a (100) semi-insulating InP substrate by organometallic vapor phase epitaxy (OMVPE). On the surface of the cladding layer, SRRs consisting of Ti and Au layers were prepared using electron-beam lithography (EBL) and lift-off process.

The fabrication process is illustrated in Fig. 10 with both cross-sectional and plan views. The process flow was as follows: a resist layer of polymethyl methacrylate (PMMA) was first spin coated onto the InP cladding layer. Following spin coating, EBL was used to write a desired SRR array pattern onto the resist. The exposed areas of the resist were dissolved during development with xylene (Fig. 10(b)), resulting in a mask for the subsequent metalevaporation process (Fig. 10(c)). Subsequently, the unexposed resist was removed along with the metal on top during lift-off with acetone (Fig. 10(d)). Figure 11 shows oblique scanning electron microscope (SEM) images of the SRR array fabricated according to this procedure. The SRRs were made of 5-nm-thick titanium and 20-nm-thick gold, and the dimensions of the individual SRR were designed on the basis of the simulation results shown in Section 3.2. In addition to experimental samples with *4-cut* SRRs (Fig. 11(a)), we also made control samples with SRRs consisting of *2-cut* Au/Ti square rings (Fig. 11(b)) with the same side length as that of the *4-cut* SRR. This *2-cut* SRR has a resonant frequency far lower than 193 THz, about 100 THz, so it does not interact with 1.5-μm light.

After the SRR array was formed, a 75-nm-thick SiO2 film was deposited on the wafer by using plasma-enhanced chemical vapor deposition. Following the spin coating of PMMA, EBL was used again to write a 1 × 1 MMI pattern onto the resist (Fig. 10(e)). The width and length of the MMI were set to 15 μm and 660 μm. Finally, the exposed regions of the SiO2 film and InP cladding layer were etched by using buffered HF and reactive ion etching (RIE), respectively, with a mixture gas of CH4 and H2 (Fig. 10(f)).

Figures 12(a) and 12(b) show the oblique and cross-sectional SEM images of the MMI region with 300×300-nm SRRs. In this study, the SiO2 layer was not removed to prevent damages to the nanoscale SRR, which can be observed in Fig. 12(b). A thinner cladding layer (420 nm in Fig. 12(b)) is preferable to obtain large magnetic interactions even though it increases the propagation loss because the optical field coupled to the SRR metal is larger. The light phase is shifted by the magnetic interaction, but this can be neglected compared to the effect of the abovementioned propagation loss. Figure 12(c) shows the magnified plan of the trial device observed using an optical microscope. We made SRRs with different sizes from 300×300 to 550×550 nm (inside size of the square SRR ring). Both the width and gap of the SRR metal region were set to 75 nm.

In the following optical measurements, we observe the magnetic interactions of the propagating light and SRRs in the device. As described in Section 2, if magnetic interactions occur between the SRRs and light, the effective permeability of the SRR array becomes nonunity, i.e., large positive or negative values. At the same time, the imaginary part of the permeability increases from 0 to a finite value, and this implies that light is absorbed in the

To move one step closer to the development of actual advanced optical-communication devices using the concept of metamaterials, we fabricated a trial device to confirm the magnetic response of a metamaterial comprising SRRs arrayed on a GaInAsP/InP 1 × 1 MMI coupler. The trial device was fabricated as follows. An undoped Ga0.25In0.75As0.54P0.46 core layer (λg = 1.22 μm, 200-nm thick) and an undoped InP cladding layer (420-nm thick) were grown on a (100) semi-insulating InP substrate by organometallic vapor phase epitaxy (OMVPE). On the surface of the cladding layer, SRRs consisting of Ti and Au layers were

The fabrication process is illustrated in Fig. 10 with both cross-sectional and plan views. The process flow was as follows: a resist layer of polymethyl methacrylate (PMMA) was first spin coated onto the InP cladding layer. Following spin coating, EBL was used to write a desired SRR array pattern onto the resist. The exposed areas of the resist were dissolved during development with xylene (Fig. 10(b)), resulting in a mask for the subsequent metalevaporation process (Fig. 10(c)). Subsequently, the unexposed resist was removed along with the metal on top during lift-off with acetone (Fig. 10(d)). Figure 11 shows oblique scanning electron microscope (SEM) images of the SRR array fabricated according to this procedure. The SRRs were made of 5-nm-thick titanium and 20-nm-thick gold, and the dimensions of the individual SRR were designed on the basis of the simulation results shown in Section 3.2. In addition to experimental samples with *4-cut* SRRs (Fig. 11(a)), we also made control samples with SRRs consisting of *2-cut* Au/Ti square rings (Fig. 11(b)) with the same side length as that of the *4-cut* SRR. This *2-cut* SRR has a resonant frequency far

After the SRR array was formed, a 75-nm-thick SiO2 film was deposited on the wafer by using plasma-enhanced chemical vapor deposition. Following the spin coating of PMMA, EBL was used again to write a 1 × 1 MMI pattern onto the resist (Fig. 10(e)). The width and length of the MMI were set to 15 μm and 660 μm. Finally, the exposed regions of the SiO2 film and InP cladding layer were etched by using buffered HF and reactive ion etching

Figures 12(a) and 12(b) show the oblique and cross-sectional SEM images of the MMI region with 300×300-nm SRRs. In this study, the SiO2 layer was not removed to prevent damages to the nanoscale SRR, which can be observed in Fig. 12(b). A thinner cladding layer (420 nm in Fig. 12(b)) is preferable to obtain large magnetic interactions even though it increases the propagation loss because the optical field coupled to the SRR metal is larger. The light phase is shifted by the magnetic interaction, but this can be neglected compared to the effect of the abovementioned propagation loss. Figure 12(c) shows the magnified plan of the trial device observed using an optical microscope. We made SRRs with different sizes from 300×300 to 550×550 nm (inside size of the square SRR ring). Both the width and gap of the SRR metal

In the following optical measurements, we observe the magnetic interactions of the propagating light and SRRs in the device. As described in Section 2, if magnetic interactions occur between the SRRs and light, the effective permeability of the SRR array becomes nonunity, i.e., large positive or negative values. At the same time, the imaginary part of the permeability increases from 0 to a finite value, and this implies that light is absorbed in the

prepared using electron-beam lithography (EBL) and lift-off process.

lower than 193 THz, about 100 THz, so it does not interact with 1.5-μm light.

(RIE), respectively, with a mixture gas of CH4 and H2 (Fig. 10(f)).

region were set to 75 nm.

**3.3 Device fabrication and measurement** 

device. Since it is possible to know the occurrence of the interaction by measuring the propagation loss of light in the device, we primarily measured the transmission and absorption of light in the device.

Fig. 10. Fabrication process for metamaterial MMI devices. SRR array was prepared using electron-beam lithography and lift-off process.

In the measurement, light was sent from a tunable laser to the device through a polarization controller. The wavelength was changed in a range of 1420-1575 nm. To clarify the effect of the magnetic interaction, we took the difference between the transmission intensity for the experimental samples (with *4-cut* SRRs) and that for the control samples (with *2-cut* SRRs). This difference shows an intrinsic change in transmission intensity induced by the SRR resonance without including parasitic factors such as wavelength-dependent ohmic loss in SRR metal, lensed-fiber coupling loss, and wavelength-dependent propagation characteristics in the MMI.

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 229

a b

(c) Fig. 12. (a) Oblique views of completed device and (b) cross-sectional view of 300×300-nm SRRs buried in SiO2 layer; (c) Plan view of GaInAsP/InP 1 × 1 MMI coupler with SRR array

The equivalent permeability and permittivity of the SRR array layer are a function of wavelength. To know their values exactly, we must know both intrinsic absorption loss and phase shift of propagating light in the device. However, we cannot extract each separately from the transmission data shown in Fig. 13. So then, to cope with the problem, we have recently proposed a measurement method that uses a Mach-Zehnder interferometer (MZI) and successfully retrieved constitutive parameters for the SRR array. The following Section

observed with optical microscopy.

provides the outline of the results.

Fig. 11. (a) Enlarged oblique views of 4-cut SRRs, and (b) 2-cut SRRs observed with scanning electron microscopy.

Figure 13 plots the measured intensity difference for devices with a SRR size of (a) 300×300 nm2, (b) 350×350 nm2, (c) 400×400 nm2, and (d) 500×500 nm2 as a function of wavelength. The magnetic interaction was observed clearly in the device with 350×350-nm2 SRRs (see Fig. 13(b)). That is, the intensity difference, induced by the SRR resonance, showed its peak at a wavelength of 1500 nm. The peak shifted to a shorter wavelength with smaller SRRs (Fig. 13(a)) and longer wavelength with larger SRRs (Fig. 13(c)), and both were out of this measurement range. With 500×500-nm2 SRRs, the intensity difference was almost 0 at this wavelength range (Fig. 13(d)), which showed that no SRR resonance occurred at 1.5-μm wavelength. These wavelength-dependent and SRR-size-dependent transmission characteristics show that the magnetic field of light interacted successfully with the SRRs to produce magnetic resonance at optical frequency. In contrast, no intensity change was observed for the TM mode. This polarization-wavelength dependent absorption is positive proof that the magnetic interaction was successfully established in our device for the TEmode light. In this manner, we can realize *non-unity* permeability in InP-based photonic devices by using the SRR metamaterial

Furthermore, for the TE mode, a magnetic field perpendicular to the axis of the split-ring, by virtue of Ampere's law, created a circulating current via the charge accumulation at the gap (see Section 2 for details). Due to the presence of gaps, the resulting charge distribution was asymmetric; this results in charge accumulation around the capacitive gaps and induces an electric dipole moment. On the other hand, for the TM mode, an electric field is present parallel to the two symmetric sides of the split-ring, whereas in the approximation of the thin metallization, the current in the perpendicular directions was negligible. In addition, the size of the ring was smaller than the incident wavelength, and this ensured that the variations in the electric field between the two sides were also negligible for a first approximation. Consequently, the charge distribution resulting from this incidence was symmetric and did not generate a circulating current.

a b

Fig. 11. (a) Enlarged oblique views of 4-cut SRRs, and (b) 2-cut SRRs observed with scanning

Figure 13 plots the measured intensity difference for devices with a SRR size of (a) 300×300 nm2, (b) 350×350 nm2, (c) 400×400 nm2, and (d) 500×500 nm2 as a function of wavelength. The magnetic interaction was observed clearly in the device with 350×350-nm2 SRRs (see Fig. 13(b)). That is, the intensity difference, induced by the SRR resonance, showed its peak at a wavelength of 1500 nm. The peak shifted to a shorter wavelength with smaller SRRs (Fig. 13(a)) and longer wavelength with larger SRRs (Fig. 13(c)), and both were out of this measurement range. With 500×500-nm2 SRRs, the intensity difference was almost 0 at this wavelength range (Fig. 13(d)), which showed that no SRR resonance occurred at 1.5-μm wavelength. These wavelength-dependent and SRR-size-dependent transmission characteristics show that the magnetic field of light interacted successfully with the SRRs to produce magnetic resonance at optical frequency. In contrast, no intensity change was observed for the TM mode. This polarization-wavelength dependent absorption is positive proof that the magnetic interaction was successfully established in our device for the TEmode light. In this manner, we can realize *non-unity* permeability in InP-based photonic

Furthermore, for the TE mode, a magnetic field perpendicular to the axis of the split-ring, by virtue of Ampere's law, created a circulating current via the charge accumulation at the gap (see Section 2 for details). Due to the presence of gaps, the resulting charge distribution was asymmetric; this results in charge accumulation around the capacitive gaps and induces an electric dipole moment. On the other hand, for the TM mode, an electric field is present parallel to the two symmetric sides of the split-ring, whereas in the approximation of the thin metallization, the current in the perpendicular directions was negligible. In addition, the size of the ring was smaller than the incident wavelength, and this ensured that the variations in the electric field between the two sides were also negligible for a first approximation. Consequently, the charge distribution resulting from this incidence was

electron microscopy.

devices by using the SRR metamaterial

symmetric and did not generate a circulating current.

a b

(c)

Fig. 12. (a) Oblique views of completed device and (b) cross-sectional view of 300×300-nm SRRs buried in SiO2 layer; (c) Plan view of GaInAsP/InP 1 × 1 MMI coupler with SRR array observed with optical microscopy.

The equivalent permeability and permittivity of the SRR array layer are a function of wavelength. To know their values exactly, we must know both intrinsic absorption loss and phase shift of propagating light in the device. However, we cannot extract each separately from the transmission data shown in Fig. 13. So then, to cope with the problem, we have recently proposed a measurement method that uses a Mach-Zehnder interferometer (MZI) and successfully retrieved constitutive parameters for the SRR array. The following Section provides the outline of the results.

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 231

**4. Permeability retrieval in III-V semiconductor-based waveguide device with** 

The accurate permeability values in waveguide-based photonic devices is a very important factor, since it mostly determines their performance. Accurate permeability data, however, cannot be extracted from the simple transmission data of the previous devices because the transmission data also includes both the effect of SRR's effective permittivity and permeability. In this Section, we have proposed a measurement method that uses a Mach-Zehnder interferometer (MZI) and successfully retrieved constitutive parameters for the SRR array.

Our MZI device is shown in Fig. 14(d). It consists of two 3-dB couplers and two arms made with GaInAsP/InP ridge waveguides, with a metal SRR array attached on one of the arms. For TE-mode input light with the SRR-resonance frequency, the SRR array interacts with the light, thereby behaving as a material with a complex refractive-index. The real part of the refractive-index mainly affects the phase of traveling light, thereby changing the phase difference at the output coupler. The imaginary part mainly causes the propagation loss of light in the arm. Therefore, the transmittance of the MZI is determined by the phase

Fig. 14. GaInAsP/InP straight ridge waveguide without SRRs (a) and with 2-cut SRRs (b); Mach-Zehnder Interferometer consisting of GaInAsP/InP waveguides and (c) 2-cut / (d) 4-

**metamaterials** 

difference and propagation loss.

cut SRR array attached on one arm.

Fig. 13. (Color online) Transmission-intensity difference (or transmission spectra) for devices with SRR size of (a) 300×300 nm2, (b) 350×350 nm2, (c) 400×400 nm2, and (d) 500×500 nm2 as a function of wavelength from 1420 to 1575 nm, measured for TE-mode light.

8

8

6

6

4

4

Transmission intensity difference (dB)

Transmission intensity difference (dB)

2

2

0

0


6

6

8

4

4

Transmission intensity difference (dB)

Transmission intensity difference (dB)

2

2

0

0




1400 1440 1480 1520 1560 1600 Wavelength (nm)

1400 1440 1480 1520 1560 1600 Wavelength (nm)

1400 1420 1480 1520 1560 1600 Wavelength (nm)

Resonance frequency Out of range

Frequency (THz)

214.3 200 187.5

SRR size : 500×500 nm

1400 1420 1480 1520 1560 1600 Wavelength (nm)

SRR size : 350×350 nm

Frequency (THz)

214.3 200 187.5

Resonance peak = 1500 nm

8

8

6

6

4

4

Transmission intensity difference (dB)

Transmission intensity difference (dB)

2

2

0

0


8

8

6

6

4

4

Transmission intensity difference (dB)

Transmission intensity difference (dB)

2

2

0

0




1400 1440 1480 1520 1560 1600 Wavelength (nm)

1400 1440 1480 1520 1560 1600 Wavelength (nm)

1400 1420 1480 1520 1560 1600 Wavelength (nm)

Resonance peak > 1565 nm

1400 1420 1480 1520 1560 1600 Wavelength (nm)

Frequency (THz)

214.3 200 187.5 8

Resonance peak < 1420 nm

SRR size : 400×400 nm

SRR size : 300×300 nm

Frequency (THz) 214.3 200 187.5

a b

c d Fig. 13. (Color online) Transmission-intensity difference (or transmission spectra) for devices with SRR size of (a) 300×300 nm2, (b) 350×350 nm2, (c) 400×400 nm2, and (d) 500×500 nm2 as

a function of wavelength from 1420 to 1575 nm, measured for TE-mode light.

#### **4. Permeability retrieval in III-V semiconductor-based waveguide device with metamaterials**

The accurate permeability values in waveguide-based photonic devices is a very important factor, since it mostly determines their performance. Accurate permeability data, however, cannot be extracted from the simple transmission data of the previous devices because the transmission data also includes both the effect of SRR's effective permittivity and permeability. In this Section, we have proposed a measurement method that uses a Mach-Zehnder interferometer (MZI) and successfully retrieved constitutive parameters for the SRR array.

Our MZI device is shown in Fig. 14(d). It consists of two 3-dB couplers and two arms made with GaInAsP/InP ridge waveguides, with a metal SRR array attached on one of the arms. For TE-mode input light with the SRR-resonance frequency, the SRR array interacts with the light, thereby behaving as a material with a complex refractive-index. The real part of the refractive-index mainly affects the phase of traveling light, thereby changing the phase difference at the output coupler. The imaginary part mainly causes the propagation loss of light in the arm. Therefore, the transmittance of the MZI is determined by the phase difference and propagation loss.

Fig. 14. GaInAsP/InP straight ridge waveguide without SRRs (a) and with 2-cut SRRs (b); Mach-Zehnder Interferometer consisting of GaInAsP/InP waveguides and (c) 2-cut / (d) 4 cut SRR array attached on one arm.

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 233

Figure 16(b) shows the output intensity for MZIs with 4-cut SRRs and that for without SRRs. Their difference shows the intensity change induced by the phase shift and the absorption loss of light in the MZI. That is, the difference shows an intrinsic change in transmission intensity induced by the SRR resonance without including parasitic factors such as metal absorption loss in the SRRs and lensed-fiber coupling loss in the measurement system.

> Frequency (THz) 210 205 200 195 190

> > a

With 2-cut SRR array

Without SRR array

With 2-cut SRR array

MZI waveguide

With 4-cut SRR array

Straight Ridge waveguide

1400 1450 1500 1550 1600

With 4-cut SRR array *Magnetic resonance* 

*loss*

*Intensity change induced by phase shift and absorption loss*

*Metal absorption loss*

Wavelength (nm)

Frequency (THz) 210 205 200 195 190

b Fig. 16. Output intensity from devices with 4-cut SRRs (blue lines), 2-cut SRRs (res lines) and without SRRs (black lines) as a function of wavelength, measured for (a) straight ridge

Wavelength (nm)

1400 1450 1500 1550 1600

waveguides and (b) MZIs.

Transmission intensity (dB)

0















Transmission intensity (dB)

The propagation loss in the arms can be inferred from the transmission data of straight waveguides with/without the SRR array shown in Figs. 14(a) and 14(b). Therefore, the effective permeability and permittivity of the SRR array can be calculated, using measured data for the transmittance of the MZI.

An actual MZI was made for measurement at 1.5-μm optical communication wavelength. Epitaxial layer structures were the same as those of our previous device. On the surface of the device, an SRR array (consisting of 10-nm thick Ti and 40-nm thick Au) was formed using electron-beam lithography (EBL) and a lift-off process. Figure 15(a) shows the oblique images of the SRR array observed with a scanning electron microscope (SEM). The 4-cut SRR was used because it has a high resonant frequency owing to its small gap capacitance as stated in Section 2.

The size of the SRR was designed for use at 1.5-μm band frequency (193 THz). After the formation of the SRR array, a SiO2 mask (100-nm thick) for the MZI pattern was formed on the device with plasma-enhanced chemical-vapor-deposition and EBL. With the SiO2 mask, the MZI structure was formed using CH4/H2 reactive ion etching. Figure 15(b) shows the SEM image of the arms with/without the SRR array. The length of the SRR array along the arm was set to 500 μm. In addition to these experimental samples, straight ridge waveguides with/without the SRRs were made. We also prepared control samples with SRRs consisting of *2-cut* square rings with the same size as that of the *4-cut* SRR. As mentioned in Section 3, the *2-cut* SRR has a resonant frequency far higher than 193 THz, so it has no interaction with 1.5-μm light.

Fig. 15. SRRs and arms of MZI: (a) oblique view of 4-cut SRRs, and (b) two arms, observed with scanning electron microscopy.

Figure 16(a) shows the output intensity for straight waveguides with 4-cut SRR array (blue dots), without SRRs (black dots), and with 2-cut SRRs (red dots). The difference between the curves without SRRs and that with 2-cut SRRs corresponds to a loss caused by light absorption of the SRR metal. The difference between the 2-cut SRRs and 4-cut SRRs shows the loss caused by the magnetic interaction between the SRRs and light. The 4-cut SRRs resonated at about 1510-1520 nm and showed the maximum loss at this wavelength.

The propagation loss in the arms can be inferred from the transmission data of straight waveguides with/without the SRR array shown in Figs. 14(a) and 14(b). Therefore, the effective permeability and permittivity of the SRR array can be calculated, using measured

An actual MZI was made for measurement at 1.5-μm optical communication wavelength. Epitaxial layer structures were the same as those of our previous device. On the surface of the device, an SRR array (consisting of 10-nm thick Ti and 40-nm thick Au) was formed using electron-beam lithography (EBL) and a lift-off process. Figure 15(a) shows the oblique images of the SRR array observed with a scanning electron microscope (SEM). The 4-cut SRR was used because it has a high resonant frequency owing to its small gap capacitance as

The size of the SRR was designed for use at 1.5-μm band frequency (193 THz). After the formation of the SRR array, a SiO2 mask (100-nm thick) for the MZI pattern was formed on the device with plasma-enhanced chemical-vapor-deposition and EBL. With the SiO2 mask, the MZI structure was formed using CH4/H2 reactive ion etching. Figure 15(b) shows the SEM image of the arms with/without the SRR array. The length of the SRR array along the arm was set to 500 μm. In addition to these experimental samples, straight ridge waveguides with/without the SRRs were made. We also prepared control samples with SRRs consisting of *2-cut* square rings with the same size as that of the *4-cut* SRR. As mentioned in Section 3, the *2-cut* SRR has a resonant frequency far higher than 193 THz, so it

a b Fig. 15. SRRs and arms of MZI: (a) oblique view of 4-cut SRRs, and (b) two arms, observed

Figure 16(a) shows the output intensity for straight waveguides with 4-cut SRR array (blue dots), without SRRs (black dots), and with 2-cut SRRs (red dots). The difference between the curves without SRRs and that with 2-cut SRRs corresponds to a loss caused by light absorption of the SRR metal. The difference between the 2-cut SRRs and 4-cut SRRs shows the loss caused by the magnetic interaction between the SRRs and light. The 4-cut SRRs

resonated at about 1510-1520 nm and showed the maximum loss at this wavelength.

data for the transmittance of the MZI.

has no interaction with 1.5-μm light.

with scanning electron microscopy.

stated in Section 2.

Figure 16(b) shows the output intensity for MZIs with 4-cut SRRs and that for without SRRs. Their difference shows the intensity change induced by the phase shift and the absorption loss of light in the MZI. That is, the difference shows an intrinsic change in transmission intensity induced by the SRR resonance without including parasitic factors such as metal absorption loss in the SRRs and lensed-fiber coupling loss in the measurement system.

Fig. 16. Output intensity from devices with 4-cut SRRs (blue lines), 2-cut SRRs (res lines) and without SRRs (black lines) as a function of wavelength, measured for (a) straight ridge waveguides and (b) MZIs.

Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 235

Realizing *non-unity* permeability at optical frequencies can be expected to lead us to advanced optical-communication devices based on novel operation principles. To move one step closer to this goal, in this paper, we have demonstrated that the permeability in semiconductor photonic devices can be controlled using the concept of metamaterials.

As an actual example, we fabricated a GaInAsP/InP MMI device combined with an SRR array that operated as a metamaterial layer. The operation wavelength was set as 1.55 μm. The transmission characteristics of this metamaterial MMI device strongly depended on the polarization and wavelength of input light. This shows that the SRR array layer interacted with the magnetic field of light and produced magnetic resonance at optical frequencies. After that, to know constitutive parameters in the device exactly, we have proposed a measurement method that uses a MZI and successfully retrieved the accurate permeability value for the SRR array. The permeability exhibited a resonance at 200 THz, and the real part of the relative permeability changed from +2.2 to -0.3 in the vicinity of this frequency. Our results show the feasibility of III-V semiconductor-based waveguide photonic devices combined with metamaterials. This would be useful in the development of novel optical-

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**5. Conclusion** 

communication devices.

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**6. References** 

On simple condition that the electric and magnetic fields are constant in the x-direction, that is, ∂x = 0, the wave equation in each layer of the device is given by Eq. (7) in Section 2.

$$\left(\partial\_y \, ^2 E\_x + \left(k\_0 \, ^2 \varepsilon\_x \mu\_z - \frac{\mu\_z}{\mu\_y} \mathcal{J}^2\right) E\_x = 0\right) \tag{12}$$

To extract the permittivity *εx* of the SRR array without resonance, we used the difference between the transmission intensity of the straight waveguides with *2-cut* SRRs and that of the control device without SRRs. The permittivity *εx* of the SRR array can be calculated, using the transfer-matrix method with Eq. (12) (*μy* and *μz* in Eq. (12) are equal to 1 because the SRRs have no resonance). After that, the real and imaginary parts of permeability *μy* of the SRR array was retrieved, using the obtained permittivity *εx* and the transmission intensity ratio (= (2-cut SRRs)/(4-cut SRRs)) for the straight waveguide and that for the MZI. In calculation, the effective thickness of the SRR layer was set to 350 nm.

Figure 17 shows the retrieved permeability (real and imaginary parts) of the SRR array as a function of frequency. The permeability exhibited a resonance at 200 THz, and the real part of the relative permeability changed from +2.2 to -0.3 in the vicinity of this frequency. This results show the feasibility of semiconductor-based photonic devices combined with metamaterials.

Fig. 17. Retrieved effective permeability (real and imaginary parts) of SRR array on MZI waveguide, plotted as a function of frequency.

#### **5. Conclusion**

234 Metamaterial

On simple condition that the electric and magnetic fields are constant in the x-direction, that is, ∂x = 0, the wave equation in each layer of the device is given by Eq. (7) in Section 2.

> 22 2 <sup>0</sup> 0 *<sup>z</sup> y x xz <sup>x</sup> y*

In calculation, the effective thickness of the SRR layer was set to 350 nm.

metamaterials.

4

3

2

1

0


Retrieved permeability

εμ

*Ek E* μ

 ∂+ − =

To extract the permittivity *εx* of the SRR array without resonance, we used the difference between the transmission intensity of the straight waveguides with *2-cut* SRRs and that of the control device without SRRs. The permittivity *εx* of the SRR array can be calculated, using the transfer-matrix method with Eq. (12) (*μy* and *μz* in Eq. (12) are equal to 1 because the SRRs have no resonance). After that, the real and imaginary parts of permeability *μy* of the SRR array was retrieved, using the obtained permittivity *εx* and the transmission intensity ratio (= (2-cut SRRs)/(4-cut SRRs)) for the straight waveguide and that for the MZI.

Figure 17 shows the retrieved permeability (real and imaginary parts) of the SRR array as a function of frequency. The permeability exhibited a resonance at 200 THz, and the real part of the relative permeability changed from +2.2 to -0.3 in the vicinity of this frequency. This results show the feasibility of semiconductor-based photonic devices combined with

1600 1550 1500 1450 1400

Wavelength (nm)

Fig. 17. Retrieved effective permeability (real and imaginary parts) of SRR array on MZI

185 190 195 200 205 210 215

Frequency (THz)

waveguide, plotted as a function of frequency.

Real part of permeability *μ<sup>y</sup>*  β

(12)

Imaginary part of permeability *μ<sup>y</sup>*

μ

Realizing *non-unity* permeability at optical frequencies can be expected to lead us to advanced optical-communication devices based on novel operation principles. To move one step closer to this goal, in this paper, we have demonstrated that the permeability in semiconductor photonic devices can be controlled using the concept of metamaterials.

As an actual example, we fabricated a GaInAsP/InP MMI device combined with an SRR array that operated as a metamaterial layer. The operation wavelength was set as 1.55 μm. The transmission characteristics of this metamaterial MMI device strongly depended on the polarization and wavelength of input light. This shows that the SRR array layer interacted with the magnetic field of light and produced magnetic resonance at optical frequencies. After that, to know constitutive parameters in the device exactly, we have proposed a measurement method that uses a MZI and successfully retrieved the accurate permeability value for the SRR array. The permeability exhibited a resonance at 200 THz, and the real part of the relative permeability changed from +2.2 to -0.3 in the vicinity of this frequency. Our results show the feasibility of III-V semiconductor-based waveguide photonic devices combined with metamaterials. This would be useful in the development of novel opticalcommunication devices.

#### **6. References**


Non-Unity Permeability in InP-Based Photonic Device Combined with Metamaterial 237

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photonic circuits: Comparison of monolithic integration techniques (review


**10** 

*China* 

**Electromagnetic Response and Broadband** 

Metamaterial (Smith, Pendry et al. 2004) interacts with electromagnetic waves in a resonant manner, affording us new giving rise to a new route for subwavelength photonic devices, such as compact antenna substrate (Li, Hang et al. 2005) and subwavelength resonant cavity (Zhou, Li et al. 2005; Li, Hao et al. 2006). Planar-type configuration,thanks to layer-by-layer fabrication technique, has been recognized as the most efficient and easiest way for the realization of metamaterials from microwave, terahertz, and optical regimes. And it is interesting to note that, in contrast to the common notion that the operational bandwidth of metamaterial is usually very narrow due to the local resonance nature, our recent studies show that planar metamaterial can be broadband in functionality (Wei, Cao et al. 2010; Wei,

It is worth noting that the studies on plasmonics and metamaterials are heuristic and beneficial to each other. Surface plasmon polaritons (SPPs) modulate light waves at the metal-dielectric interface with wavelength much smaller than that in free space (Raether 1988),which enables the control of light in a subwavelength scale for nanophotonic devices (Barnes, Dereux et al. 2003). SPPs with large coherent length are useful in many areas, including optical processing, quantum information (Kamli, Moiseev et al. 2008) and novel light-matter interactions (Vasa, Pomraenke et al. 2008). The enhancement of local fields by SPPs is particularly crucial to absorption enhancement (Andrew, Kitson et al. 1997), nonlinear optical amplification (Coutaz, Neviere et al. 1985; Tsang 1996) and weak signal probing (Kneipp, Wang et al. 1997; Nie & Emory 1997). Although SPP only exists in the visible and near-infrared regimes where free conduction-band electrons on a metal surface are driven by external fields, its analogue can be found in other frequencies where surface charge-density wave does not exists. With induced surface current oscillations on an array of metallic building blocks (Pendry, Holden et al. 1996; Pendry, Holden et al. 1999; Sievenpiper, Zhang et al. 1999; Yen, Padilla et al. 2004; Hibbins, Evans et al. 2005; Liu, Genov et al. 2006; Lockyear, Hibbins et al. 2009), a metamaterial surface can manipulate electromagnetic waves in a similar way as SPPs. Such spoof SPPs or surface resonance states

We will summarize our recent studies on planar metamaterials covering the modal expansion theory (Sheng, Stepleman et al. 1982; Lalanne, Hugonin et al. 2000; Wei, Fu et al. 2010; Wei, Li et al. 2010; Wei, Cao et al. 2010; Wei, Cao et al. 2011), the broadband enhanced

on a meta-surface can be tuned by geometric parameters.

**1. Introduction**

Cao et al. 2011).

**Utilities of Planar Metamaterials** 

Hongqiang Li and Zeyong Wei *Physics Department, Tongji University* 

[39] L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," *IEEE J. Lightwave Technol.*, vol. 13, no. 4, pp. 615–627 (1995).

### **Electromagnetic Response and Broadband Utilities of Planar Metamaterials**

Hongqiang Li and Zeyong Wei *Physics Department, Tongji University China* 

#### **1. Introduction**

238 Metamaterial

[39] L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on

pp. 615–627 (1995).

self-imaging: principles and applications," *IEEE J. Lightwave Technol.*, vol. 13, no. 4,

Metamaterial (Smith, Pendry et al. 2004) interacts with electromagnetic waves in a resonant manner, affording us new giving rise to a new route for subwavelength photonic devices, such as compact antenna substrate (Li, Hang et al. 2005) and subwavelength resonant cavity (Zhou, Li et al. 2005; Li, Hao et al. 2006). Planar-type configuration,thanks to layer-by-layer fabrication technique, has been recognized as the most efficient and easiest way for the realization of metamaterials from microwave, terahertz, and optical regimes. And it is interesting to note that, in contrast to the common notion that the operational bandwidth of metamaterial is usually very narrow due to the local resonance nature, our recent studies show that planar metamaterial can be broadband in functionality (Wei, Cao et al. 2010; Wei, Cao et al. 2011).

It is worth noting that the studies on plasmonics and metamaterials are heuristic and beneficial to each other. Surface plasmon polaritons (SPPs) modulate light waves at the metal-dielectric interface with wavelength much smaller than that in free space (Raether 1988),which enables the control of light in a subwavelength scale for nanophotonic devices (Barnes, Dereux et al. 2003). SPPs with large coherent length are useful in many areas, including optical processing, quantum information (Kamli, Moiseev et al. 2008) and novel light-matter interactions (Vasa, Pomraenke et al. 2008). The enhancement of local fields by SPPs is particularly crucial to absorption enhancement (Andrew, Kitson et al. 1997), nonlinear optical amplification (Coutaz, Neviere et al. 1985; Tsang 1996) and weak signal probing (Kneipp, Wang et al. 1997; Nie & Emory 1997). Although SPP only exists in the visible and near-infrared regimes where free conduction-band electrons on a metal surface are driven by external fields, its analogue can be found in other frequencies where surface charge-density wave does not exists. With induced surface current oscillations on an array of metallic building blocks (Pendry, Holden et al. 1996; Pendry, Holden et al. 1999; Sievenpiper, Zhang et al. 1999; Yen, Padilla et al. 2004; Hibbins, Evans et al. 2005; Liu, Genov et al. 2006; Lockyear, Hibbins et al. 2009), a metamaterial surface can manipulate electromagnetic waves in a similar way as SPPs. Such spoof SPPs or surface resonance states on a meta-surface can be tuned by geometric parameters.

We will summarize our recent studies on planar metamaterials covering the modal expansion theory (Sheng, Stepleman et al. 1982; Lalanne, Hugonin et al. 2000; Wei, Fu et al. 2010; Wei, Li et al. 2010; Wei, Cao et al. 2010; Wei, Cao et al. 2011), the broadband enhanced

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 241

For a transverse magnetic (TM) polarized incidence from the free semi-space at *z<0* (region I), the total magnetic fields in region I and region II (*z>h*), which are along x direction, can be expressed in terms of incidence, reflection and transmission coefficients D0, Rm and Tm, as

1 0 ,0 , ( ) <sup>0</sup> exp( ) exp( ) exp sin *<sup>z</sup> <sup>m</sup> z m*

3 ,0 *m zm* exp( )exp sin ( )

angle. In region II of metallic lamellar gratings, the magnetic field can be written as the expansion coefficients *<sup>l</sup> a* and *<sup>l</sup> b* of forward and backward guided aperture modes, as

<sup>2</sup> ( ) , ,

where ( ) *<sup>l</sup> g x* is in-plane evelope function of waveguide mode of metallic layer. Under the as

cos , ( ) 2 2

By applying boundary continuum conditions at *z* = 0 and *z h* = − of the metal-air interfaces,

*D R ab D R ab ua u b T R ua u b T R*

χ

0 0

( )

Ω + =+ −= −

> 1 1

+ =Ω +

− =−

− −

( )

metallic layers and Bloch modesin dlectric layers or free space, and can be expressed as

−

−

−

−

*m l p*

χ

 Ω = <sup>=</sup> 

/ 2

The coefficients *<sup>l</sup> a* and *<sup>l</sup> b* of forward and backwardwaveguide modecan be derived by

/ 2 \* / 2 , / 2 \* / 2 / 2 / 2 , / 2

*l l <sup>p</sup> <sup>p</sup> iG x <sup>l</sup> <sup>p</sup> l m <sup>p</sup> iG x iG x p*

−

*<sup>p</sup> iG x <sup>l</sup> <sup>p</sup>*

*m*

( )

χ

− << <sup>=</sup>

*else*

( )

 

( )

*l m*, are the overlap integral of the projection between waveguide modes of

( )

*m*

() ()

*g x g x dx*

*e g x dx*

*m m*

*e e dx*

*g x e dx*

−

*H g x a jk z b jk z*

( ) exp exp( ) *l l zl l zl*

*l ww x when x*

*H T jk z ik x*

= +−

θ

θ

(5)

is the incident

(3)

(4)

θ

= −+ (2)

*x x*

<sup>=</sup> (1)

*H D jk z R jk z ik x*

*m*

*<sup>z</sup> <sup>j</sup> k* denotes *th m* order of Bloch wave vector in region i (i=1,2,3),

*m*

0

∞ =

*l*

sumption of PEC for metalis, it can be expressed analytically as

0,

 π

*l x*

*g x w*

Where , *i*

we can obtain

where Ω*m l*, ,

Solving EQ.4, as

χ

+∞ =−∞

+∞ =−∞ 

transmission (Wei, Cao et al. 2011), negative refraction and subwavelength imaging (Wei, Cao et al. 2010), and the coherent control of spontaneous emission radiations in a wide frequency range (Wei, Li et al. 2010).

#### **2. Broadband response of planar metamaterial**

We introduce briefly the modal expansion method developed for multi-layered planar metamaterials in 2.1, and discuss the broadband enhanced transmission through holey metallic multi-layers (Wei, Cao et al. 2011), broadband negative refraction and subwavelength imaging in fishnet stacked metamaterial (Wei, Cao et al. 2010)in 2.2. In Section 2.3, we examine the properties of surface resonance states at a dielectricmetamaterial interface that exhibit magnetic response to the incident waves and strong local field enhancement (Wei, Li et al. 2010). We will show that a thin metamaterial slab, with a thickness much smaller than the operational wavelength, supports delocalized magnetic surface resonance states with a long coherent length in a wide range of frequencies. Operating in a broad frequency range, these spatially coherent SPPs are surface resonance states with quasi-TEM modes guided in the dielectric layer that are weakly coupled to free space, and the coupling strength can be controlled by tuning structural parameters while the frequency can be controlled by varying structural and material parameters. The high fidelity of these surface resonance states results in directional absorptivity or emissivity, which is angle-dependent with respect to frequency. These surface resonance states can give highly directional absorptivity and emissivity, and may thus help to realize interesting effects such as spatially coherent thermal emission, low-threshold plasmon lasing and sensitive photoelectric detection (Cao, Wei et al. 2011).

#### **2.1 A method of modal expansion for planar metamaterials**

The MEM is advantageous for analysis of electromagnetic transportation in planar metamaterial with layered geometry. The essence of MEM is to expand local EM fields in each layer as a series of in-plane envelope functions of eigenmodes. Let's demonstrate the formalism of MEM by solving the transmission spectra through one-dimensional metallic lamellar gratings with a thickness of *h*, as shown in Fig. 1. For simplicity, metals are treated as perfectly electric conductors (PEC) and EM waves only exist in apertures with the metallic layer.

Fig. 1. Schematic of a one-dimensional metallic lamella grating.

transmission (Wei, Cao et al. 2011), negative refraction and subwavelength imaging (Wei, Cao et al. 2010), and the coherent control of spontaneous emission radiations in a wide

We introduce briefly the modal expansion method developed for multi-layered planar metamaterials in 2.1, and discuss the broadband enhanced transmission through holey metallic multi-layers (Wei, Cao et al. 2011), broadband negative refraction and subwavelength imaging in fishnet stacked metamaterial (Wei, Cao et al. 2010)in 2.2. In Section 2.3, we examine the properties of surface resonance states at a dielectricmetamaterial interface that exhibit magnetic response to the incident waves and strong local field enhancement (Wei, Li et al. 2010). We will show that a thin metamaterial slab, with a thickness much smaller than the operational wavelength, supports delocalized magnetic surface resonance states with a long coherent length in a wide range of frequencies. Operating in a broad frequency range, these spatially coherent SPPs are surface resonance states with quasi-TEM modes guided in the dielectric layer that are weakly coupled to free space, and the coupling strength can be controlled by tuning structural parameters while the frequency can be controlled by varying structural and material parameters. The high fidelity of these surface resonance states results in directional absorptivity or emissivity, which is angle-dependent with respect to frequency. These surface resonance states can give highly directional absorptivity and emissivity, and may thus help to realize interesting effects such as spatially coherent thermal emission, low-threshold plasmon lasing and sensitive photo-

The MEM is advantageous for analysis of electromagnetic transportation in planar metamaterial with layered geometry. The essence of MEM is to expand local EM fields in each layer as a series of in-plane envelope functions of eigenmodes. Let's demonstrate the formalism of MEM by solving the transmission spectra through one-dimensional metallic lamellar gratings with a thickness of *h*, as shown in Fig. 1. For simplicity, metals are treated as perfectly electric conductors (PEC) and EM waves only exist in apertures with the

frequency range (Wei, Li et al. 2010).

electric detection (Cao, Wei et al. 2011).

metallic layer.

**2.1 A method of modal expansion for planar metamaterials** 

Fig. 1. Schematic of a one-dimensional metallic lamella grating.

**2. Broadband response of planar metamaterial** 

For a transverse magnetic (TM) polarized incidence from the free semi-space at *z<0* (region I), the total magnetic fields in region I and region II (*z>h*), which are along x direction, can be expressed in terms of incidence, reflection and transmission coefficients D0, Rm and Tm, as

$$H\_1 = \left[D\_0 \exp(jk\_{z,0}z) + \sum\_{n=-\ast}^{+\ast} R\_n \exp(-jk\_{z,n}z)\right] \exp\left(ik\_0 \sin\theta x\right)$$

$$H\_3 = \sum\_{n=-\ast}^{+\ast} T\_n \exp(jk\_{z,n}z) \exp\left(ik\_0 \sin\theta x\right) \tag{1}$$

Where , *i <sup>z</sup> <sup>j</sup> k* denotes *th m* order of Bloch wave vector in region i (i=1,2,3), θ is the incident angle. In region II of metallic lamellar gratings, the magnetic field can be written as the expansion coefficients *<sup>l</sup> a* and *<sup>l</sup> b* of forward and backward guided aperture modes, as

$$H\_2 = \sum\_{l=0}^{\bullet} \mathbf{g}\_l(\mathbf{x}) \left[ a\_l \exp\left( -j k\_{\varepsilon,l} z \right) + b\_l \exp(j k\_{\varepsilon,l} z) \right] \tag{2}$$

where ( ) *<sup>l</sup> g x* is in-plane evelope function of waveguide mode of metallic layer. Under the as sumption of PEC for metalis, it can be expressed analytically as

$$\mathbf{g}\_{l}(\mathbf{x}) = \begin{cases} \cos\left(\frac{\pi l}{\mathbf{w}\_{\times}} \mathbf{x}\right), & \text{when} \quad -\frac{\mathbf{w}\_{\times}}{2} < \mathbf{x} < \frac{\mathbf{w}\_{\times}}{2} \\ \mathbf{0}, & \text{else} \end{cases} \tag{3}$$

By applying boundary continuum conditions at *z* = 0 and *z h* = − of the metal-air interfaces, we can obtain

$$\begin{cases} \underline{\mathfrak{Q}}(D\_0 + \vec{R}) = \vec{a} + \vec{b} \\ D\_0 - \vec{R} = \underline{\mathcal{X}}(\vec{a} - \vec{b}) \\ \mu \vec{a} + \boldsymbol{\mu}^{-1} \vec{b} = \underline{\mathfrak{Q}}(\vec{T} + \vec{R}) \\ \underline{\mathcal{X}}(\boldsymbol{u}\vec{a} - \boldsymbol{\mu}^{-1}\vec{b}) = \vec{T} - \vec{R} \end{cases} \tag{4}$$

where Ω*m l*, , χ*l m*, are the overlap integral of the projection between waveguide modes of metallic layers and Bloch modesin dlectric layers or free space, and can be expressed as

$$\begin{cases} \mathbf{Q}\_{m,l} = \frac{\int\_{-p/2}^{p/2} e^{iG\_{\mathbf{x}}\mathbf{x}} \mathbf{g}\_l^\star(\mathbf{x}) d\mathbf{x}}{\int\_{-p/2}^{p/2} \mathbf{g}\_l(\mathbf{x}) \mathbf{g}\_l^\star(\mathbf{x}) d\mathbf{x}}\\\\ \mathcal{Z}\_{l,m} = \frac{\int\_{-p/2}^{p/2} \mathbf{g}\_l(\mathbf{x}) e^{-iG\_{\mathbf{x}}\mathbf{x}} d\mathbf{x}}{\int\_{-p/2}^{p/2} e^{-iG\_{\mathbf{x}}\mathbf{x}} e^{iG\_{\mathbf{x}}\mathbf{x}} d\mathbf{x}} \end{cases} \tag{5}$$

The coefficients *<sup>l</sup> a* and *<sup>l</sup> b* of forward and backwardwaveguide modecan be derived by Solving EQ.4, as

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 243

which is several orders faster than any other numerical methods. The method can, in principle, be generalized to a layered metallo-dielectric structure perforated with arbitrary shaped apertures as the in-plane functions of eigenmodes within each layer can always be

Extraordinary optical transmission (EOT) through metallic film perforated with subwavelength hole arrays has attracted considerable attentions since the pioneering study by T.W. Ebbesen and his coworkers (Ebbesen, Lezec et al. 1998; Ghaemi, Thio et al. 1998). Substantial efforts have been devoted to exploring the physical origin of EOT, both theoretically and experimentally, due to the appealing prospect in related applications (Martin-Moreno, Garcia-Vidal et al. 2001; de Abajo & Saenz 2005; Gay, Alloschery et al. 2006; Liu & Lalanne 2008; Xiao, Jinbo et al. 2010; Bahk, Park et al. 2011). Previous studies extensively investigated the EOT effects arising from the resonant tunneling of surface plasmon polaritons (SPPs) (Raether 1988; Barnes, Dereux et al. 2003) through the perforated metallic film. The frequency of such an EOT transmission peak is not only scaled to the period of hole arrays, but also very sensitive to the incident angle as the resonant tunneling occurs via the in-plane Bragg-scattering channels. Very recently, similar phenomena of the EOT through cascaded metallic multi-layers, which are perforated with one-dimensional gratings or two-dimensional hole arrays, have also been brought into attention (Miyamaru & Hangyo 2005; Ye & Zhang 2005; Chan, Marcet et al. 2006; Tang, Peng et al. 2007; Ortuno, Garcia-Meca et al. 2009; Marcet, Hang et al. 2010; Zhou, Huang et al. 2010). The resonant coupling among the SPP modes on different layers can be tuned by the spacing distance and lateral displacement of hole arrays at different layers, leading to tunable transmission peaks and zeros in spectra. It is worth noting that, when the slit size is very large or some kinds of specific apertures are adopted, the waveguide resonant modes of a slit or aperture can also give rise to the phenomena of EOT by allowing electromagnetic waves to propagate through the metallic slab. The cut-off wavelength of guided resonance modes (Baida, Van Labeke et al. 2004; Fan, Zhang et al. 2005; Fan, Zhang et al. 2005; van der Molen, Klein Koerkamp et al. 2005; Wen, Zhou et al. 2005; Wei, Fu et al. 2010) is primarily determined by the geometry of slits or apertures, and thus can be much longer than the array period. Under this circumstance, the EOT can also occur at a rather low frequency which is not scaled to the array period,and is robust against the structure disorder (Ruan & Qiu 2006). To the best of our knowledge, the EOT of metallic multi-layers arising from guided resonance modes has

Here, we investigate the enhanced transmission of metallic multi-layers perforated with periodic arrays of coaxial annular apertures (CAAs). Modal expansion method (MEM) is developed to semi-analytically deal with the electromagnetic properties of the multilayered system. We show that the hybridization of guided resonance modes of CAAs in adjacent layers dramatically extends an enhanced transmission peak into a broad passband that is nearly reflectionless. The passband gets more and more broadened with sharper edges when the system contains more metallic layers. In contrast, these results can not be observed when the wave propagation is dictated by evanescent coupling of SPP modes (Miyamaru & Hangyo 2005; Ye & Zhang 2005; Chan, Marcet et al. 2006; Tang, Peng et al. 2007; Ortuno, Garcia-Meca

solved by a standard algorithm for eigenvectors of a two-dimensional system.

**coaxial annular apertures** 

not yet been investigated before.

**2.2 Broadband transparency from stacked metallic multi-layers perforated with** 

$$\begin{cases} \vec{a} = \frac{M\_b^{-1}}{M\_b^{-1} \bullet M\_a - M\_d^{-1} \bullet M\_c} 2\underline{\Omega}\vec{D} \\\\ \vec{b} = \frac{M\_a^{-1}}{M\_a^{-1} \bullet M\_b - M\_c^{-1} \bullet M\_d} 2\underline{\Omega}\vec{D} \end{cases} \tag{6}$$

where

$$\begin{cases} M\_{\circ} = I + \underline{\Omega}\underline{\chi} \\ M\_{\circ} = I - \underline{\Omega}\underline{\chi} \\ M\_{\circ} = [I - \underline{\Omega}\underline{\chi}]\mu \\ M\_{\circ} = [I + \underline{\Omega}\underline{\chi}]\mu \end{cases} \tag{7}$$

One advantage of MEM is that the problem can be solved without solving the inverse matrix of Ω*m l*, , χ*l m*, so that the order of plane waves and that of waveguide modes are not necessarily be the same. At wavelength much larger than the array period, an *l* ≠ 0 high order mode is evanescent and contribute little to transmission and reflection as the z component of wavevector is a large imaginary number.And the calculation is quickly convergent by adopting only a few waveguide modes.We can also see from Fig.2 that the calculated transmission spectra converge qucikly with only 7 plane waves considered as well.

Fig. 2. Calculated 0th-order transmittance through the grating with different numbers of plane wave considered.

It is noticeable that the inter-layer coupling of EM waves can be analytically dealt with projection integral between the in-plane eigenmode functions of two adjacent layers. Under the treatment of MEM, a three-dimensional EM calculation will be simplified to a problem in two dimension. Thus the semi-analytical method is much faster than the conventional numerical simulations such as finite-difference-in-time-domain (FDTD) method, finiteelement method etc. For metallic gratings, holey mesh, coaxial and split-ring structures, the method is quickly convergent by adopting only one or a few guided modes of metallic layers. The results shown in Fig.2 can be accomplished on an ordinary PC within a second,

> [ ] [ ]

χ

χ

χ

χ

*l m*, so that the order of plane waves and that of waveguide modes are not necessarily

*M I u M I u*

One advantage of MEM is that the problem can be solved without solving the inverse matrix

be the same. At wavelength much larger than the array period, an *l* ≠ 0 high order mode is evanescent and contribute little to transmission and reflection as the z component of wavevector is a large imaginary number.And the calculation is quickly convergent by adopting only a few waveguide modes.We can also see from Fig.2 that the calculated

*b ba dc a ab cd*

= Ω

= Ω

<sup>−</sup>

−

transmission spectra converge qucikly with only 7 plane waves considered as well.

Fig. 2. Calculated 0th-order transmittance through the grating with different numbers of

It is noticeable that the inter-layer coupling of EM waves can be analytically dealt with projection integral between the in-plane eigenmode functions of two adjacent layers. Under the treatment of MEM, a three-dimensional EM calculation will be simplified to a problem in two dimension. Thus the semi-analytical method is much faster than the conventional numerical simulations such as finite-difference-in-time-domain (FDTD) method, finiteelement method etc. For metallic gratings, holey mesh, coaxial and split-ring structures, the method is quickly convergent by adopting only one or a few guided modes of metallic layers. The results shown in Fig.2 can be accomplished on an ordinary PC within a second,

*a b c d*

*M I M I*

= +Ω

= −Ω

 = −Ω = +Ω

where

of Ω*m l*, ,

χ

plane wave considered.

*<sup>M</sup> a D MM MM <sup>M</sup> b D MM MM*

 

− − − − − −

2

2

(6)

(7)

which is several orders faster than any other numerical methods. The method can, in principle, be generalized to a layered metallo-dielectric structure perforated with arbitrary shaped apertures as the in-plane functions of eigenmodes within each layer can always be solved by a standard algorithm for eigenvectors of a two-dimensional system.

#### **2.2 Broadband transparency from stacked metallic multi-layers perforated with coaxial annular apertures**

Extraordinary optical transmission (EOT) through metallic film perforated with subwavelength hole arrays has attracted considerable attentions since the pioneering study by T.W. Ebbesen and his coworkers (Ebbesen, Lezec et al. 1998; Ghaemi, Thio et al. 1998). Substantial efforts have been devoted to exploring the physical origin of EOT, both theoretically and experimentally, due to the appealing prospect in related applications (Martin-Moreno, Garcia-Vidal et al. 2001; de Abajo & Saenz 2005; Gay, Alloschery et al. 2006; Liu & Lalanne 2008; Xiao, Jinbo et al. 2010; Bahk, Park et al. 2011). Previous studies extensively investigated the EOT effects arising from the resonant tunneling of surface plasmon polaritons (SPPs) (Raether 1988; Barnes, Dereux et al. 2003) through the perforated metallic film. The frequency of such an EOT transmission peak is not only scaled to the period of hole arrays, but also very sensitive to the incident angle as the resonant tunneling occurs via the in-plane Bragg-scattering channels. Very recently, similar phenomena of the EOT through cascaded metallic multi-layers, which are perforated with one-dimensional gratings or two-dimensional hole arrays, have also been brought into attention (Miyamaru & Hangyo 2005; Ye & Zhang 2005; Chan, Marcet et al. 2006; Tang, Peng et al. 2007; Ortuno, Garcia-Meca et al. 2009; Marcet, Hang et al. 2010; Zhou, Huang et al. 2010). The resonant coupling among the SPP modes on different layers can be tuned by the spacing distance and lateral displacement of hole arrays at different layers, leading to tunable transmission peaks and zeros in spectra. It is worth noting that, when the slit size is very large or some kinds of specific apertures are adopted, the waveguide resonant modes of a slit or aperture can also give rise to the phenomena of EOT by allowing electromagnetic waves to propagate through the metallic slab. The cut-off wavelength of guided resonance modes (Baida, Van Labeke et al. 2004; Fan, Zhang et al. 2005; Fan, Zhang et al. 2005; van der Molen, Klein Koerkamp et al. 2005; Wen, Zhou et al. 2005; Wei, Fu et al. 2010) is primarily determined by the geometry of slits or apertures, and thus can be much longer than the array period. Under this circumstance, the EOT can also occur at a rather low frequency which is not scaled to the array period,and is robust against the structure disorder (Ruan & Qiu 2006). To the best of our knowledge, the EOT of metallic multi-layers arising from guided resonance modes has not yet been investigated before.

Here, we investigate the enhanced transmission of metallic multi-layers perforated with periodic arrays of coaxial annular apertures (CAAs). Modal expansion method (MEM) is developed to semi-analytically deal with the electromagnetic properties of the multilayered system. We show that the hybridization of guided resonance modes of CAAs in adjacent layers dramatically extends an enhanced transmission peak into a broad passband that is nearly reflectionless. The passband gets more and more broadened with sharper edges when the system contains more metallic layers. In contrast, these results can not be observed when the wave propagation is dictated by evanescent coupling of SPP modes (Miyamaru & Hangyo 2005; Ye & Zhang 2005; Chan, Marcet et al. 2006; Tang, Peng et al. 2007; Ortuno, Garcia-Meca

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 245

( , ) ' ( ) ( ) ' ( ) ( ) sin( ) *<sup>l</sup> ll ll pl ll j l <sup>g</sup> N Tr J T J Tr N T l*

and ( , ) ' ( ) ' ( ) ' ( ) ' ( ) cos( ) *<sup>l</sup> ll ll pl ll f*

are the *th l* order modal functions of radial and angular components in aperture with ( ) *<sup>l</sup> J x* and ( ) *Nl x* being the *th l* order Bessel and Neumann functions, *Tl* refers to the root of the equation '( ) '( ) '( ) '( ) 0 *l l ll J TR N Tr J Tr N TR* − = . By adopting EQ. (8) as expressions of EM fields in metallic layers and plane-waves as those in dielectric layers, we perform MEM to resolve the electromagnetic problems in the multilayered system. The method is quickly convergent by considering only 2 or 3 lowest guided resonance modes of CAAs. A higher order

imaginary number. Three guided modes ( *l* =1,2,3 ) in CAAs and 11 11 × orders of planewave basis in dielectric layers are adopted in our calculations. The results are very accurate (solid lines in Fig.4) and in good agreement with the measurements (circular dots in Fig.4).

Fig. 4. Transmission spectra through the models with (a) n=1, (b) n=2, (c) n=3, (d) n=10 metallic layers. Solid lines for calculated results by Modal expansion method (MEM),

We see from Fig. 4 (a) that there exists a transmission peak for the n=1 sample at 8.7GHz *Af* = due to the excitation of guided TE11 resonance mode in CAAs. We also see from Figs. 4 (b) and 4 (c) that there are two transmission peaks at 9.1GHz *Bf* = , and 12.3GHz *Cf* = for the *n* = 2 sample, three peaks at 8.2GHz *Df* = , 11.64GHz *Ef* = and 12.35GHz *Ff* = for the *n* = 3 sample. Figure 4 (d) presents the calculated transmission spectra of an n=10 model system. It means that, with the increase of metallic layers, more

More calculations show that, for the *n* = 2 sample, at an on-resonance frequency 9.1GHz *Bf* = or 12.3GHz *Cf* = where transmissivity is nearly unity, the spatial distribution of electric fields [see Figs. 5 (a) and 5 (b)] are symmetric or anti-symmetric about the *xy* plane. And the transmitted waves possess a phase difference of 0 (in phase) or

(out phase) with respect to the incident waves. Therefore the peaks at *Bf* and *Cf* , derived

circular dots for measured results in microwave regime.

transmission peaks emerge, giving rise to a broad transparent band.

resonance mode contributes little to the interlayer coupling as its wave vector

ρ

= −

= − *j T N Tr J T J Tr N T l*

 ρ

 ρ

> ρφ

φ

β

*<sup>l</sup>* is a large

π

ωμ

ρ

ωμ

ρ φ

> ρ φ

et al. 2009; Marcet, Hang et al. 2010; Zhou, Huang et al. 2010). Measured transmission spectra are in good agreement with calculations for the model systems with different metallic layers. The broadening and varied fine structures of the EOT passband with the increase of metallic layers,can be understood intuitively by a physical picture of mode splitting of coupled atoms. The passband of the enhanced transmission for a system with only two or three metallic layers, covering a wide frequency range with sharp band-edges, can be estimated by calculated dispersion diagram under the assumption of infinite metallic layers.

A model system with *n* metallic layers perforated with square arrays of CAAs is of our interest. Figure 3 presents the front-view photo and schematic configuration of a sample with three thin metallic layers ( *n* = 3 ) and two sandwiched dielectric space layers. The aperture arrays deposited on different layers are aligned with no displacement in *xy* plane. The geometric parameters are the lattice constant *p* =10mm of square arrays, the outer radius *R* = 4.8mm and inner radius *r* = 3.8mm of CAAs, and the thickness *t* = 0.035mm of metallic layer respectively. Each dielectric layer has a thickness of *h* =1.575mm and a permittivity of 2.65 *<sup>r</sup>* ε= .

Fig. 3. (a) Top-view photo and (b) 3D schematic of our sample with three metallic layers (n = 3). The metallic layers are perforated with coaxial annular apertures (CAAs).

Under assumption of perfect electric conductor (PEC) for metals, the electromagnetic wave fields within a metallic layer only exist in apertures. In cylindrical coordinate system, the radial and angular field components *E*ρ and *E*φ inside an aperture of the metallic layer can be analytically expanded by the superposition of guided resonance modes of the aperture, as

$$E\_{\rho}(\rho,\phi,z) = \sum\_{l=1}^{\bullet} (a\_l e^{-i\beta\_l z} + b\_l e^{i\beta\_l z}) \mathbf{g}\_l(\rho,\phi)$$

$$E\_{\phi}(\rho,\phi,z) = \sum\_{l=1}^{\bullet} (a\_l e^{-i\beta\_l z} + b\_l e^{i\beta\_l z}) f\_l(\rho,\phi) \,, \tag{8}$$

where *<sup>l</sup> a* and *<sup>l</sup> b* are the coefficients of forward and backward guided waves inside the CAAs,

et al. 2009; Marcet, Hang et al. 2010; Zhou, Huang et al. 2010). Measured transmission spectra are in good agreement with calculations for the model systems with different metallic layers. The broadening and varied fine structures of the EOT passband with the increase of metallic layers,can be understood intuitively by a physical picture of mode splitting of coupled atoms. The passband of the enhanced transmission for a system with only two or three metallic layers, covering a wide frequency range with sharp band-edges, can be estimated by calculated

A model system with *n* metallic layers perforated with square arrays of CAAs is of our interest. Figure 3 presents the front-view photo and schematic configuration of a sample with three thin metallic layers ( *n* = 3 ) and two sandwiched dielectric space layers. The aperture arrays deposited on different layers are aligned with no displacement in *xy* plane. The geometric parameters are the lattice constant *p* =10mm of square arrays, the outer radius *R* = 4.8mm and inner radius *r* = 3.8mm of CAAs, and the thickness *t* = 0.035mm of metallic layer respectively. Each dielectric layer has a thickness of *h* =1.575mm and a

Fig. 3. (a) Top-view photo and (b) 3D schematic of our sample with three metallic layers (n = 3). The metallic layers are perforated with coaxial annular apertures (CAAs).

analytically expanded by the superposition of guided resonance modes of the aperture, as

<sup>∞</sup> <sup>−</sup> =

<sup>∞</sup> <sup>−</sup> =

1

1

*l E z ae be f* β

*l E z ae be g* β

ρ

ρ ρ φ

φ ρ φ

 and *E*φ

Under assumption of perfect electric conductor (PEC) for metals, the electromagnetic wave fields within a metallic layer only exist in apertures. In cylindrical coordinate system, the radial

> (,,) ( ) (,) *l l iz iz l ll*

(,,) ( ) (,) *l l iz iz l ll*

where *<sup>l</sup> a* and *<sup>l</sup> b* are the coefficients of forward and backward guided waves inside the

 β

 β

= +

ρ φ

ρ φ

= + , (8)

inside an aperture of the metallic layer can be

dispersion diagram under the assumption of infinite metallic layers.

permittivity of 2.65 *<sup>r</sup>* ε= .

and angular field components *E*

CAAs,

$$\begin{aligned} \mathbf{g}\_{l}(\rho,\phi) &= \frac{j\alpha\mu l}{\rho} \Big[ N'\_{l}(T\_{l}r)J\_{l}(T\_{l}\rho) - J'\_{\rho}(T\_{l}r)N\_{l}(T\_{l}\rho) \Big] \sin(l\phi) \\ \text{and} \quad f\_{l}(\rho,\phi) &= j\alpha\mu T \Big[ N'\_{l}(T\_{l}r)J'\_{l}(T\_{l}\rho) - J'\_{\rho}(T\_{l}r)N'\_{l}(T\_{l}\rho) \Big] \cos(l\phi) \end{aligned}$$

are the *th l* order modal functions of radial and angular components in aperture with ( ) *<sup>l</sup> J x* and ( ) *Nl x* being the *th l* order Bessel and Neumann functions, *Tl* refers to the root of the equation '( ) '( ) '( ) '( ) 0 *l l ll J TR N Tr J Tr N TR* − = . By adopting EQ. (8) as expressions of EM fields in metallic layers and plane-waves as those in dielectric layers, we perform MEM to resolve the electromagnetic problems in the multilayered system. The method is quickly convergent by considering only 2 or 3 lowest guided resonance modes of CAAs. A higher order resonance mode contributes little to the interlayer coupling as its wave vector β*<sup>l</sup>* is a large imaginary number. Three guided modes ( *l* =1,2,3 ) in CAAs and 11 11 × orders of planewave basis in dielectric layers are adopted in our calculations. The results are very accurate (solid lines in Fig.4) and in good agreement with the measurements (circular dots in Fig.4).

Fig. 4. Transmission spectra through the models with (a) n=1, (b) n=2, (c) n=3, (d) n=10 metallic layers. Solid lines for calculated results by Modal expansion method (MEM), circular dots for measured results in microwave regime.

We see from Fig. 4 (a) that there exists a transmission peak for the n=1 sample at 8.7GHz *Af* = due to the excitation of guided TE11 resonance mode in CAAs. We also see from Figs. 4 (b) and 4 (c) that there are two transmission peaks at 9.1GHz *Bf* = , and 12.3GHz *Cf* = for the *n* = 2 sample, three peaks at 8.2GHz *Df* = , 11.64GHz *Ef* = and 12.35GHz *Ff* = for the *n* = 3 sample. Figure 4 (d) presents the calculated transmission spectra of an n=10 model system. It means that, with the increase of metallic layers, more transmission peaks emerge, giving rise to a broad transparent band.

More calculations show that, for the *n* = 2 sample, at an on-resonance frequency 9.1GHz *Bf* = or 12.3GHz *Cf* = where transmissivity is nearly unity, the spatial distribution of electric fields [see Figs. 5 (a) and 5 (b)] are symmetric or anti-symmetric about the *xy* plane. And the transmitted waves possess a phase difference of 0 (in phase) or π (out phase) with respect to the incident waves. Therefore the peaks at *Bf* and *Cf* , derived

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 247

Fig. 6. Transmission spectra through the models with (a) n=1, (b) n=2, (c) n=3, (d) n=10 metallic layers. Solid lines for calculated results by Modal expansion method (MEM),

**2.3 Broadband negative refraction from stacked fishnet metamaterial** 

This work reports for the first time that enhanced transmission peak can be broadened through stacked metallic multi-layers perforated with CAAs. Taking advantage of the excitation of guided resonance modes of CAAs and interlayer coupling, the enhanced transmission of such a system with only three metallic layers can span a wide frequency range covering about 60% of the central frequency. The broadband utility shall have enormous potential applications in optoelectronics, telecommunication and image

Since J.B. Pendry proposed perfect lens (Pendry 2000) using left-handed materials (Veselago 1968), sustained attentions have been drawn to the negative-index metamaterial (NIM) with simultaneously negative permittivity and permeability. The NIM, comprising of subwavelength metallic resonant units, has been designed and realized in both the microwave (Shelby, Smith et al. 2001) and optical regime (Liu, Guo et al. 2008). Negative refraction and subwavelength imaging with NIMs have great application potentials in photonic devices (Grbic & Eleftheriades 2004; Belov, Hao et al. 2006; Wiltshire, Pendry et al. 2006; Freire, Marques et al. 2008; Silveirinha, Fernandes et al. 2008; Silveirinha, Medeiros et al. 2010). Among various types of NIMs, one most promising candidate is the so-called fishnet NIM which comprises of alternating metal/dielectric layers perforated with twodimensional array of holes (Beruete, Campillo et al. 2007; Beruete, Sorolla et al. 2007; Beruete, Navarro-Cia et al. 2008; Navarro-Cia, Beruete et al. 2008; Navarro-Cia, Beruete et al. 2009). The simple structure also provides a feasible solution for optical NIM (Dolling, Enkrich et al. 2006; Zhang, Fan et al. 2006; Dolling, Wegener et al. 2007; Valentine, Zhang et

In most of the previous studies on fishnet NIMs (Dolling, Enkrich et al. 2006; Zhang, Fan et al. 2006; Beruete, Campillo et al. 2007; Beruete, Sorolla et al. 2007; Dolling, Wegener et al.

circular dots for measured results in microwave regime.

al. 2008; Ku & Brueck 2009; Ku, Zhang et al. 2009).

processing.

from the peak at *Af* of the *n* =1 model, come from the excitation and hybridization of the TE11 guided resonance modes in apertures at different metallic layers as a results of mode splitting of coupled apertures (or meta-atoms). Further more, the anti-symmetric mode at 12.3GHz *Cf* = of the *n* = 2 model splits into two modes of the *n* = 3 model: spatial field distribution of the one at 11.64GHz *Ef* = reveals that the incident and outgoing waves are out phase to each other [Fig. 5 (d)] and it is on the opposite for the other at 12.35GHz *Ff* = [Fig.5 (e)], while the resonant mode at the lowest frequency 8.2GHz *Df* = retains a symmetric feature in field distribution [Fig. 5 (c)] and inherits the in-phase signature from the symmetric mode at *Bf* of the *n* = 2 model.

Fig. 5. Spatial distribution of electric fields in the xz plane at on-resonance frequencies of (a) fB = 9.1GHz, (b) fC = 12.3GHz for the n = 2 model , and (c) fD = 8:2GHz, (d)fE = 11.64GHz, (e) fF = 12.35GHz for the n = 3 model.

Figure 6 (a) presents the dispersion relation of bulk material periodically constructed with layered CAAs. The band structure is calculated with MEM algorithm assuming periodic boundary conditions along the z axis. The process of mode splitting from *n* =1 to *n* = 3 , as shown in Fig. 6 (b), depicts the evolution of the enhanced transmission feature from a single transmission peak to a broad passband. It is interesting that the passband between 6.77GHz *bf* = and 12.7GHz *<sup>t</sup> f* = shown in Fig. 6 (b), predicting the passband of the n=10 model quite well, is also a good measure of the bandwidth of the n=3 sample. The total bandwidth is about 60% of the central frequency. In contrast, the EOT observed in multilayered systems of previous studies demonstrates a peak lineshape in spectra as it arises from the resonant tunneling of SPP modes among metallic films instead of guided resonance modes. And the broad passband we observed is not sensitive to the incident angle (not shown), while it is on the contrary when the SPP modes dominate.

from the peak at *Af* of the *n* =1 model, come from the excitation and hybridization of the TE11 guided resonance modes in apertures at different metallic layers as a results of mode splitting of coupled apertures (or meta-atoms). Further more, the anti-symmetric mode at 12.3GHz *Cf* = of the *n* = 2 model splits into two modes of the *n* = 3 model: spatial field distribution of the one at 11.64GHz *Ef* = reveals that the incident and outgoing waves are out phase to each other [Fig. 5 (d)] and it is on the opposite for the other at 12.35GHz *Ff* = [Fig.5 (e)], while the resonant mode at the lowest frequency 8.2GHz *Df* = retains a symmetric feature in field distribution [Fig. 5 (c)] and inherits the in-phase signature from

Fig. 5. Spatial distribution of electric fields in the xz plane at on-resonance frequencies of (a) fB = 9.1GHz, (b) fC = 12.3GHz for the n = 2 model , and (c) fD = 8:2GHz, (d)fE = 11.64GHz, (e)

Figure 6 (a) presents the dispersion relation of bulk material periodically constructed with layered CAAs. The band structure is calculated with MEM algorithm assuming periodic boundary conditions along the z axis. The process of mode splitting from *n* =1 to *n* = 3 , as shown in Fig. 6 (b), depicts the evolution of the enhanced transmission feature from a single transmission peak to a broad passband. It is interesting that the passband between 6.77GHz *bf* = and 12.7GHz *<sup>t</sup> f* = shown in Fig. 6 (b), predicting the passband of the n=10 model quite well, is also a good measure of the bandwidth of the n=3 sample. The total bandwidth is about 60% of the central frequency. In contrast, the EOT observed in multilayered systems of previous studies demonstrates a peak lineshape in spectra as it arises from the resonant tunneling of SPP modes among metallic films instead of guided resonance modes. And the broad passband we observed is not sensitive to the incident angle

(not shown), while it is on the contrary when the SPP modes dominate.

the symmetric mode at *Bf* of the *n* = 2 model.

fF = 12.35GHz for the n = 3 model.

Fig. 6. Transmission spectra through the models with (a) n=1, (b) n=2, (c) n=3, (d) n=10 metallic layers. Solid lines for calculated results by Modal expansion method (MEM), circular dots for measured results in microwave regime.

This work reports for the first time that enhanced transmission peak can be broadened through stacked metallic multi-layers perforated with CAAs. Taking advantage of the excitation of guided resonance modes of CAAs and interlayer coupling, the enhanced transmission of such a system with only three metallic layers can span a wide frequency range covering about 60% of the central frequency. The broadband utility shall have enormous potential applications in optoelectronics, telecommunication and image processing.

#### **2.3 Broadband negative refraction from stacked fishnet metamaterial**

Since J.B. Pendry proposed perfect lens (Pendry 2000) using left-handed materials (Veselago 1968), sustained attentions have been drawn to the negative-index metamaterial (NIM) with simultaneously negative permittivity and permeability. The NIM, comprising of subwavelength metallic resonant units, has been designed and realized in both the microwave (Shelby, Smith et al. 2001) and optical regime (Liu, Guo et al. 2008). Negative refraction and subwavelength imaging with NIMs have great application potentials in photonic devices (Grbic & Eleftheriades 2004; Belov, Hao et al. 2006; Wiltshire, Pendry et al. 2006; Freire, Marques et al. 2008; Silveirinha, Fernandes et al. 2008; Silveirinha, Medeiros et al. 2010). Among various types of NIMs, one most promising candidate is the so-called fishnet NIM which comprises of alternating metal/dielectric layers perforated with twodimensional array of holes (Beruete, Campillo et al. 2007; Beruete, Sorolla et al. 2007; Beruete, Navarro-Cia et al. 2008; Navarro-Cia, Beruete et al. 2008; Navarro-Cia, Beruete et al. 2009). The simple structure also provides a feasible solution for optical NIM (Dolling, Enkrich et al. 2006; Zhang, Fan et al. 2006; Dolling, Wegener et al. 2007; Valentine, Zhang et al. 2008; Ku & Brueck 2009; Ku, Zhang et al. 2009).

In most of the previous studies on fishnet NIMs (Dolling, Enkrich et al. 2006; Zhang, Fan et al. 2006; Beruete, Campillo et al. 2007; Beruete, Sorolla et al. 2007; Dolling, Wegener et al.

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 249

contain the 0th order Bloch component which is the transverse electromagnetic (TEM) mode that is always orthogonal to the local modes of air hole. In this situation, the free photons in the dielectric are the only choice for the 0 *<sup>z</sup> k* = states as no evanescent couplings happen via the breathing air holes. However the lowest branch ( 0 *<sup>z</sup> k* ≠ ) along the ΓM and XΜ directions (red solid line in Fig.8) evidently deviates from the light line in dielectric. The 0 *<sup>z</sup> k* ≠ states on this branch originate from the evanescent coupling between the adjacent slab waveguides via the TE10 mode of holes (noting that the overlap integral between a high order of guided Bloch mode and local mode of air hole is not zero). Figure 9 (a) presents the charts of equi-frequency surface (EFS) analysis for the 0 *<sup>y</sup> k* = states to further reveal the characteristics of this band. All curves in Fig. 9 (a) are in a hyperbolic-like lineshape, which indicates that all-angle negative refraction occurs in the *x*ˆ*z*ˆ plane at low frequency regime starting from zero. At lower frequency the curves in Fig. 9 (a) become much more flat, which means that the waves are also strongly collimated inside the structure along the direction parallel to the metal/dielectric layers. A numerical proof of negative refraction is shown in Fig. 9 (b). In our FDTD simulations, a monochromatic one-way Gaussian beam in the *x*ˆ*z*ˆ plane with a frequency at 11GHz is incident from upside at an incident angle of 30o. The fishnet model is stacked with 500 metal/dielectric layers along *z* direction. Given the periodicity of hole arrays and the incidence configuration, 60 periods along x direction and one period along y direction are adopted for the metal/dielectric layers in space domain. The negative refraction is clearly shown in Fig. 9 (b) with the magnetic field distribution in *xz* plane. The black arrows denote the directions of energy flow in the free space and fishnet structure. A refraction angle of -16.2o, retrieved from the refracted direction of the energy flow or the negative Goos-Hanchen shift alternatively, is in good agreement with the estimate in EFS analysis. We also see from Fig. 9 (b) that almost no reflection occurs as the incidence can easily propagates inside the structure by coupling with the guided Bloch modes in the slab

waveguide channels.

Fig. 8. The dispersion diagram of the stacked fishnet metamaterial.

2007; Beruete, Navarro-Cia et al. 2008; Navarro-Cia, Beruete et al. 2008; Valentine, Zhang et al. 2008; Ku & Brueck 2009; Ku, Zhang et al. 2009; Navarro-Cia, Beruete et al. 2009), the light waves are incident on the top interface of the metal/dielectric multi-layers. Thus the light waves can not penetrate into the structure below the cut-off frequency of air holes, and the negative index was retrieved only within in a narrow frequency range above the cut-off. In this section, a different incidence configuration is employed by impinging the light waves on the sidewall interface of fishnet NIM that is perpendicular to the metal/dielectric multilayers. As the uniformly spaced holey metallic layers of fishnet NIM constitute a multiple of slab waveguide channels filled with dielectric spacer layers, the incidence configuration of this kind enables us to fully exploit the optical properties of the fishnet NIM in the long wavelength limit. We show that the evanescent coupling between the slab waveguides gives rise to all-angle negative refraction and sub-wavelength imaging in a wide frequency range starting from zero.

Figure 7 schematically illustrates the structure of our stacked fishnet metamaterial and the incidence configuration. The metal/dielectric layers are lying in *x*ˆˆ*y* plane. The square arrays of air holes perforated on metallic layers are aligned along z axis without lateral displacement in *x*ˆˆ*y* plane. The period of the hole array, the thickness of metallic layer and dielectric layer are *p=6.0mm*, *t=0.035mm* and *h=1.575mm* respectively. The line width of metallic strips along x direction *w=0.2mm* is the same as that along y direction, and the size of square holes is *a*=*p-g=5.8mm*. The dielectric constant of the dielectric layer is *<sup>r</sup>* ε =2.55. The EM incidence waves are propagating in the *x*ˆ*z*ˆ plane with an incident angle of θ.

Fig. 7. The schematic of stacked fishnet metamaterial. The red and blue arrows refer to the directions of electric field *E* and magnetic field *H* . The plane in gray color denotes the incident plane.

Figure 8 presents the calculated dispersion diagram along the *Γ* (0,0,0)→*X* (0.5,0,0), *Γ* (0,0,0)→M (0.5,0,0.5) and M (0.5,0,0.5)→*X* (0.5,0,0) directions. The blue dashed line refers to the light line in dielectric. We notice that, the lowest branch along the ΓX direction ( 0 *<sup>z</sup> k* = ) precisely reproduces the light line in dielectric. This is understandable with the help of modal expansion method. Detailed calculations show that these 0 *<sup>z</sup> k* = states only

2007; Beruete, Navarro-Cia et al. 2008; Navarro-Cia, Beruete et al. 2008; Valentine, Zhang et al. 2008; Ku & Brueck 2009; Ku, Zhang et al. 2009; Navarro-Cia, Beruete et al. 2009), the light waves are incident on the top interface of the metal/dielectric multi-layers. Thus the light waves can not penetrate into the structure below the cut-off frequency of air holes, and the negative index was retrieved only within in a narrow frequency range above the cut-off. In this section, a different incidence configuration is employed by impinging the light waves on the sidewall interface of fishnet NIM that is perpendicular to the metal/dielectric multilayers. As the uniformly spaced holey metallic layers of fishnet NIM constitute a multiple of slab waveguide channels filled with dielectric spacer layers, the incidence configuration of this kind enables us to fully exploit the optical properties of the fishnet NIM in the long wavelength limit. We show that the evanescent coupling between the slab waveguides gives rise to all-angle negative refraction and sub-wavelength imaging in a wide frequency range

Figure 7 schematically illustrates the structure of our stacked fishnet metamaterial and the incidence configuration. The metal/dielectric layers are lying in *x*ˆˆ*y* plane. The square arrays of air holes perforated on metallic layers are aligned along z axis without lateral displacement in *x*ˆˆ*y* plane. The period of the hole array, the thickness of metallic layer and dielectric layer are *p=6.0mm*, *t=0.035mm* and *h=1.575mm* respectively. The line width of metallic strips along x direction *w=0.2mm* is the same as that along y direction, and the size

ε

θ.

=2.55. The

of square holes is *a*=*p-g=5.8mm*. The dielectric constant of the dielectric layer is *<sup>r</sup>*

Fig. 7. The schematic of stacked fishnet metamaterial. The red and blue arrows refer to the

Figure 8 presents the calculated dispersion diagram along the *Γ* (0,0,0)→*X* (0.5,0,0), *Γ* (0,0,0)→M (0.5,0,0.5) and M (0.5,0,0.5)→*X* (0.5,0,0) directions. The blue dashed line refers to the light line in dielectric. We notice that, the lowest branch along the ΓX direction ( 0 *<sup>z</sup> k* = ) precisely reproduces the light line in dielectric. This is understandable with the help of modal expansion method. Detailed calculations show that these 0 *<sup>z</sup> k* = states only

. The plane in gray color denotes the

and magnetic field *H*

EM incidence waves are propagating in the *x*ˆ*z*ˆ plane with an incident angle of

starting from zero.

directions of electric field *E*

incident plane.

contain the 0th order Bloch component which is the transverse electromagnetic (TEM) mode that is always orthogonal to the local modes of air hole. In this situation, the free photons in the dielectric are the only choice for the 0 *<sup>z</sup> k* = states as no evanescent couplings happen via the breathing air holes. However the lowest branch ( 0 *<sup>z</sup> k* ≠ ) along the ΓM and XΜ directions (red solid line in Fig.8) evidently deviates from the light line in dielectric. The 0 *<sup>z</sup> k* ≠ states on this branch originate from the evanescent coupling between the adjacent slab waveguides via the TE10 mode of holes (noting that the overlap integral between a high order of guided Bloch mode and local mode of air hole is not zero). Figure 9 (a) presents the charts of equi-frequency surface (EFS) analysis for the 0 *<sup>y</sup> k* = states to further reveal the characteristics of this band. All curves in Fig. 9 (a) are in a hyperbolic-like lineshape, which indicates that all-angle negative refraction occurs in the *x*ˆ*z*ˆ plane at low frequency regime starting from zero. At lower frequency the curves in Fig. 9 (a) become much more flat, which means that the waves are also strongly collimated inside the structure along the direction parallel to the metal/dielectric layers. A numerical proof of negative refraction is shown in Fig. 9 (b). In our FDTD simulations, a monochromatic one-way Gaussian beam in the *x*ˆ*z*ˆ plane with a frequency at 11GHz is incident from upside at an incident angle of 30o. The fishnet model is stacked with 500 metal/dielectric layers along *z* direction. Given the periodicity of hole arrays and the incidence configuration, 60 periods along x direction and one period along y direction are adopted for the metal/dielectric layers in space domain. The negative refraction is clearly shown in Fig. 9 (b) with the magnetic field distribution in *xz* plane. The black arrows denote the directions of energy flow in the free space and fishnet structure. A refraction angle of -16.2o, retrieved from the refracted direction of the energy flow or the negative Goos-Hanchen shift alternatively, is in good agreement with the estimate in EFS analysis. We also see from Fig. 9 (b) that almost no reflection occurs as the incidence can easily

propagates inside the structure by coupling with the guided Bloch modes in the slab waveguide channels.

Fig. 8. The dispersion diagram of the stacked fishnet metamaterial.

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 251

holes. If our findings are applicable in optical regime, the most field energy shall propagate outside the lossy metal film with anti-symmetric field distribution, and low loss is expected. The picture may be helpful as well to explain the low loss measured in a recent experiment

One important application of all-angle negative refraction is flat lens. The imaging performance of our stacked fishnet metamaterial is examined by the brute-force FDTD numerical simulations. As shown in Fig.10, a monochromatic point source with a frequency at 11GHz is located 15mm away from the surface at the left side of the fishnet structure. The snap shot shown in Fig. 10 (a) clearly indicates a high-quality image achieved at another side of the structure about 15mm away from the interface. The image resolution can be checked by the normalized magnetic field profile at image plane. As illustrated in Fig. 10 (b), along z axis, the full width at half maximum (FWHM) of the field profile is 10mm about one-third of the wavelength. The FWHM at a lower frequency still remains at about 10mm, leading to a better resolution in subwavelength scale along the z direction. But a longer

about fishnet optical NIMs (Zhang, Fan et al. 2006; Valentine, Zhang et al. 2008).

structure is required due to strong collimation effect at lower frequency.

Fig. 10. (a) The snap shot of magnetic field distribution in the incident plane. The model, stacked with 80 metal/dielectric multi-layers along z axis, has 20 periods along x direction and one period along y direction. (b) Normalized magnetic field profile at the source plane

In conclusion, the fishnet metamaterials can operate as plasmonic waveguide arrays. The broadband negative refraction, subwavelength imaging in the long wavelength limit have great potentials for photonic devices in microwave, THz and even in the optical regimes.

**2.4 Spatially coherent surface resonant states derived from magnetic meta-surface**  Surface plasmon polaritons (SPPs) can modulate light waves at the metal-dielectric interface with wavelength much smaller than that in free space (Raether 1988),which enables the control of light in a subwavelength scale for nanophotonic devices (Barnes, Dereux et al. 2003). SPPs with large coherent length are useful in many areas, including optical processing, quantum information (Kamli, Moiseev et al. 2008) and novel light-matter

(black solid line) and the image plane (red solid line) as a function of the z aixs

Fig. 9. (a) The charts of EFS analysis for ky=0 states respect to *<sup>z</sup> k* and *<sup>x</sup> k* . (b) The FDTD simulations on the magnetic field distribution in the problem domain. The one-way Gaussian beam is about 70mm away from the top interface of our fishnet model.

The optical properties of such a system can be described with the coupled wave equation (Haus & Molter-Orr 1983; Eisenberg, Silberberg et al. 2000; Pertsch, Zentgraf et al. 2002) by considering the coupling between the *nth* waveguide channel and its nearest neighbors, the (*n-1)th*, (*n+1)th* waveguide channels, as:

$$\mathrm{Li}\frac{da\_n(\mathbf{x})}{d\mathbf{x}} + \beta a\_n(\mathbf{x}) + \mathrm{C}[a\_{n+1}(\mathbf{x}) + a\_{n-1}(\mathbf{x})] = \mathbf{0} \tag{9}$$

Where ( ) *<sup>n</sup> a x* denotes the wave fields in the *nth* slab waveguide, C is the coupling coefficient, and β is the propagation constant of free photons in dielectric. Under the periodic boundary condition along z direction, the dispersion of the system takes the form as

$$k\_{\pm} = \beta + 2C\cos(k\_{\pm}p) \tag{10}$$

where *<sup>x</sup> k* and *<sup>z</sup> k* are the vector components along the *x* and *z* directions. At 0 *<sup>z</sup> k* = , the coupling coefficient *C* = 0 is zero (as aforementioned no evanescent coupling occurs) and we have *<sup>x</sup> k* = β which is rightly the light line in the dielectric. While *C* is always negative when 0 *<sup>z</sup> k* ≠ in the limit of long wavelength [which can be deduced from the charts in Fig. 8 (a)], giving rise to all-angle negative refraction. We note that the silver/dielectric multilayered structure also supports all-angle negative refraction in a certain optical frequency regime under the same incidence configuration of our study (Fan, Wang et al. 2006). The long range SPPs play an important role for the negative refraction. We also note that, at long wavelength limit, a holey metallic surface can be homogenized into a single-negative medium with electric response in the form of Drude model (Pendry, Martin-Moreno et al. 2004). The plasmon frequency is rightly the cut-off frequency of air holes. Thus it is reasonable for us to consider the stacked fishnet metamaterial as an artificial plasmonic waveguide array. The negative coefficient C implies that for the eigenstates on the lowest branch, the spatial field distributions are anti-symmetric with respect to the plane of air

Fig. 9. (a) The charts of EFS analysis for ky=0 states respect to *<sup>z</sup> k* and *<sup>x</sup> k* . (b) The FDTD simulations on the magnetic field distribution in the problem domain. The one-way Gaussian beam is about 70mm away from the top interface of our fishnet model.

(*n-1)th*, (*n+1)th* waveguide channels, as:

and β

we have *<sup>x</sup> k* =

β

*da x*

*dx*

β

condition along z direction, the dispersion of the system takes the form as

The optical properties of such a system can be described with the coupled wave equation (Haus & Molter-Orr 1983; Eisenberg, Silberberg et al. 2000; Pertsch, Zentgraf et al. 2002) by considering the coupling between the *nth* waveguide channel and its nearest neighbors, the

> ( ) ( ) [ ( ) ( )] 0 *<sup>n</sup> n nn*

Where ( ) *<sup>n</sup> a x* denotes the wave fields in the *nth* slab waveguide, C is the coupling coefficient,

2 cos( ) *x z k C kp* = + β

where *<sup>x</sup> k* and *<sup>z</sup> k* are the vector components along the *x* and *z* directions. At 0 *<sup>z</sup> k* = , the coupling coefficient *C* = 0 is zero (as aforementioned no evanescent coupling occurs) and

when 0 *<sup>z</sup> k* ≠ in the limit of long wavelength [which can be deduced from the charts in Fig. 8 (a)], giving rise to all-angle negative refraction. We note that the silver/dielectric multilayered structure also supports all-angle negative refraction in a certain optical frequency regime under the same incidence configuration of our study (Fan, Wang et al. 2006). The long range SPPs play an important role for the negative refraction. We also note that, at long wavelength limit, a holey metallic surface can be homogenized into a single-negative medium with electric response in the form of Drude model (Pendry, Martin-Moreno et al. 2004). The plasmon frequency is rightly the cut-off frequency of air holes. Thus it is reasonable for us to consider the stacked fishnet metamaterial as an artificial plasmonic waveguide array. The negative coefficient C implies that for the eigenstates on the lowest branch, the spatial field distributions are anti-symmetric with respect to the plane of air

++ + =

is the propagation constant of free photons in dielectric. Under the periodic boundary

*i a x Ca x a x*

1 1

which is rightly the light line in the dielectric. While *C* is always negative

+ − (9)

(10)

holes. If our findings are applicable in optical regime, the most field energy shall propagate outside the lossy metal film with anti-symmetric field distribution, and low loss is expected. The picture may be helpful as well to explain the low loss measured in a recent experiment about fishnet optical NIMs (Zhang, Fan et al. 2006; Valentine, Zhang et al. 2008).

One important application of all-angle negative refraction is flat lens. The imaging performance of our stacked fishnet metamaterial is examined by the brute-force FDTD numerical simulations. As shown in Fig.10, a monochromatic point source with a frequency at 11GHz is located 15mm away from the surface at the left side of the fishnet structure. The snap shot shown in Fig. 10 (a) clearly indicates a high-quality image achieved at another side of the structure about 15mm away from the interface. The image resolution can be checked by the normalized magnetic field profile at image plane. As illustrated in Fig. 10 (b), along z axis, the full width at half maximum (FWHM) of the field profile is 10mm about one-third of the wavelength. The FWHM at a lower frequency still remains at about 10mm, leading to a better resolution in subwavelength scale along the z direction. But a longer structure is required due to strong collimation effect at lower frequency.

Fig. 10. (a) The snap shot of magnetic field distribution in the incident plane. The model, stacked with 80 metal/dielectric multi-layers along z axis, has 20 periods along x direction and one period along y direction. (b) Normalized magnetic field profile at the source plane (black solid line) and the image plane (red solid line) as a function of the z aixs

In conclusion, the fishnet metamaterials can operate as plasmonic waveguide arrays. The broadband negative refraction, subwavelength imaging in the long wavelength limit have great potentials for photonic devices in microwave, THz and even in the optical regimes.

#### **2.4 Spatially coherent surface resonant states derived from magnetic meta-surface**

Surface plasmon polaritons (SPPs) can modulate light waves at the metal-dielectric interface with wavelength much smaller than that in free space (Raether 1988),which enables the control of light in a subwavelength scale for nanophotonic devices (Barnes, Dereux et al. 2003). SPPs with large coherent length are useful in many areas, including optical processing, quantum information (Kamli, Moiseev et al. 2008) and novel light-matter

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 253

his co-workers numerically and experimentally proved that the ultra-thin MDM structures can resonantly absorb or transmit radiations at low frequency limit (Hibbins, Sambles et al. 2004). They addressed that the central frequencies of absorption peaks are independent from the incident angle with an interpretation of Farby-Perrot resonant mode (EQ. 1 in Ref. 22). The same group further explored the angle-independent absorption, as the main scenario of the incremental work, by measuring the flat bands of surface wave dispersion in the visible (Hibbins, Murray et al. 2006) as well as the microwave region (Brown, Hibbins et al. 2008). In contrast, we find that the structures with proper design also supports very narrow absorption peaks which are sensitiveto the incident angle and obviously do not satisfy to the

Fig. 12. Absorption spectra under TM-polarized incidence (a) as a function of frequency at

It is worth noting that an angle-independent peak is quite different from an angledependent one in physics origin. The former, investigated in Refs. 22-24, comes from the localized surface resonance states, while the latter, found by us, comes from the collective surface resonance states. Mode analysis presents An an intuitive picture for the formation of these collective surface states. When high order quasi-TEM modes are dominant components of the guided waves inside the dielectric layer they will assign phase correlation to the outgoing waves emitted from the air slits of grating, thus are very crucial to the formation of collective response. Weak enough both the leakage from dielectric layer to air slits and the material absorption, the spatial coherence of surface resonant states will survive. As the interaction between the structure and the incident waves will excite quasi-TEM modes inside the dielectric layer, the magnetic induction must be parallel to the MDM surfaces if it exists. Thus a surface resonant state on a MDM structure is usually magnetic in nature. Our findings about spatially coherent surface resonance states are original compared to the common knowledge, and have great potentials in coherent control of SPPs as well as

54.3*THz* (solid line, g=0.2*μm*), 40*THz* (dashed line, g=0.2*μm*) and 40*THz* (doted line,

= (g=0.2*μm*) and (b) as a function of incident angle at

Fabry-Perot resonance condition suggested in the previous studies.

incident angles of 0 ,5 ,20 ,30 *oo o o* θ

thermal emission radiations.

g=0.1*μm*)

interactions (Vasa, Pomraenke et al. 2008). The enhancement of local fields by SPPs is particularly important as it opens a new route to absorption enhancement (Andrew, Kitson et al. 1997), nonlinear optical amplification (Coutaz, Neviere et al. 1985; Tsang 1996) as well as weak signal probing (Kneipp, Wang et al. 1997; Nie & Emory 1997). As the properties of SPP are pretty much determined by the natural (plasmon) resonance frequency, there is not much room for us to adjust the SPP response for practical applications. With induced surface current oscillations on an array of metallic building blocks (Pendry, Holden et al. 1996; Pendry, Holden et al. 1999; Sievenpiper, Zhang et al. 1999; Yen, Padilla et al. 2004; Hibbins, Evans et al. 2005; Liu, Genov et al. 2006; Lockyear, Hibbins et al. 2009), metamaterial surfaces can manipulate electromagnetic waves in a similar way as SPPs. Such SPPs or surface resonance states on structured metallic surfaces are tunable by geometric parameters.

Here, we examine the properties of surface resonance states at a dielectric-metamaterial interface that exhibit magnetic response to the incident waves and strong local field enhancement. We will see that these surface resonance states can give highly directional absorptivity and emissivity, and may thus help to realize interesting effects such as spatially coherent thermal emission. As the structure is very simple, it can be fabricated down to the IR and optical regime (Grigorenko, Geim et al. 2005; Shalaev 2007; Boltasseva & Shalaev 2008).

Fig. 11. Schematic picture of the magnetic metamaterial slab

We will show that a thin metamaterial slab, with a thickness much smaller than the operational wavelength, supports delocalized magnetic surface resonance states with a long coherent length in a wide range of frequencies. Operating in a broad frequency range, these spatially coherent SPPs are surface resonance states with quasi-TEM modes guided in the dielectric layer that are weakly coupled to free space, and the coupling strength can be controlled by tuning structural parameters while the frequency can be controlled by varying structural and material parameters. The high fidelity of these surface resonance states results in directional absorptivity or emissivity, which is angle-dependent with respect to frequency. Finite-difference-in-time-domain (FDTD) simulations verify that the highly directional emissivity from the slab persists in the presence of structural disorder in the grating layer.

Such metal-dielectric-metal (MDM) structures were recognized as artificial magnetic surfaces with high impedance by the end of last century [12]. The incident waves induce surface current solenoids on the unit cells of the ultra-thin high-impedance surface, giving rise to magnetic susceptibilities. The magnetic response can be described with an effective permeability in Lorentz type (Sievenpiper, Zhang et al. 1999; Zhou, Wen et al. 2003). After the concept of metamaterial being proposed (Engheta & Ziolkowski 2006), P. Alastair and

interactions (Vasa, Pomraenke et al. 2008). The enhancement of local fields by SPPs is particularly important as it opens a new route to absorption enhancement (Andrew, Kitson et al. 1997), nonlinear optical amplification (Coutaz, Neviere et al. 1985; Tsang 1996) as well as weak signal probing (Kneipp, Wang et al. 1997; Nie & Emory 1997). As the properties of SPP are pretty much determined by the natural (plasmon) resonance frequency, there is not much room for us to adjust the SPP response for practical applications. With induced surface current oscillations on an array of metallic building blocks (Pendry, Holden et al. 1996; Pendry, Holden et al. 1999; Sievenpiper, Zhang et al. 1999; Yen, Padilla et al. 2004; Hibbins, Evans et al. 2005; Liu, Genov et al. 2006; Lockyear, Hibbins et al. 2009), metamaterial surfaces can manipulate electromagnetic waves in a similar way as SPPs. Such SPPs or surface resonance states on structured metallic surfaces are tunable by geometric

Here, we examine the properties of surface resonance states at a dielectric-metamaterial interface that exhibit magnetic response to the incident waves and strong local field enhancement. We will see that these surface resonance states can give highly directional absorptivity and emissivity, and may thus help to realize interesting effects such as spatially coherent thermal emission. As the structure is very simple, it can be fabricated down to the IR and optical regime (Grigorenko, Geim et al. 2005; Shalaev 2007; Boltasseva & Shalaev

We will show that a thin metamaterial slab, with a thickness much smaller than the operational wavelength, supports delocalized magnetic surface resonance states with a long coherent length in a wide range of frequencies. Operating in a broad frequency range, these spatially coherent SPPs are surface resonance states with quasi-TEM modes guided in the dielectric layer that are weakly coupled to free space, and the coupling strength can be controlled by tuning structural parameters while the frequency can be controlled by varying structural and material parameters. The high fidelity of these surface resonance states results in directional absorptivity or emissivity, which is angle-dependent with respect to frequency. Finite-difference-in-time-domain (FDTD) simulations verify that the highly directional emissivity from the slab persists in the presence of structural disorder in the

Such metal-dielectric-metal (MDM) structures were recognized as artificial magnetic surfaces with high impedance by the end of last century [12]. The incident waves induce surface current solenoids on the unit cells of the ultra-thin high-impedance surface, giving rise to magnetic susceptibilities. The magnetic response can be described with an effective permeability in Lorentz type (Sievenpiper, Zhang et al. 1999; Zhou, Wen et al. 2003). After the concept of metamaterial being proposed (Engheta & Ziolkowski 2006), P. Alastair and

Fig. 11. Schematic picture of the magnetic metamaterial slab

parameters.

2008).

grating layer.

his co-workers numerically and experimentally proved that the ultra-thin MDM structures can resonantly absorb or transmit radiations at low frequency limit (Hibbins, Sambles et al. 2004). They addressed that the central frequencies of absorption peaks are independent from the incident angle with an interpretation of Farby-Perrot resonant mode (EQ. 1 in Ref. 22). The same group further explored the angle-independent absorption, as the main scenario of the incremental work, by measuring the flat bands of surface wave dispersion in the visible (Hibbins, Murray et al. 2006) as well as the microwave region (Brown, Hibbins et al. 2008). In contrast, we find that the structures with proper design also supports very narrow absorption peaks which are sensitiveto the incident angle and obviously do not satisfy to the Fabry-Perot resonance condition suggested in the previous studies.

Fig. 12. Absorption spectra under TM-polarized incidence (a) as a function of frequency at incident angles of 0 ,5 ,20 ,30 *oo o o* θ = (g=0.2*μm*) and (b) as a function of incident angle at 54.3*THz* (solid line, g=0.2*μm*), 40*THz* (dashed line, g=0.2*μm*) and 40*THz* (doted line, g=0.1*μm*)

It is worth noting that an angle-independent peak is quite different from an angledependent one in physics origin. The former, investigated in Refs. 22-24, comes from the localized surface resonance states, while the latter, found by us, comes from the collective surface resonance states. Mode analysis presents An an intuitive picture for the formation of these collective surface states. When high order quasi-TEM modes are dominant components of the guided waves inside the dielectric layer they will assign phase correlation to the outgoing waves emitted from the air slits of grating, thus are very crucial to the formation of collective response. Weak enough both the leakage from dielectric layer to air slits and the material absorption, the spatial coherence of surface resonant states will survive. As the interaction between the structure and the incident waves will excite quasi-TEM modes inside the dielectric layer, the magnetic induction must be parallel to the MDM surfaces if it exists. Thus a surface resonant state on a MDM structure is usually magnetic in nature. Our findings about spatially coherent surface resonance states are original compared to the common knowledge, and have great potentials in coherent control of SPPs as well as thermal emission radiations.

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 255

We can obtain the coefficients // <sup>0</sup> (, ) *mt fk* and // <sup>0</sup> (, ) *mr fk* of the *th <sup>m</sup>* guided and reflected waves by applying the boundary continuity conditions for the tangential components of electromagnetic wave fields (over the slits) at the interfaces *z h* = and *z ht* = + . Given that surface resonance modes are intrinsic response, we can also assign zero to the incident plane wave and apply the boundary continuity conditions for the tangential components of wave fields to derive the eigen-value equations. A surface resonance state can be determined by searching a zero value / minimum of eigen-equation determinant in the reciprocal space provided that it is non-radiative/radiative with infinite/finite life time below/above light

0

σ

θ

 (from 2.3*<sup>o</sup>* θ

θ

FWHM is reduced if the gap size is smaller, as shown with the dashed and dotted lines in

surface resonant modes can be controlled by the gap-period ratio *g* / *p* . It is worth noting that, although *ky=0* is assumed for the calculated results shown in Fig. 12, the angle-dependent

To quantitatively characterize the formation of these spatially coherent surface resonance states, we employ the eigenmode expansion method to calculate the surface resonance

 = − to 2.3*<sup>o</sup>* θ

*k*

= −

*k*

0 0 ( , ) 1 Re( ) | ( , ) | *<sup>m</sup> I z I m*

*A k r k*

= and

ω

ε

*m z*

which includes the contributions from all Bloch orders of reflected waves. As a consequence,

calculated frequency regime. In Fig. 12 (a), we present the absorption spectra at various incident angles. The spectra exhibit a low and broad peak at *13.2*THz which is almost independent of the incident angle, while the other absorption peaks at higher frequencies are narrow and sensitive to the incident angle with a maximum absorption approaching 100%. The slab thus acts as an all-angle absorber at *13.2*THz, but exhibits sharp angle-selective absorption peaks at higher frequencies. Shown as solid and dashed lines in Fig. 12 (b), the sharp angular dependence of absorption coefficients (note that the vertical axis is in log-scale) at 40.0*THz* and 54.3THz implicitly implies the existence of spatially coherent surface resonance states. The angle-dependent absorption peaks become lower and disappear gradually with the increase of the material loss. This presents a way to realize nearly perfect absorption with weakly absorptive materials by coherent surface resonance states. The coherent length of a surface resonance state can be estimated by the ratio of the wavelength

 gives information about the surface resonance states as well as the emissivity properties as governed by Kirchhoff's law (Greffet & Nieto-Vesperinas 1998). We shall assume that the dielectric spacer layer is slightly dissipative by assigning a complex

2

= 66.93 / *S m* [ <sup>2</sup>

<sup>0</sup> Im( ) 10 *III <sup>r</sup>* ε

of the absorption peak (Greffet, Carminati

λ

*m* at 40THz, which means that the coherent length of the

, the angular FWHM of the

(not shown in figure). The angular

= ) gives rise to a coherent

for the surface resonance

 εε<sup>−</sup> ≈ ] in the

 ω

line in free space.

permittivity <sup>0</sup> / *III r* ε εε

<sup>0</sup> *A k*(,) ω

λ

length

λθ

Fig.12 (b) for *g* = 0.2

We derived the absorption spectra of the slab

 σ ω= + *i* with 2.2 *<sup>r</sup>*

and the full width at half maximum (FWHM) Δ

 λ

*m* and *g* = 0.1

corresponding absorption peak 4.6*<sup>o</sup>* Δ =

μ

 μ

state at 50.22THz and <sup>0</sup> *k p* = 0.01 /

et al. 2002). For example, for the Γ<sup>4</sup> state at 54.3THz and <sup>0</sup> *k* = 0

θ

π

absorption peaks are readily obtained for any specific incident angle.

/ 68.5 12.4 Δ= ≈ *m* . The coherent length is about 220

μ

with 0.26*<sup>o</sup>* Δ =

Our model system is schematically illustrated in Fig. 11. Lying on the *x*ˆˆ*y* plane, the slab comprises an upper layer of a metallic lamellar grating with thickness *t*, a dielectric spacer layer as a slab waveguide with thickness *h* and a metallic ground plane. The metallic strips are separated by a small air gap *g*, giving rise to a period of *p=a+g* for the lamellar grating. The geometric parameters of our model are *t m* = 0.2μ , *h m* = 0.8μ , *a m* = 3.8μ , *g* = 0.2μ*m* and *pag m* =+ = 4.0μ . Each metallic strip together with the ground plane beneath it constitutes a planar resonant cavity as the building block that gives magnetic responses at cavity resonances (Sievenpiper, Zhang et al. 1999; Lockyear, Hibbins et al. 2009). As the metallic grating is along the *x*ˆ direction, the guided waves in the dielectric layer (at 0 < <*z h* in region III) shall always couple to the incident waves with a non-zero component 0 *Ex* ≠ of electric field.

As a first step, we consider a transverse magnetic (TM) polarized incident plane wave in the free semi-space (at *z ht* > + in region I). The electric field *E* lies in *x*ˆ*z*ˆ plane, the magnetic field *H* is along y axis and the in-plane wave vector is <sup>0</sup> ˆ *<sup>x</sup> <sup>x</sup> k ke* = ( <sup>0</sup> *<sup>y</sup> <sup>k</sup>* <sup>=</sup> ). The total magnetic fields in region I and in region III can be written in terms of the reflection coefficients *mr* and the guided Bloch wave coefficients *mt* , as

$$\begin{split} H^I\_\chi(\vec{r}, z) &= \mathcal{S}\_{m, 0} e^{-ik\_{z\_m}^I z} e^{ik\_x x} + \sum\_m r\_m e^{ik\_{z\_m}^I z} e^{i(k\_x + 2m\pi / p)x} \\ H\_\chi^{\text{III}}(\vec{r}, z) &= \sum\_m [t\_m e^{ik\_{z\_m}^{\text{III}} z} + t\_m e^{-ik\_{z\_m}^{\text{III}} (z - 2h)}] e^{i(k\_x + 2m\pi / p)x}, \end{split} \tag{11}$$

where the term ,0 *I zm x ik z ik x m* δ *e e* <sup>−</sup> denotes the incident plane wave with *<sup>m</sup>*,0 δ being the Kronecker function and *m* being the Bloch order; ( 2 /) *<sup>x</sup> ik m px e* <sup>+</sup> π denotes wave component of the *mth* Bloch eigenmode in the semi-free space (region I) and the dielectric layer (region III) with respect to ˆ( 2 /) *m x k xk m p* = + π . *mk* is the in-plane wave vector and ˆ 2 / *G x mp <sup>m</sup>* = ⋅ π is the *mth* reciprocal lattice vector.

$$k\_{z\_n}^{'} = \sqrt{\varepsilon\_0 \mu\_0 \alpha^2 - |\vec{k\_m}|^2} \quad \text{and} \ k\_{z\_n}^{'0'} = \sqrt{\varepsilon\_m \mu\_0 \alpha^2 - |\vec{k\_m}|^2}$$

are the *z* components of wave vector for the *mth* order Bloch eigenmode in region I and region III respectively. <sup>0</sup> ε and *III* ε are the permittivity of the vacuum and the dielectric , μ0 is the vacuum permeability. In general, we also derived the method for a plane wave incidence with any specific wavevector and any specific polarization.

We shall mainly consider infrared frequencies, at which the metals can be well approximated as perfectly electric conductors (PEC). The EM fields at *h zht* ≤≤+ in region II are squeezed inside the air gaps, in which the magnetic fields can be expressed in terms of the expansion coefficients *<sup>l</sup> a* and *<sup>l</sup> b* of forward and backward guided waves, as:

$$\|H\_{\boldsymbol{y}}^{H}(\vec{r},\boldsymbol{z}) = \sum\_{l} [a\_{l}e^{-iq\_{l}(\boldsymbol{z}-\boldsymbol{h}-\boldsymbol{\ell})} + b\_{l}e^{iq\_{l}(\boldsymbol{z}-\boldsymbol{h})}]\mathbf{g}\_{l}(\boldsymbol{x}),\tag{12}$$

where ( ) cos[ / ( / 2)] *<sup>l</sup> g x l gx g* = + π , ( 0,1,..., ,...) *l n* = is the in-plane distribution of guided mode α*<sup>l</sup>* running over all air gaps. 2 2 0 0 ( /) *<sup>l</sup> q lg* = − ε μω π is the z component of wave vector for the *lth* guided modeα*l* .

We can obtain the coefficients // <sup>0</sup> (, ) *mt fk* and // <sup>0</sup> (, ) *mr fk* of the *th <sup>m</sup>* guided and reflected waves by applying the boundary continuity conditions for the tangential components of electromagnetic wave fields (over the slits) at the interfaces *z h* = and *z ht* = + . Given that surface resonance modes are intrinsic response, we can also assign zero to the incident plane wave and apply the boundary continuity conditions for the tangential components of wave fields to derive the eigen-value equations. A surface resonance state can be determined by searching a zero value / minimum of eigen-equation determinant in the reciprocal space provided that it is non-radiative/radiative with infinite/finite life time below/above light line in free space.

We derived the absorption spectra of the slab

254 Metamaterial

Our model system is schematically illustrated in Fig. 11. Lying on the *x*ˆˆ*y* plane, the slab comprises an upper layer of a metallic lamellar grating with thickness *t*, a dielectric spacer layer as a slab waveguide with thickness *h* and a metallic ground plane. The metallic strips are separated by a small air gap *g*, giving rise to a period of *p=a+g* for the lamellar grating. The

μ

planar resonant cavity as the building block that gives magnetic responses at cavity resonances (Sievenpiper, Zhang et al. 1999; Lockyear, Hibbins et al. 2009). As the metallic grating is along the *x*ˆ direction, the guided waves in the dielectric layer (at 0 < <*z h* in region III) shall always

As a first step, we consider a transverse magnetic (TM) polarized incident plane wave in the

fields in region I and in region III can be written in terms of the reflection coefficients *mr* and

(,) [ ] ,

<sup>−</sup> denotes the incident plane wave with *<sup>m</sup>*,0

π

eigenmode in the semi-free space (region I) and the dielectric layer (region III) with respect

are the *z* components of wave vector for the *mth* order Bloch eigenmode in region I and

is the vacuum permeability. In general, we also derived the method for a plane wave

We shall mainly consider infrared frequencies, at which the metals can be well approximated as perfectly electric conductors (PEC). The EM fields at *h zht* ≤≤+ in region II are squeezed inside the air gaps, in which the magnetic fields can be expressed in terms of

> ( ) () (,) [ ] ( ), *l l II iq z h t iq z h y l ll*

> > π

*III III z z mmx*

− − +

is the in-plane wave vector and ˆ 2 / *G x mp <sup>m</sup>* = ⋅

and 2 2

*III*

*m III ik z ik z h ik m px*

*I I z z m x m x*

<sup>−</sup> <sup>+</sup>

*I ik z ik x ik z ik m px*

couple to the incident waves with a non-zero component 0 *Ex* ≠ of electric field.

is along y axis and the in-plane wave vector is <sup>0</sup> ˆ *<sup>x</sup> <sup>x</sup> k ke* =

,0

δ

*y mm m*

*y m m*

= +

2 2

the expansion coefficients *<sup>l</sup> a* and *<sup>l</sup> b* of forward and backward guided waves, as:

0 0 ( /) *<sup>l</sup> q lg* = − ε μω

*l*

0 0 | | *<sup>m</sup>*

incidence with any specific wavevector and any specific polarization.

*z m k k* = − ε μω

> and *III* ε

= +

*H r z e e re e*

*H rz te te e*

, *h m* = 0.8

. Each metallic strip together with the ground plane beneath it constitutes a

μ

( 2 /)

π

π

δ

denotes wave component of the *mth* Bloch

π

is the z component of wave vector for

( 2) ( 2 /)

<sup>0</sup> | | *<sup>m</sup>*

are the permittivity of the vacuum and the dielectric ,

*H r z ae be g x* − −− <sup>−</sup> = + (12)

, ( 0,1,..., ,...) *l n* = is the in-plane distribution of guided mode

*z III <sup>m</sup> k k* = − ε μω

(11)

, *a m* = 3.8

μ

, *g* = 0.2

lies in *x*ˆ*z*ˆ plane, the magnetic

being the Kronecker

μ0

is the *mth*

( <sup>0</sup> *<sup>y</sup> <sup>k</sup>* <sup>=</sup> ). The total magnetic

μ*m* and

geometric parameters of our model are *t m* = 0.2

the guided Bloch wave coefficients *mt* , as

*I zm x ik z ik x*

function and *m* being the Bloch order; ( 2 /) *<sup>x</sup> ik m px e* <sup>+</sup>

*I*

ε

*<sup>l</sup>* running over all air gaps. 2 2

*m* δ*e e*

π. *mk*

free semi-space (at *z ht* > + in region I). The electric field *E*

(,)

*pag m* =+ = 4.0

where the term ,0

to ˆ( 2 /) *m x k xk m p* = +

reciprocal lattice vector.

region III respectively. <sup>0</sup>

where ( ) cos[ / ( / 2)] *<sup>l</sup> g x l gx g* = + π

> α*l* .

the *lth* guided mode

α

field *H*  μ

$$\mathcal{A}(\vec{k\_{\boldsymbol{\alpha}}},\boldsymbol{\alpha}) = \mathbf{l} - \sum\_{m} \text{Re}(\frac{k\_{z\_{m}}^{I}}{k\_{z\_{\boldsymbol{\alpha}}}^{I}}) \|r\_{m}(\vec{k\_{\boldsymbol{\alpha}}},\boldsymbol{\alpha})\|^{2}$$

which includes the contributions from all Bloch orders of reflected waves. As a consequence, <sup>0</sup> *A k*(,) ω gives information about the surface resonance states as well as the emissivity properties as governed by Kirchhoff's law (Greffet & Nieto-Vesperinas 1998). We shall assume that the dielectric spacer layer is slightly dissipative by assigning a complex permittivity <sup>0</sup> / *III r* ε εε σ ω = + *i* with 2.2 *<sup>r</sup>* ε = andσ = 66.93 / *S m* [ <sup>2</sup> <sup>0</sup> Im( ) 10 *III <sup>r</sup>* ε εε <sup>−</sup> ≈ ] in the calculated frequency regime. In Fig. 12 (a), we present the absorption spectra at various incident angles. The spectra exhibit a low and broad peak at *13.2*THz which is almost independent of the incident angle, while the other absorption peaks at higher frequencies are narrow and sensitive to the incident angle with a maximum absorption approaching 100%. The slab thus acts as an all-angle absorber at *13.2*THz, but exhibits sharp angle-selective absorption peaks at higher frequencies. Shown as solid and dashed lines in Fig. 12 (b), the sharp angular dependence of absorption coefficients (note that the vertical axis is in log-scale) at 40.0*THz* and 54.3THz implicitly implies the existence of spatially coherent surface resonance states. The angle-dependent absorption peaks become lower and disappear gradually with the increase of the material loss. This presents a way to realize nearly perfect absorption with weakly absorptive materials by coherent surface resonance states. The coherent length of a surface resonance state can be estimated by the ratio of the wavelength λ and the full width at half maximum (FWHM) Δθ of the absorption peak (Greffet, Carminati et al. 2002). For example, for the Γ<sup>4</sup> state at 54.3THz and <sup>0</sup> *k* = 0 , the angular FWHM of the corresponding absorption peak 4.6*<sup>o</sup>* Δ = θ (from 2.3*<sup>o</sup>* θ = − to 2.3*<sup>o</sup>* θ = ) gives rise to a coherent length λθ μ λ / 68.5 12.4 Δ= ≈ *m* . The coherent length is about 220λ for the surface resonance state at 50.22THz and <sup>0</sup> *k p* = 0.01 / π with 0.26*<sup>o</sup>* Δ = θ (not shown in figure). The angular FWHM is reduced if the gap size is smaller, as shown with the dashed and dotted lines in Fig.12 (b) for *g* = 0.2μ*m* and *g* = 0.1μ*m* at 40THz, which means that the coherent length of the surface resonant modes can be controlled by the gap-period ratio *g* / *p* . It is worth noting that, although *ky=0* is assumed for the calculated results shown in Fig. 12, the angle-dependent absorption peaks are readily obtained for any specific incident angle.

To quantitatively characterize the formation of these spatially coherent surface resonance states, we employ the eigenmode expansion method to calculate the surface resonance

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 257

Although only TEM guided modes are allowed to be excited in the thin MDM slab within the frequency of our interest, the *B*<sup>2</sup> states are quite different from the *B*3 and *B*4 states in field patterns inside the dielectric layer. We see from Figs. 14 (b), 14 (c) ,14 (e)and 14 (f) that for a *B*3 or *B*<sup>4</sup> state, there are nodes and anti-nodes in field patterns, while for the *B*2 state, the magnetic field is almost uniformly distributed. Calculations on local field enhancement inside the dielectric slab resolve the puzzle. Black solid line in Fig. 15 presents the normalized magnetic field | | *H* inside the dielectric with respect to that of incidence

line] contributes the most at 13.2THz and the least at 50.22THz and 54.3THz; while it is just opposite for the contributions in combination from the two high order Fourier components with *m=±1* [red solid line]. Figure 15 also indicates that the enhancement of local field of an excited *B*3 or *B*4 state can be ten times larger than that of an excited *B*2 state, the enhancement factor at 50.22THz is about 100 times, while it is only 10 times at 13.2THz.

Fig. 15. Magnetic field H inside the dielectric layer normalized to that of incidence H under

line: only the 0th Bloch order considered; Blue line: summation of -1st and +1st Bloch orders

We see from Fig. 13 (b) that the surface resonance dispersion of the slab comes from the interaction between the magnetic resonances and the (folded) light lines *L*1 (for dielectrics) and *L*2 (for air) grazing on the interfaces. In the limit of a small gap-period ratio ( *g p*/ 0.05 = for example), our system is weakly Bragg-scattered, and as such, when a surface resonance state on branches *B*3 or *B*4 is excited, the induced wave fields inside the dielectric

the *B*3 and *B*4 states have high fidelity even though they are leaky modes, as most of their Bloch wavefunction components lying outside the free space light line. As the air gaps of the metallic grating serve to couple the electromagnetic waves of region I and region III, the quality factor of a resonance state can be estimated with the overlap integral between the

= 2 . Black line: all Bloch orders of TEM guided modes included; Red

> in the air gap and the dominant Bloch waves | *<sup>i</sup>*

*st* ± Bloch orders. For that reason,

*mk* >

= . The Fourier component in *m=0* order [blue solid

<sup>0</sup> | | *H* under an incident angle of 2*<sup>o</sup>*

an incident angle of <sup>o</sup>

of TEM guided modes.

θ

fundamental waveguide mode <sup>0</sup> |

of region III are guided quasi-TEM modes dominated by 1

α

( *i I III* = , ) in region I or region III for the coupling coefficients

θ

dispersion (in the limit of no material loss) as shown in Fig. 13 (b).The *B*1 surface resonance states lie below the light line *L*2 (magenta dashed line), and thus are non-radiative as evanescent modes. The surface resonances labeled as *B*2 originate from the coupling of the fundamental magnetic resonance modes of the metal strip structure with the free space light line *L*<sup>2</sup> . The surface resonances *B*<sup>3</sup> and *B*<sup>4</sup> are harmonic modes of the magnetic resonances that hybridizes with the guided mode inside the dielectric layer. The calculated reflection phase difference between the 0th order reflected and incident electric field, as shown in Fig. 13 (a) for normal incidence (red line), and 2*<sup>o</sup>* incidence (blue line), clearly shows that the resonances are magnetic in nature when the surface resonances intersect the zone center at Γ2 (13.2THz ) and Γ4 (54.3THz) as the reflection phase is zero like what a magnetic conductor surface does to the incident waves. The state Γ<sup>3</sup> , invisible in the reflection phase spectrum under normal incidence [red solid line in Fig. 13 (a)], is a dark state as its eigenmode is in mirror symmetry about the *yz* plane and can not couple with free space photons. While the other *B*3 states can couple with external light under oblique incidence [see the blue line in Fig. 13 (a)]. For example, there exists in-phase reflection at frequency 50.22THz under an incident angle of 2o, corresponding a *B*3 state at frequency 50.22THz and <sup>0</sup> *k p* ≈ 0.02 / π .

The angle-independent absorption peak at 13.2THz is due to the *B*2 mode, which is only weakly dispersive near the zone center. The more dispersive *B*3 and *B*4 modes are accountable for the incident-angle sensitive absorption in the higher frequencies in Fig. 12 (a). The field patterns in Figs. 14 (a)-14 (c) present the spatial distribution of the real part of magnetic fields excited by the incident plane waves with incident angles 0*<sup>o</sup>* , 2*<sup>o</sup>* and 0*<sup>o</sup>* for the three surface resonance states on *B*<sup>2</sup> , *B*3 and *B*<sup>4</sup> respectively, and the corresponding vector diagrams of electric fields are shown in Figs 14 (d)-14 (f). We can see clearly that the electric fields reach maximum in strength at the slab upper surface, and exponentially decay along the surface normal into the free space. This is precisely a picture of SPP modes. The field patterns comes from the coincidence of the evanescent wave components in high Bloch orders at both sides of metallic grating.

Fig. 14. Spatial distributions of magnetic fields and electric fields in the xz plane for Γ₂ state at <sup>2</sup> *f*<sup>Γ</sup> =13.2THz, <sup>o</sup> θ = 0 [ (a) and (c)], a state on B₃ at f=50.22THz, <sup>o</sup> θ = 2 [ (b) and (e)]and Γ<sup>4</sup> state at <sup>4</sup> *f*<sup>Γ</sup> =54.3THz, <sup>o</sup> θ= 0 [ (c) and (f)]

dispersion (in the limit of no material loss) as shown in Fig. 13 (b).The *B*1 surface resonance states lie below the light line *L*2 (magenta dashed line), and thus are non-radiative as evanescent modes. The surface resonances labeled as *B*2 originate from the coupling of the fundamental magnetic resonance modes of the metal strip structure with the free space light line *L*<sup>2</sup> . The surface resonances *B*<sup>3</sup> and *B*<sup>4</sup> are harmonic modes of the magnetic resonances that hybridizes with the guided mode inside the dielectric layer. The calculated reflection phase difference between the 0th order reflected and incident electric field, as shown in Fig. 13 (a) for normal incidence (red line), and 2*<sup>o</sup>* incidence (blue line), clearly shows that the resonances are magnetic in nature when the surface resonances intersect the zone center at Γ2 (13.2THz ) and Γ4 (54.3THz) as the reflection phase is zero like what a magnetic conductor surface does to the incident waves. The state Γ<sup>3</sup> , invisible in the reflection phase spectrum under normal incidence [red solid line in Fig. 13 (a)], is a dark state as its eigenmode is in mirror symmetry about the *yz* plane and can not couple with free space photons. While the other *B*3 states can couple with external light under oblique incidence [see the blue line in Fig. 13 (a)]. For example, there exists in-phase reflection at frequency 50.22THz under an incident angle of 2o,

The angle-independent absorption peak at 13.2THz is due to the *B*2 mode, which is only weakly dispersive near the zone center. The more dispersive *B*3 and *B*4 modes are accountable for the incident-angle sensitive absorption in the higher frequencies in Fig. 12 (a). The field patterns in Figs. 14 (a)-14 (c) present the spatial distribution of the real part of magnetic fields excited by the incident plane waves with incident angles 0*<sup>o</sup>* , 2*<sup>o</sup>* and 0*<sup>o</sup>* for the three surface resonance states on *B*<sup>2</sup> , *B*3 and *B*<sup>4</sup> respectively, and the corresponding vector diagrams of electric fields are shown in Figs 14 (d)-14 (f). We can see clearly that the electric fields reach maximum in strength at the slab upper surface, and exponentially decay along the surface normal into the free space. This is precisely a picture of SPP modes. The field patterns comes from the coincidence of the evanescent wave components in high Bloch

Fig. 14. Spatial distributions of magnetic fields and electric fields in the xz plane for Γ₂ state

= 0 [ (a) and (c)], a state on B₃ at f=50.22THz, <sup>o</sup>

π .

θ

= 2 [ (b) and (e)]and Γ<sup>4</sup>

corresponding a *B*3 state at frequency 50.22THz and <sup>0</sup> *k p* ≈ 0.02 /

orders at both sides of metallic grating.

at <sup>2</sup> *f*<sup>Γ</sup> =13.2THz, <sup>o</sup>

state at <sup>4</sup> *f*<sup>Γ</sup> =54.3THz, <sup>o</sup>

θ

θ

= 0 [ (c) and (f)]

Although only TEM guided modes are allowed to be excited in the thin MDM slab within the frequency of our interest, the *B*<sup>2</sup> states are quite different from the *B*3 and *B*4 states in field patterns inside the dielectric layer. We see from Figs. 14 (b), 14 (c) ,14 (e)and 14 (f) that for a *B*3 or *B*<sup>4</sup> state, there are nodes and anti-nodes in field patterns, while for the *B*2 state, the magnetic field is almost uniformly distributed. Calculations on local field enhancement inside the dielectric slab resolve the puzzle. Black solid line in Fig. 15 presents the normalized magnetic field | | *H* inside the dielectric with respect to that of incidence <sup>0</sup> | | *H* under an incident angle of 2*<sup>o</sup>* θ = . The Fourier component in *m=0* order [blue solid line] contributes the most at 13.2THz and the least at 50.22THz and 54.3THz; while it is just opposite for the contributions in combination from the two high order Fourier components with *m=±1* [red solid line]. Figure 15 also indicates that the enhancement of local field of an excited *B*3 or *B*4 state can be ten times larger than that of an excited *B*2 state, the enhancement factor at 50.22THz is about 100 times, while it is only 10 times at 13.2THz.

Fig. 15. Magnetic field H inside the dielectric layer normalized to that of incidence H under an incident angle of <sup>o</sup> θ = 2 . Black line: all Bloch orders of TEM guided modes included; Red line: only the 0th Bloch order considered; Blue line: summation of -1st and +1st Bloch orders of TEM guided modes.

We see from Fig. 13 (b) that the surface resonance dispersion of the slab comes from the interaction between the magnetic resonances and the (folded) light lines *L*1 (for dielectrics) and *L*2 (for air) grazing on the interfaces. In the limit of a small gap-period ratio ( *g p*/ 0.05 = for example), our system is weakly Bragg-scattered, and as such, when a surface resonance state on branches *B*3 or *B*4 is excited, the induced wave fields inside the dielectric of region III are guided quasi-TEM modes dominated by 1 *st* ± Bloch orders. For that reason, the *B*3 and *B*4 states have high fidelity even though they are leaky modes, as most of their Bloch wavefunction components lying outside the free space light line. As the air gaps of the metallic grating serve to couple the electromagnetic waves of region I and region III, the quality factor of a resonance state can be estimated with the overlap integral between the fundamental waveguide mode <sup>0</sup> |α > in the air gap and the dominant Bloch waves | *<sup>i</sup> mk* > ( *i I III* = , ) in region I or region III for the coupling coefficients

Electromagnetic Response and Broadband Utilities of Planar Metamaterials 259

distributions or other types as well) independently to the width of metallic strips and the center positions of air gaps to introduce a 4% (standard deviation) structural disorder. The slab has a lateral size of 60 periods along the *x*ˆ direction. A total of 1200 point sources with random phases are placed at the mesh points inside the dielectric layer. Directional emissions of a wide range of frequencies above 34THz are confirmed by the simulation. The 4% structural disorder has little impact on the directional emissivity. Fig. 16 (a), 16 (c) and 16 (e) show the far-field emission patterns in the *x*ˆ*z*ˆ plane (H-plane) at 40.0THz, 54.3THz and 58.0THz. The inset in Fig. 16 (c) is a control calculation in which the top metal gratings are removed, so that there is just a dielectric layer with random phase sources above a metal ground plane. The directivity of emission from the random sources is lost. Fig. 16 (b), 16 (d) and 16 (e) present the absorptivity (under plane wave incidence) as a function of in-plane wave-vector (solid angle) at these frequencies. The strong angle selectivity of the absorption is evident, and by Kirchhoff's law, the thermal emission should also be highly directional, which is a direct consequence of the good spatial coherence of the surface resonance states. As shown in Figs. 16 (b), 16 (d) and 16 (f), the absorption/emission peaks generally trace out an arc in the *kx-ky* plane, but near 54.3*THz* [Fig. 16 (d)], the dominant emission beam is restricted to a small region near the zone center. This is because the Γ<sup>4</sup> state is a minimum point if we consider the band structure in the *kx-ky* plane. That means that at 54.3*THz*, we can obtain a directional emission beam not just in the H-plane, but in all directions, although the

Fig. 16. Radiation patterns in the H-plane (calculated by FDTD) and absorptivity (calculated by mode expansion method) as a function of in-plane wavevectors at f=40.0THz[ (a) and (b)], f=54.3THz[ (c) and (d)] and f=58.0THz[ (e) and (f)]. In FDTD simulations, 1200 point sources with random phases are placed at the mesh points inside the dielectric layer. A 4% structural disorder is included in the80μmsimulation cell, which accounts for the slight asymmetry of the radiation patterns, but also demonstrates the robustness of the angle selectivity with respect to disorder.The inset in (c) is a control calculation in which the top metal gratings are removed, so that there is just a dielectric layer with random phase sources above a metal ground plane. The directivity of emission from the random sources is lost.

structure is periodic in only one direction.

$$\begin{split} C\_{m}^{l} &= \frac{1}{\mathcal{W}\_{\boldsymbol{x}}} \int\_{0}^{\rho} \frac{k\_{zm}^{l}}{\sqrt{\mathcal{E}\_{r}} \sqrt{k\_{\boldsymbol{x}}^{\boldsymbol{z}} + k\_{zm}^{l}^{\boldsymbol{z}}}} \, e^{-i(k\_{\boldsymbol{x}} + G\_{m})\boldsymbol{\chi}} \, \mathbf{g}\_{\boldsymbol{\eta}}(\mathbf{x}) d\mathbf{x} \\ &= \frac{1}{\sqrt{\mathcal{E}\_{r}}} \frac{k\_{zm}^{l}}{\sqrt{k\_{\boldsymbol{x}}^{\boldsymbol{z}} + k\_{zm}^{l}^{\boldsymbol{z}}}} \sin c [\frac{(k\_{\boldsymbol{x}} + G\_{m})\mathbf{g}}{2}], \end{split} \tag{13}$$

when the air gap width *g p* is satisfied. For the *B*2 states, the major Fourier component of the wavefunction is | <sup>0</sup> *<sup>i</sup> <sup>k</sup>* <sup>&</sup>gt; in zero order, and as <sup>0</sup> *III <sup>z</sup> k* is generally not small, <sup>0</sup> *III C* is usually very large according to Eq. 3, and the *B*2 states leak out easily. The states on branches *B*<sup>3</sup> and *B*4 have major Fourier components in *m=±1* order, and as they are asymptotic to the (folded) dielectric light lines L1, the absolute value of *III zm k* ( *m* = +1for 0 *<sup>x</sup> k* < or *m* = −1for 0 *<sup>x</sup> k* > ) is very small, resulting in the small coupling coefficients <sup>1</sup> *III C*<sup>−</sup> or <sup>1</sup> *III C*<sup>+</sup> . The *B3* and *B4* modes have to travel a long distance before they leak out. They have a long life time and a good spatial coherence. It also explains why the state Γ<sup>3</sup> , a state precisely superposing on folded light line <sup>1</sup> *L*' in dielectric layer, is dark to the incident plane wave as 0 *III zm k* = .

Different from *B*3 and *B*4 states, the *B*2 states have a major Fourier component in *m=0* order which directly couples to the free space photons. As a consequence, the *B*2 states, forming a flat band far away from the light line *L*2 when <sup>0</sup> *k* is small, are localized with resonant frequency scaled by local geometry of unit cell. The high mode fidelity of a *B*3 or *B*4 state also gives rise to much more intense local field compared to the *B*2 states. As shown in Fig. 15, the induced local field is 100 times stronger than the incident field for the state on *B*<sup>3</sup> ; while it is only 10 times stronger for Γ<sup>2</sup> , and this is consistent with the absorptivity shown in Fig. 12 (a). In addition, the coherent length can be adjusted by the gap width as the kernel <sup>0</sup> | | *<sup>m</sup> <sup>i</sup>* α *r k* is proportional to the gap-period ratio *<sup>g</sup>* / *<sup>p</sup>* . More calculations demonstrate that the angular FWHM of the absorption peak is reduced from 0.26*<sup>o</sup>* to 0.16*<sup>o</sup>* when the gap is decreased from 0.2μ*m* to 0.1μ*m* , corresponding to a coherent length of 358λ.

We note that most of the attentions in previous studies have been devoted on the localized B2 states (Hibbins, Sambles et al. 2004; Hibbins, Murray et al. 2006; Brown, Hibbins et al. 2008; Diem, Koschny et al. 2009). While the spatially coherent surface resonance states will lead us into a new vision about coherent control of emission radiations. J.-J. Greffet and coworkers showed that highly directional and spatially coherent thermal emission can be obtained by etching a periodic grating structure into a SiC surface (Le Gall, Olivier et al. 1997; Carminati & Greffet 1999; Greffet, Carminati et al. 2002; Marquier, Joulain et al. 2004). The magnetic resonant modes in our system can do the same, as will be demonstrated below. Our system has the advantage that the operational frequency is tunable by changing the structural parameters, and the operational bandwidth is wide. In addition, our structure supports all-angle functionality for some specific range of frequencies as shown in Fig. 16, although it is periodic only in one direction.

We performed finite-difference-in-time-domain (FDTD) simulations to emulate the emissions from a slab containing point sources with random phases using the same configuration parameters aforementioned. We purposely put disorder in structure to test the robustness of the phenomena. We assigned two Gaussian distributions (they can be uniform

<sup>0</sup> <sup>0</sup> 2 2

1 () sin [ ], <sup>2</sup>

*zm x m*

*k k Gg <sup>c</sup>*

when the air gap width *g p* is satisfied. For the *B*2 states, the major Fourier component of

very large according to Eq. 3, and the *B*2 states leak out easily. The states on branches *B*<sup>3</sup> and *B*4 have major Fourier components in *m=±1* order, and as they are asymptotic to the (folded) dielectric light lines L1, the absolute value of *III*

( *m* = +1for 0 *<sup>x</sup> k* < or *m* = −1for 0 *<sup>x</sup> k* > ) is very small, resulting in the small coupling

out. They have a long life time and a good spatial coherence. It also explains why the state Γ<sup>3</sup> , a state precisely superposing on folded light line <sup>1</sup> *L*' in dielectric layer, is dark to

Different from *B*3 and *B*4 states, the *B*2 states have a major Fourier component in *m=0* order which directly couples to the free space photons. As a consequence, the *B*2 states, forming a

frequency scaled by local geometry of unit cell. The high mode fidelity of a *B*3 or *B*4 state also gives rise to much more intense local field compared to the *B*2 states. As shown in Fig. 15, the induced local field is 100 times stronger than the incident field for the state on *B*<sup>3</sup> ; while it is only 10 times stronger for Γ<sup>2</sup> , and this is consistent with the absorptivity shown in Fig. 12 (a). In addition, the coherent length can be adjusted by the gap width as the kernel

 is proportional to the gap-period ratio *<sup>g</sup>* / *<sup>p</sup>* . More calculations demonstrate that the angular FWHM of the absorption peak is reduced from 0.26*<sup>o</sup>* to 0.16*<sup>o</sup>* when the gap is

We note that most of the attentions in previous studies have been devoted on the localized B2 states (Hibbins, Sambles et al. 2004; Hibbins, Murray et al. 2006; Brown, Hibbins et al. 2008; Diem, Koschny et al. 2009). While the spatially coherent surface resonance states will lead us into a new vision about coherent control of emission radiations. J.-J. Greffet and coworkers showed that highly directional and spatially coherent thermal emission can be obtained by etching a periodic grating structure into a SiC surface (Le Gall, Olivier et al. 1997; Carminati & Greffet 1999; Greffet, Carminati et al. 2002; Marquier, Joulain et al. 2004). The magnetic resonant modes in our system can do the same, as will be demonstrated below. Our system has the advantage that the operational frequency is tunable by changing the structural parameters, and the operational bandwidth is wide. In addition, our structure supports all-angle functionality for some specific range of frequencies as shown in Fig. 16,

We performed finite-difference-in-time-domain (FDTD) simulations to emulate the emissions from a slab containing point sources with random phases using the same configuration parameters aforementioned. We purposely put disorder in structure to test the robustness of the phenomena. We assigned two Gaussian distributions (they can be uniform

*m* , corresponding to a coherent length of 358

*<sup>k</sup> <sup>C</sup> e g x dx <sup>w</sup> k k*

*<sup>i</sup> <sup>p</sup> <sup>i</sup> zm ik G x*

− + = +

<sup>1</sup> ( )

2 2

*<sup>i</sup> <sup>k</sup>* <sup>&</sup>gt; in zero order, and as <sup>0</sup>

<sup>+</sup> <sup>=</sup> +

*k k*

ε

*zm k* = .

flat band far away from the light line *L*2 when <sup>0</sup> *k*

the wavefunction is | <sup>0</sup>

*III C*<sup>−</sup> or <sup>1</sup>

the incident plane wave as 0 *III*

μ

although it is periodic only in one direction.

*m* to 0.1

μ

coefficients <sup>1</sup>

<sup>0</sup> | | *<sup>m</sup> <sup>i</sup>* α*r k*

decreased from 0.2

*i r x zm*

*<sup>m</sup> <sup>i</sup> <sup>x</sup> r x zm i*

ε

( )

*III*

*III C*<sup>+</sup> . The *B3* and *B4* modes have to travel a long distance before they leak

*<sup>z</sup> k* is generally not small, <sup>0</sup>

is small, are localized with resonant

λ.

*x m*

(13)

*zm k*

*III C* is usually

distributions or other types as well) independently to the width of metallic strips and the center positions of air gaps to introduce a 4% (standard deviation) structural disorder. The slab has a lateral size of 60 periods along the *x*ˆ direction. A total of 1200 point sources with random phases are placed at the mesh points inside the dielectric layer. Directional emissions of a wide range of frequencies above 34THz are confirmed by the simulation. The 4% structural disorder has little impact on the directional emissivity. Fig. 16 (a), 16 (c) and 16 (e) show the far-field emission patterns in the *x*ˆ*z*ˆ plane (H-plane) at 40.0THz, 54.3THz and 58.0THz. The inset in Fig. 16 (c) is a control calculation in which the top metal gratings are removed, so that there is just a dielectric layer with random phase sources above a metal ground plane. The directivity of emission from the random sources is lost. Fig. 16 (b), 16 (d) and 16 (e) present the absorptivity (under plane wave incidence) as a function of in-plane wave-vector (solid angle) at these frequencies. The strong angle selectivity of the absorption is evident, and by Kirchhoff's law, the thermal emission should also be highly directional, which is a direct consequence of the good spatial coherence of the surface resonance states. As shown in Figs. 16 (b), 16 (d) and 16 (f), the absorption/emission peaks generally trace out an arc in the *kx-ky* plane, but near 54.3*THz* [Fig. 16 (d)], the dominant emission beam is restricted to a small region near the zone center. This is because the Γ<sup>4</sup> state is a minimum

point if we consider the band structure in the *kx-ky* plane. That means that at 54.3*THz*, we can obtain a directional emission beam not just in the H-plane, but in all directions, although the structure is periodic in only one direction.

Fig. 16. Radiation patterns in the H-plane (calculated by FDTD) and absorptivity (calculated by mode expansion method) as a function of in-plane wavevectors at f=40.0THz[ (a) and (b)], f=54.3THz[ (c) and (d)] and f=58.0THz[ (e) and (f)]. In FDTD simulations, 1200 point sources with random phases are placed at the mesh points inside the dielectric layer. A 4% structural disorder is included in the80μmsimulation cell, which accounts for the slight asymmetry of the radiation patterns, but also demonstrates the robustness of the angle selectivity with respect to disorder.The inset in (c) is a control calculation in which the top metal gratings are removed, so that there is just a dielectric layer with random phase sources above a metal ground plane. The directivity of emission from the random sources is lost.

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We note that there are other schemes to realize coherent thermal radiations, such as by utilizing three-dimensional photonic crystals (Laroche, Carminati et al. 2006) or onedimensional photonic crystal cavities (Lee, Fu et al. 2005). Our metamaterial slab presents a route to achieve linearly polarized coherent thermal emission radiations in a wide frequency range which can be tuned by adjusting structural parameters and material parameters.

In summary, we proposed a simple metamaterial slab structure that possesses spatially coherent magnetic surface resonance states in a broad range of frequencies. These states facilitate nearly perfect absorption in a thin metamaterial slab containing slightly absorptive materials. As the absorption spectrum is very narrow and sensitive to incident angle, the slab should support directional thermal emission. Direct FDTD simulation with random-phase sources corroborates the existence of strong angular emissivity even in the presence of structural disorder. As the surface resonances originate from artificial resonators, the operational frequency and the response can be tuned by varying the structural configurations. Our findings constitute a simple solution for coherent control of thermal emissions, optical antennas, infrared or THz spectroscopy as well as photon detector.

#### **3. Conclusion**

We have shown that a strategy of stacking a multiple layers of holey metallic slabs can give rise to wide transparency band. The multi-layered structure, well known as fishnet metamaterial, also supports broadband negative refraction and sub-wavelength imaging provided that the light waves are incident on its sidewall interface. We also show that coherent control of spontaneous emission radiations can be realizing in a wide frequency range by utilizing spatially coherent magnetic surface resonance states of a magnetic metasurface. The modal expansion method, developed for magnetic meta-surface and multilayered holey metallic slabs, is very fruitful for semi-analytical interpretation on the behind physics picture of planar metamaterials.

#### **4. Acknowledgment**

H. Li thanks Prof. C.T. Chan, Prof. H. Chen and Prof. D. Z. Zhang for the collaboration and helpful discussion. This work was supported by NSFC (No. 11174221, 10974144, 60678046, 10574099), CNKBRSF (Grant No. 2011CB922001), the National 863 Program of China (No.2006AA03Z407), NCET (07-0621), STCSM and SHEDF (No. 06SG24).

#### **5. References**


We note that there are other schemes to realize coherent thermal radiations, such as by utilizing three-dimensional photonic crystals (Laroche, Carminati et al. 2006) or onedimensional photonic crystal cavities (Lee, Fu et al. 2005). Our metamaterial slab presents a route to achieve linearly polarized coherent thermal emission radiations in a wide frequency range which can be tuned by adjusting structural parameters and material parameters.

In summary, we proposed a simple metamaterial slab structure that possesses spatially coherent magnetic surface resonance states in a broad range of frequencies. These states facilitate nearly perfect absorption in a thin metamaterial slab containing slightly absorptive materials. As the absorption spectrum is very narrow and sensitive to incident angle, the slab should support directional thermal emission. Direct FDTD simulation with random-phase sources corroborates the existence of strong angular emissivity even in the presence of structural disorder. As the surface resonances originate from artificial resonators, the operational frequency and the response can be tuned by varying the structural configurations. Our findings constitute a simple solution for coherent control of thermal emissions, optical antennas, infrared or THz spectroscopy as well as photon

We have shown that a strategy of stacking a multiple layers of holey metallic slabs can give rise to wide transparency band. The multi-layered structure, well known as fishnet metamaterial, also supports broadband negative refraction and sub-wavelength imaging provided that the light waves are incident on its sidewall interface. We also show that coherent control of spontaneous emission radiations can be realizing in a wide frequency range by utilizing spatially coherent magnetic surface resonance states of a magnetic metasurface. The modal expansion method, developed for magnetic meta-surface and multilayered holey metallic slabs, is very fruitful for semi-analytical interpretation on the behind

H. Li thanks Prof. C.T. Chan, Prof. H. Chen and Prof. D. Z. Zhang for the collaboration and helpful discussion. This work was supported by NSFC (No. 11174221, 10974144, 60678046, 10574099), CNKBRSF (Grant No. 2011CB922001), the National 863 Program of China

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Zhang, W. & Zheludev, N. I. (2009). Terahertz metamaterial with asymmetric transmission. *Physical Review B* Vol. 80 No. 15, (Oct) pp. 153104,ISSN 1098-0121 Smith, D. R.; Pendry, J. B. & Wiltshire, M. C. K. (2004). Metamaterials and negative refractive index. *Science* Vol. 305 No. 5685, (Aug 6) pp. 788-792,ISSN 0036-8075 Taflove, A. & Hagness, S. C. (2000). Computational Electrodynamics: The Finite-Difference

Wang, M. (2007). Coupling of surface plasmons in nanostructured metal/dielectric multilayers with subwavelength hole arrays. *Physical Review B* Vol. 76 No. 19,


**11** 

*France* 

**Design and Characterization of Metamaterials** 

Metamaterials have attracted considerable interests (Shelby, 2001, Yen, 2004, Smith, 2004, Linden, 2004, Zhang, 2004) because of their unusual electromagnetic properties (Veselago, 1968) and because of their potential applications such as invisibility cloaks (Leonhardt, 2006, Pendry, 2006, Schurig, 2006, Cai, 2007, Gaillot, 2008, Kante, 2008), the so-called perfect lenses (Pendry, 2000) and gradient index (GRIN) lenses. For example, perfect lenses require the use of Left-Handed (LH) metamaterials (Smith, 2000) having a negative refractive index, which

from near zero values to several tenths. Traditionally, these properties are achieved by the use of a combination of split-ring resonators (Pendry, 1999) and metallic wires (Pendry, 1996), with periods much smaller than the wavelength of the electromagnetic wave, such that the medium can be considered homogeneous. Lately, pairs of finite-length wires (cut wires pairs) (Shalaev, 2005) have been proposed not only to replace the conventional splitring resonators (SRRs) to produce a negative magnetic permeability under normal to plane incidence, but also lead to a negative refractive index *n* in the optical regime. However in a recent review paper (Shalaev, 2007), Shalaev stated that it is very difficult to achieve a negative refractive index with exclusively wire pairs and that the negative index value observed in the ref. (Shalaev, 2005), was accomplished in part because of the significant contribution from the imaginary part of the permeability. Nevertheless, the negative index from only cut wire and plate pairs has never been verified elsewhere (Dolling, 2005). Instead, continuous wires have been combined to the cut wire pairs to produce simultaneously a negative permittivity to lead to a negative index in the microwave domain (Zhou, 2006a). Zhou *et al*. also theoretically proposed a left-handed material using only cut wire pairs by increasing the equivalent capacitance between two consecutive short wire pairs so as to adjust the electric resonance frequency (Zhou, 2006b) This increase of capacitance can only be obtained by strongly reducing the spacing between two consecutive wires, which is quite difficult to achieve at high frequencies. These cited results concern mainly the microwave domain. In the optical regime, infrared and visible domains, the main problem concern the design and the characterization of metamaterials made of unit cells

. Invisibility cloaks require adjustable positive permeability and permittivity

can be produced by a simultaneously negative electric permittivity

**1. Introduction**

permeability

μ

**for Optical and Radio Communications** 

André de Lustrac, Shah Nawaz Burokur, Boubacar Kanté,

Alexandre Sellier and Dylan Germain

*Institut d'Electronique Fondamentale, Univ. Paris-Sud, CNRS UMR 8622, Orsay,* 

ε

and magnetic


### **Design and Characterization of Metamaterials for Optical and Radio Communications**

André de Lustrac, Shah Nawaz Burokur, Boubacar Kanté,

Alexandre Sellier and Dylan Germain *Institut d'Electronique Fondamentale, Univ. Paris-Sud, CNRS UMR 8622, Orsay, France* 

#### **1. Introduction**

268 Metamaterial

Xiao, X. A.; Jinbo, W.; Sasagawa, Y.; Miyamaru, F.; Zhang, M. Y.; Takeda, M. W.; Qiu, C. Y.;

Ye, Y. H. & Zhang, J. Y. (2005). Enhanced light transmission through cascaded metal films

Ye, Y. Q. & He, S. (2010). 90 degrees polarization rotator using a bilayered chiral

Yen, T. J.; Padilla, W. J.; Fang, N.; Vier, D. C.; Smith, D. R.; Pendry, J. B.; Basov, D. N. &

Zhang, S.; Fan, W.; Panoiu, N.; Malloy, K.; Osgood, R. & Brueck, S. (2006). Optical negative-

Zhou, L.; Huang, C. P.; Wu, S.; Yin, X. G.; Wang, Y. M.; Wang, Q. J. & Zhu, Y. Y. (2010).

Zhou, L.; Wen, W.; Chan, C. & Sheng, P. (2003). Multiband subwavelength magnetic

Zhu, B.; Feng, Y. J.; Zhao, J. M.; Huang, C. & Jiang, T. A. (2010). Switchable metamaterial

*Physics Letters* Vol. 97 No. 1, (Jul 5) pp. 011905-011903,ISSN 0003-6951 Zhou, L.; Li, H. Q.; Qin, Y. Q.; Wei, Z. Y. & Chan, C. T. (2005). Directive emissions from

18564,ISSN 1094-4087

1523,ISSN 0146-9592

3259,ISSN 0003-6951

17) pp. 203501,ISSN 0003-6951

303 No. 5663, (Mar 5) pp. 1494-1496,ISSN 0036-8075

(MAR 7 2005) pp. 101101,ISSN 0003-6951

*Express* Vol. 14 (JUL 24 2006) pp. 6778-6787,ISSN 1094-4087

*Letters* Vol. 97 No. 5, (Aug 2) pp. 051906,ISSN 0003-6951

Wen, W. J. & Sheng, P. (2010). Resonant terahertz transmissions through metal hole array on silicon substrate. *Optics Express* Vol. 18 No. 18, (Aug 30) pp. 18558-

perforated with periodic hole arrays. *Optics Letters* Vol. 30 No. 12, (Jun 15) pp. 1521-

metamaterial with giant optical activity. *Applied Physics Letters* Vol. 96 No. 20, (May

Zhang, X. (2004). Terahertz magnetic response from artificial materials. *Science* Vol.

index bulk metamaterials consisting of 2D perforated metal-dielectric stacks. *Optics* 

Enhanced optical transmission through metal-dielectric multilayer gratings. *Applied* 

subwavelength metamaterial-based cavities. *Applied Physics Letters* Vol. 86 No. 10,

reflectors based on fractals. *Applied Physics Letters* Vol. 83 (OCT 20 2003) pp. 3257-

reflector/absorber for different polarized electromagnetic waves. *Applied Physics* 

Metamaterials have attracted considerable interests (Shelby, 2001, Yen, 2004, Smith, 2004, Linden, 2004, Zhang, 2004) because of their unusual electromagnetic properties (Veselago, 1968) and because of their potential applications such as invisibility cloaks (Leonhardt, 2006, Pendry, 2006, Schurig, 2006, Cai, 2007, Gaillot, 2008, Kante, 2008), the so-called perfect lenses (Pendry, 2000) and gradient index (GRIN) lenses. For example, perfect lenses require the use of Left-Handed (LH) metamaterials (Smith, 2000) having a negative refractive index, which can be produced by a simultaneously negative electric permittivity ε and magnetic permeability μ. Invisibility cloaks require adjustable positive permeability and permittivity from near zero values to several tenths. Traditionally, these properties are achieved by the use of a combination of split-ring resonators (Pendry, 1999) and metallic wires (Pendry, 1996), with periods much smaller than the wavelength of the electromagnetic wave, such that the medium can be considered homogeneous. Lately, pairs of finite-length wires (cut wires pairs) (Shalaev, 2005) have been proposed not only to replace the conventional splitring resonators (SRRs) to produce a negative magnetic permeability under normal to plane incidence, but also lead to a negative refractive index *n* in the optical regime. However in a recent review paper (Shalaev, 2007), Shalaev stated that it is very difficult to achieve a negative refractive index with exclusively wire pairs and that the negative index value observed in the ref. (Shalaev, 2005), was accomplished in part because of the significant contribution from the imaginary part of the permeability. Nevertheless, the negative index from only cut wire and plate pairs has never been verified elsewhere (Dolling, 2005). Instead, continuous wires have been combined to the cut wire pairs to produce simultaneously a negative permittivity to lead to a negative index in the microwave domain (Zhou, 2006a). Zhou *et al*. also theoretically proposed a left-handed material using only cut wire pairs by increasing the equivalent capacitance between two consecutive short wire pairs so as to adjust the electric resonance frequency (Zhou, 2006b) This increase of capacitance can only be obtained by strongly reducing the spacing between two consecutive wires, which is quite difficult to achieve at high frequencies. These cited results concern mainly the microwave domain. In the optical regime, infrared and visible domains, the main problem concern the design and the characterization of metamaterials made of unit cells

Design and Characterization of Metamaterials for Optical and Radio Communications 271

controlled by adjusting either the spacing or the alignment of paired cut-wires. Using an asymmetric alignment, an inverted hybridization scheme, where the asymmetric mode is at a higher frequency than the symmetric mode, is predicted and thus more favorable for obtaining negative refraction. The first experimental demonstration of a negative refraction metamaterial exclusively based on paired cut-wires in the microwave range is reported.

Fig. 1. (a) Schematic of the symmetric cut-wire pair. (b) Hybridization scheme of the two coupled dipoles. (c) Transmission spectra calculated at normal incidence for a periodic array of cut-wires (red) and of paired cut-wires (blue), respectively (px= 1.2 µm; py= 200 nm; w= 30 nm; L= 600 nm; hsub= 100 nm). The 30 nm thick gold cut-wires are described using a Drude model whose parameters can be found in [Kante, 2008a]. The dielectric spacer (SiO2)

The plasmon hybridization scheme was introduced by Prodan et al. [Prodan, 2003] who gave an intuitive electromagnetic analogue of molecular orbital theory. Such a scheme has largely been used by the metamaterial community especially for simplifying metamaterial designs at optical wavelengths [Shalaev, 2005, Liu, 2008]. The coupling of two electric dipoles facing each other has thus been exploited to mimic magnetic atoms and alter the effective magnetic permeability of metamaterials in the optical range. While a magnetic activity was indeed obtained from metamaterials comprised of metallic dipoles [Shalaev, 2005, Liu, 2008], negative refraction was reported only in the pioneering demonstration by Shalaev et al. [Shalaev, 2005] who used a periodic array of cut-wire pairs. In order to unambiguously achieve negative refraction, the magnetic activity must actually occur within a frequency band in which the electric permittivity is negative. For this purpose, one solution consists of associating magnetic "atoms" (coupled metallic dipoles) to a broadband "electric plasma" (continuous wires) in the same structure. Many authors have used this solution either in the microwave [Zhou, 2006, Guven, 2006] or in the optical regime [Liu, 2008], thereby contributing to the development of the so-called fishnet structure. We propose another solution based on the control of the coupling between metallic dipoles in such a way that the symmetric and anti-symmetric bands have a sufficient overlap. The coupling strength is varied either by changing the distance between coupled dipoles or by

Fig. 1(a) shows the rectangular unit cell of the studied 2D structure in the case where the coupled metallic dipoles (cut-wires) are vertically aligned. This structure is henceforth referred to as the symmetric cut-wire structure to distinguish it from the asymmetric

permittivity is εsub=2.25.

breaking the symmetry of the structure.

with nanometric dimensions. At this scale, the control and the engineering of the electromagnetic properties of metamaterials are closely linked to the easiness of the fabrication. This easiness is more and more important with the simplicity of the geometrical shapes of the unit cell of the metamaterials. We show that we can change the conventional shape of the split ring resonator of J. Pendry to simple coupled nanowires, keeping the same electromagnetic properties (Linden, 2004, Enkrich, 2005, Kante, 2008a, Burokur, 2009a).

In the first part of this chapter we investigate numerically and experimentally the electromagnetic properties of cut wire pairs metamaterials where the symmetry between the wires on opposite faces is voluntarily broken along the **E**-field direction (Sellier, 2009, Kante, 2009a, Burokur, 2009a). It is reported that in this case the electric resonance of the cut wire pairs can occur at lower frequencies than the magnetic resonance, which is in contrast with the symmetrical configuration. This lower electrical resonance frequency allows realizing a common frequency region where the permeability and the permittivity are simultaneously negative. This claim is verified numerically and experimentally in the microwave domain and indications on designing negative refractive index from structures composed of only cut wire pairs are given.

Then, we investigate numerically and experimentally the reflection and transmission spectra for an obliquely incident plane wave on the asymmetric structure. It is reported that a diffraction threshold appears in E-plane (plane containing vectors E and k), that is the (-1,0) mode starts to propagate (Burokur, 2009b). Besides, resonances in E-plane shift in frequency with increasing oblique incidence. However in H-plane (plane containing vectors H and k), the structure is diffractionless and independent of the incidence and therefore the negative index is maintained in a wide angular range. These statements are verified numerically and experimentally in the microwave domain.

On the other hand, we show that it is possible to engineer the resonances of metamaterial in the infrared domain (Kante, 2008b, Kante, 2009b). We present an experimental and numerical analysis of the infrared response of metamaterials made of split ring resonators (SRR) and continuous nanowires deposited on silicon when the geometry of the SRRs is gradually altered. The impact of the geometric transformation of the SRRs on the spectra of the composite metamaterial is measured in the 1.5-15 μm wavelength range for the two field polarizations under normal to plane propagation. We show experimentally and numerically that tuning the SRRs towards elementary cut wires translates in a predictable manner the wavelength response of the artificial material. We also analyze coupling effects between the SRRs and the continuous nanowires for different spacing between them. The results of our study are expected to provide useful guidelines for the design of optical devices using metamaterials on silicon.

#### **2. Cut wire pairs metamaterials with broken symmetry at microwave frequencies**

#### **2.1 Plasmon hybrization**

In this chapter, we show that negative dielectric permittivity and negative magnetic permeability can be simultaneously achieved by appropriately controlling the coupling strength between paired cut-wires of adjacent layers. The coupling strength is itself

with nanometric dimensions. At this scale, the control and the engineering of the electromagnetic properties of metamaterials are closely linked to the easiness of the fabrication. This easiness is more and more important with the simplicity of the geometrical shapes of the unit cell of the metamaterials. We show that we can change the conventional shape of the split ring resonator of J. Pendry to simple coupled nanowires, keeping the same electromagnetic properties (Linden, 2004, Enkrich, 2005, Kante, 2008a, Burokur, 2009a).

In the first part of this chapter we investigate numerically and experimentally the electromagnetic properties of cut wire pairs metamaterials where the symmetry between the wires on opposite faces is voluntarily broken along the **E**-field direction (Sellier, 2009, Kante, 2009a, Burokur, 2009a). It is reported that in this case the electric resonance of the cut wire pairs can occur at lower frequencies than the magnetic resonance, which is in contrast with the symmetrical configuration. This lower electrical resonance frequency allows realizing a common frequency region where the permeability and the permittivity are simultaneously negative. This claim is verified numerically and experimentally in the microwave domain and indications on designing negative refractive index from structures composed of only cut

Then, we investigate numerically and experimentally the reflection and transmission spectra for an obliquely incident plane wave on the asymmetric structure. It is reported that a diffraction threshold appears in E-plane (plane containing vectors E and k), that is the (-1,0) mode starts to propagate (Burokur, 2009b). Besides, resonances in E-plane shift in frequency with increasing oblique incidence. However in H-plane (plane containing vectors H and k), the structure is diffractionless and independent of the incidence and therefore the negative index is maintained in a wide angular range. These statements are verified numerically and

On the other hand, we show that it is possible to engineer the resonances of metamaterial in the infrared domain (Kante, 2008b, Kante, 2009b). We present an experimental and numerical analysis of the infrared response of metamaterials made of split ring resonators (SRR) and continuous nanowires deposited on silicon when the geometry of the SRRs is gradually altered. The impact of the geometric transformation of the SRRs on the spectra of the composite metamaterial is measured in the 1.5-15 μm wavelength range for the two field polarizations under normal to plane propagation. We show experimentally and numerically that tuning the SRRs towards elementary cut wires translates in a predictable manner the wavelength response of the artificial material. We also analyze coupling effects between the SRRs and the continuous nanowires for different spacing between them. The results of our study are expected to provide useful guidelines for the design of optical devices using

**2. Cut wire pairs metamaterials with broken symmetry at microwave** 

In this chapter, we show that negative dielectric permittivity and negative magnetic permeability can be simultaneously achieved by appropriately controlling the coupling strength between paired cut-wires of adjacent layers. The coupling strength is itself

wire pairs are given.

metamaterials on silicon.

**2.1 Plasmon hybrization** 

**frequencies** 

experimentally in the microwave domain.

controlled by adjusting either the spacing or the alignment of paired cut-wires. Using an asymmetric alignment, an inverted hybridization scheme, where the asymmetric mode is at a higher frequency than the symmetric mode, is predicted and thus more favorable for obtaining negative refraction. The first experimental demonstration of a negative refraction metamaterial exclusively based on paired cut-wires in the microwave range is reported.

Fig. 1. (a) Schematic of the symmetric cut-wire pair. (b) Hybridization scheme of the two coupled dipoles. (c) Transmission spectra calculated at normal incidence for a periodic array of cut-wires (red) and of paired cut-wires (blue), respectively (px= 1.2 µm; py= 200 nm; w= 30 nm; L= 600 nm; hsub= 100 nm). The 30 nm thick gold cut-wires are described using a Drude model whose parameters can be found in [Kante, 2008a]. The dielectric spacer (SiO2) permittivity is εsub=2.25.

The plasmon hybridization scheme was introduced by Prodan et al. [Prodan, 2003] who gave an intuitive electromagnetic analogue of molecular orbital theory. Such a scheme has largely been used by the metamaterial community especially for simplifying metamaterial designs at optical wavelengths [Shalaev, 2005, Liu, 2008]. The coupling of two electric dipoles facing each other has thus been exploited to mimic magnetic atoms and alter the effective magnetic permeability of metamaterials in the optical range. While a magnetic activity was indeed obtained from metamaterials comprised of metallic dipoles [Shalaev, 2005, Liu, 2008], negative refraction was reported only in the pioneering demonstration by Shalaev et al. [Shalaev, 2005] who used a periodic array of cut-wire pairs. In order to unambiguously achieve negative refraction, the magnetic activity must actually occur within a frequency band in which the electric permittivity is negative. For this purpose, one solution consists of associating magnetic "atoms" (coupled metallic dipoles) to a broadband "electric plasma" (continuous wires) in the same structure. Many authors have used this solution either in the microwave [Zhou, 2006, Guven, 2006] or in the optical regime [Liu, 2008], thereby contributing to the development of the so-called fishnet structure. We propose another solution based on the control of the coupling between metallic dipoles in such a way that the symmetric and anti-symmetric bands have a sufficient overlap. The coupling strength is varied either by changing the distance between coupled dipoles or by breaking the symmetry of the structure.

Fig. 1(a) shows the rectangular unit cell of the studied 2D structure in the case where the coupled metallic dipoles (cut-wires) are vertically aligned. This structure is henceforth referred to as the symmetric cut-wire structure to distinguish it from the asymmetric

Design and Characterization of Metamaterials for Optical and Radio Communications 273

close interaction change. As a result the repulsive force becomes attractive and vice versa. Correspondingly, the symmetric mode becomes the low-energy mode while the asymmetric mode is shifted to higher frequencies. It is evident that this inversion process is impossible

Controlling the coupling between metallic dipoles thus allows the two plasmon resonances to be engineered. When the magnetic and electric modes are very close together, the bands of negative permeability and negative permittivity overlap, and a negative refraction material is obtained. More generally, the design of true negative index metamaterials can be

Fig. 2. (a) Asymmetric cut-wire pair with the three degrees of freedom for the control of the

Fig. 3. Influence of the coupling strength on the transmission spectra of a bi-layer structure

permittivity is εsub = 3.9. (a) variation of the dielectric spacer (or substrate) thickness hsub; (b)

In this part, we present a systematic study of the cut wires structure presented in Fig. 4, derived from the previous structures, under normal-to-plane incidence in the microwave domain. Numerical simulations performed using the FEM based software HFSS are run to

(px = 19 mm; py = 9.5 mm; w = 0.3 mm; L = 9.5 mm; hsub = 1.2 mm). The substrate

variation of the longitudinal shift dx; (c) variation of the lateral shift dy.

**2.2 Monolayer double-face structure** 

coupling strength: hsub, dx and dy. (b) Inverted hybridization scheme.

achieved by appropriate design of the three degrees of freedom hsub, dx and dy.

in the case of a lateral dy displacement of the dipoles (Fig. 3(c)).

structure discussed later (see fig. 2). Both structures consist of 2D periodic arrays of metallic cut-wires separated by a dielectric spacer [Shalaev, 2005]. The electromagnetic wave should propagate normally to the layers with the electric field parallel to the longest side of dipolar elements. The structure can be described in terms of effective parameters as long as the cutwire width w and spacer thickness hsub are much smaller than the wavelength [Shalaev, 2005]. Two series of calculations were carried out using a finite element simulation package (HFSS from Ansys), one for symmetric structures and the other for asymmetric structures. The effective parameters were obtained from the calculated transmission and reflection coefficients [Gundogdu, 2008].

The first series of calculations were performed to compare the electromagnetic response of a symmetric cut-wire bi-layer (blue curve in Fig. 1(c)) to that of a cut-wire monolayer (red curve in Fig. 1(c)). As it is evident in the figure, only one resonance is observed for the single-face cut-wire structure in the frequency range of interest. This resonance corresponds to the fundamental cut-wire dipolar mode, which in the optical regime can also be interpreted in terms of a localized plasmon resonance [Smith, 2002]. Collective electronic excitations, also called surface plasmons, are indeed the main mechanism at short wavelengths. For the double-face cut-wire structure, the coupling between paired cut-wires lifts the degeneracy of the single cut-wire mode, which hybridizes into two plasmon modes. One mode is symmetric and corresponds to in-phase current oscillations, while the other is anti-symmetric and corresponds to out-of-phase current oscillations. For a symmetric cutwire pair with a vertical alignment of the two cut-wires, the anti-symmetric mode is the lowenergy (-frequency) mode since attractive forces are present in the system. Conversely, repellent forces are produced in the case of the symmetric mode that is therefore the highfrequency mode. The stronger the coupling (the smaller the spacing between the dipoles), the larger the frequency difference between the two modes. The evolution of the transmission spectra with the thickness of the dielectric spacer (or substrate) hsub is illustrated in Fig. 3(a) in the case of a structure designed to operate in the microwave regime. Similar results were obtained for the structure in Fig. 1(c).

A second series of calculations was performed to analyze the influence of a vertical misalignment of metallic dipoles at a fixed spacer (or substrate) thickness. For this purpose, the cut-wire layers were shifted from each other in the horizontal XY plane (Fig. 2(a)) thus breaking the symmetry of the cut-wire structure. The relative displacements dx and dy in the X and Y directions respectively were used as parameters. In the example in the microwave regime, the substrate thickness was chosen to be equal to that of commercially available epoxy dielectric boards (1.2 mm). For this thickness and a vertical alignment of paired cut-wires (dx = dy = 0), the calculated transmission spectrum in Fig. 3(a) revealed a pronounced frequency separation between the symmetric (electric) and anti-symmetric (magnetic) modes. Figs. 3(b) and 3(c) show the evolution of the transmission spectrum for non-zero values of the longitudinal (dx) and lateral (dy) displacements, respectively. Quite surprisingly, as previously reported by A. Christ et al. [Christ, 2007] for the control of Fano resonances in a plasmonic lattice of continuous wires, symmetry breaking can invert the hybridization scheme due to modified Coulomb interactions (Fig. 2(b)) resulting in the symmetric resonance occurring at a lower frequency than the anti-symmetric one. The Coulomb forces in our system result from the interaction of charges located at the cut-wire ends. When the longitudinal shift (dx) is progressively increased, the signs of the charges in

structure discussed later (see fig. 2). Both structures consist of 2D periodic arrays of metallic cut-wires separated by a dielectric spacer [Shalaev, 2005]. The electromagnetic wave should propagate normally to the layers with the electric field parallel to the longest side of dipolar elements. The structure can be described in terms of effective parameters as long as the cutwire width w and spacer thickness hsub are much smaller than the wavelength [Shalaev, 2005]. Two series of calculations were carried out using a finite element simulation package (HFSS from Ansys), one for symmetric structures and the other for asymmetric structures. The effective parameters were obtained from the calculated transmission and reflection

The first series of calculations were performed to compare the electromagnetic response of a symmetric cut-wire bi-layer (blue curve in Fig. 1(c)) to that of a cut-wire monolayer (red curve in Fig. 1(c)). As it is evident in the figure, only one resonance is observed for the single-face cut-wire structure in the frequency range of interest. This resonance corresponds to the fundamental cut-wire dipolar mode, which in the optical regime can also be interpreted in terms of a localized plasmon resonance [Smith, 2002]. Collective electronic excitations, also called surface plasmons, are indeed the main mechanism at short wavelengths. For the double-face cut-wire structure, the coupling between paired cut-wires lifts the degeneracy of the single cut-wire mode, which hybridizes into two plasmon modes. One mode is symmetric and corresponds to in-phase current oscillations, while the other is anti-symmetric and corresponds to out-of-phase current oscillations. For a symmetric cutwire pair with a vertical alignment of the two cut-wires, the anti-symmetric mode is the lowenergy (-frequency) mode since attractive forces are present in the system. Conversely, repellent forces are produced in the case of the symmetric mode that is therefore the highfrequency mode. The stronger the coupling (the smaller the spacing between the dipoles), the larger the frequency difference between the two modes. The evolution of the transmission spectra with the thickness of the dielectric spacer (or substrate) hsub is illustrated in Fig. 3(a) in the case of a structure designed to operate in the microwave

A second series of calculations was performed to analyze the influence of a vertical misalignment of metallic dipoles at a fixed spacer (or substrate) thickness. For this purpose, the cut-wire layers were shifted from each other in the horizontal XY plane (Fig. 2(a)) thus breaking the symmetry of the cut-wire structure. The relative displacements dx and dy in the X and Y directions respectively were used as parameters. In the example in the microwave regime, the substrate thickness was chosen to be equal to that of commercially available epoxy dielectric boards (1.2 mm). For this thickness and a vertical alignment of paired cut-wires (dx = dy = 0), the calculated transmission spectrum in Fig. 3(a) revealed a pronounced frequency separation between the symmetric (electric) and anti-symmetric (magnetic) modes. Figs. 3(b) and 3(c) show the evolution of the transmission spectrum for non-zero values of the longitudinal (dx) and lateral (dy) displacements, respectively. Quite surprisingly, as previously reported by A. Christ et al. [Christ, 2007] for the control of Fano resonances in a plasmonic lattice of continuous wires, symmetry breaking can invert the hybridization scheme due to modified Coulomb interactions (Fig. 2(b)) resulting in the symmetric resonance occurring at a lower frequency than the anti-symmetric one. The Coulomb forces in our system result from the interaction of charges located at the cut-wire ends. When the longitudinal shift (dx) is progressively increased, the signs of the charges in

regime. Similar results were obtained for the structure in Fig. 1(c).

coefficients [Gundogdu, 2008].

close interaction change. As a result the repulsive force becomes attractive and vice versa. Correspondingly, the symmetric mode becomes the low-energy mode while the asymmetric mode is shifted to higher frequencies. It is evident that this inversion process is impossible in the case of a lateral dy displacement of the dipoles (Fig. 3(c)).

Controlling the coupling between metallic dipoles thus allows the two plasmon resonances to be engineered. When the magnetic and electric modes are very close together, the bands of negative permeability and negative permittivity overlap, and a negative refraction material is obtained. More generally, the design of true negative index metamaterials can be achieved by appropriate design of the three degrees of freedom hsub, dx and dy.

Fig. 2. (a) Asymmetric cut-wire pair with the three degrees of freedom for the control of the coupling strength: hsub, dx and dy. (b) Inverted hybridization scheme.

Fig. 3. Influence of the coupling strength on the transmission spectra of a bi-layer structure (px = 19 mm; py = 9.5 mm; w = 0.3 mm; L = 9.5 mm; hsub = 1.2 mm). The substrate permittivity is εsub = 3.9. (a) variation of the dielectric spacer (or substrate) thickness hsub; (b) variation of the longitudinal shift dx; (c) variation of the lateral shift dy.

#### **2.2 Monolayer double-face structure**

In this part, we present a systematic study of the cut wires structure presented in Fig. 4, derived from the previous structures, under normal-to-plane incidence in the microwave domain. Numerical simulations performed using the FEM based software HFSS are run to

Design and Characterization of Metamaterials for Optical and Radio Communications 275

Fig. 5. Calculated and measured reflection (S11) and transmission (S21) responses of the

peak and a dip is respectively observed in the transmission and reflection phase.

Figure 5 shows the calculated (continuous lines) and measured (dashed lines) S-parameters of the metamaterial for a monolayer configuration. There is a very good qualitative agreement between simulations and measurements. The calculated and measured magnitudes of S21 presented in Fig. 5(a) show clearly two resonance dips, the first one at 9.58 GHz and a second one at 11.39 GHz. We can note in Fig. 5(b) that a change in sign occurs for the transmission phase at the first resonance dip. At the second resonance dip, a

Using the retrieval procedure described in [Nicholson, 1970], based on the inversion of the reflection and transmission coefficients, the effective parameters of the double-face metamaterial structure are extracted. The metamaterial has a period very small compared to the wavelength λ (less than λ/20) in the propagation direction. The propagation of the electromagnetic wave travelling along this direction is dominated by this deep subwavelength period and not by the in-plane period ax or ay. There is only a single propagating mode in the negative-index frequency region, justifying the description of the

The extracted permittivity ε, permeability µ and refractive index n are shown in the various parts of Fig. 6. Two extraction procedures have been performed: the first one uses the calculated S-parameters and the second one is based on the measured S-parameters. As illustrated by the extracted parameters from the calculated and measured S-parameters, the cut wires structure shows firstly an electric resonance at the first resonance dip observed at 9.58 GHz in Fig. 6. This electric resonance exhibits values going negative for the real part of the permittivity in the vicinity of the resonance. Secondly a magnetic resonance with negative values appears at the right hand side of the second resonance dip at 11.5 GHz. Around the same frequency, the real part of the permittivity is still negative. The extracted real part of the refractive index is therefore negative around 11.5 GHz which is the frequency of the LH peak. However, we can also notice that the zero value for the ε response is very close to 13 GHz where a full transmission band is observed in Fig. 6(a). This frequency constitutes the frequency of the RH transmission peak. We can therefore deduce that this RH transmission

metamaterial for a single layer: (a) magnitude, and (b) phase.

cut wires metamaterial with an effective index [Valentine, 2008].

peak is due to an impedance matching between the structure and vacuum.

show and understand the electromagnetic behavior of the design. A single layer of the metamaterial is characterized by reflection and transmission measurements. The retrieved parameters show simultaneous resonances in the permittivity and permeability responses leading to a negative index of refraction.

Fig. 4. Unit cell of: cut wires structure under normal-to-plane propagation (ax = 9.5 mm, ay = 19 mm, w = 0.3 mm, l = 9.5 mm). The inserts show the direction and the polarization of the wave.

The cut wires metamaterial illustrated by its unit cell in Fig. 4 is employed to operate in the microwave regime. It consists of a double-face structure composed of periodic cut wires of finite length. The structure is printed on both faces of an epoxy dielectric board of thickness t = 1.2 mm and of relative permittivity εr = 3.9. For the different samples reported here, the width of the cut wires denoted by w is 0.3 mm. The length of the cut wires is l = 9.5 mm and the unit cell size in the x and y direction is respectively ax = 9.5 mm and ay = 19 mm. These dimensions have been optimized to operate around 10 GHz and remain the same throughout the section.

The reflection and transmission spectra of the metamaterial are calculated using HFSS by applying the necessary periodic boundary conditions on the unit cell. Several samples of the structure consisting of 10 × 5 cells on a 120 mm × 120 mm epoxy surface are fabricated using conventional commercial chemical etching technique. Measurements are done in an anechoic chamber using an Agilent 8722ES network analyzer and two X-band horn antennas. In the transmission measurements, the plane waves are incident normal to the prototype surface and a calibration to the transmission in free space (the metamaterial sample is removed) between the two horn antennas is done. The reflection measurements are done by placing the emitting and receiving horn antennas on the same side of the prototype and inclined with an angle of about 5° with respect to the normal on the prototype surface. The calibration for the reflection is done using a sheet of copper as reflecting mirror.

show and understand the electromagnetic behavior of the design. A single layer of the metamaterial is characterized by reflection and transmission measurements. The retrieved parameters show simultaneous resonances in the permittivity and permeability responses

Fig. 4. Unit cell of: cut wires structure under normal-to-plane propagation (ax = 9.5 mm, ay = 19 mm, w = 0.3 mm, l = 9.5 mm). The inserts show the direction and the polarization of

The cut wires metamaterial illustrated by its unit cell in Fig. 4 is employed to operate in the microwave regime. It consists of a double-face structure composed of periodic cut wires of finite length. The structure is printed on both faces of an epoxy dielectric board of thickness t = 1.2 mm and of relative permittivity εr = 3.9. For the different samples reported here, the width of the cut wires denoted by w is 0.3 mm. The length of the cut wires is l = 9.5 mm and the unit cell size in the x and y direction is respectively ax = 9.5 mm and ay = 19 mm. These dimensions have been optimized to operate around 10 GHz and remain the same

The reflection and transmission spectra of the metamaterial are calculated using HFSS by applying the necessary periodic boundary conditions on the unit cell. Several samples of the structure consisting of 10 × 5 cells on a 120 mm × 120 mm epoxy surface are fabricated using conventional commercial chemical etching technique. Measurements are done in an anechoic chamber using an Agilent 8722ES network analyzer and two X-band horn antennas. In the transmission measurements, the plane waves are incident normal to the prototype surface and a calibration to the transmission in free space (the metamaterial sample is removed) between the two horn antennas is done. The reflection measurements are done by placing the emitting and receiving horn antennas on the same side of the prototype and inclined with an angle of about 5° with respect to the normal on the prototype surface. The calibration for the reflection is done using a sheet of copper as

leading to a negative index of refraction.

the wave.

throughout the section.

reflecting mirror.

Fig. 5. Calculated and measured reflection (S11) and transmission (S21) responses of the metamaterial for a single layer: (a) magnitude, and (b) phase.

Figure 5 shows the calculated (continuous lines) and measured (dashed lines) S-parameters of the metamaterial for a monolayer configuration. There is a very good qualitative agreement between simulations and measurements. The calculated and measured magnitudes of S21 presented in Fig. 5(a) show clearly two resonance dips, the first one at 9.58 GHz and a second one at 11.39 GHz. We can note in Fig. 5(b) that a change in sign occurs for the transmission phase at the first resonance dip. At the second resonance dip, a peak and a dip is respectively observed in the transmission and reflection phase.

Using the retrieval procedure described in [Nicholson, 1970], based on the inversion of the reflection and transmission coefficients, the effective parameters of the double-face metamaterial structure are extracted. The metamaterial has a period very small compared to the wavelength λ (less than λ/20) in the propagation direction. The propagation of the electromagnetic wave travelling along this direction is dominated by this deep subwavelength period and not by the in-plane period ax or ay. There is only a single propagating mode in the negative-index frequency region, justifying the description of the cut wires metamaterial with an effective index [Valentine, 2008].

The extracted permittivity ε, permeability µ and refractive index n are shown in the various parts of Fig. 6. Two extraction procedures have been performed: the first one uses the calculated S-parameters and the second one is based on the measured S-parameters. As illustrated by the extracted parameters from the calculated and measured S-parameters, the cut wires structure shows firstly an electric resonance at the first resonance dip observed at 9.58 GHz in Fig. 6. This electric resonance exhibits values going negative for the real part of the permittivity in the vicinity of the resonance. Secondly a magnetic resonance with negative values appears at the right hand side of the second resonance dip at 11.5 GHz. Around the same frequency, the real part of the permittivity is still negative. The extracted real part of the refractive index is therefore negative around 11.5 GHz which is the frequency of the LH peak. However, we can also notice that the zero value for the ε response is very close to 13 GHz where a full transmission band is observed in Fig. 6(a). This frequency constitutes the frequency of the RH transmission peak. We can therefore deduce that this RH transmission peak is due to an impedance matching between the structure and vacuum.

(g) (h)

Fig. 6. Extracted electromagnetic properties of the cut wires metamaterial using the simulated and experimental data of Fig. 4: (a)-(c) real parts, and (d)-(f) imaginary parts of the permittivity ε, of the permeability μ and the refraction index n. The shaded yellow area delineates the frequency region where the measured real parts of ε and μ are simultaneously

negative. Measurement of the wave propagation trough a prism of 6.3°: (g) E field

11.2GHz: the optical index is negative, around -2.3.

6(c) using the retrieval procedure described in [Nicholson, 1970].

**2.3 Stacking of layers** 

cartography at 8.08GHz: the optical index is positive, around 5.6 (h) E field cartography at

Since the real part of n (n') is given by n' = ε'z' - ε''z'' from n = εz and z = √(μ/ε), the imaginary parts of the permittivity (ε'') and the permeability (µ'') also accounts for n'. Therefore, a negative real part of n can be accomplished without having ε' and µ' simultaneously negative. This can happen only if ε'' and µ'' are sufficiently large compared to ε' and µ'. A wider negative n' frequency band is observed due to the dispersion of the fabricated prototype. The shaded yellow area in Fig. 6 highlights the frequency region where the measured real parts of the permittivity (ε') and the permeability (µ') are simultaneously negative to emphasize the desired measured negative values of n'. Concerning the imaginary parts, a very good qualitative agreement is observed between calculations and experiments. We shall note that the imaginary part of n (n'') is very low in the negative n' frequency region. Figures 6(g) and 6(h) show the measurements of the near electric field through a prism with an angle of 6.3°. Figure 6(g) shows the E field cartography at 8.08 GHz: the optical index is positive and found to be around 5.6. Figure 6(h) shows the E field cartography at 11.2 GHz: the optical index is negative and calculated to be around -2.3. These values agree very well with the extracted values of the optical index calculated from the measurements of the reflexion and transmission coefficients of the figure

Stacking multiple layers of LH materials may be useful in many practical applications such as subwavelength imaging [Wang, 2007, Ziolkowski, 2003] and directive antennas [Burokur, 2005, Shelby, 2001]. It is obvious that the effective properties obtained from the inversion method on a monolayer give a good idea about the effective properties of the metamaterial. However, other effect such as inter-layer coupling must be taken into account because it

Fig. 6. Extracted electromagnetic properties of the cut wires metamaterial using the simulated and experimental data of Fig. 4: (a)-(c) real parts, and (d)-(f) imaginary parts of the permittivity ε, of the permeability μ and the refraction index n. The shaded yellow area delineates the frequency region where the measured real parts of ε and μ are simultaneously negative. Measurement of the wave propagation trough a prism of 6.3°: (g) E field cartography at 8.08GHz: the optical index is positive, around 5.6 (h) E field cartography at 11.2GHz: the optical index is negative, around -2.3.

Since the real part of n (n') is given by n' = ε'z' - ε''z'' from n = εz and z = √(μ/ε), the imaginary parts of the permittivity (ε'') and the permeability (µ'') also accounts for n'. Therefore, a negative real part of n can be accomplished without having ε' and µ' simultaneously negative. This can happen only if ε'' and µ'' are sufficiently large compared to ε' and µ'. A wider negative n' frequency band is observed due to the dispersion of the fabricated prototype. The shaded yellow area in Fig. 6 highlights the frequency region where the measured real parts of the permittivity (ε') and the permeability (µ') are simultaneously negative to emphasize the desired measured negative values of n'. Concerning the imaginary parts, a very good qualitative agreement is observed between calculations and experiments. We shall note that the imaginary part of n (n'') is very low in the negative n' frequency region. Figures 6(g) and 6(h) show the measurements of the near electric field through a prism with an angle of 6.3°. Figure 6(g) shows the E field cartography at 8.08 GHz: the optical index is positive and found to be around 5.6. Figure 6(h) shows the E field cartography at 11.2 GHz: the optical index is negative and calculated to be around -2.3. These values agree very well with the extracted values of the optical index calculated from the measurements of the reflexion and transmission coefficients of the figure 6(c) using the retrieval procedure described in [Nicholson, 1970].

#### **2.3 Stacking of layers**

Stacking multiple layers of LH materials may be useful in many practical applications such as subwavelength imaging [Wang, 2007, Ziolkowski, 2003] and directive antennas [Burokur, 2005, Shelby, 2001]. It is obvious that the effective properties obtained from the inversion method on a monolayer give a good idea about the effective properties of the metamaterial. However, other effect such as inter-layer coupling must be taken into account because it

Design and Characterization of Metamaterials for Optical and Radio Communications 279

Fig. 8. Extracted material properties for different number of layers. (a) Re(ε), (b) Re(µ), and

(c) Re(n).

affects the material properties of the structure. Therefore, two, three and four layers of the designed bi-layered metamaterial are stacked with a 1 mm air spacing between each layer as presented in Fig. 7(a). Numerical simulations are run to show the expected performances of a bulk metamaterial composed of multiple layers. The transmission spectra for the different number of layers are presented in Fig. 7(b).

Fig. 7. (a) A bulk metamaterial composed of four layers interleaved with 1 mm air spacing, and (b) transmission spectra for different number of layers.

From the spectra of Fig 7(b), we can note that the frequency of the first transmission dip remains constant with an increasing number of layers while the second dip shifts slightly towards higher frequencies. However, peaks and valleys appear at lower frequencies suggesting a coupling mechanism between consecutive layers. The number of these peaks and valleys increases with an increasing number of layers as shown in Fig. 7(b). The transmission spectra together with the corresponding reflection spectra are used for the extraction of the material properties presented in Fig. 8. It should be noted that the first transmission dip in the single layer case corresponds to an electric resonance where ε' exhibits negative values. However, other ε' < 0 frequency bands can be observed in Fig. 8(a) for the multiple layers cases due to the valleys noted in the transmission spectra. And, since the magnitude of the transmission dips decreases with the number of layers, the magnitude of ε' also decreases as shown in Fig. 8(a). At higher frequencies near 12 GHz, a magnetic resonance is also observed for multiple layers as for the single layer case. However the magnitude tends to decrease while the number of layers increases (Fig. 8(b)). For more than two layers, μ' exhibits only positive values at the resonance near 12 GHz. Besides, another magnetic resonance with μ' < 0 can be observed at lower frequencies with simultaneously ε' < 0 when more than one layer is used. So even if the μ' < 0 frequency band disappears at the second transmission dip due to the μ' > 0, a negative index band is observed at lower frequencies as shown in Fig. 8(c). This negative refractive index results from the coupling mechanism created when several layers of the double-face structure are stacked. The negative index frequency band widens when the number of layers increases.

affects the material properties of the structure. Therefore, two, three and four layers of the designed bi-layered metamaterial are stacked with a 1 mm air spacing between each layer as presented in Fig. 7(a). Numerical simulations are run to show the expected performances of a bulk metamaterial composed of multiple layers. The transmission spectra for the different

Fig. 7. (a) A bulk metamaterial composed of four layers interleaved with 1 mm air spacing,

From the spectra of Fig 7(b), we can note that the frequency of the first transmission dip remains constant with an increasing number of layers while the second dip shifts slightly towards higher frequencies. However, peaks and valleys appear at lower frequencies suggesting a coupling mechanism between consecutive layers. The number of these peaks and valleys increases with an increasing number of layers as shown in Fig. 7(b). The transmission spectra together with the corresponding reflection spectra are used for the extraction of the material properties presented in Fig. 8. It should be noted that the first transmission dip in the single layer case corresponds to an electric resonance where

ε

8(a) for the multiple layers cases due to the valleys noted in the transmission spectra. And, since the magnitude of the transmission dips decreases with the number of layers,

GHz, a magnetic resonance is also observed for multiple layers as for the single layer case. However the magnitude tends to decrease while the number of layers increases (Fig. 8(b)).

μ

index band is observed at lower frequencies as shown in Fig. 8(c). This negative refractive index results from the coupling mechanism created when several layers of the double-face structure are stacked. The negative index frequency band widens when the number of

' also decreases as shown in Fig. 8(a). At higher frequencies near 12

' < 0 when more than one layer is used. So even if the

' exhibits only positive values at the resonance near 12 GHz.

ε'

μ' < 0

' > 0, a negative

' < 0 frequency bands can be observed in Fig.

' < 0 can be observed at lower frequencies

μ

and (b) transmission spectra for different number of layers.

μ

frequency band disappears at the second transmission dip due to the

exhibits negative values. However, other

Besides, another magnetic resonance with

ε

ε

the magnitude of

For more than two layers,

with simultaneously

layers increases.

number of layers are presented in Fig. 7(b).

Fig. 8. Extracted material properties for different number of layers. (a) Re(ε), (b) Re(µ), and (c) Re(n).

Design and Characterization of Metamaterials for Optical and Radio Communications 281

2 2 cos sin *yxz z x* ε μ μ

Figures 9(d)-(f) show that n responses remain mostly unchanged compared to the 0° case

Measured reflection and transmission coefficients are compared to simulated ones in Fig. 10 for the E-plane. Apart from one sharp feature on each spectrum, simulations and measurements agree qualitatively. This peak which is much sharper than either the antisymmetric or the symmetric resonance, shifts with incidence angles from 12.47 GHz at 15° to 10.5 GHz at 30° and 9.22 GHz at 45°. It is the manifestation on the specular order (0,0) of a diffraction threshold, namely that of the (0,-1) diffracted order. At these frequencies the (0,- 1) diffracted order transits from evanescent to propagating, appearing at grazing incidence. Diffraction thresholds frequencies are calculated in two ways, with HFSS and using (2) from grating theory taking into account the fact that there is no wave vector component along x in

<sup>2</sup> <sup>2</sup> 2 2

*k k m n mn Z c c ac a*

=± − =± − − + ∈

Due to respective values of ax = 9.5 mm and ay = 19 mm, it can be seen that the (0,-1) order is the first diffracted order to become propagating. Results for diffraction thresholds frequencies are summarized in Table I. By comparison, in the H-plane even at an angle of 45° the (0,-1) diffracted order is above 18 GHz. To overcome the appearance of a diffraction threshold, a triangular lattice as shown in the inset of Fig. 10(d) is proposed to replace the square one used in the figure 10(a-c). In the triangular lattice, every other cell along x is laterally displaced by 4.75 mm along y. In this lattice, diffraction threshold frequencies are

> ( ) <sup>2</sup> <sup>2</sup> 2 2

*k k m n n mn Z c c ac a*

=± − =± − − − + ∈

Computed reflection and transmission spectra presented in Figs. 10(d)-(f) confirm the rejection of the diffraction threshold above 13 GHz. Besides, a shift in frequency can be noted for both anti-symmetric and symmetric resonances. This shift is seen to be much stronger than for any resonances in the H-plane and the detuning of both electric and magnetic resonances with respect to the incidence angle θ leads to a loss of the frequency overlap, hence the negative index, above 20°. As it can be noted particularly for θ = 15°, the magnetic resonance shifts towards lower frequencies for the triangular lattice (10.43 GHz) compared to the square one (11.02 GHz). This is most probably due to extra capacitive

In summary, we presented the dependence of resonances and retrieved effective index on the incident angle in recently proposed asymmetric cut wire pairs. No change has been observed for oblique incidence in the H-plane. However for the E-plane, a diffraction threshold appears for the square lattice rendering the introduction of effective parameters

2 2 <sup>2</sup> sin , *r r*

 πω

 πω

2 2 sin , *r r*

*x y*

 θ

*x y*

θ

 π

 π

(2)

(3)

θ

(1)

θ + μ

μ

studied in [Burokur, 2009, Sellier, 2009].

2 2 2 //

2 2 2 //

 ω

> ε

coupling between y-displaced wires on same face of the dielectric board.

ω

ε

εε

 ω

ω

⊥

E-plane.

given by (3):

⊥

#### **2.4 Incidence dependence of the negative index**

Finally, we investigate numerically and experimentally the reflection and transmission spectra for an obliquely incident plane wave on the asymmetric structure. Three different angles, namely 15°, 30° and 45° in both H- and E-planes of the square lattice are studied in simulations and experiments. Measured reflection and transmission coefficients are compared to simulated ones in Fig. 9 for the H-plane. There is qualitative agreement between simulations and measurements. Calculated and measured magnitudes of S21 show clearly two resonance dips, an electric one at 9.5 GHz and a magnetic one at 11.5 GHz. These two resonances are found to be independent of the incidence angle in the H-plane as shown in Figs. 9(a)-(c).

Fig. 9. Oblique incidence (15°, 30° and 45°) in H-plane. (a)-(c) Computed and measured reflection and transmission coefficients. (d)-(f) Real part of effective index n.

To retrieve effective parameters at oblique incidence, the retrieval procedure in [Smith, 2002] has to be modified and anisotropy has to be addressed [Burokur, 2009b]. Indeed cut wire pairs represent a biaxial anisotropic media whose principal axis are along x, y and z. Consequently 2x2 transfer matrices used in normal incidence are no longer sufficient and the full 4x4 transfer matrix accounting for coupling of s- and p-waves should be considered [Yeh, 1998]. However, since the electric field in our case is always along y independently of θ, cross-polarization terms do not arise and therefore we only use one 2x2 matrix for swaves. In this case the effective index is given by

Finally, we investigate numerically and experimentally the reflection and transmission spectra for an obliquely incident plane wave on the asymmetric structure. Three different angles, namely 15°, 30° and 45° in both H- and E-planes of the square lattice are studied in simulations and experiments. Measured reflection and transmission coefficients are compared to simulated ones in Fig. 9 for the H-plane. There is qualitative agreement between simulations and measurements. Calculated and measured magnitudes of S21 show clearly two resonance dips, an electric one at 9.5 GHz and a magnetic one at 11.5 GHz. These two resonances are found to be independent of the incidence angle in the H-plane as shown

Fig. 9. Oblique incidence (15°, 30° and 45°) in H-plane. (a)-(c) Computed and measured

To retrieve effective parameters at oblique incidence, the retrieval procedure in [Smith, 2002] has to be modified and anisotropy has to be addressed [Burokur, 2009b]. Indeed cut wire pairs represent a biaxial anisotropic media whose principal axis are along x, y and z. Consequently 2x2 transfer matrices used in normal incidence are no longer sufficient and the full 4x4 transfer matrix accounting for coupling of s- and p-waves should be considered [Yeh, 1998]. However, since the electric field in our case is always along y independently of θ, cross-polarization terms do not arise and therefore we only use one 2x2 matrix for s-

reflection and transmission coefficients. (d)-(f) Real part of effective index n.

waves. In this case the effective index is given by

**2.4 Incidence dependence of the negative index** 

in Figs. 9(a)-(c).

$$\frac{\mathfrak{e}\_y \mu\_x \mu\_z}{\mu\_z \cos^2 \theta + \mu\_x \sin^2 \theta} \tag{1}$$

Figures 9(d)-(f) show that n responses remain mostly unchanged compared to the 0° case studied in [Burokur, 2009, Sellier, 2009].

Measured reflection and transmission coefficients are compared to simulated ones in Fig. 10 for the E-plane. Apart from one sharp feature on each spectrum, simulations and measurements agree qualitatively. This peak which is much sharper than either the antisymmetric or the symmetric resonance, shifts with incidence angles from 12.47 GHz at 15° to 10.5 GHz at 30° and 9.22 GHz at 45°. It is the manifestation on the specular order (0,0) of a diffraction threshold, namely that of the (0,-1) diffracted order. At these frequencies the (0,- 1) diffracted order transits from evanescent to propagating, appearing at grazing incidence. Diffraction thresholds frequencies are calculated in two ways, with HFSS and using (2) from grating theory taking into account the fact that there is no wave vector component along x in E-plane.

$$k\_{\perp} = \pm \sqrt{\frac{\alpha^2}{c^2} \varepsilon\_r - k\_{\perp \prime}^2} = \pm \sqrt{\frac{\alpha^2}{c^2} \varepsilon\_r - \left( m \frac{2\pi}{a\_x} \right)^2} - \left( \frac{\alpha}{c} \sin \theta + n \frac{2\pi}{a\_y} \right)^2 \quad m, n \in \mathbb{Z} \tag{2}$$

Due to respective values of ax = 9.5 mm and ay = 19 mm, it can be seen that the (0,-1) order is the first diffracted order to become propagating. Results for diffraction thresholds frequencies are summarized in Table I. By comparison, in the H-plane even at an angle of 45° the (0,-1) diffracted order is above 18 GHz. To overcome the appearance of a diffraction threshold, a triangular lattice as shown in the inset of Fig. 10(d) is proposed to replace the square one used in the figure 10(a-c). In the triangular lattice, every other cell along x is laterally displaced by 4.75 mm along y. In this lattice, diffraction threshold frequencies are given by (3):

$$k\_{\perp} = \pm \sqrt{\frac{\alpha^2}{c^2} \varepsilon\_r - k\_{\perp \prime}^2} = \pm \sqrt{\frac{\alpha^2}{c^2} \varepsilon\_r - \left( (2m - n) \frac{2\pi}{a\_x} \right)^2} - \left( \frac{\alpha}{c} \sin \theta + n \frac{2\pi}{a\_y} \right)^2 \quad m, n \in \mathbb{Z} \tag{3}$$

Computed reflection and transmission spectra presented in Figs. 10(d)-(f) confirm the rejection of the diffraction threshold above 13 GHz. Besides, a shift in frequency can be noted for both anti-symmetric and symmetric resonances. This shift is seen to be much stronger than for any resonances in the H-plane and the detuning of both electric and magnetic resonances with respect to the incidence angle θ leads to a loss of the frequency overlap, hence the negative index, above 20°. As it can be noted particularly for θ = 15°, the magnetic resonance shifts towards lower frequencies for the triangular lattice (10.43 GHz) compared to the square one (11.02 GHz). This is most probably due to extra capacitive coupling between y-displaced wires on same face of the dielectric board.

In summary, we presented the dependence of resonances and retrieved effective index on the incident angle in recently proposed asymmetric cut wire pairs. No change has been observed for oblique incidence in the H-plane. However for the E-plane, a diffraction threshold appears for the square lattice rendering the introduction of effective parameters

Design and Characterization of Metamaterials for Optical and Radio Communications 283

Recently, a theoretical study showed that optical resonances in both SRR arrays and cut wire arrays could be interpreted in terms of plasmon resonances [Rockstuhl, 2006]. A gradual shift in the SRR resonance frequencies was predicted when reducing the length of SRR legs to the point where each SRR was transformed into a single wire piece. The magnetic and electric properties of these modified versions of SRRs were theoretically investigated in [Zhou, 2007]. Following these theoretical studies, we present here an experimental and numerical analysis of the infrared response of metamaterials made of continuous nanowires and split ring resonators where the geometry is gradually altered. The metamaterial structure is fabricated on low-doped silicon. The impact of the geometric transformation of the SRRs on the spectra of the composite metamaterial is measured in the 20-400 THz frequency range (i.e., in the 1.5 - 15 μm wavelength range) for the two field polarizations under normal to plane propagation. Coupling effects between the SRRs and the continuous nanowires are analyzed for different spacings between them. The results of our study are expected to provide useful guidelines for the design and engineering of negative index

**3.1.1 Design, fabrication, characterization and modeling of metamaterial structures**  Four structures consisting of a two-dimensional periodic array of gold nanowires and gold SRRs were fabricated on a 280μm thick silicon substrate (Fig. 11). The fabrication steps included e-beam lithography, high vacuum electron beam evaporation of 5 nm thick titanium and 40 nm thick gold films, and a lift-off process. It is worthwhile noticing that all structures were fabricated in the same run, thereby allowing a meaningful comparison of their optical characteristics. As seen in Fig. 11, the four structures only differ in the shape of SRRs, which are gradually transformed into simple cut wires from structure 1 to structure 4. In the intermediate cases of structures 2 and 3, SRRs appear to be U-shaped with smaller legs than in the standard case of structure 1. Except for this resonator shape, all the other geometrical parameters of the four structures are identical. In each case, the lattice period is ~ 600nm, the width of all wires (continuous and discontinuous) is ~ 50nm, and the continuous wires are parallel to the SRR bases with a separation of 115 ± 20 nm between each continuous wire and the closest SRR base. The SRR gap width in structure 1 is ~100 nm while the length of the two SRR legs is ~ 280nm. This length is reduced to ~190 and 110 nm in structures 2 and 3, respectively. The scanning electron microscope (SEM) images reported in Fig. 11 (middle row) illustrate the excellent regularity of the four fabricated structures.

The transmission and reflection spectra of the fabricated structures were measured under normal-to-plane incidence with a FTIR (Fourier Transformed InfraRed spectrometer) BioRad FTS 60 equipped with a Cassegrain microscope. The FTIR beam was polarized using a KRS5 polarizer adapted to the wavelength region from ~ 1.5 to 15μm (i.e. to frequencies varying from 20 to 200 THz). A diaphragm was used in such a way as to produce a light spot smaller than 100 × 100 μm2 onto the sample (i.e. smaller than the surface of each periodic structure). Measurements were performed for two field polarizations of the incident beam, the parallel polarization with the illuminating electric field parallel to both the continuous wires and the SRR gaps and the perpendicular polarization with the electric field perpendicular to the

**3. Infrared metamaterials and plasmons hybridization** 

**3.1 Engineering resonances in infrared metamaterials** 

metamaterials on silicon.

meaningless. A triangular lattice has therefore been proposed to avoid having the diffraction threshold below the electric and magnetic resonances. A detuning of both resonances has been observed leading to a lack of resonance frequencies overlap above 20°.

Fig. 10. Oblique incidence (15°, 30° and 45°) in E-plane. (a)-(c) Computed and measured reflection and transmission coefficients for the square lattice. (d)-(f) Computed and measured reflection and transmission coefficients for the triangular lattice.


Table 1. Computed numerical and analytical diffraction threshold frequencies (GHz)

#### **3. Infrared metamaterials and plasmons hybridization**

#### **3.1 Engineering resonances in infrared metamaterials**

282 Metamaterial

meaningless. A triangular lattice has therefore been proposed to avoid having the diffraction threshold below the electric and magnetic resonances. A detuning of both resonances has

Fig. 10. Oblique incidence (15°, 30° and 45°) in E-plane. (a)-(c) Computed and measured reflection and transmission coefficients for the square lattice. (d)-(f) Computed and

= 0° 15.78 15.79 22.31 22.33

= 15° 12.58 12.54 19.15 19.15

= 30° 10.54 10.53 17.31 17.32

= 45° 9.24 9.25 16.35 16.35

Table 1. Computed numerical and analytical diffraction threshold frequencies (GHz)

HFSS (triangular lattice) Mode (0,-1)

Grating (triangular lattice) Mode (0,-1)

measured reflection and transmission coefficients for the triangular lattice.

Grating (square lattice) Mode (0,-1)

HFSS (square lattice) Mode (0,-1)

θ

θ

θ

θ

been observed leading to a lack of resonance frequencies overlap above 20°.

Recently, a theoretical study showed that optical resonances in both SRR arrays and cut wire arrays could be interpreted in terms of plasmon resonances [Rockstuhl, 2006]. A gradual shift in the SRR resonance frequencies was predicted when reducing the length of SRR legs to the point where each SRR was transformed into a single wire piece. The magnetic and electric properties of these modified versions of SRRs were theoretically investigated in [Zhou, 2007]. Following these theoretical studies, we present here an experimental and numerical analysis of the infrared response of metamaterials made of continuous nanowires and split ring resonators where the geometry is gradually altered. The metamaterial structure is fabricated on low-doped silicon. The impact of the geometric transformation of the SRRs on the spectra of the composite metamaterial is measured in the 20-400 THz frequency range (i.e., in the 1.5 - 15 μm wavelength range) for the two field polarizations under normal to plane propagation. Coupling effects between the SRRs and the continuous nanowires are analyzed for different spacings between them. The results of our study are expected to provide useful guidelines for the design and engineering of negative index metamaterials on silicon.

#### **3.1.1 Design, fabrication, characterization and modeling of metamaterial structures**

Four structures consisting of a two-dimensional periodic array of gold nanowires and gold SRRs were fabricated on a 280μm thick silicon substrate (Fig. 11). The fabrication steps included e-beam lithography, high vacuum electron beam evaporation of 5 nm thick titanium and 40 nm thick gold films, and a lift-off process. It is worthwhile noticing that all structures were fabricated in the same run, thereby allowing a meaningful comparison of their optical characteristics. As seen in Fig. 11, the four structures only differ in the shape of SRRs, which are gradually transformed into simple cut wires from structure 1 to structure 4. In the intermediate cases of structures 2 and 3, SRRs appear to be U-shaped with smaller legs than in the standard case of structure 1. Except for this resonator shape, all the other geometrical parameters of the four structures are identical. In each case, the lattice period is ~ 600nm, the width of all wires (continuous and discontinuous) is ~ 50nm, and the continuous wires are parallel to the SRR bases with a separation of 115 ± 20 nm between each continuous wire and the closest SRR base. The SRR gap width in structure 1 is ~100 nm while the length of the two SRR legs is ~ 280nm. This length is reduced to ~190 and 110 nm in structures 2 and 3, respectively. The scanning electron microscope (SEM) images reported in Fig. 11 (middle row) illustrate the excellent regularity of the four fabricated structures.

The transmission and reflection spectra of the fabricated structures were measured under normal-to-plane incidence with a FTIR (Fourier Transformed InfraRed spectrometer) BioRad FTS 60 equipped with a Cassegrain microscope. The FTIR beam was polarized using a KRS5 polarizer adapted to the wavelength region from ~ 1.5 to 15μm (i.e. to frequencies varying from 20 to 200 THz). A diaphragm was used in such a way as to produce a light spot smaller than 100 × 100 μm2 onto the sample (i.e. smaller than the surface of each periodic structure). Measurements were performed for two field polarizations of the incident beam, the parallel polarization with the illuminating electric field parallel to both the continuous wires and the SRR gaps and the perpendicular polarization with the electric field perpendicular to the

Design and Characterization of Metamaterials for Optical and Radio Communications 285

= 6.478 x 1013 s-1 (fc =10.3 THz). Actually, the collision frequency can be considered to a certain extent as a fit parameter. An increase of the collision frequency results in higher absorption losses of the structures, while it does not change the spectral positions of resonances. The value reported above for ωc is 2.6 times larger than in bulk gold. This increase is supposed to account for additional scattering experienced by electrons at the

The results of our measurements and simulations are shown in Fig. 12 for the four structures. Results for the parallel polarization are gathered in the series of figures from (a) to (d). Those for the perpendicular polarization are gathered in the series of figures from (e) to (h). In all these figures, resonances manifest themselves as reflection maxima correlated with transmission minima. A slow decrease (resp. increase) of the transmission level (resp. reflection level) is also observed at low frequencies for the parallel polarization. This latter evolution can be readily attributed to the plasmon-like band associated to the periodic array

For the parallel polarization and the structures with the SRRs of larger sizes (structures 1 and 2), two resonances are observed within the spectral window of measurements [Figs. 12(a) and 12(b)]. Only the first resonance is experimentally observed for structures 3 and 4. The SRR resonances actually shift towards higher and higher frequencies as the whole SRR length is decreased. In the same time, their amplitude becomes smaller and smaller. For the first resonance, the maximum reflection Rmax decreases from ~0.63 to ~0.5 while the minimum reflection Rmin remains close to ~0.35 [Fig. 11(b)]. Accordingly, the minimum transmission Tmin increases from ~0.12 to ~0.25 while the maximum transmission Tmax remains close to ~0.6 [Fig. 11(a)]. It is worthwhile noticing that the values of R and T out of resonance correspond to those expected for a single face of silicon wafer partially covered with ~10-15% of highly reflecting metal. Whereas (R+T) approaches unity in this case (Rmin+Tmax~0.95), its smaller value at resonance (Rmax+Tmin~0.75) clearly indicates the presence of dissipative losses in metallic elements. The frequency positions of resonances, the values of R and T out of resonance, the values of (R+T) in general as well as the shapes of experimental curves in Figs. 12(a) and 12(b) are very well reproduced by numerical simulations [Figs. 12(c) and 12(d)]. The only discrepancy between experiments and simulations stems from the smaller amplitudes of resonances measured in experiments, especially those at high frequencies. The second resonance predicted at ~180 THz (λ ~ 1.7μm) for structure 3 is even not resolved in the experiments. Actually, minute deviations of the geometry from unit cell to unit cell and particularly residual surface roughness of the SRRs and continuous wires can explain the damping and inhomogeneous broadening of

resonances as well as the increasing importance of these effects at high frequencies.

For the perpendicular polarization and for structures 1, 2 and 3, a single resonance is observed within the spectral range of measurements [Figs. 12(e) and 12(f)]. In contrast, no resonance is detected for the structure with cut wires. The evolution of the SRR resonance with the SRR length is actually similar to that observed for the parallel polarization. Smaller SRR lengths simultaneously lead to higher resonance frequencies and smaller resonance amplitudes. A good agreement is found between experimental results and numerical simulations [Figs. 12(g) and 12(h)]. Previous remarks made for the parallel polarization

metal surfaces.

**3.1.2 Results of measurements and simulations** 

of continuous wires [Pendry, 1998].

continuous wires and SRR gaps. The measured transmission spectra were normalized versus the transmission of an unprocessed part of the silicon substrate. The measured reflection spectra were normalized versus the reflection of a 40 nm thick gold film deposited on silicon.


Fig. 11. Schematic representations (top) and scanning electron microscope images (middle and bottom) of the four metamaterial structures fabricated on silicon. The top and bottom pictures show the elementary unit cell of each structure. From left to right, the length of SRR legs is reduced to the point where the SRR is transformed into a single wire piece. The middle pictures show the regularity achieved in the fabrication of the periodic arrays of nanowires and split ring resonators

Numerical simulations of the spectral responses of the four structures were performed with a finite element software (HFSS, 2006). Periodic boundary conditions were applied to the lateral sides of the elementary lattice cell (Fig. 11). The silicon substrate was assumed to be lossless with a constant permittivity equal to 11.9. A Drude model was used to simulate the permittivity and loss tangent of the gold wires:

$$\varepsilon\left(\phi\right) = 1 - \frac{\alpha\_p^2}{o\left(o + io\_c\right)}\tag{4}$$

where ωp and ωc are the plasma and collision frequency of the gold film, respectively. The values of ωp and ωc chosen in the simulations were: ωp =1.367 x 1016s-1 (fp =2176 THz) and ω<sup>c</sup>

= 6.478 x 1013 s-1 (fc =10.3 THz). Actually, the collision frequency can be considered to a certain extent as a fit parameter. An increase of the collision frequency results in higher absorption losses of the structures, while it does not change the spectral positions of resonances. The value reported above for ωc is 2.6 times larger than in bulk gold. This increase is supposed to account for additional scattering experienced by electrons at the metal surfaces.

#### **3.1.2 Results of measurements and simulations**

284 Metamaterial

continuous wires and SRR gaps. The measured transmission spectra were normalized versus the transmission of an unprocessed part of the silicon substrate. The measured reflection spectra were normalized versus the reflection of a 40 nm thick gold film deposited

Fig. 11. Schematic representations (top) and scanning electron microscope images (middle and bottom) of the four metamaterial structures fabricated on silicon. The top and bottom pictures show the elementary unit cell of each structure. From left to right, the length of SRR legs is reduced to the point where the SRR is transformed into a single wire piece. The middle pictures show the regularity achieved in the fabrication of the periodic arrays of

Numerical simulations of the spectral responses of the four structures were performed with a finite element software (HFSS, 2006). Periodic boundary conditions were applied to the lateral sides of the elementary lattice cell (Fig. 11). The silicon substrate was assumed to be lossless with a constant permittivity equal to 11.9. A Drude model was used to simulate the

( ) ( )

where ωp and ωc are the plasma and collision frequency of the gold film, respectively. The values of ωp and ωc chosen in the simulations were: ωp =1.367 x 1016s-1 (fp =2176 THz) and ω<sup>c</sup>

ε ω 1 *<sup>p</sup>*

ωω

2

ω

*c i*

= − <sup>+</sup> (4)

 ω

nanowires and split ring resonators

permittivity and loss tangent of the gold wires:

on silicon.

The results of our measurements and simulations are shown in Fig. 12 for the four structures. Results for the parallel polarization are gathered in the series of figures from (a) to (d). Those for the perpendicular polarization are gathered in the series of figures from (e) to (h). In all these figures, resonances manifest themselves as reflection maxima correlated with transmission minima. A slow decrease (resp. increase) of the transmission level (resp. reflection level) is also observed at low frequencies for the parallel polarization. This latter evolution can be readily attributed to the plasmon-like band associated to the periodic array of continuous wires [Pendry, 1998].

For the parallel polarization and the structures with the SRRs of larger sizes (structures 1 and 2), two resonances are observed within the spectral window of measurements [Figs. 12(a) and 12(b)]. Only the first resonance is experimentally observed for structures 3 and 4. The SRR resonances actually shift towards higher and higher frequencies as the whole SRR length is decreased. In the same time, their amplitude becomes smaller and smaller. For the first resonance, the maximum reflection Rmax decreases from ~0.63 to ~0.5 while the minimum reflection Rmin remains close to ~0.35 [Fig. 11(b)]. Accordingly, the minimum transmission Tmin increases from ~0.12 to ~0.25 while the maximum transmission Tmax remains close to ~0.6 [Fig. 11(a)]. It is worthwhile noticing that the values of R and T out of resonance correspond to those expected for a single face of silicon wafer partially covered with ~10-15% of highly reflecting metal. Whereas (R+T) approaches unity in this case (Rmin+Tmax~0.95), its smaller value at resonance (Rmax+Tmin~0.75) clearly indicates the presence of dissipative losses in metallic elements. The frequency positions of resonances, the values of R and T out of resonance, the values of (R+T) in general as well as the shapes of experimental curves in Figs. 12(a) and 12(b) are very well reproduced by numerical simulations [Figs. 12(c) and 12(d)]. The only discrepancy between experiments and simulations stems from the smaller amplitudes of resonances measured in experiments, especially those at high frequencies. The second resonance predicted at ~180 THz (λ ~ 1.7μm) for structure 3 is even not resolved in the experiments. Actually, minute deviations of the geometry from unit cell to unit cell and particularly residual surface roughness of the SRRs and continuous wires can explain the damping and inhomogeneous broadening of resonances as well as the increasing importance of these effects at high frequencies.

For the perpendicular polarization and for structures 1, 2 and 3, a single resonance is observed within the spectral range of measurements [Figs. 12(e) and 12(f)]. In contrast, no resonance is detected for the structure with cut wires. The evolution of the SRR resonance with the SRR length is actually similar to that observed for the parallel polarization. Smaller SRR lengths simultaneously lead to higher resonance frequencies and smaller resonance amplitudes. A good agreement is found between experimental results and numerical simulations [Figs. 12(g) and 12(h)]. Previous remarks made for the parallel polarization

Design and Characterization of Metamaterials for Optical and Radio Communications 287

Fig. 12. Measured and simulated transmission/reflection spectra of the four structures depicted in Fig. 10. Curves in red, blue, green and black are for the 1st, 2nd, 3rd and 4th structures, respectively. – (a) and (b): transmission and reflection spectra measured for the parallel polarization (incident electric field parallel to the SRR gap). – (c) and (d): numerical simulations corresponding to (a) and (b), respectively. – (e) and (f): transmission and reflection spectra measured for the perpendicular polarization (incident electric field

perpendicular to the SRR gap). – (g) and (h): numerical simulations corresponding to (e) and

At this stage, it is interesting to compare the frequency positions of resonances reported in Figs. 12 to those reported in previous works for similar structures with gold SRRs on glass substrate. For instance, the first resonance calculated in [Rockstuhl, 2006] for the parallel polarization and U–shaped SRRs with 400 nm long base and 190 nm long legs was found to be close to 3800 cm–1, i.e. close to 115 THz instead of 65 THz measured in our experiments for U-shaped SRRs with similar sizes [Figs. 12(a), 12(b), and Fig. 13(b)]. The first resonance calculated in [15] for the parallel polarization and U-shaped SRRs with the same base but with 110 nm legs was found to be near 4800 cm–1, i.e. near 144 THz instead of 80 THz measured in our experiments [Figs. 12(a), 12(b) and Fig. 13(c)]. Actually, all the mode frequencies calculated in [15] are 1.7- 1.9 times higher than those reported in this work, whatever the resonance order and the field polarization are. Approximately the same ratio is obtained when comparing the first resonance measured in [10] for SRRs of standard shape

(f), respectively.

**3.1.3 Frequency positions of resonances** 

apply to the perpendicular polarization. Minute deviations from unit cell to unit cell and surface roughness of metallic elements are likely to explain the broader resonances with smaller amplitudes observed in the experiments. The second resonances predicted for structures 1, 2 and 3 in the frequency region from 170 to 200 THz [Figs. 12(g) and 12(h) manifest themselves only as smooth maxima (resp. minima) in the measured reflection (resp. transmission) spectra of Fig. 12(f) (resp. Fig. 12(e)]. Supplementary measurements between 200 and 250 THz (not shown here) did not reveal any other resonance.

Figure 13 shows the distribution of the electric field calculated at the bottom surface of metallic elements for each of the resonant modes observed in Fig. 12. These results are in agreement with previous calculations reported by Rockstuhl et al. [Rockstuhl, 2006]. When the incident field is polarized parallel to the SRR gap, resonant modes possess an odd number of nodes along the entire SRR. This number is equal to one for the first resonance, while it is equal to three for the second resonance. The first resonance also identifies to the so-called LC resonance as defined in previous studies at microwave and far infrared frequencies [Katsarakis, 2004, Katsarakis, 2005]. It simply identifies to the dipolar mode in the case of the cut wire. When the incident field is polarized perpendicular to the gap, resonant plasmon modes possess an even number of nodes. The first resonant mode in this polarization thus exhibits one additional node as compared to the first resonance in the parallel polarization. This in turn requires higher energies of the light field to excite this mode. The frequency of the first resonant mode in the perpendicular polarization is typically two times higher than that of the first resonant mode in the parallel polarization.

apply to the perpendicular polarization. Minute deviations from unit cell to unit cell and surface roughness of metallic elements are likely to explain the broader resonances with smaller amplitudes observed in the experiments. The second resonances predicted for structures 1, 2 and 3 in the frequency region from 170 to 200 THz [Figs. 12(g) and 12(h) manifest themselves only as smooth maxima (resp. minima) in the measured reflection (resp. transmission) spectra of Fig. 12(f) (resp. Fig. 12(e)]. Supplementary measurements

Figure 13 shows the distribution of the electric field calculated at the bottom surface of metallic elements for each of the resonant modes observed in Fig. 12. These results are in agreement with previous calculations reported by Rockstuhl et al. [Rockstuhl, 2006]. When the incident field is polarized parallel to the SRR gap, resonant modes possess an odd number of nodes along the entire SRR. This number is equal to one for the first resonance, while it is equal to three for the second resonance. The first resonance also identifies to the so-called LC resonance as defined in previous studies at microwave and far infrared frequencies [Katsarakis, 2004, Katsarakis, 2005]. It simply identifies to the dipolar mode in the case of the cut wire. When the incident field is polarized perpendicular to the gap, resonant plasmon modes possess an even number of nodes. The first resonant mode in this polarization thus exhibits one additional node as compared to the first resonance in the parallel polarization. This in turn requires higher energies of the light field to excite this mode. The frequency of the first resonant mode in the perpendicular polarization is typically two times higher than that of the first resonant mode in the parallel polarization.

between 200 and 250 THz (not shown here) did not reveal any other resonance.

Fig. 12. Measured and simulated transmission/reflection spectra of the four structures depicted in Fig. 10. Curves in red, blue, green and black are for the 1st, 2nd, 3rd and 4th structures, respectively. – (a) and (b): transmission and reflection spectra measured for the parallel polarization (incident electric field parallel to the SRR gap). – (c) and (d): numerical simulations corresponding to (a) and (b), respectively. – (e) and (f): transmission and reflection spectra measured for the perpendicular polarization (incident electric field perpendicular to the SRR gap). – (g) and (h): numerical simulations corresponding to (e) and (f), respectively.

#### **3.1.3 Frequency positions of resonances**

At this stage, it is interesting to compare the frequency positions of resonances reported in Figs. 12 to those reported in previous works for similar structures with gold SRRs on glass substrate. For instance, the first resonance calculated in [Rockstuhl, 2006] for the parallel polarization and U–shaped SRRs with 400 nm long base and 190 nm long legs was found to be close to 3800 cm–1, i.e. close to 115 THz instead of 65 THz measured in our experiments for U-shaped SRRs with similar sizes [Figs. 12(a), 12(b), and Fig. 13(b)]. The first resonance calculated in [15] for the parallel polarization and U-shaped SRRs with the same base but with 110 nm legs was found to be near 4800 cm–1, i.e. near 144 THz instead of 80 THz measured in our experiments [Figs. 12(a), 12(b) and Fig. 13(c)]. Actually, all the mode frequencies calculated in [15] are 1.7- 1.9 times higher than those reported in this work, whatever the resonance order and the field polarization are. Approximately the same ratio is obtained when comparing the first resonance measured in [10] for SRRs of standard shape

Design and Characterization of Metamaterials for Optical and Radio Communications 289

Fig. 13. Magnitude of the normal electric field component (|Ez|) calculated at the bottom surface of the metallic elements for each of the resonant plasmon modes observed in the different spectra of Fig. 11. The different colours, from blue to red correspond to increasing magnitudes of the field component. Modes are classified according to the resonance order and to the polarization of the incident electric field. As expected, both the energy and the

Calculations above were repeated for structures with U-shaped SRRs as structures 2 and 3 and for structures with cut wires as structure 4 (Fig. 11). The same behavior was found in the case of structures with U-shaped SRRs. The frequency position of the first resonance was not modified when the distance between SRRs and continuous wires was varied. The second resonance split into two components for small values of d. In contrast, the structures with cut wires did not exhibit the same "robustness" of the first resonance against coupling

number of field nodes increase with the resonance order.

effects.

(~100 THz) to that reported in Fig. 12(a) (~60 THz) for standard SRRs with the same total length (lm ~960 nm). This ratio is actually of the same order of magnitude than the ratio between the refractive index of glass and that of silicon. This suggests that in our case the electromagnetic field at resonance largely extends into the silicon substrate.

#### **3.1.4 Coupling effects between continuous wires and SRRs**

In the previous sections, it has been implicitly assumed that the presence of continuous wires had no influence on the resonant response of the structures except for a slow decrease of the transmission observed at low frequencies for the parallel polarization. However, a careful examination of the field distributions in Fig. 13 indicates that, at least for the second resonance, the electromagnetic field extends well in the region comprised between the SRR and the closest wire. We thus performed a numerical analysis to investigate in more detail the possible existence of coupling effects between the SRRs and continuous wires. For this purpose, the spectral responses of the four structures were calculated for different distances d between the SRR base and the closest wire. They were also compared to the spectral response of a periodic array of SRRs only.

Results of our calculations are shown in Figs. 14(a) and 14(b) for structure 1 with three values of d, for a periodic array of SRRs without continuous wires and for a periodic array of wires without SRRs. The dimensions of SRRs are the same for the first four structures. The lattice period is identical for all the structures. Calculations are performed within the same spectral range as in Fig. 12, and the incident electric field is polarized parallel to the continuous wires and/or to the SRR gaps. Figure 14(a) represents the transmission spectra calculated for the different structures. Figure 14(b) shows the electric field distributions calculated for the different resonances observed in Fig. 14(a). As seen in Fig. 14(a), the position of the first SRR resonance is not modified by the presence of the continuous wires whatever the separation between SRRs and wires is. Only the shape of the resonance is modified, and it becomes asymmetric with the presence of the wires.

This asymmetry mainly results from the fact that the optical response of the wires [pink dashed curve in Fig. 14(a)] adds to that of the SRRs. A weak coupling between SRRs and wires only occurs at the smallest separations between the two metallic elements as shown from the calculated distribution of the electric field at the first SRR resonance (Fig. 14(b), second column).

The evolution of the second SRR resonance is quite different. For a sufficiently large separation between the SRRs and wires (d ≥ 100 nm), the frequency position of this second resonance is still rather independent of the presence of the wires. This justifies our previous interpretations concerning the results of Fig. 12, where the different spectra were obtained for d ≈ 130 nm. However, for small separations between the SRRs and wires, strong coupling effects exist, which lead to a splitting of the second SRR resonance into two components [Fig. 14(a)]. These components are well separated for the smallest value of d (black dashed curve in Fig. 14(a), d = 10 nm). As seen in Fig. 14(b) (second and third columns), the modal field of the low frequency component is found to be concentrated in the region between the SRR base and the closest wire. That of the high frequency component is rather concentrated in the SRR legs.

(~100 THz) to that reported in Fig. 12(a) (~60 THz) for standard SRRs with the same total length (lm ~960 nm). This ratio is actually of the same order of magnitude than the ratio between the refractive index of glass and that of silicon. This suggests that in our case the

In the previous sections, it has been implicitly assumed that the presence of continuous wires had no influence on the resonant response of the structures except for a slow decrease of the transmission observed at low frequencies for the parallel polarization. However, a careful examination of the field distributions in Fig. 13 indicates that, at least for the second resonance, the electromagnetic field extends well in the region comprised between the SRR and the closest wire. We thus performed a numerical analysis to investigate in more detail the possible existence of coupling effects between the SRRs and continuous wires. For this purpose, the spectral responses of the four structures were calculated for different distances d between the SRR base and the closest wire. They were also compared to the spectral

Results of our calculations are shown in Figs. 14(a) and 14(b) for structure 1 with three values of d, for a periodic array of SRRs without continuous wires and for a periodic array of wires without SRRs. The dimensions of SRRs are the same for the first four structures. The lattice period is identical for all the structures. Calculations are performed within the same spectral range as in Fig. 12, and the incident electric field is polarized parallel to the continuous wires and/or to the SRR gaps. Figure 14(a) represents the transmission spectra calculated for the different structures. Figure 14(b) shows the electric field distributions calculated for the different resonances observed in Fig. 14(a). As seen in Fig. 14(a), the position of the first SRR resonance is not modified by the presence of the continuous wires whatever the separation between SRRs and wires is. Only the shape of the resonance is

This asymmetry mainly results from the fact that the optical response of the wires [pink dashed curve in Fig. 14(a)] adds to that of the SRRs. A weak coupling between SRRs and wires only occurs at the smallest separations between the two metallic elements as shown from the calculated distribution of the electric field at the first SRR resonance (Fig. 14(b),

The evolution of the second SRR resonance is quite different. For a sufficiently large separation between the SRRs and wires (d ≥ 100 nm), the frequency position of this second resonance is still rather independent of the presence of the wires. This justifies our previous interpretations concerning the results of Fig. 12, where the different spectra were obtained for d ≈ 130 nm. However, for small separations between the SRRs and wires, strong coupling effects exist, which lead to a splitting of the second SRR resonance into two components [Fig. 14(a)]. These components are well separated for the smallest value of d (black dashed curve in Fig. 14(a), d = 10 nm). As seen in Fig. 14(b) (second and third columns), the modal field of the low frequency component is found to be concentrated in the region between the SRR base and the closest wire. That of the high frequency component

electromagnetic field at resonance largely extends into the silicon substrate.

**3.1.4 Coupling effects between continuous wires and SRRs** 

modified, and it becomes asymmetric with the presence of the wires.

response of a periodic array of SRRs only.

second column).

is rather concentrated in the SRR legs.

Fig. 13. Magnitude of the normal electric field component (|Ez|) calculated at the bottom surface of the metallic elements for each of the resonant plasmon modes observed in the different spectra of Fig. 11. The different colours, from blue to red correspond to increasing magnitudes of the field component. Modes are classified according to the resonance order and to the polarization of the incident electric field. As expected, both the energy and the number of field nodes increase with the resonance order.

Calculations above were repeated for structures with U-shaped SRRs as structures 2 and 3 and for structures with cut wires as structure 4 (Fig. 11). The same behavior was found in the case of structures with U-shaped SRRs. The frequency position of the first resonance was not modified when the distance between SRRs and continuous wires was varied. The second resonance split into two components for small values of d. In contrast, the structures with cut wires did not exhibit the same "robustness" of the first resonance against coupling effects.

Design and Characterization of Metamaterials for Optical and Radio Communications 291

In the previous structures, SRRs were associated with continuous wires to obtain negative index at infrared wavelengths. In this chapter we will study asymmetric cut-wire pairs to obtain also negative index. Metallic nanostructures can be regarded as elementary circuits including nano-capacitors, inductors or resistors [Engheta, 2007]. The simplest resonator that can be imagined is a dipole consisting of a simple metallic cut-wire. Coupling two such oscillators lead to two eigenmodes with opposite symmetry. The virtual current loop of the anti- symmetric mode is now recognized as a mean to create artificial magnetism at optical frequencies [Grigorenko, 2005, Shalaev, 2005]. Fig. 15 shows the structure under consideration. It consists of a periodic array of paired cut-wires separated by a dielectric layer. For simplicity the spacer has been taken to be silicon dioxide (SiO2) with a dielectric permittivity εr = 2.25 and a thickness of 100 nm. The surrounding medium is air with εr = 1. The relevant polarization of the impinging light is given in Fig. 15(a) with the electric field parallel to the longest side of the cut- wires. Transmission spectra are presented in Fig. 15(b). All the simulations are done using a commercial finite element code (HFSS, 2006). An array of isolated cut-wires is actually found when the separation distance between cut-wires is large: only one resonance (the dipolar mode) is observed in this case (red curve in Fig. 15(b)). This response can be interpreted in terms of a localized plasmon resonance [Rockstuhl, 2006, Kante, 2008]. When the separation distance between cut-wire pairs is progressively diminished, the response of the paired system is modified due to the

Following the plasmon hybridization concept [Kreibig, 1981, Prodan, 2003], coupling effects lift the degeneracy of the single cut-wire mode, thus leading to two distinct plasmons modes [Liu, 2007, Kante 2009], the anti-symmetric and symmetric modes as shown in Fig. 15(b) (blue curve). The symmetric mode with larger restoring force is at a higher energy than the anti-symmetric mode. The main idea of our work is to invert this process. The symmetric mode, which easily couples to incident light, corresponds to a wide rejection band with negative permittivity. The anti-symmetric mode, which is difficult to excite due to the opposite contributions of the two dipoles, manifests itself as a small transmission dip with negative permeability. Achieving such a resonance at a higher frequency than the symmetric mode will ease the overlap condition as proposed recently in [Kante, 2009, Sellier, 2009]. For this purpose, one solution is to break the symmetry of the cut-wire pair. This is achieved in Fig. 15(d) by displacing one of two cut-

Fig. 16(a) and (b) shows the evolutions of the resonant modes and transmission spectra with the longitudinal shift dx between the two cut-wires of each pair. Simulations are performed in the infrared domain. As a major result, for a sufficiently large displacement dx (here dx > 350 nm), the hybridization scheme is inverted with the symmetric mode at a lower frequency than the anti-symmetric mode. The two hybridization schemes are presented in Fig. 15(c) and (e), respectively. Fig. 16(d) clearly shows that a negative refraction regime is obtained (between 145 and 160 THz for dx = 600 nm) for the inverted scheme while the index of refraction remains always positive (dx = 0 nm) for the normal hybridization case (Fig. 16(c)). This extends to infrared frequencies previous results reported by the authors at

**3.2.1 Negative refractive index in optical asymmetric cut-wire pairs** 

**3.2 Optical asymmetric cut-wire pairs** 

interaction between its elementary constituents.

wires in the direction of the electric field.

The resonant mode split into two distinct components when the separation between cut wires and continuous wires was smaller than 50 nm. This different behavior can be simply explained by the fact that unlike structures with true SRRs, the field of the first resonant mode is obviously concentrated in the close neighbourhood of the continuous wires, i.e. in the cut wires themselves.

Coupling effects were also investigated from numerical simulations for the perpendicular polarization. We only considered the first SRR resonance since it was the only one clearly observed in the experiments (Fig. 12). The frequency position of this resonance was found to be rather independent of the presence of continuous wires whatever the separation between the SRRs and wires was. However, a small splitting of the resonance was observed for very small separations (d = 10 nm) in the case of U-shaped SRRs with small legs (structure 3).

#### **3.1.5 Metamaterials with negative refraction on silicon**

It is now well established that the use of an array of continuous metallic wires allows obtaining a negative permittivity over the whole plasmon-like band when the electric field is polarized parallel to the wires [Pendry, 1998]. Figures 12 and 14(a) presently show that this band can extend well up to near-infrared frequencies for a sufficiently small period of the wire lattice. On the other hand, it has been demonstrated that an array of metallic SRRs can exhibit a magnetic response in the optical domain with a negative permeability at certain frequencies [Enkrich, 2005]. However, this situation only occurs at SRR resonances and for an oblique or grazing incidence, i.e. for an incident magnetic field with a non-zero component along the SRR axis. An additional condition is that the resonant plasmon mode must possess an odd number of field nodes along the SRR [Shalaev, 2005]. Concerning this latter aspect, our experimental results confirm that the first resonant mode, the so-called LC resonance, is by far the most exploitable due both to its strength and to its robustness against parasitic coupling effects. They also show that its frequency position can be finely tuned by adjusting the total length of SRRs and using for instance U-shaped SRRs. One solution to achieve a magnetic response at normal incidence with respect to the sample plane consists in using a stack of SRR layers [Liu, 2008] or simpler, a stack of cut-wires as originally proposed in [Shalaev, 2005]. Coupling between adjacent SRRs or between adjacent cut-wires leads to the formation of hybridized plasmon modes of opposite symmetry. Anti-symmetric plasmon modes can exhibit a magnetic response, and lead to a negative permeability in certain frequency regions. Our experimental results in Fig. 12 show that the resonance associated to the dipolar mode of cut-wires is well pronounced for the fabricated structures. Coupling between two such modes in a multilayer stack should thus allow obtaining a magnetic response at normal incidence. One advantage in using stacked cut-wires instead of stacked SRRs stems from the possibility of achieving more easily a magnetic response at (high) near- infrared frequencies. This is all the more true when metallic nanostructures are fabricated on a high permittivity substrate such as silicon. All the plasmon resonances are shifted to low frequencies, and the realization of very-small-size SRRs operating at telecommunication wavelengths on silicon would require pushing the lithographic techniques to their present limits.

#### **3.2 Optical asymmetric cut-wire pairs**

290 Metamaterial

The resonant mode split into two distinct components when the separation between cut wires and continuous wires was smaller than 50 nm. This different behavior can be simply explained by the fact that unlike structures with true SRRs, the field of the first resonant mode is obviously concentrated in the close neighbourhood of the continuous wires, i.e. in

Coupling effects were also investigated from numerical simulations for the perpendicular polarization. We only considered the first SRR resonance since it was the only one clearly observed in the experiments (Fig. 12). The frequency position of this resonance was found to be rather independent of the presence of continuous wires whatever the separation between the SRRs and wires was. However, a small splitting of the resonance was observed for very small separations (d = 10 nm) in the case of U-shaped SRRs with small

It is now well established that the use of an array of continuous metallic wires allows obtaining a negative permittivity over the whole plasmon-like band when the electric field is polarized parallel to the wires [Pendry, 1998]. Figures 12 and 14(a) presently show that this band can extend well up to near-infrared frequencies for a sufficiently small period of the wire lattice. On the other hand, it has been demonstrated that an array of metallic SRRs can exhibit a magnetic response in the optical domain with a negative permeability at certain frequencies [Enkrich, 2005]. However, this situation only occurs at SRR resonances and for an oblique or grazing incidence, i.e. for an incident magnetic field with a non-zero component along the SRR axis. An additional condition is that the resonant plasmon mode must possess an odd number of field nodes along the SRR [Shalaev, 2005]. Concerning this latter aspect, our experimental results confirm that the first resonant mode, the so-called LC resonance, is by far the most exploitable due both to its strength and to its robustness against parasitic coupling effects. They also show that its frequency position can be finely tuned by adjusting the total length of SRRs and using for instance U-shaped SRRs. One solution to achieve a magnetic response at normal incidence with respect to the sample plane consists in using a stack of SRR layers [Liu, 2008] or simpler, a stack of cut-wires as originally proposed in [Shalaev, 2005]. Coupling between adjacent SRRs or between adjacent cut-wires leads to the formation of hybridized plasmon modes of opposite symmetry. Anti-symmetric plasmon modes can exhibit a magnetic response, and lead to a negative permeability in certain frequency regions. Our experimental results in Fig. 12 show that the resonance associated to the dipolar mode of cut-wires is well pronounced for the fabricated structures. Coupling between two such modes in a multilayer stack should thus allow obtaining a magnetic response at normal incidence. One advantage in using stacked cut-wires instead of stacked SRRs stems from the possibility of achieving more easily a magnetic response at (high) near- infrared frequencies. This is all the more true when metallic nanostructures are fabricated on a high permittivity substrate such as silicon. All the plasmon resonances are shifted to low frequencies, and the realization of very-small-size SRRs operating at telecommunication wavelengths on silicon would require pushing the lithographic techniques to their present

the cut wires themselves.

legs (structure 3).

limits.

**3.1.5 Metamaterials with negative refraction on silicon** 

#### **3.2.1 Negative refractive index in optical asymmetric cut-wire pairs**

In the previous structures, SRRs were associated with continuous wires to obtain negative index at infrared wavelengths. In this chapter we will study asymmetric cut-wire pairs to obtain also negative index. Metallic nanostructures can be regarded as elementary circuits including nano-capacitors, inductors or resistors [Engheta, 2007]. The simplest resonator that can be imagined is a dipole consisting of a simple metallic cut-wire. Coupling two such oscillators lead to two eigenmodes with opposite symmetry. The virtual current loop of the anti- symmetric mode is now recognized as a mean to create artificial magnetism at optical frequencies [Grigorenko, 2005, Shalaev, 2005]. Fig. 15 shows the structure under consideration. It consists of a periodic array of paired cut-wires separated by a dielectric layer. For simplicity the spacer has been taken to be silicon dioxide (SiO2) with a dielectric permittivity εr = 2.25 and a thickness of 100 nm. The surrounding medium is air with εr = 1. The relevant polarization of the impinging light is given in Fig. 15(a) with the electric field parallel to the longest side of the cut- wires. Transmission spectra are presented in Fig. 15(b). All the simulations are done using a commercial finite element code (HFSS, 2006). An array of isolated cut-wires is actually found when the separation distance between cut-wires is large: only one resonance (the dipolar mode) is observed in this case (red curve in Fig. 15(b)). This response can be interpreted in terms of a localized plasmon resonance [Rockstuhl, 2006, Kante, 2008]. When the separation distance between cut-wire pairs is progressively diminished, the response of the paired system is modified due to the interaction between its elementary constituents.

Following the plasmon hybridization concept [Kreibig, 1981, Prodan, 2003], coupling effects lift the degeneracy of the single cut-wire mode, thus leading to two distinct plasmons modes [Liu, 2007, Kante 2009], the anti-symmetric and symmetric modes as shown in Fig. 15(b) (blue curve). The symmetric mode with larger restoring force is at a higher energy than the anti-symmetric mode. The main idea of our work is to invert this process. The symmetric mode, which easily couples to incident light, corresponds to a wide rejection band with negative permittivity. The anti-symmetric mode, which is difficult to excite due to the opposite contributions of the two dipoles, manifests itself as a small transmission dip with negative permeability. Achieving such a resonance at a higher frequency than the symmetric mode will ease the overlap condition as proposed recently in [Kante, 2009, Sellier, 2009]. For this purpose, one solution is to break the symmetry of the cut-wire pair. This is achieved in Fig. 15(d) by displacing one of two cutwires in the direction of the electric field.

Fig. 16(a) and (b) shows the evolutions of the resonant modes and transmission spectra with the longitudinal shift dx between the two cut-wires of each pair. Simulations are performed in the infrared domain. As a major result, for a sufficiently large displacement dx (here dx > 350 nm), the hybridization scheme is inverted with the symmetric mode at a lower frequency than the anti-symmetric mode. The two hybridization schemes are presented in Fig. 15(c) and (e), respectively. Fig. 16(d) clearly shows that a negative refraction regime is obtained (between 145 and 160 THz for dx = 600 nm) for the inverted scheme while the index of refraction remains always positive (dx = 0 nm) for the normal hybridization case (Fig. 16(c)). This extends to infrared frequencies previous results reported by the authors at

Design and Characterization of Metamaterials for Optical and Radio Communications 293

Fig. 14. (a) Transmission spectra calculated for a periodic array of SRRs (red curve), a periodic array of continuous wires (pink dashed curve) and periodic arrays of SRRs and wires with different separations between SRRs and wires: d=130 nm (blue curve), d=50 nm (green dashed curve), d=10 nm (black dashed curve). In each case, the incident electric field is polarized parallel to the gap, the SRR dimensions and lattice period as the same as for structure 1 in Fig. 10. (b) Magnitude of the normal electric field component (|Ez|) calculated at the bottom surface of the metallic elements for each of the resonant modes observed in the transmission spectra of Fig. 14 (a). The different colours, from blue to red

correspond to increasing magnitudes of the field component. Modes are classified

wire.

according to the resonance order and to the separation d between the SRR and the closest

microwave frequencies [Kante, 2009, Sellier, 2009]. As seen in Fig. 16(d), negative refraction corresponds to an overlap between the region with negative permittivity and that with negative permeability. It is worthwhile noticing that the domain of overlap with negative permittivity and permeability can be controlled through the different degrees of freedom of the structure [Kante, 2009].

#### **3.2.2 Hybridization of the localized plasmons of SRRs**

The recipe proposed above for obtaining a negative index with plasmon hybridization can be applied to any structure supporting localized plasmons. The inversion process resulted from a radical change in near field Coulomb interactions between cut-wires in each pair [Kante, 2009, Christ, 2008]. Let us consider for instance a periodic array of paired SRRs, which are fundamental building blocks in the design of metamaterials. Indeed, the use of SRRs has allowed the achievement of negative magnetic permeability, which is impossible with natural materials at high frequencies. However, obtaining a negative magnetic permeability requires the incident wave to possess a magnetic component along the SRR axis. Such a requirement is not easily fulfilled in optics [Dolling, 2006, Liu, 2009]. Moreover, the electromagnetic response of SRRs has been shown to saturate at optical frequencies [O'Brien, 2002]. In most experimental works reported so far in the infrared domain, normal incidence has been used instead of a grazing or oblique incidence [Kante, 2008]. In this situation, only the electric field can couple to the structure. SRR resonances are nothing but plasmonic resonances which can be classified into even and odd modes depending on the polarization the exciting light with respect to the structure [20]. While even modes are excited for an incident electric field perpendicular to the SRR gap, odd modes are excited for an electric field parallel to this gap. It is straightforward to see that only odd modes can lead to a magnetic moment or eventually to a negative permeability under oblique incidence.

microwave frequencies [Kante, 2009, Sellier, 2009]. As seen in Fig. 16(d), negative refraction corresponds to an overlap between the region with negative permittivity and that with negative permeability. It is worthwhile noticing that the domain of overlap with negative permittivity and permeability can be controlled through the different degrees of freedom of

The recipe proposed above for obtaining a negative index with plasmon hybridization can be applied to any structure supporting localized plasmons. The inversion process resulted from a radical change in near field Coulomb interactions between cut-wires in each pair [Kante, 2009, Christ, 2008]. Let us consider for instance a periodic array of paired SRRs, which are fundamental building blocks in the design of metamaterials. Indeed, the use of SRRs has allowed the achievement of negative magnetic permeability, which is impossible with natural materials at high frequencies. However, obtaining a negative magnetic permeability requires the incident wave to possess a magnetic component along the SRR axis. Such a requirement is not easily fulfilled in optics [Dolling, 2006, Liu, 2009]. Moreover, the electromagnetic response of SRRs has been shown to saturate at optical frequencies [O'Brien, 2002]. In most experimental works reported so far in the infrared domain, normal incidence has been used instead of a grazing or oblique incidence [Kante, 2008]. In this situation, only the electric field can couple to the structure. SRR resonances are nothing but plasmonic resonances which can be classified into even and odd modes depending on the polarization the exciting light with respect to the structure [20]. While even modes are excited for an incident electric field perpendicular to the SRR gap, odd modes are excited for an electric field parallel to this gap. It is straightforward to see that only odd modes can lead to a magnetic moment or eventually to a negative permeability

the structure [Kante, 2009].

under oblique incidence.

**3.2.2 Hybridization of the localized plasmons of SRRs** 

Fig. 14. (a) Transmission spectra calculated for a periodic array of SRRs (red curve), a periodic array of continuous wires (pink dashed curve) and periodic arrays of SRRs and wires with different separations between SRRs and wires: d=130 nm (blue curve), d=50 nm (green dashed curve), d=10 nm (black dashed curve). In each case, the incident electric field is polarized parallel to the gap, the SRR dimensions and lattice period as the same as for structure 1 in Fig. 10. (b) Magnitude of the normal electric field component (|Ez|) calculated at the bottom surface of the metallic elements for each of the resonant modes observed in the transmission spectra of Fig. 14 (a). The different colours, from blue to red correspond to increasing magnitudes of the field component. Modes are classified according to the resonance order and to the separation d between the SRR and the closest wire.

Design and Characterization of Metamaterials for Optical and Radio Communications 295

''stereometamaterial''. In contrast, rotating one SRR with respect to another did not produce any inverted hybridization scheme, while this scheme is the most appropriate one for obtaining negative refraction. In fact, split ring resonators are complex structures regarding their responses to an electromagnetic field since they support localized plasmons addressable either by the electric field or the magnetic field. To our knowledge, using the two degrees of freedom (dx, dy) in the design of the SRR pair (Fig. 17 (left graph)) to achieve a negative index of refraction has never been reported so far in the context of SRR-based

Fig. 16. (a) Transmission spectra of the two-dimensional array of asymmetric cut-wire pairs calculated for different values of the longitudinal shift (dx) between the two cut-wires in each pair. (b) Evolutions of the symmetric and anti-symmetric mode frequencies as a function of the dx shift. (c) Effective index of refraction [30] calculated for dx = 0 with a de-embedding up to

the metamaterials interfaces. (d) Same type of calculations for dx = 600 nm.

metamaterials.

Fig. 15. (a and d) Schematics of the symmetric and asymmetric cut-wire pairs, respectively. (b) Transmission of an array of un-coupled or coupled cut-wires structure with px=1.2mm, py=200nm, pz=600nm, w=30nm, L=600nm, hsub =100nm and εsub =2.25. The pieces of gold metals are described by a Drude model whose parameters can be found in [Kante, 2008]. (c and e) Hybridization scheme and inverted hybridization scheme, respectively.

Fig. 17 (red curves) presents transmission spectra of a single SRR layer designed to operate in the infrared range for two polarizations at normal incidence. When the electric field is perpendicular to the gap, only one resonance (the fundamental even mode) is observed in the frequency range of interest while two resonances are observed for the parallel polarization. These last two resonances respectively correspond to the first and second odd plasmonic modes [Rockstuhl, 2006, Kante, 2008], the fundamental mode being also called LC resonance. When two such SRR structures are brought close to each other, the localized plasmons can hybridize according to a plasmon hybridization scheme similar to the one previously described for cut-wire pairs (Fig. 15). In what follows, the hybridization scheme of an SRR pair is analyzed for both parallel and perpendicular polarization as well as for resonances of different orders. As it will be shown, a negative index of refraction can be obtained at normal incidence for a periodic structure, which is exclusively made of SRRs and based on an inverted hybridization scheme. The SRR pair forming the elementary motif of the periodic structure is depicted in Fig. 17 (left graph). As seen, the lower SRR of the pair is shifted in the x and y directions with respect to the upper one. Following our previous work on cut-wire pairs, this configuration will be simply called ''asymmetric SRR pair''. A system of coupled SRRs has been recently investigated by Liu et al. [Liu, 2009], but in a twisted configuration. These authors showed that the twist angle between vertically coupled SRRs could modify either the electric or magnetic response of the system resulting in what they called a ''stereometamaterial''. Our structure can thus be regarded as a particular

Fig. 15. (a and d) Schematics of the symmetric and asymmetric cut-wire pairs, respectively. (b) Transmission of an array of un-coupled or coupled cut-wires structure with px=1.2mm, py=200nm, pz=600nm, w=30nm, L=600nm, hsub =100nm and εsub =2.25. The pieces of gold metals are described by a Drude model whose parameters can be found in [Kante, 2008]. (c

Fig. 17 (red curves) presents transmission spectra of a single SRR layer designed to operate in the infrared range for two polarizations at normal incidence. When the electric field is perpendicular to the gap, only one resonance (the fundamental even mode) is observed in the frequency range of interest while two resonances are observed for the parallel polarization. These last two resonances respectively correspond to the first and second odd plasmonic modes [Rockstuhl, 2006, Kante, 2008], the fundamental mode being also called LC resonance. When two such SRR structures are brought close to each other, the localized plasmons can hybridize according to a plasmon hybridization scheme similar to the one previously described for cut-wire pairs (Fig. 15). In what follows, the hybridization scheme of an SRR pair is analyzed for both parallel and perpendicular polarization as well as for resonances of different orders. As it will be shown, a negative index of refraction can be obtained at normal incidence for a periodic structure, which is exclusively made of SRRs and based on an inverted hybridization scheme. The SRR pair forming the elementary motif of the periodic structure is depicted in Fig. 17 (left graph). As seen, the lower SRR of the pair is shifted in the x and y directions with respect to the upper one. Following our previous work on cut-wire pairs, this configuration will be simply called ''asymmetric SRR pair''. A system of coupled SRRs has been recently investigated by Liu et al. [Liu, 2009], but in a twisted configuration. These authors showed that the twist angle between vertically coupled SRRs could modify either the electric or magnetic response of the system resulting in what they called a ''stereometamaterial''. Our structure can thus be regarded as a particular

and e) Hybridization scheme and inverted hybridization scheme, respectively.

''stereometamaterial''. In contrast, rotating one SRR with respect to another did not produce any inverted hybridization scheme, while this scheme is the most appropriate one for obtaining negative refraction. In fact, split ring resonators are complex structures regarding their responses to an electromagnetic field since they support localized plasmons addressable either by the electric field or the magnetic field. To our knowledge, using the two degrees of freedom (dx, dy) in the design of the SRR pair (Fig. 17 (left graph)) to achieve a negative index of refraction has never been reported so far in the context of SRR-based metamaterials.

Fig. 16. (a) Transmission spectra of the two-dimensional array of asymmetric cut-wire pairs calculated for different values of the longitudinal shift (dx) between the two cut-wires in each pair. (b) Evolutions of the symmetric and anti-symmetric mode frequencies as a function of the dx shift. (c) Effective index of refraction [30] calculated for dx = 0 with a de-embedding up to the metamaterials interfaces. (d) Same type of calculations for dx = 600 nm.

Design and Characterization of Metamaterials for Optical and Radio Communications 297

Let us now consider the second odd mode. The corresponding distribution of electric charges and field nodes in SRR arms is shown in the middle graph of Fig. 17 (right insert) for a one-layer SRR array. This picture shows that this mode has a ''dipole activity'' in both the x and y directions. Charges at the extremities of CD and EF arms produce the dipolar activity along the x direction while those at the extremities of the DE arm produce the dipolar activity along the y direction. Correspondingly, in a two-layer SRR array, an inversion of the hybridization occurs whether one of the two layers is displaced along the x or the y direction (Fig. 18). In each case, the inversion stems from the near field interaction between the active dipoles in the corresponding direction. In the case of a displacement along x, the inversion results from the interaction of dipoles CD and EF with dipoles C'D' and E'F'. The primes referring to the second SRR. In the case of a displacement along y, it

Fig. 18. Transmission spectra of a periodic array of asymmetric SRRs pairs around the LC resonance for different values of dx at dy=0 (left) and for different values of dy at dx = 0 (right)

When the incident electric field is perpendicular to the gap, the SRR can be regarded as two cut-wires in parallel since the charges in the two arms GH and IH oscillate in phase (see insert in the right graph of Fig. 17). In consequence, the hybridization of localized plasmons evolves a priori as in the case of cut-wires pairs [Kante, 2009]. Therefore, only a displacement along the x direction (parallel to the dipoles GH and HI) for one of the SRRs of the asymmetric pair can lead to an inverted hybridization scheme with the possibility of a

The inversion of the hybridization scheme for dx ≠ 0 is illustrated from calculated transmission spectra in Fig. 20(a). Results of calculations for the effective index of refraction

**3.2.5 Hybridization of the fundamental even plasmonic mode (perpendicular** 

**3.2.4 Hybridization of the second odd plasmonic mode** 

results from the interaction between DE and D'E'.

**polarization)** 

negative index of refraction.

#### **3.2.3 Hybridization of the LC resonance**

The LC resonance (i.e. the fundamental odd mode) is excited at normal incidence when the electric field is parallel to the gap. This resonance has been experimentally observed from microwaves to optics. Electric charges in metallic SRR arms are mainly located near the SRR gap (insert of Fig. 17, middle graph). Fig. 18 shows the modifications of the transmission spectrum when coupling vertically two SRR layers. Curves in the left graph correspond to different values of dx at a fixed dy (dy = 0). Curves in the right graph correspond to different values of dy at fixed dx (dx = 0). As expected, in all cases, coupling between SRRs splits the LC resonance in two eigenmodes. The symmetric mode, which has the largest amplitude, is found at the highest frequency for dx = dy = 0. In principle, a longitudinal magnetic coupling also exists [Liu, 2009], but can be neglected for a qualitative understanding of the hybridization scheme in presence of electric coupling. As seen in the left graph of Fig. 17, a shift of one of the two SRRs in the direction perpendicular to the gap (dx≠0) has a weak influence. One just observes a small decrease of the mode amplitudes. The evolution is radically different when the shift occurs in the direction parallel to the gap (dy ≠ 0). An inversion of the hybridization scheme is observed for sufficiently large values of dy. The overall results obtained for the LC mode of SRRs are then very comparable to those reported for the dipolar mode of cut-wires. The LC mode exhibits indeed a dipole-like behavior.

Fig. 17. (Left) Schematics of the asymmetric SRR pair. (Middle) Transmission spectra of periodic SRR arrays at normal incidence for a field polarization parallel to the SRR gap. (Right) Transmission spectra of periodic SRR arrays for a field polarization perpendicular to the SRR gap. In the middle and right graphs, red curves are for one-SRR-layer array while blue curves are for a two-layer array of paired SRRs. Inserts show the distributions of charges and electric field nodes for the different resonances in the one-layer array. Squared SRRs are used with 700 nm side length and 200 nm gap width. The 100 nm wide conducting elements are described using the Drude model reported in [Kante, 2008]. Other parameters are: px = py = 1.4 mm, pz = 1 mm, hsub = 100 nm, εsub = 2.25.

#### **3.2.4 Hybridization of the second odd plasmonic mode**

296 Metamaterial

The LC resonance (i.e. the fundamental odd mode) is excited at normal incidence when the electric field is parallel to the gap. This resonance has been experimentally observed from microwaves to optics. Electric charges in metallic SRR arms are mainly located near the SRR gap (insert of Fig. 17, middle graph). Fig. 18 shows the modifications of the transmission spectrum when coupling vertically two SRR layers. Curves in the left graph correspond to different values of dx at a fixed dy (dy = 0). Curves in the right graph correspond to different values of dy at fixed dx (dx = 0). As expected, in all cases, coupling between SRRs splits the LC resonance in two eigenmodes. The symmetric mode, which has the largest amplitude, is found at the highest frequency for dx = dy = 0. In principle, a longitudinal magnetic coupling also exists [Liu, 2009], but can be neglected for a qualitative understanding of the hybridization scheme in presence of electric coupling. As seen in the left graph of Fig. 17, a shift of one of the two SRRs in the direction perpendicular to the gap (dx≠0) has a weak influence. One just observes a small decrease of the mode amplitudes. The evolution is radically different when the shift occurs in the direction parallel to the gap (dy ≠ 0). An inversion of the hybridization scheme is observed for sufficiently large values of dy. The overall results obtained for the LC mode of SRRs are then very comparable to those reported for the dipolar mode of cut-wires. The LC mode exhibits indeed a dipole-like

Fig. 17. (Left) Schematics of the asymmetric SRR pair. (Middle) Transmission spectra of periodic SRR arrays at normal incidence for a field polarization parallel to the SRR gap. (Right) Transmission spectra of periodic SRR arrays for a field polarization perpendicular to the SRR gap. In the middle and right graphs, red curves are for one-SRR-layer array while blue curves are for a two-layer array of paired SRRs. Inserts show the distributions of charges and electric field nodes for the different resonances in the one-layer array. Squared SRRs are used with 700 nm side length and 200 nm gap width. The 100 nm wide conducting elements are described using the Drude model reported in [Kante, 2008]. Other parameters

are: px = py = 1.4 mm, pz = 1 mm, hsub = 100 nm, εsub = 2.25.

**3.2.3 Hybridization of the LC resonance** 

behavior.

Let us now consider the second odd mode. The corresponding distribution of electric charges and field nodes in SRR arms is shown in the middle graph of Fig. 17 (right insert) for a one-layer SRR array. This picture shows that this mode has a ''dipole activity'' in both the x and y directions. Charges at the extremities of CD and EF arms produce the dipolar activity along the x direction while those at the extremities of the DE arm produce the dipolar activity along the y direction. Correspondingly, in a two-layer SRR array, an inversion of the hybridization occurs whether one of the two layers is displaced along the x or the y direction (Fig. 18). In each case, the inversion stems from the near field interaction between the active dipoles in the corresponding direction. In the case of a displacement along x, the inversion results from the interaction of dipoles CD and EF with dipoles C'D' and E'F'. The primes referring to the second SRR. In the case of a displacement along y, it results from the interaction between DE and D'E'.

Fig. 18. Transmission spectra of a periodic array of asymmetric SRRs pairs around the LC resonance for different values of dx at dy=0 (left) and for different values of dy at dx = 0 (right)

#### **3.2.5 Hybridization of the fundamental even plasmonic mode (perpendicular polarization)**

When the incident electric field is perpendicular to the gap, the SRR can be regarded as two cut-wires in parallel since the charges in the two arms GH and IH oscillate in phase (see insert in the right graph of Fig. 17). In consequence, the hybridization of localized plasmons evolves a priori as in the case of cut-wires pairs [Kante, 2009]. Therefore, only a displacement along the x direction (parallel to the dipoles GH and HI) for one of the SRRs of the asymmetric pair can lead to an inverted hybridization scheme with the possibility of a negative index of refraction.

The inversion of the hybridization scheme for dx ≠ 0 is illustrated from calculated transmission spectra in Fig. 20(a). Results of calculations for the effective index of refraction

Design and Characterization of Metamaterials for Optical and Radio Communications 299

Fig. 20. (Top) Transmission spectra of a periodic array of asymmetric SRRs pairs for vertical polarization and different values of dx at dy=0 (a) and for different values of dy at dx = 0 (b). (Bottom) Effective index of refraction (real and imaginary parts) calculated for dx = 0 and dy = 0 (c) and for dx = 700 nm, dy = 0 (d), pz = 1 mm with a de-embedding up to the

In this chapter, we have presented different metamaterials with a negative index at microwave and optical frequencies. Through numerical simulations and measurements, we have shown that it was possible to obtain a negative index by optimizing the coupling between different layers of metamaterials. We have also shown that this concept can be implemented from microwaves to optics. These works are believed to open a new way for

the design of negative refraction metamaterials from microwaves to optics.

metamaterials interfaces.

**4. Conclusions** 

are presented in Fig. 20(c) and (d) for a symmetric SRR pair (dx = dy = 0) and an asymmetric pair (dx = 700 nm, dy = 0), respectively. Results obtained for the symmetric pair are comparable to those reported in [Liu, 2008]. There with no overlap between the regions of negative epsilon and negative mu. In contrast, a negative index of refraction is clearly obtained for the asymmetric SRR pair in Fig. 20(d).

Results obtained for a transversal displacement (dy ≠ 0) of one SRR of the pair may appear to be quite surprising since three peaks are observed in the transmission spectra instead of two (Fig. 20(b)). However, a careful analysis reveals that these peaks have different origins. Three types of dipole–dipole interactions are indeed involved: GH with G'H', GH with I'H' and IH with G' H'. For an intermediate value of dy (i.e. dy = py/2 = 350 nm), the dipole– dipole interactions GH - G'H' and I'H' - IH become degenerate, thereby leading to the disappearance of one of the three peaks.

In conclusion we have numerically demonstrated that the coupling between localized plasmons in periodic arrays of paired cut-wires or SRRs can be controlled by modifying the symmetry of each individual pair. It has been shown that breaking the symmetry of cut-wire or SRR stacks can lead to a negative index of refraction. The scheme proposed here contrasts with previous designs of negative index metamaterials where two kinds of meta-atoms were mixed. Only one type of meta-atom supporting localized plasmons is used. A true negative index band is achieved provided that the coupling between localized plasmons is appropriately controlled.

Fig. 19. Transmission spectra of a periodic array of a symmetric SRRs pairs around the second odd SRR mode for different values of dx at dy=0 (left) and for different values of dy at dx = 0 (right).

Fig. 20. (Top) Transmission spectra of a periodic array of asymmetric SRRs pairs for vertical polarization and different values of dx at dy=0 (a) and for different values of dy at dx = 0 (b). (Bottom) Effective index of refraction (real and imaginary parts) calculated for dx = 0 and dy = 0 (c) and for dx = 700 nm, dy = 0 (d), pz = 1 mm with a de-embedding up to the metamaterials interfaces.

#### **4. Conclusions**

298 Metamaterial

are presented in Fig. 20(c) and (d) for a symmetric SRR pair (dx = dy = 0) and an asymmetric pair (dx = 700 nm, dy = 0), respectively. Results obtained for the symmetric pair are comparable to those reported in [Liu, 2008]. There with no overlap between the regions of negative epsilon and negative mu. In contrast, a negative index of refraction is clearly

Results obtained for a transversal displacement (dy ≠ 0) of one SRR of the pair may appear to be quite surprising since three peaks are observed in the transmission spectra instead of two (Fig. 20(b)). However, a careful analysis reveals that these peaks have different origins. Three types of dipole–dipole interactions are indeed involved: GH with G'H', GH with I'H' and IH with G' H'. For an intermediate value of dy (i.e. dy = py/2 = 350 nm), the dipole– dipole interactions GH - G'H' and I'H' - IH become degenerate, thereby leading to the

In conclusion we have numerically demonstrated that the coupling between localized plasmons in periodic arrays of paired cut-wires or SRRs can be controlled by modifying the symmetry of each individual pair. It has been shown that breaking the symmetry of cut-wire or SRR stacks can lead to a negative index of refraction. The scheme proposed here contrasts with previous designs of negative index metamaterials where two kinds of meta-atoms were mixed. Only one type of meta-atom supporting localized plasmons is used. A true negative index band is achieved provided that the coupling between localized plasmons is

Fig. 19. Transmission spectra of a periodic array of a symmetric SRRs pairs around the second odd SRR mode for different values of dx at dy=0 (left) and for different values of dy

obtained for the asymmetric SRR pair in Fig. 20(d).

disappearance of one of the three peaks.

appropriately controlled.

at dx = 0 (right).

In this chapter, we have presented different metamaterials with a negative index at microwave and optical frequencies. Through numerical simulations and measurements, we have shown that it was possible to obtain a negative index by optimizing the coupling between different layers of metamaterials. We have also shown that this concept can be implemented from microwaves to optics. These works are believed to open a new way for the design of negative refraction metamaterials from microwaves to optics.

Design and Characterization of Metamaterials for Optical and Radio Communications 301

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**12** 

*Spain* 

**Characterization of Metamaterial** 

Jordi Bonache3 and Ferran Martín3 *1Universidad Politécnica de Madrid, 2Universidad de Castilla-La Mancha, 3Universitat Autònoma de Barcelona,* 

**Transmission Lines with Coupled** 

**Resonators Through Parameter Extraction** 

Since resonant-type metamaterial transmission lines were proposed (Martín et al., 2003), this kind of transmission lines have been of significant importance in the development of new and innovative microwave devices. The small size and novel characteristics of these transmission lines based on sub-wavelength resonators allows the miniaturization and improvement of existing devices (Bonache et al., 2006a, 2006b; Gil et al., 2007a, 2007b), as well as the design of components with new functionalities (Sisó et al., 2008, 2009). Due to the complicated layouts that these designs usually involve, having an accurate equivalent circuit model is an important assist during the design process. Besides, the application of parameter extraction methods (Bonache et al., 2006c) to obtain the values of the electrical parameters of the circuit model makes possible the characterization of both the transmission line and, therefore, the microwave device. The circuit models and parameter extraction methods presented in this chapter have been widely verified and their accuracy permits even their application for automatic layout generation based on space mapping techniques (Selga et al., 2010), which is a large and useful advance in the design of such structures.

**2. Metamaterial transmission lines based on the resonant approach** 

structures based on different kinds of resonators are presented and discussed.

**2.1 Transmission lines based on Split-Ring Resonators (SRRs)** 

In this section, several structures of metamaterial transmission lines based on the resonant approach are shown. The considered structures are implemented in planar technology and consist in a host microstrip line or in coplanar wave guide (CPW) loaded with sub-wavelength resonant particles. Each structure requires the combination of resonators and other loading elements in order to achieve the intended propagation. In the following sections, several

The metamaterial transmission lines based on SRRs were proposed in 2003 by Martín et al. (Martín et al., 2003). They designed a transmission line exhibiting left-handed transmission

**1. Introduction** 

Francisco Aznar-Ballesta1, Marta Gil2, Miguel Durán-Sindreu3,


### **Characterization of Metamaterial Transmission Lines with Coupled Resonators Through Parameter Extraction**

Francisco Aznar-Ballesta1, Marta Gil2, Miguel Durán-Sindreu3, Jordi Bonache3 and Ferran Martín3 *1Universidad Politécnica de Madrid, 2Universidad de Castilla-La Mancha, 3Universitat Autònoma de Barcelona, Spain* 

#### **1. Introduction**

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(2004), Terahertz Magnetic Response from Artificial Materials, *Science* Vol.303,

Midinfrared Resonant Magnetic Nanostructures Exhibiting a Negative

Since resonant-type metamaterial transmission lines were proposed (Martín et al., 2003), this kind of transmission lines have been of significant importance in the development of new and innovative microwave devices. The small size and novel characteristics of these transmission lines based on sub-wavelength resonators allows the miniaturization and improvement of existing devices (Bonache et al., 2006a, 2006b; Gil et al., 2007a, 2007b), as well as the design of components with new functionalities (Sisó et al., 2008, 2009). Due to the complicated layouts that these designs usually involve, having an accurate equivalent circuit model is an important assist during the design process. Besides, the application of parameter extraction methods (Bonache et al., 2006c) to obtain the values of the electrical parameters of the circuit model makes possible the characterization of both the transmission line and, therefore, the microwave device. The circuit models and parameter extraction methods presented in this chapter have been widely verified and their accuracy permits even their application for automatic layout generation based on space mapping techniques (Selga et al., 2010), which is a large and useful advance in the design of such structures.

#### **2. Metamaterial transmission lines based on the resonant approach**

In this section, several structures of metamaterial transmission lines based on the resonant approach are shown. The considered structures are implemented in planar technology and consist in a host microstrip line or in coplanar wave guide (CPW) loaded with sub-wavelength resonant particles. Each structure requires the combination of resonators and other loading elements in order to achieve the intended propagation. In the following sections, several structures based on different kinds of resonators are presented and discussed.

#### **2.1 Transmission lines based on Split-Ring Resonators (SRRs)**

The metamaterial transmission lines based on SRRs were proposed in 2003 by Martín et al. (Martín et al., 2003). They designed a transmission line exhibiting left-handed transmission

Characterization of Metamaterial

resonator based on SRR (b).

CPW (a) and microstrip (b) technology.

permittivity of the structure in this case as well.

behaviour.

Transmission Lines with Coupled Resonators Through Parameter Extraction 305

capacitance of the resonator can be increased (within the technology limits), the resonance frequency of the SRR can be decreased, reducing its electrical size (Aznar et al., 2008b). This is the case of the resonators shown in Fig. 1, which are electrically smaller than the SRR.

Fig. 2. Equivalent circuit models for a SRR (a) and simplified equivalent circuit model for a

Fig. 3. Layouts of metamaterial transmission lines loaded with resonators based on SRR in

The layout of two unit cells of metamaterial transmission lines loaded with resonators based on SRR are shown in Fig. 3(a) for CPW and Fig. 3(b) for microstrip technology. In the CPW configuration, the SRRs are paired on the lower substrate side (where they are etched), beneath the slots of the structure and centred with the shunt strips. The resonators are responsible for the negative effective permeability, whereas negative effective permittivity is achieved by means of the shunt connected strips. In case the shunt strips are eliminated, the CPW loaded with resonators provides only negative permeability, showing stop-band

In the case of microstrip lines, the SRRs can be etched in pairs on the upper substrate side, adjacent to the conductor strip. The metallic vias are responsible for the negative

The layout of the CSRR is shown in Fig. 4(a). This resonator results from the application of the Babinet principle to the structure of the SRR, which leads to its complementary

**2.2 Transmission lines based on Complementary Split-Ring Resonators (CSRRs)** 

by loading a CPW structure with SRRs. Left-handed transmission is achieved when the effective permittivity and permeability of a medium are both negative, providing negative values of the phase velocity and refraction index, among other peculiarities (Veselago, 1968). This first left-handed SRR-based CPW was inspired on the medium of Smith et al., 2000. By etching SRRs in the back substrate side of the CPW, beneath the slots, and shunt connected metallic strips between the central strip and ground plane. A one dimensional effective medium with simultaneous negative permeability (due to the presence of the SRRs) and permittivity (thanks to the shunt strips) in a narrow band was achieved (Martín et al., 2003).

The resonators used in this kind of transmission lines based on the resonant approach can be SRRs or other similar resonators with different topologies based on the SRR. The layout for the SRR is shown in Fig. 1(a). In Fig. 1, other examples of these resonators, like the spiral resonator (SR, Fig. 1b) with only one metal layer, are shown. Resonators implemented with two metal layers, like the broadside coupled non-bianisotropic split ring resonator (BC-NB-SRR, Fig. 1c), the broadside coupled spiral resonator with two turns (BC-SR(2), Fig. 1d) and the broadside coupled spiral resonator with four turns (BC-SR(4), Fig. 1e) are also depicted (Marqués et al., 2003; Aznar et al., 2008b).

Fig. 1. Examples of topologies of different resonators based on the SRR (a): SR (b), BC-NB-SRR (c), BC-SR(2) (d) and BC-SR(4) (e).

The equivalent circuit model for the SRR is shown in Fig. 2(a) (Baena et al., 2005). The capacitance *C0*/2 is related with each of the two SRR halves, whereas *Ls* is the resonator selfinductance. *C0* can be obtained as *C0*=2πr*Cpul*, where *Cpul* represents the per unit length capacitance between de rings forming the resonator. Regarding *Ls*, it can be approximated to the inductance of a single ring with the average radius of the resonator and the width of the rings, *c*. Taking into account the circuit model of the resonator, its resonance frequency can be calculated as:

$$
\alpha\_0 = \frac{1}{\sqrt{L\_s C\_s}} \tag{1}
$$

The resonators based on the SRR, like the examples represented in Fig. 1 can also be modelled by a simple *L-C* resonant tank (see Fig. 2b). As long as the inductance and the

by loading a CPW structure with SRRs. Left-handed transmission is achieved when the effective permittivity and permeability of a medium are both negative, providing negative values of the phase velocity and refraction index, among other peculiarities (Veselago, 1968). This first left-handed SRR-based CPW was inspired on the medium of Smith et al., 2000. By etching SRRs in the back substrate side of the CPW, beneath the slots, and shunt connected metallic strips between the central strip and ground plane. A one dimensional effective medium with simultaneous negative permeability (due to the presence of the SRRs) and permittivity (thanks to the shunt strips) in a narrow band was achieved (Martín et al., 2003). The resonators used in this kind of transmission lines based on the resonant approach can be SRRs or other similar resonators with different topologies based on the SRR. The layout for the SRR is shown in Fig. 1(a). In Fig. 1, other examples of these resonators, like the spiral resonator (SR, Fig. 1b) with only one metal layer, are shown. Resonators implemented with two metal layers, like the broadside coupled non-bianisotropic split ring resonator (BC-NB-SRR, Fig. 1c), the broadside coupled spiral resonator with two turns (BC-SR(2), Fig. 1d) and the broadside coupled spiral resonator with four turns (BC-SR(4), Fig. 1e) are also depicted

Fig. 1. Examples of topologies of different resonators based on the SRR (a): SR (b), BC-NB-

0

ω

1 *L CS S*

The resonators based on the SRR, like the examples represented in Fig. 1 can also be modelled by a simple *L-C* resonant tank (see Fig. 2b). As long as the inductance and the

= (1)

The equivalent circuit model for the SRR is shown in Fig. 2(a) (Baena et al., 2005). The capacitance *C0*/2 is related with each of the two SRR halves, whereas *Ls* is the resonator selfinductance. *C0* can be obtained as *C0*=2πr*Cpul*, where *Cpul* represents the per unit length capacitance between de rings forming the resonator. Regarding *Ls*, it can be approximated to the inductance of a single ring with the average radius of the resonator and the width of the rings, *c*. Taking into account the circuit model of the resonator, its resonance frequency can

(Marqués et al., 2003; Aznar et al., 2008b).

SRR (c), BC-SR(2) (d) and BC-SR(4) (e).

be calculated as:

capacitance of the resonator can be increased (within the technology limits), the resonance frequency of the SRR can be decreased, reducing its electrical size (Aznar et al., 2008b). This is the case of the resonators shown in Fig. 1, which are electrically smaller than the SRR.

Fig. 2. Equivalent circuit models for a SRR (a) and simplified equivalent circuit model for a resonator based on SRR (b).

Fig. 3. Layouts of metamaterial transmission lines loaded with resonators based on SRR in CPW (a) and microstrip (b) technology.

The layout of two unit cells of metamaterial transmission lines loaded with resonators based on SRR are shown in Fig. 3(a) for CPW and Fig. 3(b) for microstrip technology. In the CPW configuration, the SRRs are paired on the lower substrate side (where they are etched), beneath the slots of the structure and centred with the shunt strips. The resonators are responsible for the negative effective permeability, whereas negative effective permittivity is achieved by means of the shunt connected strips. In case the shunt strips are eliminated, the CPW loaded with resonators provides only negative permeability, showing stop-band behaviour.

In the case of microstrip lines, the SRRs can be etched in pairs on the upper substrate side, adjacent to the conductor strip. The metallic vias are responsible for the negative permittivity of the structure in this case as well.

#### **2.2 Transmission lines based on Complementary Split-Ring Resonators (CSRRs)**

The layout of the CSRR is shown in Fig. 4(a). This resonator results from the application of the Babinet principle to the structure of the SRR, which leads to its complementary

Characterization of Metamaterial

Transmission Lines with Coupled Resonators Through Parameter Extraction 307

CSRR is obtained from the SRR. These resonators, as is shown in Figs. 6 and 7, can be implemented either in microstrip or in coplanar technology (Aznar et al., 2008b, 2009a, 2009b; Durán-Sindreu et al., 2009; Vélez et al., 2010). The equivalent circuit models of the resonators are also shown in Fig. 6. The model of the OSRR is a series LC resonator (Martel et al., 2004). The inductance *Ls* can be obtained as the inductance of a ring with the average radius of the resonator and the same width, *c*, of the rings forming the OSRR. The capacitance *C0* is the distributed edge capacitance that appears between the two concentric rings. In a similar way, the OCSRR can be modelled by means of a parallel LC resonant tank (Aznar et al., 2008b; Vélez et al., 2009), where the inductance *L0* is the inductance of the metallic strip between the slot hooks and the capacitance is that of a disk with radius *r0-c*/2 surrounded by a metallic plane separated by a distance *c*. According to this, it follows that, for identical dimensions and substrate, the resonance frequency of the OSRR or OCSRR is

Fig. 6. Examples of microstrip structures loaded with open resonators. (a) Open split-ring resonator (OSRR) and its equivalent circuit model. (b) Open complementary split-ring

As example, Fig. 7(a), shows a CPW transmission line loaded with a pair of OCSRRs. Fig.

Fig. 7. Layout of a CPW based on OCSRR (a) and based on OSRR (b). In (a) the backside strips (in dark grey) connecting the different ground plane regions are necessary to prevent

the slot mode of the CPW and the second resonance of the OCSRRs.

half the resonance frequency of the SRR or CSRR, respectively.

resonator (OCSRR) and its equivalent circuit model.

7(b) represents an OSRR-based CPW unit cell.

counterpart (Falcone et al., 2004). In the CSRR the rings are etched on a metallic surface and its electric and magnetic properties are interchanged with respect to the SRR: the CSRR can be excited by an axial time-varying electric field and exhibits negative values of the dielectric permittivity.

Fig. 4. (a) Representation of a complementary split ring resonator (CSRR); in this case the resonator is etched on a metallic surface; metallic part is represented in grey. (b) Scheme of a unit cell of a CSRR-based resonant-type metamaterial transmission line. CSRRs are etched on the ground plane (in grey) of a microstrip line just below the capacitive gaps etched on the signal strip (black).

Fig. 5. Equivalent circuit models for a CSRR (a) and simplified equivalent circuit model for a resonator based on CSRR (b).

In this case, the CSRRs are etched on the ground plane of a microstrip transmission line. The CSRRs, which provide negative permittivity, are combined with capacitive gaps etched on the signal strip just above the resonators (see Fig. 4b). The gaps are in this case responsible for the negative permeability. The equivalent circuit model of the CSRR is shown in Fig. 5(a) (Baena et al., 2005). The resonance frequency of the CSRR is roughly the same of the frequency of a SRR with the same dimensions. Many other resonators admit a complementary counterpart, like, for example the SR shown in Fig. 1(b).

#### **2.3 Transmission lines based on open resonators: Open Split-Ring Resonators (OSRRs) and Open Complementary Split-Ring Resonators (OSRRs)**

A different kind of resonators consists in open resonators. Fig. 6 shows the layouts and equivalent circuit models of the open SRR (Martel et al., 2004) and the open complementary SRR (Vélez et al., 2009). As can be seen in the layout, the OSRR is based on the SRR and is obtained by truncating the rings forming the resonator and elongating them outwards. The OCSRR can be obtained as the complementary particle of the OSRR, in a similar way as the

counterpart (Falcone et al., 2004). In the CSRR the rings are etched on a metallic surface and its electric and magnetic properties are interchanged with respect to the SRR: the CSRR can be excited by an axial time-varying electric field and exhibits negative values of the

Fig. 4. (a) Representation of a complementary split ring resonator (CSRR); in this case the resonator is etched on a metallic surface; metallic part is represented in grey. (b) Scheme of a unit cell of a CSRR-based resonant-type metamaterial transmission line. CSRRs are etched on the ground plane (in grey) of a microstrip line just below the capacitive gaps etched on

Fig. 5. Equivalent circuit models for a CSRR (a) and simplified equivalent circuit model for a

In this case, the CSRRs are etched on the ground plane of a microstrip transmission line. The CSRRs, which provide negative permittivity, are combined with capacitive gaps etched on the signal strip just above the resonators (see Fig. 4b). The gaps are in this case responsible for the negative permeability. The equivalent circuit model of the CSRR is shown in Fig. 5(a) (Baena et al., 2005). The resonance frequency of the CSRR is roughly the same of the frequency of a SRR with the same dimensions. Many other resonators admit a

complementary counterpart, like, for example the SR shown in Fig. 1(b).

**(OSRRs) and Open Complementary Split-Ring Resonators (OSRRs)** 

**2.3 Transmission lines based on open resonators: Open Split-Ring Resonators** 

A different kind of resonators consists in open resonators. Fig. 6 shows the layouts and equivalent circuit models of the open SRR (Martel et al., 2004) and the open complementary SRR (Vélez et al., 2009). As can be seen in the layout, the OSRR is based on the SRR and is obtained by truncating the rings forming the resonator and elongating them outwards. The OCSRR can be obtained as the complementary particle of the OSRR, in a similar way as the

dielectric permittivity.

the signal strip (black).

resonator based on CSRR (b).

CSRR is obtained from the SRR. These resonators, as is shown in Figs. 6 and 7, can be implemented either in microstrip or in coplanar technology (Aznar et al., 2008b, 2009a, 2009b; Durán-Sindreu et al., 2009; Vélez et al., 2010). The equivalent circuit models of the resonators are also shown in Fig. 6. The model of the OSRR is a series LC resonator (Martel et al., 2004). The inductance *Ls* can be obtained as the inductance of a ring with the average radius of the resonator and the same width, *c*, of the rings forming the OSRR. The capacitance *C0* is the distributed edge capacitance that appears between the two concentric rings. In a similar way, the OCSRR can be modelled by means of a parallel LC resonant tank (Aznar et al., 2008b; Vélez et al., 2009), where the inductance *L0* is the inductance of the metallic strip between the slot hooks and the capacitance is that of a disk with radius *r0-c*/2 surrounded by a metallic plane separated by a distance *c*. According to this, it follows that, for identical dimensions and substrate, the resonance frequency of the OSRR or OCSRR is half the resonance frequency of the SRR or CSRR, respectively.

Fig. 6. Examples of microstrip structures loaded with open resonators. (a) Open split-ring resonator (OSRR) and its equivalent circuit model. (b) Open complementary split-ring resonator (OCSRR) and its equivalent circuit model.

As example, Fig. 7(a), shows a CPW transmission line loaded with a pair of OCSRRs. Fig. 7(b) represents an OSRR-based CPW unit cell.

Fig. 7. Layout of a CPW based on OCSRR (a) and based on OSRR (b). In (a) the backside strips (in dark grey) connecting the different ground plane regions are necessary to prevent the slot mode of the CPW and the second resonance of the OCSRRs.

Characterization of Metamaterial

(Pozar, 1990)

with ( ) <sup>1</sup>

Eq. 7, Eq. 8 and

ω<sup>0</sup> *L CS S*

with

2

−

Transmission Lines with Coupled Resonators Through Parameter Extraction 309

Due to symmetry considerations and reciprocity, the admittance matrix of the circuit of Fig. 8(a) (which is a biport) must satisfy *Y*12=*Y*21 and *Y*11=*Y*22. From these matrix elements, the series (*Zs*) and shunt (*Zp*) impedances of the equivalent π-circuit model can be obtained

> () ( ) <sup>1</sup> *Z Y <sup>S</sup>* ω21

() ( ) <sup>1</sup> *Z YY <sup>P</sup>* ω

11 21

*Y*21 is inferred by grounding port 1 and obtaining the ratio between the current at port 1 and the applied voltage at port 2. *Y*11 is simply the input admittance of the biport, seen from port 1, with a short circuit at port 2. After a straightforward but tedious calculation, the elements of the admittance matrix are obtained, and by applying Eq. 2 and Eq. 3, we finally obtain

> <sup>4</sup> <sup>2</sup> 2 2 <sup>1</sup>

<sup>+</sup> =+ + − −

*L L <sup>L</sup> Zj M <sup>L</sup> <sup>M</sup> <sup>L</sup>*

( ) 2 2 *P P <sup>L</sup> Z jL*

( ) <sup>2</sup> 2 22 1

*LL L Zj L <sup>L</sup> L C*

2 2

2 2 0

*L LL <sup>C</sup> M L*

<sup>2</sup> <sup>1</sup>

′ <sup>=</sup> <sup>+</sup>

These results indicate that the circuit model of the unit cell of the left-handed lines loaded with SRRs and shunt inductors (Fig. 8a) can be formally expressed as the π circuit of Fig 8(b). These parameters are related to the parameters of the circuit of Fig. 8(a), according to

ω

ω

′ <sup>=</sup>

( ) <sup>2</sup>

*P*

ω ω

*S S*

2

*S*

*L MC*

*S S*

*S*

ω

 ω

= . Expression 4 can be rewritten as

ω

 ω −

−

= (2)

= + (3)

2 2

*S*

2

*P*

*L L*

<sup>+</sup>

*M L L*

*P S*

<sup>2</sup> <sup>2</sup>

ω

*L*

*P*

2

*P*

= + (5)

(4)

(6)

(7)

(8)

*L*

0 2

ω

ω

*S*

*P S SS*

 ′ = + ++′ <sup>−</sup> ′ ′

> 0 2 1 4

> > 4

*M*

<sup>+</sup>

*P*

*L*

+

1 2

1 2

*S P S*

1

#### **3. Equivalent circuit models for metamaterial transmission lines based on the resonant approach**

In this section, we present the equivalent circuit models for the resonant-type metamaterial transmission lines presented in section 2. These models have been widely studied and confirmed and are able to model the composite behaviour of the considered structures (Aznar et al., 2008a; Gil et al., 2006). For a given kind of structure, the circuits are independent on the employed resonator. This means that the model for structures based on SRRs would be the same in case that the employed resonators were, for example, spiral resonators, which are electrically smaller but have the same equivalent circuit as the SRRs.

#### **3.1 Equivalent circuit model for metamaterial transmission lines based on Split-Ring Resonators (SRRs)**

The behaviour of the SRR-based structures (with and without shunt connected strips) can be interpreted to the light of the lumped element equivalent circuit models of the unit cells (Aznar et al., 2008a) (see Fig. 8a). Fig. 9 shows the typical behavior of two structures based on SRRs. The first one (left-handed) includes shunt strips, whereas the second one (negativepermeability) does not. In the circuit model, *L* and *C* account for the line inductance and capacitance, respectively, *Cs* and *Ls* model the SRR, *M* is the mutual inductive coupling between the line and the SRRs, and *Lp* is the inductance of the shunt strips (in case they are included). From the transmission line approach of metamaterials (Caloz & Itoh, 2005; Marqués et al., 2008), it follows that the structure exhibits left-handed wave propagation in those regions where the series reactance and shunt susceptance are negative, whereas in case they are positive, the propagation is conventional. According to this, the model in Fig. 8(a) perfectly explains the composite behavior of the structures shown in Fig. 3 (Aznar et al., 2008c). The inductance of the shunt inductive strips *Lp* is located between the two inductances (*L/2*) that model each line section, to the left and right of the position of the shunt strips. This reflects the location of the inductive strips. The resulting model is neither a π circuit nor a T circuit. Consequently, the transmission zero frequency and the frequency where the phase shift nulls cannot be directly obtained from it. It can be demonstrated that the model of Fig. 8(a) can be transformed to a π circuit, more convenient for its study and formally identical to that of Fig. 8(b) with modified parameters.

Fig. 8. Proposed circuit model for the basic cell of the left handed CPW or microstrip structure with loaded resonators based on the SRR (a). Transformation of the model to a π circuit (b).

Due to symmetry considerations and reciprocity, the admittance matrix of the circuit of Fig. 8(a) (which is a biport) must satisfy *Y*12=*Y*21 and *Y*11=*Y*22. From these matrix elements, the series (*Zs*) and shunt (*Zp*) impedances of the equivalent π-circuit model can be obtained (Pozar, 1990)

$$Z\_{\rm S} \begin{pmatrix} \alpha \end{pmatrix} = \begin{pmatrix} Y\_{\rm 21} \end{pmatrix}^{-1} \tag{2}$$

$$Z\_P\left(\alpha\right) = \left(Y\_{11} + Y\_{21}\right)^{-1} \tag{3}$$

*Y*21 is inferred by grounding port 1 and obtaining the ratio between the current at port 1 and the applied voltage at port 2. *Y*11 is simply the input admittance of the biport, seen from port 1, with a short circuit at port 2. After a straightforward but tedious calculation, the elements of the admittance matrix are obtained, and by applying Eq. 2 and Eq. 3, we finally obtain

$$Z\_S(o) = j o \phi \left( 2 + \frac{L}{2L\_p} \right) \left[ \frac{L}{2} + M^2 \frac{1 + \frac{L}{4L\_p}}{L\_s \left( \frac{o\_0^2}{o^2} - 1 \right) - \frac{M^2}{2L\_p}} \right] \tag{4}$$

$$Z\_P\left(\phi\right) = j\phi \left(2L\_p + \frac{L}{2}\right) \tag{5}$$

with ( ) <sup>1</sup> 2 ω<sup>0</sup> *L CS S* − = . Expression 4 can be rewritten as

$$Z\_{\rm s}(o) = jo \left( 2 + \frac{L}{2L\_{\rm p}} \right) \left[ \frac{L}{2} + L\_{\rm s}' + \frac{L\_{\rm s}'}{1 - L\_{\rm s}'C\_{\rm s}'o^2} \right] \tag{6}$$

with

308 Metamaterial

**3. Equivalent circuit models for metamaterial transmission lines based on the** 

In this section, we present the equivalent circuit models for the resonant-type metamaterial transmission lines presented in section 2. These models have been widely studied and confirmed and are able to model the composite behaviour of the considered structures (Aznar et al., 2008a; Gil et al., 2006). For a given kind of structure, the circuits are independent on the employed resonator. This means that the model for structures based on SRRs would be the same in case that the employed resonators were, for example, spiral resonators, which are electrically smaller but have the same equivalent circuit as the SRRs.

**3.1 Equivalent circuit model for metamaterial transmission lines based on Split-Ring** 

The behaviour of the SRR-based structures (with and without shunt connected strips) can be interpreted to the light of the lumped element equivalent circuit models of the unit cells (Aznar et al., 2008a) (see Fig. 8a). Fig. 9 shows the typical behavior of two structures based on SRRs. The first one (left-handed) includes shunt strips, whereas the second one (negativepermeability) does not. In the circuit model, *L* and *C* account for the line inductance and capacitance, respectively, *Cs* and *Ls* model the SRR, *M* is the mutual inductive coupling between the line and the SRRs, and *Lp* is the inductance of the shunt strips (in case they are included). From the transmission line approach of metamaterials (Caloz & Itoh, 2005; Marqués et al., 2008), it follows that the structure exhibits left-handed wave propagation in those regions where the series reactance and shunt susceptance are negative, whereas in case they are positive, the propagation is conventional. According to this, the model in Fig. 8(a) perfectly explains the composite behavior of the structures shown in Fig. 3 (Aznar et al., 2008c). The inductance of the shunt inductive strips *Lp* is located between the two inductances (*L/2*) that model each line section, to the left and right of the position of the shunt strips. This reflects the location of the inductive strips. The resulting model is neither a π circuit nor a T circuit. Consequently, the transmission zero frequency and the frequency where the phase shift nulls cannot be directly obtained from it. It can be demonstrated that the model of Fig. 8(a) can be transformed to a π circuit, more convenient for its study and

formally identical to that of Fig. 8(b) with modified parameters.

Fig. 8. Proposed circuit model for the basic cell of the left handed CPW or microstrip structure with loaded resonators based on the SRR (a). Transformation of the model to a π

**resonant approach** 

**Resonators (SRRs)** 

circuit (b).

$$L\_s' = 2M^2 \mathcal{C}\_s a\_0^2 \frac{\left(1 + \frac{L}{4L\_p}\right)^2}{1 + \frac{M^2}{2L\_p L\_s}}\tag{7}$$

$$C\_{\rm s}^{\prime} = \frac{L\_{\rm s}}{2M^2 a\_0^2} \left( \frac{1 + \frac{M^2}{2L\_p L\_{\rm s}}}{1 + \frac{L}{4L\_p}} \right)^2 \tag{8}$$

These results indicate that the circuit model of the unit cell of the left-handed lines loaded with SRRs and shunt inductors (Fig. 8a) can be formally expressed as the π circuit of Fig 8(b). These parameters are related to the parameters of the circuit of Fig. 8(a), according to Eq. 7, Eq. 8 and

Characterization of Metamaterial

Transmission Lines with Coupled Resonators Through Parameter Extraction 311

In Fig. 9 we can see two configuration examples for a CPW with loaded SRRs, with and without shunt strips. Electromagnetic simulation, circuit simulation and measurement show very good agreement in both cases. Therefore, we are able to confirm that the proposed circuit model is correct and accurate. This circuit model can also be applied to the microstrip

The equivalent circuit model for the CSRR-based structure shown in Fig. 4(b) is the circuit shown in Fig. 10(a), which provides an accurate description of the behaviour of the structure (Aznar et al, 2008c, 2008d) and can be transformed into the circuit shown in Fig 10(b).

Fig. 10. Equivalent circuit model for the structure based on CSRR shown in Fig. 4(b) (a).

In the equivalent circuit (Fig. 10 a), the resonator is modelled by the resonant tank formed by *Lc* and *Cc*. The line parameters are *L* and *CL* and the gap is modelled by the π-structure formed by *Cs* and *Cf*, which take into account the series and the fringing capacitances due to the presence of the capacitive gap. The modified circuit (Fig. 10b) is perfectly able to reproduce the behaviour of the structure. Nevertheless, the equivalent circuit (Fig. 10a) can be transformed into the modified circuit (Fig. 10b), by means of the following equations:

22 *C CC <sup>g</sup>* = +*<sup>S</sup> par* (14)

*C CC*

so that the modified circuit (Fig. 10b), much simpler, can substitute the equivalent circuit (Fig. 10a) for a more straightforward work. The excellent agreement between electromagnetic and electrical simulation (employing the proposed model) can be observed

In case the host transmission line is loaded just with CSRRs, the resulting structure shows negative permittivity and, thus, stop-band behaviour. This is the case of the structure shown

ω*z*, ω *C*

*C*

in Fig. 11 (a). The notable frequencies (

section 4.1.

( ) 2 *par S par S*

*C CC par* = +*<sup>f</sup> <sup>L</sup>* (13)

<sup>+</sup> <sup>=</sup> (15)

*<sup>0</sup>*) of this circuit model are indicated below in

Modified circuit model for the structure based on CSRR shown in Fig. 4(b) (b).

structure shown in Fig. 3(b), which is in this sense, equivalent to the CPW structure.

**3.2 Equivalent circuit model for metamaterial transmission lines based on** 

**Complementary Split-Ring Resonators (CSRRs)** 

$$L' = \left(2 + \frac{L}{2L\_{\mathbb{P}}}\right)\frac{L}{2} - L\_{\mathbb{S}}'\tag{9}$$

$$L\_p' = 2L\_p + \frac{L}{2} \tag{10}$$

The transmission zero frequency ω*<sup>z</sup>* for the circuit of Fig. 8(a) can be obtained forcing *ZS(*ω*)*=∞. This gives

$$\alpha\_{\mathbb{Z}} = \alpha\_0 \sqrt{\frac{1}{1 + \frac{M^2}{2L\_p L\_s}}} \tag{11}$$

It can be observed, that the transmission zero frequency is always located below the resonance frequency of the SRRs ω*0* ( ω*z*<ω*0*). On the other hand, the frequency where φ=0, ω*S*, is obtained by forcing *ZS(*ω*)*=0. This gives

$$\alpha\_{\rm s} = \frac{1}{\sqrt{C\_{\rm s} \left(L\_{\rm s} - 2\frac{M^2}{L}\right)}}\tag{12}$$

Despite that *ZS(*ω*)*=0 is a function of *LP*, unexpectedly, ω*<sup>S</sup>* does not depend on the shunt inductance.

Fig. 9. Layouts of the considered CPW structures with SRRs and shunt strips (a) and with SRRs only (b); simulated and measured transmission coefficient *S*21 and simulated dispersion relation (c). The considered substrate is Rogers RO3010 with thickness *h*=1.27 mm and dielectric constant ε*r*=10.2. Relevant dimensions are rings width *c*=0.6 mm, distance between the rings *d*=0.2 mm, and internal radius *r*=2.4 mm. For the CPW structure, the central strip width is *W*=7 mm and the width of the slots is *G*=1.35 mm. Finally, the shunt strip width is 0.2 mm. The results of the electrical simulation with extracted parameters are depicted by using symbols. We have actually represented the modulus of the phase since it is negative for the left-handed line. Discrepancy between measurement and simulation is attributed to fabrication related tolerances.

2 2 *P P*

> 0 2 1

> > +

1

*S S <sup>M</sup> C L*

Fig. 9. Layouts of the considered CPW structures with SRRs and shunt strips (a) and with SRRs only (b); simulated and measured transmission coefficient *S*21 and simulated dispersion relation (c). The considered substrate is Rogers RO3010 with thickness *h*=1.27 mm and dielectric constant ε*r*=10.2. Relevant dimensions are rings width *c*=0.6 mm, distance between the rings *d*=0.2 mm, and internal radius *r*=2.4 mm. For the CPW structure, the central strip width is *W*=7 mm and the width of the slots is *G*=1.35 mm. Finally, the shunt strip width is 0.2 mm. The results of the electrical simulation with extracted parameters are depicted by using symbols. We have actually represented the modulus of the phase since it is negative for the left-handed line. Discrepancy between measurement and simulation is

2

 − 

*P S M L L*

2

ω

*L*

1 2

It can be observed, that the transmission zero frequency is always located below the

2 2 *<sup>S</sup> P L L L L L* ′ ′ =+ − 

(9)

(11)

(12)

*<sup>S</sup>* does not depend on the shunt

φ=0,

*<sup>L</sup> L L* ′ = + (10)

*<sup>z</sup>* for the circuit of Fig. 8(a) can be obtained forcing

*0*). On the other hand, the frequency where

2

ω

*Z*

ω ω=

*)*=0. This gives

*S*

*)*=0 is a function of *LP*, unexpectedly,

ω=

ω*0* ( ω*z*<ω

ω

The transmission zero frequency

resonance frequency of the SRRs

ω

attributed to fabrication related tolerances.

*S*, is obtained by forcing *ZS(*

Despite that *ZS(*

inductance.

*)*=∞. This gives

*ZS(*ω

ω

In Fig. 9 we can see two configuration examples for a CPW with loaded SRRs, with and without shunt strips. Electromagnetic simulation, circuit simulation and measurement show very good agreement in both cases. Therefore, we are able to confirm that the proposed circuit model is correct and accurate. This circuit model can also be applied to the microstrip structure shown in Fig. 3(b), which is in this sense, equivalent to the CPW structure.

#### **3.2 Equivalent circuit model for metamaterial transmission lines based on Complementary Split-Ring Resonators (CSRRs)**

The equivalent circuit model for the CSRR-based structure shown in Fig. 4(b) is the circuit shown in Fig. 10(a), which provides an accurate description of the behaviour of the structure (Aznar et al, 2008c, 2008d) and can be transformed into the circuit shown in Fig 10(b).

Fig. 10. Equivalent circuit model for the structure based on CSRR shown in Fig. 4(b) (a). Modified circuit model for the structure based on CSRR shown in Fig. 4(b) (b).

In the equivalent circuit (Fig. 10 a), the resonator is modelled by the resonant tank formed by *Lc* and *Cc*. The line parameters are *L* and *CL* and the gap is modelled by the π-structure formed by *Cs* and *Cf*, which take into account the series and the fringing capacitances due to the presence of the capacitive gap. The modified circuit (Fig. 10b) is perfectly able to reproduce the behaviour of the structure. Nevertheless, the equivalent circuit (Fig. 10a) can be transformed into the modified circuit (Fig. 10b), by means of the following equations:

$$\mathbf{C}\_{\mu\nu} = \mathbf{C}\_f + \mathbf{C}\_L \tag{13}$$

$$\mathcal{2C}\_{\mathcal{g}} = \mathcal{2C}\_{\mathcal{S}} + \mathcal{C}\_{\mu\nu} \tag{14}$$

$$\mathbf{C} = \frac{\mathbf{C}\_{\mu\nu} \left( 2\mathbf{C}\_{\text{S}} + \mathbf{C}\_{\mu\nu} \right)}{\mathbf{C}\_{\text{S}}} \tag{15}$$

so that the modified circuit (Fig. 10b), much simpler, can substitute the equivalent circuit (Fig. 10a) for a more straightforward work. The excellent agreement between electromagnetic and electrical simulation (employing the proposed model) can be observed in Fig. 11 (a). The notable frequencies (ω*z*, ω*<sup>0</sup>*) of this circuit model are indicated below in section 4.1.

In case the host transmission line is loaded just with CSRRs, the resulting structure shows negative permittivity and, thus, stop-band behaviour. This is the case of the structure shown

Characterization of Metamaterial

**on the resonant approach** 

**Ring Resonators (SRRs)** 

Transmission Lines with Coupled Resonators Through Parameter Extraction 313

Fig. 12. (a) Circuit model, and (b) simplified circuit model of a CPW transmission line loaded with a pair of OCSRRs. (c) Circuit model, and (d) simplified circuit model of a CPW

The transformation to simplify the circuit based on OSRR (Figs. 7b, 12 c and 12 d) is:

and the transformations to simplify the circuit based on OCSRR (Figs. 7a, 12 a and 12 b) are:

2 *P*

*P*

**4. Parameter extraction technique for metamaterial transmission lines based** 

In previous sections we have presented different metamaterial transmission lines loaded with resonators and their equivalent circuit models. In this section we present a parameter extraction technique which, properly modified, can be applied to all of them. With this technique we can determinate the parameters of the circuit model of the structure. This represents an important assist in the design process, which becomes easier and faster.

The parameter extraction method consists in the imposition of several conditions obtained either from the simulated or measured response of the considered structure. The number of imposed conditions must be enough to obtain the values of all the parameters of the circuit.

**4.1 Parameter extraction technique metamaterial transmission lines based on Split-**

The parameter extraction for the metamaterial transmission line based on SRRs is focused on the simplified equivalent circuit of the Fig. 8(b). This method was proposed in the reference (Aznar et al., 2008e). Since the number of parameters of the circuit model is five, we also need five conditions to univocally determine all parameters. From the representation of the reflection coefficient of a single unit cell, *S*11, in the Smith chart, two

2 *LL L S S* ′ = + (16)

2 2 *CCC P P* ′ = + (17)

*<sup>L</sup> <sup>L</sup>*′ <sup>=</sup> (18)

transmission line loaded with a series connected OSRR.

in Fig. 11 (b). For these structures, the circuit model would be the same as in Fig. 10(b) except for the capacitances *Cg*, which would be eliminated.

Fig. 11. Simulated (through the Agilent Momentum commercial software) frequency responses of the unit cell structures shown in the insets: (a) microstrip line loaded with CSRRs and series gaps, (b) microstrip line only loaded with CSRRs. The response that has been obtained from circuit simulation of the equivalent model with extracted parameters is also included. For the structures (a) and (b), the dimensions are: the strip line width *Wm* = 1.15 mm, the length *D* = 8 mm, and the gap width *wg* = 0.16 mm. In both cases, the dimensions for the CSRR are: the outer ring width *cout* = 0.364 mm, the inner ring width *cinn* = 0.366 mm, distance between the rings *d* = 0.24 mm, and the internal radius *r* = 2.691 mm. The considered substrate is Rogers RO3010 with the dielectric constant ε*<sup>r</sup>* = 10.2 and the thickness *h* = 1.27 mm.

#### **3.3 Equivalent circuit models for metamaterial transmission lines based on open resonators: Open Split-Ring Resonators (OSRRs) and Open Complementary Split-Ring Resonators (OSRRs)**

We will now consider the model of a CPW transmission line loaded with a pair of OCSRRs shown in Fig. 7(a). The resonator is represented by the elements *Cp* and *Lp* in Fig. 12(a). Although the particle is electrically small, it has been found that the structure exhibits certain frequency shift at resonance with respect to the frequency theoretically predicted by the equivalent circuit shown in Fig. 6. This frequency shift is expected if access lines are present. However, in the absence of access lines, we still obtain a small (although non negligible) phase shift. This means that the OCSRR-loaded CPW cannot be merely modeled as a two-port network with a shunt connected parallel resonator. To properly model the structure, we must introduce additional elements to account for the phase shift. That is, we must introduce phase shifting lines at both sides of the resonator. Such transmission line sections can be modeled through series inductances (*L*) and shunt capacitances (*C*), as depicted in Fig. 12(a). For design purposes, we can also use the simplified model depicted in Fig 12(b).

For a CPW loaded with a series connected OSRR represented in Fig. 7(b), a similar phenomenology results. Thus, to take into account these parasitic effects, we must introduce additional elements in the two port network describing the structure. A typical topology and the circuit model of these OSRR-loaded CPW transmission line sections are depicted in Fig. 12(c) and Fig. 12(d).

in Fig. 11 (b). For these structures, the circuit model would be the same as in Fig. 10(b)

Fig. 11. Simulated (through the Agilent Momentum commercial software) frequency responses of the unit cell structures shown in the insets: (a) microstrip line loaded with CSRRs and series gaps, (b) microstrip line only loaded with CSRRs. The response that has been obtained from circuit simulation of the equivalent model with extracted parameters is also included. For the structures (a) and (b), the dimensions are: the strip line width *Wm* = 1.15 mm, the length *D* = 8 mm, and the gap width *wg* = 0.16 mm. In both cases, the

dimensions for the CSRR are: the outer ring width *cout* = 0.364 mm, the inner ring width *cinn* = 0.366 mm, distance between the rings *d* = 0.24 mm, and the internal radius *r* = 2.691 mm.

**3.3 Equivalent circuit models for metamaterial transmission lines based on open resonators: Open Split-Ring Resonators (OSRRs) and Open Complementary Split-**

We will now consider the model of a CPW transmission line loaded with a pair of OCSRRs shown in Fig. 7(a). The resonator is represented by the elements *Cp* and *Lp* in Fig. 12(a). Although the particle is electrically small, it has been found that the structure exhibits certain frequency shift at resonance with respect to the frequency theoretically predicted by the equivalent circuit shown in Fig. 6. This frequency shift is expected if access lines are present. However, in the absence of access lines, we still obtain a small (although non negligible) phase shift. This means that the OCSRR-loaded CPW cannot be merely modeled as a two-port network with a shunt connected parallel resonator. To properly model the structure, we must introduce additional elements to account for the phase shift. That is, we must introduce phase shifting lines at both sides of the resonator. Such transmission line sections can be modeled through series inductances (*L*) and shunt capacitances (*C*), as depicted in Fig. 12(a). For design purposes, we can also use the simplified model depicted in

For a CPW loaded with a series connected OSRR represented in Fig. 7(b), a similar phenomenology results. Thus, to take into account these parasitic effects, we must introduce additional elements in the two port network describing the structure. A typical topology and the circuit model of these OSRR-loaded CPW transmission line sections are depicted in

ε

*<sup>r</sup>* = 10.2 and the

The considered substrate is Rogers RO3010 with the dielectric constant

thickness *h* = 1.27 mm.

Fig 12(b).

Fig. 12(c) and Fig. 12(d).

**Ring Resonators (OSRRs)** 

except for the capacitances *Cg*, which would be eliminated.

Fig. 12. (a) Circuit model, and (b) simplified circuit model of a CPW transmission line loaded with a pair of OCSRRs. (c) Circuit model, and (d) simplified circuit model of a CPW transmission line loaded with a series connected OSRR.

The transformation to simplify the circuit based on OSRR (Figs. 7b, 12 c and 12 d) is:

$$L\_{\rm s}^{\prime} = L\_{\rm s} + 2L$$

and the transformations to simplify the circuit based on OCSRR (Figs. 7a, 12 a and 12 b) are:

$$\mathbf{C}'\_{p} = \mathbf{2C}\_{p} + \mathbf{2C} \tag{17}$$

$$L\_p' = \frac{L\_p}{2} \tag{18}$$

#### **4. Parameter extraction technique for metamaterial transmission lines based on the resonant approach**

In previous sections we have presented different metamaterial transmission lines loaded with resonators and their equivalent circuit models. In this section we present a parameter extraction technique which, properly modified, can be applied to all of them. With this technique we can determinate the parameters of the circuit model of the structure. This represents an important assist in the design process, which becomes easier and faster.

The parameter extraction method consists in the imposition of several conditions obtained either from the simulated or measured response of the considered structure. The number of imposed conditions must be enough to obtain the values of all the parameters of the circuit.

#### **4.1 Parameter extraction technique metamaterial transmission lines based on Split-Ring Resonators (SRRs)**

The parameter extraction for the metamaterial transmission line based on SRRs is focused on the simplified equivalent circuit of the Fig. 8(b). This method was proposed in the reference (Aznar et al., 2008e). Since the number of parameters of the circuit model is five, we also need five conditions to univocally determine all parameters. From the representation of the reflection coefficient of a single unit cell, *S*11, in the Smith chart, two

Characterization of Metamaterial

circuit parameters of the circuit model in Fig. 8(b).

left handed cell based on a CPW structure.

expressions:

**Complementary Split-Ring Resonators (CSRRs)** 

Transmission Lines with Coupled Resonators Through Parameter Extraction 315

reflection coefficient on a Smith chart and read the susceptance seen from the ports at that frequency where *S*11 intercepts the unit conductance circle. Since this is simply the susceptance corresponding the line capacitance (provided *Lp* has been removed), we can thus univocally determine *C*. Hence, this is the fifth condition that is required to extract the

Fig. 13. Reflection coefficient on the Smith chart (a); frequency response (reflection, *S11*, and transmission, *S21*, coefficients) depicted in a decibel scale and the dispersion relation (b) for a

The technique is based on the equivalent circuit model of a CSRR-loaded transmission line shown in Fig. 10(b). This method was proposed in 2006 (Bonache et al., 2006c). The considered structures are a negative permittivity as well as a left handed microstrip line (gap capacitors are required in the latter case). In view of the models, if losses are neglected (this is reasonable in a first order approximation), two characteristic frequencies can be identified: the frequency that nulls the shunt impedance (transmission zero frequency, *fz*) and the frequency that nulls the shunt admittance (which obviously coincides with the intrinsic resonance frequency of the CSRR, *f0*). These frequencies are given by the following

( )

(24)

*L C* <sup>=</sup> (25)

*C C*

1

1 2 *C C*

π

and they can be either experimentally determined, or obtained from the simulated response of the structure. At *fz* a notch in the transmission coefficient is expected and this frequency can be accurately measured. To obtain *f0*, a representation of the transmission coefficient on a Smith chart is required (see example in Fig. 14). At this frequency the shunt path to ground

2 *<sup>Z</sup>*

0

*f*

π*LCC* <sup>=</sup> <sup>+</sup>

*f*

**4.2 Parameter extraction technique metamaterial transmission lines based on** 

conditions are obtained. Firstly, we can determine the frequency that nulls the series reactance, *fs*, from the intercept of *S*11 with the unit conductance circle (see Fig. 13 a). This is obvious since at this frequency, the real part of the admittance seen from the ports is simply the admittance of the opposite port, that is, *Y0*=(*Z0)*−1=(50Ω)−1=0.02 S. Hence, *S*11 must be allocated in the unit conductance circle at *fs*, as illustrated in the example provided in Fig. 13. This frequency is given by the following expression:

$$f\_s = \frac{1}{2\pi} \sqrt{\frac{1}{L\_s' C\_s'} + \frac{1}{L'C\_s'}}\tag{19}$$

Secondly, the susceptance of the unit cell seen from the ports at *fs*, whose value can be inferred from the Smith chart, is

$$B\left(a o\_{\rm s}\right) = \frac{CL\_{\rm p}'a o\_{\rm s}^2 - 2}{L\_{\rm p}'a o\_{\rm s}}\tag{20}$$

with 2 *S S* ω π = *f* . The next condition concerns the parallel resonator of the series branch. Namely, the resonance frequency of this resonator is given by

$$f\_Z = \frac{1}{2\pi} \sqrt{\frac{1}{L\_s' C\_s'}}\tag{21}$$

Notice that this frequency does not coincide with the intrinsic resonance frequency of the magnetically driven resonator, *fo (*which is the resonance frequency of the tank formed by *Ls* and *Cs).* The transmission zero frequency *fz (*Eq. 21) can be easily obtained from the transmission coefficient *S*21 of the unit cell since at this frequency the series branch is opened and the whole power injected from the input port is reflected back to the source. Thus, the transmission coefficient nulls (zero transmission) and *fz* can easily be identified from the representation of the transmission coefficient in a decibel scale (see example in Fig. 13 b).

Another condition can be deduced from the phase of the transmission coefficient, φ*S21*. At the frequency where φ*S21*=90º, *f*π*/2*, the electrical length of the unit cell, φ*=βl* (*β* being the phase constant and *l* the length of the unit cell), is ( ) <sup>2</sup> *f* 90º φ π = − . Since the dispersion relation of a periodic structure consisting of cascaded unit cells, as those in Fig. 8(b) is

$$\cos \phi = 1 + \frac{Z\_s(o)}{Z\_\rho(o)} \tag{22}$$

with *Zs* and *Zp* being the series and shunt impedances, respectively, of the π-circuit model, it follows that

$$Z\_S\left(\alpha\_{\pi\ell^2}\right) = -Z\_P\left(\alpha\_{\pi\ell^2}\right) \tag{23}$$

with 2 2 2 *f* ω π π π = . Expressions (19-21) and (23) are four of the five conditions needed to univocally determine the circuit parameters in Fig. 8(b). Now, by removing the shunt connected vias or strips in the layouts in Fig. 3, we can represent the corresponding

conditions are obtained. Firstly, we can determine the frequency that nulls the series reactance, *fs*, from the intercept of *S*11 with the unit conductance circle (see Fig. 13 a). This is obvious since at this frequency, the real part of the admittance seen from the ports is simply the admittance of the opposite port, that is, *Y0*=(*Z0)*−1=(50Ω)−1=0.02 S. Hence, *S*11 must be allocated in the unit conductance circle at *fs*, as illustrated in the example provided in Fig. 13.

11 1

*LC LC*

Secondly, the susceptance of the unit cell seen from the ports at *fs*, whose value can be

( ) <sup>2</sup> <sup>2</sup> *P S*

*L* ω

1 1

π

Notice that this frequency does not coincide with the intrinsic resonance frequency of the magnetically driven resonator, *fo (*which is the resonance frequency of the tank formed by *Ls* and *Cs).* The transmission zero frequency *fz (*Eq. 21) can be easily obtained from the transmission coefficient *S*21 of the unit cell since at this frequency the series branch is opened and the whole power injected from the input port is reflected back to the source. Thus, the transmission coefficient nulls (zero transmission) and *fz* can easily be identified from the representation of the transmission coefficient in a decibel scale (see example in Fig. 13 b).

*SS S*

*P S*

ω

= *f* . The next condition concerns the parallel resonator of the series branch.

*S S*

*/2*, the electrical length of the unit cell,

( )

ω

ω

 ω

= . Expressions (19-21) and (23) are four of the five conditions needed to

 π

= + ′′ ′′ (19)

′ <sup>−</sup> <sup>=</sup> ′ (20)

*L C* <sup>=</sup> ′ ′ (21)

φ

= − . Since the dispersion relation of a

= + (22)

2 2 = − ( ) (23)

φ*S21*. At

*=βl* (*β* being the phase

2 *<sup>S</sup>*

π

*S*

ω

*CL <sup>B</sup>*

2 *<sup>Z</sup>*

Another condition can be deduced from the phase of the transmission coefficient,

φ

*Z Z S P* ( ) ω

π

φ π

( ) cos 1 *<sup>S</sup> P Z Z*

with *Zs* and *Zp* being the series and shunt impedances, respectively, of the π-circuit model, it

univocally determine the circuit parameters in Fig. 8(b). Now, by removing the shunt connected vias or strips in the layouts in Fig. 3, we can represent the corresponding

*f*

*f*

Namely, the resonance frequency of this resonator is given by

π

periodic structure consisting of cascaded unit cells, as those in Fig. 8(b) is

constant and *l* the length of the unit cell), is ( ) <sup>2</sup> *f* 90º

This frequency is given by the following expression:

inferred from the Smith chart, is

with 2 *S S* ω

 π

the frequency where

follows that

with 2 2 2 *f* ω

π

 π

 π φ*S21*=90º, *f* reflection coefficient on a Smith chart and read the susceptance seen from the ports at that frequency where *S*11 intercepts the unit conductance circle. Since this is simply the susceptance corresponding the line capacitance (provided *Lp* has been removed), we can thus univocally determine *C*. Hence, this is the fifth condition that is required to extract the circuit parameters of the circuit model in Fig. 8(b).

Fig. 13. Reflection coefficient on the Smith chart (a); frequency response (reflection, *S11*, and transmission, *S21*, coefficients) depicted in a decibel scale and the dispersion relation (b) for a left handed cell based on a CPW structure.

#### **4.2 Parameter extraction technique metamaterial transmission lines based on Complementary Split-Ring Resonators (CSRRs)**

The technique is based on the equivalent circuit model of a CSRR-loaded transmission line shown in Fig. 10(b). This method was proposed in 2006 (Bonache et al., 2006c). The considered structures are a negative permittivity as well as a left handed microstrip line (gap capacitors are required in the latter case). In view of the models, if losses are neglected (this is reasonable in a first order approximation), two characteristic frequencies can be identified: the frequency that nulls the shunt impedance (transmission zero frequency, *fz*) and the frequency that nulls the shunt admittance (which obviously coincides with the intrinsic resonance frequency of the CSRR, *f0*). These frequencies are given by the following expressions:

$$f\_{\chi} = \frac{1}{2\pi\sqrt{L\_{\subset}(\mathbb{C} + \mathbb{C}\_{\subset})}}\tag{24}$$

$$f\_0 = \frac{1}{2\pi\sqrt{L\_\odot C\_\odot}}\tag{25}$$

and they can be either experimentally determined, or obtained from the simulated response of the structure. At *fz* a notch in the transmission coefficient is expected and this frequency can be accurately measured. To obtain *f0*, a representation of the transmission coefficient on a Smith chart is required (see example in Fig. 14). At this frequency the shunt path to ground

Characterization of Metamaterial

**Split-Ring Resonators (OSRRs)** 

resonance frequency of the series branch

characteristic impedance is given by

obtained:

that is,

Transmission Lines with Coupled Resonators Through Parameter Extraction 317

This subsection is focused on the parameter extraction for the structures represented in Fig. 7. The parameters of the circuit model of a CPW loaded with an OSRR (Fig. 12 d) can be extracted from the measurement or the electromagnetic simulation of the structure following a straightforward procedure (Durán-Sindreu et al., 2009a, 2009b). First of all, from the intercept of the return loss curve with the unit conductance circle in the Smith chart, we

> <sup>0</sup> 2 *ZS <sup>B</sup> <sup>C</sup>* ω=

> > 1 *ZS C LS S*

where *B* is the susceptance in the intercept point. The frequency at this intercept point is the

To determine the other two element values of this branch, one more condition is needed.

transmission) the characteristic impedance of the structure is 50Ω. In this π-circuit, the

( ) () ()

Thus, by forcing this impedance to 50Ω, the second condition results. By inverting (27) and (28), we can determine the element values of the series branch. The following results are

> 2 2 2 <sup>0</sup> <sup>0</sup> <sup>1</sup> <sup>1</sup>

> > 2 0

The parameters of the circuit model of a CPW loaded with an OCSRR (Fig. 12 b) can be extracted following a similar procedure. In this case, the intercept of the return loss curve with the unit resistance circle in the Smith chart gives the value of the series inductance

> 2 *ZP <sup>X</sup> <sup>L</sup>* ω→∞

where *X* is the reactance at the intercept point. The shunt branch resonates at this frequency,

ω *C*<sup>=</sup> ′ <sup>=</sup> ′

*<sup>Z</sup> <sup>Z</sup> <sup>C</sup> <sup>C</sup>*

=− +

1

*S*

*<sup>S</sup> <sup>Z</sup>*

() ()

 ω

2 2

*Z C*

ω  ω

2

*P S P S Z Z*

*Z Z* ω

ω

2 0

ω

This condition comes from the fact that at the reflection zero frequency

<sup>0</sup> 2

2

ω

ω

*S Z*

*S*

*L*

=

ω

*Z*

*S*

= (27)

<sup>=</sup> <sup>=</sup> ′ (28)

<sup>=</sup> <sup>+</sup> (29)

ω

(30)

(31)

= (32)

*<sup>z</sup>* (maximum

**4.3 Parameter extraction technique for metamaterial transmission lines based on open resonators: Open Split-Ring Resonators (OSRRs) and Open Complementary** 

can directly infer the value of the shunt capacitance according to

is opened, and the input impedance seen from the ports is solely formed by the series elements of the structure (*L*, for the negative permittivity line, and *L* and *Cg* for the left handed line) and the resistance of the opposite port (50 Ω). Therefore, *f0* is given by the intersection between the measured (or simulated) *S11* curve and the unit normalized resistance circle. From this result we can also obtain the impedance of the series elements at that frequency. This gives directly the value of *L* for the negative permittivity line. For the left handed line, *L* can be independently estimated from a transmission line calculator, or from the value extracted for the negative permeability line, corrected by the presence of the gap, whereas *Cg* can be determined by adjusting its value to fit the impedance value read from the Smith chart at *f0* from the simulation or experiment.

Fig. 14. (a) Layouts of the structures employed in the parameter extraction method for the CSRR-based unit cell. (b) Frequency responses of the measurement, the electromagnetic simulation and the electric simulation employing the extracted parameters. Representation of the S11 parameter for the identification of the CSRR resonance frequency (c) for the complete (LH) structure (d) for the structure without gap (ε<0). Measurement and electric simulation.

Expressions (24) and (25) are dependent on three parameters. Therefore, we cannot directly obtain the element values of the CSRR (as desired) and the coupling capacitance. To this end, we need an additional condition, namely:

$$Z\_s\left(j\alpha\_{\pi\sharp^2}\right) = -Z\_p\left(j\alpha\_{\pi\sharp^2}\right) \tag{26}$$

where *ZS* ( ) *j*ω and *ZP* ( ) *j*ω are the series and shunt impedance of the T-circuit model of the structure, respectively, and ωπ 2 is the angular frequency where the phase of the transmission coefficient (which is a measurable quantity) is ( ) <sup>21</sup> φ *S* = π 2 . Thus, from (23)–(25) we can determine the three reactive element values that contribute to the shunt impedance.

is opened, and the input impedance seen from the ports is solely formed by the series elements of the structure (*L*, for the negative permittivity line, and *L* and *Cg* for the left handed line) and the resistance of the opposite port (50 Ω). Therefore, *f0* is given by the intersection between the measured (or simulated) *S11* curve and the unit normalized resistance circle. From this result we can also obtain the impedance of the series elements at that frequency. This gives directly the value of *L* for the negative permittivity line. For the left handed line, *L* can be independently estimated from a transmission line calculator, or from the value extracted for the negative permeability line, corrected by the presence of the gap, whereas *Cg* can be determined by adjusting its value to fit the impedance value read

Fig. 14. (a) Layouts of the structures employed in the parameter extraction method for the CSRR-based unit cell. (b) Frequency responses of the measurement, the electromagnetic simulation and the electric simulation employing the extracted parameters. Representation of the S11 parameter for the identification of the CSRR resonance frequency (c) for the complete (LH) structure (d) for the structure without gap (ε<0). Measurement and electric

Expressions (24) and (25) are dependent on three parameters. Therefore, we cannot directly obtain the element values of the CSRR (as desired) and the coupling capacitance. To this

> ω

φ*S* =

 π

are the series and shunt impedance of the T-circuit model of the

π

2 is the angular frequency where the phase of the transmission

2 2 = − ( ) (26)

2 . Thus, from (23)–(25) we can

*Zj Zj S P* ( ) ω

π

determine the three reactive element values that contribute to the shunt impedance.

from the Smith chart at *f0* from the simulation or experiment.

simulation.

where *ZS* ( ) *j*

ω

structure, respectively, and

end, we need an additional condition, namely:

ω

ωπ

coefficient (which is a measurable quantity) is ( ) <sup>21</sup>

and *ZP* ( ) *j*

#### **4.3 Parameter extraction technique for metamaterial transmission lines based on open resonators: Open Split-Ring Resonators (OSRRs) and Open Complementary Split-Ring Resonators (OSRRs)**

This subsection is focused on the parameter extraction for the structures represented in Fig. 7. The parameters of the circuit model of a CPW loaded with an OSRR (Fig. 12 d) can be extracted from the measurement or the electromagnetic simulation of the structure following a straightforward procedure (Durán-Sindreu et al., 2009a, 2009b). First of all, from the intercept of the return loss curve with the unit conductance circle in the Smith chart, we can directly infer the value of the shunt capacitance according to

$$C = \frac{B}{2a\_{\mathbb{Z}\_8 \to 0}}\tag{27}$$

where *B* is the susceptance in the intercept point. The frequency at this intercept point is the resonance frequency of the series branch

$$\left. \phi^2 \right|\_{Z\_s=0} = \frac{1}{C\_s L\_s'} \tag{28}$$

To determine the other two element values of this branch, one more condition is needed. This condition comes from the fact that at the reflection zero frequency ω*<sup>z</sup>* (maximum transmission) the characteristic impedance of the structure is 50Ω. In this π-circuit, the characteristic impedance is given by

$$Z\_0\left(\left.\alpha\right>\right) = \sqrt{\frac{Z\_\rho\left(\left.\alpha\right>^2 Z\_s\left(\left.\alpha\right)\right>}{2Z\_\rho\left(\left.\alpha\right> + Z\_s\left(\left.\alpha\right)\right>}}}\tag{29}$$

Thus, by forcing this impedance to 50Ω, the second condition results. By inverting (27) and (28), we can determine the element values of the series branch. The following results are obtained:

$$C\_s = \left[\frac{\alpha\_\varkappa^2}{\alpha^2\Big|\_{Z\_s=0}} - 1\right] \left\{\frac{1}{2Z\_0^2\alpha\_\varkappa^2\mathcal{C}} + \frac{\mathcal{C}}{2}\right\} \tag{30}$$

$$L\_{\rm S}^{\prime} = \frac{1}{a\nu^2 \Big|\_{Z\_{\rm S} = 0} C\_{\rm S}^{\prime}} \tag{31}$$

The parameters of the circuit model of a CPW loaded with an OCSRR (Fig. 12 b) can be extracted following a similar procedure. In this case, the intercept of the return loss curve with the unit resistance circle in the Smith chart gives the value of the series inductance

$$L = \frac{X}{2a\Big|\_{Z\_{\mathbb{P}} \to \cdots}}\tag{32}$$

where *X* is the reactance at the intercept point. The shunt branch resonates at this frequency, that is,

Characterization of Metamaterial

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compact (<1cm2) band pass filters with wide bandwidth and high selectivity at Cband, *Proceedings of the 36th European Microwave Conference*, pp. 599-602, ISBN 2-

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handed transmission lines based on electrically small, open resonators, *Proceedings of IEEE MTT-S International Microwave Symposium*, pp. 45-48, ISBN 978-1-4244-2804-

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$$\left. \phi \phi^2 \right|\_{Z\_P \to =} = \frac{1}{L\_P' \mathcal{C}\_P'} \tag{33}$$

Finally, at the reflection zero frequency (ω*<sup>z</sup>*), the characteristic impedance, given by

$$Z\_o(\ \alpha) = \sqrt{Z\_s(\ \alpha) \left[Z\_s(\ \alpha) + 2Z\_p(\ \alpha)\right]} \tag{34}$$

must be forced to be 50Ω. From these two latter conditions, we finally obtain

$$L\_p' = \left[\frac{\alpha\_\mathcal{Z}^2}{\alpha^2\Big|\_{Z\_p \to \omega}} - 1\right] \left\{\frac{Z\_0^2}{2\alpha\_\mathcal{Z}^2 L} + \frac{L}{2}\right\} \tag{35}$$

$$C\_p' = \frac{1}{o\big|\_{Z\_p \to \dotsb} L\_p'} \tag{36}$$

and the element values are determined.

#### **5. Conclusion**

In this chapter, different kinds of resonant-type metamaterial transmission lines based on subwavelength resonators have been presented and studied. There are several types of resonators which allow their use in the implementation of this kind of artificial transmission lines. SRR-, CSRR- and open resonator-based structures have been presented and studied. The equivalent circuit models for each of the kinds of structures presented have been exposed, together with their parameter extraction methods. These methods provide the circuit model parameters, extracted from the frequency response of the structure, being a very useful design tool and allowing the corroboration of the proposed circuits as correct models for the considered structures. Furthermore, thanks to their accuracy, the parameter extraction methods have already been applied in automation procedures which allow the automatic generation of layouts for this kind of structures (Selga et al., 2010). After the imposition of certain conditions on the frequency response, which implies certain values for the circuit model parameters, the layout of the structure satisfying the required conditions can be automatically generated by means of space mapping techniques. This is a good example of the usefulness and importance of the equivalent circuit models and the parameter extraction methods.

#### **6. References**


<sup>2</sup> 1 *ZP L CP P*

<sup>0</sup> () () () () 2 *Z ZZ Z*

2 2

<sup>2</sup> <sup>2</sup> 1

*<sup>Z</sup> <sup>Z</sup> Z L <sup>L</sup>*

*P*

In this chapter, different kinds of resonant-type metamaterial transmission lines based on subwavelength resonators have been presented and studied. There are several types of resonators which allow their use in the implementation of this kind of artificial transmission lines. SRR-, CSRR- and open resonator-based structures have been presented and studied. The equivalent circuit models for each of the kinds of structures presented have been exposed, together with their parameter extraction methods. These methods provide the circuit model parameters, extracted from the frequency response of the structure, being a very useful design tool and allowing the corroboration of the proposed circuits as correct models for the considered structures. Furthermore, thanks to their accuracy, the parameter extraction methods have already been applied in automation procedures which allow the automatic generation of layouts for this kind of structures (Selga et al., 2010). After the imposition of certain conditions on the frequency response, which implies certain values for the circuit model parameters, the layout of the structure satisfying the required conditions can be automatically generated by means of space mapping techniques. This is a good example of the usefulness and importance of the equivalent circuit models and the

Aznar, F.; Bonache, J. & Martín, F., Improved circuit model for left-handed lines loaded with

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ω*L* →∞

1

*Z P*

0

ω

2 2

*L*

→∞ <sup>=</sup> ′ ′ (33)

′ <sup>=</sup> ′ (36)

(35)

*<sup>z</sup>*), the characteristic impedance, given by

 ω= *SS P* + (34)

ω

must be forced to be 50Ω. From these two latter conditions, we finally obtain

ω

ω

*P*

ω

 ωω

*P Z*

*P*

*C*

→∞ ′ =−+

ω

Finally, at the reflection zero frequency (

and the element values are determined.

parameter extraction methods.

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**6. References** 

**5. Conclusion** 


**0**

**13**

*Italy*

**Genetic Programming**

*University of Cagliari*

Luisa Deias, Giuseppe Mazzarella and Nicola Sirena

**Synthesis of Planar EBG Structures Based on**

*Frequency selective surfaces* (FSS) consist of two-dimensional periodic arrays of metal patches patterned on a dielectric substrate or apertures etched on a metal screen (Hosseini et al., 2006;

These periodic structures resonate at certain frequencies, thus ensuring filtering characteristics, exploited both in the microwave and optical region of the electromagnetic spectrum. Frequency selective surfaces have been thoroughly studied over the years (Mittra et al., 1988; Munk, 2000), and they have found new life in the past decade when *electromagnetic bandgap* (EBG) (Sievenpiper et al., 1999; Yang et al., 1999) structures were introduced, firstly under the name of photonic bandgap (PBG) materials, in analogy to the bandgaps present in electric crystals, even though no photons were involved. Some well–known EBG structures are the Uniplanar Compact Photonic Band-Gap (UC–PBG)(Yang et al., 1999) (Fig. 1) and the

Sievenpiper "mushroom" high–impedance surface (Sievenpiper et al., 1999).

Fig. 1. Uniplanar Compact Photonic Band-Gap (UC–PBG)(Yang et al., 1999)

This entirely new class of structures, encompassing FSS as one of its subclasses (planar EBG) display some very interesting new electromagnetic properties. The presence of a stopband for this structure has been theoretically and experimentally verified and exploited

**1. Introduction**

Wu, 1995; Yang et al., 1999).

**1.1 Electronic Band–Gap (EBG) structures**


### **Synthesis of Planar EBG Structures Based on Genetic Programming**

Luisa Deias, Giuseppe Mazzarella and Nicola Sirena *University of Cagliari Italy*

#### **1. Introduction**

320 Metamaterial

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metamaterial transmission lines based on complementary split-rings resonators and their applications to very wideband and compact filter design", *IEEE Transactions on Microwave Theory and Techniques*, Vol. 55, No. 6, June 2007, pp. 1296-

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filters in coplanar waveguide technology implemented by means of electrically small metamaterial-inspired open resonators, *IET Microwave Antennas and* 

#### **1.1 Electronic Band–Gap (EBG) structures**

*Frequency selective surfaces* (FSS) consist of two-dimensional periodic arrays of metal patches patterned on a dielectric substrate or apertures etched on a metal screen (Hosseini et al., 2006; Wu, 1995; Yang et al., 1999).

These periodic structures resonate at certain frequencies, thus ensuring filtering characteristics, exploited both in the microwave and optical region of the electromagnetic spectrum. Frequency selective surfaces have been thoroughly studied over the years (Mittra et al., 1988; Munk, 2000), and they have found new life in the past decade when *electromagnetic bandgap* (EBG) (Sievenpiper et al., 1999; Yang et al., 1999) structures were introduced, firstly under the name of photonic bandgap (PBG) materials, in analogy to the bandgaps present in electric crystals, even though no photons were involved. Some well–known EBG structures are the Uniplanar Compact Photonic Band-Gap (UC–PBG)(Yang et al., 1999) (Fig. 1) and the Sievenpiper "mushroom" high–impedance surface (Sievenpiper et al., 1999).

Fig. 1. Uniplanar Compact Photonic Band-Gap (UC–PBG)(Yang et al., 1999)

This entirely new class of structures, encompassing FSS as one of its subclasses (planar EBG) display some very interesting new electromagnetic properties. The presence of a stopband for this structure has been theoretically and experimentally verified and exploited

Genetic Algorithms (GA), already widely used in antenna design (Jones & Joines, 2000; Lohn et al., 2001) and more recently also applied to EBG design and optimization (Bray et al., 2006; Ge et al., 2007; Yeo et al., 2002) iteratively transform populations of mathematical objects (typically fixed–length binary character strings), each with an associated fitness value, into

Synthesis of Planar EBG Structures Based on Genetic Programming 323

In GP, unlike GA chromosomes need not be represented by bit-strings and the alteration process includes other "genetic" operators appropriate for the given structure and the given problem. We can say that GA works on the "nucleotide" (i.e. bit) level, in the sense that the antenna or EBG structure is completely defined from the beginning and only a handful of parameters remains to be optimized. The approach proposed by Koza assumes no "a priori" structure. Instead, it builds up the structure of the individuals as the procedure evolves. As a consequence, its solution space has the power of the continuum, while the GA solution space is a discrete one, so it is a very small subspace of the former. The goal of genetic programming is not simply to evolve a bit–string representation of some problem but the computer code

A fixed–length coding such as GA is rather artificial. As it cannot provide a dynamic variability in length, such a coding often causes considerable redundancy and reduces the efficiency of genetic search. In contrast, GP uses high–level building blocks of variable length. Their size and complexity can change during breeding. Moreover, the typical evolution operators work on actual physical structures rather than on sequences of bits with no intuitive link to the EGB surface shape. The enormous power of this strategy fully allows

Getting machines to produce human–like results is the reason for the existence of the fields of artificial intelligence and machine learning. Genetic programming addresses this challenge by providing a method for automatically creating a working computer program from a high–level description of the problem. This is the reason why, unlike Genetic Algorithms, Genetic Programs often deliver elegant human–like solutions not anticipated by the programmer, providing only a minimum amount of pre–supplied human knowledge,

Yet, while allowing to explore and evaluate general configurations, this approach can lead to a severely ill–conditioned synthesis problem. A suitable stabilization is therefore obtained by imposing problem–specific requirements, in our case the periodicity of the the surface elements, which is directly related to the resonant frequency, and the physical parameters, which are not so relevant to the problem. We therefore let then the Genetic Programming

Representation is a key issue in genetic programming (Koza, 1992) because the representation scheme can severely limit the window through which the system "observes" its world.

Since GP manipulates programs by applying genetic operators, a programming language such as LISP was chosen, since it allows each individual, i.e. computer program, to be manipulated as data. Any LISP S–expression can be depicted as a rooted point-labelled tree with ordered branches, as shown in Fig.2. Therefore in GP each individual is a computer program, described through the set of instructions needed to "build" it, typically implemented

strategy evaluate every possible shape in the solution space we delimited.

new populations using the Darwinian principle of natural selection.

that solves that problem.

analysis and information.

in S–expressions.

the exploration of more general shapes.

in different realizations, i.e. TEM waveguide, slow–wave planar structure and low–loss conductor–backed coplanar waveguide.

By choosing the proper geometry of the periodic surface we can shape the electromagnetic behavior of these structures and they can be made to act as a so–called electromagnetic crystal, exhibiting frequency bands inside which the propagation of electromagnetic waves is not allowed or is highly attenuated. The concept of suppressing surfaces waves on metals is not new. Surface waves can be eliminated from a metal surface over a finite frequency band by applying a periodic texture. It has been done long before EGB structures were introduced using several geometries, such as a metal sheet covered with small bumps or a corrugated metal slab.

The novelty of EBGs is the application of an array of periodic patches or apertures, i.e. lumped circuit elements, to produce a thin two–dimensional structure that must generally be described by band structure concepts, even though the thickness and periodicity are both much smaller than the operating wavelength.

These periodic structures can also be designed to act as an *artificial magnetic conductor* (AMC) or high–impedance electromagnetic ground plane over a desired (quite) narrow frequency range, corresponding to the forbidden frequency band. Hence, the key feature of these structures is the reflection of an incident plane wave with no phase reversal, unlike normal metal surfaces (Sievenpiper et al., 1999).

High–impedance surfaces are widely studied now as promising antenna substrates (Feresidis et al., 2005; Gonzalo et al., 1999; Hosseini et al., 2006). A possibility of realizing a magnetic wall near the resonant frequency of a very thin structure is very attractive, since this allows one to design low–profile antennas and enhance the performance of printed antennas. The main drawback of this strategy is the reduced bandwidth of the complete antenna, since the frequency range over which these EBG surfaces behave as an AMC is usually narrowband and fixed by their geometrical configuration. The ultimate goal is then to design and incorporate such metamaterial–substrates in antenna structures in order to improve antenna performance.

A key issue in the research field of metamaterials is then represented by the design and optimization of EBG structures. Different techniques have been investigated, among which methods of global optimization such as genetic algorithm and particle swarm optimization (PSO),(Bray et al., 2006; Ge et al., 2007; Kovács et al., 2010; Tavallaee & Rahmat-Samii, 2007; Yeo et al., 2002).

#### **1.2 Genetic programming**

*Genetic Programming* (GP) (Koza, 1992) falls into the larger class of evolutionary computations (Fogel, 2006; Michalewicz, 1992), including genetic algorithms, evolution strategies and evolutionary programming, which can be described as highly parallel probabilistic search algorithms imitating the principles of natural evolution for optimization problems, based on the idea that most real–world problems cannot be handled with binary representations. Such algorithms allow task specific knowledge emerge while solving the problem. Genetic Programming, a variant of genetic algorithms yet with marked differences, is an especially interesting form of computational problem solving.

2 Will-be-set-by-IN-TECH

in different realizations, i.e. TEM waveguide, slow–wave planar structure and low–loss

By choosing the proper geometry of the periodic surface we can shape the electromagnetic behavior of these structures and they can be made to act as a so–called electromagnetic crystal, exhibiting frequency bands inside which the propagation of electromagnetic waves is not allowed or is highly attenuated. The concept of suppressing surfaces waves on metals is not new. Surface waves can be eliminated from a metal surface over a finite frequency band by applying a periodic texture. It has been done long before EGB structures were introduced using several geometries, such as a metal sheet covered with small bumps or a corrugated

The novelty of EBGs is the application of an array of periodic patches or apertures, i.e. lumped circuit elements, to produce a thin two–dimensional structure that must generally be described by band structure concepts, even though the thickness and periodicity are both

These periodic structures can also be designed to act as an *artificial magnetic conductor* (AMC) or high–impedance electromagnetic ground plane over a desired (quite) narrow frequency range, corresponding to the forbidden frequency band. Hence, the key feature of these structures is the reflection of an incident plane wave with no phase reversal, unlike normal

High–impedance surfaces are widely studied now as promising antenna substrates (Feresidis et al., 2005; Gonzalo et al., 1999; Hosseini et al., 2006). A possibility of realizing a magnetic wall near the resonant frequency of a very thin structure is very attractive, since this allows one to design low–profile antennas and enhance the performance of printed antennas. The main drawback of this strategy is the reduced bandwidth of the complete antenna, since the frequency range over which these EBG surfaces behave as an AMC is usually narrowband and fixed by their geometrical configuration. The ultimate goal is then to design and incorporate such metamaterial–substrates in antenna structures in order to improve antenna performance. A key issue in the research field of metamaterials is then represented by the design and optimization of EBG structures. Different techniques have been investigated, among which methods of global optimization such as genetic algorithm and particle swarm optimization (PSO),(Bray et al., 2006; Ge et al., 2007; Kovács et al., 2010; Tavallaee & Rahmat-Samii, 2007;

*Genetic Programming* (GP) (Koza, 1992) falls into the larger class of evolutionary computations (Fogel, 2006; Michalewicz, 1992), including genetic algorithms, evolution strategies and evolutionary programming, which can be described as highly parallel probabilistic search algorithms imitating the principles of natural evolution for optimization problems, based on the idea that most real–world problems cannot be handled with binary representations. Such algorithms allow task specific knowledge emerge while solving the problem. Genetic Programming, a variant of genetic algorithms yet with marked differences, is an especially

conductor–backed coplanar waveguide.

much smaller than the operating wavelength.

metal surfaces (Sievenpiper et al., 1999).

metal slab.

Yeo et al., 2002).

**1.2 Genetic programming**

interesting form of computational problem solving.

Genetic Algorithms (GA), already widely used in antenna design (Jones & Joines, 2000; Lohn et al., 2001) and more recently also applied to EBG design and optimization (Bray et al., 2006; Ge et al., 2007; Yeo et al., 2002) iteratively transform populations of mathematical objects (typically fixed–length binary character strings), each with an associated fitness value, into new populations using the Darwinian principle of natural selection.

In GP, unlike GA chromosomes need not be represented by bit-strings and the alteration process includes other "genetic" operators appropriate for the given structure and the given problem. We can say that GA works on the "nucleotide" (i.e. bit) level, in the sense that the antenna or EBG structure is completely defined from the beginning and only a handful of parameters remains to be optimized. The approach proposed by Koza assumes no "a priori" structure. Instead, it builds up the structure of the individuals as the procedure evolves. As a consequence, its solution space has the power of the continuum, while the GA solution space is a discrete one, so it is a very small subspace of the former. The goal of genetic programming is not simply to evolve a bit–string representation of some problem but the computer code that solves that problem.

A fixed–length coding such as GA is rather artificial. As it cannot provide a dynamic variability in length, such a coding often causes considerable redundancy and reduces the efficiency of genetic search. In contrast, GP uses high–level building blocks of variable length. Their size and complexity can change during breeding. Moreover, the typical evolution operators work on actual physical structures rather than on sequences of bits with no intuitive link to the EGB surface shape. The enormous power of this strategy fully allows the exploration of more general shapes.

Getting machines to produce human–like results is the reason for the existence of the fields of artificial intelligence and machine learning. Genetic programming addresses this challenge by providing a method for automatically creating a working computer program from a high–level description of the problem. This is the reason why, unlike Genetic Algorithms, Genetic Programs often deliver elegant human–like solutions not anticipated by the programmer, providing only a minimum amount of pre–supplied human knowledge, analysis and information.

Yet, while allowing to explore and evaluate general configurations, this approach can lead to a severely ill–conditioned synthesis problem. A suitable stabilization is therefore obtained by imposing problem–specific requirements, in our case the periodicity of the the surface elements, which is directly related to the resonant frequency, and the physical parameters, which are not so relevant to the problem. We therefore let then the Genetic Programming strategy evaluate every possible shape in the solution space we delimited.

Representation is a key issue in genetic programming (Koza, 1992) because the representation scheme can severely limit the window through which the system "observes" its world.

Since GP manipulates programs by applying genetic operators, a programming language such as LISP was chosen, since it allows each individual, i.e. computer program, to be manipulated as data. Any LISP S–expression can be depicted as a rooted point-labelled tree with ordered branches, as shown in Fig.2. Therefore in GP each individual is a computer program, described through the set of instructions needed to "build" it, typically implemented in S–expressions.

**2. EBG design usign genetic programming**

patch (d) circular loop (e) square loops (f) tripole.

(Branch (Rettangolo 0.16589776 0.2873323 (Ruota 90.0 (Ruota -90.0 END))) (Ruota 90.0

(Branch (Branch END END) (Ruota 90.0 END))

(Ruota 90.0 (Rettangolo 0.51570845 0.25133058

has been implemented in Fortran.

END))))))))

(Branch END END)

END)))))

Tree 0:

Tree 1:

Tree 2:

Tree 3:

In the design of frequency selective surfaces, the choice of the proper element may be of utmost importance. In fact some elements are intrinsically more broad–banded or more narrow–banded than others, while some can be more easily varied by design. In literature we can find a large variety of element types, the more common are illustrated in Fig. 3. It can be now devised the great potential of Genetic Programming, since we don't really want any possible geometry to be taken into account but only feasible geometries, just like a human being would design them on a piece of paper, yet faster and with much more "fantasy".

Synthesis of Planar EBG Structures Based on Genetic Programming 325

Fig. 3. Some typical FSS unit cell geometries: (a) cross dipole (b) Jerusalem cross (c) square

Therefore the goal of the design process is to obtain the unit cell aperture geometry of the periodic surface which fulfills the desired requirements on the resonant frequency of the periodic structure, analyzed with a full–wave technique. The GP approach has been implemented in Java, while the full–wave analysis of the periodic structure for each individual

The S–expression of an individual of the population can take for example the following form:

(Rettangolo 0.6837649 0.2757175 (Rettangolo 0.7254247 0.23175128

(Ruota 90.0 (Branch (Ruota -90.0 END) (Rettangolo 0.75231904 0.2833914

(Ruota 90.0 (Ruota 90.0 (Ruota -90.0 (Rettangolo 0.19471756 0.19593427

Fig. 2. S-expression: tree architecture.

While the evolutionary "strategy" is almost standard for a variety of problems (we used ECJ, a general purpose Java-based Evolutionary Computation research system and developed at George Mason University ECLab), three elements must be defined to evolve a design, i.e. "individual":

	- **–** CROSSOVER:creates new S–expressions by exchanging sub S–expressions between two S–expressions. The sub S-expressions exchanged are selected randomly.
	- **–** MUTATION: creates a new S–expression by replacing an existing argument symbol with the other possible symbol. The argument symbol replaced is selected randomly.

In other words, representation refers to a form of data structure. Variation operations are applied to existing solutions to create new solutions. Usually, variation is based on random perturbation.

The starting point is an initial population of randomly generated computer programs composed of functions and terminals appropriate to the specified problem domain. The evolutionary strategy works in order to find the best individual, in terms of their closeness to the constraints set in the design and evaluated as a "fitness" function. This strategy let to interesting and promising results in the synthesis of EBGs, even when considering more complex structures and requirements (Deias et al., 2009a;b; 2010).

#### **2. EBG design usign genetic programming**

4 Will-be-set-by-IN-TECH

While the evolutionary "strategy" is almost standard for a variety of problems (we used ECJ, a general purpose Java-based Evolutionary Computation research system and developed at George Mason University ECLab), three elements must be defined to evolve a design, i.e.

• VARIATION OPERATORS: code that takes one or more design representations as input and

S–expressions. The sub S-expressions exchanged are selected randomly.

**–** CROSSOVER:creates new S–expressions by exchanging sub S–expressions between two

**–** MUTATION: creates a new S–expression by replacing an existing argument symbol with the other possible symbol. The argument symbol replaced is selected randomly.

In other words, representation refers to a form of data structure. Variation operations are applied to existing solutions to create new solutions. Usually, variation is based on random

The starting point is an initial population of randomly generated computer programs composed of functions and terminals appropriate to the specified problem domain. The evolutionary strategy works in order to find the best individual, in terms of their closeness to the constraints set in the design and evaluated as a "fitness" function. This strategy let to interesting and promising results in the synthesis of EBGs, even when considering more

Fig. 2. S-expression: tree architecture.

outputs a design derived from them

• REPRESENTATION SCHEME:definition of the design space

• FITNESS FUNCTION: a function that evaluates the individuals

complex structures and requirements (Deias et al., 2009a;b; 2010).

"individual":

perturbation.

In the design of frequency selective surfaces, the choice of the proper element may be of utmost importance. In fact some elements are intrinsically more broad–banded or more narrow–banded than others, while some can be more easily varied by design. In literature we can find a large variety of element types, the more common are illustrated in Fig. 3. It can be now devised the great potential of Genetic Programming, since we don't really want any possible geometry to be taken into account but only feasible geometries, just like a human being would design them on a piece of paper, yet faster and with much more "fantasy".

Fig. 3. Some typical FSS unit cell geometries: (a) cross dipole (b) Jerusalem cross (c) square patch (d) circular loop (e) square loops (f) tripole.

Therefore the goal of the design process is to obtain the unit cell aperture geometry of the periodic surface which fulfills the desired requirements on the resonant frequency of the periodic structure, analyzed with a full–wave technique. The GP approach has been implemented in Java, while the full–wave analysis of the periodic structure for each individual has been implemented in Fortran.

The S–expression of an individual of the population can take for example the following form:

```
Tree 0:
 (Branch (Rettangolo 0.16589776 0.2873323
  (Ruota 90.0 (Ruota -90.0 END))) (Ruota 90.0
    (Rettangolo 0.6837649 0.2757175 (Rettangolo 0.7254247 0.23175128
     (Ruota 90.0 (Ruota 90.0 (Ruota -90.0 (Rettangolo 0.19471756 0.19593427
       END))))))))
Tree 1:
 (Branch (Branch END END) (Ruota 90.0 END))
Tree 2:
 (Branch END END)
Tree 3:
 (Ruota 90.0 (Rettangolo 0.51570845 0.25133058
   (Ruota 90.0 (Branch (Ruota -90.0 END) (Rettangolo 0.75231904 0.2833914
    END)))))
```
Fig. 5. Flowchart.

discretization step.

The fitness function employed by our GP optimization is:

*FF* = *ω*<sup>1</sup> ∗ *ω*<sup>2</sup> ∗ *e*

frequency points evaluation, individuals with larger bandwidth to prevail.

shown in (Collin, 1985) for a different problem, namely wire antennas.

where *ω*<sup>1</sup> and *ω*<sup>2</sup> are penalty coefficients developed in order to avoid geometries with high number of discretization elements within the unit cell and allowing, when using a three

Synthesis of Planar EBG Structures Based on Genetic Programming 327

A full–wave simulation using Method of Moments is carried out on a single design, i.e. unit cell of the infinite periodic surface and individual of a set population. The starting point is a random initial population of individuals. The external parameters of the structure (i.e. substrate thickness and dielectric constant) are fixed, altogether with the periodicity of the planar EBG. The geometry of the aperture is subject to the GP optimization, while the unit square cell within which each individual/design can evolve is fixed together with its

The set of admissible solution is then composed by every geometry/aperture that can be designed in this square, built as series of segments that can evolve in every direction, with no limit on the number of segments, on their width and length and number of subsequent ramifications. We have fixed the discretization step within the unit cell in order to pose a limit not only to the computational burden but also to the geometrical spatial bandwidth of the possible solutions. In such a way we obtain a stabilizing effect on the problem, as much as

The phase of the reflection coefficient, computed at one or more frequencies depending on the goal, is then incorporated into the GP strategy for the evaluation procedure, the fitness function determining the environment within which the solutions "live" and the best

*Phase*(*fc* )

<sup>20</sup> (1)

The corresponding input to the Fortran executable is the following, i.e. a collection of rectangular patches that form the geometry of the aperture:

Patch 48 -0.0 , -0.0 , 0.4572 , 0.3048 -0.4572 , -0.0 , 0.4572 , 0.3048 -0.0 , -0.3048 , 0.4572 , 0.3048 -0.4572 , -0.3048 , 0.4572 , 0.3048 -0.3048 , -0.0 , 0.3048 , 0.4572 -0.0 , -0.0 , 0.3048 , 0.4572 -0.3048 , -0.4572 , 0.3048 , 0.4572 -0.0 , -0.4572 , 0.3048 , 0.4572 0.3048 , -0.0 , 0.15239999 , 0.762 -0.4572 , -0.0 , 0.15239999 , 0.762 ... -1.3716 , -0.0 , 0.762 , 0.3048 0.6096 , -0.0 , 0.762 , 0.3048 -0.0 , -0.0 , 0.1524 , 0.3048 -0.1524 , -0.0 , 0.1524 , 0.3048 -0.0 , -0.3048 , 0.1524 , 0.3048 -0.1524 , -0.3048 , 0.1524 , 0.3048 -0.3048 , -0.0 , 0.3048 , 0.1524 -0.0 , -0.0 , 0.3048 , 0.1524 -0.3048 , -0.1524 , 0.3048 , 0.1524 -0.0 , -0.1524 , 0.3048 , 0.1524

And the corresponding geometry of the aperture is shown in Fig.4.

Fig. 4. Aperture Geometry.

In our specific problem, we thus obtain a set of rectangular patches which describe the aperture geometry of the EBG unit cell.

The fitness criteria measure the quality of any given solution. The selection method uses the score obtained for each solution to determine which to save and which to eliminate from the population at each generation. Those solutions that survive are the "parents" of the next generation. The initialization of an evolutionary algorithm can be completely at random, or can incorporate human or other expertise about solutions that may work better than others.

Fig. 5. Flowchart.

6 Will-be-set-by-IN-TECH

The corresponding input to the Fortran executable is the following, i.e. a collection of

rectangular patches that form the geometry of the aperture:

And the corresponding geometry of the aperture is shown in Fig.4.

In our specific problem, we thus obtain a set of rectangular patches which describe the

The fitness criteria measure the quality of any given solution. The selection method uses the score obtained for each solution to determine which to save and which to eliminate from the population at each generation. Those solutions that survive are the "parents" of the next generation. The initialization of an evolutionary algorithm can be completely at random, or can incorporate human or other expertise about solutions that may work better than others.

Patch 48

...



Fig. 4. Aperture Geometry.

aperture geometry of the EBG unit cell.

The fitness function employed by our GP optimization is:

$$FF = \omega\_1 \* \omega\_2 \* e^{\frac{\text{Phase}(\mathcal{f}\_\mathcal{C})}{20}} \tag{1}$$

where *ω*<sup>1</sup> and *ω*<sup>2</sup> are penalty coefficients developed in order to avoid geometries with high number of discretization elements within the unit cell and allowing, when using a three frequency points evaluation, individuals with larger bandwidth to prevail.

A full–wave simulation using Method of Moments is carried out on a single design, i.e. unit cell of the infinite periodic surface and individual of a set population. The starting point is a random initial population of individuals. The external parameters of the structure (i.e. substrate thickness and dielectric constant) are fixed, altogether with the periodicity of the planar EBG. The geometry of the aperture is subject to the GP optimization, while the unit square cell within which each individual/design can evolve is fixed together with its discretization step.

The set of admissible solution is then composed by every geometry/aperture that can be designed in this square, built as series of segments that can evolve in every direction, with no limit on the number of segments, on their width and length and number of subsequent ramifications. We have fixed the discretization step within the unit cell in order to pose a limit not only to the computational burden but also to the geometrical spatial bandwidth of the possible solutions. In such a way we obtain a stabilizing effect on the problem, as much as shown in (Collin, 1985) for a different problem, namely wire antennas.

The phase of the reflection coefficient, computed at one or more frequencies depending on the goal, is then incorporated into the GP strategy for the evaluation procedure, the fitness function determining the environment within which the solutions "live" and the best

Fig. 7. Unit cell geometry of a Frequency Selective Surface with a dielectric substrate on ground plane, and magnetic currents equivalent to an aperture. The air–dielectric interface is

Synthesis of Planar EBG Structures Based on Genetic Programming 329

(a)

V*eq*

V*eq* V*eq*

Z<sup>1</sup>

h

Z<sup>1</sup>

h

MA1




MA2

MA3

(b)

Fig. 8. Equivalent circuits resulting from the equivalence applied to a FSS (in the spectral

the advantages of the aperture approach are far more greater (Asole et al., 2007).

of the method. The generalization to N-layer FSS is really straightforward.

PEC

A1

A2

A3

As a consequence of our aperture approach, the MoM matrix can be decoupled in the sum of "localized" admittance matrices, each one relevant to a single region. For multi–layered FSS

Let's consider the 3–layer AMC structure as shown in Fig.9, since it contains all the features

h1

Fig. 9. Geometry of a 3–layer structure and magnetic currents equivalent to the apertures

h2

h3

Z<sup>0</sup>

Z<sup>0</sup>

domain): (a) metallic patches; (b) apertures.

z

(central cell).

εr1

εr2

εr3

a PEC.

individual of a generation is selected. The evolutionary operators (reproduction, crossover and mutation) are then applied to the best individual leading to the subsequent evolution of the population and the next generation. In this way the geometrical properties are optimized to evolve the best solution (zero phase at the desired resonant frequency).

#### **3. Single and multi–layer EBG full–wave analysis**

Various techniques have been proposed in literature in order to analyze frequency selective surfaces (FSS) (Bardi et al., 2002; Bozzi & Perregrini, 1999; Harms et al., 1994; Mittra et al., 1988; Wu, 1995). The standard method of moments (MoM) approach is based on the induced electric currents on the FSS but it can be prohibitively costly from a computational point of view, if not even impractical in some cases, and this could explain why MoM is so unpopular.

Frequency Selective Surfaces (FSS) consist of two-dimensional periodic arrays of metallic patches patterned on a dielectric substrate or apertures etched on a metal screen, either entities can be isolated or connected in a rectangular grid. Hence, for any FSS, we can consider either the periodicity of the metallic patches or that of the apertures. By taking properly into account the continuity of electric or magnetic currents along adjacent cells we can conveniently adopt one of the two approaches, considering as unknowns of the problem either the electric or magnetic currents.

Fig. 6. Cross section view of a single layer FSS with dielectric substrate on ground plane.

The integral equation for the electric field (EFIE) applied to the periodic array of metal patches and the integral equation for the magnetic field (MFIE) applied to the periodic array of apertures thus represent two alternative formulations of the same problem, yielding to the same solution. The corresponding equivalent circuit for each approach is shown in Fig.8

Using a MoM approach based on the apertures, a very effective procedure can be devised (Deias & Mazzarella, 2006).

The aperture oriented approach (i.e. MFIE formulation) starts by applying the equivalence theorem (Balanis, 1996). The aperture is closed by a conductive sheet, and two unknown magnetic current densities **M***<sup>A</sup>* and −**M***<sup>A</sup>* on the opposite sides of the conductive sheet, as shown in Fig.7, are defined so that the continuity of the tangential electric field is guaranteed.

The MoM matrix is then obtained by imposing the continuity of the tangential component of the magnetic field across the aperture. The magnetic field is computed using a Green function in the spectral domain, too. But the relevant equivalent circuit, shown in Fig.8(b), allows to decouple the two regions below and above the metallization. This is a key feature not available in the conventional method based on the EFIE: in Fig.8(a) we can see that the equivalent circuit in this case bounds together the different regions.

8 Will-be-set-by-IN-TECH

individual of a generation is selected. The evolutionary operators (reproduction, crossover and mutation) are then applied to the best individual leading to the subsequent evolution of the population and the next generation. In this way the geometrical properties are optimized

Various techniques have been proposed in literature in order to analyze frequency selective surfaces (FSS) (Bardi et al., 2002; Bozzi & Perregrini, 1999; Harms et al., 1994; Mittra et al., 1988; Wu, 1995). The standard method of moments (MoM) approach is based on the induced electric currents on the FSS but it can be prohibitively costly from a computational point of view, if not even impractical in some cases, and this could explain why MoM is so unpopular. Frequency Selective Surfaces (FSS) consist of two-dimensional periodic arrays of metallic patches patterned on a dielectric substrate or apertures etched on a metal screen, either entities can be isolated or connected in a rectangular grid. Hence, for any FSS, we can consider either the periodicity of the metallic patches or that of the apertures. By taking properly into account the continuity of electric or magnetic currents along adjacent cells we can conveniently adopt one of the two approaches, considering as unknowns of the problem either the electric or

Fig. 6. Cross section view of a single layer FSS with dielectric substrate on ground plane.

The integral equation for the electric field (EFIE) applied to the periodic array of metal patches and the integral equation for the magnetic field (MFIE) applied to the periodic array of apertures thus represent two alternative formulations of the same problem, yielding to the same solution. The corresponding equivalent circuit for each approach is shown in Fig.8

Using a MoM approach based on the apertures, a very effective procedure can be devised

The aperture oriented approach (i.e. MFIE formulation) starts by applying the equivalence theorem (Balanis, 1996). The aperture is closed by a conductive sheet, and two unknown magnetic current densities **M***<sup>A</sup>* and −**M***<sup>A</sup>* on the opposite sides of the conductive sheet, as shown in Fig.7, are defined so that the continuity of the tangential electric field is guaranteed. The MoM matrix is then obtained by imposing the continuity of the tangential component of the magnetic field across the aperture. The magnetic field is computed using a Green function in the spectral domain, too. But the relevant equivalent circuit, shown in Fig.8(b), allows to decouple the two regions below and above the metallization. This is a key feature not available in the conventional method based on the EFIE: in Fig.8(a) we can see that the equivalent circuit

to evolve the best solution (zero phase at the desired resonant frequency).

**3. Single and multi–layer EBG full–wave analysis**

magnetic currents.

(Deias & Mazzarella, 2006).

in this case bounds together the different regions.

Fig. 7. Unit cell geometry of a Frequency Selective Surface with a dielectric substrate on ground plane, and magnetic currents equivalent to an aperture. The air–dielectric interface is a PEC.

Fig. 8. Equivalent circuits resulting from the equivalence applied to a FSS (in the spectral domain): (a) metallic patches; (b) apertures.

As a consequence of our aperture approach, the MoM matrix can be decoupled in the sum of "localized" admittance matrices, each one relevant to a single region. For multi–layered FSS the advantages of the aperture approach are far more greater (Asole et al., 2007).

Let's consider the 3–layer AMC structure as shown in Fig.9, since it contains all the features of the method. The generalization to N-layer FSS is really straightforward.

Fig. 9. Geometry of a 3–layer structure and magnetic currents equivalent to the apertures (central cell).

For the second discontinuity *A*2, as for any intermediate aperture, enforcing the boundary

Synthesis of Planar EBG Structures Based on Genetic Programming 331

2,*<sup>t</sup>* <sup>=</sup> **<sup>H</sup>***A*2*A*<sup>2</sup>

where *m* = 1, . . . , *Nb* and the l.h.s. term represents the self–term and the r.h.s. term the coupling terms, i.e. the effect of the magnetic currents **M***A*<sup>3</sup> and **M***A*<sup>1</sup> on the aperture *A*2.

3,*<sup>t</sup>* <sup>=</sup> **<sup>H</sup>***A*3*A*<sup>3</sup>

where *m* = 1, . . . , *Nb* and, on the l.h.s we have the self–term and the first r.h.s. term is the coupling term, i.e. the effect of the magnetic current **M***A*<sup>2</sup> on the aperture *A*3. The second term

The presence of the coupling terms is the critical point in the multi–layer formulation. The key feature of the aperture approach, as previously illustrated, relies on the decoupling of the different regions. If we have a more general structure made up of N layers, we know that when dealing with the aperture *A*1, which lies in the bottom layer, we only have to consider the effect of the magnetic current on the aperture *A*2. When considering the aperture *AN*, the last one, we only have to consider the effect of the magnetic current on the aperture *AN*−1, and when considering any intermediate aperture *Ai*, for *i* = 2, ..., *N* − 1 we have to take into account only the effect of the apertures immediately below and above, i.e. *Ai*−<sup>1</sup> and *Ai*+1.

where Y is the coefficient matrix, Mis the vector of the unknown coefficients, i.e. the magnetic current on the apertures, and T is the r.h.s, which represent the incident magnetic field on the

The solution of the above equation (12) yields the unknown induced magnetic current on the apertures. The reflection coefficient is then easily calculated from the magnetic current on the

It is important to stress out that inserting more layers will result in a larger matrix, affecting thus the overall computational time, yet for each additional layer only two blocks have to be

3,*<sup>t</sup>* <sup>+</sup> **<sup>H</sup>***A*3*A*<sup>2</sup>

**G**ˆ *<sup>A</sup>*<sup>2</sup> *<sup>A</sup>*<sup>2</sup> <sup>3</sup> · <sup>ˆ</sup> **f** *A*2 *n* · **f** *A*2 *m* =

**G**ˆ *<sup>A</sup>*<sup>3</sup> *<sup>A</sup>*<sup>3</sup> <sup>4</sup> · <sup>ˆ</sup> **f** *A*3 *n* · **f** *A*3 *m* 

Y·M = T (12)

**G**ˆ *<sup>A</sup>*<sup>1</sup> *<sup>A</sup>*<sup>2</sup> <sup>2</sup> · <sup>ˆ</sup> **f** *A*1 *n* · **f** *A*2 *m*

4,*<sup>t</sup>* + **H***inc*,*<sup>t</sup>* (10)

*<sup>A</sup>*<sup>3</sup> *<sup>m</sup>* (*x*, *<sup>y</sup>*) *dS* (11)

3,*<sup>t</sup>* (8)

(9)

**H***A*2*A*<sup>2</sup>

**G**ˆ *<sup>A</sup>*<sup>3</sup> *<sup>A</sup>*<sup>2</sup> <sup>3</sup> · <sup>ˆ</sup> **f** *A*3 *n* · **f** *<sup>A</sup>*<sup>2</sup> *<sup>m</sup>* <sup>−</sup> *Nb* ∑ *n*=1 *<sup>m</sup>A*<sup>1</sup> *<sup>n</sup> A*2 *A*2

For the last (third) discontinuity *A*3, the boundary condition leads to:

3,*<sup>t</sup>* <sup>+</sup> **<sup>H</sup>***A*2*A*<sup>3</sup>

**G**ˆ *<sup>A</sup>*<sup>3</sup> *<sup>A</sup>*<sup>3</sup> <sup>3</sup> · <sup>ˆ</sup> **f** *A*3 *n* · **f** *<sup>A</sup>*<sup>3</sup> *<sup>m</sup>* + *A*3 *A*3

**G**ˆ *<sup>A</sup>*<sup>2</sup> *<sup>A</sup>*<sup>3</sup> <sup>3</sup> · <sup>ˆ</sup> **f** *A*2 *n* · **f** *<sup>A</sup>*<sup>3</sup> *<sup>m</sup>* <sup>−</sup> *S* **H***inc* · **f**

**H***A*3*A*<sup>3</sup>

of the r.h.s represent the known terms of the MoM system.

Equations (7),(9),(11) can be written in matrix form:

2,*<sup>t</sup>* <sup>+</sup> **<sup>H</sup>***A*1*A*<sup>2</sup>

**G**ˆ *<sup>A</sup>*<sup>2</sup> *<sup>A</sup>*<sup>2</sup> <sup>2</sup> · <sup>ˆ</sup> **f** *A*2 *n* · **f** *<sup>A</sup>*<sup>2</sup> *<sup>m</sup>* + *A*2 *A*2

condition we obtain:

*Nb* ∑ *n*=1

*Nb* ∑ *n*=1 *<sup>m</sup>A*<sup>3</sup> *<sup>n</sup> A*2 *A*2

*Nb* ∑ *n*=1

= − *Nb* ∑ *n*=1 *<sup>m</sup>A*<sup>2</sup> *<sup>n</sup> A*3 *A*3

<sup>−</sup>*mA*<sup>3</sup> *<sup>n</sup> A*3 *A*3

<sup>−</sup>*mA*<sup>2</sup> *<sup>n</sup> A*2 *A*2

which leads to:

and we obtain:

last aperture *A*3.

upper layer.

computed.

The dielectric layers can be chosen with the desired physical parameter and a superstrate can also be used to cover the structure. The unit cell of the periodic printed surfaces on the different layers may be of different shape but they must have the same periodicity.

According to the equivalence theorem, each aperture is closed by a PEC, and two unknown magnetic current densities **M***Anm* and −**M***Anm* on the opposite sides are defined in order to guarantee the continuity of the tangential electric field. By the Floquet theorem, the equivalent currents on the (*n*, *m*)*th* cell of the periodic surface are connected to the central one by:

$$\mathbf{M}\_{A,nm} = e^{-j\beta nd\_x \chi} e^{-j\beta md\_y y} \mathbf{M}\_A \tag{2}$$

where **M***<sup>A</sup>* is the current in the central cell.

As a consequence, only the current in the central cell is the unknown of the problem, and need to be discretized as a linear combination of *Nb* basis functions:

$$\mathbf{M}\_A = \sum\_{n=1}^{N\_b} m\_n^A \mathbf{f}\_n \tag{3}$$

Then we obtain the MoM linear equation system by imposing the continuity of the tangential component of the magnetic field across each aperture. Again, by the Floquet theorem, this continuity needs to be forced only on the central cell (see Fig.**??**). It can be easily seen that, in the multilayered structure, we can distinguish three cases, namely an aperture lying on the first layer, on an intermediate layer or on the last layer.

For the first layer, i.e. for the discontinuity *A*1, the continuity of the transverse magnetic field can be written as:

$$\mathbf{H}\_{1,t}^{A\_1 A\_1} = \mathbf{H}\_{2,t}^{A\_1 A\_1} + \mathbf{H}\_{2,t}^{A\_2 A\_1} \tag{4}$$

which is then enforced in a weak form. A set of testing functions is selected and the boundary condition on the magnetic field is multiplied by each testing function and the result integrated over aperture. Choosing the Galerkin method, i.e. the testing functions used are the same basis functions used to express the unknown magnetic currents, we obtain for layer *A*1:

$$\int\_{A\_1} \mathbf{H}\_{1,t}^{A\_1 A\_1} \cdot \mathbf{f}\_{\mathfrak{m}} = \int\_{A\_1} \mathbf{H}\_{2,t}^{A\_1 A\_1} \cdot \mathbf{f}\_{\mathfrak{m}} + \int\_{A\_1} \mathbf{H}\_{2,t}^{A\_2 A\_1} \cdot \mathbf{f}\_{\mathfrak{m}\_{\ell}} \qquad m = 1, \dots, N\_b \tag{5}$$

which, introducing the Green Functions can be written as:

$$
\int\_{A\_1} \langle \mathbf{\hat{G}\_1^{A\_1 A\_1}}, -\mathbf{M}\_{A\_1} \rangle \cdot \mathbf{f}\_m = \int\_{A\_1} \langle \mathbf{\hat{G}\_2^{A\_1 A\_1}}, \mathbf{M}\_{A\_1} \rangle \cdot \mathbf{f}\_m + \\
$$

$$
\int\_{A\_1} \langle \mathbf{\hat{G}\_2^{A\_2 A\_1}}, \mathbf{M}\_{A\_2} \rangle \cdot \mathbf{f}\_m + \int\_{A\_1} \langle \mathbf{\hat{G}\_2^{A\_2 A\_1}}, \mathbf{M}\_{A\_2} \rangle \cdot \mathbf{f}\_m \qquad m = 1, \dots, N\_b \tag{6}
$$

Expressing the unknown magnetic currents in the form (3):

$$\sum\_{n=1}^{N\_b} -m\_n^{A\_1} \left( \int\_{A\_1} \left( \int\_{A\_1} \mathbf{G}\_1^{A\_1 A\_1} \cdot \mathbf{f}\_n^{A\_1} \right) \cdot \mathbf{f}\_m^{A\_1} + \int\_{A\_1} \left( \int\_{A\_1} \mathbf{G}\_2^{A\_1 A\_1} \cdot \mathbf{f}\_n^{A\_1} \right) \cdot \mathbf{f}\_m^{A\_1} \right)$$

$$= \sum\_{n=1}^{N\_b} m\_n^{A\_2} \int\_{A\_1} \left( \int\_{A\_1} \mathbf{\hat{G}}\_2^{A\_2 A\_1} \cdot \mathbf{f}\_n^{A\_2} \right) \cdot \mathbf{f}\_m^{A\_1}, \qquad m = 1, \dots, N\_b \tag{7}$$

where the l.h.s. term represents the self–term, i.e. the effect of the magnetic current **M***A*<sup>1</sup> on the aperture *A*<sup>1</sup> , and the r.h.s. term is the coupling term, i.e. the effect of the magnetic current **M***A*<sup>2</sup> on the aperture *A*1.

For the second discontinuity *A*2, as for any intermediate aperture, enforcing the boundary condition we obtain:

$$\mathbf{H}\_{2,t}^{A\_2 A\_2} + \mathbf{H}\_{2,t}^{A\_1 A\_2} = \mathbf{H}\_{3,t}^{A\_2 A\_2} + \mathbf{H}\_{3,t}^{A\_3 A\_2} \tag{8}$$

which leads to:

10 Will-be-set-by-IN-TECH

The dielectric layers can be chosen with the desired physical parameter and a superstrate can also be used to cover the structure. The unit cell of the periodic printed surfaces on the

According to the equivalence theorem, each aperture is closed by a PEC, and two unknown magnetic current densities **M***Anm* and −**M***Anm* on the opposite sides are defined in order to guarantee the continuity of the tangential electric field. By the Floquet theorem, the equivalent currents on the (*n*, *m*)*th* cell of the periodic surface are connected to the central one by:

<sup>−</sup>*jβndx xe*

As a consequence, only the current in the central cell is the unknown of the problem, and need

*Nb* ∑ *n*=1

Then we obtain the MoM linear equation system by imposing the continuity of the tangential component of the magnetic field across each aperture. Again, by the Floquet theorem, this continuity needs to be forced only on the central cell (see Fig.**??**). It can be easily seen that, in the multilayered structure, we can distinguish three cases, namely an aperture lying on the

For the first layer, i.e. for the discontinuity *A*1, the continuity of the transverse magnetic field

which is then enforced in a weak form. A set of testing functions is selected and the boundary condition on the magnetic field is multiplied by each testing function and the result integrated over aperture. Choosing the Galerkin method, i.e. the testing functions used are the same basis functions used to express the unknown magnetic currents, we obtain for layer *A*1:

> *A*<sup>1</sup>

where the l.h.s. term represents the self–term, i.e. the effect of the magnetic current **M***A*<sup>1</sup> on the aperture *A*<sup>1</sup> , and the r.h.s. term is the coupling term, i.e. the effect of the magnetic current

2,*<sup>t</sup>* <sup>+</sup> **<sup>H</sup>***A*2*A*<sup>1</sup>

**H***A*2*A*<sup>1</sup>

<sup>2</sup> , **M***A*<sup>1</sup> � · **f***<sup>m</sup>* +

1,*<sup>t</sup>* <sup>=</sup> **<sup>H</sup>***A*1*A*<sup>1</sup>

 *A*1 �**G**<sup>ˆ</sup> *<sup>A</sup>*<sup>1</sup> *<sup>A</sup>*<sup>1</sup>

 *A*1 �**G**<sup>ˆ</sup> *<sup>A</sup>*<sup>2</sup> *<sup>A</sup>*<sup>1</sup>

*m<sup>A</sup>*

**M***<sup>A</sup>* =

<sup>−</sup>*jβmdy <sup>y</sup>***M***<sup>A</sup>* (2)

*<sup>n</sup>* **f***<sup>n</sup>* (3)

2,*<sup>t</sup>* (4)

2,*<sup>t</sup>* · **f***m*, *m* = 1, . . . , *Nb* (5)

<sup>2</sup> , **M***A*<sup>2</sup> � · **f***m*, *m* = 1, . . . , *Nb* (6)

*<sup>A</sup>*<sup>1</sup> *<sup>m</sup>* , *<sup>m</sup>* <sup>=</sup> 1, . . . , *Nb* (7)

**G**ˆ *<sup>A</sup>*<sup>1</sup> *<sup>A</sup>*<sup>1</sup> <sup>2</sup> · <sup>ˆ</sup> **f** *A*1 *n* · **f** *A*1 *m* 

different layers may be of different shape but they must have the same periodicity.

**M***A*,*nm* = *e*

**H***A*1*A*<sup>1</sup>

**H***A*1*A*<sup>1</sup> 2,*<sup>t</sup>* · **f***<sup>m</sup>* +

**G**ˆ *<sup>A</sup>*<sup>1</sup> *<sup>A</sup>*<sup>1</sup> <sup>1</sup> · <sup>ˆ</sup> **f** *A*1 *n* · **f** *<sup>A</sup>*<sup>1</sup> *<sup>m</sup>* + *A*1 *A*1

**G**ˆ *<sup>A</sup>*<sup>2</sup> *<sup>A</sup>*<sup>1</sup> <sup>2</sup> · <sup>ˆ</sup> **f** *A*2 *n* · **f**

to be discretized as a linear combination of *Nb* basis functions:

first layer, on an intermediate layer or on the last layer.

 *A*<sup>1</sup>

<sup>1</sup> , −**M***A*<sup>1</sup> � · **f***<sup>m</sup>* =

<sup>2</sup> , **M***A*<sup>2</sup> � · **f***m*, +

Expressing the unknown magnetic currents in the form (3):

which, introducing the Green Functions can be written as:

can be written as:

 *A*<sup>1</sup>

**H***A*1*A*<sup>1</sup> 1,*<sup>t</sup>* · **f***<sup>m</sup>* =

> *A*1 �**G**<sup>ˆ</sup> *<sup>A</sup>*<sup>1</sup> *<sup>A</sup>*<sup>1</sup>

> *A*1 �**G**<sup>ˆ</sup> *<sup>A</sup>*<sup>2</sup> *<sup>A</sup>*<sup>1</sup>

*Nb* ∑ *n*=1

= *Nb* ∑ *n*=1 *<sup>m</sup>A*<sup>2</sup> *<sup>n</sup> A*1 *A*1

**M***A*<sup>2</sup> on the aperture *A*1.

<sup>−</sup>*mA*<sup>1</sup> *<sup>n</sup> A*1 *A*1

where **M***<sup>A</sup>* is the current in the central cell.

$$\sum\_{n=1}^{N\_b} -m\_n^{A\_2} \left( \int\_{A\_2} \left( \int\_{A\_2} \hat{\mathbf{G}}\_2^{A\_2 A\_2} \cdot \hat{\mathbf{f}}\_n^{A\_2} \right) \cdot \mathbf{f}\_m^{A\_2} + \int\_{A\_2} \left( \int\_{A\_2} \hat{\mathbf{G}}\_3^{A\_2 A\_2} \cdot \hat{\mathbf{f}}\_n^{A\_2} \right) \cdot \mathbf{f}\_m^{A\_2} \right) =$$

$$\sum\_{n=1}^{N\_b} m\_n^{A\_3} \int\_{A\_2} \left( \int\_{A\_2} \hat{\mathbf{G}}\_3^{A\_3 A\_2} \cdot \hat{\mathbf{f}}\_n^{A\_3} \right) \cdot \mathbf{f}\_m^{A\_2} - \sum\_{n=1}^{N\_b} m\_n^{A\_1} \int\_{A\_2} \left( \int\_{A\_2} \hat{\mathbf{G}}\_2^{A\_1 A\_2} \cdot \hat{\mathbf{f}}\_n^{A\_1} \right) \cdot \mathbf{f}\_m^{A\_2}$$

where *m* = 1, . . . , *Nb* and the l.h.s. term represents the self–term and the r.h.s. term the coupling terms, i.e. the effect of the magnetic currents **M***A*<sup>3</sup> and **M***A*<sup>1</sup> on the aperture *A*2.

For the last (third) discontinuity *A*3, the boundary condition leads to:

$$\mathbf{H}\_{3,t}^{A\_3A\_3} + \mathbf{H}\_{3,t}^{A\_2A\_3} = \mathbf{H}\_{4,t}^{A\_3A\_3} + \mathbf{H}\_{\text{inc},t} \tag{10}$$

and we obtain:

$$\sum\_{n=1}^{N\_b} -m\_n^{A\_3} \left( \int\_{A\_3} \left( \int\_{A\_3} \hat{\mathbf{G}}\_3^{A\_3 A\_3} \cdot \hat{\mathbf{f}}\_n^{A\_3} \right) \cdot \mathbf{f}\_m^{A\_3} + \int\_{A\_3} \left( \int\_{A\_3} \hat{\mathbf{G}}\_4^{A\_3 A\_3} \cdot \hat{\mathbf{f}}\_n^{A\_3} \right) \cdot \mathbf{f}\_m^{A\_3} \right)$$

$$= -\sum\_{n=1}^{N\_b} m\_n^{A\_2} \int\_{A\_3} \left( \int\_{A\_3} \hat{\mathbf{G}}\_3^{A\_2 A\_3} \cdot \hat{\mathbf{f}}\_n^{A\_2} \right) \cdot \mathbf{f}\_m^{A\_3} - \int\_S \mathbf{H}\_{\text{inc}} \cdot \mathbf{f}\_m^{A\_3} (\mathbf{x}, y) \, dS \tag{11}$$

where *m* = 1, . . . , *Nb* and, on the l.h.s we have the self–term and the first r.h.s. term is the coupling term, i.e. the effect of the magnetic current **M***A*<sup>2</sup> on the aperture *A*3. The second term of the r.h.s represent the known terms of the MoM system.

The presence of the coupling terms is the critical point in the multi–layer formulation. The key feature of the aperture approach, as previously illustrated, relies on the decoupling of the different regions. If we have a more general structure made up of N layers, we know that when dealing with the aperture *A*1, which lies in the bottom layer, we only have to consider the effect of the magnetic current on the aperture *A*2. When considering the aperture *AN*, the last one, we only have to consider the effect of the magnetic current on the aperture *AN*−1, and when considering any intermediate aperture *Ai*, for *i* = 2, ..., *N* − 1 we have to take into account only the effect of the apertures immediately below and above, i.e. *Ai*−<sup>1</sup> and *Ai*+1. Equations (7),(9),(11) can be written in matrix form:

$$
\mathcal{Y} \cdot \mathcal{M} = \mathcal{T} \tag{12}
$$

where Y is the coefficient matrix, Mis the vector of the unknown coefficients, i.e. the magnetic current on the apertures, and T is the r.h.s, which represent the incident magnetic field on the last aperture *A*3.

The solution of the above equation (12) yields the unknown induced magnetic current on the apertures. The reflection coefficient is then easily calculated from the magnetic current on the upper layer.

It is important to stress out that inserting more layers will result in a larger matrix, affecting thus the overall computational time, yet for each additional layer only two blocks have to be computed.

(a) Geometry

optimization stage cannot be known in advance.

Fig. 10. UC–EBG

Table 1. Data

Table 2. Results

the reflection coefficient is 2.2◦).

<sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> <sup>15</sup> <sup>16</sup> −200

(b) Reflection Coefficient

frequency

Synthesis of Planar EBG Structures Based on Genetic Programming 333

this approach is irrespective of the particular shape of the metallization, which in the GP

The GP optimization was launched to find the best structure, keeping as fixed parameters the periodicity, substrate dielectric constant and thickness, as shown in the following table.

**Fixed Parameters**

Periodicity (unit square cell) *dx* = 3.048 mm; *dy* = 3.048 mm

In Fig.11 and Fig.12 we can see two different runs of our optimization. And in Tab.2 it is

Itoh (lit.) 76.05◦ 20.52◦ -48.56◦ Case A 57.8◦ 2.2◦ -63.16◦ Case B 39.28◦ 0.83◦ -35.16◦ Case C 13.06◦ -6.13◦ -24.93◦

In Case A we used for the unit cell a discretization step of 0.1016mm, making the playground for the GP optimization a 30x30 grid. The penalty coefficients, both for the number of unknowns (i.e. borders of the discretized geometry) and for the bandwidth were initially not too strict, making the evolution process quickly evolve towards quite bulky structures. The resonant frequency was reached after few generations (for case A at gen. 13 the phase of

**Phase of Refl. Coeff. 14 GHz 14.2 GHz 14.4 GHz**

Discretization step 0.1016 mm (30 × 30 *grid*)

Center Frequency 14.2 GHz

Substrate dielectric constant *<sup>r</sup>* = 10.2 Substrate thickness 0.635 mm

shown how the goal is achieved, in comparison also to the reference structure.

phase of reflection coefficient [Deg]

The explicit evaluation of MoM system matrix Y requires the computation of the magnetic fields we previously introduced in (6),(8),(10). These are total magnetic fields,i.e. fields due to the magnetic currents of all the periodicity cells. Since the problem is linear, we start computing the magnetic field due to the magnetic current of a single cell. This field can be expressed in terms of a spectral domain (dyadic) Green function **G**ˆ (*u*, *v*) connecting the transverse magnetic current density on the aperture to the transverse magnetic field:

$$\mathbf{H}\_{t}^{\text{unit cell}}(\mathbf{x}, y) = \int\_{-\infty}^{+\infty} \int\_{-\infty}^{+\infty} \mathbf{\hat{G}}(u, v) \cdot \mathbf{\hat{M}}(u, v) \, e^{-j(ux + vy)} \, du \, dv \tag{13}$$

where **M** is the magnetic current on the aperture of the central cell.

We can then write the overall magnetic transverse field of the infinite planar array by summing (13) on the infinite cells of the AMC:

$$\mathbf{H}\_{l}(x,y) = \sum\_{r} \sum\_{s} e^{-jk\_{0}r d\_{x} \sin \theta \cos \phi} \ e^{-jk\_{0}s d\_{y} \sin \theta \sin \phi}.$$

$$\cdot \int\_{-\infty}^{+\infty} \int\_{-\infty}^{+\infty} \mathbf{\hat{G}}(u,v) \cdot \mathbf{\hat{M}}(u,v) \ e^{-j(ux+vy)} \, du \, dv \tag{14}$$

where *dx* and *dy* are the periodicity along the *x*–axis and the *y*–axis respectively, (*θ*, *φ*) is the direction of the incident field and *k*<sup>0</sup> is the free–space propagation constant.

Equation (14) can be simplyfied taking into account the periodicity of the structure. The fields on either side of the AMC can be expanded in terms of the Floquet space harmonics (Munk, 2000). Therefore the cell summation in (14) can be rearranged (Pozar & Schaubert, 1984) using the Poisson formula. As a result, we obtain the following compact expression:

$$\mathbf{H}\_{l}(\mathbf{x},\mathbf{y}) = \frac{4\pi^{2}}{d\_{x}d\_{\mathcal{Y}}} \sum\_{r} \sum\_{s} \hat{\mathbf{G}}(\boldsymbol{u}\_{r\star}\boldsymbol{v}\_{s}) \cdot \mathbf{\hat{M}}(\boldsymbol{u}\_{r\star}\boldsymbol{v}\_{s}) \, e^{-j(\boldsymbol{u}\_{r\star}\boldsymbol{x}+\boldsymbol{v}\_{s}\boldsymbol{y})} \tag{15}$$

where the summation is now on the Floquet modes.

Equation (15) can be used for both fields (above and below the metallization) by using the pertinent Green function, which can be derived from the equivalent circuit shown in Fig.8(b) and setting **M** = ±**M***<sup>A</sup>* (current on the pertinent layer of the central cell).

#### **4. Single–layer EBG design results**

Significative and promising results have been found when this approach was applied to a single–layer EBG structure (Deias et al., 2009a).

One of the most interesting structures that can be found in literature is the Itoh's UC–EBG structure (Yang et al., 1999), shown in Fig.10(a) consisting of a Jerusalem cross aperture.

The reference structure has periodicity *dx* = *dy* = 3.048*mm*, *h* = 0.635*mm* and *ε* = 10.2, displaying a resonant frequency at 14.25GHz. The reflection coefficient is shown in Fig.10(b).

We decided to use these parameters as constraints for our test, in order to compare the results obtained using the GP approach with Itoh's UC–EBG.

We used for the full–wave analysis of the periodic structure our previously developed Fortran code, implementing the aperture approach described in the previous section. Furthermore

Fig. 10. UC–EBG

12 Will-be-set-by-IN-TECH

The explicit evaluation of MoM system matrix Y requires the computation of the magnetic fields we previously introduced in (6),(8),(10). These are total magnetic fields,i.e. fields due to the magnetic currents of all the periodicity cells. Since the problem is linear, we start computing the magnetic field due to the magnetic current of a single cell. This field can be expressed in terms of a spectral domain (dyadic) Green function **G**ˆ (*u*, *v*) connecting the

We can then write the overall magnetic transverse field of the infinite planar array by summing

<sup>−</sup>*jk*0*rdx* sin *<sup>θ</sup>* cos *<sup>φ</sup> e*

**<sup>G</sup>**<sup>ˆ</sup> (*u*, *<sup>v</sup>*) · **<sup>M</sup>**<sup>ˆ</sup> (*u*, *<sup>v</sup>*) *<sup>e</sup>*

where *dx* and *dy* are the periodicity along the *x*–axis and the *y*–axis respectively, (*θ*, *φ*) is the

Equation (14) can be simplyfied taking into account the periodicity of the structure. The fields on either side of the AMC can be expanded in terms of the Floquet space harmonics (Munk, 2000). Therefore the cell summation in (14) can be rearranged (Pozar & Schaubert, 1984) using

Equation (15) can be used for both fields (above and below the metallization) by using the pertinent Green function, which can be derived from the equivalent circuit shown in Fig.8(b)

Significative and promising results have been found when this approach was applied to a

One of the most interesting structures that can be found in literature is the Itoh's UC–EBG structure (Yang et al., 1999), shown in Fig.10(a) consisting of a Jerusalem cross aperture.

The reference structure has periodicity *dx* = *dy* = 3.048*mm*, *h* = 0.635*mm* and *ε* = 10.2, displaying a resonant frequency at 14.25GHz. The reflection coefficient is shown in Fig.10(b). We decided to use these parameters as constraints for our test, in order to compare the results

We used for the full–wave analysis of the periodic structure our previously developed Fortran code, implementing the aperture approach described in the previous section. Furthermore

**<sup>G</sup>**<sup>ˆ</sup> (*ur*, *vs*) · **<sup>M</sup>**<sup>ˆ</sup> (*ur*, *vs*) *<sup>e</sup>*

**<sup>G</sup>**<sup>ˆ</sup> (*u*, *<sup>v</sup>*) · **<sup>M</sup>**<sup>ˆ</sup> (*u*, *<sup>v</sup>*) *<sup>e</sup>*

<sup>−</sup>*jk*0*sdy* sin *<sup>θ</sup>* sin *<sup>φ</sup>* ·

<sup>−</sup>*j*(*ux*+*vy*) *du dv* (13)

<sup>−</sup>*j*(*ux*+*vy*) *du dv* (14)

<sup>−</sup>*j*(*urx*+*vsy*) (15)

transverse magnetic current density on the aperture to the transverse magnetic field:

 +∞ −∞

*e*

direction of the incident field and *k*<sup>0</sup> is the free–space propagation constant.

the Poisson formula. As a result, we obtain the following compact expression:

<sup>∑</sup>*<sup>r</sup>* <sup>∑</sup>*<sup>s</sup>*

and setting **M** = ±**M***<sup>A</sup>* (current on the pertinent layer of the central cell).

 +∞ −∞

where **M** is the magnetic current on the aperture of the central cell.

 +∞ −∞

**<sup>H</sup>***t*(*x*, *<sup>y</sup>*) = <sup>∑</sup>*<sup>r</sup>* <sup>∑</sup>*<sup>s</sup>*

**H***unit cell*

(13) on the infinite cells of the AMC:

*<sup>t</sup>* (*x*, *y*) =

· +∞ −∞

**<sup>H</sup>***t*(*x*, *<sup>y</sup>*) = <sup>4</sup>*π*<sup>2</sup>

where the summation is now on the Floquet modes.

**4. Single–layer EBG design results**

single–layer EBG structure (Deias et al., 2009a).

obtained using the GP approach with Itoh's UC–EBG.

*dxdy*

this approach is irrespective of the particular shape of the metallization, which in the GP optimization stage cannot be known in advance.

The GP optimization was launched to find the best structure, keeping as fixed parameters the periodicity, substrate dielectric constant and thickness, as shown in the following table.


Table 1. Data

In Fig.11 and Fig.12 we can see two different runs of our optimization. And in Tab.2 it is shown how the goal is achieved, in comparison also to the reference structure.


Table 2. Results

In Case A we used for the unit cell a discretization step of 0.1016mm, making the playground for the GP optimization a 30x30 grid. The penalty coefficients, both for the number of unknowns (i.e. borders of the discretized geometry) and for the bandwidth were initially not too strict, making the evolution process quickly evolve towards quite bulky structures. The resonant frequency was reached after few generations (for case A at gen. 13 the phase of the reflection coefficient is 2.2◦).

13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6 14.8 15

Synthesis of Planar EBG Structures Based on Genetic Programming 335

[GHz]

(a) Case A (b) Case B (c) Case C

We can exploit this approach using a more complex fitness function in order to obtain, for example, larger bandwidth (Deias et al., 2009a),as shown in case B, or at more interesting

It is well known that there is a growing interest in antennas integrated with EBG surfaces for communication system applications, covering the 2.45 GHz and the 5 GHz wireless networking bands (Hung-Hsuan et al., 2007; Zhu & Langley, 2009). As proof of the effectiveness of our approach a simple EBG surface at this working frequencies was found

In Fig.16(b) we can see the reflection coefficient for the best individual shown Fig.16(a), as obtained after nine generations. The performance of our optimization is quite good leading

These results prove that the main advantage of the genetic approach is to be identified in the generalization of the "solution space", evolution being able to breed geometries that we could

to a simple and effective solution with a yet limited computational effort.

−150

Fig. 13. Reflection Coefficient.

Fig. 14. Best individual

frequencies (Deias et al., 2010).

with a yet limited computational effort.

Fixed parameters in this case are the following:

−100

case A case B case C Itoh

−50

0

Refl. Coeff. [°]

50

100

150

Fig. 12. Case B.

Fig. 11. Case A.

In Case B we then acted more stringently on the penalty coefficients. The structure remains spatially more limited, as the number of unknowns is kept reasonably low, and the bandwidth is largely improved. The number of generations in order to achieve a good result is still low.

As expected, the geometries resemble the reference one and other well–known configurations, since the only stringent requirement in this case is the resonant frequency. The fitness in this case is relatively simple, aimed almost uniquely to direct the evolution process towards a structure that resonates at the desired frequency. We then can insert an additional penalty coefficient in order to maximize the bandwidth, a secondary objective overlapped to the main goal.

Finally, as we can see in Fig.14(c), we allowed the geometry to touch the borders of the unit cell, thus considering a continuous aperture (while the geometry of the metallic patch will therefore be finite), and we found out a further improvement in the bandwidth. In Fig.15 the entire surface is shown.

Fig. 13. Reflection Coefficient.

14 Will-be-set-by-IN-TECH

(a) GEN.0 (b) GEN.4 (c) GEN.5

(d) GEN.8 (e) GEN.11 (f) GEN.12

(a) GEN.0 (b) GEN.2 (c) GEN.5

(d) GEN.6 (e) GEN.9 (f) GEN.11

In Case B we then acted more stringently on the penalty coefficients. The structure remains spatially more limited, as the number of unknowns is kept reasonably low, and the bandwidth is largely improved. The number of generations in order to achieve a good result is still low. As expected, the geometries resemble the reference one and other well–known configurations, since the only stringent requirement in this case is the resonant frequency. The fitness in this case is relatively simple, aimed almost uniquely to direct the evolution process towards a structure that resonates at the desired frequency. We then can insert an additional penalty coefficient in order to maximize the bandwidth, a secondary objective overlapped to the main

Finally, as we can see in Fig.14(c), we allowed the geometry to touch the borders of the unit cell, thus considering a continuous aperture (while the geometry of the metallic patch will therefore be finite), and we found out a further improvement in the bandwidth. In Fig.15 the

Fig. 11. Case A.

Fig. 12. Case B.

goal.

entire surface is shown.

Fig. 14. Best individual

We can exploit this approach using a more complex fitness function in order to obtain, for example, larger bandwidth (Deias et al., 2009a),as shown in case B, or at more interesting frequencies (Deias et al., 2010).

It is well known that there is a growing interest in antennas integrated with EBG surfaces for communication system applications, covering the 2.45 GHz and the 5 GHz wireless networking bands (Hung-Hsuan et al., 2007; Zhu & Langley, 2009). As proof of the effectiveness of our approach a simple EBG surface at this working frequencies was found with a yet limited computational effort.

Fixed parameters in this case are the following:

In Fig.16(b) we can see the reflection coefficient for the best individual shown Fig.16(a), as obtained after nine generations. The performance of our optimization is quite good leading to a simple and effective solution with a yet limited computational effort.

These results prove that the main advantage of the genetic approach is to be identified in the generalization of the "solution space", evolution being able to breed geometries that we could

(a) Geometry

Synthesis of Planar EBG Structures Based on Genetic Programming 337

2 2.2 2.4 2.6 2.8 3

[GHz]

(b) Reflection Coefficient

**Fixed Parameters**

Periodicity (unit square cell) *dx* = 3.048 mm; *dy* = 3.048 mm

This quite simple result proves that genetic programming can be efficiently adopted for this kind of design synthesis and optimization even in the case of complex structures consisting of multiple layers of EBG surfaces. Moreover the fitness function can be modified in order to

Discretization step 0.1524 mm (20 × 20 *grid*)

Center Frequency 2.45 GHz

Substrate dielectric constant *<sup>r</sup>* = 10.2 Substrate thickness 3.354772 mm

−180 −140 −100 −60 −20 20 60 100 140 180

take into consideration other requirements.

Fig. 16

Table 4. Data

Refl. Coeff. [°]

Fig. 15. EBG Surface (Case C)


Table 3. Data

never think of. Evolutionary computation on the other hand let us scope through geometries as though they were human–described.

#### **5. Multi–layer EBG design results**

The periodic surfaces in the single–layer configuration shown so far exhibit perfect reflection or transmission only at resonance. However, many applications, exploiting the filtering properties of frequency selective surfaces, seek for a resonant curve with a flat top and faster roll off. This goal can be achieved by using two or more periodic surfaces cascaded with dielectric slabs sandwiched in between . By using multiple layers of frequency selective surfaces as part of the substrate we operate in a manner similar to that used for designing broad–band microwave filters. With such a configuration typically we can obtain a bandwidth that is considerably larger than that of a single structure.

Genetic Programming strategy in conjunction with the flexible aperture approach previously described proved to be effective in the design and optimization of EBG surfaces in the case of a two–layer structure with the same geometry in both layers. When considering more layers or different geometries we only have to take into consideration an increased computational effort.

In the case of a two–layer configuration with the following fixed parameters:

We obtained the following geometry after two generations, each generation with a population of 150 individuals:

(a) Geometry

(b) Reflection Coefficient

#### Fig. 16

16 Will-be-set-by-IN-TECH

**Fixed Parameters**

Periodicity (unit square cell) *dx* = 40 mm; *dy* = 40 mm

Discretization step 0.8 mm (50 × 50 *grid*)

never think of. Evolutionary computation on the other hand let us scope through geometries

The periodic surfaces in the single–layer configuration shown so far exhibit perfect reflection or transmission only at resonance. However, many applications, exploiting the filtering properties of frequency selective surfaces, seek for a resonant curve with a flat top and faster roll off. This goal can be achieved by using two or more periodic surfaces cascaded with dielectric slabs sandwiched in between . By using multiple layers of frequency selective surfaces as part of the substrate we operate in a manner similar to that used for designing broad–band microwave filters. With such a configuration typically we can obtain a bandwidth

Genetic Programming strategy in conjunction with the flexible aperture approach previously described proved to be effective in the design and optimization of EBG surfaces in the case of a two–layer structure with the same geometry in both layers. When considering more layers or different geometries we only have to take into consideration an increased computational

We obtained the following geometry after two generations, each generation with a population

In the case of a two–layer configuration with the following fixed parameters:

Center Frequency 2.45 GHz

Substrate dielectric constant *<sup>r</sup>* = 1.38 Substrate thickness 2.2 mm

Fig. 15. EBG Surface (Case C)

as though they were human–described.

**5. Multi–layer EBG design results**

that is considerably larger than that of a single structure.

Table 3. Data

effort.

of 150 individuals:


#### Table 4. Data

This quite simple result proves that genetic programming can be efficiently adopted for this kind of design synthesis and optimization even in the case of complex structures consisting of multiple layers of EBG surfaces. Moreover the fitness function can be modified in order to take into consideration other requirements.

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#### Fig. 17

#### **6. Acknowledgements**

Work partially supported by Regione Autonoma della Sardegna, under contract CRP1\_511: "Valutazione e utilizzo della Genetic Programming nel progetto di strutture a radiofrequenza e microonde".

#### **7. References**

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18 Will-be-set-by-IN-TECH

(a) Geometry

9 10 11 12 13 14

[GHz]

(b) Reflection Coefficient

Work partially supported by Regione Autonoma della Sardegna, under contract CRP1\_511: "Valutazione e utilizzo della Genetic Programming nel progetto di strutture a radiofrequenza

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−200

−150

−100

−50

Refl. Coeff. [°]

Fig. 17

e microonde".

**7. References**

**6. Acknowledgements**

2): 641–644.

0

50

100

150

200


**1. Introduction** 

importance.

Berkov8, E. Semenova8 and N. Gorn8

*Dipartimento di Fisica, Perugia, Italy* 

*Garching bei Muenchen, Germany* 

*8Innovent Technology Development, Jena, Germany*

*3CNISM, Unità di Perugia and Dipartimento di Fisica, Perugia, Italy 4Dipartimento di Fisica and CNISM, Università di Ferrara, Ferrara, Italy 5Faculty of Physics, Adam Mickiewicz University, Poznan, Poland 6 H. H. Wills Physics Laboratory, Bristol, United Kingdom* 

**14** 

**Magnonic Metamaterials** 

A large proportion of the recent growth of the volume of electromagnetics research has been associated with the emergence of so called electromagnetic metamaterials1 and the discovered ability to design their unusual properties2,3 by tweaking the geometry and structure of the constituent "meta-atoms"4. For example, negative permittivity and negative permeability can be achieved, leading to negative refractive index metamaterials2. The negative permeability could be obtained via geometrical control of high frequency currents, e.g. in arrays of split ring resonators5, or alternatively one could rely on spin resonances in natural magnetic materials6,7, as was suggested by Veselago in Ref. 2. The age of nanotechnology therefore sets an intriguing quest for additional benefits to be gained by structuring natural magnetic materials into so called *magnonic* metamaterials, in which the frequency and strength of resonances based on spin waves (magnons)8 are determined by the geometry and magnetization configuration of meta-atoms. Spin waves can have frequencies of up to hundreds of GHz (in the exchange dominated regime)6-9 and have already been shown to play an important role in the high frequency magnetic response of composites10-14. Moreover, in view of the rapid advances in the field of magnonics9,15,16,, which in particular promises devices employing propagating spin waves, the appropriate design of magnonic metamaterials with properties defined with respect to propagating spin waves rather than electromagnetic waves acquires an independent and significant

\*M. Dvornik1, R.V. Mikhaylovskiy1, O. Dmytriiev1, G. Gubbiotti2,3, S. Tacchi3, M. Madami3, G. Carlotti3, F. Montoncello4, L. Giovannini4, R. Zivieri4, J.W. Klos5, M.L. Sokolovskyy5, S. Mamica5, M. Krawczyk5, M. Okuda6, J.C. Eloi6, S. Ward Jones6, W. Schwarzacher6, T. Schwarze7, F. Brandl7, D. Grundler7, D.V.

<sup>2</sup>*Istituto Officina dei Materiali del Consiglio Nazionale delle Ricerche (IOM-CNR), Sede di Perugia, c/o* 

*7Lehrstuhl fuer Physik funktionaler Schichtsysteme, Physik Department, Technische Universität München,* 

V.V. Kruglyak et al.\*

*University of Exeter, Exeter,* 

*School of Physics,* 

*United Kingdom* 


### **Magnonic Metamaterials**

### V.V. Kruglyak et al.\*

*School of Physics, University of Exeter, Exeter, United Kingdom* 

#### **1. Introduction**

20 Will-be-set-by-IN-TECH

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A large proportion of the recent growth of the volume of electromagnetics research has been associated with the emergence of so called electromagnetic metamaterials1 and the discovered ability to design their unusual properties2,3 by tweaking the geometry and structure of the constituent "meta-atoms"4. For example, negative permittivity and negative permeability can be achieved, leading to negative refractive index metamaterials2. The negative permeability could be obtained via geometrical control of high frequency currents, e.g. in arrays of split ring resonators5, or alternatively one could rely on spin resonances in natural magnetic materials6,7, as was suggested by Veselago in Ref. 2. The age of nanotechnology therefore sets an intriguing quest for additional benefits to be gained by structuring natural magnetic materials into so called *magnonic* metamaterials, in which the frequency and strength of resonances based on spin waves (magnons)8 are determined by the geometry and magnetization configuration of meta-atoms. Spin waves can have frequencies of up to hundreds of GHz (in the exchange dominated regime)6-9 and have already been shown to play an important role in the high frequency magnetic response of composites10-14. Moreover, in view of the rapid advances in the field of magnonics9,15,16,, which in particular promises devices employing propagating spin waves, the appropriate design of magnonic metamaterials with properties defined with respect to propagating spin waves rather than electromagnetic waves acquires an independent and significant importance.


<sup>\*</sup>M. Dvornik1, R.V. Mikhaylovskiy1, O. Dmytriiev1, G. Gubbiotti2,3, S. Tacchi3, M. Madami3, G. Carlotti3, F. Montoncello4, L. Giovannini4, R. Zivieri4, J.W. Klos5, M.L. Sokolovskyy5, S. Mamica5, M. Krawczyk5, M. Okuda6, J.C. Eloi6, S. Ward Jones6, W. Schwarzacher6, T. Schwarze7, F. Brandl7, D. Grundler7, D.V. Berkov8, E. Semenova8 and N. Gorn8

<sup>2</sup>*Istituto Officina dei Materiali del Consiglio Nazionale delle Ricerche (IOM-CNR), Sede di Perugia, c/o Dipartimento di Fisica, Perugia, Italy* 

*<sup>7</sup>Lehrstuhl fuer Physik funktionaler Schichtsysteme, Physik Department, Technische Universität München, Garching bei Muenchen, Germany* 

*<sup>8</sup>Innovent Technology Development, Jena, Germany*

Magnonic Metamaterials 343

Some metamaterials (e.g. arrays of voids) are simpler to consider as created by geometrical modification of previously continuous materials. This however does not exclude their treatment as if they were constructed from building blocks considered above. Also, one has to note that, from the point of view of electromagnetic waves, vacuum is a continuous material in that important sense that it supports propagation of electromagnetic waves. Therefore, even arrays of entirely disconnected blocks are metamaterials for electromagnetic

Generally, various dynamical properties of metamaterials can be considered relative to excitations with wavelength either comparable to or much greater than the characteristic size of the building blocks. The former case is associated with studies of artificial "band gap crystals", e.g. photonic17, plasmonic18, phononic19, and magnonic20 crystals. The latter case allows one to study the metamaterials as some effectively continuous media. The physical object under study is nonetheless the same in both cases. For example, metamaterials with artificial periodic modulation of the refractive index with periodicity comparable to the wavelength of electromagnetic waves in the visible range are known as photonic band gap structures, using which it is possible to enable propagation of light in particular directions, to localize it in chosen channels or zones, or even to completely prohibit its propagation. In the microwave frequency range, the same structures would behave as effectively continuous

In a particular frequency region of interest, the same metamaterials can behave differently with respect to excitations of different kinds, e.g. electromagnetic, sound or spin waves. Attractive opportunities arise from use of one excitation, with respect to which the sample behaves as a discrete structure (e. g. a band gap metamaterial), to design a resonance feature for another excitation, with respect to which the structure behaves as quasi-continuous metamaterial. For example, plasmonic resonances can be used to alter effective electromagnetic properties in THz to visible frequency range. Magnonic and LC resonances can be used for the same purpose in GHz-THz and GHz frequency ranges, respectively.

A thorough reader will probably find that, although term "metamaterial" emerged recently as a result of ever increasing pressure to spice up funding applications and publications in high profile journals, the underpinning research can be sometimes traced back to the middle or even beginning of the 20th century. Yet, one can also argue that it is the recent advances in nanofabrication and experimental tools that justify the boom by allowing researchers to revisit the established notions at reduced length scales and to apply results to modern

The main reason for electrically neutral atoms to be able to interact with electromagnetic radiation is that they themselves possess electrical structure. Indeed, each atom consists of positively charged core and negatively charged electrons. The core and electrons are not rigidly coupled, and so, the atom can be electrically polarised by and thereby interact with

However, the electric charge is not the only property responsible for the interaction of atoms or materials with electromagnetic radiation. In particular, spins and associated magnetic

**3. Spin waves and effectively continuous magnonic metamaterials** 

waves. This distinguishes electromagnetic waves from the other excitations.

materials.

technology.

an external electric field.

In this Chapter, we review the recent advances in studies of magnonic metamaterials and challenges that have to be overcome in order for the rich opportunities of exploitation of such metamaterials to be realised. We start from a discussion of the notion of a metamaterial and its relevance to areas beyond that of common electromagnetics. We introduce band gap and effectively continuous magnonic metamaterials and demonstrate their properties and functionalities, presenting either experimental or theoretical evidence as appropriate for the illustration.

#### **2. What makes a metamaterial?**

To answer the question in the title in the most general sense (i.e. going beyond the topic of more common electromagnetic metamaterials), one has to learn first what makes standard (i.e., "nature-made") *materials*. One talks about a material when many atoms or ions are joined together to collectively create a novel quality with properties that are not observed in the constituent atoms in isolation. For example, isolated atoms have discrete electronic energy levels and associated discrete electromagnetic spectra. In a material consisting of many of such atoms, each discrete electronic level is split into a continuous "electronic band", and accordingly the electromagnetic spectrum of the material is also continuous.

As a result of joining into a material, atoms also collectively acquire new properties that are less trivially connected with those of isolated atoms. Firstly, different classes of materials are formed, including e.g. dielectrics, semiconductors, or metals, ferro-, piezo-, or segnetoelectrics, dia-, para-, antiferro- or ferro-magnets, superconductors, materials with a range of different mechanical properties, etc. Secondly, new classes of waves (excitations) and their quanta (so called quasi-particles) are observed, including plasmons, magnons, phonons, excitons etc. One can also distinguish many "hybrids" of the solid state quasi-particles among themselves as well as with photons, e.g. surface plasmon-polaritons, magnetoexcitons etc.

A material does not have to have a periodic arrangement of atoms and can be amorphous or quasi-crystalline. The classification of materials and their properties often does not have strict boundaries, while the differences are often quantitative. For example, dielectrics and semiconductors are different only by the size of the band gap; the same material can be ferromagnetic below and paramagnetic above a certain temperature, etc. Some electronic energy levels can remain discrete, therefore retaining their atomistic character. Materials can also consist of rather complex building blocks, e.g. molecules, crystal unit cells (often consisting of many atoms or ions), or crystallites. The most complex nature-made (truly functional!) materials – biological tissues – consist of living cells composed of millions of atoms.

This brings us to the idea of a *metamaterial* – an artificial assembly of "man-made" building blocks with tailored properties. Living aside static properties and biological tissues, the most practical ways to "tailor" properties of the building blocks are via their geometrical shaping and compositional modulation. The former leads to confinement of solid state excitations (e.g. those listed above) and hence to formation of a discrete spectrum of the allowed modes. Brought sufficiently close together, the blocks can form a metamaterial with the discrete spectrum split into a band structure.

In this Chapter, we review the recent advances in studies of magnonic metamaterials and challenges that have to be overcome in order for the rich opportunities of exploitation of such metamaterials to be realised. We start from a discussion of the notion of a metamaterial and its relevance to areas beyond that of common electromagnetics. We introduce band gap and effectively continuous magnonic metamaterials and demonstrate their properties and functionalities, presenting either experimental or theoretical evidence as appropriate for the

To answer the question in the title in the most general sense (i.e. going beyond the topic of more common electromagnetic metamaterials), one has to learn first what makes standard (i.e., "nature-made") *materials*. One talks about a material when many atoms or ions are joined together to collectively create a novel quality with properties that are not observed in the constituent atoms in isolation. For example, isolated atoms have discrete electronic energy levels and associated discrete electromagnetic spectra. In a material consisting of many of such atoms, each discrete electronic level is split into a continuous "electronic band", and accordingly the electromagnetic spectrum of the material is also continuous.

As a result of joining into a material, atoms also collectively acquire new properties that are less trivially connected with those of isolated atoms. Firstly, different classes of materials are formed, including e.g. dielectrics, semiconductors, or metals, ferro-, piezo-, or segnetoelectrics, dia-, para-, antiferro- or ferro-magnets, superconductors, materials with a range of different mechanical properties, etc. Secondly, new classes of waves (excitations) and their quanta (so called quasi-particles) are observed, including plasmons, magnons, phonons, excitons etc. One can also distinguish many "hybrids" of the solid state quasi-particles among themselves as well as with photons, e.g. surface plasmon-polaritons, magneto-

A material does not have to have a periodic arrangement of atoms and can be amorphous or quasi-crystalline. The classification of materials and their properties often does not have strict boundaries, while the differences are often quantitative. For example, dielectrics and semiconductors are different only by the size of the band gap; the same material can be ferromagnetic below and paramagnetic above a certain temperature, etc. Some electronic energy levels can remain discrete, therefore retaining their atomistic character. Materials can also consist of rather complex building blocks, e.g. molecules, crystal unit cells (often consisting of many atoms or ions), or crystallites. The most complex nature-made (truly functional!) materials – biological tissues – consist of living cells composed of millions of

This brings us to the idea of a *metamaterial* – an artificial assembly of "man-made" building blocks with tailored properties. Living aside static properties and biological tissues, the most practical ways to "tailor" properties of the building blocks are via their geometrical shaping and compositional modulation. The former leads to confinement of solid state excitations (e.g. those listed above) and hence to formation of a discrete spectrum of the allowed modes. Brought sufficiently close together, the blocks can form a metamaterial with the discrete

illustration.

excitons etc.

atoms.

spectrum split into a band structure.

**2. What makes a metamaterial?** 

Some metamaterials (e.g. arrays of voids) are simpler to consider as created by geometrical modification of previously continuous materials. This however does not exclude their treatment as if they were constructed from building blocks considered above. Also, one has to note that, from the point of view of electromagnetic waves, vacuum is a continuous material in that important sense that it supports propagation of electromagnetic waves. Therefore, even arrays of entirely disconnected blocks are metamaterials for electromagnetic waves. This distinguishes electromagnetic waves from the other excitations.

Generally, various dynamical properties of metamaterials can be considered relative to excitations with wavelength either comparable to or much greater than the characteristic size of the building blocks. The former case is associated with studies of artificial "band gap crystals", e.g. photonic17, plasmonic18, phononic19, and magnonic20 crystals. The latter case allows one to study the metamaterials as some effectively continuous media. The physical object under study is nonetheless the same in both cases. For example, metamaterials with artificial periodic modulation of the refractive index with periodicity comparable to the wavelength of electromagnetic waves in the visible range are known as photonic band gap structures, using which it is possible to enable propagation of light in particular directions, to localize it in chosen channels or zones, or even to completely prohibit its propagation. In the microwave frequency range, the same structures would behave as effectively continuous materials.

In a particular frequency region of interest, the same metamaterials can behave differently with respect to excitations of different kinds, e.g. electromagnetic, sound or spin waves. Attractive opportunities arise from use of one excitation, with respect to which the sample behaves as a discrete structure (e. g. a band gap metamaterial), to design a resonance feature for another excitation, with respect to which the structure behaves as quasi-continuous metamaterial. For example, plasmonic resonances can be used to alter effective electromagnetic properties in THz to visible frequency range. Magnonic and LC resonances can be used for the same purpose in GHz-THz and GHz frequency ranges, respectively.

A thorough reader will probably find that, although term "metamaterial" emerged recently as a result of ever increasing pressure to spice up funding applications and publications in high profile journals, the underpinning research can be sometimes traced back to the middle or even beginning of the 20th century. Yet, one can also argue that it is the recent advances in nanofabrication and experimental tools that justify the boom by allowing researchers to revisit the established notions at reduced length scales and to apply results to modern technology.

#### **3. Spin waves and effectively continuous magnonic metamaterials**

The main reason for electrically neutral atoms to be able to interact with electromagnetic radiation is that they themselves possess electrical structure. Indeed, each atom consists of positively charged core and negatively charged electrons. The core and electrons are not rigidly coupled, and so, the atom can be electrically polarised by and thereby interact with an external electric field.

However, the electric charge is not the only property responsible for the interaction of atoms or materials with electromagnetic radiation. In particular, spins and associated magnetic

Magnonic Metamaterials 345

the attenuated total reflection technique has been successfully applied to studies of magnons in antiferromagnets63. However, this field of research is still at its infancy and is not

0.0 0.1 0.2 0.3 0.4

2

(a)

(b)

1

f (THz)

Fig. 1. (b) The effective permeability calculated for the structure shown in (a) is plotted as a function of the frequency for cases of in-plane (1) and out-of-plane (2) magnetizations. The solid and dashed lines denote the real and imaginary parts of the effective permeability, respectively. The structure under study represents an array of CoFe films with thickness of 5 nm. The filling factor of *ρ* = 0.25 is assumed. The spins are perfectly pinned at one side of

The VNA-FMR technique represents a relatively new twist in the FMR spectroscopy where VNA highlights the use of a broadband vector network analyser (VNA) operated in the GHz frequency regime. Microwaves applied to a waveguide locally excite spin waves that in turn induce a high-frequency voltage due to precessing magnetisation (Figure 2 (a)). The VNA-FMR technique measures spectra of both the amplitude and phase change of microwaves

reviewed here to any significant extent.


each film and are free at the other. (After Ref. 7)

moments of elementary particles are responsible for their interaction with an external magnetic field. Being always electrically neutral, some atoms still have a net magnetic moment and hence are called "magnetic". Generally, electromagnetic field interacts with charges more strongly than with spins, and so, the associated resonance frequencies are weaker in spin resonance experiments. However, via the quantum mechanical Pauli Exclusion Principle, the electron spin governs the order in which the electronic bands are populated by electrons, and thereby strongly affects the electromagnetic spectrum. As the other side of the same phenomenon, the Pauli Exclusion Principle and electrostatic interaction result in the exchange interaction between spins and are therefore responsible for the ordering of spins observed in e.g. ferromagnetic materials. The perturbations of the magnetic ordering are called spin waves – the central object of magnonics and magnonic metamaterials.

Spin waves exhibit most of the properties inherent to waves of other origins, including the excitation and propagation21,22,23, reflection and refraction24,25,26,27,28,29, interference and diffraction30,31,32,33,34,35, focusing and self-focusing36,37,38,39,40,41, tunnelling42,43, Doppler effect44,45,46, and formation of spin-wave envelope solitons47,48,49,50. Spin-wave quantization due to the finite size effect was discovered in thin films51,52 and more recently in laterally confined magnetic structures53,54,55,56,57,58, with their effect upon the high frequency permeability and the observation of a negative permeability discussed in Refs. 7 and 59, respectively.

The possibility of using spin waves for the design of high frequency permeability follows already from their first direct observation in cavity ferromagnetic resonance (FMR) experiments60. In the latter measurements, the precession of magnetisation60 is detected by measuring spectra of the absorption of microwaves in the cavity containing the magnetic sample under study. The spectra are determined by the density of states of spin waves that can resonantly couple to the microwave field. The very long wavelength of microwaves, as compared to the length scale of magnetic structures of interest, limits the application of the FMR technique to studies of magnonic modes with significant Fourier amplitude at nearly zero values of the wave vector52. However, this also mimics potential applications in which either the electromagnetic response of a magnonic device containing nano-structured functional magnetic elements is read out by the effectively uniform electromagnetic field, or the magnonic (meta-) material61 is supposed to absorb the incident electromagnetic radiation. Continuous magnetic materials and arrays of non-interacting magnetic elements appear preferred for such applications near the frequency of the uniform FMR. However, more sophisticated micromagnetic engineering is required to push up the frequency of operation of such materials e.g. using the exchange field62, which originates from the strongest of the magnetic interactions, rather than the uniform anisotropy or applied magnetic field. An example of using the concept can be found e.g. in Ref. 7 and is also illustrated in Figure 1.

The FMR is conventionally used to study magnetisation dynamics at frequencies up to about 100 GHz. At higher frequencies, the mismatch between the linear momentum of free space electromagnetic radiation (photon) and that of a magnon increasingly prohibits an efficient coupling. Therefore, higher frequencies require one to use different experimental and technical concepts by which to interrogate and measure e.g. THz magnons. Here, methods, known e.g. from plasmonics, might help to couple light to magnons. For example,

moments of elementary particles are responsible for their interaction with an external magnetic field. Being always electrically neutral, some atoms still have a net magnetic moment and hence are called "magnetic". Generally, electromagnetic field interacts with charges more strongly than with spins, and so, the associated resonance frequencies are weaker in spin resonance experiments. However, via the quantum mechanical Pauli Exclusion Principle, the electron spin governs the order in which the electronic bands are populated by electrons, and thereby strongly affects the electromagnetic spectrum. As the other side of the same phenomenon, the Pauli Exclusion Principle and electrostatic interaction result in the exchange interaction between spins and are therefore responsible for the ordering of spins observed in e.g. ferromagnetic materials. The perturbations of the magnetic ordering are called spin waves – the central object of magnonics and magnonic

Spin waves exhibit most of the properties inherent to waves of other origins, including the excitation and propagation21,22,23, reflection and refraction24,25,26,27,28,29, interference and diffraction30,31,32,33,34,35, focusing and self-focusing36,37,38,39,40,41, tunnelling42,43, Doppler effect44,45,46, and formation of spin-wave envelope solitons47,48,49,50. Spin-wave quantization due to the finite size effect was discovered in thin films51,52 and more recently in laterally confined magnetic structures53,54,55,56,57,58, with their effect upon the high frequency permeability and the observation of a negative permeability discussed in Refs. 7 and 59,

The possibility of using spin waves for the design of high frequency permeability follows already from their first direct observation in cavity ferromagnetic resonance (FMR) experiments60. In the latter measurements, the precession of magnetisation60 is detected by measuring spectra of the absorption of microwaves in the cavity containing the magnetic sample under study. The spectra are determined by the density of states of spin waves that can resonantly couple to the microwave field. The very long wavelength of microwaves, as compared to the length scale of magnetic structures of interest, limits the application of the FMR technique to studies of magnonic modes with significant Fourier amplitude at nearly zero values of the wave vector52. However, this also mimics potential applications in which either the electromagnetic response of a magnonic device containing nano-structured functional magnetic elements is read out by the effectively uniform electromagnetic field, or the magnonic (meta-) material61 is supposed to absorb the incident electromagnetic radiation. Continuous magnetic materials and arrays of non-interacting magnetic elements appear preferred for such applications near the frequency of the uniform FMR. However, more sophisticated micromagnetic engineering is required to push up the frequency of operation of such materials e.g. using the exchange field62, which originates from the strongest of the magnetic interactions, rather than the uniform anisotropy or applied magnetic field. An example of using the concept can be found e.g. in Ref. 7 and is also

The FMR is conventionally used to study magnetisation dynamics at frequencies up to about 100 GHz. At higher frequencies, the mismatch between the linear momentum of free space electromagnetic radiation (photon) and that of a magnon increasingly prohibits an efficient coupling. Therefore, higher frequencies require one to use different experimental and technical concepts by which to interrogate and measure e.g. THz magnons. Here, methods, known e.g. from plasmonics, might help to couple light to magnons. For example,

metamaterials.

respectively.

illustrated in Figure 1.

the attenuated total reflection technique has been successfully applied to studies of magnons in antiferromagnets63. However, this field of research is still at its infancy and is not reviewed here to any significant extent.

Fig. 1. (b) The effective permeability calculated for the structure shown in (a) is plotted as a function of the frequency for cases of in-plane (1) and out-of-plane (2) magnetizations. The solid and dashed lines denote the real and imaginary parts of the effective permeability, respectively. The structure under study represents an array of CoFe films with thickness of 5 nm. The filling factor of *ρ* = 0.25 is assumed. The spins are perfectly pinned at one side of each film and are free at the other. (After Ref. 7)

The VNA-FMR technique represents a relatively new twist in the FMR spectroscopy where VNA highlights the use of a broadband vector network analyser (VNA) operated in the GHz frequency regime. Microwaves applied to a waveguide locally excite spin waves that in turn induce a high-frequency voltage due to precessing magnetisation (Figure 2 (a)). The VNA-FMR technique measures spectra of both the amplitude and phase change of microwaves

Magnonic Metamaterials 347

pulses and controls their arrival time relative to the pump. By changing the optical path of the probe pulse one can trace the time evolution of the excited dynamics. By scanning the position of the optical probe on the surface of the sample, one acquires images of the dynamic magnetisation with a spatial resolution of down to 250 nm in real space67-70, and is suitable for studying both continuous and nanostructured samples, as demonstrated in Figure 3. The temporal resolution of TRSKM can be well on the sub-ps time scale, therefore offering the detection of spin waves in the THz frequency regime. The TRSKM performs a 3D vectorial analysis of the time dependent magnetization71 and is therefore phase sensitive. Alternatively, one can combine the magneto-optical detection with a VNA-FMR setup to image spin wave modes in the frequency rather than time domain72.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Fig. 3. The fast Fourier transform (FFT) power spectrum calculated from a time resolved Kerr signal acquired from the center of a 4 x 4 μm2 array of 40 x 80 nm2 stadium shaped ferromagnetic elements at a bias magnetic field of 197 Oe is shown on a logarithmic scale together with the fit to a Lorentzian 3-peak function. The inset shows images of the modes confined within the entire array and corresponding to the peak frequencies identified from the fit. The darker shades of gray correspond to greater mode amplitude. With respect to the

long wavelength spin wave modes, the array acts as a continuous element made of a magnonic metamaterial. Such arrays will also act as metamaterials with respect to

Frequency (GHz)

3.0 GHz 3.6 GHz 4.2 GHz

1E-7

microwaves. After Ref. 61.

1E-6

Data

Fit with 3 Lorentzian peaks

**H**bias

 Peak 1 (3.0 GHz) Peak 2 (3.6 GHz) Peak 3 (4.2 GHz)

1E-5

1E-4

FFT power (arb. units)

1E-3

0.01

passing through a magnetic sample integrated with the waveguide64,65,66. The geometrical parameters of the waveguide determine the spatial distribution of the rf magnetic field and therefore the wavelength spectrum addressed by the microwave field. Hence, the VNA-FMR can be also referred to as a "near field" FMR. Due to the large penetration depth of microwaves, both thin film and bulk samples can be successfully investigated using this technique.

Fig. 2. (a) Sketch of a coplanar waveguide (CPW) integrated to an antidot lattice prepared from a thin Permalloy film. The CPW consists of three metallic leads (ground-signal-ground leads). Adjusting the dimensions of the CPW allows one to vary the profile of the magnetic field generated by microwave current *i*rf supplied by the VNA. This defines the wave vector transferred to the sample. The same CPW picks up the voltage induced by precessing spins. The CPW is isolated from the ferromagnet by an insulating layer (not shown). (b) Simulated spatial distributions of spin precession amplitudes reflecting two different standing spinwave excitations. The film is assumed to be 26 nm thick. The period (hole diameter) is 490 (240) nm. A magnetic field of a few 10 mT is applied in horizontal direction. The mode pattern shown on the right belongs to a localized mode that has frequency higher than that of the extended mode shown on the left. (After Ref. 91) Bright colours correspond to large amplitudes. The holes are shown in white.

An experimental insight into the structure of magnonic modes in nanostructured magnonic metamaterials, e.g. such those shown in Figure 2, can be achieved with state of art dynamic magnetic imaging techniques, e.g. the time-resolved scanning Kerr microscopy (TRSKM)61. In a TRSKM experiment, the sample is pumped so as to excite spin waves, with the pump stimulus being both repetitive and coherent, i.e. having a well-defined phase with respect to the probe beam. To probe, one uses ultrashort optical

passing through a magnetic sample integrated with the waveguide64,65,66. The geometrical parameters of the waveguide determine the spatial distribution of the rf magnetic field and therefore the wavelength spectrum addressed by the microwave field. Hence, the VNA-FMR can be also referred to as a "near field" FMR. Due to the large penetration depth of microwaves, both thin film and bulk samples can be successfully investigated using this

Fig. 2. (a) Sketch of a coplanar waveguide (CPW) integrated to an antidot lattice prepared from a thin Permalloy film. The CPW consists of three metallic leads (ground-signal-ground leads). Adjusting the dimensions of the CPW allows one to vary the profile of the magnetic field generated by microwave current *i*rf supplied by the VNA. This defines the wave vector transferred to the sample. The same CPW picks up the voltage induced by precessing spins. The CPW is isolated from the ferromagnet by an insulating layer (not shown). (b) Simulated spatial distributions of spin precession amplitudes reflecting two different standing spinwave excitations. The film is assumed to be 26 nm thick. The period (hole diameter) is 490 (240) nm. A magnetic field of a few 10 mT is applied in horizontal direction. The mode pattern shown on the right belongs to a localized mode that has frequency higher than that of the extended mode shown on the left. (After Ref. 91) Bright colours correspond to large

An experimental insight into the structure of magnonic modes in nanostructured magnonic metamaterials, e.g. such those shown in Figure 2, can be achieved with state of art dynamic magnetic imaging techniques, e.g. the time-resolved scanning Kerr microscopy (TRSKM)61. In a TRSKM experiment, the sample is pumped so as to excite spin waves, with the pump stimulus being both repetitive and coherent, i.e. having a well-defined phase with respect to the probe beam. To probe, one uses ultrashort optical

technique.

(a)

(b)

amplitudes. The holes are shown in white.

pulses and controls their arrival time relative to the pump. By changing the optical path of the probe pulse one can trace the time evolution of the excited dynamics. By scanning the position of the optical probe on the surface of the sample, one acquires images of the dynamic magnetisation with a spatial resolution of down to 250 nm in real space67-70, and is suitable for studying both continuous and nanostructured samples, as demonstrated in Figure 3. The temporal resolution of TRSKM can be well on the sub-ps time scale, therefore offering the detection of spin waves in the THz frequency regime. The TRSKM performs a 3D vectorial analysis of the time dependent magnetization71 and is therefore phase sensitive. Alternatively, one can combine the magneto-optical detection with a VNA-FMR setup to image spin wave modes in the frequency rather than time domain72.

Fig. 3. The fast Fourier transform (FFT) power spectrum calculated from a time resolved Kerr signal acquired from the center of a 4 x 4 μm2 array of 40 x 80 nm2 stadium shaped ferromagnetic elements at a bias magnetic field of 197 Oe is shown on a logarithmic scale together with the fit to a Lorentzian 3-peak function. The inset shows images of the modes confined within the entire array and corresponding to the peak frequencies identified from the fit. The darker shades of gray correspond to greater mode amplitude. With respect to the long wavelength spin wave modes, the array acts as a continuous element made of a magnonic metamaterial. Such arrays will also act as metamaterials with respect to microwaves. After Ref. 61.

Magnonic Metamaterials 349

(*n*DE) modes characterized by nodal planes parallel to the applied magnetic field **H**, Backward-like (*n*BA) modes with nodal planes perpendicular to **H** and *n*End-Modes (*n*EM)

Figure 4 (a) shows a scanning electron microscopy (SEM) image of the studied sample together with a reference frame and the directions of *q* and **H**. In Figure 4 (b) black lines denote frequencies of magnonic modes for *q* = 0, while the frequency curves corresponding

**H**

Fig. 4. (a) SEM image of the sample: Permalloy rectangular dots have lateral dimensions 715 × 450 nm2 and interdot separation Δ = 55 nm. A reference frame with the direction of **H** along the *y-*axis (easy axis) and of the wave vector *q* is also shown. (b) Calculated frequency behaviour vs. interdot separation for the sample of dots 715 x 450 nm2 in the Voigt geometry for an applied magnetic field of magnitude *H* = 1 kOe. Full black lines: frequencies at *q* = 0.

As expected, for large interdot separations, each mode is characterised by a single frequency value and frequency is independent of *q*. On decreasing the separation, interdot coupling gives rise to the appearance of bands. Due to the effect of stray magnetic field within each band the frequency of the collective modes depends on *q.* The largest band width is that of the *F* mode that has the largest stray field at any separation. Another significant feature is the

and the 2DE mode is smaller for *q* = 0. As a matter of fact, for small separation the energetic

almost the same as that of the 1-DE mode. The difference is represented by the band gap between the two modes at the 1BZ boundary. As an example the band width for a small interdot separation Δ = 10 nm of the most representative modes shown in panel (b) was estimated. Calculated band width of the *F* mode turned out to be about 3.8 GHz, the largest one among those of most representative modes, but note that also the other collective modes of the spectrum have an important calculated band width in this limit (larger than 0.8 GHz).

π

π/ *a*.

narrowing of the band gaps, as Δ→ 0, either at *q* = 0 or at *q*BZ =

gap between the *F* and the 1DE mode is smaller for *q*BZ =

cost required to excite the F mode at *q*BZ =

*q*

π

/ *a* with *a* the periodicity are indicated

2DE

1DE

*F*

1BA

0 300 600

Separation Δ (nm)

/ *a*. In particular, the band

/ *a*, while that between the 1DE

8

π

/ *a*, namely in the anti-phase configuration, is

π

10

Frequency (GHz)

*x*

12

14

(b)

localized at the edges of each dot with nodal planes of the DE type.

to the edge of the first Brillouin zone (1BZ) at *q*BZ =

*<sup>y</sup>* (a)

Dashed red lines: frequencies at *q* =

by dashed red lines83.

#### **4. Band gap magnonic metamaterials**

Periodically modulated magnetic materials have been shown to form magnonic crystals, i.e., a magnetic analogue of photonic crystals. Indeed, the spin wave spectrum is modified by patterning73 and shows a tailored band structure in periodic magnetic materials74. The band spectrum consists of bands of allowed magnonic states and forbidden-frequency gaps ("band gaps"), in which there are no allowed magnonic states. One of the first attempts to study the propagation of spin waves in periodic magnetic structures was made by Elachi75. Nowadays, the number of studies on this topic has surged and continues to grow at a fast pace.

The recent advances in the studies of the band gap magnonic metamaterials are associated with advances in the Brillouin light scattering (BLS) technique, which has proved to be a very powerful tool for the investigation of magnetization dynamics in magnonic structures76. Thanks to the wave vector conservation in the magnon–photon interaction, one has the possibility to measure the dispersion relation (frequency versus wave vector) of spin waves. In particular, the BLS technique is suitable for measuring the magnonic band gap dispersion, provided that the periodicity of the magnonic crystal is such that the Brillouin zone (BZ) boundary lies in the accessible wave vector range (up to 2.2x105 rad/cm). The magnonic dispersion can be measured in different scattering geometries that differ by the relative orientation of the exchanged wave vector and direction of the applied magnetic field. So planar 1D magnonic crystals formed by arrays of closely spaced Permalloy stripes of identical77 or alternating width78 were studied, showing the existence of tuneable band gaps. Furthermore, alternating stripes of two different magnetic materials were studied in Ref. 79. In addition, it has recently been shown that, using a large aperture microscope objective, BLS can be used as a scanning probe technique, therefore permitting the map-out of the spatial distribution of magnonic normal modes with a lateral resolution of down to a few hundred nanometres80,81.

The studied sample consist of long chains of Permalloy rectangular dots with rounded corners and lateral dimensions of 715 x 450 nm2, thickness of 40 nm, edge-to-edge separation Δ = 55 nm. The magnetic material was deposited on Silicon substrate at room temperature. In our calculations, each dot was divided into cells with size Δ*<sup>x</sup>* × Δ*y* × Δ*z* = 5 nm × 5 nm × 40 nm. The ground state was obtained using a micromagnetic code. The magnetization was assumed to be uniform in each cell and to precess around its equilibrium direction along effective field **H**eff. Contributions arising from the external (Zeeman), demagnetising and exchange fields were included in **H**eff. The following magnetic parameters obtained from the fit to the BLS frequencies of the Damon-Eshbach mode in the reference 40 nm thick continuous Permalloy film were used: saturation magnetization 4π *M*s = 9 kG, γ/2π = 2.94 GHz/kOe, and *A* = 1.1 10-6 erg/cm with *A* being the exchange stiffness constant.

First, the collective dynamics was studied in the Voigt geometry, namely with the wave vector *q* perpendicular to the applied magnetic field **H** and with *q* = *K* , where *K* is the Bloch wave vector.

The collective modes of the array of dots were studied by using the Dynamical Matrix Method (DMM) extended to periodic magnetic systems82. On the basis of the number and the direction of nodal planes *n* = 0,1,2,.. of the dynamic magnetization inside each dot, we have classified collective modes into: the *F* mode with no nodal planes, Damon-Eshbach-like

Periodically modulated magnetic materials have been shown to form magnonic crystals, i.e., a magnetic analogue of photonic crystals. Indeed, the spin wave spectrum is modified by patterning73 and shows a tailored band structure in periodic magnetic materials74. The band spectrum consists of bands of allowed magnonic states and forbidden-frequency gaps ("band gaps"), in which there are no allowed magnonic states. One of the first attempts to study the propagation of spin waves in periodic magnetic structures was made by Elachi75. Nowadays, the number of studies on this topic has surged and continues to grow at a fast

The recent advances in the studies of the band gap magnonic metamaterials are associated with advances in the Brillouin light scattering (BLS) technique, which has proved to be a very powerful tool for the investigation of magnetization dynamics in magnonic structures76. Thanks to the wave vector conservation in the magnon–photon interaction, one has the possibility to measure the dispersion relation (frequency versus wave vector) of spin waves. In particular, the BLS technique is suitable for measuring the magnonic band gap dispersion, provided that the periodicity of the magnonic crystal is such that the Brillouin zone (BZ) boundary lies in the accessible wave vector range (up to 2.2x105 rad/cm). The magnonic dispersion can be measured in different scattering geometries that differ by the relative orientation of the exchanged wave vector and direction of the applied magnetic field. So planar 1D magnonic crystals formed by arrays of closely spaced Permalloy stripes of identical77 or alternating width78 were studied, showing the existence of tuneable band gaps. Furthermore, alternating stripes of two different magnetic materials were studied in Ref. 79. In addition, it has recently been shown that, using a large aperture microscope objective, BLS can be used as a scanning probe technique, therefore permitting the map-out of the spatial distribution of magnonic normal modes with a lateral resolution of down to a

The studied sample consist of long chains of Permalloy rectangular dots with rounded corners and lateral dimensions of 715 x 450 nm2, thickness of 40 nm, edge-to-edge separation Δ = 55 nm. The magnetic material was deposited on Silicon substrate at room temperature. In our calculations, each dot was divided into cells with size Δ*<sup>x</sup>* × Δ*y* × Δ*z* = 5 nm × 5 nm × 40 nm. The ground state was obtained using a micromagnetic code. The magnetization was assumed to be uniform in each cell and to precess around its equilibrium direction along effective field **H**eff. Contributions arising from the external (Zeeman), demagnetising and exchange fields were included in **H**eff. The following magnetic parameters obtained from the fit to the BLS frequencies of the Damon-Eshbach mode in the reference 40 nm thick

First, the collective dynamics was studied in the Voigt geometry, namely with the wave vector *q* perpendicular to the applied magnetic field **H** and with *q* = *K* , where *K* is the Bloch

The collective modes of the array of dots were studied by using the Dynamical Matrix Method (DMM) extended to periodic magnetic systems82. On the basis of the number and the direction of nodal planes *n* = 0,1,2,.. of the dynamic magnetization inside each dot, we have classified collective modes into: the *F* mode with no nodal planes, Damon-Eshbach-like

π

 *M*s = 9 kG, γ/2

π= 2.94

continuous Permalloy film were used: saturation magnetization 4

GHz/kOe, and *A* = 1.1 10-6 erg/cm with *A* being the exchange stiffness constant.

**4. Band gap magnonic metamaterials** 

pace.

few hundred nanometres80,81.

wave vector.

(*n*DE) modes characterized by nodal planes parallel to the applied magnetic field **H**, Backward-like (*n*BA) modes with nodal planes perpendicular to **H** and *n*End-Modes (*n*EM) localized at the edges of each dot with nodal planes of the DE type.

Figure 4 (a) shows a scanning electron microscopy (SEM) image of the studied sample together with a reference frame and the directions of *q* and **H**. In Figure 4 (b) black lines denote frequencies of magnonic modes for *q* = 0, while the frequency curves corresponding to the edge of the first Brillouin zone (1BZ) at *q*BZ = π / *a* with *a* the periodicity are indicated by dashed red lines83.

Fig. 4. (a) SEM image of the sample: Permalloy rectangular dots have lateral dimensions 715 × 450 nm2 and interdot separation Δ = 55 nm. A reference frame with the direction of **H** along the *y-*axis (easy axis) and of the wave vector *q* is also shown. (b) Calculated frequency behaviour vs. interdot separation for the sample of dots 715 x 450 nm2 in the Voigt geometry for an applied magnetic field of magnitude *H* = 1 kOe. Full black lines: frequencies at *q* = 0. Dashed red lines: frequencies at *q* =π/ *a*.

As expected, for large interdot separations, each mode is characterised by a single frequency value and frequency is independent of *q*. On decreasing the separation, interdot coupling gives rise to the appearance of bands. Due to the effect of stray magnetic field within each band the frequency of the collective modes depends on *q.* The largest band width is that of the *F* mode that has the largest stray field at any separation. Another significant feature is the narrowing of the band gaps, as Δ→ 0, either at *q* = 0 or at *q*BZ = π / *a*. In particular, the band gap between the *F* and the 1DE mode is smaller for *q*BZ = π / *a*, while that between the 1DE and the 2DE mode is smaller for *q* = 0. As a matter of fact, for small separation the energetic cost required to excite the F mode at *q*BZ = π / *a*, namely in the anti-phase configuration, is almost the same as that of the 1-DE mode. The difference is represented by the band gap between the two modes at the 1BZ boundary. As an example the band width for a small interdot separation Δ = 10 nm of the most representative modes shown in panel (b) was estimated. Calculated band width of the *F* mode turned out to be about 3.8 GHz, the largest one among those of most representative modes, but note that also the other collective modes of the spectrum have an important calculated band width in this limit (larger than 0.8 GHz).

Magnonic Metamaterials 351

depth of modulation of the exchange parameter has a drastic effect upon the position and width of the band gaps. Collective dynamics of lattices of magnetic vortices was studied in Refs. 87,88. FMR and time resolved scanning Kerr microscopy (TRSKM) were used to study localisation of spin waves in an array of antidots formed in a metallic ferromagnetic film in Ref. 89. VNA-FMR measurements and micromagnetic simulations were used to demonstrate a control of spin wave transmission through a similar array of antidots by an

The BLS technique has been exploited to achieve a complete mapping of the spin-wave dispersion curves, along the principal symmetry directions of the first BZ, for a 2D magnonic crystal consisting of a square array of 50 nm thick NiFe disks (Figure 6 (a)). The disks have a diameter d = 600 nm and are arranged in a square matrix with the edge-to-edge interdot separation of 55 nm (period *a* = 655 nm). This corresponds to a square first BZ of side 2π/*a* = 2x4.8·104 rad/cm. The spin-wave frequency dispersion was studied along the principal directions of the first BZ, i.e. ΓX, ΓY, XM and YM, as shown in Figure 6 (b), for

In Figure 6 (c), the dispersion curves of the most representative modes are shown along the symmetry directions of the first BZ, showing a very good agreement between experimental points and calculated curves. Since the magnetic modes maintain a symmetry character similar to those found for the isolated dot, they can be labelled in the same way. Depending on the number *m* (*n*) of nodal lines perpendicular (parallel) to the direction of the magnetization (*x* direction), the modes are named as backward-volume-like modes *m*-BA (Damon-Eshbach-like *n*-DE), while modes with mixed character are denoted as *m*-BA×*n*-DE. The mode without nodal lines is classified as the fundamental mode (F), while modes with dynamic magnetization localized at the ends of the particle92 are labelled as *n*-EM, depending on the number of nodes *n*. The mode type, i.e. the two indices *m* and *n*, together

Figure 6 (c) shows that several modes exhibit an appreciable dispersion and are therefore propagating modes, while other modes show a frequency that remains almost constant with the wave vector, at least within the frequency resolution of the experiment. The width of each magnonic band (allowed miniband) is proportional to the mean square dynamic magnetization inside a single dot as found by DMM approach. Hence, the dispersion is largest for the F mode, while it decreases rapidly for the higher order modes, which are characterised by an increasing number of oscillations within the single dot. Starting from the Γ point, the measured frequency of the F mode increases (decreases) along the ΓY (ΓX) direction, reaching its maximum (minimum) at the Y (X) point. This behaviour reflects the properties of the dipole-exchange spin waves in the reference continuous film (open squares in Figure 6 (c)). However, the F mode is significantly downshifted with respect to the spin waves in the continuous film, which is due to the static demagnetizing field in the array along the applied field direction (*x*-direction). As a consequence of this in-plane anisotropy induced by the applied field, one can see that the frequency of the DE mode in the continuous film coincides near the Y-point with that of the 1-DE mode of the array. At the same time, the backward volume spin wave in the continuous film has a frequency that is in the middle of the band gap of the array, i.e. quite far from that of the corresponding 1-BA mode at X-point. In the latter point, an overlap between the F and 1-BA bands is observed.

external field H = 1.0 kOe applied along the [10] direction of the disk array.

with the Bloch wave vector *K*, uniquely identifies the excitation.

external magnetic field in Refs. 90,91 (Figure 2).

Finally, the investigation was completed by assuming the Bloch wave vector parallel to the applied magnetic field84. This geometry corresponds to the so called backward volume magnetostatic spin-wave (BWVMS) geometry. In Figure 5, the dependence of the spin-wave frequencies on Δ calculated at *q =* 0 is compared to the case of *q* = π / *a* , i.e. at the edge of the 1BZ. As it can be seen, for separation values of about 300 nm, the frequencies at the centre and at the edge of the 1BZ are almost degenerate and both tend to the value of the mode frequency of an isolated dot. On reducing the value of Δ, however, the dynamic dipolar magnetic coupling becomes strong enough to remove the degeneracy. This leads to the appearance of magnonics bands whose widths increase with decreasing the interdot distance. Also in this geometry there is a narrowing of band gaps, as Δ → 0, between given couples of adjacent modes, either at *q* = 0 or at *q*BZ = π/ *a*.

Fig. 5. Calculated mode frequencies behavior vs. interdot separation in the BWVMS geometry, i.e. for *q* parallel to **H**. Black lines: frequencies for *q* = 0. Dashed (red) lines: frequencies for *q*BZ = π / *a*. A magnetic field of intensity *H* = 1.5 kOe was applied along the *x*-axis (hard axis) for the sample shown in Fig. 4 (a). The wave vector *q* was along the chains. The direction of *q* and **H** is also shown.

In particular, among the modes shown, there is a narrowing at *q*BZ = π / *a* between the 1BA and the *F* mode. Interestingly, for Δ < 100 nm, the frequency increment of the 0EM is much more accentuated than that of collective modes in the higher part of the spectrum. This behaviour (which also concerns the others *n*-EM, not shown here) is related to the fact that the *n*EM are localized at some portions (typically one corner) of the adjacent edges of neighbouring dots.

2D magnonic crystals were proposed in Refs. 85,86, where the spectrum of dipole-exchange spin waves propagating in the plane of a magnonic crystal was discussed. The magnonic crystal consisted of periodically arranged infinitely long ferromagnetic cylinders embedded in a matrix of a different ferromagnetic material. The position and width of band gaps in the magnonic spectrum were investigated as a function of the period of the structure and the depth of modulation depth ("contrast") of the magnetic parameters. It was found that the

Finally, the investigation was completed by assuming the Bloch wave vector parallel to the applied magnetic field84. This geometry corresponds to the so called backward volume magnetostatic spin-wave (BWVMS) geometry. In Figure 5, the dependence of the spin-wave

1BZ. As it can be seen, for separation values of about 300 nm, the frequencies at the centre and at the edge of the 1BZ are almost degenerate and both tend to the value of the mode frequency of an isolated dot. On reducing the value of Δ, however, the dynamic dipolar magnetic coupling becomes strong enough to remove the degeneracy. This leads to the appearance of magnonics bands whose widths increase with decreasing the interdot distance. Also in this geometry there is a narrowing of band gaps, as Δ → 0, between given

0 100 200 300

Separation Δ (nm)

Fig. 5. Calculated mode frequencies behavior vs. interdot separation in the BWVMS geometry, i.e. for *q* parallel to **H**. Black lines: frequencies for *q* = 0. Dashed (red) lines:

In particular, among the modes shown, there is a narrowing at *q*BZ =

*x*-axis (hard axis) for the sample shown in Fig. 4 (a). The wave vector *q* was along the chains.

and the *F* mode. Interestingly, for Δ < 100 nm, the frequency increment of the 0EM is much more accentuated than that of collective modes in the higher part of the spectrum. This behaviour (which also concerns the others *n*-EM, not shown here) is related to the fact that the *n*EM are localized at some portions (typically one corner) of the adjacent edges of

2D magnonic crystals were proposed in Refs. 85,86, where the spectrum of dipole-exchange spin waves propagating in the plane of a magnonic crystal was discussed. The magnonic crystal consisted of periodically arranged infinitely long ferromagnetic cylinders embedded in a matrix of a different ferromagnetic material. The position and width of band gaps in the magnonic spectrum were investigated as a function of the period of the structure and the depth of modulation depth ("contrast") of the magnetic parameters. It was found that the

π/ *a*.

> **1DE** *F*

**1BA**

**0EM**

/ *a*. A magnetic field of intensity *H* = 1.5 kOe was applied along the

π

/ *a* , i.e. at the edge of the

*q*

π

/ *a* between the 1BA

frequencies on Δ calculated at *q =* 0 is compared to the case of *q* =

couples of adjacent modes, either at *q* = 0 or at *q*BZ =

2

π

The direction of *q* and **H** is also shown.

4

6

8

Frequency (GHz)

frequencies for *q*BZ =

neighbouring dots.

10

12

14

depth of modulation of the exchange parameter has a drastic effect upon the position and width of the band gaps. Collective dynamics of lattices of magnetic vortices was studied in Refs. 87,88. FMR and time resolved scanning Kerr microscopy (TRSKM) were used to study localisation of spin waves in an array of antidots formed in a metallic ferromagnetic film in Ref. 89. VNA-FMR measurements and micromagnetic simulations were used to demonstrate a control of spin wave transmission through a similar array of antidots by an external magnetic field in Refs. 90,91 (Figure 2).

The BLS technique has been exploited to achieve a complete mapping of the spin-wave dispersion curves, along the principal symmetry directions of the first BZ, for a 2D magnonic crystal consisting of a square array of 50 nm thick NiFe disks (Figure 6 (a)). The disks have a diameter d = 600 nm and are arranged in a square matrix with the edge-to-edge interdot separation of 55 nm (period *a* = 655 nm). This corresponds to a square first BZ of side 2π/*a* = 2x4.8·104 rad/cm. The spin-wave frequency dispersion was studied along the principal directions of the first BZ, i.e. ΓX, ΓY, XM and YM, as shown in Figure 6 (b), for external field H = 1.0 kOe applied along the [10] direction of the disk array.

In Figure 6 (c), the dispersion curves of the most representative modes are shown along the symmetry directions of the first BZ, showing a very good agreement between experimental points and calculated curves. Since the magnetic modes maintain a symmetry character similar to those found for the isolated dot, they can be labelled in the same way. Depending on the number *m* (*n*) of nodal lines perpendicular (parallel) to the direction of the magnetization (*x* direction), the modes are named as backward-volume-like modes *m*-BA (Damon-Eshbach-like *n*-DE), while modes with mixed character are denoted as *m*-BA×*n*-DE. The mode without nodal lines is classified as the fundamental mode (F), while modes with dynamic magnetization localized at the ends of the particle92 are labelled as *n*-EM, depending on the number of nodes *n*. The mode type, i.e. the two indices *m* and *n*, together with the Bloch wave vector *K*, uniquely identifies the excitation.

Figure 6 (c) shows that several modes exhibit an appreciable dispersion and are therefore propagating modes, while other modes show a frequency that remains almost constant with the wave vector, at least within the frequency resolution of the experiment. The width of each magnonic band (allowed miniband) is proportional to the mean square dynamic magnetization inside a single dot as found by DMM approach. Hence, the dispersion is largest for the F mode, while it decreases rapidly for the higher order modes, which are characterised by an increasing number of oscillations within the single dot. Starting from the Γ point, the measured frequency of the F mode increases (decreases) along the ΓY (ΓX) direction, reaching its maximum (minimum) at the Y (X) point. This behaviour reflects the properties of the dipole-exchange spin waves in the reference continuous film (open squares in Figure 6 (c)). However, the F mode is significantly downshifted with respect to the spin waves in the continuous film, which is due to the static demagnetizing field in the array along the applied field direction (*x*-direction). As a consequence of this in-plane anisotropy induced by the applied field, one can see that the frequency of the DE mode in the continuous film coincides near the Y-point with that of the 1-DE mode of the array. At the same time, the backward volume spin wave in the continuous film has a frequency that is in the middle of the band gap of the array, i.e. quite far from that of the corresponding 1-BA mode at X-point. In the latter point, an overlap between the F and 1-BA bands is observed.

Magnonic Metamaterials 353

of *k*eff helps to understand the frequency dispersion of the magnonic crystal, because, in the limit of a continuous medium, *k*eff becomes the real wave vector of the spin excitation of the continuous film, whose dispersion curves have the following properties. The mode frequency increases (decreases) when the modulus of the wave vector increases in a direction perpendicular (parallel) to the applied field, corresponding to the MSSW

To realize band gap magnonic metamaterials with a band structure of higher in-plane symmetry arrays of circular nanomagnets have been considered where an out-of-plane magnetic field stabilizes the so-called vortex state in each of the unit cells of the periodic nanodisk lattice. The dipolar interaction via nanoscale air gaps leads to allowed minibands and forbidden frequency gaps which exhibit a four-fold symmetry in in-plane directions if nanodisks are arranged in a square lattice 94. This higher symmetry goes beyond the magnonic crystals where an in-plane field governs the symmetry of the band structure.

The knowledge of the physical mechanisms which govern the dynamical behaviour of nanoscale magnetic elements is of fundamental importance for understanding the general properties of metamaterials. Because of that, the theoretical derivations of the frequency and the spatial profile of magnonic modes become necessary to gain a physical understanding of the processes observed at a macroscopic level95-97. Several analytical and numerical methods are used to derive the profile of normal modes. Analytical models require some preliminary assumptions (approximations) regarding the mode profiles98,99. Numerical tools based on micromagnetic simulations has been developed for solving the equation of motion in the time domain and to successively perform a Fourier analysis of that output signal100,101. The same equation can also be solved in the frequency domain. The Dynamical Matrix Method (DMM)102 belongs to the latter approach, in which the sample is subdivided into cells and the linearized Landau–Lifshitz equation of motion is recast as a generalized eigenvalue problem, which is numerically solved by means of a finite-element method. Band structures of periodic composites can then be calculated with the help of the Bloch theorem, which reduces the number of independent variables of a periodic system to that of the

The band structures of spin-waves in materials with discrete translational symmetry can be calculated also by the plane wave method (PWM). The PWM is a popular tool commonly used for studying electronic, photonic and phononic crystals because of its conceptual simplicity and applicability to any type of lattice and shape of scattering centers. The method was also adapted to the calculations of the magnonic band structures and is constantly improved, with its field of application extending to new problems85,103. Recently, the PWM has also been used for the calculation of spin-wave spectra of 1D magnonic crystals of finite thickness104 and 2D thin-film magnonic crystals105. Only very recently, the PWM has been employed for the first time for calculating spin-wave spectra of 2D antidot arrays based on a

The 2D magnonic crystals composed of two ferromagnetic materials in thin film geometry were studied theoretically in Refs.105,107. The plane wave method with supercell formulation was used to study the edge effect in magnonic crystals in Ref. 105. It was shown that localisation at the edges of corners of the 2D magnonic crystals with finite lateral

square lattice with a good agreement with experimental results obtained106.

**5. Theory of band gap magnonic metamaterials** 

(MSBVW) geometry.

corresponding unit cell82.

Fig. 6. (a) SEM image of the array of Permalloy disks is shown. BLS spectra were measured applying the external field H along the [10] array direction (x-axis) and changing the qx and qy components of the in-plane transferred wave-vector **q**. (b) Surface BZ of the 2D periodic array is shown. The behaviour of the dynamic magnetization of the fundamental mode is schematically shown in four point of the BZ for a 3x3 sub-matrix of dots. The two different colours represent out-of-plane dynamic magnetization of opposite sign. (c) Measured frequencies (dots) are shown as a function of the spin-wave wave vector along the principal direction of the first BZ, for an external magnetic field H = 1.0 kOe. The calculated dispersion curves of the most significant modes are also reported. Bold, solid or dotted lines refer to modes whose calculated cross section is comparable, smaller than 1/10 or smaller than 1/100 with respect to that of the F mode, respectively. For the sake of comparison, the experimental dispersion of the DE mode of the unpatterned film is also reported as open squares.

In Ref. 93 the dispersion of different modes was interpreted in terms of effective wave vector *k*eff introduced as an auxiliary variable that includes and replaces the band index, i.e. the mode type, and the Bloch wave vector. It can be defined in the extended zone scheme as:

$$\mathbf{k}^{\rm eff} = \mathbf{K} + P(m) \left[ m + \frac{1 - P(m)}{2} \right] \frac{\pi}{a} \hat{\mathbf{x}} + P(n) \left[ n + \frac{1 - P(n)}{2} \right] \frac{\pi}{a} \hat{\mathbf{y}} \tag{1}$$

where P(*i*) is the parity function (=+1,-1 for *i* even or odd, respectively), *m*-BA×*n*-DE is the mode type, and *K* is assumed to vary in the reduced BZ. The effective wave vector represents the overall oscillation of the magnetization in the array, taking into account the oscillation within the dot due to the mode character (second and third terms in Eq. (1)) and the change between adjacent dots due to the Bloch wave vector (first term). The introduction

Fig. 6. (a) SEM image of the array of Permalloy disks is shown. BLS spectra were measured applying the external field H along the [10] array direction (x-axis) and changing the qx and qy components of the in-plane transferred wave-vector **q**. (b) Surface BZ of the 2D periodic array is shown. The behaviour of the dynamic magnetization of the fundamental mode is schematically shown in four point of the BZ for a 3x3 sub-matrix of dots. The two different colours represent out-of-plane dynamic magnetization of opposite sign. (c) Measured frequencies (dots) are shown as a function of the spin-wave wave vector along the principal direction of the first BZ, for an external magnetic field H = 1.0 kOe. The calculated dispersion curves of the most significant modes are also reported. Bold, solid or dotted lines refer to modes whose calculated cross section is comparable, smaller than 1/10 or smaller than 1/100 with respect to that of the F mode, respectively. For the sake of comparison, the experimental

dispersion of the DE mode of the unpatterned film is also reported as open squares.

In Ref. 93 the dispersion of different modes was interpreted in terms of effective wave vector *k*eff introduced as an auxiliary variable that includes and replaces the band index, i.e. the mode type, and the Bloch wave vector. It can be defined in the extended zone scheme as:

eff 1 () 1 () <sup>ˆ</sup> ( ) <sup>ˆ</sup> 2 2

=+ + + + *k K P m P n P(m) m Pn n*

where P(*i*) is the parity function (=+1,-1 for *i* even or odd, respectively), *m*-BA×*n*-DE is the mode type, and *K* is assumed to vary in the reduced BZ. The effective wave vector represents the overall oscillation of the magnetization in the array, taking into account the oscillation within the dot due to the mode character (second and third terms in Eq. (1)) and the change between adjacent dots due to the Bloch wave vector (first term). The introduction

 − − π

*a a*

 π

**x y** (1)

of *k*eff helps to understand the frequency dispersion of the magnonic crystal, because, in the limit of a continuous medium, *k*eff becomes the real wave vector of the spin excitation of the continuous film, whose dispersion curves have the following properties. The mode frequency increases (decreases) when the modulus of the wave vector increases in a direction perpendicular (parallel) to the applied field, corresponding to the MSSW (MSBVW) geometry.

To realize band gap magnonic metamaterials with a band structure of higher in-plane symmetry arrays of circular nanomagnets have been considered where an out-of-plane magnetic field stabilizes the so-called vortex state in each of the unit cells of the periodic nanodisk lattice. The dipolar interaction via nanoscale air gaps leads to allowed minibands and forbidden frequency gaps which exhibit a four-fold symmetry in in-plane directions if nanodisks are arranged in a square lattice 94. This higher symmetry goes beyond the magnonic crystals where an in-plane field governs the symmetry of the band structure.

#### **5. Theory of band gap magnonic metamaterials**

The knowledge of the physical mechanisms which govern the dynamical behaviour of nanoscale magnetic elements is of fundamental importance for understanding the general properties of metamaterials. Because of that, the theoretical derivations of the frequency and the spatial profile of magnonic modes become necessary to gain a physical understanding of the processes observed at a macroscopic level95-97. Several analytical and numerical methods are used to derive the profile of normal modes. Analytical models require some preliminary assumptions (approximations) regarding the mode profiles98,99. Numerical tools based on micromagnetic simulations has been developed for solving the equation of motion in the time domain and to successively perform a Fourier analysis of that output signal100,101. The same equation can also be solved in the frequency domain. The Dynamical Matrix Method (DMM)102 belongs to the latter approach, in which the sample is subdivided into cells and the linearized Landau–Lifshitz equation of motion is recast as a generalized eigenvalue problem, which is numerically solved by means of a finite-element method. Band structures of periodic composites can then be calculated with the help of the Bloch theorem, which reduces the number of independent variables of a periodic system to that of the corresponding unit cell82.

The band structures of spin-waves in materials with discrete translational symmetry can be calculated also by the plane wave method (PWM). The PWM is a popular tool commonly used for studying electronic, photonic and phononic crystals because of its conceptual simplicity and applicability to any type of lattice and shape of scattering centers. The method was also adapted to the calculations of the magnonic band structures and is constantly improved, with its field of application extending to new problems85,103. Recently, the PWM has also been used for the calculation of spin-wave spectra of 1D magnonic crystals of finite thickness104 and 2D thin-film magnonic crystals105. Only very recently, the PWM has been employed for the first time for calculating spin-wave spectra of 2D antidot arrays based on a square lattice with a good agreement with experimental results obtained106.

The 2D magnonic crystals composed of two ferromagnetic materials in thin film geometry were studied theoretically in Refs.105,107. The plane wave method with supercell formulation was used to study the edge effect in magnonic crystals in Ref. 105. It was shown that localisation at the edges of corners of the 2D magnonic crystals with finite lateral

Magnonic Metamaterials 355

The elliptical deformation of cylindrical dots in 2D magnonic crystals was investigated in Ref. 107 by means of the plane wave method. The use of rods in the shape of elliptic cylinders as scattering centres in 2D magnonic crystals implies the introduction of two additional structural parameters: the cross-sectional ellipticity of the rods and the angle of their rotation in the plane perpendicular to the rod axis (the plane of spin-wave propagation). In contrast to the lattice constant, a change of which will strongly modify the magnonic spectrum, these new parameters allow fine tuning of the width and position of the bands and band gaps. For specific in-plane rotation angles, changing the rod ellipticity will modify the position, width, and number of bands (Figure 8). Thus, an appropriate use of rods of elliptical cross section offers additional possibilities in the design of magnonic

Fig. 8. Fine tuning of the magnonic band structure in the thin film 2D magnonic crystal. (a) Schematic view of the 2D magnonic crystal under study, section in the plane of periodicity. (b) High-symmetry path over the 2D Brillouin zone for ellipses arranged in a square lattice. (c)-(d) The lowest magnonic gaps (shaded) vs. rod ellipticity for 2D YIG/Fe magnonic crystals with two angles of the in-plane rotation of the rods: (c) α = 0° and (d) α = 45°. The other parameters are: lattice constant 10 nm, filling fraction 0.5, film thickness 50 nm, and

3D band gap and effectively continuous metamaterials are the least studied objects in magnonics, due to both increased difficulty of their theoretical treatment and currently limited outlook for their fabrication and experimental investigation. Collective spin wave modes in 3D arrays of ferromagnetic particles in non-magnetic matrices were studied in Refs. 109,110. Magnonic band structure of 3D all-ferromagnetic magnonic crystals was calculated by Krawczyk and Puszkarski111,112. Here, again the depth of modulation of magnetic parameters is essential to generate magnonic bands and forbidden-frequency gaps

An example of magnonic band structure for a 3D magnonic crystal resulting from the numerical solution of Landau-Lifshitz equations with the plane-wave method is shown in

filters with precisely adjusted passbands and stopbands.

external magnetic field 0.1 T.

of significant width.

extension is possible. The PWM was powerful to remodel the magnonic miniband formation in short-period antidot arrays as well as the tunable metamaterial properties of large-period antidot arrays prepared in Ni80Fe20 106,108.

Fig. 7. The magnonic spectrum of periodic slab of finite thickness composed of the Ni inclusions embedded in Fe matrix (a) is strongly dependent on the filling fraction: the ratio of inclusion volume to the volume of unit cell, and depend on the lattice type. The maximum of the width of the first magnonic gap is reached for the intermediate values of the filling fraction (b). Note that the first absolute magnonic gap is wider for the triangular lattice (c) than for the square lattice (d).

The results of a comparative study of 2D magnonic crystals with square and triangular lattice of the Ni inclusions in Fe matrix are shown in Figure 7. The magnonic band structures for triangular and square lattices are shown in Figure 7 (c) and (d), respectively. We find that in the range of small lattice constants the triangular arrangement of cylindrical dots support opening of a magnonic gap. This gap exists for a greater range of filling fraction values and is much wider for a triangular lattice (see Figure 7 (b)). This is similar to the results of similar studies of photonic and phononic crystals. An increase of the lattice constant results in changes in mutual relation between the exchange and magnetostatic interactions. This leads to lowering frequencies of spin waves and decreasing the gap width up to closing it. In the magnetostatic regime, i.e. when the magnetostatic interactions dominate, the nonuniformity of the demagnetizing field is crucial for low frequency spin waves, especially for edge modes.

extension is possible. The PWM was powerful to remodel the magnonic miniband formation in short-period antidot arrays as well as the tunable metamaterial properties of large-period

width of the 1st gap

(b)

ff=0.55

0.0 0.2 0.4 0.6 0.8 1.0

(d)

(c)

filling fraction

X M X' M

 X X'

a=50nm 2c=5nm H =50mT <sup>0</sup> <sup>0</sup>

 0

Y H0

X

10

q q

Fig. 7. The magnonic spectrum of periodic slab of finite thickness composed of the Ni inclusions embedded in Fe matrix (a) is strongly dependent on the filling fraction: the ratio

maximum of the width of the first magnonic gap is reached for the intermediate values of the filling fraction (b). Note that the first absolute magnonic gap is wider for the triangular

The results of a comparative study of 2D magnonic crystals with square and triangular lattice of the Ni inclusions in Fe matrix are shown in Figure 7. The magnonic band structures for triangular and square lattices are shown in Figure 7 (c) and (d), respectively. We find that in the range of small lattice constants the triangular arrangement of cylindrical dots support opening of a magnonic gap. This gap exists for a greater range of filling fraction values and is much wider for a triangular lattice (see Figure 7 (b)). This is similar to the results of similar studies of photonic and phononic crystals. An increase of the lattice constant results in changes in mutual relation between the exchange and magnetostatic interactions. This leads to lowering frequencies of spin waves and decreasing the gap width up to closing it. In the magnetostatic regime, i.e. when the magnetostatic interactions dominate, the nonuniformity of the demagnetizing field is crucial for low frequency spin

of inclusion volume to the volume of unit cell, and depend on the lattice type. The

20

30

antidot arrays prepared in Ni80Fe20 106,108.

a

ff=0.40

lattice (c) than for the square lattice (d).

waves, especially for edge modes.

X

Y H0

<sup>K</sup> <sup>M</sup> <sup>K</sup>' <sup>M</sup>' <sup>0</sup>

Z X Y H0

<sup>f</sup> <sup>r</sup>

(c) (d)

M(t)

m(t)

MS

2c

(a)

10

20

30

The elliptical deformation of cylindrical dots in 2D magnonic crystals was investigated in Ref. 107 by means of the plane wave method. The use of rods in the shape of elliptic cylinders as scattering centres in 2D magnonic crystals implies the introduction of two additional structural parameters: the cross-sectional ellipticity of the rods and the angle of their rotation in the plane perpendicular to the rod axis (the plane of spin-wave propagation). In contrast to the lattice constant, a change of which will strongly modify the magnonic spectrum, these new parameters allow fine tuning of the width and position of the bands and band gaps. For specific in-plane rotation angles, changing the rod ellipticity will modify the position, width, and number of bands (Figure 8). Thus, an appropriate use of rods of elliptical cross section offers additional possibilities in the design of magnonic filters with precisely adjusted passbands and stopbands.

Fig. 8. Fine tuning of the magnonic band structure in the thin film 2D magnonic crystal. (a) Schematic view of the 2D magnonic crystal under study, section in the plane of periodicity. (b) High-symmetry path over the 2D Brillouin zone for ellipses arranged in a square lattice. (c)-(d) The lowest magnonic gaps (shaded) vs. rod ellipticity for 2D YIG/Fe magnonic crystals with two angles of the in-plane rotation of the rods: (c) α = 0° and (d) α = 45°. The other parameters are: lattice constant 10 nm, filling fraction 0.5, film thickness 50 nm, and external magnetic field 0.1 T.

3D band gap and effectively continuous metamaterials are the least studied objects in magnonics, due to both increased difficulty of their theoretical treatment and currently limited outlook for their fabrication and experimental investigation. Collective spin wave modes in 3D arrays of ferromagnetic particles in non-magnetic matrices were studied in Refs. 109,110. Magnonic band structure of 3D all-ferromagnetic magnonic crystals was calculated by Krawczyk and Puszkarski111,112. Here, again the depth of modulation of magnetic parameters is essential to generate magnonic bands and forbidden-frequency gaps of significant width.

An example of magnonic band structure for a 3D magnonic crystal resulting from the numerical solution of Landau-Lifshitz equations with the plane-wave method is shown in

Magnonic Metamaterials 357

Using the method from Refs. 112 and 113, we performed calculations for 3D magnonic crystals based on magnetoferritin crystals (mFT) described in the next section. In the calculations, we assumed mFT spheres (diameter 8 nm) in fcc lattice (with lattice constant 14 nm) immersed in Co. Assuming magnetic and structural parameters taken from literature for dehydrated mFT crystals and cobalt, we showed in Figure 10 (a) that a wide absolute magnonic band gap should exist in the magnonic spectrum of the structure, well above 100 GHz. It means that replacing the protein shell in magnetoferritin crystals with ferromagnetic metals should allow for opening the magnonic band gap. The gap is absolute and wide, and

Fig. 10. (a) The magnonic band structure of fcc magnonic crystal with lattice constant of 14 nm is shown. The magnonic crystal is composed of magnetoferritin spheres of diameter 8 nm embedded in Co. The external bias magnetic field 0.1 T is applied along z-axis. (b) First Brillouin zone of the fcc lattice, with the path along which we calculate the magnonic band

The magnonic dispersion can also be calculated from the results of micromagnetic simulations by Fourier transforming them in both temporal and spatial dimensions into the reciprocal space. For 1D samples, the method was realised in Ref. 16, and then used in a number of further studies114,115. So, Figure 11 shows the magnonic dispersion of a stack of dipolarly coupled magnetic nanoelements, studied in Ref. 115. The sign of the magnonic dispersion along the stacking direction is determined by the spatial character and ellipticity of the corresponding modes of an isolated nanoelement. Moreover, there exists a critical value of the ellipticity at which the sign of the magnonic dispersion changes from negative to positive in a discontinuous way. The discovered effect suggests a novel way of tailoring the dispersion of collective spin waves in magnonic bandgap metamaterials, unique to

Reprogrammable dynamic response has been demonstrated through different remanent states of planar arrays of nanomagnets116. The reconfiguration of a 1D magnonic crystal has recently been demonstrated via variation of the orientation of neighbouring ferromagnetic nanowires from a parallel to anti-parallel magnetic states (Figure 12)117,118. Experiments and simulations have shown that spin waves propagating perpendicular to the long axis of such coupled nanowires experience different artificial magnonic band structures in

configurations (a) and (b). Magnonic dispersions have thus become reprogrammable.

so, it is very promising for application of 3D magnonic crystals.

(a) (b)

structure in (a).

magnonics.

Figure 9. The assumed value of the simple cubic (sc) lattice constant was 10 nm; the magnetic parameters of the matrix material were close to those of YIG, and the magnetic parameters of the ferromagnetic material of the spheres corresponded to iron. Two magnonic gaps in the resulting spectrum were observed. The first magnonic gap is delimited by the two lowest spin-wave excitations, one localized in the Fe spheres and the other in the matrix branches I and II, respectively (Figure 9 (b) and (c)). Among three cubic structures studied in Ref. 112 the magnonic crystal with face centred cubic (fcc) lattice is most suitable for gap opening.

In Ref. 113, a detailed study of all the possible combinations of 3D magnonic crystal component materials from: Co, Ni, Fe, and Py for spheres and matrix was performed to find optimal material configurations for which either absolute or partial magnonic gaps occur in the magnonic spectrum of the 3D magnonic crystal with hexagonal structure. Among the MCs considered in this study, an absolute magnonic gap is obtained in a crystal with Ni spheres embedded in Fe.

Fig. 9. (a) The magnonic band structure of a simple cubic (sc) magnonic crystal with a lattice constant of 10 nm. The magnonic crystal is composed of Fe spheres of radius 26.28 Å disposed in sites of the sc lattice and embedded in YIG. Blue circles indicate the beginnings of the first two branches at point at which profiles of spin-waves are shown in (b) and (c). (b) and (c) Profiles of squared dynamic magnetization component in two adjacent planes, (002) and (001) right and left column, respectively. The planes are shown in the inset on the top-right. White colour corresponds to maximum values of amplitudes dynamic magnetization. Reproduced from Ref. 112. Copyright 2008, American Physical Society.

Figure 9. The assumed value of the simple cubic (sc) lattice constant was 10 nm; the magnetic parameters of the matrix material were close to those of YIG, and the magnetic parameters of the ferromagnetic material of the spheres corresponded to iron. Two magnonic gaps in the resulting spectrum were observed. The first magnonic gap is delimited by the two lowest spin-wave excitations, one localized in the Fe spheres and the other in the matrix branches I and II, respectively (Figure 9 (b) and (c)). Among three cubic structures studied in Ref. 112 the magnonic crystal with face centred cubic (fcc) lattice is

In Ref. 113, a detailed study of all the possible combinations of 3D magnonic crystal component materials from: Co, Ni, Fe, and Py for spheres and matrix was performed to find optimal material configurations for which either absolute or partial magnonic gaps occur in the magnonic spectrum of the 3D magnonic crystal with hexagonal structure. Among the MCs considered in this study, an absolute magnonic gap is obtained in a crystal with Ni

Fig. 9. (a) The magnonic band structure of a simple cubic (sc) magnonic crystal with a lattice constant of 10 nm. The magnonic crystal is composed of Fe spheres of radius 26.28 Å disposed in sites of the sc lattice and embedded in YIG. Blue circles indicate the

beginnings of the first two branches at point at which profiles of spin-waves are shown in (b) and (c). (b) and (c) Profiles of squared dynamic magnetization component in two adjacent planes, (002) and (001) right and left column, respectively. The planes are shown in the inset on the top-right. White colour corresponds to maximum values of amplitudes dynamic magnetization. Reproduced from Ref. 112. Copyright 2008, American Physical

most suitable for gap opening.

spheres embedded in Fe.

Society.

Using the method from Refs. 112 and 113, we performed calculations for 3D magnonic crystals based on magnetoferritin crystals (mFT) described in the next section. In the calculations, we assumed mFT spheres (diameter 8 nm) in fcc lattice (with lattice constant 14 nm) immersed in Co. Assuming magnetic and structural parameters taken from literature for dehydrated mFT crystals and cobalt, we showed in Figure 10 (a) that a wide absolute magnonic band gap should exist in the magnonic spectrum of the structure, well above 100 GHz. It means that replacing the protein shell in magnetoferritin crystals with ferromagnetic metals should allow for opening the magnonic band gap. The gap is absolute and wide, and so, it is very promising for application of 3D magnonic crystals.

Fig. 10. (a) The magnonic band structure of fcc magnonic crystal with lattice constant of 14 nm is shown. The magnonic crystal is composed of magnetoferritin spheres of diameter 8 nm embedded in Co. The external bias magnetic field 0.1 T is applied along z-axis. (b) First Brillouin zone of the fcc lattice, with the path along which we calculate the magnonic band structure in (a).

The magnonic dispersion can also be calculated from the results of micromagnetic simulations by Fourier transforming them in both temporal and spatial dimensions into the reciprocal space. For 1D samples, the method was realised in Ref. 16, and then used in a number of further studies114,115. So, Figure 11 shows the magnonic dispersion of a stack of dipolarly coupled magnetic nanoelements, studied in Ref. 115. The sign of the magnonic dispersion along the stacking direction is determined by the spatial character and ellipticity of the corresponding modes of an isolated nanoelement. Moreover, there exists a critical value of the ellipticity at which the sign of the magnonic dispersion changes from negative to positive in a discontinuous way. The discovered effect suggests a novel way of tailoring the dispersion of collective spin waves in magnonic bandgap metamaterials, unique to magnonics.

Reprogrammable dynamic response has been demonstrated through different remanent states of planar arrays of nanomagnets116. The reconfiguration of a 1D magnonic crystal has recently been demonstrated via variation of the orientation of neighbouring ferromagnetic nanowires from a parallel to anti-parallel magnetic states (Figure 12)117,118. Experiments and simulations have shown that spin waves propagating perpendicular to the long axis of such coupled nanowires experience different artificial magnonic band structures in configurations (a) and (b). Magnonic dispersions have thus become reprogrammable.

Magnonic Metamaterials 359

Fig. 12. Two different remanent magnetic configurations of a 1D magnonic crystal formed by interacting ferromagnetic nanowires are shown for (a) parallel and (b) anti-parallel alignment of neighbouring nanowires. In (b), the magnetic unit cell of the magnonic crystal

The theory of the effective properties of magnonic metamaterials in situations when they behave as "effectively continuous" can often be derived as the long wavelength limit of the corresponding band gap theory. The latter can however be quite complex, so that it becomes more practical and useful to develop the effectively continuous theory without a reference to the band gap one. In the case of the effective permeability of magnetic composites and metamaterials, the majority of analytical models employ the so-called macrospin approximation, in which each magnetic inclusion within a non-magnetic matrix is considered as a single giant spin and is therefore characterized by a single magnetic resonance. However, it is well known that the spin wave spectrum of magnetic nanostructures and nano-elements has a complex structure, featuring series of resonances due to spatially non-uniform spin wave modes54-56,61,64,67-69,71,92-99,115,119. Each of the resonances is expected to contribute to the susceptibility tensor of the magnetic constituents and correspondingly to the permeability tensor of the whole metamaterial. The resonance frequencies can be controlled and reconfigured by the external magnetic119-121 and electric122,123 fields, and the same functionalities should therefore be inherited by the

A method of calculation of the effective permeability that takes full account of the complex spectrum of the metamaterial's individual magnetic constituents has been demonstrated in

is twice as large as the geometrical one, leading to zone folding effects of magnon

**6. Theory of effectively continuous magnonic metamaterias** 

(b)

(a)

dispersions in the reciprocal space [117].

magnonic metamaterials.

Fig. 11. The magnonic dispersion is shown for a stack of 240 stadium shaped magnetic elements with dimensions of 100x50x10 nm3. The inset schematically shows the studied sample and geometry of the problem. After Ref. 115.

**H**b

*x*

50 nm

*z*

*y*

**h**e


rad/cm)

0

sample and geometry of the problem. After Ref. 115.

Wavevector (106

Fig. 11. The magnonic dispersion is shown for a stack of 240 stadium shaped magnetic elements with dimensions of 100x50x10 nm3. The inset schematically shows the studied

5

10

15

20

Frequency (GHz)

25

30

35

40

Fig. 12. Two different remanent magnetic configurations of a 1D magnonic crystal formed by interacting ferromagnetic nanowires are shown for (a) parallel and (b) anti-parallel alignment of neighbouring nanowires. In (b), the magnetic unit cell of the magnonic crystal is twice as large as the geometrical one, leading to zone folding effects of magnon dispersions in the reciprocal space [117].

### **6. Theory of effectively continuous magnonic metamaterias**

The theory of the effective properties of magnonic metamaterials in situations when they behave as "effectively continuous" can often be derived as the long wavelength limit of the corresponding band gap theory. The latter can however be quite complex, so that it becomes more practical and useful to develop the effectively continuous theory without a reference to the band gap one. In the case of the effective permeability of magnetic composites and metamaterials, the majority of analytical models employ the so-called macrospin approximation, in which each magnetic inclusion within a non-magnetic matrix is considered as a single giant spin and is therefore characterized by a single magnetic resonance. However, it is well known that the spin wave spectrum of magnetic nanostructures and nano-elements has a complex structure, featuring series of resonances due to spatially non-uniform spin wave modes54-56,61,64,67-69,71,92-99,115,119. Each of the resonances is expected to contribute to the susceptibility tensor of the magnetic constituents and correspondingly to the permeability tensor of the whole metamaterial. The resonance frequencies can be controlled and reconfigured by the external magnetic119-121 and electric122,123 fields, and the same functionalities should therefore be inherited by the magnonic metamaterials.

A method of calculation of the effective permeability that takes full account of the complex spectrum of the metamaterial's individual magnetic constituents has been demonstrated in

Magnonic Metamaterials 361

μyz

DMM



0.0

0.4

μzz

Fig. 14. The real and imaginary parts of the four components of the effective permeability tensor are shown as functions of frequency for the metamaterial depicted in Figure 13. The filling factor is 2.48% and the constant external magnetic field is *H*bias = 933 Oe. The field is applied in the plane of the layers along the x axis. The insets show the spatial profiles of the

In the context of nanostructured magnonic metamaterials, the main nanomanufacturing challenge is to fabricate large-scale periodic structures consisting of or containing magnetic materials precisely and controllably tailored at the nanometre scale. Being at the limit of current lithographic tools, the challenge requires bottom-up technologies be exploited instead. For example, protein based colloidal crystallisation techniques can be used to

There are a number of possible ways to achieve a periodic 3-dimensional magnetic structure. Apart from top-down lithography, which is generally limited to structures with a height much less than their in-plane dimensions, self-assembly offers a number of promising routes. For example, a number of researchers have used colloidal crystallization to generate periodic 3-dimensional arrays of magnetic nanoparticles129. A variation of this approach, which has the advantage that it separates nanoparticle functionality and array formation, exploits the ability of certain proteins both to act as templates for nanoparticle formation and to crystallize. The first demonstration of this method used the ubiquitous iron

0.8

1.2

6 7 8 9 10 11 12

Im part

Re part

Re part

Frequency (GHz)

6 7 8 9 10 11 12

Frequency (GHz)

ampl. phase

Im part

6 7 8 9 10 11 12

Frequency (GHz)

OOMMF

Re part

Im part

MicroMagus

6 7 8 9 10 11 12

Re part

**7. Bottom-up technologies for metamaterials** 

produce macroscopic 3D ordered magnetic arrays127,128.

Im part

Frequency (GHz)

mode amplitude (top) and phase (bottom) for the two dominant modes.





μzy

0.0

0.4


0

μyy

2

4

Ref. 59. In this method, the susceptibility tensor of an isolated inclusion is calculated numerically and then used as an input to an analytical expression (a so-called "mixing rule") for the permeability of the whole metamaterial. Finding the susceptibility tensor of the isolated inclusion is a standard problem for micromagnetics, and can be addressed using a number of different approaches. For example, full-scale numerical micromagnetic simulations could be performed using one of the available micromagnetic packages (e.g. Nmag124, OOMMF125, or MicroMagus126). Alternatively, the dynamical matrix method, which has already been introduced above, can be used a form modified to facilitate the susceptibility calculations. The results of the application of the method to a model metamaterial representing an array of magnetic nanodisks embedded into a non-magnetic matrix (inset in Figure 13) is shown in Figure 14. Figure 13 shows the region of geometrical parameters of such a metamaterial, in which one of the components of the permeability tensor becomes negative within a certain frequency range. The method also presents a useful way by which to compare the different micromagnetic methods in order to evaluate the accuracy to be expected from micromagnetic simulations. In particular, we find that the results produced by the state-of-art micromagnetic simulations agree with each other within an error bar of about 5%, which has to be taken into account when micromagnetic calculations are used to model experimental data.

Fig. 13. The range of values of the in-plane edge-to-edge separation (b) and the distance between layers (c) in which μ*yy* component of the permeability becomes negative near the frequencies of the dominant magnonic resonances. The inset shows the geometry of the metamaterial consisting of magnetic discs in a non-magnetic matrix. The discs are located in nodes of a hexagonal lattice. The disk diameter is *d =* 195 nm, the in-plane edge-to-edge separation is *b* = 20 nm, the distance between the layers is *c* = 140 nm and is much greater than disk thickness *l* = 5 nm.

Ref. 59. In this method, the susceptibility tensor of an isolated inclusion is calculated numerically and then used as an input to an analytical expression (a so-called "mixing rule") for the permeability of the whole metamaterial. Finding the susceptibility tensor of the isolated inclusion is a standard problem for micromagnetics, and can be addressed using a number of different approaches. For example, full-scale numerical micromagnetic simulations could be performed using one of the available micromagnetic packages (e.g. Nmag124, OOMMF125, or MicroMagus126). Alternatively, the dynamical matrix method, which has already been introduced above, can be used a form modified to facilitate the susceptibility calculations. The results of the application of the method to a model metamaterial representing an array of magnetic nanodisks embedded into a non-magnetic matrix (inset in Figure 13) is shown in Figure 14. Figure 13 shows the region of geometrical parameters of such a metamaterial, in which one of the components of the permeability tensor becomes negative within a certain frequency range. The method also presents a useful way by which to compare the different micromagnetic methods in order to evaluate the accuracy to be expected from micromagnetic simulations. In particular, we find that the results produced by the state-of-art micromagnetic simulations agree with each other within an error bar of about 5%, which has to be taken into account when micromagnetic

50 100 150 200

*yy* component of the permeability becomes negative near the

b (nm)

(Filling factor ~1.6 %)

Fig. 13. The range of values of the in-plane edge-to-edge separation (b) and the distance

frequencies of the dominant magnonic resonances. The inset shows the geometry of the metamaterial consisting of magnetic discs in a non-magnetic matrix. The discs are located in nodes of a hexagonal lattice. The disk diameter is *d =* 195 nm, the in-plane edge-to-edge separation is *b* = 20 nm, the distance between the layers is *c* = 140 nm and is much greater

Negative μ Positiveμ

calculations are used to model experimental data.

50

μ

100

150

c (nm)

between layers (c) in which

than disk thickness *l* = 5 nm.

200

250

Fig. 14. The real and imaginary parts of the four components of the effective permeability tensor are shown as functions of frequency for the metamaterial depicted in Figure 13. The filling factor is 2.48% and the constant external magnetic field is *H*bias = 933 Oe. The field is applied in the plane of the layers along the x axis. The insets show the spatial profiles of the mode amplitude (top) and phase (bottom) for the two dominant modes.

#### **7. Bottom-up technologies for metamaterials**

In the context of nanostructured magnonic metamaterials, the main nanomanufacturing challenge is to fabricate large-scale periodic structures consisting of or containing magnetic materials precisely and controllably tailored at the nanometre scale. Being at the limit of current lithographic tools, the challenge requires bottom-up technologies be exploited instead. For example, protein based colloidal crystallisation techniques can be used to produce macroscopic 3D ordered magnetic arrays127,128.

There are a number of possible ways to achieve a periodic 3-dimensional magnetic structure. Apart from top-down lithography, which is generally limited to structures with a height much less than their in-plane dimensions, self-assembly offers a number of promising routes. For example, a number of researchers have used colloidal crystallization to generate periodic 3-dimensional arrays of magnetic nanoparticles129. A variation of this approach, which has the advantage that it separates nanoparticle functionality and array formation, exploits the ability of certain proteins both to act as templates for nanoparticle formation and to crystallize. The first demonstration of this method used the ubiquitous iron

Magnonic Metamaterials 363

excitations in the magnetoferritin crystals when exploring the metamaterials properties. Using a 20 µm wide inner conductor, we excite spin waves with a distribution of wave vectors **k** given by the current distribution through the CPW135. Experiments are carried out in a cryogenic setup allowing us to perform measurements over a wide temperature range with an external field of up to 2.5 T. The field is applied in a direction perpendicular to the plane of the CPW. This direction is chosen so that torque τ ~ **M** x **h**rf is maximised

We start our discussion from room temperature data taken at 290 K (Figure 16). We measure the dynamic response as a function of applied external field. We observe pronounced absorption with a linear dependency indicated by the white dashed line. Interestingly, the resonance starts to appear at fields larger than 0.1 T, whereas at 0 T there is no resonance observed. We attribute the observed behaviour to the paramagnetic response of the protein crystals, in which the magnetization vectors of the nanoparticles are randomly aligned at zero field. An anisotropy term does not seem to be present to provoke a non-zero resonance frequency at small field. The individual nanoparticles show superparamagnetic behaviour,

Ferritin crystals

Fig. 16. (Top) Microscopy image of the 9 mm long coplanar wave guide containing 50 magnetoferritin crystals as shown in Figure 15. (Bottom) Spectroscopy data taken at 290 K (left) and 5 K (right) as a function of perpendicular magnetic field. Dark colour indicates spin-wave excitation. The broken lines are guides for the eyes reflecting both the field

The data at 5 K show a different behaviour, as we observe a clear absorption peak even at zero field. Here, the resonance is measured to be at 6 GHz. For fields larger than about 0.5 T, we observe a linear dependence of the resonance frequency with the applied field as well,

(about 50 crystals) Coplanar

250 µm

waveguide contact pads

when **M** (**h**rf) is out-of-plane (in-plane). Torque τ excites the spin waves.

which is expected for particles of this size.

dependency at 290 K to facilitate comparison.

Coplanar waveguide (9 mm long)

storage protein ferritin as a template for the growth of ferrimagnetic magnetite-maghemite nanoparticles127,130. Ferritin consists of a spherical protein shell with an outer diameter of 12 nm and an inner diameter of 8 nm, and the *in vivo* incorporation of Fe into apoferritin (ferritin without its mineral core) is achieved by the oxidation of Fe2+ ions transported through its ion channels.

Following magnetite nanoparticle synthesis, the magnetoferritin (this is the name given to ferritin containing a synthetic ferrimagnetic core) is purified by ion-exchange chromatography, and then passed through a stainless-steel-packed column in a uniform 0.7- 0.8 T magnetic field to separate any protein including non- or poorly-magnetic nanoparticles from the magnetoferritin. In a further purification stage size-exclusion chromatography is used to separate magnetoferritin monomers from dimers and oligomers before crystallization, which used the sitting-drop vapour diffusion technique and Cd2+ as a crystallization agent. A schematic of this process is shown in Figure 15.

Fig. 15. (a) Schematic showing how crystallizing a protein used as a template for nanoparticle growth gives a periodic 3D array of nanoparticles. (b) Optical image showing magnetoferritin crystals. Each crystal is a periodic 3-dimensional array of magnetite nanoparticles.

To study the metamaterials properties of several magnetoferritin crystals in parallel, fifty of the as-prepared crystals were mounted on a coplanar waveguide (CPW) in order to perform all electrical spin-wave spectroscopy131,132. In contrast to earlier studies on nanoparticle arrays where a microwave cavity at a fixed frequency was used133,134, the CPW-based technique allows one to measure over a broad frequency range. The CPW with the magnetoferritin crystals on top of it is shown in Figure 16. Using a VNA, which is connected to the CPW via microwave probe tips, we apply a microwave current to the CPW. This provokes a microwave magnetic field **h**rf around the inner conductor of the CPW. Considering the frequencies ranging from 10 MHz up to 26.5 GHz, we address spin

storage protein ferritin as a template for the growth of ferrimagnetic magnetite-maghemite nanoparticles127,130. Ferritin consists of a spherical protein shell with an outer diameter of 12 nm and an inner diameter of 8 nm, and the *in vivo* incorporation of Fe into apoferritin (ferritin without its mineral core) is achieved by the oxidation of Fe2+ ions transported

Following magnetite nanoparticle synthesis, the magnetoferritin (this is the name given to ferritin containing a synthetic ferrimagnetic core) is purified by ion-exchange chromatography, and then passed through a stainless-steel-packed column in a uniform 0.7- 0.8 T magnetic field to separate any protein including non- or poorly-magnetic nanoparticles from the magnetoferritin. In a further purification stage size-exclusion chromatography is used to separate magnetoferritin monomers from dimers and oligomers before crystallization, which used the sitting-drop vapour diffusion technique and Cd2+ as a

Nanoparticle

(a)

Protein molecule

Fig. 15. (a) Schematic showing how crystallizing a protein used as a template for

nanoparticle growth gives a periodic 3D array of nanoparticles. (b) Optical image showing magnetoferritin crystals. Each crystal is a periodic 3-dimensional array of magnetite

To study the metamaterials properties of several magnetoferritin crystals in parallel, fifty of the as-prepared crystals were mounted on a coplanar waveguide (CPW) in order to perform all electrical spin-wave spectroscopy131,132. In contrast to earlier studies on nanoparticle arrays where a microwave cavity at a fixed frequency was used133,134, the CPW-based technique allows one to measure over a broad frequency range. The CPW with the magnetoferritin crystals on top of it is shown in Figure 16. Using a VNA, which is connected to the CPW via microwave probe tips, we apply a microwave current to the CPW. This provokes a microwave magnetic field **h**rf around the inner conductor of the CPW. Considering the frequencies ranging from 10 MHz up to 26.5 GHz, we address spin

crystallization agent. A schematic of this process is shown in Figure 15.

(b)

through its ion channels.

nanoparticles.

excitations in the magnetoferritin crystals when exploring the metamaterials properties. Using a 20 µm wide inner conductor, we excite spin waves with a distribution of wave vectors **k** given by the current distribution through the CPW135. Experiments are carried out in a cryogenic setup allowing us to perform measurements over a wide temperature range with an external field of up to 2.5 T. The field is applied in a direction perpendicular to the plane of the CPW. This direction is chosen so that torque τ ~ **M** x **h**rf is maximised when **M** (**h**rf) is out-of-plane (in-plane). Torque τ excites the spin waves.

We start our discussion from room temperature data taken at 290 K (Figure 16). We measure the dynamic response as a function of applied external field. We observe pronounced absorption with a linear dependency indicated by the white dashed line. Interestingly, the resonance starts to appear at fields larger than 0.1 T, whereas at 0 T there is no resonance observed. We attribute the observed behaviour to the paramagnetic response of the protein crystals, in which the magnetization vectors of the nanoparticles are randomly aligned at zero field. An anisotropy term does not seem to be present to provoke a non-zero resonance frequency at small field. The individual nanoparticles show superparamagnetic behaviour, which is expected for particles of this size.

Fig. 16. (Top) Microscopy image of the 9 mm long coplanar wave guide containing 50 magnetoferritin crystals as shown in Figure 15. (Bottom) Spectroscopy data taken at 290 K (left) and 5 K (right) as a function of perpendicular magnetic field. Dark colour indicates spin-wave excitation. The broken lines are guides for the eyes reflecting both the field dependency at 290 K to facilitate comparison.

The data at 5 K show a different behaviour, as we observe a clear absorption peak even at zero field. Here, the resonance is measured to be at 6 GHz. For fields larger than about 0.5 T, we observe a linear dependence of the resonance frequency with the applied field as well,

Magnonic Metamaterials 365

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In order to use 3D nanoparticle assemblies as all-magnetic metamaterials, it would be relevant that researchers combine the protein based nano-manufacturing with advanced 3D material deposition techniques such as atomic layer deposition (ALD)136 and electrodeposition tailored for use with multiple magnetic materials. From this allferromagnetic 3D magnonic metamaterials might result. ALD film growth is self-limited, thereby achieving atomic scale control of the deposition. Recently, ALD was used to deposit ferromagnetic thin-films (such as Ni, Co, Fe3O4) into deep-etched trenches and membranes137,138,139. The complementary topology is also possible by, e.g., conformal coating of templates consisting of tailored nanowires140. Such possibilities make ALD a promising tool by which to fabricate 3D magnonic devices. Electrodeposition is also very well suited for deposition into complex templates141. It is fast and thereby suitable for scaling-up to produce large numbers of devices. For example, arrays of cylindrical magnetic nanowires deposited electrochemically within porous membranes142,143,144 have attracted much attention due to their potential for use as microwave145 and THz13 devices.

#### **8. Conclusions**

The modern research on fundamental properties of materials is increasingly driven by their anticipated potential for technological applications. In this Chapter, we have reviewed research that has been conducted within the MAGNONICS project funded by the European Commission to reveal the potential of magnonic metamaterials nanoscale building blocks of which are made of magnetic materials. A particular attention is devoted to the use of spin wave resonances tailored in magnonic crystals for design of novel features in the electromagnetic properties in the GHz-THz frequency range. These entirely new electromagnetic metamaterials could be designed to tune the transmission, absorption and reflection of electromagnetic radiation in the GHz-THz frequency range. Such magnonic metamaterials could also find their use within magnonic filters and logic gates.

The field of magnonics and magnonic metamaterials is very young. However, the growing community of magnonics researchers has already demonstrated that they are up to the challenges existing in the magnonics science and technology. In particular, this Chapter demonstrates important advances recently achieved in the direction of the development of effectively continuous and band gap magnonic metamaterials.

#### **9. Acknowledgments**

We thank S. Neusser for providing simulation results, P. Berberich, G. Duerr, R. Huber, J. Topp, and T. Rapp for experimental support. The authors gratefully acknowledge funding received from the European Community's Seventh Framework Programme (FP7/2007-2013) under Grant Agreements no 233552 (DYNAMAG) and 228673 (MAGNONICS), from the Engineering and Physical Sciences Research Council (EPSRC) of United Kingdom (project EP/E055087/1), from the German excellence cluster "Nanosystems Initiative Munich (NIM)".

#### **10. References**

364 Metamaterial

but it is slightly shifted towards higher frequencies if compared to the room temperature measurement (white dashed line). The resonance frequency at zero field implies an

In order to use 3D nanoparticle assemblies as all-magnetic metamaterials, it would be relevant that researchers combine the protein based nano-manufacturing with advanced 3D material deposition techniques such as atomic layer deposition (ALD)136 and electrodeposition tailored for use with multiple magnetic materials. From this allferromagnetic 3D magnonic metamaterials might result. ALD film growth is self-limited, thereby achieving atomic scale control of the deposition. Recently, ALD was used to deposit ferromagnetic thin-films (such as Ni, Co, Fe3O4) into deep-etched trenches and membranes137,138,139. The complementary topology is also possible by, e.g., conformal coating of templates consisting of tailored nanowires140. Such possibilities make ALD a promising tool by which to fabricate 3D magnonic devices. Electrodeposition is also very well suited for deposition into complex templates141. It is fast and thereby suitable for scaling-up to produce large numbers of devices. For example, arrays of cylindrical magnetic nanowires deposited electrochemically within porous membranes142,143,144 have attracted much

The modern research on fundamental properties of materials is increasingly driven by their anticipated potential for technological applications. In this Chapter, we have reviewed research that has been conducted within the MAGNONICS project funded by the European Commission to reveal the potential of magnonic metamaterials nanoscale building blocks of which are made of magnetic materials. A particular attention is devoted to the use of spin wave resonances tailored in magnonic crystals for design of novel features in the electromagnetic properties in the GHz-THz frequency range. These entirely new electromagnetic metamaterials could be designed to tune the transmission, absorption and reflection of electromagnetic radiation in the GHz-THz frequency range. Such magnonic

The field of magnonics and magnonic metamaterials is very young. However, the growing community of magnonics researchers has already demonstrated that they are up to the challenges existing in the magnonics science and technology. In particular, this Chapter demonstrates important advances recently achieved in the direction of the development of

We thank S. Neusser for providing simulation results, P. Berberich, G. Duerr, R. Huber, J. Topp, and T. Rapp for experimental support. The authors gratefully acknowledge funding received from the European Community's Seventh Framework Programme (FP7/2007-2013) under Grant Agreements no 233552 (DYNAMAG) and 228673 (MAGNONICS), from the Engineering and Physical Sciences Research Council (EPSRC) of United Kingdom (project EP/E055087/1), from the German excellence cluster "Nanosystems Initiative Munich

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attention due to their potential for use as microwave145 and THz13 devices.

metamaterials could also find their use within magnonic filters and logic gates.

effectively continuous and band gap magnonic metamaterials.

**8. Conclusions** 

**9. Acknowledgments** 

(NIM)".


Magnonic Metamaterials 367

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**Section 3** 

**The Applications of Metamaterials** 

