**Section 1**

**The Physics of Metamaterials** 

**0**

**1**

*Canada*

**Novel Electromagnetic Phenomena in Graphene**

**and Subsequent Microwave Devices Enabled by**

Metamaterials, i.e. artificial materials with properties not found in nature (Caloz & Itoh, 2006; Marqués et al., 2008), have gained a lot of attention in the past decade. The virtually unlimited freedom in tailoring their constitutive parameters created hopes for the realization of unique devices, such as subwavelength lenses and invisibility cloaks, which were essentially impossible to achieve with classical materials. However, the implementation of such devices has been hindered by the inherent limitation of conventional metamaterials: non-perfect homogeneity due to the periodicity being insufficiently smaller than the wavelength, high insertion and thermal losses due to the conduction properties of the constituent elements and narrow bandwidth caused by the strongly resonant nature of the metamaterials constituent elements. Furthermore, the complex 3D structure of bulk metamaterials makes them unsuitable as microwave substrates and superstrates for planar devices where integration is a critical issue. Therefore, there is a need to develop a new generation of metamaterials with higher compactness and more homogeneity, exploiting the recent advances of nanotechnology, for instance in the fields of nanowires, nanopolymers,

carbon nanotubes and graphene (Dragoman & Dragoman, 2009; Hanson, 2007).

enhanced-functionality microwave devices.

Graphene is a 2D material consisting of carbon atoms in a 2D honeycomb lattice (Geim & Novoselov, 2007). It was first synthesized from graphite in the form of micro-flakes in 2004 (Novoselov et al., 2004) and since then it has been gaining a continuously increasing interest. Its gap-less linear band-structure results in unique phenomena, such as high electron mobility, ambipolarity, non-zero minimum conductivity and anomalous half-integer quantum Hall effect (Neto et al., 2009). Until now, a lot of research has been devoted to the design of electronic transport devices, such as transistors and non-linear components, mainly exploiting the high mobility and the ambipolar field effect of graphene (Lin et al., 2009; Wang et al., 2009). Recently, it has been demonstrated that graphene also possesses unique electromagnetic properties, such as low loss surface waves (Koppens et al., 2011; Mikhailov & Ziegler, 2007; Vakil & Engheta, 2011), huge Faraday rotation (Crassee et al., 2011; Sounas & Caloz, 2011b) and super-confined edge magnetoplasmons (Mishchenko et al., 2010; Sounas & Caloz, 2011a). These properties, combined with the ambipolar field effect which provides enhanced control capabilities of graphene's conductivity, open a new path towards super-compact and

**1. Introduction**

**Multi-Scale Metamaterials**

*École Polytechnique de Montréal*

Dimitrios L. Sounas and Christophe Caloz

### **Novel Electromagnetic Phenomena in Graphene and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials**

Dimitrios L. Sounas and Christophe Caloz *École Polytechnique de Montréal Canada*

#### **1. Introduction**

Metamaterials, i.e. artificial materials with properties not found in nature (Caloz & Itoh, 2006; Marqués et al., 2008), have gained a lot of attention in the past decade. The virtually unlimited freedom in tailoring their constitutive parameters created hopes for the realization of unique devices, such as subwavelength lenses and invisibility cloaks, which were essentially impossible to achieve with classical materials. However, the implementation of such devices has been hindered by the inherent limitation of conventional metamaterials: non-perfect homogeneity due to the periodicity being insufficiently smaller than the wavelength, high insertion and thermal losses due to the conduction properties of the constituent elements and narrow bandwidth caused by the strongly resonant nature of the metamaterials constituent elements. Furthermore, the complex 3D structure of bulk metamaterials makes them unsuitable as microwave substrates and superstrates for planar devices where integration is a critical issue. Therefore, there is a need to develop a new generation of metamaterials with higher compactness and more homogeneity, exploiting the recent advances of nanotechnology, for instance in the fields of nanowires, nanopolymers, carbon nanotubes and graphene (Dragoman & Dragoman, 2009; Hanson, 2007).

Graphene is a 2D material consisting of carbon atoms in a 2D honeycomb lattice (Geim & Novoselov, 2007). It was first synthesized from graphite in the form of micro-flakes in 2004 (Novoselov et al., 2004) and since then it has been gaining a continuously increasing interest. Its gap-less linear band-structure results in unique phenomena, such as high electron mobility, ambipolarity, non-zero minimum conductivity and anomalous half-integer quantum Hall effect (Neto et al., 2009). Until now, a lot of research has been devoted to the design of electronic transport devices, such as transistors and non-linear components, mainly exploiting the high mobility and the ambipolar field effect of graphene (Lin et al., 2009; Wang et al., 2009). Recently, it has been demonstrated that graphene also possesses unique electromagnetic properties, such as low loss surface waves (Koppens et al., 2011; Mikhailov & Ziegler, 2007; Vakil & Engheta, 2011), huge Faraday rotation (Crassee et al., 2011; Sounas & Caloz, 2011b) and super-confined edge magnetoplasmons (Mishchenko et al., 2010; Sounas & Caloz, 2011a). These properties, combined with the ambipolar field effect which provides enhanced control capabilities of graphene's conductivity, open a new path towards super-compact and enhanced-functionality microwave devices.


(Dirac point).

*E*/*t*

0

PEC

energy-wavevector dispersion relation takes the form

5 -5

*<sup>K</sup>*� *<sup>K</sup>*

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

*kya kxa*

(a)

*z* = 0

electrostatic voltage is applied between graphene and the electrode.

0

<sup>5</sup> Novel Electromagnetic Phenomena in Graphene

Fig. 1. Energy-wavenumber dispersion diagram of electrons in graphene. (a) Dispersion diagram in the entire Brillouin zone. (b) Dispersion diagram around the *K* symmetry point

*ε<sup>r</sup> d*

Fig. 2. Structure for controlling the chemical potential of graphene. A conducting electrode is placed at the bottom of graphene, forming with graphene a parallel plate-capacitor. An

As already mentioned, the *K* and *K*� points are of significant importance in graphene. Using a Taylor expansion around these points in Equation 1 and keeping only the first order term, the

where **<sup>q</sup>** <sup>=</sup> **<sup>k</sup>** <sup>−</sup> **kK**,**K**� and *vF* <sup>=</sup> <sup>3</sup>*ta*/(2¯*h*) <sup>≈</sup> 106 m/s is the Fermi velocity. Equation 3 is similar to the dispersion relation *E* = *hck* ¯ of photons in vacuum and indicates that electrons in graphene behave like zero-mass particles (Dirac fermions) with an energy-independent velocity. As a result electrons in graphene exhibit quantum phenomena, such as the

Therefore, *ns*, and subsequently *μc*, can be controlled via *Vg* (varying *Vg* either provides electrons to graphene or depletes graphene from electrons). This way of controlling the carrier density is ofter called electrical doping, in contrast to chemical doping, where the carrier density is controlled by the density of impurity atoms introduced into the graphene lattice.

5

*y*

1.5 2 2.5



*E*/*t*

0 0.5 1 1.5

> 3 0.5 1 1.5 2

*K*

*kya kxa*

(b)

*Vg*

*x*

*E*(**q**) = ±*hv*¯ *<sup>F</sup>*|**q**|, (3)

*z*

The chapter is organized as follows. Section 2 provides a brief overview of the electronic band structure of graphene, which is responsible for most of its unique properties. Section 3 introduces the surface conductivity model of graphene in the presence of a static magnetic field and gives a physical interpretation of the conductivity expressions. Section 4 constitutes the core of the chapter providing a theoretical analysis of Faraday rotation in graphene. In Section 5 two applications based on the gyrotropic properties of graphene, namely a circular waveguide Faraday rotator and a spatial isolator, are proposed. Section 6 describes a practical implementation of such devices through the multiscale metamaterials concept. Finally, Section 7 summarizes the chapter.

#### **2. Electronic band structure of graphene**

The unique properties of graphene originate from its electronic band structure. Using a tight-binding approximation and considering only the nearest-neighbor terms, the energy-wavevector dispersion relation of electrons in graphene is found as (Neto et al., 2009)

$$E\_{\pm}(k\_{\mathbf{x}},k\_{\mathbf{y}}) = \pm t \sqrt{3 + 2 \cos \left(\sqrt{3}k\_{\mathbf{y}}a\right) + 4 \cos \left(\frac{\sqrt{3}}{2}k\_{\mathbf{y}}a\right) \cos \left(\frac{3}{2}k\_{\mathbf{x}}a\right)}\tag{1}$$

where *t* = 2.8 eV is the nearest-neighbor hopping energy, *a* = 1.42 Å is the interatomic distance, *kx* and *ky* are the *x* and *y* wavevector components, respectively, and the plus and minus signs refer to the upper (conduction) and the lower (valence) bands, respectively. Figs 1(a) and 1(b) plot *E* versus *kx* and *ky* in the entire Brillouin zone and around the *K* symmetry point, respectively. The *K* and *K*� points are very important, since the energy at these points, which is 0, is the chemical potential<sup>1</sup> of intrinsic graphene. A zero chemical potential means a completely full valence band and a completely empty conduction band, hence minimal conductivity.

The first thing one can observe from Fig. 1 is the absence of an energy gap between the valence and the conduction bands. Because of this, graphene is usually referred to as a semi-metal<sup>2</sup> or a zero-gap semiconductor3. The zero gap allows changing the type of conduction charge carriers from electrons to holes and vice versa by shifting the chemical potential from positive (conduction band) to negative (valence band) values, a phenomenon known as ambipolarity. The chemical potential can be easily tuned by an electrostatic voltage between graphene and an electrode parallel to it, as it is shown in Fig. 2. Assuming that the surface carrier density of graphene is *ns*, the electric dispacement on both sizes of graphene is oriented perpendicularly to graphene, along the *z* axis, and it has a value *Dz* = (*nse*/2)sgn(*z*). The electric field in the substrate is then *Ez* = *Dz*/(*εrε*0) = −*nse*/(2*εrε*0). Integrating *Ez* from graphene (*z* = 0) to the PEC (*z* = −*d*) plane, the apllied voltage is found as

$$V\_{\mathcal{S}} = \frac{n\_{\text{s}}ed}{2\varepsilon\_{r}\varepsilon\_{0}}.\tag{2}$$

<sup>1</sup> The chemical potential is the energy level with 0.5 probability of being fully occupied by electrons (the Fermi-Dirac distribution is 0.5). At zero temperature and under zero applied field (equilibrium), the chemical potential is equal to the Fermi level.

<sup>2</sup> Metals have their conduction band penetrating into their valence band.

<sup>3</sup> Semiconductors have a non-zero gap between the valence and the conduction bands.

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

The chapter is organized as follows. Section 2 provides a brief overview of the electronic band structure of graphene, which is responsible for most of its unique properties. Section 3 introduces the surface conductivity model of graphene in the presence of a static magnetic field and gives a physical interpretation of the conductivity expressions. Section 4 constitutes the core of the chapter providing a theoretical analysis of Faraday rotation in graphene. In Section 5 two applications based on the gyrotropic properties of graphene, namely a circular waveguide Faraday rotator and a spatial isolator, are proposed. Section 6 describes a practical implementation of such devices through the multiscale metamaterials concept.

The unique properties of graphene originate from its electronic band structure. Using a tight-binding approximation and considering only the nearest-neighbor terms, the energy-wavevector dispersion relation of electrons in graphene is found as (Neto et al., 2009)

where *t* = 2.8 eV is the nearest-neighbor hopping energy, *a* = 1.42 Å is the interatomic distance, *kx* and *ky* are the *x* and *y* wavevector components, respectively, and the plus and minus signs refer to the upper (conduction) and the lower (valence) bands, respectively. Figs 1(a) and 1(b) plot *E* versus *kx* and *ky* in the entire Brillouin zone and around the *K* symmetry point, respectively. The *K* and *K*� points are very important, since the energy at these points, which is 0, is the chemical potential<sup>1</sup> of intrinsic graphene. A zero chemical potential means a completely full valence band and a completely empty conduction band,

The first thing one can observe from Fig. 1 is the absence of an energy gap between the valence and the conduction bands. Because of this, graphene is usually referred to as a semi-metal<sup>2</sup> or a zero-gap semiconductor3. The zero gap allows changing the type of conduction charge carriers from electrons to holes and vice versa by shifting the chemical potential from positive (conduction band) to negative (valence band) values, a phenomenon known as ambipolarity. The chemical potential can be easily tuned by an electrostatic voltage between graphene and an electrode parallel to it, as it is shown in Fig. 2. Assuming that the surface carrier density of graphene is *ns*, the electric dispacement on both sizes of graphene is oriented perpendicularly to graphene, along the *z* axis, and it has a value *Dz* = (*nse*/2)sgn(*z*). The electric field in the substrate is then *Ez* = *Dz*/(*εrε*0) = −*nse*/(2*εrε*0). Integrating *Ez* from graphene (*z* = 0) to the

> *Vg* <sup>=</sup> *nsed* 2*εrε*<sup>0</sup>

<sup>1</sup> The chemical potential is the energy level with 0.5 probability of being fully occupied by electrons (the Fermi-Dirac distribution is 0.5). At zero temperature and under zero applied field (equilibrium), the

+ 4 cos

 <sup>√</sup><sup>3</sup> <sup>2</sup> *kya*

 cos 3 2 *kxa* 

. (2)

, (1)

√ 3*kya* 

Finally, Section 7 summarizes the chapter.

*E*±(*kx*, *ky*) = ±*t*

PEC (*z* = −*d*) plane, the apllied voltage is found as

chemical potential is equal to the Fermi level.

<sup>2</sup> Metals have their conduction band penetrating into their valence band.

<sup>3</sup> Semiconductors have a non-zero gap between the valence and the conduction bands.

hence minimal conductivity.

<sup>3</sup> <sup>+</sup> 2 cos

**2. Electronic band structure of graphene**

Fig. 1. Energy-wavenumber dispersion diagram of electrons in graphene. (a) Dispersion diagram in the entire Brillouin zone. (b) Dispersion diagram around the *K* symmetry point (Dirac point).

Therefore, *ns*, and subsequently *μc*, can be controlled via *Vg* (varying *Vg* either provides electrons to graphene or depletes graphene from electrons). This way of controlling the carrier density is ofter called electrical doping, in contrast to chemical doping, where the carrier density is controlled by the density of impurity atoms introduced into the graphene lattice.

Fig. 2. Structure for controlling the chemical potential of graphene. A conducting electrode is placed at the bottom of graphene, forming with graphene a parallel plate-capacitor. An electrostatic voltage is applied between graphene and the electrode.

As already mentioned, the *K* and *K*� points are of significant importance in graphene. Using a Taylor expansion around these points in Equation 1 and keeping only the first order term, the energy-wavevector dispersion relation takes the form

$$E(\mathbf{q}) = \pm \hbar v\_F |\mathbf{q}|\_{\prime} \tag{3}$$

where **<sup>q</sup>** <sup>=</sup> **<sup>k</sup>** <sup>−</sup> **kK**,**K**� and *vF* <sup>=</sup> <sup>3</sup>*ta*/(2¯*h*) <sup>≈</sup> 106 m/s is the Fermi velocity. Equation 3 is similar to the dispersion relation *E* = *hck* ¯ of photons in vacuum and indicates that electrons in graphene behave like zero-mass particles (Dirac fermions) with an energy-independent velocity. As a result electrons in graphene exhibit quantum phenomena, such as the

be given in the next section. It is known from solid state physics that electrons are distributed

<sup>7</sup> Novel Electromagnetic Phenomena in Graphene

which represents the probability of finding an electron at the energy level *E*. In Equation 5 *μc* is the chemical potential, *kB* is the Boltzmann constant and *T* the temperature in *K*. Fig. 4 plots *fd*(*E*) for *μ<sup>c</sup>* > 0, that is a partially filled conduction band. The electron and hole densities are

*<sup>N</sup>*(*E*) = <sup>2</sup>*<sup>E</sup>*

is the density of states. The density of states is the same for electrons and holes in graphene, due to the band-structure symmetry around the *E* = 0 plane. The effective total carrier density is then *ns* = *nse* − *nsh*. Note that in intrinsic graphene *nse* = *nsh* and, therefore, *ns* = 0.

where Li2 is the polylogarithm function of second order (Abramowitz & Stegun, 1964). Inserting this relation into (2) provides the relation indicateing the required biasing voltage for a desired chemical potential. If *μ<sup>c</sup>* � *kBT*, that is for highly doped graphene, Equation 8

> *ns* <sup>=</sup> *<sup>μ</sup>*<sup>2</sup> *c πh*¯ <sup>2</sup>*v*<sup>2</sup> *F*

The essentially 2D structure of graphene makes surface conductivity the most natural appropriate quantity to model its electrical properties. When graphene is biased with a static magnetic field **B**<sup>0</sup> perpendicular to its plane, conductivity takes a tensorial form, which may be deduced by considering the motion of an electron under an electric field **E** in the graphene plane, as illustrated in Fig. 5. For simplicity, the electron is considered to be initially at rest. Let us first examine the case **E** = *Ex***x**ˆ. The electric field exerts a force **F***<sup>e</sup>* = −*e***x**ˆ on the electron, which accelerates it in the −*x* direction. As long as the electron acquires a non-zero velocity, a magnetic force **F***<sup>m</sup>* = −*e***v** × **B**<sup>0</sup> along the −*y* direction is exerted on it and deflects it towards the −*y* direction. The motion of the electron is, therefore, a combination of two simpler motions, one along the −*x* axis and one along the −*y* axis, and the electric current has two components, one along the +*x* direction and one along the +*y* direction4. For small

<sup>4</sup> The current is opposite to the electron's velocity, due to the negative charge of the electron.

*πh*¯ <sup>2</sup>*v*<sup>2</sup> *F*

<sup>1</sup> <sup>+</sup> *<sup>e</sup>*(*E*−*μc*)/(*kBT*) , (5)

*fd*(*E*)*N*(*E*)*dE* (6a)

(7)

, (8)

. (9)

<sup>−</sup><sup>∞</sup> [<sup>1</sup> <sup>−</sup> *fd*(*E*)] *<sup>N</sup>*(*E*)*dE*, (6b)

in the energy levels of any material according to the Fermi-Dirac distribution

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

*nse* =

*nsh* =

 ∞ 0

0

Inserting Equation 5 into Equations 6 and then taking their difference yields

 Li2 −*e* <sup>−</sup> *<sup>μ</sup><sup>c</sup> kBT* − Li2 −*e μc kBT*

*ns* <sup>=</sup> <sup>2</sup>(*kBT*)<sup>2</sup> *πh*¯ <sup>2</sup>*v*<sup>2</sup> *F*

**3. Conductivity of magnetically biased graphene**

then

where

takes the asymptotic form

*fd*(*E*) = <sup>1</sup>

half-integer quantum Hall effect and the Klein paradox, normally only encountered in high energy relativistic particles.

Another important consequence of the graphene band structure, and more specifically of the double valley degeneracy associated with the K and K' point, is the extremely low phonon scattering of electrons. This induces an upper limit of 200, 000 cm2/Vs in the graphene mobility at room temperature and more than 1, 000, 000 cm2/Vs at low temperature! However, the practical mobility is much lower (it can be so low as 1, 000 cm2/Vs), due to various defects in the graphene lattice, such as impurities and wrinkles, and due to scattering from the substrate. Except for phonon scattering, which is related to the nature of the material and therefore imposes an upper bound on mobility, the scattering factors mainly depend on the production process. Therefore, improved production processes are expected to significantly increase the mobility to much higher values.

Biasing graphene with a static magnetic field **B**<sup>0</sup> perpendicular to its plane results in quantization of the band diagram, as illustrated in Fig. 3. This phenomenon is called Landau quantization and it exists in all semiconductors. However, contrary to semiconductors with a parabolic band dispersion, the energy levels (Landau levels) in graphene are non-uniformly spaced, as a result of the linear dispersion close to the *K* and *K*� points. Furthermore, the 0th order Landau level has a zero energy, irrespectively of the applied magnetic field, which is the reason for the observation of the quantum Hall effect in graphene even at room temperature. The energy of the *n*th Landau level in graphene is

$$E\_{\rm nl} = \sqrt{n}L\_{\prime} \tag{4}$$

where *L* = 2¯*heB*0*v*<sup>2</sup> *<sup>F</sup>* is the Landau energy scale.

*E*

*EN*<sup>+</sup><sup>1</sup>

Fig. 3. Energy-wavenumber dispersion diagram in graphene, biased with a static magnetic field perpendicular to its plane. The dispersion diagram is quantized in the discrete energy levels *E*0, *E*±1, . . ..

Before closing this introductory section, let us provide the relation between the carrier density and the chemical potential, since the latter enters in all the conductivity expression that will be given in the next section. It is known from solid state physics that electrons are distributed in the energy levels of any material according to the Fermi-Dirac distribution

$$f\_d(E) = \frac{1}{1 + e^{(E - \mu\_c)/(k\_B T)}},\tag{5}$$

which represents the probability of finding an electron at the energy level *E*. In Equation 5 *μc* is the chemical potential, *kB* is the Boltzmann constant and *T* the temperature in *K*. Fig. 4 plots *fd*(*E*) for *μ<sup>c</sup>* > 0, that is a partially filled conduction band. The electron and hole densities are then

$$m\_{\mathfrak{sl}} = \int\_0^\infty f\_d(E) N(E) dE \tag{6a}$$

$$m\_{\rm sh} = \int\_{-\infty}^{0} \left[1 - f\_d(E)\right] N(E) dE\_\prime \tag{6b}$$

where

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

half-integer quantum Hall effect and the Klein paradox, normally only encountered in high

Another important consequence of the graphene band structure, and more specifically of the double valley degeneracy associated with the K and K' point, is the extremely low phonon scattering of electrons. This induces an upper limit of 200, 000 cm2/Vs in the graphene mobility at room temperature and more than 1, 000, 000 cm2/Vs at low temperature! However, the practical mobility is much lower (it can be so low as 1, 000 cm2/Vs), due to various defects in the graphene lattice, such as impurities and wrinkles, and due to scattering from the substrate. Except for phonon scattering, which is related to the nature of the material and therefore imposes an upper bound on mobility, the scattering factors mainly depend on the production process. Therefore, improved production processes are expected to significantly

Biasing graphene with a static magnetic field **B**<sup>0</sup> perpendicular to its plane results in quantization of the band diagram, as illustrated in Fig. 3. This phenomenon is called Landau quantization and it exists in all semiconductors. However, contrary to semiconductors with a parabolic band dispersion, the energy levels (Landau levels) in graphene are non-uniformly spaced, as a result of the linear dispersion close to the *K* and *K*� points. Furthermore, the 0th order Landau level has a zero energy, irrespectively of the applied magnetic field, which is the reason for the observation of the quantum Hall effect in graphene even at room temperature.

*En* <sup>=</sup> <sup>√</sup>*nL*, (4)

*E*1,..., *EN*−<sup>2</sup>

Fig. 3. Energy-wavenumber dispersion diagram in graphene, biased with a static magnetic field perpendicular to its plane. The dispersion diagram is quantized in the discrete energy

Before closing this introductory section, let us provide the relation between the carrier density and the chemical potential, since the latter enters in all the conductivity expression that will

−*E*1,..., −*EN*−<sup>2</sup>

*k*

energy relativistic particles.

where *L* =

levels *E*0, *E*±1, . . ..

2¯*heB*0*v*<sup>2</sup>

increase the mobility to much higher values.

The energy of the *n*th Landau level in graphene is

*<sup>F</sup>* is the Landau energy scale.

*E*

*E*0

−*EN*−<sup>1</sup> −*EN* −*EN*<sup>+</sup><sup>1</sup>

*EN*−<sup>1</sup> *EN EN*<sup>+</sup><sup>1</sup>

$$N(E) = \frac{2E}{\pi \hbar^2 v\_F^2} \tag{7}$$

is the density of states. The density of states is the same for electrons and holes in graphene, due to the band-structure symmetry around the *E* = 0 plane. The effective total carrier density is then *ns* = *nse* − *nsh*. Note that in intrinsic graphene *nse* = *nsh* and, therefore, *ns* = 0. Inserting Equation 5 into Equations 6 and then taking their difference yields

$$n\_s = \frac{2(k\_B T)^2}{\pi \hbar^2 v\_F^2} \left[ \text{Li}\_2 \left( -e^{-\frac{\mu\_c}{k\_B T}} \right) - \text{Li}\_2 \left( -e^{\frac{\mu\_c}{k\_B T}} \right) \right],\tag{8}$$

where Li2 is the polylogarithm function of second order (Abramowitz & Stegun, 1964). Inserting this relation into (2) provides the relation indicateing the required biasing voltage for a desired chemical potential. If *μ<sup>c</sup>* � *kBT*, that is for highly doped graphene, Equation 8 takes the asymptotic form

$$n\_s = \frac{\mu\_c^2}{\pi \hbar^2 v\_F^2}.\tag{9}$$

#### **3. Conductivity of magnetically biased graphene**

The essentially 2D structure of graphene makes surface conductivity the most natural appropriate quantity to model its electrical properties. When graphene is biased with a static magnetic field **B**<sup>0</sup> perpendicular to its plane, conductivity takes a tensorial form, which may be deduced by considering the motion of an electron under an electric field **E** in the graphene plane, as illustrated in Fig. 5. For simplicity, the electron is considered to be initially at rest. Let us first examine the case **E** = *Ex***x**ˆ. The electric field exerts a force **F***<sup>e</sup>* = −*e***x**ˆ on the electron, which accelerates it in the −*x* direction. As long as the electron acquires a non-zero velocity, a magnetic force **F***<sup>m</sup>* = −*e***v** × **B**<sup>0</sup> along the −*y* direction is exerted on it and deflects it towards the −*y* direction. The motion of the electron is, therefore, a combination of two simpler motions, one along the −*x* axis and one along the −*y* axis, and the electric current has two components, one along the +*x* direction and one along the +*y* direction4. For small

<sup>4</sup> The current is opposite to the electron's velocity, due to the negative charge of the electron.

where

is the conductivity tensor of graphene and

scattering rate Γ, *σxx* and *σyx* are given by

*<sup>σ</sup>xx*(*ω*, *<sup>B</sup>*0) = *<sup>e</sup>*2*v*<sup>2</sup>

*<sup>σ</sup>yx*(*ω*, *<sup>B</sup>*0) = <sup>−</sup> *<sup>e</sup>*2*v*<sup>2</sup>

×  *<sup>σ</sup>*¯¯ = *<sup>σ</sup>xx* ¯¯*It* + *<sup>σ</sup>yx* ¯¯*Jt* (13)

¯¯*It* = **<sup>x</sup>**ˆ**x**<sup>ˆ</sup> + **<sup>y</sup>**ˆ**y**ˆ, (14a) ¯¯*Jt* <sup>=</sup> **<sup>y</sup>**<sup>ˆ</sup> **<sup>x</sup>**<sup>ˆ</sup> <sup>−</sup> **<sup>x</sup>**ˆ**y**ˆ. (14b)

Closed-form expressions for *σxx* and *σyx* have been obtained in (Gusynin et al., 2009) through a quantum mechanical analysis which involves the Kubo formula (Kubo, 1957). Assuming an energy independent scattering mechanism, which is equivalent to considering an average

<sup>9</sup> Novel Electromagnetic Phenomena in Graphene

∞ ∑ *n*=0

<sup>×</sup> *fd*(*En*) <sup>−</sup> *fd*(*En*+1) + *fd*(−*En*+1) <sup>−</sup> *fd*(−*En*) (*En*+<sup>1</sup> <sup>−</sup> *En*)<sup>2</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>(*<sup>ω</sup>* <sup>−</sup> *<sup>j</sup>*2Γ)<sup>2</sup>

(*En*+<sup>1</sup> <sup>−</sup> *En*)<sup>2</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>(*<sup>ω</sup>* <sup>−</sup> *<sup>j</sup>*2Γ)<sup>2</sup> + (*En* → −*En*)

A physical explanation of Equation 15 may be provided with the help of Fig. 6. Each of the summation terms in Equation 15 corresponds to an electron transition between two Landau levels. Two types of transitions exist: intraband, between levels in the same band, and interband between levels of different bands. From all the possible intraband transitions, the only ones allowed by the selection rules are these from level *n* to *n* + 1 in the conduction band and from level −*n* − 1 to −*n* in the valence band. Similarly, the only allowed interband transitions are from level −*n* to *n* + 1 and from −*n* − 1 to *n*. The photon energy needed for a transition between the levels *E*initial and *E*final is *E*final − *E*initial. A photon with this energy corresponds to an electromagnetic wave of frequency (*E*final − *E*initial)/¯*h*. The probability of such a transition is proportional to the difference between the probability *fd*(*E*initial) of the initial level being full and the probability *fd*(*E*final) of the final level being empty. The transitions with the highest probabilities are those crossing *μc*, across which the largest difference in *fd*(*E*) exists, emphasized by thicker arrows in Fig. 6. Therefore, assuming a finite temperature and that *μ<sup>c</sup>* lies between the levels *EN* and *EN*+1, the interband transitions with the lowest energy are the ones that involve the −*N* and *N* + 1 levels or the −*N* − 1 and *N* levels. As a result, interband transitions occur essentially at frequencies ¯*hω*inter ≥ *EN* + *EN*+1. Assume now that *μ<sup>c</sup>* � *L*. For the magnetic field value of 1 T, used in the chapter, *L* = 0.036 eV, and the condition *μ<sup>c</sup>* � *L* is thus largely satisfied if *μ<sup>c</sup>* > 1 eV. Then *EN*+<sup>1</sup> > *μ<sup>c</sup>* � *L*, which through Equation 4 yields *N* � 1 and *EN* ≈ *EN*+<sup>1</sup> ≈ *μ<sup>c</sup>* (due to the <sup>√</sup>*<sup>n</sup>* compression factor in *En*), resulting in ¯*hω*inter <sup>≥</sup> <sup>2</sup>*μc*. In most of practical situations, *μ<sup>c</sup>* ≥ 0.05 eV, so that ¯*hω*inter ≥ 0.15 PHz. Therefore, all the results of the chapter, which pertain to microwave and millimeter-wave operation, are derived by considering only the

1

*En*+<sup>1</sup> − *En*

+(*En* → −*En*)} , (15a)

[ *fd*(*En*) − *fd*(*En*+1) − *fd*(−*En*+1) + *fd*(−*En*)]

. (15b)

*<sup>F</sup>*|*eB*0|*h*¯(*ω* − *j*2Γ) −*jπ*

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

∞ ∑ *n*=0

1

*<sup>F</sup>eB*<sup>0</sup> *π*

Fig. 4. Fermi-Dirac distribution for *μ<sup>c</sup>* > 0. The red area corresponds to free electrons and the blue to free holes (electrons and holes that can conduct).

**E**, when nonlinear effects are negligible, the current can thus be related to the electric field through

$$\mathbf{J} = \sigma\_{\mathbf{xx}} E\_{\mathbf{x}} \mathbf{\hat{x}} + \sigma\_{\mathbf{yx}} E\_{\mathbf{x}} \mathbf{\hat{y}}.\tag{10}$$

The proportionality coefficients *σxx* and *σxy* correspond to the longitudinal (parallel to **E**) and transverse (perpendicular to **E**) conductivities, respectively, which depend on the band structure of the material, the frequency of **E** and **B**0. Following a similar analysis for **E** = *Ey***y**ˆ, which is the case depicted in Fig. 5(b), and assuming that graphene exhibits the same properties in all its directions, we derive

$$\mathbf{J} = -\sigma\_{yx} E\_{\mathbf{y}} \mathbf{\hat{x}} + \sigma\_{\mathbf{xx}} E\_{\mathbf{x}} \mathbf{\hat{y}}.\tag{11}$$

Equations 10 and 11 can be combined into the single equation

Fig. 5. Motion of an electron on graphene in an electric field and a static magnetic field. The electron, initially accelerated in the direction of the electric field, is deflected in the direction perpendicular to this field, due to the static magnetic field. Two electric current components, parallel and perpendicular to the electric field, are subsequently generated. (a) Case of the electric field along the *x*-axis. (b) Case of the electric field along the *y*-axis.

where

6 Will-be-set-by-IN-TECH

electrons

blue to free holes (electrons and holes that can conduct).

same properties in all its directions, we derive

Equations 10 and 11 can be combined into the single equation

**E**

**F***m*

through

*μc*

*E*

0 0.5 1

Fig. 4. Fermi-Dirac distribution for *μ<sup>c</sup>* > 0. The red area corresponds to free electrons and the

**E**, when nonlinear effects are negligible, the current can thus be related to the electric field

The proportionality coefficients *σxx* and *σxy* correspond to the longitudinal (parallel to **E**) and transverse (perpendicular to **E**) conductivities, respectively, which depend on the band structure of the material, the frequency of **E** and **B**0. Following a similar analysis for **E** = *Ey***y**ˆ, which is the case depicted in Fig. 5(b), and assuming that graphene exhibits the

**<sup>F</sup>***<sup>e</sup>* **<sup>F</sup>***<sup>m</sup>*

*y*

*z* <sup>−</sup> <sup>−</sup>*<sup>e</sup> <sup>e</sup>*

(a) (b)

Fig. 5. Motion of an electron on graphene in an electric field and a static magnetic field. The electron, initially accelerated in the direction of the electric field, is deflected in the direction perpendicular to this field, due to the static magnetic field. Two electric current components, parallel and perpendicular to the electric field, are subsequently generated. (a) Case of the

*x*

*Jx*

*Jy*

electric field along the *x*-axis. (b) Case of the electric field along the *y*-axis.

holes *fd*(*E*)

**J** = *σxxEx***x**ˆ + *σyxEx***y**ˆ. (10)

**J** = −*σyxEy***x**ˆ + *σxxEx***y**ˆ. (11)

**<sup>E</sup> <sup>F</sup>***<sup>e</sup>*

**<sup>B</sup> <sup>B</sup>**<sup>0</sup> <sup>0</sup>

*Jx*

**<sup>J</sup>** <sup>=</sup> *<sup>σ</sup>*¯¯ · **<sup>E</sup>**, (12)

*Jy*

$$
\tilde{\sigma} = \sigma\_{\text{xx}} \bar{\mathbf{I}}\_{\text{t}} + \sigma\_{\text{yx}} \bar{\mathbf{I}}\_{\text{t}} \tag{13}
$$

is the conductivity tensor of graphene and

$$
\vec{I}\_l = \mathbf{\hat{x}}\hat{\mathbf{x}} + \mathbf{\hat{y}}\hat{\mathbf{y}}\_l \tag{14a}
$$

$$
\bar{\bar{f}}\_t = \mathbf{\hat{y}}\hat{\mathbf{x}} - \mathbf{\hat{x}}\hat{\mathbf{y}}.\tag{14b}
$$

Closed-form expressions for *σxx* and *σyx* have been obtained in (Gusynin et al., 2009) through a quantum mechanical analysis which involves the Kubo formula (Kubo, 1957). Assuming an energy independent scattering mechanism, which is equivalent to considering an average scattering rate Γ, *σxx* and *σyx* are given by

$$\begin{split} \sigma\_{\text{xx}}(\omega, B\_0) &= \frac{\sigma^2 v\_F^2 |eB\_0| \hbar (\omega - j2\Gamma)}{-j\pi} \sum\_{n=0}^{\infty} \left\{ \frac{1}{E\_{n+1} - E\_n} \right. \\ &\times \frac{f\_d(E\_n) - f\_d(E\_{n+1}) + f\_d(-E\_{n+1}) - f\_d(-E\_n)}{(E\_{n+1} - E\_n)^2 - \hbar^2 (\omega - j2\Gamma)^2} \\ &+ (E\_n \to -E\_n) \} . \end{split} \tag{15a}$$

$$
\sigma\_{\rm yX}(\omega, B\_0) = -\frac{e^2 \upsilon\_F^2 e B\_0}{\pi} \sum\_{n=0}^{\infty} \left[ f\_d(E\_n) - f\_d(E\_{n+1}) - f\_d(-E\_{n+1}) + f\_d(-E\_n) \right]
$$

$$
\times \left[ \frac{1}{(E\_{n+1} - E\_n)^2 - \hbar^2(\omega - j2\Gamma)^2} + (E\_n \to -E\_n) \right]. \tag{15b}
$$

A physical explanation of Equation 15 may be provided with the help of Fig. 6. Each of the summation terms in Equation 15 corresponds to an electron transition between two Landau levels. Two types of transitions exist: intraband, between levels in the same band, and interband between levels of different bands. From all the possible intraband transitions, the only ones allowed by the selection rules are these from level *n* to *n* + 1 in the conduction band and from level −*n* − 1 to −*n* in the valence band. Similarly, the only allowed interband transitions are from level −*n* to *n* + 1 and from −*n* − 1 to *n*. The photon energy needed for a transition between the levels *E*initial and *E*final is *E*final − *E*initial. A photon with this energy corresponds to an electromagnetic wave of frequency (*E*final − *E*initial)/¯*h*. The probability of such a transition is proportional to the difference between the probability *fd*(*E*initial) of the initial level being full and the probability *fd*(*E*final) of the final level being empty. The transitions with the highest probabilities are those crossing *μc*, across which the largest difference in *fd*(*E*) exists, emphasized by thicker arrows in Fig. 6. Therefore, assuming a finite temperature and that *μ<sup>c</sup>* lies between the levels *EN* and *EN*+1, the interband transitions with the lowest energy are the ones that involve the −*N* and *N* + 1 levels or the −*N* − 1 and *N* levels. As a result, interband transitions occur essentially at frequencies ¯*hω*inter ≥ *EN* + *EN*+1.

Assume now that *μ<sup>c</sup>* � *L*. For the magnetic field value of 1 T, used in the chapter, *L* = 0.036 eV, and the condition *μ<sup>c</sup>* � *L* is thus largely satisfied if *μ<sup>c</sup>* > 1 eV. Then *EN*+<sup>1</sup> > *μ<sup>c</sup>* � *L*, which through Equation 4 yields *N* � 1 and *EN* ≈ *EN*+<sup>1</sup> ≈ *μ<sup>c</sup>* (due to the <sup>√</sup>*<sup>n</sup>* compression factor in *En*), resulting in ¯*hω*inter <sup>≥</sup> <sup>2</sup>*μc*. In most of practical situations, *μ<sup>c</sup>* ≥ 0.05 eV, so that ¯*hω*inter ≥ 0.15 PHz. Therefore, all the results of the chapter, which pertain to microwave and millimeter-wave operation, are derived by considering only the

<sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup> <sup>104</sup> -4

Im{*σxx*}

Im{*σyx*}

*ω<sup>c</sup>* 2*π*

Re{*σxx*}

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

<sup>11</sup> Novel Electromagnetic Phenomena in Graphene

Re{*σyx*}

Drude model Kubo formula

frequency (GHz)

Fig. 7. Graphene conductivity versus frequency computed from the exact expressions 15

When a plasma, such as a semiconductor, is biased with a static magnetic field it exhibits Faraday rotation at frequencies above the cyclotron frequency. The reason why Faraday rotation is only observed above the cyclotron resonance is that below the resonance the plasma is impenetrable. Graphene, being a semiconductor, is also expected to exhibit Faraday rotation (Crassee et al., 2011; Sounas & Caloz, 2011b). However, contrary to semiconductors, Faraday rotation in graphene exists also below the cyclotron resonance, as a result of its super-thin, semi-transparent nature. This section theoretically analyzes Faraday rotation in graphene. The results will be used in the next section where possible applications of this phenomenon

Consider a graphene sheet coinciding with the *z* = 0 plane in free space and biased with a static magnetic field **B**<sup>0</sup> = *B*0**z**ˆ and a plane wave impinging normally on the sheet from medium 1 towards medium 2, as illustrated in Fig. 8. The electric field of the incident, reflected

<sup>0</sup> *<sup>e</sup>*−*jk*0*<sup>z</sup>*

*<sup>z</sup>*=<sup>0</sup> <sup>=</sup> *<sup>σ</sup>*¯¯ · **<sup>E</sup>**tran

, (19a)

<sup>0</sup> *<sup>e</sup>jk*0*z*, (19b)

<sup>0</sup> *<sup>e</sup>*−*jk*0*z*. (19c)

*<sup>z</sup>*=<sup>0</sup> = **0**, (20a)

*<sup>z</sup>*=0, (20b)

**E**inc = **E**inc

**E**ref = **E**ref

**E**tran = **E**tran

**<sup>E</sup>**tran <sup>−</sup> **<sup>E</sup>**inc <sup>−</sup> **<sup>E</sup>**ref

**<sup>H</sup>**tran <sup>−</sup> **<sup>H</sup>**inc <sup>−</sup> **<sup>H</sup>**ref

<sup>0</sup> , we will use the boundary conditions

(circles) and from the approximate expressions 16 (lines), for *μ* = 5000 cm2/Vs,


0

2

conductivity

*ns* = 1013 cm−2, *B*<sup>0</sup> = 1 T and *T* = 300 K.

**4. Faraday rotation in graphene**

and transmitted waves then read

<sup>0</sup> and **<sup>E</sup>**tran

**z**ˆ × 

**z**ˆ × 

will be suggested.

In order to find **E**ref

 (mS)

4

6

8

Fig. 6. Electron transitions in graphene when illuminated by an electromagnetic wave of frequency *ω*. The red arrows represent the allowed intraband (right-hand side) and interband (left-hand side) electron transitions between the energy levels. A photon can be absorbed by graphene if its energy ¯*hω* coincides with the energy of any of the allowed electron transitions. The probability of an electron transition depends on the probability of the initial level to be full and the probability of the final level to be empty. The probability of a level with energy *E* to be full is given by the Fermi-Dirac distribution *fd*(*E*) (orange curve on the right).

intraband terms in Equation 15. Under the condition *N* � 1, the Landau levels around *μc*, which produce non-negligible intraband transitions, lie very close to each other. It can be shown (Kubo, 1957) that in this case *σxx* and *σyx* follow the Drude model form

$$
\sigma\_{\rm xx}(\omega, \theta\_0) = \sigma\_0 \frac{1 + j\omega \tau}{(\omega\_c \tau)^2 + (1 + j\omega \tau)^2} \tag{16a}
$$

$$
\sigma\_{yx}(\omega\_\prime B\_0) = \sigma\_0 \frac{\omega\_\varepsilon \tau}{(\omega\_\varepsilon \tau)^2 + (1 + j\omega \tau)^2} \tag{16b}
$$

where

$$
\sigma\_0 = \frac{2e^2 \tau}{\pi \hbar^2} k\_B T \ln\left(2 \cosh\frac{\mu\_c}{2k\_B T}\right) \tag{17}
$$

is the DC conductivity of graphene, *τ* = 1/(2Γ) is the scattering time and *ω<sup>c</sup>* is the cyclotron frequency. The latter corresponds to the difference between the levels *EN* and *EN*+1,

$$
\omega\_c = \frac{E\_{N+1} - E\_N}{\hbar} = \frac{L}{\hbar \left(\sqrt{N+1} + \sqrt{N}\right)} \approx \frac{L^2}{2\hbar\mu\_c} = \frac{eB\_0 v\_F^2}{\mu\_c}.\tag{18}
$$

For the *B*<sup>0</sup> values used in the chapter, *μ<sup>c</sup>* � *L* also implies *μ<sup>c</sup>* � *kBT* (*kBT* = 0.026 eV at 300 K), and Equation 17 subsequently simplifies to *σ*<sup>0</sup> = *nseμ*, where *ns* = *μ*<sup>2</sup> *<sup>c</sup>*/*πh*¯ <sup>2</sup>*v*<sup>2</sup> *<sup>F</sup>* is the carrier density and *μ* = *eτv*<sup>2</sup> *<sup>F</sup>*/*μ<sup>c</sup>* the mobility. Figure 7 plots *σxx* and *σyx* versus frequency using both Equation 15 and 16, for *B*<sup>0</sup> = 1 T, *μ* = 5000 cm2/Vs, *ns* = 1013 cm−<sup>2</sup> and *T* = 300 K, corresponding to values widely used throughout the chapter. Excellent agreement between the two models is observed in the frequency range shown.

Fig. 7. Graphene conductivity versus frequency computed from the exact expressions 15 (circles) and from the approximate expressions 16 (lines), for *μ* = 5000 cm2/Vs, *ns* = 1013 cm−2, *B*<sup>0</sup> = 1 T and *T* = 300 K.

#### **4. Faraday rotation in graphene**

8 Will-be-set-by-IN-TECH

*E*1,..., *EN*−<sup>2</sup>

Fig. 6. Electron transitions in graphene when illuminated by an electromagnetic wave of frequency *ω*. The red arrows represent the allowed intraband (right-hand side) and interband (left-hand side) electron transitions between the energy levels. A photon can be absorbed by graphene if its energy ¯*hω* coincides with the energy of any of the allowed electron transitions. The probability of an electron transition depends on the probability of the initial level to be full and the probability of the final level to be empty. The probability of a level with energy *E* to be full is given by the Fermi-Dirac distribution *fd*(*E*) (orange curve

intraband terms in Equation 15. Under the condition *N* � 1, the Landau levels around *μc*, which produce non-negligible intraband transitions, lie very close to each other. It can be

is the DC conductivity of graphene, *τ* = 1/(2Γ) is the scattering time and *ω<sup>c</sup>* is the cyclotron

<sup>√</sup>*<sup>N</sup>* <sup>+</sup> <sup>1</sup> <sup>+</sup> <sup>√</sup>*<sup>N</sup>*

For the *B*<sup>0</sup> values used in the chapter, *μ<sup>c</sup>* � *L* also implies *μ<sup>c</sup>* � *kBT* (*kBT* = 0.026 eV at 300 K),

both Equation 15 and 16, for *B*<sup>0</sup> = 1 T, *μ* = 5000 cm2/Vs, *ns* = 1013 cm−<sup>2</sup> and *T* = 300 K, corresponding to values widely used throughout the chapter. Excellent agreement between

1 + *jωτ*

*ωcτ*

2 cosh *<sup>μ</sup><sup>c</sup>*

2*kBT*

 <sup>≈</sup> *<sup>L</sup>*<sup>2</sup> 2¯*hμc*

*<sup>F</sup>*/*μ<sup>c</sup>* the mobility. Figure 7 plots *σxx* and *σyx* versus frequency using

(*ωcτ*)<sup>2</sup> + (<sup>1</sup> <sup>+</sup> *<sup>j</sup>ωτ*)<sup>2</sup> , (16a)

(*ωcτ*)<sup>2</sup> + (<sup>1</sup> <sup>+</sup> *<sup>j</sup>ωτ*)<sup>2</sup> , (16b)

<sup>=</sup> *eB*0*v*<sup>2</sup> *F μc*

*<sup>c</sup>*/*πh*¯ <sup>2</sup>*v*<sup>2</sup>

(17)

. (18)

*<sup>F</sup>* is the carrier

shown (Kubo, 1957) that in this case *σxx* and *σyx* follow the Drude model form

*σxx*(*ω*, *B*0) = *σ*<sup>0</sup>

*σyx*(*ω*, *B*0) = *σ*<sup>0</sup>

*<sup>σ</sup>*<sup>0</sup> <sup>=</sup> <sup>2</sup>*e*2*<sup>τ</sup>*

*<sup>ω</sup><sup>c</sup>* <sup>=</sup> *EN*+<sup>1</sup> <sup>−</sup> *EN*

the two models is observed in the frequency range shown.

*<sup>π</sup>h*¯ <sup>2</sup> *kBT* ln

frequency. The latter corresponds to the difference between the levels *EN* and *EN*+1,

*<sup>h</sup>*¯ <sup>=</sup> *<sup>L</sup> h*¯

and Equation 17 subsequently simplifies to *σ*<sup>0</sup> = *nseμ*, where *ns* = *μ*<sup>2</sup>

−*E*1,..., −*EN*−<sup>2</sup>

*μc*

*h*¯ *ω*

*<sup>k</sup> fd*(*E*)

*E E*

*EN EN*+<sup>1</sup>

*E*<sup>0</sup> = 0

−*EN* −*EN*+<sup>1</sup>

on the right).

where

density and *μ* = *eτv*<sup>2</sup>

When a plasma, such as a semiconductor, is biased with a static magnetic field it exhibits Faraday rotation at frequencies above the cyclotron frequency. The reason why Faraday rotation is only observed above the cyclotron resonance is that below the resonance the plasma is impenetrable. Graphene, being a semiconductor, is also expected to exhibit Faraday rotation (Crassee et al., 2011; Sounas & Caloz, 2011b). However, contrary to semiconductors, Faraday rotation in graphene exists also below the cyclotron resonance, as a result of its super-thin, semi-transparent nature. This section theoretically analyzes Faraday rotation in graphene. The results will be used in the next section where possible applications of this phenomenon will be suggested.

Consider a graphene sheet coinciding with the *z* = 0 plane in free space and biased with a static magnetic field **B**<sup>0</sup> = *B*0**z**ˆ and a plane wave impinging normally on the sheet from medium 1 towards medium 2, as illustrated in Fig. 8. The electric field of the incident, reflected and transmitted waves then read

$$\mathbf{E}^{\text{inc}} = \mathbf{E}\_0^{\text{inc}} e^{-jk\_0 z},\tag{19a}$$

$$\mathbf{E}^{\text{ref}} = \mathbf{E}\_0^{\text{ref}} e^{j k\_0 z} \,\tag{19b}$$

$$\mathbf{E}^{\text{tran}} = \mathbf{E}\_0^{\text{tran}} \mathbf{e}^{-jk\_0 z}. \tag{19c}$$

In order to find **E**ref <sup>0</sup> and **<sup>E</sup>**tran <sup>0</sup> , we will use the boundary conditions

$$\left. \mathbf{\hat{z}} \times \left( \mathbf{E^{tran}} - \mathbf{E^{inc}} - \mathbf{E^{ref}} \right) \right|\_{z=0} = \mathbf{0},\tag{20a}$$

$$\left. \dot{\mathbf{z}} \times \left( \mathbf{H}^{\text{tran}} - \mathbf{H}^{\text{inc}} - \mathbf{H}^{\text{ref}} \right) \right|\_{z=0} = \left. \vec{\sigma} \cdot \mathbf{E}^{\text{tran}} \right|\_{z=0'} \tag{20b}$$

**E***<sup>i</sup>* **E***<sup>r</sup>*

Fig. 8. Magnetically biased graphene sheet in the *z* = 0 plane. A plane wave impinges

Similarly, the transmission coefficient for a right-handed (RH) circularly polarized wave is

*<sup>T</sup>*RH <sup>=</sup> <sup>2</sup>

where *σRH* = *σxx* + *jσyx*. Equations 29 and 30 reveal that *T*LH � *T*RH, a phenomenon known as circular birefringence, with Faraday rotation being one of its most important consequences. Fig. 9 plots the amplitude and phase of *T*LH and *T*RH versus frequency for a graphene sheet with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>μ</sup><sup>c</sup>* <sup>=</sup> 0.37 eV and *<sup>B</sup>*<sup>0</sup> <sup>=</sup> 1 T. Two things can be observed: a) the amplitudes of *T*LH and *T*RH are almost the same far from the cyclotron resonance and b) there is a huge phase difference between *T*LH and *T*RH below the cyclotron resonance. These two facts result in a large Faraday rotation below the cyclotron resonance. To prove this statement,

(**x**<sup>ˆ</sup> <sup>+</sup> *<sup>j</sup>***y**ˆ) + <sup>1</sup>

<sup>2</sup> (**x**<sup>ˆ</sup> <sup>+</sup> *<sup>j</sup>***y**ˆ) + *<sup>T</sup>*RH

If |*T*LH|≈|*T*RH|, as in the case far below and above from the cyclotron resonance, Equation 32

**<sup>x</sup>**<sup>ˆ</sup> cos <sup>Δ</sup>*<sup>ϕ</sup>* 2 

where *ϕ*LH = arg{*T*LH}, *ϕ*RH = arg{*T*RH}, *ϕ*av = (*ϕ*LH + *ϕ*RH)/2 and Δ*ϕ* = *ϕ*LH − *ϕ*RH.

*<sup>θ</sup>* <sup>=</sup> <sup>Δ</sup>*<sup>ϕ</sup>*

2

*<sup>e</sup>jϕ*LH (**x**<sup>ˆ</sup> <sup>+</sup> *<sup>j</sup>***y**ˆ) + *<sup>e</sup>jϕ*RH (**x**<sup>ˆ</sup> <sup>−</sup> *<sup>j</sup>***y**ˆ)

2 + *η*0*σ*RH

normally on graphene.

consider an *x*-polarized incident wave **E**inc

The transmitted wave is then

becomes

circularly polarized wave and a LH circularly polarized wave as

**<sup>E</sup>**inc <sup>=</sup> <sup>1</sup> 2

**<sup>E</sup>**tran <sup>=</sup> *<sup>T</sup>*LH

<sup>2</sup> *<sup>e</sup>jϕ*av

**<sup>E</sup>**tran <sup>=</sup> <sup>|</sup>*T*LH<sup>|</sup> 2

Since Δ*ϕ* > 0 (Fig. 9) the field has been rotated by an angle

<sup>=</sup> <sup>|</sup>*T*LH<sup>|</sup>

**B**0

*x*

<sup>0</sup> = **x**ˆ. Such a wave can be decomposed into a RH

2

<sup>2</sup> (34)

<sup>−</sup> **<sup>y</sup>**<sup>ˆ</sup> sin <sup>Δ</sup>*<sup>ϕ</sup>*

(**x**ˆ − *j***y**ˆ). (31)

<sup>2</sup> (**x**<sup>ˆ</sup> <sup>−</sup> *<sup>j</sup>***y**ˆ). (32)

, (33)

, (30)

*y z*

**E***t*

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

<sup>13</sup> Novel Electromagnetic Phenomena in Graphene

*θ*

which hold in the graphene plane. The magnetic fields **H**inc, **H**ref and **H**tran are related to the corresponding electric fields through

$$\mathbf{H}^{\text{inc}} = \frac{\hat{\mathbf{z}} \times \mathbf{E}^{\text{inc}}}{\eta\_0},\tag{21a}$$

$$\mathbf{H}^{\text{ref}} = -\frac{\hat{\mathbf{z}} \times \mathbf{E}^{\text{inc}}}{\eta\_0},\tag{21b}$$

$$\mathbf{H}^{\text{tran}} = \frac{\mathbf{\hat{z}} \times \mathbf{E}^{\text{inc}}}{\eta\_0} \,\tag{21c}$$

where *η*<sup>0</sup> is the free-space wave impedance. Inserting Equation 19 into 20a yields

$$\mathbf{E}\_0^{\text{tran}} - \mathbf{E}\_0^{\text{inc}} - \mathbf{E}\_0^{\text{tran}} = \mathbf{0}.\tag{22}$$

Similarly, inserting Equation 21 into 20b, one gets

$$-\mathbf{E}\_0^{\text{tran}} + \mathbf{E}\_0^{\text{inc}} - \mathbf{E}\_0^{\text{ref}} = \eta\_0 \mathbf{\tilde{r}} \cdot \mathbf{E}\_0^{\text{tran}}.\tag{23}$$

Solving Equations 22 and 20b simultaneously with respect to **E**tran <sup>0</sup> gives

$$\mathbf{E}\_0^{\text{tran}} = 2\left(2\overline{I}\_t + \eta\_0 \overline{\sigma}\right)^{-1} \cdot \mathbf{E}\_0^{\text{inc}}.\tag{24}$$

Therefore, the transmission dyadic for incidence from medium 1 towards medium 2 is

$$\bar{T}\_{21} = 2\left(2\bar{I}\_t + \eta\_0 \bar{\sigma}\right)^{-1} = \frac{2}{(2 + \eta\_0 \sigma\_{xx})^2 + (\eta\_0 \sigma\_{yx})^2} \left[ (2 + \eta\_0 \sigma\_{xx})\bar{I}\_t - \eta\_0 \sigma\_{yx}\bar{\bar{l}}\_t \right],\tag{25}$$

where the dyadic identity (Lindell, 1996)

$$\left(a\vec{I}\_{l} + b\vec{f}\_{l}\right)^{-1} = \frac{1}{a^{2} + b^{2}} \left(a\vec{I}\_{l} - b\vec{f}\_{l}\right),\tag{26}$$

has been used. Through a similar analysis, it can be shown that the transmission dyadic for incidence from medium 2 towards medium 1 is

$$
\vec{T}\_{12} = \vec{T}\_{21\prime} \tag{27}
$$

a condition which, as it will be shown later, indicates non-reciprocity.

Consider now the specific case of a left-handed (LH) circularly polarized incident wave with respect to the *z*-axis **E**inc LH,0 = **x**ˆ + *j***y**ˆ. The transmitted wave is then, according to Equation 25,

$$\mathbf{E}\_{\rm LH,0}^{\rm tran} = \mathbf{\tilde{T}}\_{\rm 21} \cdot \mathbf{E}\_{\rm LH,0}^{\rm inc} = \frac{2}{2 + \eta\_0 \sigma\_{\rm LH}} \mathbf{E}\_{\rm LH,0}^{\rm inc} \tag{28}$$

where *σ*LH = *σxx* − *jσyx*. Therefore, in the case of a LH circularly polarized wave the transmission coefficient simplifies to

$$T\_{\rm LH} = \frac{2}{2 + \eta\_0 \sigma\_{\rm LH}}.\tag{29}$$

Fig. 8. Magnetically biased graphene sheet in the *z* = 0 plane. A plane wave impinges normally on graphene.

Similarly, the transmission coefficient for a right-handed (RH) circularly polarized wave is

$$T\_{\rm RH} = \frac{2}{2 + \eta\_0 \sigma\_{\rm RH}},\tag{30}$$

where *σRH* = *σxx* + *jσyx*. Equations 29 and 30 reveal that *T*LH � *T*RH, a phenomenon known as circular birefringence, with Faraday rotation being one of its most important consequences.

Fig. 9 plots the amplitude and phase of *T*LH and *T*RH versus frequency for a graphene sheet with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>μ</sup><sup>c</sup>* <sup>=</sup> 0.37 eV and *<sup>B</sup>*<sup>0</sup> <sup>=</sup> 1 T. Two things can be observed: a) the amplitudes of *T*LH and *T*RH are almost the same far from the cyclotron resonance and b) there is a huge phase difference between *T*LH and *T*RH below the cyclotron resonance. These two facts result in a large Faraday rotation below the cyclotron resonance. To prove this statement, consider an *x*-polarized incident wave **E**inc <sup>0</sup> = **x**ˆ. Such a wave can be decomposed into a RH circularly polarized wave and a LH circularly polarized wave as

$$\mathbf{E}^{\text{inc}} = \frac{1}{2}(\hat{\mathbf{x}} + j\hat{\mathbf{y}}) + \frac{1}{2}(\hat{\mathbf{x}} - j\hat{\mathbf{y}}).\tag{31}$$

The transmitted wave is then

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

which hold in the graphene plane. The magnetic fields **H**inc, **H**ref and **H**tran are related to the

**<sup>H</sup>**inc <sup>=</sup> **<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>**inc

**<sup>H</sup>**ref <sup>=</sup> <sup>−</sup> **<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>**inc

**<sup>H</sup>**tran <sup>=</sup> **<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>**inc

where *η*<sup>0</sup> is the free-space wave impedance. Inserting Equation 19 into 20a yields

**E**tran <sup>0</sup> <sup>−</sup> **<sup>E</sup>**inc

<sup>0</sup> <sup>+</sup> **<sup>E</sup>**inc

<sup>=</sup> <sup>2</sup>

−<sup>1</sup>

*T* ¯¯ <sup>12</sup> = *T* ¯¯

Therefore, the transmission dyadic for incidence from medium 1 towards medium 2 is

<sup>−</sup> **<sup>E</sup>**tran

Solving Equations 22 and 20b simultaneously with respect to **E**tran

**E**tran <sup>0</sup> = 2

Similarly, inserting Equation 21 into 20b, one gets

<sup>2</sup> ¯¯*It* + *<sup>η</sup>*0*σ*¯¯

incidence from medium 2 towards medium 1 is

where the dyadic identity (Lindell, 1996)

−<sup>1</sup>

**E**tran LH,0 = *T* ¯¯ <sup>21</sup> · **<sup>E</sup>**inc

*<sup>a</sup>* ¯¯*It* + *<sup>b</sup>* ¯¯*Jt*

a condition which, as it will be shown later, indicates non-reciprocity.

*T* ¯¯ <sup>21</sup> = 2 

respect to the *z*-axis **E**inc

transmission coefficient simplifies to

*η*0

*η*0

<sup>0</sup> <sup>−</sup> **<sup>E</sup>**tran

<sup>0</sup> <sup>=</sup> *<sup>η</sup>*0*σ*¯¯ · **<sup>E</sup>**tran

−<sup>1</sup>

· **<sup>E</sup>**inc

*<sup>a</sup>* ¯¯*It* <sup>−</sup> *<sup>b</sup>* ¯¯*Jt*

LH,0 = **x**ˆ + *j***y**ˆ. The transmitted wave is then, according to Equation 25,

**E**inc

2 + *η*0*σ*LH

<sup>0</sup> <sup>−</sup> **<sup>E</sup>**ref

<sup>2</sup> ¯¯*It* <sup>+</sup> *<sup>η</sup>*0*σ*¯¯

(2 + *η*0*σxx*)<sup>2</sup> + (*η*0*σyx*)<sup>2</sup>

<sup>=</sup> <sup>1</sup> *a*<sup>2</sup> + *b*<sup>2</sup>

has been used. Through a similar analysis, it can be shown that the transmission dyadic for

Consider now the specific case of a left-handed (LH) circularly polarized incident wave with

where *σ*LH = *σxx* − *jσyx*. Therefore, in the case of a LH circularly polarized wave the

*<sup>T</sup>*LH <sup>=</sup> <sup>2</sup>

LH,0 <sup>=</sup> <sup>2</sup>

2 + *η*0*σ*LH

*η*0

, (21a)

, (21b)

, (21c)

<sup>0</sup> = **0**. (22)

<sup>0</sup> gives

(<sup>2</sup> <sup>+</sup> *<sup>η</sup>*0*σxx*)¯¯*It* <sup>−</sup> *<sup>η</sup>*0*σyx* ¯¯*Jt*

21, (27)

<sup>0</sup> . (23)

<sup>0</sup> . (24)

, (26)

LH,0, (28)

. (29)

, (25)

corresponding electric fields through

$$\mathbf{E}^{\text{tran}} = \frac{T\_{\text{LH}}}{2} (\hat{\mathbf{x}} + j\hat{\mathbf{y}}) + \frac{T\_{\text{RH}}}{2} (\hat{\mathbf{x}} - j\hat{\mathbf{y}}).\tag{32}$$

If |*T*LH|≈|*T*RH|, as in the case far below and above from the cyclotron resonance, Equation 32 becomes

$$\begin{split} \mathbf{E}^{\text{tran}} &= \frac{|T\_{\text{LH}}|}{2} \left[ e^{j\varphi\_{\text{LH}}} (\hat{\mathbf{x}} + j\hat{\mathbf{y}}) + e^{j\varphi\_{\text{RH}}} (\hat{\mathbf{x}} - j\hat{\mathbf{y}}) \right] \\ &= \frac{|T\_{\text{LH}}|}{2} e^{j\varphi\_{\text{eV}}} \left[ \hat{\mathbf{x}} \cos\left(\frac{\Delta\varphi}{2}\right) - \hat{\mathbf{y}} \sin\left(\frac{\Delta\varphi}{2}\right) \right], \end{split} \tag{33}$$

where *ϕ*LH = arg{*T*LH}, *ϕ*RH = arg{*T*RH}, *ϕ*av = (*ϕ*LH + *ϕ*RH)/2 and Δ*ϕ* = *ϕ*LH − *ϕ*RH. Since Δ*ϕ* > 0 (Fig. 9) the field has been rotated by an angle

$$
\theta = \frac{\Delta \varphi}{2} \tag{34}
$$

*μc* (eV)

<sup>15</sup> Novel Electromagnetic Phenomena in Graphene

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

Fig. 10. Rotation angle (color plot) and transmission amplitude (contour plot) versus *μc* and *B*<sup>0</sup> for a normally incident plane wave on a graphene sheet in free space, for the parameters

an opposite charge than electrons and, therefore, provide an opposite transverse current than electrons, hence an opposite rotation direction, as illustrated in Fig. 11. This property is of significant practical importance, since it allows the control of the rotation direction via an electrostatic potential applied to graphene, as shown in Fig. 2, while keeping the magnetic field constant. Note that the static magnetic field is usually provided by permanent magnets

**E E**

**F***e* **F***e*

*x*

(a) (b) Fig. 11. Reversal of the direction of cyclotron rotation resulting from reversal of the type of charge carriers. (a) Case of electrons carriers. (b) Case of hole carriers, where the the

Although neglected until now, the fact that *σxx* and *σyx* are not purely real or purely imaginary numbers results in the transmitted wave being elliptically instead of linearly polarized. Fig. 12 plots *<sup>θ</sup>* and the axial ratio of the transmitted wave versus frequency for *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *μ<sup>c</sup>* = 0.37 eV, *B*<sup>0</sup> = 1 T and *T* = 300 K. For *ω* � *ωc*, when *σxx* and *σyx* are almost purely real, the axial ratio is very high, indicating a linearly polarized wave. As frequency increases, the axial ratio decreases and it becomes minimum close to the cyclotron resonance, where

*Jy*

**B**<sup>0</sup> **B**<sup>0</sup>

*Jx*

*e*

**F***m* **F***m*

*y*

*z*

*Jx*

transverse current has been reversed due to the reversal of the charge sign.

*Jy*

*B*0 (T)

*<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>f</sup>* <sup>=</sup> 30 GHz, and *<sup>T</sup>* <sup>=</sup> 300 K.

and therefore it is extremely difficult to invert.

−*e*

in the RH direction with respect to the +*z*-direction (the direction of the magnetic field). For *ω* � *ωc*, where *σxx* and *σyx* are almost purely real, the phases of *T*LH and *T*RH are found from Equations 29 and 30 as

$$\varphi\_{\rm LH} = \tan^{-1} \left( \frac{\eta\_0 \sigma\_{\rm yx}}{2 + \eta\_0 \sigma\_{\rm xx}} \right) \tag{35a}$$

$$\varphi\_{\rm RH} = -\tan^{-1}\left(\frac{\eta\_0 \sigma\_{\rm yx}}{2 + \eta\_0 \sigma\_{\rm xx}}\right). \tag{35b}$$

Thus, for *ω* � *ωc*, the rotation angle is

$$\theta = \tan^{-1} \left( \frac{\eta\_0 \sigma\_{yx}}{2 + \eta\_0 \sigma\_{xx}} \right). \tag{36}$$

Fig. 9. Transmission coefficient for normal incidence of a LH and a RH circularly polarized wave on a graphene sheet in free space with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>μ</sup><sup>c</sup>* <sup>=</sup> 0.37 eV and *<sup>B</sup>*<sup>0</sup> <sup>=</sup> 1 T.

Fig. 10 presents *θ* and the transmission amplitude (|*T*LH|≈|*T*RH|) versus *μ<sup>c</sup>* and *B*<sup>0</sup> for a graphene sheet with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s at 30 GHz and *<sup>T</sup>* <sup>=</sup> 300 K. The rotation angle clearly depends on both *μ<sup>c</sup>* and *B*0, providing two degrees of freedom in controlling it. Regarding the *μ<sup>c</sup>* dependence, *θ* → 0 as *μ<sup>c</sup>* → 0. This is expected from the fact that for *μ<sup>c</sup>* = 0 there are very few electrons in the conduction band, hence the conductivity is very small and the interaction between the electromagnetic waves and the material is negligible. As *μc* increases, *θ* also increases up to a specific *μ<sup>c</sup>* value, which depends on *B*0. As *μ<sup>c</sup>* further increases, *θ* decreases and tends to 0 for very large *μc*, because the conductivity becomes very large and the material behaves like a perfect electric conductor. Concerning the *B*<sup>0</sup> dependence, Fig. 10 indicates that *θ* increases along a line of constant transmission amplitude. For *μc* fixed, *θ* increases linearly with *B*<sup>0</sup> when *ωcτ* � 1, it becomes maximum when *ωcτ* ≈ 1 and it decreases inversely proportionally with *B*<sup>0</sup> when *ωcτ* � 1.

In addition to controlling the amount of rotation, we can also control its direction via *μc*, exploiting the ambipolar properties of graphene. Specifically, by inverting *μc* from positive to negative values, the type of charge carriers changes from electrons to holes. Holes have

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

in the RH direction with respect to the +*z*-direction (the direction of the magnetic field). For *ω* � *ωc*, where *σxx* and *σyx* are almost purely real, the phases of *T*LH and *T*RH are found from

> *η*0*σyx* 2 + *η*0*σxx*

 *η*0*σyx* 2 + *η*0*σxx*

<sup>10</sup> −2 <sup>10</sup> −1 <sup>10</sup><sup>0</sup> <sup>10</sup><sup>1</sup> 0.3

Fig. 9. Transmission coefficient for normal incidence of a LH and a RH circularly polarized wave on a graphene sheet in free space with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>μ</sup><sup>c</sup>* <sup>=</sup> 0.37 eV and *<sup>B</sup>*<sup>0</sup> <sup>=</sup> 1 T.

Fig. 10 presents *θ* and the transmission amplitude (|*T*LH|≈|*T*RH|) versus *μ<sup>c</sup>* and *B*<sup>0</sup> for a graphene sheet with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s at 30 GHz and *<sup>T</sup>* <sup>=</sup> 300 K. The rotation angle clearly depends on both *μ<sup>c</sup>* and *B*0, providing two degrees of freedom in controlling it. Regarding the *μ<sup>c</sup>* dependence, *θ* → 0 as *μ<sup>c</sup>* → 0. This is expected from the fact that for *μ<sup>c</sup>* = 0 there are very few electrons in the conduction band, hence the conductivity is very small and the interaction between the electromagnetic waves and the material is negligible. As *μc* increases, *θ* also increases up to a specific *μ<sup>c</sup>* value, which depends on *B*0. As *μ<sup>c</sup>* further increases, *θ* decreases and tends to 0 for very large *μc*, because the conductivity becomes very large and the material behaves like a perfect electric conductor. Concerning the *B*<sup>0</sup> dependence, Fig. 10 indicates that *θ* increases along a line of constant transmission amplitude. For *μc* fixed, *θ* increases linearly with *B*<sup>0</sup> when *ωcτ* � 1, it becomes maximum when *ωcτ* ≈ 1 and it decreases inversely

In addition to controlling the amount of rotation, we can also control its direction via *μc*, exploiting the ambipolar properties of graphene. Specifically, by inverting *μc* from positive to negative values, the type of charge carriers changes from electrons to holes. Holes have


*ω*/*ωc*

 *η*0*σyx* 2 + *η*0*σxx*


(35a)

. (35b)

. (36)

−20

−7.5

phase (deg)

5

17.5

30

*ϕ*LH = tan−<sup>1</sup>

*<sup>ϕ</sup>*RH <sup>=</sup> <sup>−</sup> tan−<sup>1</sup>

*θ* = tan−<sup>1</sup>

Equations 29 and 30 as

Thus, for *ω* � *ωc*, the rotation angle is

0.475

0.65

amplitude

proportionally with *B*<sup>0</sup> when *ωcτ* � 1.

0.825

1

*ϕ*LH

*ϕ*RH

Fig. 10. Rotation angle (color plot) and transmission amplitude (contour plot) versus *μc* and *B*<sup>0</sup> for a normally incident plane wave on a graphene sheet in free space, for the parameters *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>f</sup>* <sup>=</sup> 30 GHz, and *<sup>T</sup>* <sup>=</sup> 300 K.

an opposite charge than electrons and, therefore, provide an opposite transverse current than electrons, hence an opposite rotation direction, as illustrated in Fig. 11. This property is of significant practical importance, since it allows the control of the rotation direction via an electrostatic potential applied to graphene, as shown in Fig. 2, while keeping the magnetic field constant. Note that the static magnetic field is usually provided by permanent magnets and therefore it is extremely difficult to invert.

Fig. 11. Reversal of the direction of cyclotron rotation resulting from reversal of the type of charge carriers. (a) Case of electrons carriers. (b) Case of hole carriers, where the the transverse current has been reversed due to the reversal of the charge sign.

Although neglected until now, the fact that *σxx* and *σyx* are not purely real or purely imaginary numbers results in the transmitted wave being elliptically instead of linearly polarized. Fig. 12 plots *<sup>θ</sup>* and the axial ratio of the transmitted wave versus frequency for *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *μ<sup>c</sup>* = 0.37 eV, *B*<sup>0</sup> = 1 T and *T* = 300 K. For *ω* � *ωc*, when *σxx* and *σyx* are almost purely real, the axial ratio is very high, indicating a linearly polarized wave. As frequency increases, the axial ratio decreases and it becomes minimum close to the cyclotron resonance, where

*x*

1

of 2*θ*, where *θ* is the rotation angle for a single pass through graphene.

be polarized along the *x* axis and propagates along the +*z* direction.

**E**h,inc

<sup>11</sup> the first root of *J*�

**E***h*<sup>−</sup>

11 **E***h*<sup>+</sup>

Fig. 13. Non-reciprocity in graphene. A wave normally impinging on graphene from medium 1 to medium 2 and transmitted back to medium 1 undergoes a total rotation angle

dimensions much smaller than the dimensions of the ferrite devices, can be achieved.

where <sup>∇</sup>*<sup>t</sup>* is the transverse (on the *<sup>ρ</sup>* <sup>−</sup> *<sup>ϕ</sup>* plane) del operator and *<sup>ψ</sup>*11(*ρ*, *<sup>ϕ</sup>*) = *<sup>J</sup>*1(*k<sup>h</sup>*

The proposed circular waveguide Faraday rotator consists of a graphene sheet loading the cross section of a circular waveguide, as shown in Fig. 14. The graphene sheet is perpendicular to the waveguide axis and biased with an axial static magnetic field **B**0. The waveguide has a radius *a* and it is excited in its dominant H11 (TE11) mode. The incident mode is assumed to

mode can be analyzed into two degenerate orthogonal modes, a LH circularly polarized mode

*<sup>T</sup>*,11 = **z**ˆ × ∇*<sup>t</sup>* × *ψ*<sup>−</sup>

*<sup>T</sup>*,11 <sup>=</sup> **<sup>z</sup>**<sup>ˆ</sup> × ∇*<sup>t</sup>* <sup>×</sup> *<sup>ψ</sup>*<sup>+</sup>

*<sup>T</sup>*,11 (*ρ*, *ϕ*) = **z**ˆ × ∇*<sup>t</sup>* × *ψ*11(*ρ*, *ϕ*), (37)

<sup>1</sup>(*x*). Using cos *<sup>ϕ</sup>* = (*ej<sup>ϕ</sup>* <sup>+</sup> *<sup>e</sup>*−*jϕ*)/2, the incident

<sup>11</sup> (38a)

<sup>11</sup>, (38b)

*<sup>c</sup>*,11*ρ*) cos *ϕ*,

In this section, two applications of the Faraday rotation effect in graphene will be presented, namely a circular waveguide polarization rotator and a non-reciprocal spatial isolator. Although such devices can be realized with conventional gyrotropic media, such as ferrites, graphene offers increased tunability and electric rotation reversal, and possibly other benefits (such as high heat sink) still to be investigated. The purpose of the section is to show the feasibility of using graphene in such types of devices and not to provide an optimized design. Therefore, the dimensions of the proposed devices are comparable to their ferrite counterparts. However, we believe that by a proper design, e.g. by using a stacked structure composed of alternating graphene sheets and dielectric layers, highly compact devices, with

<sup>17</sup> Novel Electromagnetic Phenomena in Graphene

2

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

**B**0

*y*

*z*

**5.1 Circular waveguide faraday rotator**

The transverse electric field of the mode is

<sup>11</sup>/*a* and *x*�

and a RH circularly polarized one H<sup>+</sup>

**5. Applications**

with *k<sup>h</sup>*

H− 11 *<sup>c</sup>*,11 = *x*�

Fig. 12. Rotation angle, computed by Equation 34, and axial ratio, computed from Equation 25 and Balanis (2005), versus frequency for a normally incident plane wave on a graphene sheet in free space, for the parameters *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>μ</sup><sup>c</sup>* <sup>=</sup> 0.37 eV, *<sup>B</sup>*<sup>0</sup> <sup>=</sup> 1 T, and *T* = 300 K.

there is a strong interaction between the material and the circular polarization of the same handedness as the cyclotron motion of electrons. For a further increase of frequency this "resonance" between the circularly polarized waves and the electrons diminishes gradually and the axial ratio increases.

Fig. 12 also shows that Faraday rotation in graphene is an extremely broadband phenomenon below the resonance (in the case of Fig. 12 from 0 to 200 GHz), as a result of the non-dispersive characteristics of conductivity below the resonance (Fig. 7). It has to be stressed that this frequency-independent behavior is different from the corresponding found in ferrites above the ferromagnetic resonance (Lax & Button, 1962), since the former is due to the non-dispersive characteristics of the material while the latter is due to the increase of the electrical length with frequency.

The direction of Faraday rotation in graphene, as in any magneto-optical material, does not depend on the propagation direction but on the direction of the applied static magnetic field. This can be directly seen from *T* ¯¯ <sup>21</sup> = *T* ¯¯ <sup>12</sup> (Equation 27), which states that for an incident wave of a given polarization the polarization of the transmitted wave is always the same, irrespectively of the propagation direction. Thus, graphene is a non-reciprocal material. In order to demonstrate this statement, consider a plane wave impinging on graphene from medium 1 to medium 2 and then reflected back to medium 1 without changing its polarization (e.g. reflection by a PMC wall after graphene), as illustrated in Fig. 13. As the wave passes through graphene from medium 1 towards medium 2, its polarization is rotated by an angle *θ* in the LH direction with respect to the +*z* direction (the biasing magnetic field direction). As the wave then passes again through graphene from medium 2 back to medium 1, its polarization is again rotated by an angle *θ* in the LH direction with respect to the +*z* axis. Therefore, the polarization of the wave transmitted from medium 1 to medium 2 and then back from medium 2 to medium 1 has undergone an overall rotation of 2*θ*, which is a manifestation of non-reciprocity.

Fig. 13. Non-reciprocity in graphene. A wave normally impinging on graphene from medium 1 to medium 2 and transmitted back to medium 1 undergoes a total rotation angle of 2*θ*, where *θ* is the rotation angle for a single pass through graphene.

#### **5. Applications**

14 Will-be-set-by-IN-TECH

102

frequency (GHz)

there is a strong interaction between the material and the circular polarization of the same handedness as the cyclotron motion of electrons. For a further increase of frequency this "resonance" between the circularly polarized waves and the electrons diminishes gradually

Fig. 12 also shows that Faraday rotation in graphene is an extremely broadband phenomenon below the resonance (in the case of Fig. 12 from 0 to 200 GHz), as a result of the non-dispersive characteristics of conductivity below the resonance (Fig. 7). It has to be stressed that this frequency-independent behavior is different from the corresponding found in ferrites above the ferromagnetic resonance (Lax & Button, 1962), since the former is due to the non-dispersive characteristics of the material while the latter is due to the increase of the

The direction of Faraday rotation in graphene, as in any magneto-optical material, does not depend on the propagation direction but on the direction of the applied static magnetic field.

wave of a given polarization the polarization of the transmitted wave is always the same, irrespectively of the propagation direction. Thus, graphene is a non-reciprocal material. In order to demonstrate this statement, consider a plane wave impinging on graphene from medium 1 to medium 2 and then reflected back to medium 1 without changing its polarization (e.g. reflection by a PMC wall after graphene), as illustrated in Fig. 13. As the wave passes through graphene from medium 1 towards medium 2, its polarization is rotated by an angle *θ* in the LH direction with respect to the +*z* direction (the biasing magnetic field direction). As the wave then passes again through graphene from medium 2 back to medium 1, its polarization is again rotated by an angle *θ* in the LH direction with respect to the +*z* axis. Therefore, the polarization of the wave transmitted from medium 1 to medium 2 and then back from medium 2 to medium 1 has undergone an overall rotation of 2*θ*, which is a

¯¯ <sup>21</sup> = *T* ¯¯

Equation 25 and Balanis (2005), versus frequency for a normally incident plane wave on a graphene sheet in free space, for the parameters *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>μ</sup><sup>c</sup>* <sup>=</sup> 0.37 eV, *<sup>B</sup>*<sup>0</sup> <sup>=</sup> 1 T,

103

*ω<sup>c</sup>* 2*π* 104

<sup>12</sup> (Equation 27), which states that for an incident

5

10

15

axial ratio (dB)

20

25

30

35

100


0

5

rotation angle (deg)

and *T* = 300 K.

and the axial ratio increases.

electrical length with frequency.

This can be directly seen from *T*

manifestation of non-reciprocity.

10

15

20

101

Fig. 12. Rotation angle, computed by Equation 34, and axial ratio, computed from

In this section, two applications of the Faraday rotation effect in graphene will be presented, namely a circular waveguide polarization rotator and a non-reciprocal spatial isolator. Although such devices can be realized with conventional gyrotropic media, such as ferrites, graphene offers increased tunability and electric rotation reversal, and possibly other benefits (such as high heat sink) still to be investigated. The purpose of the section is to show the feasibility of using graphene in such types of devices and not to provide an optimized design. Therefore, the dimensions of the proposed devices are comparable to their ferrite counterparts. However, we believe that by a proper design, e.g. by using a stacked structure composed of alternating graphene sheets and dielectric layers, highly compact devices, with dimensions much smaller than the dimensions of the ferrite devices, can be achieved.

#### **5.1 Circular waveguide faraday rotator**

The proposed circular waveguide Faraday rotator consists of a graphene sheet loading the cross section of a circular waveguide, as shown in Fig. 14. The graphene sheet is perpendicular to the waveguide axis and biased with an axial static magnetic field **B**0. The waveguide has a radius *a* and it is excited in its dominant H11 (TE11) mode. The incident mode is assumed to be polarized along the *x* axis and propagates along the +*z* direction.

The transverse electric field of the mode is

$$\mathbf{E}\_{\rm T,11}^{h,\rm inc}(\rho\_\prime\,\varphi) = \mathbf{\hat{z}} \times \nabla\_\mathbf{t} \times \psi\_{11}(\rho\_\prime\,\varphi),\tag{37}$$

where <sup>∇</sup>*<sup>t</sup>* is the transverse (on the *<sup>ρ</sup>* <sup>−</sup> *<sup>ϕ</sup>* plane) del operator and *<sup>ψ</sup>*11(*ρ*, *<sup>ϕ</sup>*) = *<sup>J</sup>*1(*k<sup>h</sup> <sup>c</sup>*,11*ρ*) cos *ϕ*, with *k<sup>h</sup> <sup>c</sup>*,11 = *x*� <sup>11</sup>/*a* and *x*� <sup>11</sup> the first root of *J*� <sup>1</sup>(*x*). Using cos *<sup>ϕ</sup>* = (*ej<sup>ϕ</sup>* <sup>+</sup> *<sup>e</sup>*−*jϕ*)/2, the incident mode can be analyzed into two degenerate orthogonal modes, a LH circularly polarized mode H− 11

$$\mathbf{E}\_{T,11}^{h-} = \mathbf{\hat{z}} \times \nabla\_t \times \boldsymbol{\psi}\_{11}^{-} \tag{38a}$$

and a RH circularly polarized one H<sup>+</sup> 11

$$\mathbf{E}\_{T,11}^{\hbar+} = \hat{\mathbf{z}} \times \nabla\_{\mathbf{t}} \times \boldsymbol{\psi}\_{11'}^{+} \tag{38b}$$

Inserting Equations 39 into 40a and using the orthogonality conditions (Collin, 1990)

<sup>19</sup> Novel Electromagnetic Phenomena in Graphene

 *S* **E***e*<sup>−</sup> *<sup>T</sup>*,1*<sup>n</sup>* · **<sup>E</sup>***e*<sup>+</sup>

 *S* **E***e*<sup>−</sup> *<sup>T</sup>*,1*<sup>n</sup>* · **<sup>E</sup>***h*<sup>+</sup>

*<sup>T</sup>*,1*<sup>m</sup>* = 0, *n* �= *m* (42)

*<sup>T</sup>*,1*<sup>m</sup>* = 0, (43)

<sup>1</sup>*<sup>n</sup>* , (44a)

<sup>1</sup>*<sup>n</sup>* . (44b)

, (45a)

, (45b)

, (46a)

. (46b)

, (47a)

*<sup>T</sup>*,1*<sup>m</sup>* =

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

*<sup>T</sup>*,1*<sup>m</sup>* =

*Rh*<sup>−</sup>

1*nσxxTh*<sup>−</sup>

1*nσxxTe*<sup>−</sup>

<sup>1</sup>*<sup>n</sup>* <sup>+</sup> *<sup>δ</sup>*1*<sup>n</sup>* <sup>=</sup> *<sup>T</sup>h*<sup>−</sup>

*Re*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* <sup>=</sup> *<sup>T</sup>e*<sup>−</sup>

<sup>1</sup>*<sup>n</sup>* <sup>+</sup> *<sup>Z</sup><sup>h</sup>*

<sup>1</sup>*<sup>n</sup>* <sup>+</sup> *<sup>Z</sup><sup>e</sup>*

, *J*

, *J*

In the derivation of Equation 45 the identity ¯¯*Jt* · **<sup>v</sup>** <sup>=</sup> **<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>v</sup>**, which holds for any vector **<sup>v</sup>**, has been used. Omitting the proof, which can be found in (Sounas & Caloz, 2011d), we directly

*he*

*ee*

, *<sup>J</sup>*

<sup>1</sup>*nσyx*

<sup>1</sup>*nσyx*

∞ ∑ *m*=0

∞ ∑ *m*=0

Equations 48 constitute a linear system of equations, the solution of which gives *<sup>T</sup>h*/*e*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* . One

 *J hh nmT<sup>h</sup>*<sup>−</sup> <sup>1</sup>*<sup>m</sup>* + *J he nmT<sup>e</sup>*<sup>−</sup> 1*m* 

 *J eh nmT<sup>h</sup>*<sup>−</sup> <sup>1</sup>*<sup>m</sup>* + *J ee nmT<sup>e</sup>*<sup>−</sup> 1*m* 

<sup>1</sup>*nσyx*

<sup>1</sup>*nσyx*

*he nm* =

*ee nm* =

∞ ∑ *m*=0

∞ ∑ *m*=0  *J hh nmT<sup>h</sup>*<sup>−</sup> <sup>1</sup>*<sup>m</sup>* + *J he nmT<sup>e</sup>*<sup>−</sup> 1*m* 

 *J eh nmT<sup>h</sup>*<sup>−</sup> <sup>1</sup>*<sup>m</sup>* + *J ee nmT<sup>e</sup>*<sup>−</sup> 1*m* 

 *S* **E***h*<sup>+</sup> 1*n* · **<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>***e*<sup>−</sup> 1*m dS*

 *S* **E***e*<sup>+</sup> 1*n* · **<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>***e*<sup>−</sup> 1*m dS*

*nm* <sup>=</sup> <sup>2</sup>(*x*�

<sup>1</sup>*n*)<sup>2</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup> 1*m* (*x*�

 (*x*�  *S* **E***h*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* · **<sup>E</sup>***h*<sup>−</sup> <sup>1</sup>*<sup>n</sup> dS*

 *S* **E***e*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* · **<sup>E</sup>***e*<sup>−</sup> <sup>1</sup>*<sup>n</sup> dS*

1*n*)<sup>2</sup>

*nm* = 0, (47b)

<sup>1</sup>(*x*), respectively. Solving Equations 44 with

<sup>1</sup>*n*)<sup>2</sup> − <sup>1</sup>

= 2*δ*1*n*, (48a)

= 0. (48b)

 *S* **E***h*<sup>−</sup> *<sup>T</sup>*,1*<sup>n</sup>* · **<sup>E</sup>***h*<sup>+</sup>

 *S* **E***h*<sup>−</sup> *<sup>T</sup>*,1*<sup>n</sup>* · **<sup>E</sup>***e*<sup>+</sup>

with *S* the waveguide cross section, one obtains

Similarly, introducing Equation 41 into 40b yields

<sup>1</sup>*<sup>n</sup>* <sup>−</sup> *<sup>R</sup>e*<sup>−</sup>

<sup>1</sup>*<sup>n</sup>* <sup>=</sup> *<sup>Z</sup><sup>h</sup>*

<sup>1</sup>*<sup>n</sup>* <sup>=</sup> *<sup>Z</sup><sup>e</sup>*

**<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>***h*<sup>−</sup> 1*m dS*

**<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>***h*<sup>−</sup> 1*m dS*

give the final expressions of the *J* coefficients of Equations 46

(*x*�

1*nσxx*

<sup>1</sup>*nσxx*) *<sup>T</sup>e*<sup>−</sup>

<sup>1</sup>*<sup>n</sup>* in a similar manner.

<sup>1</sup>*m*)<sup>2</sup> − <sup>1</sup>

, *J*

<sup>1</sup>*<sup>n</sup>* are the *n*th roots of *J*1(*x*) and *J*�

*Th*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* <sup>+</sup> *<sup>Z</sup><sup>h</sup>*

respect to *<sup>R</sup>h*/*e*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* and introducing the result into Equations 45 we get

<sup>1</sup>*<sup>n</sup>* <sup>+</sup> *<sup>Z</sup><sup>e</sup>*

<sup>1</sup>*<sup>n</sup>* <sup>+</sup> *<sup>δ</sup>*1*<sup>n</sup>* <sup>−</sup> *<sup>R</sup>h*<sup>−</sup>

<sup>−</sup>*Te*<sup>−</sup>

 *S* **E***h*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* · **<sup>E</sup>***h*<sup>−</sup> <sup>1</sup>*<sup>n</sup> dS*

 *S* **E***e*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* · **<sup>E</sup>***e*<sup>−</sup> <sup>1</sup>*<sup>n</sup> dS*

 *S* **E***h*<sup>+</sup> 1*n* · 

 *S* **E***e*<sup>+</sup> 1*n* · 

*nm* <sup>=</sup> <sup>−</sup> *<sup>j</sup>*<sup>2</sup> (*x*� <sup>1</sup>*n*)<sup>2</sup> − <sup>1</sup>

> 2 + *Z<sup>h</sup>*

> > (2 + *Z<sup>e</sup>*

 *J he mn*<sup>∗</sup>

<sup>−</sup>*Th*<sup>−</sup>

*J hh nm* =

*J eh nm* =

*J hh*

*J eh nm* = −

where *x*1*<sup>n</sup>* and *x*�

can calculate *Th*/*e*<sup>+</sup>

where

Fig. 14. Circular cylindrical waveguide of radius *a* loaded by a graphene sheet perpendicular to its axis. The sheet is placed at the *z* = 0 plane and biased by a perpendicular static magnetic field *B*0. The waveguide is excited in its dominant H11 (TE11) mode.

both with amplitude equal to 1/2. In Equation 38 *ψ*± <sup>11</sup> <sup>=</sup> *<sup>J</sup>*1(*k<sup>h</sup> <sup>c</sup>*,11*ρ*)*e*∓*jϕ*. Since any pair of modes with *e*−*j<sup>ϕ</sup>* and *ej<sup>ϕ</sup>* azimuthal dependencies are orthogonal to each other and because of the phase matching condition in the graphene plane, any of these modes creates reflected and transmitted modes with the same azimuthal dependence. Therefore, the H− <sup>11</sup> mode will excite the H− <sup>1</sup>*<sup>n</sup>* and E<sup>−</sup> <sup>1</sup>*<sup>n</sup>* modes, while the H<sup>+</sup> <sup>11</sup> mode will excite the H<sup>+</sup> <sup>1</sup>*<sup>n</sup>* ans E<sup>+</sup> <sup>1</sup>*<sup>n</sup>* modes. Next, we focus on the H− <sup>11</sup> mode. The analysis for the H<sup>+</sup> <sup>11</sup> is completely similar.

The transverse reflected and transmitted fields in the graphene plane (*z* = 0 plane) read

$$\mathbf{E}\_T^{-\text{ref}} = \sum\_{n=1}^{\infty} \left( R\_{1n}^{h-} \mathbf{E}\_{T,1n}^{h-} + R\_{1n}^{e-} \mathbf{E}\_{T,1n}^{e-} \right) \, \text{} \tag{39a}$$

$$\mathbf{E}\_T^{-\text{tran}} = \sum\_{n=1}^{\infty} \left( T\_{1n}^{l-} \mathbf{E}\_{T,1n}^{l-} + T\_{1n}^{\varepsilon-} \mathbf{E}\_{T,1n}^{\varepsilon-} \right) \, \tag{39b}$$

respectively. For the computation of the coefficients *Rh*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* , *<sup>R</sup>e*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* , *<sup>T</sup>h*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* and *<sup>T</sup>e*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* , the boundary conditions

$$\left. \hat{\mathbf{z}} \times \left( \mathbf{E}\_T^{\text{tran}} - \mathbf{E}\_T^{\text{inc}} - \mathbf{E}\_T^{\text{ref}} \right) \right|\_{z=0} = 0,\tag{40a}$$

$$\left. \hat{\mathbf{z}} \times \left( \mathbf{H}\_T^{\text{tran}} - \mathbf{H}\_T^{\text{inc}} - \mathbf{H}\_T^{\text{ref}} \right) \right|\_{z=0} = \left. \vec{\sigma} \cdot \mathbf{E}\_T^{\text{tran}} \right|\_{z=0} \tag{40b}$$

should be used. The reflected and transmitted magnetic fields in the graphene plane are given by

$$\mathbf{H}\_T^{-\text{ref}} = -\sum\_{n=1}^{\infty} \left( R\_{1n}^{h-} \frac{\hat{\mathbf{z}} \times \mathbf{E}\_{T,1n}^{h-}}{Z\_{1n}^h} + R\_{1n}^{e-} \frac{\hat{\mathbf{z}} \times \mathbf{E}\_{T,1n}^{e-}}{Z\_{1n}^e} \right),\tag{41a}$$

$$\mathbf{E}\_{T}^{-\text{ran}} = \sum\_{n=1}^{\infty} \left( T\_{1n}^{\text{lr}-} \frac{\hat{\mathbf{z}} \times \mathbf{E}\_{T,1n}^{\text{lr}-}}{Z\_{1n}^{\text{lr}}} + T\_{1n}^{\varepsilon-} \frac{\hat{\mathbf{z}} \times \mathbf{E}\_{T,1n}^{\varepsilon-}}{Z\_{1n}^{\varepsilon}} \right), \tag{41b}$$

where *Z<sup>h</sup>* <sup>1</sup>*<sup>n</sup>* <sup>=</sup> *<sup>η</sup>*0(*k*0/*β<sup>h</sup>* <sup>1</sup>*n*) and *<sup>Z</sup><sup>e</sup>* <sup>1</sup>*<sup>n</sup>* <sup>=</sup> *<sup>η</sup>*0(*β<sup>e</sup>* <sup>1</sup>*n*/*k*0) are the wave impedances of the H1*<sup>n</sup>* and E1*<sup>n</sup>* modes, respectively, and *β<sup>h</sup>* <sup>1</sup>*<sup>n</sup>* and *<sup>β</sup><sup>e</sup>* <sup>1</sup>*<sup>n</sup>* are the corresponding propagation numbers.

Inserting Equations 39 into 40a and using the orthogonality conditions (Collin, 1990)

$$\int\_{S} \mathbf{E}\_{T,1n}^{h-} \cdot \mathbf{E}\_{T,1m}^{h+} = \int\_{S} \mathbf{E}\_{T,1n}^{e-} \cdot \mathbf{E}\_{T,1m}^{e+} = 0, \quad n \neq m \tag{42}$$

$$\int\_{S} \mathbf{E}\_{T,1n}^{h-} \cdot \mathbf{E}\_{T,1m}^{\varepsilon+} = \int\_{S} \mathbf{E}\_{T,1n}^{\varepsilon-} \cdot \mathbf{E}\_{T,1m}^{h+} = 0,\tag{43}$$

with *S* the waveguide cross section, one obtains

$$R\_{1n}^{\hbar-} + \delta\_{1n} = T\_{1n}^{\hbar-} \, , \tag{44a}$$

$$R\_{1n}^{\varepsilon-} = T\_{1n}^{\varepsilon-}.\tag{44b}$$

Similarly, introducing Equation 41 into 40b yields

$$-T\_{1n}^{\hbar-} + \delta\_{1n} - R\_{1n}^{\hbar-} = Z\_{1n}^{\hbar} \sigma\_{xx} T\_{1n}^{\hbar-} + Z\_{1n}^{\hbar} \sigma\_{yx} \sum\_{m=0}^{\infty} \left( f\_{nm}^{\hbar\hbar} T\_{1m}^{\hbar-} + f\_{nm}^{\hbar\varepsilon} T\_{1m}^{\epsilon-} \right), \tag{45a}$$

$$-T\_{1n}^{\varepsilon-} - R\_{1n}^{\varepsilon-} = Z\_{1n}^{\varepsilon} \sigma\_{\mathbf{x}\mathbf{x}} T\_{1n}^{\varepsilon-} + Z\_{1n}^{\varepsilon} \sigma\_{\mathbf{y}\mathbf{x}} \sum\_{m=0}^{\infty} \left( f\_{nm}^{\varepsilon\hbar} T\_{1m}^{\hbar-} + f\_{nm}^{\varepsilon\varepsilon} T\_{1m}^{\varepsilon-} \right), \tag{45b}$$

where

16 Will-be-set-by-IN-TECH

Fig. 14. Circular cylindrical waveguide of radius *a* loaded by a graphene sheet perpendicular

modes with *e*−*j<sup>ϕ</sup>* and *ej<sup>ϕ</sup>* azimuthal dependencies are orthogonal to each other and because of the phase matching condition in the graphene plane, any of these modes creates reflected

*z*

<sup>11</sup> <sup>=</sup> *<sup>J</sup>*1(*k<sup>h</sup>*

<sup>11</sup> is completely similar.

<sup>1</sup>*<sup>n</sup>* **<sup>E</sup>***e*<sup>−</sup> *T*,1*n* 

<sup>1</sup>*<sup>n</sup>* **<sup>E</sup>***e*<sup>−</sup> *T*,1*n* 

<sup>1</sup>*<sup>n</sup>* , *<sup>R</sup>e*<sup>−</sup>

<sup>=</sup> *<sup>σ</sup>*¯¯ · **<sup>E</sup>**tran *T z*=0

> **<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>***e*<sup>−</sup> *T*,1*n Ze* 1*n*

**<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>***e*<sup>−</sup> *T*,1*n Ze* 1*n*

<sup>1</sup>*<sup>n</sup>* are the corresponding propagation numbers.

<sup>1</sup>*n*/*k*0) are the wave impedances of the H1*<sup>n</sup>* and E1*<sup>n</sup>*

+ *Re*<sup>−</sup> 1*n*

+ *Te*<sup>−</sup> 1*n*

<sup>1</sup>*<sup>n</sup>* , *<sup>T</sup>h*<sup>−</sup>

<sup>1</sup>*<sup>n</sup>* and *<sup>T</sup>e*<sup>−</sup>

= 0, (40a)

<sup>11</sup> mode will excite the H<sup>+</sup>

*<sup>T</sup>*,1*<sup>n</sup>* <sup>+</sup> *<sup>R</sup>e*<sup>−</sup>

*<sup>T</sup>*,1*<sup>n</sup>* <sup>+</sup> *<sup>T</sup>e*<sup>−</sup>

*<sup>c</sup>*,11*ρ*)*e*∓*jϕ*. Since any pair of

, (39a)

, (39b)

<sup>1</sup>*<sup>n</sup>* ans E<sup>+</sup>

<sup>11</sup> mode will

<sup>1</sup>*<sup>n</sup>* modes. Next,

<sup>1</sup>*<sup>n</sup>* , the boundary

, (40b)

, (41a)

, (41b)

*y*

to its axis. The sheet is placed at the *z* = 0 plane and biased by a perpendicular static magnetic field *B*0. The waveguide is excited in its dominant H11 (TE11) mode.

and transmitted modes with the same azimuthal dependence. Therefore, the H−

∞ ∑ *n*=1

∞ ∑ *n*=1  *Rh*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* **<sup>E</sup>***h*<sup>−</sup>

 *Th*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* **<sup>E</sup>***h*<sup>−</sup>

*<sup>T</sup>* <sup>−</sup> **<sup>E</sup>**ref *T z*=0

*<sup>T</sup>* <sup>−</sup> **<sup>H</sup>**ref *T z*=0

should be used. The reflected and transmitted magnetic fields in the graphene plane are given

**<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>***h*<sup>−</sup> *T*,1*n Zh* 1*n*

**<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>E</sup>***h*<sup>−</sup> *T*,1*n Zh* 1*n*

The transverse reflected and transmitted fields in the graphene plane (*z* = 0 plane) read

**B**0

*x*

both with amplitude equal to 1/2. In Equation 38 *ψ*±

<sup>1</sup>*<sup>n</sup>* modes, while the H<sup>+</sup>

**E**−ref *<sup>T</sup>* =

**E**−tran *<sup>T</sup>* =

respectively. For the computation of the coefficients *Rh*<sup>−</sup>

**z**ˆ × **E**tran *<sup>T</sup>* <sup>−</sup> **<sup>E</sup>**inc

**z**ˆ × **H**tran *<sup>T</sup>* <sup>−</sup> **<sup>H</sup>**inc

**H**−ref *<sup>T</sup>* = −

**E**−tran *<sup>T</sup>* =

<sup>1</sup>*n*) and *<sup>Z</sup><sup>e</sup>*

<sup>1</sup>*<sup>n</sup>* <sup>=</sup> *<sup>η</sup>*0(*k*0/*β<sup>h</sup>*

modes, respectively, and *β<sup>h</sup>*

∞ ∑ *n*=1

> *Th*<sup>−</sup> 1*n*

<sup>1</sup>*<sup>n</sup>* <sup>=</sup> *<sup>η</sup>*0(*β<sup>e</sup>*

∞ ∑ *n*=1

<sup>1</sup>*<sup>n</sup>* and *<sup>β</sup><sup>e</sup>*

 *Rh*<sup>−</sup> 1*n*

<sup>11</sup> mode. The analysis for the H<sup>+</sup>

excite the H−

conditions

by

where *Z<sup>h</sup>*

we focus on the H−

<sup>1</sup>*<sup>n</sup>* and E<sup>−</sup>

$$\begin{aligned} J\_{nm}^{hh} &= \frac{\int\_{\mathcal{S}} \mathbf{E}\_{1n}^{h+} \cdot \left(\hat{\mathbf{z}} \times \mathbf{E}\_{1m}^{h-}\right) d\mathcal{S}}{\int\_{\mathcal{S}} \mathbf{E}\_{1n}^{h+} \cdot \mathbf{E}\_{1n}^{h-} d\mathcal{S}}, & J\_{nm}^{he} &= \frac{\int\_{\mathcal{S}} \mathbf{E}\_{1n}^{h+} \cdot \left(\hat{\mathbf{z}} \times \mathbf{E}\_{1m}^{e-}\right) d\mathcal{S}}{\int\_{\mathcal{S}} \mathbf{E}\_{1n}^{h+} \cdot \mathbf{E}\_{1n}^{h-} d\mathcal{S}}, \\ &\int\_{\mathcal{S}} \mathbf{E}\_{1n}^{e+} \cdot \left(\hat{\mathbf{z}} \times \mathbf{E}\_{1m}^{h-}\right) d\mathcal{S}}{\int\_{\mathcal{S}} \mathbf{E}\_{1n}^{e+} \cdot \left(\hat{\mathbf{z}} \times \mathbf{E}\_{1m}^{e-}\right) d\mathcal{S}}. \end{aligned} \tag{46a}$$

$$J\_{nm}^{\varepsilon\hbar} = \frac{\int\_{S} \mathbf{E}\_{1n}^{\varepsilon+} \cdot \left(\mathbf{\hat{z}} \times \mathbf{E}\_{1m}^{\hbar-}\right) d\mathbf{S}}{\int\_{S} \mathbf{E}\_{1n}^{\varepsilon+} \cdot \mathbf{E}\_{1n}^{\varepsilon-} d\mathbf{S}}, \qquad \qquad J\_{nm}^{\varepsilon\varepsilon} = \frac{\int\_{S} \mathbf{E}\_{1n}^{\varepsilon+} \cdot \left(\mathbf{\hat{z}} \times \mathbf{E}\_{1m}^{\varepsilon-}\right) d\mathbf{S}}{\int\_{S} \mathbf{E}\_{1n}^{\varepsilon+} \cdot \mathbf{E}\_{1n}^{\varepsilon-} d\mathbf{S}}. \tag{46b}$$

In the derivation of Equation 45 the identity ¯¯*Jt* · **<sup>v</sup>** <sup>=</sup> **<sup>z</sup>**<sup>ˆ</sup> <sup>×</sup> **<sup>v</sup>**, which holds for any vector **<sup>v</sup>**, has been used. Omitting the proof, which can be found in (Sounas & Caloz, 2011d), we directly give the final expressions of the *J* coefficients of Equations 46

$$J\_{nm}^{\text{th}} = -\frac{j2}{\sqrt{\left[ (\mathbf{x}\_{1n}')^2 - 1 \right] \left[ (\mathbf{x}\_{1m}')^2 - 1 \right]}}, \qquad J\_{nm}^{\text{he}} = \frac{2(\mathbf{x}\_{1n}')^2}{\left[ (\mathbf{x}\_{1n}')^2 - \mathbf{x}\_{1m}^2 \right] \sqrt{(\mathbf{x}\_{1n}')^2 - 1}}, \tag{47a}$$

$$J\_{nm}^{\text{eff}} = -\left(J\_{nm}^{\text{loc}}\right)^{\*}, \tag{47a} \qquad \qquad \qquad J\_{nm}^{\text{cc}} = 0,\tag{47b}$$

where *x*1*<sup>n</sup>* and *x*� <sup>1</sup>*<sup>n</sup>* are the *n*th roots of *J*1(*x*) and *J*� <sup>1</sup>(*x*), respectively. Solving Equations 44 with respect to *<sup>R</sup>h*/*e*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* and introducing the result into Equations 45 we get

$$\left(2 + Z\_{1n}^{\hbar} \sigma\_{xx}\right) T\_{1n}^{\hbar -} + Z\_{1n}^{\hbar} \sigma\_{yx} \sum\_{m=0}^{\infty} \left(J\_{nm}^{\hbar \hbar} T\_{1m}^{\hbar -} + J\_{nm}^{\hbar \varepsilon} T\_{1m}^{\varepsilon -}\right) = 2\delta\_{1n} \tag{48a}$$

$$\left(2 + Z\_{1n}^{\varepsilon} \sigma\_{xx}\right) T\_{1n}^{\varepsilon-} + Z\_{1n}^{\varepsilon} \sigma\_{yx} \sum\_{m=0}^{\infty} \left(f\_{nm}^{\varepsilon \hbar} T\_{1m}^{\hbar -} + f\_{nm}^{\varepsilon \varepsilon} T\_{1m}^{\varepsilon -}\right) = 0. \tag{48b}$$

Equations 48 constitute a linear system of equations, the solution of which gives *<sup>T</sup>h*/*e*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* . One can calculate *Th*/*e*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* in a similar manner.

1.2 1.4 1.6 1.8 2 2.2

waveguide - exact

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

waveguide - approximate free space - normal incidence

<sup>21</sup> Novel Electromagnetic Phenomena in Graphene

frequency (GHz)

As a second application of the gyrotropic properties of graphene, we propose the spatial isolator of Fig. 16 (Sounas & Caloz, 2011c). It consists of a magnetically biased graphene sheet in the middle of two wire grids, which are rotated by an angle of 45◦ with respect to each other. The structure allows propagation of a *y*-polarized wave from medium 1 towards

The operation principle of the structure is the same as of the ferrite-based non-reciprocal spatial isolator presented in (Parsa et al., 2010). A *y*-polarized wave propagating along the −*z* direction passes through the wire grid at *z* = *d*1, it is then rotated by 45◦ in the LH direction with respect to the *z*-axis due to graphene and it eventually passes through the wire grid at *z* = *d*2. On the other hand, a wave propagating along the +*z* direction is directly blocked by the grid at *z* = *d*<sup>2</sup> if it is polarized parallel to the wires of this grid. If it is perpendicularly polarized to the wires of this grid, it passes through this grid, it is then rotated by 45◦ in the LH direction as it passes through graphene and it is eventually blocked by the grid at *z* = *d*1. The structure can be analyzed by using the transmission line model of Fig. 17. In this model graphene and the wire grids are represented by shunt dyadic admittances and the space between graphene and the grids by transmission line sections with wave impedance *η*<sup>0</sup> and propagation number *k*0. The shunt admittance which models the graphene sheet is simply the tensorial conductivity of graphene (Equation 13). The shunt admittance which models the

medium 2, while it blocks any wave propagating from medium 2 towards medium 1.

*Y* ¯¯

*Yg*� <sup>=</sup> *<sup>j</sup>η*−<sup>1</sup> 0 *λ d*

Fig. 15. Frequency dependence of the rotation angle and the transmission amplitude for a waveguide, computed exactly by Equations 48 and 49 and approximately by Equations 50 and 51, for a waveguide with *<sup>a</sup>* <sup>=</sup> 8 cm and a graphene sheet with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s,

0

<sup>1</sup> <sup>=</sup> *Yg*�**x**ˆ**x**<sup>ˆ</sup> <sup>+</sup> *Yg*⊥**y**ˆ**y**ˆ, (53)

(54a)

1

sin *<sup>π</sup><sup>w</sup>* 2*d*

ln

0.1

0.2

transmission

amplitude

0.3

0.4

0.5

0.6

0.7

14

*μ<sup>c</sup>* = 0.3 eV, *B*<sup>0</sup> = 1 T and *T* = 300 K.

**5.2 Spatial isolator**

grid at *z* = *d*<sup>1</sup> is

where

16

18

rotation angle (deg)

20

22

24

26

28

If the operation frequency is between the cutoff frequencies of the dominant mode and the first higher order mode, only the dominant mode exists far away from the graphene discontinuity, since all the higher order modes are evanescent. Then, similar to the case of plane wave incidence on graphene in free space, the rotation angle of the transmitted mode is

$$
\theta = \frac{\varrho^- - \varrho^+}{2},
\tag{49}
$$

where here *<sup>ϕ</sup>*<sup>±</sup> <sup>=</sup> arg{*Th*<sup>±</sup> <sup>11</sup> }.

An approximate closed-form expression for *θ* can be obtained by considering only the dominant mode terms in Equations 48. The solution of Equations 48 reads then

$$T\_{11}^{h\pm} = \frac{2}{(2 + Z\_{11}^h \sigma\_{\rm xx}) \pm j a Z\_{11}^h \sigma\_{\rm yx}} \, ^\prime \tag{50}$$

with *α* = 2/[(*x*� <sup>11</sup>)<sup>2</sup> <sup>−</sup> <sup>1</sup>]. Subsequently, the rotation angle is found as

$$\theta = \tan^{-1} \left( \frac{a Z\_{11}^h \sigma\_{yx}}{2 + Z\_{11}^h \sigma\_{xx}} \right). \tag{51}$$

The frequency dependence of the rotation angle and the transmission amplitude, computed exactly via Equation 49 and approximately via Equation 51, are presented in Fig. 15 for a waveguide with *<sup>a</sup>* <sup>=</sup> 8 cm and a graphene sheet with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>μ</sup><sup>c</sup>* <sup>=</sup> 0.3 eV and *<sup>B</sup>*<sup>0</sup> <sup>=</sup> 1 T. The rotation angle and the transmission amplitude for normal incidence on graphene in free space, computed through Equation 36, are also plotted for comparison. Very good agreement between the exact (Equation 49) and the approximate (Equation 51) formulas is observed.

Fig. 15 shows that *θ* decreases as frequency increases. This may be understood from the frequency behavior of *Z<sup>h</sup>* <sup>11</sup>, which tends to ∞ at the cutoff of the H11 mode and increases towards *η*<sup>0</sup> as frequency increases. At the H11 mode cutoff frequency, *θ* takes its maximum value

$$\theta = \tan^{-1} \left( a \frac{\sigma\_{yx}}{\sigma\_{xx}} \right). \tag{52}$$

However, the transmission amplitude close to cutoff is very small. On the other hand, one would expect that at *ω* → ∞, when the H11 mode propagates almost parallel to the waveguide axis and *Z<sup>h</sup>* <sup>11</sup> → *η*0, *θ* would tend to its free space value (Equation 36), whereas, as seen from Equation 51, it tends to a smaller value5. A physical explanation can be given as follows. As the H11 wave impinges on graphene, it produces a transverse current component, which is responsible for the polarization rotation. However, this current does not perfectly match the transverse pattern of the mode and therefore only partially contributes to the transmitted field amplitude, thus providing a smaller rotation angle than in free space. This mismatch is also responsible for the excitation of the higher order modes in the waveguide environment.

Fig. 15. Frequency dependence of the rotation angle and the transmission amplitude for a waveguide, computed exactly by Equations 48 and 49 and approximately by Equations 50 and 51, for a waveguide with *<sup>a</sup>* <sup>=</sup> 8 cm and a graphene sheet with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *μ<sup>c</sup>* = 0.3 eV, *B*<sup>0</sup> = 1 T and *T* = 300 K.

#### **5.2 Spatial isolator**

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

If the operation frequency is between the cutoff frequencies of the dominant mode and the first higher order mode, only the dominant mode exists far away from the graphene discontinuity, since all the higher order modes are evanescent. Then, similar to the case of plane wave

*<sup>θ</sup>* <sup>=</sup> *<sup>ϕ</sup>*<sup>−</sup> <sup>−</sup> *<sup>ϕ</sup>*<sup>+</sup>

An approximate closed-form expression for *θ* can be obtained by considering only the

*<sup>α</sup>Z<sup>h</sup>*

The frequency dependence of the rotation angle and the transmission amplitude, computed exactly via Equation 49 and approximately via Equation 51, are presented in Fig. 15 for a waveguide with *<sup>a</sup>* <sup>=</sup> 8 cm and a graphene sheet with *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s, *<sup>μ</sup><sup>c</sup>* <sup>=</sup> 0.3 eV and *<sup>B</sup>*<sup>0</sup> <sup>=</sup> 1 T. The rotation angle and the transmission amplitude for normal incidence on graphene in free space, computed through Equation 36, are also plotted for comparison. Very good agreement between the exact (Equation 49) and the approximate (Equation 51) formulas is

Fig. 15 shows that *θ* decreases as frequency increases. This may be understood from the

towards *η*<sup>0</sup> as frequency increases. At the H11 mode cutoff frequency, *θ* takes its maximum

However, the transmission amplitude close to cutoff is very small. On the other hand, one would expect that at *ω* → ∞, when the H11 mode propagates almost parallel to the waveguide

Equation 51, it tends to a smaller value5. A physical explanation can be given as follows. As the H11 wave impinges on graphene, it produces a transverse current component, which is responsible for the polarization rotation. However, this current does not perfectly match the transverse pattern of the mode and therefore only partially contributes to the transmitted field amplitude, thus providing a smaller rotation angle than in free space. This mismatch is also responsible for the excitation of the higher order modes in the waveguide environment.

<sup>11</sup> → *η*0, *θ* would tend to its free space value (Equation 36), whereas, as seen from

 *α σyx σxx* 

*θ* = tan−<sup>1</sup>

<sup>11</sup>*σxx*) <sup>±</sup> *<sup>j</sup>αZ<sup>h</sup>*

<sup>11</sup>*σyx* 2 + *Z<sup>h</sup>* <sup>11</sup>*σxx* <sup>11</sup>*σyx*

<sup>11</sup>, which tends to ∞ at the cutoff of the H11 mode and increases

<sup>2</sup> , (49)

, (50)

. (51)

. (52)

incidence on graphene in free space, the rotation angle of the transmitted mode is

dominant mode terms in Equations 48. The solution of Equations 48 reads then

<sup>11</sup> <sup>=</sup> <sup>2</sup> (2 + *Z<sup>h</sup>*

<sup>11</sup>)<sup>2</sup> <sup>−</sup> <sup>1</sup>]. Subsequently, the rotation angle is found as

*θ* = tan−<sup>1</sup>

*Th*<sup>±</sup>

where here *<sup>ϕ</sup>*<sup>±</sup> <sup>=</sup> arg{*Th*<sup>±</sup>

with *α* = 2/[(*x*�

observed.

value

axis and *Z<sup>h</sup>*

<sup>5</sup> *α* < 1.

frequency behavior of *Z<sup>h</sup>*

<sup>11</sup> }.

As a second application of the gyrotropic properties of graphene, we propose the spatial isolator of Fig. 16 (Sounas & Caloz, 2011c). It consists of a magnetically biased graphene sheet in the middle of two wire grids, which are rotated by an angle of 45◦ with respect to each other. The structure allows propagation of a *y*-polarized wave from medium 1 towards medium 2, while it blocks any wave propagating from medium 2 towards medium 1.

The operation principle of the structure is the same as of the ferrite-based non-reciprocal spatial isolator presented in (Parsa et al., 2010). A *y*-polarized wave propagating along the −*z* direction passes through the wire grid at *z* = *d*1, it is then rotated by 45◦ in the LH direction with respect to the *z*-axis due to graphene and it eventually passes through the wire grid at *z* = *d*2. On the other hand, a wave propagating along the +*z* direction is directly blocked by the grid at *z* = *d*<sup>2</sup> if it is polarized parallel to the wires of this grid. If it is perpendicularly polarized to the wires of this grid, it passes through this grid, it is then rotated by 45◦ in the LH direction as it passes through graphene and it is eventually blocked by the grid at *z* = *d*1.

The structure can be analyzed by using the transmission line model of Fig. 17. In this model graphene and the wire grids are represented by shunt dyadic admittances and the space between graphene and the grids by transmission line sections with wave impedance *η*<sup>0</sup> and propagation number *k*0. The shunt admittance which models the graphene sheet is simply the tensorial conductivity of graphene (Equation 13). The shunt admittance which models the grid at *z* = *d*<sup>1</sup> is

$$
\tilde{Y}\_1 = Y\_{\mathcal{S}\parallel}\mathbf{\hat{x}}\mathbf{\hat{x}} + Y\_{\mathcal{S}\perp}\mathbf{\hat{y}}\mathbf{\hat{y}},\tag{53}
$$

where

$$Y\_{\mathcal{S}\parallel} = j\eta\_0^{-1} \frac{\lambda}{d} \frac{1}{\ln\left[\sin\left(\frac{\pi w}{2d}\right)\right]}\tag{54a}$$

The ABCD matrix of the structure, through which the transmission coefficients will be

<sup>23</sup> Novel Electromagnetic Phenomena in Graphene

*jη*−<sup>1</sup>

¯¯*<sup>I</sup>* <sup>−</sup>(*A*¯¯ <sup>+</sup> *<sup>η</sup>*0*<sup>B</sup>*

<sup>−</sup>*η*<sup>0</sup> ¯¯*<sup>I</sup>* <sup>−</sup>(*<sup>C</sup>*

 *S* ¯¯ <sup>21</sup> · **y**ˆ 

cos(*k*0*d*2) *jη*<sup>0</sup> sin(*k*0*d*2)

<sup>0</sup> sin(*k*0*d*2) cos(*k*0*d*2)

¯¯ <sup>+</sup> *<sup>η</sup>*0*D*¯¯ )

¯¯)

<sup>−</sup><sup>1</sup> ·

<sup>−</sup> 20 log10

Fig. 18 presents isolation in dB (color plot), the transmission loss from medium 1 to medium 2 in dB (continuous lines contour plot) and the rotation angle of graphene in free space (dashed lines contour plot) versus *<sup>μ</sup><sup>c</sup>* and *<sup>B</sup>*<sup>0</sup> for *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s. The results are derived for *d*<sup>1</sup> = *d*<sup>1</sup> = *λ*/4, which has been found to be the condition of the highest isolation. As an example of how to read Fig. 18, the point indicated by the circle corresponds to *B*<sup>0</sup> = 0.61 T, *μ<sup>c</sup>* = 0.11 eV, *I* = 20 dB, a 3 dB transmission loss from medium 1 towards medium 2 and 9◦ of rotation angle for graphene in free space. One observes that the isolation can be so high as 20 dB even when the rotation angle of graphene in free space is so low as 10◦. This contrasts with the ferrite-based isolator, where a 45◦ rotation angle is always required. This surprising behavior is attributed to the semi-transparent nature of graphene which forces the wave to follow a complex path between the grids, passing several times through graphene. The condition for which this complex path leads to constructive interference for propagation along the +*z* direction and destructive interference for propagation along the −*z* direction is *d*<sup>1</sup> = *d*<sup>2</sup> = *λ*/4. The main reason for the relatively high loss (3 dB) for transmission along the +*z* direction is the graphene loss, which is increased by the large effective wave path in the space between the grids. This loss can be reduced by using graphene with higher mobility

One of the most significant advantages of the proposed isolator is the control of the isolation direction via the chemical potential, exploiting the ambipolar properties of graphene. In particular, by inverting the chemical potential, the direction of Faraday rotation provided by graphene also inverts, which in turn inverts the isolation direction. Remember that the chemical potential can be easily controlled via an electrostatic voltage applied between graphene and an electrode parallel to graphene (Fig. 2). In such a way it could be possible to fabricate a non-reciprocal antenna radome, which dynamically converts an antenna from

As already pointed out, the structures presented in Section 5 are essentially "proofs of concept" and do not follow an optimal design. For example, a practical circular waveguide

cos(*k*0*d*1) *jη*<sup>0</sup> sin(*k*0*d*1)

<sup>0</sup> sin(*k*0*d*1) cos(*k*0*d*1)

 · ¯¯*I* 0 ¯¯ *Y* ¯¯ 2 ¯¯*I* 

<sup>−</sup> ¯¯*<sup>I</sup> <sup>A</sup>*

<sup>−</sup>*η*<sup>0</sup> ¯¯*<sup>I</sup> <sup>C</sup>*

 *S* ¯¯ 12 ·  · ¯¯*I* 0 ¯¯ *<sup>σ</sup>*¯¯1 ¯¯*<sup>I</sup>* 

¯¯ <sup>−</sup> *<sup>η</sup>*0*<sup>B</sup>* ¯¯

¯¯ <sup>−</sup> *<sup>η</sup>*0*D*¯¯

**<sup>x</sup>**<sup>ˆ</sup> <sup>+</sup> **<sup>y</sup>**<sup>ˆ</sup> <sup>√</sup> 2  . (56)

. (58)

(57)

computed, is

The S-matrix is then

 *A* ¯¯ *B* ¯¯ *C* ¯¯ *D*¯¯ = ¯¯*I* 0 ¯¯ *Y* ¯¯ 1 ¯¯*I* ·

·

 *S* ¯¯ =

and the isolation may be subsequently computed as

*I* = 20 log10

(lower scattering) and properly adjusting the structure parameters.

transmitting to receiving and vice versa through a proper control signal.

**6. Enabling of graphene gyrotropy with metamaterials**

*jη*−<sup>1</sup>

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

Fig. 16. Graphene-based non-reciprocal spatial isolator, consisting of a graphene sheet and two wire grids, one with wires parallel to the *x*-axis and the other with wires tilted by −45◦ with respect to the *x*-axis. A *y*-polarized wave propagating along the −*z*-direction passes through the isolator, while any wave propagating along the *z*-direction does not.

and

$$Y\_{\mathcal{S}^\perp} = -j\eta\_0^{-1} \frac{4d}{\lambda} \ln\left\{ \sin\left[\frac{\pi(d-w)}{2d}\right] \right\} \tag{54b}$$

are the parallel and the perpendicular to the wires admittances of the grid, respectively (Tretyakov, 2003). In Equation 54 *w* is the wire width and *d* is the periodicity of the grid. The shunt admittance of the grid at *z* = *d*<sup>2</sup> is

$$\bar{Y}\_2 = Y\_{\mathcal{g}\parallel} \hat{\mathbf{f}} \mathbf{f} + Y\_{\mathcal{g}\perp} (\hat{\mathbf{z}} \times \hat{\mathbf{f}}) (\hat{\mathbf{z}} \times \hat{\mathbf{f}}),\tag{55}$$

where ˆ**t** = (**x**ˆ + **y**ˆ)/ <sup>√</sup>2 is the unit vector parallel to the wires.

Fig. 17. Transmission line model of the proposed graphene-based non-reciprocal spatial isolator.

The ABCD matrix of the structure, through which the transmission coefficients will be computed, is

$$
\begin{bmatrix} \vec{A} \cdot \vec{B} \\ \vec{C} \cdot \vec{D} \end{bmatrix} = \begin{bmatrix} \vec{I} \cdot \vec{0} \\ \vec{Y}\_1 \cdot \vec{I} \end{bmatrix} \cdot \begin{bmatrix} \cos(k\_0 d\_1) & j\eta\_0 \sin(k\_0 d\_1) \\ j\eta\_0^{-1} \sin(k\_0 d\_1) & \cos(k\_0 d\_1) \end{bmatrix} \cdot \begin{bmatrix} \vec{I} \cdot \vec{0} \\ \vec{\sigma}\_1 \cdot \vec{I} \end{bmatrix}
$$

$$
\cdot \begin{bmatrix} \cos(k\_0 d\_2) & j\eta\_0 \sin(k\_0 d\_2) \\ j\eta\_0^{-1} \sin(k\_0 d\_2) & \cos(k\_0 d\_2) \end{bmatrix} \cdot \begin{bmatrix} \bar{I} \cdot \vec{0} \\ \bar{Y}\_2 \cdot \bar{I} \end{bmatrix} \cdot \tag{56}
$$

The S-matrix is then

20 Will-be-set-by-IN-TECH

*d*1

through the isolator, while any wave propagating along the *z*-direction does not.

<sup>√</sup>2 is the unit vector parallel to the wires.

*Yg*<sup>⊥</sup> <sup>=</sup> <sup>−</sup>*jη*−<sup>1</sup> 0 4*d <sup>λ</sup>* ln sin 

> *Y* ¯¯

*Y* ¯¯

The shunt admittance of the grid at *z* = *d*<sup>2</sup> is

Fig. 16. Graphene-based non-reciprocal spatial isolator, consisting of a graphene sheet and two wire grids, one with wires parallel to the *x*-axis and the other with wires tilted by −45◦ with respect to the *x*-axis. A *y*-polarized wave propagating along the −*z*-direction passes

are the parallel and the perpendicular to the wires admittances of the grid, respectively (Tretyakov, 2003). In Equation 54 *w* is the wire width and *d* is the periodicity of the grid.

<sup>1</sup> *σ*¯¯ *Y*

Fig. 17. Transmission line model of the proposed graphene-based non-reciprocal spatial

<sup>2</sup> *k*0, *η*<sup>0</sup> *k*0, *η*<sup>0</sup> *k*0, *η*<sup>0</sup> *k*0, *η*<sup>0</sup>

1 2

*d*<sup>1</sup> *d*<sup>2</sup>

*π*(*d* − *w*) 2*d*

<sup>2</sup> <sup>=</sup> *Yg*�ˆ**t**ˆ**<sup>t</sup>** <sup>+</sup> *Yg*⊥(**z**<sup>ˆ</sup> <sup>×</sup> <sup>ˆ</sup>**t**)(**z**<sup>ˆ</sup> <sup>×</sup> <sup>ˆ</sup>**t**), (55)

¯¯

(54b)

*d*<sup>2</sup> **B**<sup>0</sup>

2

and

where ˆ**t** = (**x**ˆ + **y**ˆ)/

isolator.

*x*

*z*

1

*y*

$$
\begin{bmatrix} \vec{\mathcal{S}} \end{bmatrix} = \begin{bmatrix} \vec{I} & -(\vec{A} + \eta\_0 \vec{B}) \\ -\eta\_0 \vec{I} - (\vec{\mathcal{C}} + \eta\_0 \vec{D}) \end{bmatrix}^{-1} \cdot \begin{bmatrix} -\vec{I} & \vec{A} - \eta\_0 \vec{B} \\ -\eta\_0 \vec{I} \cdot \vec{C} - \eta\_0 \vec{D} \end{bmatrix} \tag{57}
$$

and the isolation may be subsequently computed as

$$I = 20\log\_{10}\left|\bar{\tilde{S}}\_{21} \cdot \hat{\mathfrak{y}}\right| - 20\log\_{10}\left|\bar{\tilde{S}}\_{12} \cdot \frac{\hat{\mathfrak{x}} + \hat{\mathfrak{y}}}{\sqrt{2}}\right|.\tag{58}$$

Fig. 18 presents isolation in dB (color plot), the transmission loss from medium 1 to medium 2 in dB (continuous lines contour plot) and the rotation angle of graphene in free space (dashed lines contour plot) versus *<sup>μ</sup><sup>c</sup>* and *<sup>B</sup>*<sup>0</sup> for *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s. The results are derived for *d*<sup>1</sup> = *d*<sup>1</sup> = *λ*/4, which has been found to be the condition of the highest isolation. As an example of how to read Fig. 18, the point indicated by the circle corresponds to *B*<sup>0</sup> = 0.61 T, *μ<sup>c</sup>* = 0.11 eV, *I* = 20 dB, a 3 dB transmission loss from medium 1 towards medium 2 and 9◦ of rotation angle for graphene in free space. One observes that the isolation can be so high as 20 dB even when the rotation angle of graphene in free space is so low as 10◦. This contrasts with the ferrite-based isolator, where a 45◦ rotation angle is always required. This surprising behavior is attributed to the semi-transparent nature of graphene which forces the wave to follow a complex path between the grids, passing several times through graphene. The condition for which this complex path leads to constructive interference for propagation along the +*z* direction and destructive interference for propagation along the −*z* direction is *d*<sup>1</sup> = *d*<sup>2</sup> = *λ*/4. The main reason for the relatively high loss (3 dB) for transmission along the +*z* direction is the graphene loss, which is increased by the large effective wave path in the space between the grids. This loss can be reduced by using graphene with higher mobility (lower scattering) and properly adjusting the structure parameters.

One of the most significant advantages of the proposed isolator is the control of the isolation direction via the chemical potential, exploiting the ambipolar properties of graphene. In particular, by inverting the chemical potential, the direction of Faraday rotation provided by graphene also inverts, which in turn inverts the isolation direction. Remember that the chemical potential can be easily controlled via an electrostatic voltage applied between graphene and an electrode parallel to graphene (Fig. 2). In such a way it could be possible to fabricate a non-reciprocal antenna radome, which dynamically converts an antenna from transmitting to receiving and vice versa through a proper control signal.

#### **6. Enabling of graphene gyrotropy with metamaterials**

As already pointed out, the structures presented in Section 5 are essentially "proofs of concept" and do not follow an optimal design. For example, a practical circular waveguide

Verdet constant (rotation per unit thickness and unit magnetic field). Such an artificial material could be also used in the spatial isolator to reduce significantly its size. In a next nanoscale level, ferromagnetic nanowires membranes (Carignan et al., 2011), which are transparent to normally incident waves, could be introduced between the successive graphene layers, as illustrated in Fig. 19(b) to provide the necessary static magnetic field, making the material self-biased. Finally, boron nitride (BN) could be selected as the separating dielectric between the graphene sheets, since as it has been recently shown, graphene sandwiched between BN layers exhibits extremely high mobility, sometimes approaching that of suspended graphene (Mayorov et al., 2011). It is therefore clear that by exploiting metamaterials concept in the micro, nano and atomic scale simultaneously (Caloz et al., n.d.), novel material with

<sup>25</sup> Novel Electromagnetic Phenomena in Graphene

and Subsequent Microwave Devices Enabled by Multi-Scale Metamaterials

The gyrotropic properties of magnetically biased graphene have been analyzed. Graphene exhibits strong non-reciprocity and extremely broadband Faraday rotation at microwave frequencies, below the cyclotron resonance. Two degrees of freedom may be used to control the amount and the direction of rotation, namely the chemical potential (via an applied static voltage) and the static magnetic field. This allows full tuning of Faraday rotation via the chemical potential, while keeping the magnetic field constant, which is usually provided by permanent magnets and is thus difficult to control. Two applications of graphene gyrotropy have been proposed: a circular waveguide Faraday rotator and a spatial isolator. A practical realization of such devices with unprecedented characteristics, such as super compactness, is possible through the multi-scale (micro, nano and atomic scales) metamaterial concept.

Abramowitz, M. & Stegun, I. A. (1964). *Handbook of mathematical functions with formulas, graphs,*

Caloz, C., Carignan, L.-P., Ménard, D. & Yelon, A. (n.d.). The concept of multi-scale

Caloz, C. & Itoh, T. (2006). *Electromagnetic Metamaterials: Transmission Line Theory, and Microwave Applications: The Engineering Approach*, John Wiley & Sons. Carignan, L.-P., Yelon, A., Ménard, D. & Caloz, C. (2011). Ferromagnetic nanowire

Crassee, I., Levallois, J., Walter, A. L., Ostler, M., Bostwick, A., Rotenberg, E., Seyller, T.,

Dragoman, M. & Dragoman, D. (2009). *Nanoelectronics: principles and devices*, Artech House. Geim, A. K. & Novoselov, K. S. (2007). The rise of graphene, *Nature Materials* 6: 183–191. Gusynin, V. P., Sharapov, S. G. & Carbotte, J. P. (2009). On the universal ac optical background

Koppens, F. H. L., Chang, D. E. & García de Abajo, F. J. (2011). Graphene plasmonics: A platform for strong light–matter interactions, *Nano Letters* 11(8): 3370–3377.

metamaterials: theory and applications, *IEEE Trans. Microwave Theory Tech.*

van der Marel, D. & Kuzmenko, A. B. (2011). Giant faraday rotation in single- and

*and mathematical tables*, Vol. 55, Courier Dover publications. Balanis, C. A. (2005). *Antenna theory: analysis and design*, John Wiley & Sons.

unprecedented properties can be created.

**7. Conclusions**

**8. References**

metamaterials.

59(10): 2568–2586.

Collin, R. E. (1990). *Field theory of guided waves*, IEEE Press.

multilayer graphene, *Nature Phys.* 7: 48–51.

Hanson, G. W. (2007). *Fundamentals of Nanoelectronics*, Prentice Hall.

in graphene, *New J. Phys.* 11: 095013.

Fig. 18. Color plot: isolation in dB achieved by the proposed graphene-based non-reciprocal spatial isolator versus *<sup>μ</sup><sup>c</sup>* and *<sup>B</sup>*<sup>0</sup> for *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s. Solid lines contour plot: power transmission coefficient in dB for the wave allowed to pass through the isolator. Dashed lines contour plot: polarization rotation angle of the graphene sheet in free space.

Faraday rotator, which is a necessary component in a waveguide isolator or a gyrator, requires a rotation angle of at least 90◦. Although the rotation angle provided by a single graphene sheet is huge considering its one-atom thickness, it is quite far from the value required in a waveguide isolator. In order to achieve a rotation angle so high as 90◦ it is necessary to stack several graphene sheets. The material and the spacing between the sheets as well as the rotation angle provided by each sheet must be carefully chosen so that the transmission through such a structure be maximum. In such a way, a microscale metamaterial, similar to this illustrated in Fig. 19(a), can be conceived. Such a metamaterial provides a gradual rotation of the wave polarization, thus mimicking a ferrite medium with, however, a much larger

Fig. 19. Graphene-based multiscale metamaterials, which exhibit a ferrite-type response, however, with a higher rotation power than ferrites. (a) Micro-scale metamaterial consisting of a period stack of graphene sheets and dielectric layers. (b) Multiscale metamaterial consisting of a periodic stack of graphene sheets, ferromagnetic nanowires (FMNW) membranes and dielectric layers. FMNW metamaterial membranes are used for biasing graphene.

Verdet constant (rotation per unit thickness and unit magnetic field). Such an artificial material could be also used in the spatial isolator to reduce significantly its size. In a next nanoscale level, ferromagnetic nanowires membranes (Carignan et al., 2011), which are transparent to normally incident waves, could be introduced between the successive graphene layers, as illustrated in Fig. 19(b) to provide the necessary static magnetic field, making the material self-biased. Finally, boron nitride (BN) could be selected as the separating dielectric between the graphene sheets, since as it has been recently shown, graphene sandwiched between BN layers exhibits extremely high mobility, sometimes approaching that of suspended graphene (Mayorov et al., 2011). It is therefore clear that by exploiting metamaterials concept in the micro, nano and atomic scale simultaneously (Caloz et al., n.d.), novel material with unprecedented properties can be created.

#### **7. Conclusions**

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

Fig. 18. Color plot: isolation in dB achieved by the proposed graphene-based non-reciprocal spatial isolator versus *<sup>μ</sup><sup>c</sup>* and *<sup>B</sup>*<sup>0</sup> for *<sup>τ</sup>* <sup>=</sup> 1.84 <sup>×</sup> <sup>10</sup>−<sup>13</sup> s. Solid lines contour plot: power transmission coefficient in dB for the wave allowed to pass through the isolator. Dashed lines

Faraday rotator, which is a necessary component in a waveguide isolator or a gyrator, requires a rotation angle of at least 90◦. Although the rotation angle provided by a single graphene sheet is huge considering its one-atom thickness, it is quite far from the value required in a waveguide isolator. In order to achieve a rotation angle so high as 90◦ it is necessary to stack several graphene sheets. The material and the spacing between the sheets as well as the rotation angle provided by each sheet must be carefully chosen so that the transmission through such a structure be maximum. In such a way, a microscale metamaterial, similar to this illustrated in Fig. 19(a), can be conceived. Such a metamaterial provides a gradual rotation of the wave polarization, thus mimicking a ferrite medium with, however, a much larger

FMNWs

Fig. 19. Graphene-based multiscale metamaterials, which exhibit a ferrite-type response, however, with a higher rotation power than ferrites. (a) Micro-scale metamaterial consisting of a period stack of graphene sheets and dielectric layers. (b) Multiscale metamaterial consisting of a periodic stack of graphene sheets, ferromagnetic nanowires (FMNW) membranes and dielectric layers. FMNW metamaterial membranes are used for biasing

(b)

contour plot: polarization rotation angle of the graphene sheet in free space.

(a)

graphene.

The gyrotropic properties of magnetically biased graphene have been analyzed. Graphene exhibits strong non-reciprocity and extremely broadband Faraday rotation at microwave frequencies, below the cyclotron resonance. Two degrees of freedom may be used to control the amount and the direction of rotation, namely the chemical potential (via an applied static voltage) and the static magnetic field. This allows full tuning of Faraday rotation via the chemical potential, while keeping the magnetic field constant, which is usually provided by permanent magnets and is thus difficult to control. Two applications of graphene gyrotropy have been proposed: a circular waveguide Faraday rotator and a spatial isolator. A practical realization of such devices with unprecedented characteristics, such as super compactness, is possible through the multi-scale (micro, nano and atomic scales) metamaterial concept.

#### **8. References**


Collin, R. E. (1990). *Field theory of guided waves*, IEEE Press.


Geim, A. K. & Novoselov, K. S. (2007). The rise of graphene, *Nature Materials* 6: 183–191.


**2** 

*Lithuania* 

**Electrodynamical Analysis** 

**of Open Lossy Metamaterial** 

**Waveguide and Scattering Structures** 

L. Nickelson, S. Asmontas, T. Gric, J. Bucinskas and A. Bubnelis *State Research Institute Center for Physical Sciences and Technology, Vilnius,* 

Large stream of articles devoted to the study of metamaterial waveguide and metamaterial scattering (reflecting) structures points that there is a need for development devices possessing unique characteristics, as multifunctionality, reconfigurability, certain frequency bandwidth, ability to operate at high-powers and high-radiation conditions. The importance of diffraction problems for scattering structures is based on their great practical utility for many applications, such as reflector antennas, the analysis of structures in open space, electromagnetic (EM) defence of structures, the scattering modeling for remote sensing purposes, high frequency telecommunications, computer network, invisibility cloaks technology and radar systems (Li et.al., 2011; Zhou et.al.; 2011, Zhu et al., 2010; Mirza et al.,

The technological potential of metamaterials for developing novel devices offers a very promising alternative that could potentially overcome the limitations of current technology. The metamaterial waveguide and scattering structures can operate as different devices that possess different specific qualities as well. In order to create a new microwave device it is necessary to know the main electrodynamical characteristics of metamaterial structures on

Here are presented electrical field distributions and dispersion characteristics of open metamaterial waveguides in subsections 1-3 and numerical analysis of the scattered and absorbed microwave powers of the layered metamaterial cylinders in the subsection 4.

Here the open (without conductor screen) square metamaterial waveguides with sizes 5x5·mm2 and 4x4· mm2 are investigated by our algorithms that were created on the base of the Singular Integral equations' (SIE) method (Nickelson & Sugurov, 2005; Nickelson et al., InTech2011). Due to the fact that metamaterial is a substance with losses we have determined the complex roots of the dispersion equation by using of the Muller method in

**2. Analyses of electrical field and dispersion characteristics of square** 

2009; Abdalla & Hu, 2009; Engheta & Ziolkowski, 2005).

the basis of which the device is supposed to be created.

**metamaterial lossy waveguides by the SIE method** 

our researches (Nickelson et al., InTech2011).

**1. Introduction**


### **Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures**

L. Nickelson, S. Asmontas, T. Gric, J. Bucinskas and A. Bubnelis *State Research Institute Center for Physical Sciences and Technology, Vilnius, Lithuania* 

#### **1. Introduction**

24 Will-be-set-by-IN-TECH

26 Metamaterial

Kubo, R. (1957). Statistical-mechanical theory of irreversible processes. I. general theory

Lin, Y.-M., Jenkins, K. A., Valdes-Garcia, A., Small, J. P., Farmer, D. B. & Avouris, P. (2009). Operation of graphene transistors at gigahertz frequencies, *Nano Lett.* 9: 422–426.

Marqués, R., Martín, F. & Sorolla, M. (2008). *Metamaterials with Negative Parameters: Theory,*

Mayorov, A. S., Gorbachev, R. V., Morozov, S. V., Britnell, L., Jalil, R., Ponomarenko, L. A.,

Mikhailov, S. A. & Ziegler, K. (2007). New electromagnetic mode in graphene, *Phys. Rev. Lett.*

Mishchenko, E. G., Shytov, A. V. & Silvestrov, P. G. (2010). Guided plasmons in graphene *p*-*n*

Neto, A. H. C., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. (2009). The electronic

Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D., Zhang, Y., Dubonos, S. V., Grigorieva,

Parsa, A., Kodera, T. & Caloz, C. (2010). Ferrite based non-reciprocal radome, generalized

Sounas, D. L. & Caloz, C. (2011a). Edge surface modes in magnetically-biased

Sounas, D. L. & Caloz, C. (2011b). Electromagnetic non-reciprocity and gyrotropy of graphene,

Sounas, D. L. & Caloz, C. (2011c). Graphene-based non-reciprocal spatial isolator, *Proc. IEEE International Symposium on Antennas and Propagation, Spokane, WA, (APS 2011)*. Sounas, D. L. & Caloz, C. (2011d). Gyrotropy and non-reciprocity of graphene for microwave

Vakil, A. & Engheta, N. (2011). Transformation optics using Graphene, *Science*

Wang, H., Nezich, D., Kong, J. & Palacios, T. (2009). Graphene frequency multipliers, *IEEE*

chemically-doped graphene strips, *App. Plys. Lett.* . under review.

applications, *IEEE Trans. Microw. Theory Tech.* . under review. Tretyakov, S. (2003). *Analytical modelling in applied electromagnetics*, Artech House, Inc.

I. V. & Firsov, A. A. (2004). Electric field effect in atomically thin carbon films, *Science*

scattering matrix analysis and experimental demonstration, *IEEE Trans. Antennas*

Blake, P., Novoselov, K. S., Watanabe, K., Taniguchi, T. & Geim, A. K. (2011). Micrometer-scale ballistic transport in encapsulated graphene at room temperature,

Lax, B. & Button, K. J. (1962). *Microwave Ferrites and Ferrimagnetics*, McGraw-Hill.

Lindell, I. V. (1996). *Methods for electromagnetic field analysis*, IEEE Press.

*Design and Microwave Applications*, John Wiley & Sons.

properties of graphene, *Rev. Mod. Phys.* 81: 109–162.

12(6): 570–586.

99: 016803.

306: 666–669.

*Propag.* . in press.

*Appl. Phys. Lett.* 98: 021911.

332(6035): 1291–1294.

*Electron Device Lett.* 30: 547–549.

*Nano Lett.* 11(6): 2396–2399.

junctions, *Phys. Rev. Lett.* 104: 156806.

and simple applications to magnetic and conduction problems, *J. Phys. Soc. Japan*

Large stream of articles devoted to the study of metamaterial waveguide and metamaterial scattering (reflecting) structures points that there is a need for development devices possessing unique characteristics, as multifunctionality, reconfigurability, certain frequency bandwidth, ability to operate at high-powers and high-radiation conditions. The importance of diffraction problems for scattering structures is based on their great practical utility for many applications, such as reflector antennas, the analysis of structures in open space, electromagnetic (EM) defence of structures, the scattering modeling for remote sensing purposes, high frequency telecommunications, computer network, invisibility cloaks technology and radar systems (Li et.al., 2011; Zhou et.al.; 2011, Zhu et al., 2010; Mirza et al., 2009; Abdalla & Hu, 2009; Engheta & Ziolkowski, 2005).

The technological potential of metamaterials for developing novel devices offers a very promising alternative that could potentially overcome the limitations of current technology. The metamaterial waveguide and scattering structures can operate as different devices that possess different specific qualities as well. In order to create a new microwave device it is necessary to know the main electrodynamical characteristics of metamaterial structures on the basis of which the device is supposed to be created.

Here are presented electrical field distributions and dispersion characteristics of open metamaterial waveguides in subsections 1-3 and numerical analysis of the scattered and absorbed microwave powers of the layered metamaterial cylinders in the subsection 4.

#### **2. Analyses of electrical field and dispersion characteristics of square metamaterial lossy waveguides by the SIE method**

Here the open (without conductor screen) square metamaterial waveguides with sizes 5x5·mm2 and 4x4· mm2 are investigated by our algorithms that were created on the base of the Singular Integral equations' (SIE) method (Nickelson & Sugurov, 2005; Nickelson et al., InTech2011). Due to the fact that metamaterial is a substance with losses we have determined the complex roots of the dispersion equation by using of the Muller method in our researches (Nickelson et al., InTech2011).

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 29

from 105 GHz till 115 GHz. This feature could be used in practice for creation of feeder lines

and specific devices that require low distortions in the signal transmission.

(a) (b)

metamaterial has εmet = -10.83-i0.02 and μmet = 0.5 – i 0.01 at f = 110 GHz.

mm2 metamaterial waveguide. (a) – f= 95 GHz, (b) – f=110 GHz.

metamaterial is the important feature of this kind metamaterial.

solutions of considered electrodynamical problems.

section.

Fig. 2.2. The 3D electric field distribution of the main mode propagating in the square 5x5

The 3D electric field distributions of the main mode at f = 95 GHz and 110 GHz are shown in Fig. 2.2. The metamaterial has εmet = -23.75-i 18.75 and μmet = 1.75+i 6.25 at f = 95 GHz. The

We see that the waveguide losses are large at f = 95 GHz and they are low at f = 110 GHz (Fig. 2.1(b)). We present here the electric field line distribution at these two frequencies in order to compare how the losses influence on the field picture. The calculations of the electric fields in this section were fulfilled at the approximately 10000 points in every cross-

The metamaterial is an epsilon-negative media at f = 95 GHz and f = 110 GHz. In Fig. 2.2(a) we see the electric field is very small in the center of the waveguide cross-section. Such distribution can be explained by the large loss at f = 95 GHz and the EM wave does not deeply penetrate into the metamaterial. The refractive index of single negative metamaterial and the transverse propagation constants of waveguide made of the metamaterial are imaginary numbers. For this reason the electric field concentrates near the waveguide border. The tendency of the field to concentrate at the interface of the single negative

The behaviour of the EM field components when approaching to the apex of the waveguide cross-section contour (waveguide edge) are an important point of an electrodynamical solution. We would like to note that the condition on the waveguide edge is satisfied in our

Examining the electric field lines near the upper right corner of the square waveguide we would like to note that the electric field lines are directed counter-clockwise on the right side of the corner and the lines are directed clockwise on the left side of the corner. The electric

Our computer programmes are written in MATLAB. They let us to investigate open absorptive waveguides with different shapes of cross-section (Gric & Nickelson, 2011). We have used the values of the complex relative permittivity εmet and the complex relative permeability μmet of metamaterial from (Penciu et al, 2006) in suctions 1 and 2.

#### **2.1 Numerical investigations of square 5x5 mm2 metamaterial waveguide**

The dispersion characteristics and the 3D electric field distributions are presented here (Figs 1.1–1.3). The characteristics are shown in Fig. 2.1 (Gric et al., 2010). The main wave (mode) is denoted with black points; the first higher mode is denoted with circles. The values of εmet and μmet are different at every frequency. The real part of the permittivity is always negative at the all frequency range 75-115 GHz. The imaginary part of the permittivity is negative approximately when 90 ≤ *f* ≤ 100 GHz. The real part of the permeability is negative when 100 ≤ *f* ≤ 105 GHz. The imaginary part of the permeability is always equal to zero or a positive number at the all mentioned frequency range. In the Fig. 2.1 are presented dependencies of the complex propagation constant h = h'- i h" on the frequency.

Fig. 2.1. Dispersion characteristics of square 5x5 mm2 metamaterial waveguide (a) – the dependence of the normalized propagation constant*,* (b) – the dependence of the attenuation constant on frequencies.

In Fig. 2.1(a) we see that dependencies of the normalized propagation constant h'/k on the frequency, where h'=2·π/λ and λ is the wavelength of microwave in the metamaterial waveguide, k=2 π f/c, k is the wavenumber in vacuum, f is an operating frequency, c is the speed of light in vacuum. The curves of the main mode and the first higher mode are not smooth. The magnitudes h'/k <1 for both modes (Fig. 2.1(a)). It means that the main and the first higher modes are the fast waveguide waves. In Fig. 2.1(b) are shown dependencies of the waveguide attenuation constant (losses) h" on the frequency. We see that the values of the main and higher mode losses are commensurate and the losses are not higher in the frequency range. The loss maximums of the main mode and the first higher mode are slightly shifted. The maximum of main mode losses is 0.32 dB/mm at f = 87.5 GHz and the wavelength of this mode is equal to 4.2·mm. The maximum of the first higher mode is 0.36 dB/mm at f = 90 GHz and the wavelength of this mode is equal to 4·mm (Fig. 2.1). It is important to remark that the losses of the main mode are very low at the frequency range

Our computer programmes are written in MATLAB. They let us to investigate open absorptive waveguides with different shapes of cross-section (Gric & Nickelson, 2011). We have used the values of the complex relative permittivity εmet and the complex relative

The dispersion characteristics and the 3D electric field distributions are presented here (Figs 1.1–1.3). The characteristics are shown in Fig. 2.1 (Gric et al., 2010). The main wave (mode) is denoted with black points; the first higher mode is denoted with circles. The values of εmet and μmet are different at every frequency. The real part of the permittivity is always negative at the all frequency range 75-115 GHz. The imaginary part of the permittivity is negative approximately when 90 ≤ *f* ≤ 100 GHz. The real part of the permeability is negative when 100 ≤ *f* ≤ 105 GHz. The imaginary part of the permeability is always equal to zero or a positive number at the all mentioned frequency range. In the Fig. 2.1 are presented

> 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

h'', dB/mm

75 85 95 105 115 f, GHz

 **metamaterial waveguide** 

permeability μmet of metamaterial from (Penciu et al, 2006) in suctions 1 and 2.

dependencies of the complex propagation constant h = h'- i h" on the frequency.

**2.1 Numerical investigations of square 5x5 mm2**

75 85 95 105 115 f, GHz

attenuation constant on frequencies.

(a) (b)

Fig. 2.1. Dispersion characteristics of square 5x5 mm2 metamaterial waveguide

(a) – the dependence of the normalized propagation constant*,* (b) – the dependence of the

In Fig. 2.1(a) we see that dependencies of the normalized propagation constant h'/k on the frequency, where h'=2·π/λ and λ is the wavelength of microwave in the metamaterial waveguide, k=2 π f/c, k is the wavenumber in vacuum, f is an operating frequency, c is the speed of light in vacuum. The curves of the main mode and the first higher mode are not smooth. The magnitudes h'/k <1 for both modes (Fig. 2.1(a)). It means that the main and the first higher modes are the fast waveguide waves. In Fig. 2.1(b) are shown dependencies of the waveguide attenuation constant (losses) h" on the frequency. We see that the values of the main and higher mode losses are commensurate and the losses are not higher in the frequency range. The loss maximums of the main mode and the first higher mode are slightly shifted. The maximum of main mode losses is 0.32 dB/mm at f = 87.5 GHz and the wavelength of this mode is equal to 4.2·mm. The maximum of the first higher mode is 0.36 dB/mm at f = 90 GHz and the wavelength of this mode is equal to 4·mm (Fig. 2.1). It is important to remark that the losses of the main mode are very low at the frequency range

0.75 0.77 0.79 0.81 0.83 0.85 0.87 0.89 0.91 0.93 0.95

h'/k

from 105 GHz till 115 GHz. This feature could be used in practice for creation of feeder lines and specific devices that require low distortions in the signal transmission.

Fig. 2.2. The 3D electric field distribution of the main mode propagating in the square 5x5 mm2 metamaterial waveguide. (a) – f= 95 GHz, (b) – f=110 GHz.

The 3D electric field distributions of the main mode at f = 95 GHz and 110 GHz are shown in Fig. 2.2. The metamaterial has εmet = -23.75-i 18.75 and μmet = 1.75+i 6.25 at f = 95 GHz. The metamaterial has εmet = -10.83-i0.02 and μmet = 0.5 – i 0.01 at f = 110 GHz.

We see that the waveguide losses are large at f = 95 GHz and they are low at f = 110 GHz (Fig. 2.1(b)). We present here the electric field line distribution at these two frequencies in order to compare how the losses influence on the field picture. The calculations of the electric fields in this section were fulfilled at the approximately 10000 points in every crosssection.

The metamaterial is an epsilon-negative media at f = 95 GHz and f = 110 GHz. In Fig. 2.2(a) we see the electric field is very small in the center of the waveguide cross-section. Such distribution can be explained by the large loss at f = 95 GHz and the EM wave does not deeply penetrate into the metamaterial. The refractive index of single negative metamaterial and the transverse propagation constants of waveguide made of the metamaterial are imaginary numbers. For this reason the electric field concentrates near the waveguide border. The tendency of the field to concentrate at the interface of the single negative metamaterial is the important feature of this kind metamaterial.

The behaviour of the EM field components when approaching to the apex of the waveguide cross-section contour (waveguide edge) are an important point of an electrodynamical solution. We would like to note that the condition on the waveguide edge is satisfied in our solutions of considered electrodynamical problems.

Examining the electric field lines near the upper right corner of the square waveguide we would like to note that the electric field lines are directed counter-clockwise on the right side of the corner and the lines are directed clockwise on the left side of the corner. The electric

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 31

(a) (b)

5x5 mm2 metamaterial waveguide. a) – f= 95 GHz, b) – f= 110 GHz.

and relatively large at this frequency.

**2.2 Numerical investigations of square 4x4 mm2**

(Gric et al., 2010). We used the values of ε<sup>r</sup>

very small at the frequency 103.75 GHz.

(Figs 2.2-2.3).

Fig. 2.3. The 3D electric field distribution of the first higher mode propagating in the square

In Fig. 2.3(a) we see that the electric field at f = 95 GHz for the first higher modes is also concentrated near the metamaterial borders and the strongest field is at two diagonal corners of the cross-section, i.e. at the right upper corner and the left bottom corner. There is a strong asymmetry of the electric field distribution on the perimeter of waveguide. It happened probably by reason that the real and imaginary parts of permittivity are negative

In Figs 2.2(b)–2.3 (b) we see that the electric field distributions of the main and first higher modes have a more homogeneous picture in the waveguide cross-section at f = 110 GHz. This happened because the electric field penetrates deeper into the metamaterial at this frequency because the waveguide loss is small at f = 110 GHz (Fig. 2.1(b)). We can also see here some asymmetry of electric field lines on the waveguide cross-section. The projections of the vector electric fields on the waveguide sidewalls are depicted along the waveguide

The dispersion characteristics of the square metamaterial waveguide are presented in Fig.2.4

the normalized propagation constant h'/k is shown. In Fig. 2.4(a) we see that the magnitude h' is less than the wave number *k* in the frequency range 85-108 GHz. It means that at these frequencies the main and the first higher modes are fast waves. The wavelengths of both modes differ slightly. The losses of both propagating modes in the metamaterial square waveguide change in very complicated way when the frequency increases. Mode losses have several maximums and minimums. We see that the losses of the main mode become a

m and μ<sup>r</sup>

 **metamaterial waveguide** 

m from (Penciu et al, 2006). In Fig. 2.4(a)

field lines near the left bottom corner of the square waveguide are distributed in similar way. The lines are directed counter-clockwise on the right side of the corner and they are directed clockwise on the left side of the corner. The electric field lines are diverging of the left bottom corner while the lines are converging to the right upper corner. For this reason, we believe that there is an increased density of electric lines in the upper corner and the weaker density in the bottom corner (Fig. 2.2(a)).

The same effect of no uniformity in the electric field distribution on the waveguide perimeter we can see for the circular waveguide (see below in section 2, Figs 2.2, 2.3).


Table 2.1. The EM field components of the main mode at the point with coordinates *x* = 4 *mm* and *y* = 4 *mm* when *f*=95 GHz and *f*=110 *GHz*


Table 2.2. The EM field components of the first higher mode at the point with coordinates x = 4 mm and y = 4 mm when f = 95 GHz and f = 110 GHz

In tables 2.1 and 2.2 we demonstrate the values of complex EM field components of the main mode and the first higher mode at two frequencies. On the base of the table data we can say, that the both modes on these frequencies are hybrid modes.

We do not classified the waveguide modes here in the usual way, e.g. the hybrid magnetic HEmn or the hybrid electric EHmn modes because the kind of mode of strong lossy waveguides may change when we change the frequency [Nickelson et al., 2011; Asmontas et al., 2010].

field lines near the left bottom corner of the square waveguide are distributed in similar way. The lines are directed counter-clockwise on the right side of the corner and they are directed clockwise on the left side of the corner. The electric field lines are diverging of the left bottom corner while the lines are converging to the right upper corner. For this reason, we believe that there is an increased density of electric lines in the upper corner and the

The same effect of no uniformity in the electric field distribution on the waveguide

f = 95 GHz, square waveguide 5x5 mm2 Ez [V/m] Ex [V/m] Ey [V/m] 3.765·10-1 - i 6.411·10-1 1.5099-i 2.1941 -0.9943+i 1.9673 Hz [A/m] Hx [A/m] Hy [A/m] -3.39·10-2 + i 7.51·10-2 -7·10-4 + i1.1·10-3 2.6·10-3 – i 3.9·10-3 f = 110 GHz Ez [V/m] Ex [V/m] Ey [V/m] -1.2766·10-5 – i 1.6592·10-5 0.0134 + i 0.0211 -0.0291 + i 0.0702 Hz [A/m] Hx [A/m] Hy [A/m] -3.0·10-3 - i 2.9·10-3 1.3048·10-4 –i 3.0558·10-4 5.9501 10-5+i 1.0936·10-4

perimeter we can see for the circular waveguide (see below in section 2, Figs 2.2, 2.3).

Table 2.1. The EM field components of the main mode at the point with coordinates

Table 2.2. The EM field components of the first higher mode at the point with coordinates x = 4 mm and y = 4 mm when f = 95 GHz and f = 110 GHz

that the both modes on these frequencies are hybrid modes.

al., 2010].

f = 95 GHz, square waveguide 5x5 mm2 Ez [V/m] Ex [V/m] Ey [V/m] 3.932·10-1 - i6.181·10-1 1.3730 - i1.8577 -9.535·10-1+ i 1.6598 Hz [A/m] Hx [A/m] Hy [A/m] -3.20·10-2+ i6.36·10-2 -8·10-4 + i1.4·10-3 2.5·10-3- i3.4·10-3 f = 110 GHz Ez [V/m] Ex [V/m] Ey [V/m] 1.3083·10-6-i1.2678·10-5 -4.6·10-3 + i2.30·10-2 -2.63·10-2 + i4.31·10-2 Hz [A/m] Hx [A/m] Hy [A/m] -1.9·10-3 - i2.4·10-3 1.0715·10-4 - i1.7759·10-4 -2.7720·10-5+i1.0632·10-4

In tables 2.1 and 2.2 we demonstrate the values of complex EM field components of the main mode and the first higher mode at two frequencies. On the base of the table data we can say,

We do not classified the waveguide modes here in the usual way, e.g. the hybrid magnetic HEmn or the hybrid electric EHmn modes because the kind of mode of strong lossy waveguides may change when we change the frequency [Nickelson et al., 2011; Asmontas et

weaker density in the bottom corner (Fig. 2.2(a)).

*x* = 4 *mm* and *y* = 4 *mm* when *f*=95 GHz and *f*=110 *GHz*

Fig. 2.3. The 3D electric field distribution of the first higher mode propagating in the square 5x5 mm2 metamaterial waveguide. a) – f= 95 GHz, b) – f= 110 GHz.

In Fig. 2.3(a) we see that the electric field at f = 95 GHz for the first higher modes is also concentrated near the metamaterial borders and the strongest field is at two diagonal corners of the cross-section, i.e. at the right upper corner and the left bottom corner. There is a strong asymmetry of the electric field distribution on the perimeter of waveguide. It happened probably by reason that the real and imaginary parts of permittivity are negative and relatively large at this frequency.

In Figs 2.2(b)–2.3 (b) we see that the electric field distributions of the main and first higher modes have a more homogeneous picture in the waveguide cross-section at f = 110 GHz. This happened because the electric field penetrates deeper into the metamaterial at this frequency because the waveguide loss is small at f = 110 GHz (Fig. 2.1(b)). We can also see here some asymmetry of electric field lines on the waveguide cross-section. The projections of the vector electric fields on the waveguide sidewalls are depicted along the waveguide (Figs 2.2-2.3).

#### **2.2 Numerical investigations of square 4x4 mm2 metamaterial waveguide**

The dispersion characteristics of the square metamaterial waveguide are presented in Fig.2.4 (Gric et al., 2010). We used the values of ε<sup>r</sup> m and μ<sup>r</sup> m from (Penciu et al, 2006). In Fig. 2.4(a) the normalized propagation constant h'/k is shown. In Fig. 2.4(a) we see that the magnitude h' is less than the wave number *k* in the frequency range 85-108 GHz. It means that at these frequencies the main and the first higher modes are fast waves. The wavelengths of both modes differ slightly. The losses of both propagating modes in the metamaterial square waveguide change in very complicated way when the frequency increases. Mode losses have several maximums and minimums. We see that the losses of the main mode become a very small at the frequency 103.75 GHz.

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 33

Here the open circular metamaterial waveguide is investigated by the partial area method (Nickelson et al., 2008). The presentation of longitudinal components of the electric Ezm and magnetic Hzm fields that satisfy to the Maxwell's equations in the metamaterial medium is in

<sup>⊥</sup> r)exp(imφ), Hzm =B1Jm(k <sup>+</sup>

where A1, B1 are unknown arbitrary amplitudes*,* Jm is the Bessel function of the m−th order,

where A2, B2 are unknown arbitrary amplitudes, Hm is the Hankel function of the m−th

As far as the circular waveguide is researched in the cylindrical coordinate system we have to satisfy the boundary conditions for two components of the electric field (Eφ, Ez) and the magnetic field (Hφ, Hz) on the cylindrical interface metamaterial-air. The condition at

hollow-core waveguide, when a hole is surrounded by a metamaterial medium. After substitution of expressions (1) and (2) in the transverse components expressed in terms of the longitudinal components (Kong, 2008) we obtain the expressions of all transverse EM field components. The result of solution is the dispersion equation in the determinant form. We determine complex roots of the dispersion equation by using of the Muller method.

**3.1 Investigations of the circular metamaterial waveguide (r=2.5 mm) by the partial** 

waveguide could be used as a narrowband filter at frequencies 102-102.5 GHz.

with black points and the first higher mode is denoted with circles.

We discovered the particularity in the electric field distribution on the cross-section of the open circular metamaterial waveguide at the operating frequency 95 GHz. We find that this

The circular metamaterial waveguide with r=2.5 mm was researched. The dispersion characteristics and the 3D electric and magnetic field distributions were calculated (see Figs 2.1-2.5). The dispersion characteristics are presented in Fig. 3.1. The main mode is denoted

In Fig. 3.1 (a) we see that the normalized propagation constants of the main and the first higher modes are fairly smooth except only one protrusion at frequencies between 97 GHz and 102 GHz. There is a large peak of the main mode losses at frequency f = 101.25 GHz. At this frequency metamaterial is double negative with εr,met = -9.17 -i0.83 and μr,met = -0.75. We see that losses of the main mode are very small at the frequency ranges 75-100 GHz and 102.5-115 GHz. While the losses of the first higher mode are significantly higher at this frequency ranges

magnetic field Hza components that satisfy to Maxwell's equations in air are:

<sup>⊥</sup> r)·exp(imφ), Hz

<sup>⊥</sup> is the transverse propagation constant of the metamaterial medium, r is the radius of the circular metamaterial waveguide, m is the azimuthal index characterizing azimuthal variations of the field, φ is the azimuthal angle. The presentation the electric field Eza and the

a =B2Hm(k <sup>−</sup>

<sup>⊥</sup> is the transverse propagation constant of air medium.

<sup>⊥</sup> changes on the opposite one for the metamaterial

<sup>⊥</sup> r)exp(imφ) (3.1)

<sup>⊥</sup> r)·exp(imφ) (3.2)

**3. Investigations of the circular waveguides by the partial area method** 

the form:

k +

Ez

Eza =A2Hm(k <sup>−</sup>

infinity also is satisfied. The sign of k <sup>−</sup>

order and the second kind, k <sup>−</sup>

**area method** 

m =A1Jm(k <sup>+</sup>

Fig. 2.4. The dispersion characteristics of the square 4x4 mm2 metamaterial waveguide (a) – the dependence of the normalized propagation constant, (b) - the dependence of the attenuation constant on frequencies.

In Fig. 2.5 we see that the electric field is weak inside of square metamaterial waveguide of size 4x4 mm2 at f=92.5 GHz. The electric field lines concentrate in the two diagonal waveguide corners in other way in the comparison with the waveguide of size 5x5 mm2 at f=95 GHz.

Fig. 2.5. The 3D vector electric field distribution of the main mode propagating in the open square 4x4 mm2 metamaterial waveguide at f=92.5 GHz.

The 3D electric field distribution of the main mode at the frequency 92.5 GHz is depicted in Fig. 2.5. The permittivity of the metamaterial at f=92.5 GHz is -35-i2.5 and the permeability is 2.25+i0.25. The metamaterial is a single negative matter. The calculations of the electric fields were fulfilled at the approximately 160 points in every cross-section (Fig 2.5).

h'', dB/mm


(a) (b)

In Fig. 2.5 we see that the electric field is weak inside of square metamaterial waveguide of size 4x4 mm2 at f=92.5 GHz. The electric field lines concentrate in the two diagonal waveguide corners in other way in the comparison with the waveguide of size 5x5 mm2 at

Fig. 2.5. The 3D vector electric field distribution of the main mode propagating in the open

The 3D electric field distribution of the main mode at the frequency 92.5 GHz is depicted in Fig. 2.5. The permittivity of the metamaterial at f=92.5 GHz is -35-i2.5 and the permeability is 2.25+i0.25. The metamaterial is a single negative matter. The calculations of the electric

fields were fulfilled at the approximately 160 points in every cross-section (Fig 2.5).

square 4x4 mm2 metamaterial waveguide at f=92.5 GHz.

Fig. 2.4. The dispersion characteristics of the square 4x4 mm2 metamaterial waveguide (a) – the dependence of the normalized propagation constant, (b) - the dependence of the

85 90 95 100 105 110 f, GHz

0.67 0.72 0.77 0.82 0.87 0.92 0.97 1.02 1.07 1.12

f=95 GHz.

h'/k

85 90 95 100 105 110 f, GHz

attenuation constant on frequencies.

### **3. Investigations of the circular waveguides by the partial area method**

Here the open circular metamaterial waveguide is investigated by the partial area method (Nickelson et al., 2008). The presentation of longitudinal components of the electric Ez m and magnetic Hz m fields that satisfy to the Maxwell's equations in the metamaterial medium is in the form:

$$\mathbf{E}\_{\mathbf{z}}\mathbf{m} = \mathbf{A}\_{\text{J}}\mathbf{J}\_{\text{m}}(\mathbf{k}\_{\perp}^{\mathrm{+}}\mathbf{r})\exp(\mathbf{imq}), \quad \mathbf{H}\_{\text{z}}\mathbf{m} = \mathbf{B}\_{\text{J}}\mathbf{J}\_{\text{m}}(\mathbf{k}\_{\perp}^{\mathrm{+}}\mathbf{r})\exp(\mathbf{imq})\tag{3.1}$$

where A1, B1 are unknown arbitrary amplitudes*,* Jm is the Bessel function of the m−th order, k + <sup>⊥</sup> is the transverse propagation constant of the metamaterial medium, r is the radius of the circular metamaterial waveguide, m is the azimuthal index characterizing azimuthal variations of the field, φ is the azimuthal angle. The presentation the electric field Eza and the magnetic field Hza components that satisfy to Maxwell's equations in air are:

$$\mathbf{E\_{z^{\mathfrak{a}}} = \mathbf{A\_{2}H\_{m}(k\_{\perp}{}^{\-}\mathbf{r})\exp(im\mathbf{q})}, \quad \mathbf{H\_{z^{\mathfrak{a}}} = \mathbf{B\_{2}H\_{m}(k\_{\perp}{}^{-}\mathbf{r})\exp(im\mathbf{q})}\tag{3.2}$$

where A2, B2 are unknown arbitrary amplitudes, Hm is the Hankel function of the m−th order and the second kind, k <sup>−</sup> <sup>⊥</sup> is the transverse propagation constant of air medium.

As far as the circular waveguide is researched in the cylindrical coordinate system we have to satisfy the boundary conditions for two components of the electric field (Eφ, Ez) and the magnetic field (Hφ, Hz) on the cylindrical interface metamaterial-air. The condition at infinity also is satisfied. The sign of k <sup>−</sup> <sup>⊥</sup> changes on the opposite one for the metamaterial hollow-core waveguide, when a hole is surrounded by a metamaterial medium. After substitution of expressions (1) and (2) in the transverse components expressed in terms of the longitudinal components (Kong, 2008) we obtain the expressions of all transverse EM field components. The result of solution is the dispersion equation in the determinant form. We determine complex roots of the dispersion equation by using of the Muller method.

#### **3.1 Investigations of the circular metamaterial waveguide (r=2.5 mm) by the partial area method**

We discovered the particularity in the electric field distribution on the cross-section of the open circular metamaterial waveguide at the operating frequency 95 GHz. We find that this waveguide could be used as a narrowband filter at frequencies 102-102.5 GHz.

The circular metamaterial waveguide with r=2.5 mm was researched. The dispersion characteristics and the 3D electric and magnetic field distributions were calculated (see Figs 2.1-2.5). The dispersion characteristics are presented in Fig. 3.1. The main mode is denoted with black points and the first higher mode is denoted with circles.

In Fig. 3.1 (a) we see that the normalized propagation constants of the main and the first higher modes are fairly smooth except only one protrusion at frequencies between 97 GHz and 102 GHz. There is a large peak of the main mode losses at frequency f = 101.25 GHz. At this frequency metamaterial is double negative with εr,met = -9.17 -i0.83 and μr,met = -0.75. We see that losses of the main mode are very small at the frequency ranges 75-100 GHz and 102.5-115 GHz. While the losses of the first higher mode are significantly higher at this frequency ranges

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 35

(a) (b) Fig. 3.2. The 3D electric field distributions of the main mode of circular metamaterial waveguide with r=2.5 mm at f = 95 GHz. (a) – the electric field strength lines outside the waveguide (b) – the 143 times were increased electric field strength lines inside the

In In Figs 3.2–3.5 we see that the electric field is irregular on the waveguide perimeter of the cross-section while the cross-section of the waveguide is a circle. We see that and the most part of the electric field localizes on the border and outside of the waveguide. The electric field is strongest outside the waveguide when φ is 0 or π radians. The electric field has the minimum values and the electric field lines are directed clockwise or counter-clockwise to the right and left of the points with φ equal to π/2 or 3π/2 radians (Figs 3.2(a)–3.5(a)). We see that at the points when the electric field outside of the metamaterial waveguide has the maximum value the field inside of the waveguide is minimal. In Figs 3.2(b)–3.5(b) we see that the maximum electric field inside the metamaterial waveguide is when φ is equal to π/2 or 3π/2 radians. The length of the circular waveguide in z-direction (the Figs 3.2– 3.5) is three times longer than wavelength of microwave in the waveguide in our calculations.

*f* = 95 *GHz,* waveguide diameter 5 mm

1.6760·10-4 -i3.3526·10-4 3.5399·10-5-i1.2553·10-5 -2.7112·10-5-i1.2042·10-6




*Ez* [V/m] *Ex* [V/m] *Ey* [V/m]

*Ez* [V/m] *Ex* [V/m] *Ey* [V/m]

*Hz Hx Hy*

*Hz Hx Hy*

coordinates r = 2 mm , φ = 45° when f=95 and f=110 GHz

waveguide.

*f* = 110 *GHz*

(Fig. 3.1(b)). Therefore the investigated circular metamaterial waveguide can be used as a filter at the frequencies 100-102.5 GHz and as a one mode lossless waveguide at the frequency ranges 75-100 GHz and 102.5-115 GHz. As we can see the first higher mode is a quickly attenuated wave because it has very large losses in all frequency range.

Fig. 3.1. The dispersion characteristics of the circular metamaterial waveguide with r=2.5 mm (a) – the dependence of the normalized propagation constant, (b) – the dependence of the attenuation constant on frequencies.

The 3D electric field distributions of the main mode were calculated at frequencies 95 GHz and 110 GHz (Fig. 3.2 and 3.3) as well as the first higher mode at the same frequencies (Fig. 3.4 and 3.5). The electric field inside the circular metamaterial waveguide is much smaller than outside of waveguide for this reason we have increased the electric field strength lines inside of the waveguide in order to see them (Figs 3.2(b) – 3.5(b)).

Because the metamaterial has more losses at 95 GHz than at 110 GHz, so the electric field inside of waveguide is weaker at 95 GHz (see tables 3.1, 3.2) compare to the inner electric field at 110 GHz. On this reason we increased the electric field strength lines at 95 GHz in the 143 times and at 110 GHz in the 14 time. The calculations of the electric fields were fulfilled at the approximately 10000 points in every cross-section.


Table 3.1. The electromagnetic field components of the main mode at the point with coordinates r = 2 mm, φ = 45° when f = 95 and f = 110 GHz

(Fig. 3.1(b)). Therefore the investigated circular metamaterial waveguide can be used as a filter at the frequencies 100-102.5 GHz and as a one mode lossless waveguide at the frequency ranges 75-100 GHz and 102.5-115 GHz. As we can see the first higher mode is a quickly

> 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Fig. 3.1. The dispersion characteristics of the circular metamaterial waveguide with r=2.5 mm (a) – the dependence of the normalized propagation constant, (b) – the dependence of

The 3D electric field distributions of the main mode were calculated at frequencies 95 GHz and 110 GHz (Fig. 3.2 and 3.3) as well as the first higher mode at the same frequencies (Fig. 3.4 and 3.5). The electric field inside the circular metamaterial waveguide is much smaller than outside of waveguide for this reason we have increased the electric field strength lines

Because the metamaterial has more losses at 95 GHz than at 110 GHz, so the electric field inside of waveguide is weaker at 95 GHz (see tables 3.1, 3.2) compare to the inner electric field at 110 GHz. On this reason we increased the electric field strength lines at 95 GHz in the 143 times and at 110 GHz in the 14 time. The calculations of the electric fields were

*f* = 95 *GHz,* waveguide diameter 5mm


2.5111·10-7-i2.0158·10-7 -2.3843·10-8-i2.8570·10-8 2.5496·10-6 +i1.8364·10-7

0.0067 +i0.0067 -0.0027 + i0.0027 -7.8928·10-4 -i7.8928·10-4

3.1409·10-12 -i1.1109·10-5 -3.3354·10-6-i3.3354·10-6 7.1930·10-5 -i7.1930·10-5

h'',dB/mm

75 85 95 105 115 f, GHz

attenuated wave because it has very large losses in all frequency range.

(a) (b)

inside of the waveguide in order to see them (Figs 3.2(b) – 3.5(b)).

fulfilled at the approximately 10000 points in every cross-section.

*Ez* [V/m] *Ex* [V/m] *Ey* [V/m]

*Hz* [A/m] *Hx* [A/m] *Hy* [A/m]

*Ez* [V/m] *Ex* [V/m] *Ey* [V/m]

*Hz* [A/m] *Hx* [A/m] *Hy* [A/m]

with coordinates r = 2 mm, φ = 45° when f = 95 and f = 110 GHz

Table 3.1. The electromagnetic field components of the main mode at the point

75 85 95 105 115 f, GHz

the attenuation constant on frequencies.

0.66 0.71 0.76 0.81 0.86 0.91 0.96 1.01 1.06 1.11 1.16

*f* = 110 *GHz*

h'/k

Fig. 3.2. The 3D electric field distributions of the main mode of circular metamaterial waveguide with r=2.5 mm at f = 95 GHz. (a) – the electric field strength lines outside the waveguide (b) – the 143 times were increased electric field strength lines inside the waveguide.

In In Figs 3.2–3.5 we see that the electric field is irregular on the waveguide perimeter of the cross-section while the cross-section of the waveguide is a circle. We see that and the most part of the electric field localizes on the border and outside of the waveguide. The electric field is strongest outside the waveguide when φ is 0 or π radians. The electric field has the minimum values and the electric field lines are directed clockwise or counter-clockwise to the right and left of the points with φ equal to π/2 or 3π/2 radians (Figs 3.2(a)–3.5(a)). We see that at the points when the electric field outside of the metamaterial waveguide has the maximum value the field inside of the waveguide is minimal. In Figs 3.2(b)–3.5(b) we see that the maximum electric field inside the metamaterial waveguide is when φ is equal to π/2 or 3π/2 radians. The length of the circular waveguide in z-direction (the Figs 3.2– 3.5) is three times longer than wavelength of microwave in the waveguide in our calculations.


Table 3.2. The EM components of the first higher mode with coordinates at the point with coordinates r = 2 mm , φ = 45° when f=95 and f=110 GHz

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 37

GHz and 110 GHz are small (please, compare with losses of square waveguide (Fig.2.1)), for this reason the electric field amplitudes vary slightly in the longitudinal direction. Comparing Figs 3.4 and 3.5 we see that the larger is the electrical field outside of the waveguide at the point with a certain angle φ the smaller is the electrical field inside of waveguide at the point with the same φ. The last statement is true for all the investigated cases. The electric field amplitude of the first higher mode became smaller in longitudinal direction with increasing of coordinate z (Figs 3.4(b) and 3.5(b)). We observe in this case, the fast wave attenuation. It happened because the losses of the first higher mode at 95 GHz and

The electrical field inside of the waveguide is very small at all frequencies. However the

We would like to note that the boundary conditions on the border of waveguide are satisfied, i.e. the tangential components of electric and magnetic fields are equal on the interface air-metamaterial. For this reason when the electric field lines has a tangential character outside the waveguide (Figs 3.2(a), 3.3(a)) the same character of tangential components has to be on the interface of the metamaterial side. The direction of electric field

(a) (b) Fig. 3.5. The 3D electric field distributions of the first higher mode of circular metamaterial waveguide with r=2.5 mm at f = 110 GHz (a) – the electric field strength lines inside the waveguide (b) – the 14 times were increased electric field strength lines outside the

Comparing dispersion characteristics of square and circular waveguides (Figs 2.1 and 3.1) we see that they are different due to the boundary conditions, which have a strong influence on the dispersion characteristics in our frequency range. As an additional example, the dispersion characteristic (2πf· SQRT(εr,met μr,met) of plane EM wave propagating in the same metamaterial only when the one has the infinite dimensions is strongly different in comparison with the waveguide dispersion characteristics. The relatively small losses of the EM wave in the infinite

metamaterial are only at the frequencies between 102.5 GHz and 105 GHz.

waveguide.

110 GHz are enough large in comparison with the main mode (Fig. 3.1(b)).

observable electric field strength lines appear at the waveguide boundary.

lines changes with removing deeper in the metamaterial from the interface.

Fig. 3.3. The 3D electric field distributions of the main of circular metamaterial waveguide with r=2.5 mm at f = 110 GHz. (a) – the electric field strength lines outside the waveguide (b) – the 14 times were increased electric field strength lines inside the waveguide

Fig. 3.4. The 3D electric field distributions of the first higher mode of circular metamaterial waveguide with r=2.5 mm at f = 95 GHz (a) – the electric field strength lines outside the waveguide (b) – the 143 times were increased electric field strength lines inside the waveguide

We can see that the electric field along the waveguide changes periodically (Figs 3.2–3.5). Comparing Figs 3.2(a) and 3.3(a) we see that the main mode' electrical field at 110 GHz is large and have a little different distribution in longitudinal direction in comparison with the electrical field at 95 GHz. Since the waveguide losses of the main mode at frequencies 95

(a) (b) Fig. 3.3. The 3D electric field distributions of the main of circular metamaterial waveguide with r=2.5 mm at f = 110 GHz. (a) – the electric field strength lines outside the waveguide

(a) (b) Fig. 3.4. The 3D electric field distributions of the first higher mode of circular metamaterial waveguide with r=2.5 mm at f = 95 GHz (a) – the electric field strength lines outside the waveguide (b) – the 143 times were increased electric field strength lines inside the

We can see that the electric field along the waveguide changes periodically (Figs 3.2–3.5). Comparing Figs 3.2(a) and 3.3(a) we see that the main mode' electrical field at 110 GHz is large and have a little different distribution in longitudinal direction in comparison with the electrical field at 95 GHz. Since the waveguide losses of the main mode at frequencies 95

waveguide

(b) – the 14 times were increased electric field strength lines inside the waveguide

GHz and 110 GHz are small (please, compare with losses of square waveguide (Fig.2.1)), for this reason the electric field amplitudes vary slightly in the longitudinal direction. Comparing Figs 3.4 and 3.5 we see that the larger is the electrical field outside of the waveguide at the point with a certain angle φ the smaller is the electrical field inside of waveguide at the point with the same φ. The last statement is true for all the investigated cases. The electric field amplitude of the first higher mode became smaller in longitudinal direction with increasing of coordinate z (Figs 3.4(b) and 3.5(b)). We observe in this case, the fast wave attenuation. It happened because the losses of the first higher mode at 95 GHz and 110 GHz are enough large in comparison with the main mode (Fig. 3.1(b)).

The electrical field inside of the waveguide is very small at all frequencies. However the observable electric field strength lines appear at the waveguide boundary.

We would like to note that the boundary conditions on the border of waveguide are satisfied, i.e. the tangential components of electric and magnetic fields are equal on the interface air-metamaterial. For this reason when the electric field lines has a tangential character outside the waveguide (Figs 3.2(a), 3.3(a)) the same character of tangential components has to be on the interface of the metamaterial side. The direction of electric field lines changes with removing deeper in the metamaterial from the interface.

Fig. 3.5. The 3D electric field distributions of the first higher mode of circular metamaterial waveguide with r=2.5 mm at f = 110 GHz (a) – the electric field strength lines inside the waveguide (b) – the 14 times were increased electric field strength lines outside the waveguide.

Comparing dispersion characteristics of square and circular waveguides (Figs 2.1 and 3.1) we see that they are different due to the boundary conditions, which have a strong influence on the dispersion characteristics in our frequency range. As an additional example, the dispersion characteristic (2πf· SQRT(εr,met μr,met) of plane EM wave propagating in the same metamaterial only when the one has the infinite dimensions is strongly different in comparison with the waveguide dispersion characteristics. The relatively small losses of the EM wave in the infinite metamaterial are only at the frequencies between 102.5 GHz and 105 GHz.

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 39

In Fig. 3.6 (b) losses of the main mode are shown. In Fig. 3.6 (c) losses of the first higher mode are presented. We show the losses of the modes in the different scales because the losses are not commensurate. We see (Fig. 3.6 (a)) that the main mode is a slow mode and the first higher mode can be a slow mode or a fast mode dependent on the frequency range. There are two frequency ranges 75-97.5 GHz and 100-115 GHz when losses of the main mode are extremely small. On the other side there are frequency ranges, for example f=95 -

101.25 GHz and f=107.5-115GHz when losses of the first higher mode are large.

(a) (b) Fig. 3.7. The 3D vector electric field distributions of the main mode of circular metamaterial

We have calculated the 3D vector electric field distributions of the main mode propagating in the open circular metamaterial waveguide. The calculation was fulfilled inside and outside the waveguide in 1500 points. The electric field distributions were calculated at frequency f=95 GHz. At this frequency the metamaterial is single-negative. At this

In Fig. 3.7(a) an enlarged picture of the electric field lines inside the metamaterial waveguide is shown. We see that the strongest electric field is in the thin surface layer which is located at the interface metamaterial-air. In Fig. 3.7(b) the electric field lines inside and outside of the metamaterial waveguide are shown. In Fig. 3.7(b) we see that the electric field inside the waveguide is significantly weaker than outside it. We also clearly see that the electric field

The dispersion characteristics of the metamaterial hollow-core waveguide with the radius of

In Fig. 3.8 (a) dispersion characteristics of the main and the first higher modes are presented. Both modes are the fast ones. Their electromagnetic energy concentrates in the hollow-core

the hole (in the metamaterial medium) equal to 2 mm are presented in Fig. 3.8.

waveguide with r=2mm (a) – inside; (b) – inside and outside it.

distributions are periodically repeated in the longitudinal direction.

frequency εr,met = -23.75- i18.75 and μr,met = 1.75+ i1.625.

**3.2.2 The metamaterial hollow-core waveguide** 

We would like to draw attention to the fact that the feature of the irregular distribution of electric field lines in the cross-section of square and circular metamaterial waveguides at 95 GHz is very similar.

#### **3.2 Investigations of the circular metamaterial waveguide (r = 2 mm) by the partial area method**

#### **3.2.1 The metamaterial rod waveguide**

We have investigated circular metamaterial waveguides by our algorithm that was created using the partial area method (Nickelson et al., 2008).

In Fig. 3.6 the dispersion characteristics of the metamaterial waveguide are presented. In Fig. 3.6 (a) the normalized propagation constant h'/k of the main mode and the first higher mode is shown. The main mode is denoted with black points and the first higher mode is denoted with circles.

Fig. 3.6. The dispersion characteristics of the circular metamaterial waveguide with r=2 mm (a) – the dependence of the normalized propagation constant, (b), (c) – the dependence of the attenuation constant on frequencies.

We would like to draw attention to the fact that the feature of the irregular distribution of electric field lines in the cross-section of square and circular metamaterial waveguides at 95

We have investigated circular metamaterial waveguides by our algorithm that was created

In Fig. 3.6 the dispersion characteristics of the metamaterial waveguide are presented. In Fig. 3.6 (a) the normalized propagation constant h'/k of the main mode and the first higher mode is shown. The main mode is denoted with black points and the first higher mode is

> 75 85 95 105 115 f, GHz

> > (a)

Fig. 3.6. The dispersion characteristics of the circular metamaterial waveguide with r=2 mm (a) – the dependence of the normalized propagation constant, (b), (c) – the dependence of

0.8

75 85 95 105 115 f, GHz

2.8

4.8

6.8

h'', dB/mm

8.8

10.8

**3.2 Investigations of the circular metamaterial waveguide (r = 2 mm) by the partial** 

GHz is very similar.

denoted with circles.

**3.2.1 The metamaterial rod waveguide** 

using the partial area method (Nickelson et al., 2008).

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

(b) (c)

the attenuation constant on frequencies.

h'/k

**area method** 

In Fig. 3.6 (b) losses of the main mode are shown. In Fig. 3.6 (c) losses of the first higher mode are presented. We show the losses of the modes in the different scales because the losses are not commensurate. We see (Fig. 3.6 (a)) that the main mode is a slow mode and the first higher mode can be a slow mode or a fast mode dependent on the frequency range.

There are two frequency ranges 75-97.5 GHz and 100-115 GHz when losses of the main mode are extremely small. On the other side there are frequency ranges, for example f=95 - 101.25 GHz and f=107.5-115GHz when losses of the first higher mode are large.

Fig. 3.7. The 3D vector electric field distributions of the main mode of circular metamaterial waveguide with r=2mm (a) – inside; (b) – inside and outside it.

We have calculated the 3D vector electric field distributions of the main mode propagating in the open circular metamaterial waveguide. The calculation was fulfilled inside and outside the waveguide in 1500 points. The electric field distributions were calculated at frequency f=95 GHz. At this frequency the metamaterial is single-negative. At this frequency εr,met = -23.75- i18.75 and μr,met = 1.75+ i1.625.

In Fig. 3.7(a) an enlarged picture of the electric field lines inside the metamaterial waveguide is shown. We see that the strongest electric field is in the thin surface layer which is located at the interface metamaterial-air. In Fig. 3.7(b) the electric field lines inside and outside of the metamaterial waveguide are shown. In Fig. 3.7(b) we see that the electric field inside the waveguide is significantly weaker than outside it. We also clearly see that the electric field distributions are periodically repeated in the longitudinal direction.

#### **3.2.2 The metamaterial hollow-core waveguide**

The dispersion characteristics of the metamaterial hollow-core waveguide with the radius of the hole (in the metamaterial medium) equal to 2 mm are presented in Fig. 3.8.

In Fig. 3.8 (a) dispersion characteristics of the main and the first higher modes are presented. Both modes are the fast ones. Their electromagnetic energy concentrates in the hollow-core

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 41

In Fig. 3.9 we see that the electric field inside the air hole is significantly weaker than outside in the metamaterial media. We also clearly see that the electric field distributions are periodically repeated in the longitudinal direction. As the waveguide losses exist only in the metamaterial and the most part of EM energy propagates into air hole then we see no decrease in the amplitude of the electric field with a change in coordinate z (please, compare

Fig. 3.9. The 3D vector electric field distribution of the main mode of the hollow-core

1. The open lossy metamaterial waveguides with different shapes of the cross-section were investigated by using of our computer programs that have written in MATLAB language. The computer codes were based on the method of singular integral equations

2. We have calculated the dispersion characteristics (the propagation and attenuation constants) at the frequency range 75-115 GHz as well as the 2D and 3D electromagnetic field distributions. We took the electromagnetic parameters of metamaterial close to practice. Our computer algorithms can be useful working out microwave devices on the

3. We discovered the special feature of the open lossy metamaterial waveguides. Propagation and attenuation constants depend on the waveguide sizes in an unpredictable complex manner. E.g., the circular metamaterial waveguide with r=2.5

4. The microwave signals propagating on the open metamaterial waveguides (r=2.5 mm) are absorbed at the narrow 2.5 GHz frequency range. This waveguide can be used as a band-stop filter at f1=100-102.5 GHz when the losses in the passing frequencies 75-100,

metamaterial waveguide with r=2 mm at frequency 95 GHz.

base of waveguides made of strong lossy materials.

102.5-115 GHz are 100 times less than losses at f1.

**4. Conclusions on sections 2&3** 

and the partial area method.

mm (Fig. 3.1) and r=2 mm (Fig. 3.6).

with Figs 3.4(b), 3.5(b)).

air area. There are the frequency ranges where modes propagate with very small losses. We can see that losses of the main mode in the frequency range 75-90 GHz and 104-115 are very low and they can be large in the frequency range 91.25-103.75 GHz. The first high mode' losses change abruptly. The losses can be very low at some frequencies approximately 75- 87 GHz, 106 and 111 GHz.

Losses of the first high mode are low and this mode can easily propagate and it can modulate the amplitude of the main mode in devices that were created on the base of the metamaterial waveguide with r=2 mm. There is also good possibility to create a devise on the base of the first high mode in the range f=95-105 GHz.

We have calculated the 3D vector electric field distributions of the main mode propagating in the hollow-core metamaterial waveguide. The electric field distributions were calculated at frequency f = 95 GHz. The calculations of the electric fields were fulfilled at the approximately 10000 points in every cross-section.

Fig. 3.8. The dispersion characteristics of the hollow-core metamaterial waveguide with r=2 mm (a) – the dependence of the normalized propagation constant, (b)& (c) – the dependence of the attenuation constant on frequencies.

In Fig. 3.9 we see that the electric field inside the air hole is significantly weaker than outside in the metamaterial media. We also clearly see that the electric field distributions are periodically repeated in the longitudinal direction. As the waveguide losses exist only in the metamaterial and the most part of EM energy propagates into air hole then we see no decrease in the amplitude of the electric field with a change in coordinate z (please, compare with Figs 3.4(b), 3.5(b)).

Fig. 3.9. The 3D vector electric field distribution of the main mode of the hollow-core metamaterial waveguide with r=2 mm at frequency 95 GHz.

#### **4. Conclusions on sections 2&3**

40 Metamaterial

air area. There are the frequency ranges where modes propagate with very small losses. We can see that losses of the main mode in the frequency range 75-90 GHz and 104-115 are very low and they can be large in the frequency range 91.25-103.75 GHz. The first high mode' losses change abruptly. The losses can be very low at some frequencies approximately 75- 87

Losses of the first high mode are low and this mode can easily propagate and it can modulate the amplitude of the main mode in devices that were created on the base of the metamaterial waveguide with r=2 mm. There is also good possibility to create a devise on

We have calculated the 3D vector electric field distributions of the main mode propagating in the hollow-core metamaterial waveguide. The electric field distributions were calculated at frequency f = 95 GHz. The calculations of the electric fields were fulfilled at the

> 75 85 95 105 115 f, GHz

> > (a)

h'',dB/mm

(b) (c)

Fig. 3.8. The dispersion characteristics of the hollow-core metamaterial waveguide with r=2 mm (a) – the dependence of the normalized propagation constant, (b)& (c) – the dependence

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

> 75 85 95 105 115 f, GHz

the base of the first high mode in the range f=95-105 GHz.

0 0.2 0.4 0.6 0.8 1 1.2

h'/k

approximately 10000 points in every cross-section.

of the attenuation constant on frequencies.

GHz, 106 and 111 GHz.


Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 43

this reason there were given only the real parts of the permittivity εr,ij = (εxx, εxx, εzz) and

The tensor components of the relative permittivity and the relative permeability are

epxx/ ω2, εzz=1- ω<sup>2</sup>

mpxx/ ω2, μzz=1- ω<sup>2</sup>

here ω=2πf is the angular frequency of microwaves, f is the operating frequency. The electric plasma frequencies of metamaterial are ωepxx= 2πfepxx, fepxx=(12)1/2 GHz, ωepzz=2πfepzz, fepzz=2.5 GHz. The magnetic plasma frequencies of metamaterial are ωmpxx= 2πfmpxx GHz, fmpxx = (6)1/2 GHz and ωmpzz=2πfmpzz GHz, fmpzz=2 GHz. The values of angular frequencies are taken from (Liu et al., 2007). The magnitudes of tensor components εxx, εzz, μxx, μzz are

(a) (b) Fig. 5.1. Dependences of the relative (a) permittivity and (b) permeability tensor components

We see that the permittivity components εxx and εzz have negative values from 1.5 to ~3.5 GHz and from 1.5 to ~2.5 GHz, respectively. The permeability components μxx and μzz have negative values from 1.5 to~2.5 GHz and from 1.5 to ~2 GHz, respectively. We realize that all tensor components are negative at the frequency range from 1.5 GHz to ~2 GHz. Absolute values of tensor components are less than 1 at the frequency range from ~2.5 GHz to 4 GHz. The values of tensor components become equal to zero at the operating frequency f equal to the metamaterial electric fepxx=3.46 GHz, fepzz= 2.5 GHz or magnetic fmpxx =2.45 GHz, fmpzz=2

In Figs. 5.2–5.7 are shown dispersion characteristics (phase constants) of open circular waveguide made of the uniaxial electrically and magnetically anisotropic metamaterial. The calculations are performed for the left-handed (extraordinary) circularly polarized microwaves when exp(+imφ), m=0, 1, 2,…is the wave (mode) azimuthal periodicity index, φ is the azimuthal coordinate. We investigated only modes with the index m=1, because it is

real numbers. In Fig. 5.1(a,b) are presented their dependencies on the frequency.

epzz/ ω2, (5.1)

mpzz/ ω2, (5.2)

permeability μr, ij =(μxx, μxx, μzz)tensor components.

described by following formulae (Liu et al., 2007):

of the metamaterial on the frequency.

GHz plasma frequencies.

εxx=1- ω<sup>2</sup>

μxx=1- ω<sup>2</sup>

5. Investigated EM fields of square and circular waveguides have irregular distributions on the waveguide perimeter. Investigated EM field lines for the main and first higher modes of the squire waveguides are concentrated near the metamaterial borders and the strongest field is at two diagonal corners of the cross-section. The electric field lines are diverging of the one corner while the lines are converging at the other corner.

#### **5. Dispersion characteristic analysis of circular anisotropic metamaterial waveguide with the effective metamaterial permittivity and permeability near to zero**

#### **5.1 Introduction**

In the past several years many specialist focused on the experimental and theoretical investigations of the zero-refractive index (or zero-index) metamaterials. These metamaterials attracted researches due to their unconventional constitutive parameters and anomalous effects to work out novel electromagnetic devices. Zero- index metamaterial may have the epsilon-near-zero (ENZ) or mu-near-zero (MNZ) properties simultaneously or in turn, one after another at different frequencies.

Zero-index metamaterials are dispersive (electromagnetic parameters dependence on a frequency) media. The constitutive parameters of anisotropic dispersive metamaterials can be described by expressions that involve the plasma frequencies. The metamaterial on frequencies near to plasma resonances is called a plasmonic metamaterial.

Zero-index metamaterials are used in different devices as a transformer to achieve the perfect impedance match between two waveguides with a negligible reflection or to improve the transmission through a waveguide bend as well as for the matching of waveguide structure impedance with the free space impedance (when the metamaterial epsilon and mu are simultaneously very close to zero). Plasmonic metamaterial provides manipulating of the antenna phase fronts and enhancing the antenna radiation directivity. In a Zero-index metamaterial waveguide can be observed a super-tunneling effect. ENZ metamaterials may allow reducing of waveguide sizes and can be used as a frequency selective surface (Bai et al., 2010; Ko&Lee, 2010; Lopez-Garcia et al., 2011; Luo et al., 2011; Wang & Huang, 2010; Oraizi et al. 2009; Zhou et al., 2009, Liu et al., 2008).

#### **5.2 Analysis and simulation of phase constant dependencies**

Here we presented the phase constant (real part of the waveguide longitudinal propagation constant) of propagating modes on the circular anisotropic metamaterial waveguide when the metamaterial permittivity and permeability may take values close to zero at certain frequencies. Further we call a plasmonic waveguide.

The solution of Maxwell's equations for the circular anisotropic metamaterial waveguide was carried out by the partial area method (Nickelson et al., 2009). The computer program for the dispersion characteristic calculations has created in MATLAB language. Computer program allows take into account a very large material attenuation (Nickelson et al., 2011; Asmontas et al., 2010). In this section constitutive parameters of the uniaxial electrically and magnetically anisotropic metamaterial were taken from the article (Liu et al., 2007). In the mentioned article was considered an anisotropic dispersive lossless metamaterial slab. For

5. Investigated EM fields of square and circular waveguides have irregular distributions on the waveguide perimeter. Investigated EM field lines for the main and first higher modes of the squire waveguides are concentrated near the metamaterial borders and the strongest field is at two diagonal corners of the cross-section. The electric field lines

are diverging of the one corner while the lines are converging at the other corner.

**5. Dispersion characteristic analysis of circular anisotropic metamaterial waveguide with the effective metamaterial permittivity and permeability near** 

In the past several years many specialist focused on the experimental and theoretical investigations of the zero-refractive index (or zero-index) metamaterials. These metamaterials attracted researches due to their unconventional constitutive parameters and anomalous effects to work out novel electromagnetic devices. Zero- index metamaterial may have the epsilon-near-zero (ENZ) or mu-near-zero (MNZ) properties simultaneously or in

Zero-index metamaterials are dispersive (electromagnetic parameters dependence on a frequency) media. The constitutive parameters of anisotropic dispersive metamaterials can be described by expressions that involve the plasma frequencies. The metamaterial on

Zero-index metamaterials are used in different devices as a transformer to achieve the perfect impedance match between two waveguides with a negligible reflection or to improve the transmission through a waveguide bend as well as for the matching of waveguide structure impedance with the free space impedance (when the metamaterial epsilon and mu are simultaneously very close to zero). Plasmonic metamaterial provides manipulating of the antenna phase fronts and enhancing the antenna radiation directivity. In a Zero-index metamaterial waveguide can be observed a super-tunneling effect. ENZ metamaterials may allow reducing of waveguide sizes and can be used as a frequency selective surface (Bai et al., 2010; Ko&Lee, 2010; Lopez-Garcia et al., 2011; Luo et al., 2011;

Here we presented the phase constant (real part of the waveguide longitudinal propagation constant) of propagating modes on the circular anisotropic metamaterial waveguide when the metamaterial permittivity and permeability may take values close to zero at certain

The solution of Maxwell's equations for the circular anisotropic metamaterial waveguide was carried out by the partial area method (Nickelson et al., 2009). The computer program for the dispersion characteristic calculations has created in MATLAB language. Computer program allows take into account a very large material attenuation (Nickelson et al., 2011; Asmontas et al., 2010). In this section constitutive parameters of the uniaxial electrically and magnetically anisotropic metamaterial were taken from the article (Liu et al., 2007). In the mentioned article was considered an anisotropic dispersive lossless metamaterial slab. For

frequencies near to plasma resonances is called a plasmonic metamaterial.

Wang & Huang, 2010; Oraizi et al. 2009; Zhou et al., 2009, Liu et al., 2008).

**5.2 Analysis and simulation of phase constant dependencies** 

frequencies. Further we call a plasmonic waveguide.

**to zero** 

**5.1 Introduction** 

turn, one after another at different frequencies.

this reason there were given only the real parts of the permittivity εr,ij = (εxx, εxx, εzz) and permeability μr, ij =(μxx, μxx, μzz)tensor components.

The tensor components of the relative permittivity and the relative permeability are described by following formulae (Liu et al., 2007):

$$\varepsilon\_{\infty} = 1 \text{ - } \alpha^2 \varepsilon\_{\text{eprx}} \text{ / } \alpha^2, \quad \varepsilon\_{\text{zz}} = 1 \text{ - } \alpha^2 \varepsilon\_{\text{eprx}} \text{ / } \alpha^2. \tag{5.1}$$

$$
\mu\_{\rm xx} = 1 \text{ - } \alpha^2 \mu\_{\rm mpx} \text{/ } \alpha^2, \quad \mu\_{\rm xz} = 1 \text{ - } \alpha^2 \mu\_{\rm mpxz} \text{/ } \alpha^2. \tag{5.2}
$$

here ω=2πf is the angular frequency of microwaves, f is the operating frequency. The electric plasma frequencies of metamaterial are ωepxx= 2πfepxx, fepxx=(12)1/2 GHz, ωepzz=2πfepzz, fepzz=2.5 GHz. The magnetic plasma frequencies of metamaterial are ωmpxx= 2πfmpxx GHz, fmpxx = (6)1/2 GHz and ωmpzz=2πfmpzz GHz, fmpzz=2 GHz. The values of angular frequencies are taken from (Liu et al., 2007). The magnitudes of tensor components εxx, εzz, μxx, μzz are real numbers. In Fig. 5.1(a,b) are presented their dependencies on the frequency.

Fig. 5.1. Dependences of the relative (a) permittivity and (b) permeability tensor components of the metamaterial on the frequency.

We see that the permittivity components εxx and εzz have negative values from 1.5 to ~3.5 GHz and from 1.5 to ~2.5 GHz, respectively. The permeability components μxx and μzz have negative values from 1.5 to~2.5 GHz and from 1.5 to ~2 GHz, respectively. We realize that all tensor components are negative at the frequency range from 1.5 GHz to ~2 GHz. Absolute values of tensor components are less than 1 at the frequency range from ~2.5 GHz to 4 GHz.

The values of tensor components become equal to zero at the operating frequency f equal to the metamaterial electric fepxx=3.46 GHz, fepzz= 2.5 GHz or magnetic fmpxx =2.45 GHz, fmpzz=2 GHz plasma frequencies.

In Figs. 5.2–5.7 are shown dispersion characteristics (phase constants) of open circular waveguide made of the uniaxial electrically and magnetically anisotropic metamaterial. The calculations are performed for the left-handed (extraordinary) circularly polarized microwaves when exp(+imφ), m=0, 1, 2,…is the wave (mode) azimuthal periodicity index, φ is the azimuthal coordinate. We investigated only modes with the index m=1, because it is

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 45

separated by a larger distance from other eigenmodes. The vertical branch of the left lateral mode is located at the magnetic plasma fmpzz frequency equal to 2 GHz. We can distinguish also the right lateral dispersion branch of the package. The mode with this dispersion characteristic is also more specific one. i.e. this mode is separated by a larger distance from other modes. The vertical branch of this mode is located about 2.7 GHz and shifted on the

higher frequency side with increasing of a radius.

(a) (b)

(a) (b)

metamaterial waveguide with (a) r=7 mm and (b) r=10 mm.

dispersion curves fan out from a point with a value equal to fmpxx.

Fig. 5.4. The phase constant dependencies of propagating modes on the anisotropic

A dense bunch of dispersion curves located between the extreme left and right curves that were previously described. The number of curves increases rapidly at increasing of waveguide radius. It is interesting to note that all dispersion branches of the dense bunch are within the frequency band of 2-2.5 GHz. Apparently the dense bunch of dispersion characteristics related to plasma fmpzz and fepzz frequencies. The cutoff frequencies of all dispersion characteristics of the dense bunch are the same and equal to f~2.46 GHz. The

metamaterial waveguide with (a) r=3 mm and (b) r =5 mm.

Fig. 5.3. The phase constant dependencies of propagating modes on the anisotropic

known the main mode of open circular dielectric waveguide specify by m=1 and the main mode of a dielectric waveguide is a hybrid mode HE11 (Nickelson et al., 2009).

Here is presented the phase constant h' (the real part of longitudinal propagation constant) dependencies of plasmonic metamaterial waveguides with radii r equal to 0.1 mm, 1 mm, 3 mm, 5 mm, 7 mm, 10 mm and 15 mm. The phase constant h' is equal to 2π/λw, where λw is the wavelength of certain mode. Our aim is to investigate how an increase in the plasmonic waveguide radius affects on the eigenmode numbers, mode cutoff frequencies and a shape of dispersion characteristics.

The analysis of Figs 5.2-5.5 shows that there are three main frequency areas where localize dispersion curves.

A shape all dispersion characteristics Figs 5.2-5.5 are unusual in the comparison with traditional dispersion characteristics of open cylindrical waveguides made of dielectrics, semiconductors or magnetoactive semiconductor plasma (Nickelson et al., 2011; Asmontas et al., 2009; Nickelson et al., 2009). Because the dispersion characteristic branches of analyzed plasmonic waveguides are vertical.

Fig. 5.2. The phase constant dependencies of propagating modes on the anisotropic metamaterial waveguide with (a) r=0.1 mm and (b) r=1 mm.

We see that there is a single mode with the cutoff frequency close to f=1.5 GHz. The cutoff frequency of this mode shifted in the direction of lower frequencies with increasing of the waveguide radius. This first single mode is special one because the mode does not match any of plasma fepxx, fepzz, fmpxx, fmpzz frequencies. We can observe how a shape of the dispersion characteristic changes in the vicinity of the cutoff frequency.

We would like to draw your attention to the fact that the anisotropic metamaterial is described by the negative tensor components εxx, εzz, μxx, μzz in the frequencies less than 2 GHz (see formulae 5.1 and 5.2). It is mean that the first mode propagates in the waveguide when the metamaterial is double negative (DN). This wave is particularly important because small changes in frequency produce large changes in phase.

We can watch a package of dispersion branches closed to cutoff frequency 2.5 GHz. We see that the left lateral dispersion branch of the package is a special eigenmode, i.e. this one is

known the main mode of open circular dielectric waveguide specify by m=1 and the main

Here is presented the phase constant h' (the real part of longitudinal propagation constant) dependencies of plasmonic metamaterial waveguides with radii r equal to 0.1 mm, 1 mm, 3 mm, 5 mm, 7 mm, 10 mm and 15 mm. The phase constant h' is equal to 2π/λw, where λw is the wavelength of certain mode. Our aim is to investigate how an increase in the plasmonic waveguide radius affects on the eigenmode numbers, mode cutoff frequencies and a shape

The analysis of Figs 5.2-5.5 shows that there are three main frequency areas where localize

A shape all dispersion characteristics Figs 5.2-5.5 are unusual in the comparison with traditional dispersion characteristics of open cylindrical waveguides made of dielectrics, semiconductors or magnetoactive semiconductor plasma (Nickelson et al., 2011; Asmontas et al., 2009; Nickelson et al., 2009). Because the dispersion characteristic branches of

mode of a dielectric waveguide is a hybrid mode HE11 (Nickelson et al., 2009).

(a) (b)

dispersion characteristic changes in the vicinity of the cutoff frequency.

metamaterial waveguide with (a) r=0.1 mm and (b) r=1 mm.

small changes in frequency produce large changes in phase.

Fig. 5.2. The phase constant dependencies of propagating modes on the anisotropic

We see that there is a single mode with the cutoff frequency close to f=1.5 GHz. The cutoff frequency of this mode shifted in the direction of lower frequencies with increasing of the waveguide radius. This first single mode is special one because the mode does not match any of plasma fepxx, fepzz, fmpxx, fmpzz frequencies. We can observe how a shape of the

We would like to draw your attention to the fact that the anisotropic metamaterial is described by the negative tensor components εxx, εzz, μxx, μzz in the frequencies less than 2 GHz (see formulae 5.1 and 5.2). It is mean that the first mode propagates in the waveguide when the metamaterial is double negative (DN). This wave is particularly important because

We can watch a package of dispersion branches closed to cutoff frequency 2.5 GHz. We see that the left lateral dispersion branch of the package is a special eigenmode, i.e. this one is

of dispersion characteristics.

analyzed plasmonic waveguides are vertical.

dispersion curves.

separated by a larger distance from other eigenmodes. The vertical branch of the left lateral mode is located at the magnetic plasma fmpzz frequency equal to 2 GHz. We can distinguish also the right lateral dispersion branch of the package. The mode with this dispersion characteristic is also more specific one. i.e. this mode is separated by a larger distance from other modes. The vertical branch of this mode is located about 2.7 GHz and shifted on the higher frequency side with increasing of a radius.

Fig. 5.3. The phase constant dependencies of propagating modes on the anisotropic metamaterial waveguide with (a) r=3 mm and (b) r =5 mm.

Fig. 5.4. The phase constant dependencies of propagating modes on the anisotropic metamaterial waveguide with (a) r=7 mm and (b) r=10 mm.

A dense bunch of dispersion curves located between the extreme left and right curves that were previously described. The number of curves increases rapidly at increasing of waveguide radius. It is interesting to note that all dispersion branches of the dense bunch are within the frequency band of 2-2.5 GHz. Apparently the dense bunch of dispersion characteristics related to plasma fmpzz and fepzz frequencies. The cutoff frequencies of all dispersion characteristics of the dense bunch are the same and equal to f~2.46 GHz. The dispersion curves fan out from a point with a value equal to fmpxx.

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 47

(a) (b)

(a) (b)

Fig. 5.7. Frequency branches of phase constants of waveguide with r=15 mm at frequencies:

1. The open anisotropic metamaterial waveguides with seven different radii were investigated by using of our computer programs that have written in MATLAB

2. We discovered the anomalous dispersion of the analyzed plasmonic waveguide

3. We find a mode with the cutoff frequency close to f=1.5 GHz with some anomalous features, e.g. the small changes in the frequency produce the very large changes in

language. The algorithm is based on the partial area method.

phase. This property could be useful in practical realizations.

(a) 1.9-2.5 GHz and (b) 2.3-2.45 GHz.

eigenmodes (Figs. 3.2-3.5).

**6. Conclusions** 

Fig. 5.6. Low frequency branches of phase constants of open plasmonic metamaterial waveguide with r=15 mm at frequencies: (a) f=0.005-0.06 GHz and (b) f=1.35-1.55 GHz.

Second dense bunch of dispersion curves is at the electric plasma frequency fepxx~3.46 GHz. The number of curves increases rapidly at increasing of waveguide radius. All dispersion characteristics are within the frequency band of 2.5 GHz and 3.46 GHz.

Fig. 5.5. The phase constant dependencies of propagating eigenmodes on the open plasmonic metamaterial waveguide with (a) r=10 mm (b) r=15 mm.

The dispersion characteristics on the right side of the bunch are more vertical. The greatest number of modes can be excited at the electric plasma frequency fepxx~3.46 GHz in the comparison with other plasma frequencies. The cutoff frequencies of dispersion characteristics of this dense bunch are the same and equal to f~3.46 GHz. We did not find the plasmonic metamaterial waveguide eigenmodes in the frequency range from 3.5 GHz till 2000 GHz.

We expanded the searching of eigenmodes on frequencies below f=1 GHz for the plasmonic waveguide with r= 15 mm. In Fig. 5.5 (b) is shown eigenmodes dispersion characteristics of plasmonic metamaterial waveguide in the frequency range from 5 MHz till 3.5 GHz. We present here more detailed calculations of the dispersion characteristics of the plasmonic waveguide eigenmodes with the radius equal to 15 mm (Figs. 5.6 and 5.7). Here are presented the new dispersion characteristic branches of waveguide eigenmodes in the band of frequency from 5 MHz till 600 MHz (Fig. 5.6 (a)). We see that the dispersion curves have the clear expressed cutoff frequencies and they have an opposite slope in the comparison with dispersion curves of open ordinary waveguides (Nickelson et al., 2011; Asmontas et al., 2010).

It should be stressed that these very low frequency dispersion characteristics have obtained by solving of Maxwell's equations with certain boundary conditions. These low frequency modes are also the metamaterial waveguide eigenmodes.

In the Fig. 5.6 (b) is shown the dispersion characteristic of a singular mode on a larger scale. We see that this mode from 1.35 till ~1.45 GHz is a static mode, because no dependence on the frequency. We can observe the anomaly dispersion closed to f~ 1.45 GHz and the very strong dispersion in the frequency band over ~1.47 GHz. We can see the dispersion curves in the area of cutoff frequencies on a larger scale 1.9-2.5 GHz in Figs 5.7(a) and (b). We can note the anomalous dispersion hook of eigenmodes in the band f=1.9-2 GHz.

Fig. 5.6. Low frequency branches of phase constants of open plasmonic metamaterial waveguide with r=15 mm at frequencies: (a) f=0.005-0.06 GHz and (b) f=1.35-1.55 GHz.

Fig. 5.7. Frequency branches of phase constants of waveguide with r=15 mm at frequencies: (a) 1.9-2.5 GHz and (b) 2.3-2.45 GHz.

#### **6. Conclusions**

46 Metamaterial

Second dense bunch of dispersion curves is at the electric plasma frequency fepxx~3.46 GHz. The number of curves increases rapidly at increasing of waveguide radius. All dispersion

characteristics are within the frequency band of 2.5 GHz and 3.46 GHz.

(a) (b)

plasmonic metamaterial waveguide with (a) r=10 mm (b) r=15 mm.

modes are also the metamaterial waveguide eigenmodes.

till 2000 GHz.

2010).

Fig. 5.5. The phase constant dependencies of propagating eigenmodes on the open

The dispersion characteristics on the right side of the bunch are more vertical. The greatest number of modes can be excited at the electric plasma frequency fepxx~3.46 GHz in the comparison with other plasma frequencies. The cutoff frequencies of dispersion characteristics of this dense bunch are the same and equal to f~3.46 GHz. We did not find the plasmonic metamaterial waveguide eigenmodes in the frequency range from 3.5 GHz

We expanded the searching of eigenmodes on frequencies below f=1 GHz for the plasmonic waveguide with r= 15 mm. In Fig. 5.5 (b) is shown eigenmodes dispersion characteristics of plasmonic metamaterial waveguide in the frequency range from 5 MHz till 3.5 GHz. We present here more detailed calculations of the dispersion characteristics of the plasmonic waveguide eigenmodes with the radius equal to 15 mm (Figs. 5.6 and 5.7). Here are presented the new dispersion characteristic branches of waveguide eigenmodes in the band of frequency from 5 MHz till 600 MHz (Fig. 5.6 (a)). We see that the dispersion curves have the clear expressed cutoff frequencies and they have an opposite slope in the comparison with dispersion curves of open ordinary waveguides (Nickelson et al., 2011; Asmontas et al.,

It should be stressed that these very low frequency dispersion characteristics have obtained by solving of Maxwell's equations with certain boundary conditions. These low frequency

In the Fig. 5.6 (b) is shown the dispersion characteristic of a singular mode on a larger scale. We see that this mode from 1.35 till ~1.45 GHz is a static mode, because no dependence on the frequency. We can observe the anomaly dispersion closed to f~ 1.45 GHz and the very strong dispersion in the frequency band over ~1.47 GHz. We can see the dispersion curves in the area of cutoff frequencies on a larger scale 1.9-2.5 GHz in Figs 5.7(a) and (b). We can

note the anomalous dispersion hook of eigenmodes in the band f=1.9-2 GHz.


Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 49

The complex metamaterial permittivity is *εmet* = ε'met - iε"met = *│εmet*│exp(-i δmet). The complex permeability is μmet = μ'met-iμ"met = *│εmet*│exp(-i δmet). The metamaterial εmet and μmet are the same for the single-layer and twelve-layer cylinders. The module of the metamaterial permittivity is |εmet|=20 and the metamaterial permeability is |μmet|=2. The phase of metamaterial permittivity is δε,met=0.7854 [rad] and the phase of permeability is δμ

We present here the absorbed power by the acrylic-glass layer (Fig. 7.3) and the absorbed power by the metamaterial (Fig. 7.4) for the single-layer cylinder. The total absorbed power by single-layered cylinder is equal to the sum of absorbed powers by metamaterial and acrylic-glass. And we show here the total absorbed power by all layers of the twelve-layered

In Figs 7.2-7.6 are presented the averaged scattered or absorbed power values per oscillation

The absorbed and scattered power calculations were fulfilled using by formula (28) in (Bucinskas et al., 2010)]. The integral of the formula has a positive sign when we calculate the scattered power. And this integral has a negative sign when we calculate the absorbed power. For this reason the scattered power has a positive value and the absorbed power has

We presented here the dependencies of scattered and absorbed powers of the single-layered cylinder on signs of ε'met , ε"met, μ'met and μ"met, i.e. when metamaterial is double positive

**7.1 Numerical analysis of the scattered and absorbed microwave power of the single-**

In this subsection we investigate the scattered and absorbed powers of the incident microwave by the cylinder consists a core of metamaterial which is covered with a singlelayer of acrylic-glass. The external cylinder radius is R1=2 mm and the cylinder core has

Nowadays there is a huge interest to the composite materials with untraditional values of the complex permittivity *εmet* and the complex permeability *μmet.* We analyzed here the scattered and absorbed powers for four versions of hypothetic metamaterial parameter signs. The complex metamaterial permittivity *εmet*=s1*│εmet*│exp(-s2iδmet,ε) was taken for two combinations of signs, when s1=s2=±1. It is known that each metamaterial is intended for use in a specific frequency range and has a specific value of the effective permittivity and permeability at the certain frequency. For this reason we took the absolute values of real and imaginary parts of permittivity *εmet* and permeability *μmet* constant at all frequencies in our calculations. And the major impact makes the sizes' relation of wavelength and cylinder

In figures 7.2-7.4 of this section are analyzed how the signs of the complex metamaterial permittivity and permeability influence on the scattered and absorbed powers when the plane perpendicularly or parallel polarized microwave impinges on the metamaterial-glass

(DP), *ε-* single negative (SN) or μ *-* single negative (SN), double negative (DN).

,met=0.6981 [rad] for the single-layer and twelve layer cylinders.

period for the unit length of the metamaterial-glass cylinder.

radius R2=1.8 mm, so the thickness of the glass layer is 0.2 mm.

cylinder.

**layer cylinder** 

layers.

cylinder (Fig. 7.1).

a negative one (see Figs 7.2-7.6).

#### **7. Microwave scattering and absorption by layered metamaterial-glass cylinders**

In this section we are going to give our calculation results for a single-layered cylinder and a twelve layered cylinder. The single–layered cylinder consists of a metamaterial core which is coated with an acrylic-glass layer. The twelve-layered cylinder consists of a conductor core which is covered with 12 metamaterial and acrylic-glass alternately layers. The acrylic-glass material is an external layer of each cylinder. The calculation of the scattered (reflected) and absorbed powers are based on the rigorous solution of scattering boundary problem (Nickelson & Bucinskas, 2011). The solution of mentioned electrodynamical problems and expressions of absorbed and scattered powers are given in article (Bucinskas et al., 2010).

The number and thickness of layers is not limited in the presented algorithm. The central core of multilayered cylinder can be made of different isotropic materials as a metamaterial, a ceramic matter or a semiconductor as well as of a perfect conductor. The isotropic coated layers can be of strongly lossy (absorbed) materials.

The signs of the complex permittivity and the complex permeability can be negative or positive in different combinations.

Here are presented the scattered and absorbed power of layered cylinder dependent on the hypothetic metamaterial permittivity and permeability signs and losses. We used for calculations our computer programs which are written in FORTRAN language.

The extern radius of both (single- and twelve-layered) cylinders is the same and equal to 2 mm. We show here our results only in the frequency range from 1 till 120 GHz. We present dependencies of the scattered and absorbed powers by the cylinders at the incident perpendicularly (the angle ψ = 0o, Fig. 7.1) and parallel (the angle ψ = 90o, Fig. 7.1) polarized microwaves. An incident angle of microwave is θ= 90o here.

Fig. 7.1. *N*-layered metamaterial-glass cylinder model and designations

We admitted that acrylic-glass material is a non-dispersive and weakly lossy one with the complex permittivity *εg* = εg'-iεg" = *│εg*│exp(-i δg) = 3.8 – *i* 0.0005, i.e. the phase of the complex glass permittivity is δg=arctan(*εg*'/*εg*") =1.3·10-4 [rad] and the glass permeability is equal to μg =1.

In this section we are going to give our calculation results for a single-layered cylinder and a twelve layered cylinder. The single–layered cylinder consists of a metamaterial core which is coated with an acrylic-glass layer. The twelve-layered cylinder consists of a conductor core which is covered with 12 metamaterial and acrylic-glass alternately layers. The acrylic-glass material is an external layer of each cylinder. The calculation of the scattered (reflected) and absorbed powers are based on the rigorous solution of scattering boundary problem (Nickelson & Bucinskas, 2011). The solution of mentioned electrodynamical problems and expressions of absorbed and scattered powers are given in article (Bucinskas et al., 2010).

The number and thickness of layers is not limited in the presented algorithm. The central core of multilayered cylinder can be made of different isotropic materials as a metamaterial, a ceramic matter or a semiconductor as well as of a perfect conductor. The isotropic coated

The signs of the complex permittivity and the complex permeability can be negative or

Here are presented the scattered and absorbed power of layered cylinder dependent on the hypothetic metamaterial permittivity and permeability signs and losses. We used for

The extern radius of both (single- and twelve-layered) cylinders is the same and equal to 2 mm. We show here our results only in the frequency range from 1 till 120 GHz. We present dependencies of the scattered and absorbed powers by the cylinders at the incident perpendicularly (the angle ψ = 0o, Fig. 7.1) and parallel (the angle ψ = 90o, Fig. 7.1) polarized

*k*

θ

*nz k*

*nx*

*O z <sup>y</sup> <sup>x</sup>* ~

*RN R R*<sup>1</sup> <sup>2</sup>

~

calculations our computer programs which are written in FORTRAN language.

INCIDENT MICROWAVE

GLASS

*E*0

Fig. 7.1. *N*-layered metamaterial-glass cylinder model and designations

ψ

METAMATERIAL

We admitted that acrylic-glass material is a non-dispersive and weakly lossy one with the complex permittivity *εg* = εg'-iεg" = *│εg*│exp(-i δg) = 3.8 – *i* 0.0005, i.e. the phase of the complex glass permittivity is δg=arctan(*εg*'/*εg*") =1.3·10-4 [rad] and the glass permeability is

*ny*

layers can be of strongly lossy (absorbed) materials.

microwaves. An incident angle of microwave is θ= 90o here.

positive in different combinations.

~

~

equal to μg =1.

**7. Microwave scattering and absorption by layered metamaterial-glass** 

**cylinders** 

The complex metamaterial permittivity is *εmet* = ε'met - iε"met = *│εmet*│exp(-i δmet). The complex permeability is μmet = μ'met-iμ"met = *│εmet*│exp(-i δmet). The metamaterial εmet and μmet are the same for the single-layer and twelve-layer cylinders. The module of the metamaterial permittivity is |εmet|=20 and the metamaterial permeability is |μmet|=2. The phase of metamaterial permittivity is δε,met=0.7854 [rad] and the phase of permeability is δμ ,met=0.6981 [rad] for the single-layer and twelve layer cylinders.

We present here the absorbed power by the acrylic-glass layer (Fig. 7.3) and the absorbed power by the metamaterial (Fig. 7.4) for the single-layer cylinder. The total absorbed power by single-layered cylinder is equal to the sum of absorbed powers by metamaterial and acrylic-glass. And we show here the total absorbed power by all layers of the twelve-layered cylinder.

In Figs 7.2-7.6 are presented the averaged scattered or absorbed power values per oscillation period for the unit length of the metamaterial-glass cylinder.

The absorbed and scattered power calculations were fulfilled using by formula (28) in (Bucinskas et al., 2010)]. The integral of the formula has a positive sign when we calculate the scattered power. And this integral has a negative sign when we calculate the absorbed power. For this reason the scattered power has a positive value and the absorbed power has a negative one (see Figs 7.2-7.6).

We presented here the dependencies of scattered and absorbed powers of the single-layered cylinder on signs of ε'met , ε"met, μ'met and μ"met, i.e. when metamaterial is double positive (DP), *ε-* single negative (SN) or μ *-* single negative (SN), double negative (DN).

#### **7.1 Numerical analysis of the scattered and absorbed microwave power of the singlelayer cylinder**

In this subsection we investigate the scattered and absorbed powers of the incident microwave by the cylinder consists a core of metamaterial which is covered with a singlelayer of acrylic-glass. The external cylinder radius is R1=2 mm and the cylinder core has radius R2=1.8 mm, so the thickness of the glass layer is 0.2 mm.

Nowadays there is a huge interest to the composite materials with untraditional values of the complex permittivity *εmet* and the complex permeability *μmet.* We analyzed here the scattered and absorbed powers for four versions of hypothetic metamaterial parameter signs. The complex metamaterial permittivity *εmet*=s1*│εmet*│exp(-s2iδmet,ε) was taken for two combinations of signs, when s1=s2=±1. It is known that each metamaterial is intended for use in a specific frequency range and has a specific value of the effective permittivity and permeability at the certain frequency. For this reason we took the absolute values of real and imaginary parts of permittivity *εmet* and permeability *μmet* constant at all frequencies in our calculations. And the major impact makes the sizes' relation of wavelength and cylinder layers.

In figures 7.2-7.4 of this section are analyzed how the signs of the complex metamaterial permittivity and permeability influence on the scattered and absorbed powers when the plane perpendicularly or parallel polarized microwave impinges on the metamaterial-glass cylinder (Fig. 7.1).

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 51

In Fig. 7.3 is presented the absorbed microwave power Wa,glass by the coated glass layer at two microwave polarizations. We see that the behavior of the microwave power absorption strongly depends on the signs s1, s2, s3, s4 of the permittivity and the permeability. The absorption is especially different at the higher frequencies. We see that the absorption power

In Fig. 7.3(b) is presented the absorbed power by the coated glass layer for the incident parallel polarized microwave. There are some small distortion "hooks" of the absorbed power at the low frequencies. The absorbed power by glass layer increases with increasing of frequencies (curves 1, 3, 4) for frequencies that are larger than 20 GHz. The absorbed power is approximately constant when the SN metamaterial core permittivity and



(a) (b)

Fig. 7.4. Absorbed power by metamaterial core on the frequency of incident (a) -


0.0 *<sup>W</sup>*a,met, W/m

(a) (b)

Fig. 7.3. Absorbed power by the acrylic-glass layer on the frequency of incident (a) -



0.0

*W*a,glass, W/m

0 30 60 90 120

*f*, GHz

0 30 60 90 120

*f*, GHz

of the glass layer is larger at higher frequencies.

permeability have signs s1=s2=+1, s3=s4=-1.

*W*a,glass, W/m






0.0

*W*a,met, W/m



0.0

0 30 60 90 120

*f*, GHz

perpendicular and (b) - parallel polarized microwaves.

 1 2 3 4

0 30 60 90 120

*f*, GHz

perpendicular and (b) - parallel polarized microwaves

Designations in Figs 7.2-7.4 correspond: the curve 1 is for a DP material when s1=s2=s3= s4=+1 (line with black squares); the curve 2 is for a SN material when s1=s2=+1, s3= s4=-1 (line with empty squares); the curve 3 is for a SN material when s1=s2=-1, s3= s4=+1 (line with black triangulars); the curve 4 is for a DN material when s1=s2=s3=s4=-1 (line with empty triangulars).

The scattered and absorbed dependences of the metamaterial-glass cylinder when the incident microwave has the perpendicular or parallel polarization are shown in Figs 7.2-7.4. In Fig. 7.2 is presented the dependence of total scattered power WS on the microwave frequency *f* at two polarizations. We see that the character of curves for all metamaterial sign versions (curves 1-4) is the same at the perpendicular polarization. A comparison of curves 1-4 (Fig. 7.2(a)) shows that only curve 3 that describes by the metamaterial permittivity and permeability signs s1=s2=-1, s3=s4=+1 is the most different in comparison with other three cases. At the beginning the scattered power grows till the maximum value after that decreases till the minimum and later increases again with increasing of frequency. The total scattered power maximums of all curves are in the frequency range about 44-53 GHz. Curves 1 and 4 practically coincide with each other. The lowest scattered power is for the curve 3 at the frequencies about 1-35 GHz and the total scattered power minimum exists for the curve 2 approximately at the frequency 80 GHz.

Fig. 7.2. Scattered power of metamaterial-glass cylinder on the frequency of incident (a) perpendicular and (b) - parallel polarized microwaves.

The total scattered power of the incident perpendicularly and parallel polarized microwaves (Fig. 7.2(a, b)) are different. The scattered microwave power curves for the incident parallel polarized microwave have two maximums in the frequency range 1-120 GHz. The first position of the scattered power maxima are in the narrow frequency interval about 10-15 GHz and the second position of the maxima is in the interval 85-100 GHz. The largest scattering is at the lower frequencies. The maximum scattering is higher for the incident parallel polarized microwave in comparison with the incident perpendicularly polarized one. While the total scattered power curves for the incident perpendicularly polarized microwave (Fig. 7.2(a)) have only one maximum.

Designations in Figs 7.2-7.4 correspond: the curve 1 is for a DP material when s1=s2=s3= s4=+1 (line with black squares); the curve 2 is for a SN material when s1=s2=+1, s3= s4=-1 (line with empty squares); the curve 3 is for a SN material when s1=s2=-1, s3= s4=+1 (line with black triangulars); the curve 4 is for a DN material when s1=s2=s3=s4=-1 (line with

The scattered and absorbed dependences of the metamaterial-glass cylinder when the incident microwave has the perpendicular or parallel polarization are shown in Figs 7.2-7.4. In Fig. 7.2 is presented the dependence of total scattered power WS on the microwave frequency *f* at two polarizations. We see that the character of curves for all metamaterial sign versions (curves 1-4) is the same at the perpendicular polarization. A comparison of curves 1-4 (Fig. 7.2(a)) shows that only curve 3 that describes by the metamaterial permittivity and permeability signs s1=s2=-1, s3=s4=+1 is the most different in comparison with other three cases. At the beginning the scattered power grows till the maximum value after that decreases till the minimum and later increases again with increasing of frequency. The total scattered power maximums of all curves are in the frequency range about 44-53 GHz. Curves 1 and 4 practically coincide with each other. The lowest scattered power is for the curve 3 at the frequencies about 1-35 GHz and the total scattered power minimum exists for

0.0

(a) (b) Fig. 7.2. Scattered power of metamaterial-glass cylinder on the frequency of incident (a) -

The total scattered power of the incident perpendicularly and parallel polarized microwaves (Fig. 7.2(a, b)) are different. The scattered microwave power curves for the incident parallel polarized microwave have two maximums in the frequency range 1-120 GHz. The first position of the scattered power maxima are in the narrow frequency interval about 10-15 GHz and the second position of the maxima is in the interval 85-100 GHz. The largest scattering is at the lower frequencies. The maximum scattering is higher for the incident parallel polarized microwave in comparison with the incident perpendicularly polarized one. While the total scattered power curves for the incident perpendicularly polarized

1.0x10-5

2.0x10-5 *<sup>W</sup>* <sup>s</sup>

, W/m

0 30 60 90 120

*f*, GHz

empty triangulars).

0.0

3.0x10-6

6.0x10-6

9.0x10-6

1.2x10-5

*W* <sup>s</sup> , W/m

the curve 2 approximately at the frequency 80 GHz.

0 30 60 90 120

*f*, GHz

perpendicular and (b) - parallel polarized microwaves.

microwave (Fig. 7.2(a)) have only one maximum.

In Fig. 7.3 is presented the absorbed microwave power Wa,glass by the coated glass layer at two microwave polarizations. We see that the behavior of the microwave power absorption strongly depends on the signs s1, s2, s3, s4 of the permittivity and the permeability. The absorption is especially different at the higher frequencies. We see that the absorption power of the glass layer is larger at higher frequencies.

In Fig. 7.3(b) is presented the absorbed power by the coated glass layer for the incident parallel polarized microwave. There are some small distortion "hooks" of the absorbed power at the low frequencies. The absorbed power by glass layer increases with increasing of frequencies (curves 1, 3, 4) for frequencies that are larger than 20 GHz. The absorbed power is approximately constant when the SN metamaterial core permittivity and permeability have signs s1=s2=+1, s3=s4=-1.

Fig. 7.3. Absorbed power by the acrylic-glass layer on the frequency of incident (a) perpendicular and (b) - parallel polarized microwaves.

Fig. 7.4. Absorbed power by metamaterial core on the frequency of incident (a) perpendicular and (b) - parallel polarized microwaves

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 53



*W*

a, W/m




0.0

30 60 90 120

*f*, GHz

1

2

30 60 90 120

(a) (b)

comparison with the single-layered cylinder (Figs 7.2(b) and 7.5(b)).

GHz till 120 GHz in a comparison with the single-layered one.

Fig. 7.6. Total absorbed power by the twelve layered metamaterial-glass cylinder on the frequency of incident (a) - perpendicular and (b) - parallel polarized microwaves.

The comparison of scattered power of single- and twelve-layered cylinders shows that dependencies for the perpendicular polarized microwave are very similar. The scattering power maximum is at ~45 GHz and the minimum is at ~80 GHz (Figs 7.2(a) and 7.5(a)).

The scattered powers of single- and twelve-layered cylinders for the parallel polarized microwave are different (Figs 7.2(b) and 7.5(b)). We see that the scattered power of singlelayered cylinder has only two maximums in the considered frequency range while the twelve-layered cylinder dependency has four maximums. The first scattered power maximum of twelve-layered cylinder has a sharp pike and is more than twice larger in

Comparing Figs 7.3(a), 7.4(a) and 7.6(a) for perpendicular polarized microwave we see that the absorbed power determined mainly due to metamaterial losses. The absorbed power extremums of the twelve layered cylinder (Fig. 7.6 (a)) shifted to the higher frequencies in the comparison with single-layered cylinder (Fig. 7.4(a)) for the DP

Comparing curves for single- and twelve–layered cylinders (Fig. 7.4(a), curve 4 and Fig. 7.6 (a), curve 2) when the metamaterial is SN we can note that their lineaments are different at

The dependencies of absorbed parallel polarized microwave power for single-layered and twelve-layered cylinders in general are alike. The first minimum of absorbed power is shifted from ~5 GHz (Fig. 7.4(b)) till 10 GHz (Fig. 7.6 (b)) i.e. with growing of the number of layers the minimum shifted to the side of higher frequencies. The absorbed power dependency of twelve layered cylinder has a wavy behavior in the frequency range from 20

*f*, GHz

 1 2


metamaterial (curve 2).

the low frequencies.

*W*

a, W/m

0.0

In Fig. 7.4(a) is given the absorbed power by the metamaterial core of the cylinder for the perpendicular microwave polarization. We see that the Wa,met magnitudes have the pronounced wave-like nature dependent on the frequency. The metamaterial core absorption has the largest absolute value when the metamaterial permittivity has some negative values and the permeability has some positive values (curve 3). In Fig. 7.4(b) is given the metamaterial core absorbed power Wa,met for the incident parallel polarized microwave. We see that the absorbed powers have maximum values at about 5 GHz and their values vary slightly after 20 GHz with increasing of frequency. The absorption by the metamaterial is the largest at the low frequencies (curves 1 and 2) when the metamaterial has the positive permittivity. The comparison of absorbed powers in figures 7.3 and 7.4 shows that dependencies are absolutely different.

#### **7.2 Numerical analysis of the scattered and absorbed microwave power of twelvelayer cylinder**

In this subsection we present the total scattered power and the total absorbed power of incident microwave by the cylinder that consists of conductor core covered with 12 metamaterial and glass alternately layers. Designations in Figs 7.5 and 7.6 correspond: the curve 1 is for a DP material when s1=s2=s3= s4=+1 (line with black squares) and the curve 2 is for a DN material when s1=s2=s3= s4=-1 (line with empty triangulars).

In Figs 7.5 and 7.6 the permittivities and permeabilities metamaterial and acrylic-glass are the same as in the previous subsection. We see that the total scattered and absorbed powers strongly dependent on the polarization of incident wave. The characteristics of perpendicular polarized wave (Figs 7.5(a) and 7.6(a)) and the parallel polarized wave (Figs 7.5(b) and 7.6(b)) are completely different. Particularly noticeable the correlation between the total scattered and absorbed powers for the parallel polarized wave of the twelvelayered cylinder. We see that extremums of the total scattered and absorbed powers coincide in the frequency scale. For example, when the scattered power of cylinder has a maximum (Fig. 7.5(b)) then the absorbed power has a minimum (Fig. 7.6(b)) at f~10 GHz.

Fig. 7.5. Total scattered power of twelve layer metamaterial-glass cylinder on the frequency of incident (a) - perpendicular and (b) - parallel polarized microwaves

In Fig. 7.4(a) is given the absorbed power by the metamaterial core of the cylinder for the perpendicular microwave polarization. We see that the Wa,met magnitudes have the pronounced wave-like nature dependent on the frequency. The metamaterial core absorption has the largest absolute value when the metamaterial permittivity has some negative values and the permeability has some positive values (curve 3). In Fig. 7.4(b) is given the metamaterial core absorbed power Wa,met for the incident parallel polarized microwave. We see that the absorbed powers have maximum values at about 5 GHz and their values vary slightly after 20 GHz with increasing of frequency. The absorption by the metamaterial is the largest at the low frequencies (curves 1 and 2) when the metamaterial has the positive permittivity. The comparison of absorbed powers in figures 7.3 and 7.4

**7.2 Numerical analysis of the scattered and absorbed microwave power of twelve-**

for a DN material when s1=s2=s3= s4=-1 (line with empty triangulars).

30 60 90 120

(a) (b)

of incident (a) - perpendicular and (b) - parallel polarized microwaves

1

2

*f*, GHz

In this subsection we present the total scattered power and the total absorbed power of incident microwave by the cylinder that consists of conductor core covered with 12 metamaterial and glass alternately layers. Designations in Figs 7.5 and 7.6 correspond: the curve 1 is for a DP material when s1=s2=s3= s4=+1 (line with black squares) and the curve 2 is

In Figs 7.5 and 7.6 the permittivities and permeabilities metamaterial and acrylic-glass are the same as in the previous subsection. We see that the total scattered and absorbed powers strongly dependent on the polarization of incident wave. The characteristics of perpendicular polarized wave (Figs 7.5(a) and 7.6(a)) and the parallel polarized wave (Figs 7.5(b) and 7.6(b)) are completely different. Particularly noticeable the correlation between the total scattered and absorbed powers for the parallel polarized wave of the twelvelayered cylinder. We see that extremums of the total scattered and absorbed powers coincide in the frequency scale. For example, when the scattered power of cylinder has a maximum (Fig. 7.5(b)) then the absorbed power has a minimum (Fig. 7.6(b)) at f~10 GHz.

0.0

1.0x10-4

2.0x10-4

*W*

Fig. 7.5. Total scattered power of twelve layer metamaterial-glass cylinder on the frequency

s, W/m

3.0x10-4

4.0x10-4

30 60 90 120

*f*, GHz

 1 2

shows that dependencies are absolutely different.

**layer cylinder** 

0.0

2.0x10-6

4.0x10-6

*W*

s, W/m

6.0x10-6

8.0x10-6

Fig. 7.6. Total absorbed power by the twelve layered metamaterial-glass cylinder on the frequency of incident (a) - perpendicular and (b) - parallel polarized microwaves.

The comparison of scattered power of single- and twelve-layered cylinders shows that dependencies for the perpendicular polarized microwave are very similar. The scattering power maximum is at ~45 GHz and the minimum is at ~80 GHz (Figs 7.2(a) and 7.5(a)).

The scattered powers of single- and twelve-layered cylinders for the parallel polarized microwave are different (Figs 7.2(b) and 7.5(b)). We see that the scattered power of singlelayered cylinder has only two maximums in the considered frequency range while the twelve-layered cylinder dependency has four maximums. The first scattered power maximum of twelve-layered cylinder has a sharp pike and is more than twice larger in comparison with the single-layered cylinder (Figs 7.2(b) and 7.5(b)).

Comparing Figs 7.3(a), 7.4(a) and 7.6(a) for perpendicular polarized microwave we see that the absorbed power determined mainly due to metamaterial losses. The absorbed power extremums of the twelve layered cylinder (Fig. 7.6 (a)) shifted to the higher frequencies in the comparison with single-layered cylinder (Fig. 7.4(a)) for the DP metamaterial (curve 2).

Comparing curves for single- and twelve–layered cylinders (Fig. 7.4(a), curve 4 and Fig. 7.6 (a), curve 2) when the metamaterial is SN we can note that their lineaments are different at the low frequencies.

The dependencies of absorbed parallel polarized microwave power for single-layered and twelve-layered cylinders in general are alike. The first minimum of absorbed power is shifted from ~5 GHz (Fig. 7.4(b)) till 10 GHz (Fig. 7.6 (b)) i.e. with growing of the number of layers the minimum shifted to the side of higher frequencies. The absorbed power dependency of twelve layered cylinder has a wavy behavior in the frequency range from 20 GHz till 120 GHz in a comparison with the single-layered one.

Electrodynamical Analysis of Open Lossy Metamaterial Waveguide and Scattering Structures 55

Feng, T. H.; Li, Y.H.; Jiang, H. T.; Li W. X., Yang F.; Dong X. P.& Chen H. (2011) Tunable

Gric, T. & Nickelson, L. 2011. Electrodynamical investigation of the photonic waveguide

Gric, T.; Nickelson, L. & Asmontas, S. (2010). Electrodynamical characteristic particularity of

Gric, T.; Ašmontas, S. & Nickelson, L. (2010). 3D vector electric field distributions and

Ko, S.-T. & Lee, J.-H. (2010). Dual Property of Mu-near-zero to Epsilon-near-zero material, *J.of the Korean Physical Society*, Vol. 57, No. 1, pp. 51-54, ISSN 03744884. Kong, J.A. (2008). *Electromagnetic wave theory,* 1016 p., EMW Publishing, ISBN 0-9668143-9-8,

Li, M.-H.; Yang, H.-L.; Hou, X.-W.; Tian, Y.& Hou, D.-Y. (2011). Perfect metamaterial

Liu, S.-H.; Liang, C.-H.; Ding, W.; Chen, L. & Pan, W.-T. (2007). Electromagnetic wave

Lopez-Garcia, B.; Murthy, D. V. B. & Corona-Chavez, A. (2011). Half mode microwave

12-16 Sept. 2011, The Electromagnetics Academy, Cambridge, MA 02138. Liu, R.; Cheng, Q.; Hand, T.; Mock J.J.; Cui T. J.; Cummer S. A. &Smit, D. R.(2008)

Mirza, O.; Sabas, J. N.; Shi, S. & Prather, D. W. (2009) Experimental demonstration of

Mirzavand, R.; Honarbakhsh, B.; Abdipour, A. & Tavakoli A. (2009). Metamaterial-based

Nickelson, L.& Bucinskas, J. (2011). Microwave diffraction dependencies of a conductor

absorber with dual bands, *Progress In Electromagnetics Research,* PIER108, 37-49,

propagation through a slab waveguide of uniaxially anisotropic dispersive metamaterial, *Progress In Electromagnetics Research*, PIER76, pp. 467–475, ISSN 1070-

filters based on epsilon near zero and mu near zero concepts. *Progress In Electromagnetics Research,* Vol. 113, pp. 379-393*,* ISSN 1070-4698, E-ISSN 1559-8985. Luo, J.; Chen, H.; Lai, Y.; Xu, P. & Gao L. (2011) Anomalous transmission properties of epsilon-

near-zero metamaterials. *Proceedings of* Progress In Electromagnetics Research Symposium, pp. 1299-1302, ISSN 1559-9450, ISBN 978-1-934142-18-9, Suzhou, China,

Experimental Demonstration of Electromagnetic Tunneling Through an Epsilon-Near-Zero Metamaterial at Microwave Frequencies, Physical Review Letters, PRL

metamaterialbased phase modulation, *Progress In Electromagnetics Research,* PIER

phase shifters for ultra wide-band applications. *J. of Electromagn. Waves and Appl.,* 

cylinder coated with twelve glass and semiconductor layers on the *n*-Si specific resistivity. *Electronics and Electrical Engineering,* Vol. 115, No. 9, pp. 47-50, ISSN

*Research,* PIER109, pp. 361-379, ISSN 1070-4698, E-ISSN 1559-8985.

4698, E-ISSN 1559-8985.

14-16 June 2010, JUSIDA*,* Vilnius*.* 

Cambridge, Massachusetts, USA.

ISSN 1070-4698, E-ISSN 1559-8985.

100, pp. 023903-(1-4), ISSN 0031-9007.

Vol. 23, pp. 1489–1496, ISSN 0920–5071.

1392-1215.

93, pp. 1–12, ISSN 1070-4698, E-ISSN 1559-8985.

4698, E-ISSN 1559-8985.

single-negative metamaterials based on microstrip transmission line with varactor diodes loading, *Progress In Electromagnetics Research,* PIER 120, 35-50, ISSN 1070-

structure. *Electronics and Electrical Engineering*, Vol. 111, No. 5, pp. 1-4, ISSN 1392-1215.

open metamaterial square and circular waveguides. *Progress In Electromagnetics* 

dispersion characteristics of open rectangular and circular metamaterial waveguides, *Proceedings of 4th Microwave & Radar Week: 18th Int. Conf. on Microwaves, Radar, and Wireless Communicatins MIKON-2010,* Vol. 2, pp. 578–581, ISBN 978-9955-690-20-7, IEEE Catalog Number CFP10784-PRT, Vilnius, Lithuania,

#### **8. Conclusions**


#### **9. References**


1. We found that the scattered power dependences have wave behaviors. The minimal scattering from the metamaterial-glass cylinder are observed for the every metamaterial with some sign combinations of the permittivity and the permeability at the special

2. We found that the largest absorbed power by the coated acrylic-glass layer is observed for the case when the metamaterial is a single negative material with the negative permittivity. The absorbed power of the glass layer increases with increasing of

4. The comparison of single- and twelve- layered cylinder characteristics shows that the absorbed power extremums shift to the direction of higher frequencies when the

Abdalla, M. A.& Hu, Z. (2009). Multi-band functional tunable lh impedance transformer, *J.* 

Asmontas, S.; Nickelson, L.; Bubnelis, A.; Martavicius, R. & Skudutis, J. (2010). Hybrid Mode

Asmontas, S.; Nickelson, L.; Gric, T. & Galwas, B. A. (2009). Solution of Maxwell's Equations

Asmontas, S.; Nickelson, L. & Plonis, D. (2009). Dependences of propagation constants of

Bai, J.; Shi, S. & Prather, D. W. (2010). Analysis of epsilon-near-zero metamaterial super-

Bucinskas, J.; Nickelson, L. & Sugurovas, V. (2010). Microwave scattering and absorption by

Bucinskas, J.; Nickelson, L. & Sugurovas, V. (2010). Microwave diffraction characteristic

Engheta, N. & Ziolkowski R.W. (2005). A positive future for double-negative metamaterials.

*Electromagnetics Research M,* Vol. 14, pp. 113-121, E-ISSN 1559-8985.

*Research,* PIER105, pp. 103–118, ISSN 1070-4698, E-ISSN 1559-8985.

*Engineering,* Vol. 94, No. 6, pp. 57-60, ISSN 1392-1215.

Dispersion Characteristic Dependencies of Cylindrical Dipolar Glass Waveguides on Temperatures, *Electronics and Electrical Engineering*, Vol. 106, No. 10, pp. 83-86,

by the Partial Area Method for the Electrodynamical Analyses of Open Lossy Metamaterial Waveguides, *in Proceedings of the International Conference Differential Equations and their Applicati*ons. *DETA‒2009,* pp. 17–20, ISBN 978-9955-25-747-9, Panevezys, Lithuania*,* 10-12 Sept. 2009*,* TECHNOLOGIJA, Kaunas University of

cylindrical n-Si rod on the material specific resistivity, *Electronics and Electrical* 

tunneling using cascaded ultra-narrow waveguide channels. *Progress In* 

a multilayered lossy metamaterial-glass cylinder, *Progress In Electromagnetics* 

analysis of 2D multilayered uniaxial anisotropic cylinder, *Progress In Electromagnetics Research*, PIER109, pp. 175-190, ISSN 1070-4698, E-ISSN 1559-8985. Chen, H.; Wu, B.-I. & Kong, J. A. (2006) Review of electromagnetic theory in left-handed

materials *J. of Electromagn. Waves and Appl.,* Vol. 20, No. 15, pp. 2137–2151, ISSN

*IEEE Transactions on microwave theory and techniques,* Vol. 53, No. 4, pp. 1535- 1556,

*of Electromagn. Waves and Appl.,* Vol. 23, pp. 39–47*,* ISSN 0920–5071.

frequency in the range 1-120 GHz for both microwave polarizations (Figs 7.2). 3. The metamaterial core absorbed power of the parallel polarized incident microwave has the minimum value at low frequencies and slightly dependent on the frequency at the

**8. Conclusions** 

**9. References** 

frequency range (Figs 7.2).

range 20-120 GHz (Fig. 7.4(b)).

ISSN 1392-1215.

Technology, Kaunas.

0920–5071.

ISSN 0018–9480.

number of layers becomes larger (Figs 7.4 and 7.6).


**1. Introduction**

Metamaterial is a very fresh concept in modern photonics, which are referred to a new class of electromagnetic media whose permittivity or permeability is beyond traditional values. Fifty years ago, a kind of new materials whose permittivity *�* and permeability *μ* are simultaneously negative was theoretically predicted to possess a negative refractive index *n* with many unusual properties[1]. In last decade, negative-*n* metallic resonating composites and two dimensional (2D) isotropic negative-*n* material have been constructed[2, 3], and negative light refraction was observed[4]. The unconventional properties of such materials, such as the evanescent wave could be amplified by negative-*n* so that the sub-wavelength resolution could be achieved[5], have drawn an increasing amount of attention in both science and engineering[6]. After negative-*n* material, more such unconventional materials are found, so that a new concept "metamaterial" is generated, which termed for the effective medium with very special permittivity *�eff* , or permeability *�eff* , or both, over a certain finite frequency band. Such physical media are composed of distinct elements (photonic atoms) which are generally made of sub-wavelength metallic structure and their size scale is much smaller than the wavelengths in the frequency range of interest. Thus, the effective composite media could be considered homogeneous at the wavelengths under consideration. Since their abnormal properties and related totally new phenomena can even go beyond the traditional physical

Xunya Jiang, Zheng Liu, Wei Li, Zixian Liang, Penjun Yao, Xulin Lin,

*State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of* 

**The Group Velocity Picture of** 

Xiaogang Zhang, Yongliang Zhang and Lina Shi

*Microsystem and Information Technology, CAS, Shanghai* 

**Metamaterial Systems** 

*China* 

**3**

However, from the beginning of the metamaterial research, there are many arguments for a lot topics, such as, Pendry's famous pioneer work of superlens[5] was commended several times. One main reason of so many arguments is that the light *beams*in different metamaterials seem to be too strange (even weird) to be acceptable. So it is natural to argue whether these beams could be real. Another main reason is a general weakness of current metamaterial studies which mainly focus on the single frequency properties and neglect the dispersion. Actually these two reasons are related. We know that the dispersion , in the frame of classical electrodynamic, means the electromagnetic response of the material to the external field, and plays the key role in the metamaterial abnormal properties. For these strange beams, such as negative refraction beams, with dispersion we can obtain the group velocity (energy velocity) which determine the beams propagating direction. So the group velocity should be the basic picture for us to understand these strange beams and help us to design related devices. More

limit, metamaterial becomes one of hottest topics in modern photonics.


### **The Group Velocity Picture of Metamaterial Systems**

Xunya Jiang, Zheng Liu, Wei Li, Zixian Liang, Penjun Yao, Xulin Lin, Xiaogang Zhang, Yongliang Zhang and Lina Shi *State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, CAS, Shanghai China* 

#### **1. Introduction**

56 Metamaterial

Nickelson, L.; Gric, T. & Asmontas, S. (2011). Chapter 6: Electrodynamical modelling of

Nickelson, L.; Bubnelis, A.; Baskys, A. & Navickas, R. (2011). The magnetoactive *p*-Ge rod

Nickelson, L.; Asmontas, S.; Malisauskas, V. & Martavicius, R. (2009). The dependence of

Nickelson, L.; Gric, T.; Asmontas, S. & Martavicius, R. (2009) Analyses of the gyroelectric

Nickelson, L.; Gric, T.; Asmontas, S. & Martavicius, R. (2008). Electrodynamical analyses of

Nickelson, L.; Galwas, B.A.; Gric, T. & Ašmontas, S. (2008). Electric field distributions in the

Nickelson, L. & Shugurov, V. (2005). *Singular integral equations' method for the analysis of* 

Oraizi, H. & Abdolali, A. (2009) Some aspects of radio wave propagation in double zero

Wang, B. &. Huang, K.-M (2010). Shaping the radiation pattern with mu and epsilon-near-

Zhou, H.; Ding, F.; Jin, Y. & He, S. L. (2011). Terahertz metamaterial modulators based on

Zhou, H.; Pei,Z.; Qu, S.; Zhang, S. & Wang, J. (2009) A planar zero-index metamaterial for

Zhu, B.; Wang, Z.; Huang, C.; Feng, Y.; Zhao, J. & Jiang, T. (2010) Polarization insensitive

PIER101, pp. 231-239, ISSN 1070-4698, E-ISSN 1559-8985.

*Electromagn. Waves and Appl.*, Vol. 23, pp. 1957–1968*,* ISSN 0920–5071. Penciu, R.S.; Kafesaki, M.; Gundogdu, T.F.; Economou, E.N. & Soukoulis, C.M. (2006).

*Electrical Engineering*, Vol. 82, No. 2, pp. 3-8, ISSN 1392-1215.

May 2008, MDruk, Warszawa, Poland.

Leiden-Boston, The Netherlands.

ISSN 1070-4698, E-ISSN 1559-8985.

4698, E-ISSN 1559-8985.

307-201-2, Rijeka, Croatia.

51, ISSN 0022-3778.

June – 2 July 2009.

1215.

4410.

0920–5071.

open cylindrical and rectangular carbide waveguides, In: *Properties and applications of Silicon Carbide*, (Ed. Rosario Gerhardt), 535 p., pp 115-141, InTech, ISBN 978-953-

waveguide loss analysis on the concentration of two component hole charge carriers, *Electronics and Electrical Engineering,* Vol. 110, No. 4, pp. 53-56, ISSN 1392-

open cylindrical magnetoactive *p*-Ge and *p*-Si plasma waveguide mode cutoff frequencies on hole concentrations, *Journal of Plasma Physics,* Vol. 75, No. 1, pp. 35–

plasma rod waveguide, *Proceedings of 17th IEEE International Pulsed Power Conference*, pp. 727-730, IEEE Catalog Number: CFP09PPC-DVD, Library of Congress: 2009901215, DC USA, ISBN 978-1-4244-4065-8, Washington, DC, USA**,** 29

dielectric and metamaterial hollow-core cylindrical waveguides. *Electronics and* 

cross sections of the metamaterial hollow-core and rod waveguides, *Proceedings of 17th International Conference on Microwaves, Radar and Wireless Communication MIKON-2008,* Vol. 2, pp. 497-500, ISBN 978-1424-431-22-9, Wrocław, Poland, 19-21

*microwave structures*, 348 p., VSP Brill Academic Publishers, ISBN 90-6764-410-2,

metamaterials having the real parts of epsilon and mu equal to zero, *J. of* 

Theoretical study of left-handed behavior of composite metamaterials. *Photonics and Nanostructures – Fundamentals and Applications*, Vol. 4, pp. 12-16, ISSN 1569-

zero metamaterials. *Progress In Electromagnetics Research,* PIER 106, pp. 107-119,

absorption, *Progress In Electromagnetics Research,* PIER119, pp. 449-460, ISSN 1070-

directive emission, *J. of Electromagn. Waves and Appl.,* Vol. 23, pp. 953–962, ISSN

metamaterial absorber with wide incident angle, *Progress In Electromagnetics,* 

Metamaterial is a very fresh concept in modern photonics, which are referred to a new class of electromagnetic media whose permittivity or permeability is beyond traditional values. Fifty years ago, a kind of new materials whose permittivity *�* and permeability *μ* are simultaneously negative was theoretically predicted to possess a negative refractive index *n* with many unusual properties[1]. In last decade, negative-*n* metallic resonating composites and two dimensional (2D) isotropic negative-*n* material have been constructed[2, 3], and negative light refraction was observed[4]. The unconventional properties of such materials, such as the evanescent wave could be amplified by negative-*n* so that the sub-wavelength resolution could be achieved[5], have drawn an increasing amount of attention in both science and engineering[6]. After negative-*n* material, more such unconventional materials are found, so that a new concept "metamaterial" is generated, which termed for the effective medium with very special permittivity *�eff* , or permeability *�eff* , or both, over a certain finite frequency band. Such physical media are composed of distinct elements (photonic atoms) which are generally made of sub-wavelength metallic structure and their size scale is much smaller than the wavelengths in the frequency range of interest. Thus, the effective composite media could be considered homogeneous at the wavelengths under consideration. Since their abnormal properties and related totally new phenomena can even go beyond the traditional physical limit, metamaterial becomes one of hottest topics in modern photonics.

However, from the beginning of the metamaterial research, there are many arguments for a lot topics, such as, Pendry's famous pioneer work of superlens[5] was commended several times. One main reason of so many arguments is that the light *beams*in different metamaterials seem to be too strange (even weird) to be acceptable. So it is natural to argue whether these beams could be real. Another main reason is a general weakness of current metamaterial studies which mainly focus on the single frequency properties and neglect the dispersion. Actually these two reasons are related. We know that the dispersion , in the frame of classical electrodynamic, means the electromagnetic response of the material to the external field, and plays the key role in the metamaterial abnormal properties. For these strange beams, such as negative refraction beams, with dispersion we can obtain the group velocity (energy velocity) which determine the beams propagating direction. So the group velocity should be the basic picture for us to understand these strange beams and help us to design related devices. More

range, which can be used as a removable memory or a light trap. All theoretical results are

The Group Velocity Picture of Metamaterial Systems 59

In the third section we propose general evanescent-mode-sensing methods to probe the quantum electrodynamics (QED) vacuum polarization. The methods are based on the phase change and the energy time delay of evanescent wave caused by small dissipation. From our methods, high sensitivity can be achieved even though the external field, realizable in

In the forth section the image field of the negative-index superlens with the quasi-monochromatic random source is discussed, and dramatic temporal-coherence gain of the image in the numerical simulation is observed, even if there is almost no reflection and no frequency filtering effects. From the new physical picture, a theory is constructed to obtain the image field and demonstrate that the temporal coherence gain is from different "group" retarded time of different optical paths. Our theory agrees excellently with the numerical simulation and strict Green's function method. These study should have important consequences in the coherence studies in the related systems and the design of

In the fifth section, the dynamical processes of dispersive cloak by finite-difference time-domain numerical simulation are carried out. It is found that there is a strong scattering process before achieving the stable state and its time length can be tuned by the dispersive strength. Poynting-vector directions show that the stable cloaking state is constructed locally while an intensity front sweeps through the cloak. Deeper studies demonstrate that the group velocity tangent component *Vg<sup>θ</sup>* is the dominant factor in the process. This study is helpful not only for clear physical pictures but also for designing better cloaks to defend passive radars. In the sixth section, the limitation of the electromagnetic cloak with dispersive material is investigated based on causality. The results show that perfect invisibility can not be achieved because of the dilemma that either the group velocity *Vg* diverges or a strong absorption is imposed on the cloaking material. It is an intrinsic conflict which originates from the demand of causality. However, the total cross section can really be reduced through the approach of coordinate transformation. A simulation of finite-difference time-domain method

**2. Hyper-interface, the bridge between radiative wave and evanescent wave**

Many new phenomena are observed at the interfaces between meta-material and common dielectric material, such as the negative refraction which is found at the left-handed material(LHM) surface More interestingly, the evanescent wave (EW) could be amplified at LHM interface so that the super-resolution could be achieved[14]. Besides the LHM, there is another class of anisotropic metamaterial, so called "hyperbolic medium"(HM), in which one of the diagonal permittivity tensor components is negative and results in a hyperbolic dispersion. For convenience, we call the interface between a HM and a common dielectric material as "hyper-interface"(HI). Some surprising electromagnetic properties of HI are intensively studied recently[15, 16]. For instance, HI can convert the EW into the radiative wave (RW) so that the sub-wavelength information could be observed at far-field, which is called "hyper-lens" effect[15]. Very recently it is found that when HI is perpendicular to one

contemporary experiments, is much smaller than the Schwinger critical field.

demonstrated by finite-difference time-domain simulation.

novel devices.

is performed to validate the analysis.

In the last section, we give a summary of our works.

seriously, the dispersion is related with some very basic limitations of this world, *e*.*g*. the causality limitation that the group velocity in metamaterial should be less than the vacuum light speed. If our design of devices is based on the metamaterials which violate these basic limitations, the design surely can not work since such metamaterial could not exist in this world. From this view, the group velocity picture is not only needed in understanding and explanation, but also required in research of some topics and design of devices. For instance, in the research of the limitation of the cloak[7, 8] and abnormal phenomena on the interface of the hyperbolic metamaterial[9], if we neglect the dispersion of material, from group velocity picture we will immediately find that we have fallen into a superluminal trap, since the energy velocity in such artificial systems is divergent. So, the group velocity picture can help us avoid such traps.

As a basic value for revealing the complex propagating process, abnormal group velocity has been studied for decades. In the early 1960's the group velocity in material has been studied by Brillouin[10] and the group velocity in strongly scattering media is investigated by J. H. Page, Ping Sheng *et al.* in 1996[11], which indicate that the physical origin of the remarkably low velocities of propagation lies in the renormalization of the effective medium by strong resonant scattering. So far, metamaterials generally are composite of "photonic atoms" which can scatter light coherently. And all abnormal properties of metamaterials, *e.g.* these strange beams, are from these complex coherent scattering. Another byproduct of these scattering is the (abnormal) group velocity. In other words, the strange beam and the abnormal group velocity are two sides of a same coin. Further more, with some abnormal group velocity, such as the extremely slow light, we can design new signal-processing devices or new detecting devices. Hence, exploring the group velocity in metamaterial is very vital for revealing mechanism and the design of the real optical devices.

The numerical simulation takes an important role in research for modern photonics. For metamaterial, since the difficulties of experimental realization, the numerical tools become very essential for researchers. But, in some frequency domain simulation softwares, the dispersion is neglected totally. As we discussed above, we think such softwares can misleading researchers to some imaginary metamaterial which can not exist in this world. Such as for cloaking study, these softwares could present perfect invisibility very easily, but from our study[7, 8], that is misleading one. We strongly recommend the time-domain softwares, such as finite-difference time-domain (FDTD) method or finite-element time-domain(FETD) method. Their simulating results are much more convincible since they are generally with physical dispersion in the simulation and fit for metamaterial studies.

This paper is organized as following. The first section is the introduction in which we generally introduce the group velocity picture of the metamaterial study. As we have discussed, the group velocity is the key for understanding these abnormal properties of metamaterials and also can help us to avoid some traps of basic physical limit. We have also commended the softwares fit for metamaterial studies.

In the second section, the optical properties of the interface between hyperbolic meta-material (with anisotropic hyperbolic dispersion) and common dielectric is investigated. With material dispersion, a comprehensive theory is constructed, and the hyperlens effect that the evanescent wave can be converted into the radiative wave is confirmed. At the inverse process of hyperlens, we find a novel mechanism to compress and stop (slow) light at wide frequency 2 Will-be-set-by-IN-TECH

seriously, the dispersion is related with some very basic limitations of this world, *e*.*g*. the causality limitation that the group velocity in metamaterial should be less than the vacuum light speed. If our design of devices is based on the metamaterials which violate these basic limitations, the design surely can not work since such metamaterial could not exist in this world. From this view, the group velocity picture is not only needed in understanding and explanation, but also required in research of some topics and design of devices. For instance, in the research of the limitation of the cloak[7, 8] and abnormal phenomena on the interface of the hyperbolic metamaterial[9], if we neglect the dispersion of material, from group velocity picture we will immediately find that we have fallen into a superluminal trap, since the energy velocity in such artificial systems is divergent. So, the group velocity picture can help us avoid

As a basic value for revealing the complex propagating process, abnormal group velocity has been studied for decades. In the early 1960's the group velocity in material has been studied by Brillouin[10] and the group velocity in strongly scattering media is investigated by J. H. Page, Ping Sheng *et al.* in 1996[11], which indicate that the physical origin of the remarkably low velocities of propagation lies in the renormalization of the effective medium by strong resonant scattering. So far, metamaterials generally are composite of "photonic atoms" which can scatter light coherently. And all abnormal properties of metamaterials, *e.g.* these strange beams, are from these complex coherent scattering. Another byproduct of these scattering is the (abnormal) group velocity. In other words, the strange beam and the abnormal group velocity are two sides of a same coin. Further more, with some abnormal group velocity, such as the extremely slow light, we can design new signal-processing devices or new detecting devices. Hence, exploring the group velocity in metamaterial is very vital

The numerical simulation takes an important role in research for modern photonics. For metamaterial, since the difficulties of experimental realization, the numerical tools become very essential for researchers. But, in some frequency domain simulation softwares, the dispersion is neglected totally. As we discussed above, we think such softwares can misleading researchers to some imaginary metamaterial which can not exist in this world. Such as for cloaking study, these softwares could present perfect invisibility very easily, but from our study[7, 8], that is misleading one. We strongly recommend the time-domain softwares, such as finite-difference time-domain (FDTD) method or finite-element time-domain(FETD) method. Their simulating results are much more convincible since they are generally with physical dispersion in the simulation and fit for

This paper is organized as following. The first section is the introduction in which we generally introduce the group velocity picture of the metamaterial study. As we have discussed, the group velocity is the key for understanding these abnormal properties of metamaterials and also can help us to avoid some traps of basic physical limit. We have

In the second section, the optical properties of the interface between hyperbolic meta-material (with anisotropic hyperbolic dispersion) and common dielectric is investigated. With material dispersion, a comprehensive theory is constructed, and the hyperlens effect that the evanescent wave can be converted into the radiative wave is confirmed. At the inverse process of hyperlens, we find a novel mechanism to compress and stop (slow) light at wide frequency

for revealing mechanism and the design of the real optical devices.

also commended the softwares fit for metamaterial studies.

such traps.

metamaterial studies.

range, which can be used as a removable memory or a light trap. All theoretical results are demonstrated by finite-difference time-domain simulation.

In the third section we propose general evanescent-mode-sensing methods to probe the quantum electrodynamics (QED) vacuum polarization. The methods are based on the phase change and the energy time delay of evanescent wave caused by small dissipation. From our methods, high sensitivity can be achieved even though the external field, realizable in contemporary experiments, is much smaller than the Schwinger critical field.

In the forth section the image field of the negative-index superlens with the quasi-monochromatic random source is discussed, and dramatic temporal-coherence gain of the image in the numerical simulation is observed, even if there is almost no reflection and no frequency filtering effects. From the new physical picture, a theory is constructed to obtain the image field and demonstrate that the temporal coherence gain is from different "group" retarded time of different optical paths. Our theory agrees excellently with the numerical simulation and strict Green's function method. These study should have important consequences in the coherence studies in the related systems and the design of novel devices.

In the fifth section, the dynamical processes of dispersive cloak by finite-difference time-domain numerical simulation are carried out. It is found that there is a strong scattering process before achieving the stable state and its time length can be tuned by the dispersive strength. Poynting-vector directions show that the stable cloaking state is constructed locally while an intensity front sweeps through the cloak. Deeper studies demonstrate that the group velocity tangent component *Vg<sup>θ</sup>* is the dominant factor in the process. This study is helpful not only for clear physical pictures but also for designing better cloaks to defend passive radars.

In the sixth section, the limitation of the electromagnetic cloak with dispersive material is investigated based on causality. The results show that perfect invisibility can not be achieved because of the dilemma that either the group velocity *Vg* diverges or a strong absorption is imposed on the cloaking material. It is an intrinsic conflict which originates from the demand of causality. However, the total cross section can really be reduced through the approach of coordinate transformation. A simulation of finite-difference time-domain method is performed to validate the analysis.

In the last section, we give a summary of our works.

#### **2. Hyper-interface, the bridge between radiative wave and evanescent wave**

Many new phenomena are observed at the interfaces between meta-material and common dielectric material, such as the negative refraction which is found at the left-handed material(LHM) surface More interestingly, the evanescent wave (EW) could be amplified at LHM interface so that the super-resolution could be achieved[14]. Besides the LHM, there is another class of anisotropic metamaterial, so called "hyperbolic medium"(HM), in which one of the diagonal permittivity tensor components is negative and results in a hyperbolic dispersion. For convenience, we call the interface between a HM and a common dielectric material as "hyper-interface"(HI). Some surprising electromagnetic properties of HI are intensively studied recently[15, 16]. For instance, HI can convert the EW into the radiative wave (RW) so that the sub-wavelength information could be observed at far-field, which is called "hyper-lens" effect[15]. Very recently it is found that when HI is perpendicular to one

H<sup>h</sup>

θ k<sup>x</sup>

kt

ki

Fig. 1. The frequency contour of HM and isotropic dielectric material in *k* space. The inset:

In the HM region and the isotropic dielectric region the fields can be expressed uniformly as

the scattered fields from HI; *ξ* = *x*, *y*; *σ* = *h*, *d* for the HM region or dielectric region. Since the translation symmetry of flat HI, the wave-vector parallel component *kx* is continuous at

To explore the transmission and reflection properties of HI, we define a scattering matrix

*sz*)*<sup>T</sup>* <sup>=</sup> *<sup>S</sup>*2×<sup>2</sup> (*H<sup>h</sup>*

*ix*)*�*/(*c<sup>h</sup>*

*<sup>i</sup>*(*s*)*<sup>x</sup>* <sup>=</sup> <sup>−</sup>˜

respectively. From Eq.(3) we can easily get the reflection and transmission coefficients across the HI from upper to down, or inverse. For the case of hyperlens that the wave is incident

*sx* <sup>−</sup> *<sup>c</sup><sup>h</sup>*

*<sup>x</sup>* + *<sup>α</sup>*2))1/2 + ˜

*ixH<sup>σ</sup> iz* + *<sup>c</sup><sup>σ</sup> sxH<sup>σ</sup>*

*νξ*/*H<sup>σ</sup>*

<sup>−</sup> (*ky* cos *<sup>θ</sup>* <sup>+</sup> *kx* sin *<sup>θ</sup>*)<sup>2</sup>

<sup>|</sup>*�*1<sup>|</sup> = ( *<sup>ω</sup>*

*iyH<sup>σ</sup> iz* + *<sup>c</sup><sup>σ</sup> syH<sup>σ</sup>*

*<sup>ν</sup><sup>z</sup>* with *ν* = *i*,*s* for the incident fields to HI or

*ix�* <sup>+</sup> ˜ *kd y*)/(*c<sup>h</sup>*

*<sup>y</sup>*), and *<sup>S</sup>*<sup>22</sup> <sup>=</sup> <sup>−</sup>2˜

*i*(*s*)*x* , *c<sup>d</sup>* *kσ νξ* <sup>=</sup> *<sup>k</sup><sup>σ</sup>*

*iz*)*<sup>T</sup>* (3)

/*�*, where factors *α*, *γ* are defined as

*kxαγ*)/*α*<sup>2</sup> are uniquely determined by Eq.(2),

*kd*

*sx�* + ˜ *kd <sup>y</sup>*), *S*<sup>12</sup> =

*<sup>i</sup>*(*s*)*<sup>x</sup>* are worked out to be

*<sup>i</sup>*(*s*)*<sup>y</sup>* = ±

*kd iy*/(*c<sup>h</sup>*

*sz*) + **ey**(*c<sup>σ</sup>*

*iz*, *<sup>H</sup><sup>d</sup>*

*sx�* + ˜ *kd*

*kd i*(*s*)*y*

<sup>2</sup> ; *<sup>γ</sup>* = (|*�*1<sup>|</sup> <sup>+</sup> *�*2) sin 2*θ*/2*α*, and the values of ˜

*<sup>c</sup>* )<sup>2</sup> (2)

*sz*)]*eikxx*−*iω<sup>t</sup>* where

*νξ*/*k*<sup>0</sup> where

*sx�* <sup>−</sup> ˜ *kd iy*).

> *�* <sup>−</sup> ˜ *k*2 *x*

the schematic figure of our model.

*iz* + *<sup>H</sup><sup>σ</sup>*

the coefficients are defined as *c<sup>σ</sup>*

*sz*)*eikxx*−*iω<sup>t</sup>*

*sx�* <sup>−</sup> ˜ *kd*

*α*<sup>2</sup> + ˜

**H***<sup>σ</sup>* = **ez**(*H<sup>σ</sup>*

*k*<sup>0</sup> = *ω*/*c*.

(−*c<sup>h</sup>*

and ˜ *kh i*(*s*)*y*

*ch*

*ix�* <sup>+</sup> ˜ *kd iy*)/(*c<sup>h</sup>*

*<sup>i</sup>*(*s*)*<sup>x</sup>* = (−˜

*kh i*(*s*)*y*

*<sup>α</sup>* = (*�*<sup>1</sup> sin2 *<sup>θ</sup>* <sup>−</sup> *�*<sup>2</sup> cos<sup>2</sup> *<sup>θ</sup>*)

and ˜ *kh*

then the most general frequency contour in *k* space of HM is:

(*kx* cos *<sup>θ</sup>* <sup>−</sup> *ky* sin *<sup>θ</sup>*)<sup>2</sup> *�*2

associated with the incident fields and out-going fields at HI.

(*H<sup>h</sup> sz*, *<sup>H</sup><sup>d</sup>*

where the superscript *<sup>T</sup>* means the matrix transpose, *<sup>S</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup>(*c<sup>h</sup>*

*iy*), *<sup>S</sup>*<sup>21</sup> = (*c<sup>h</sup>*

*kxαγ*)/(|*�*1|*�*2); *<sup>c</sup><sup>d</sup>*

*k*2

From the standard boundary conditions, the coefficients *c<sup>h</sup>*

1

*<sup>i</sup>*(*s*)*<sup>y</sup>* = (±(|*�*1|*�*2(˜

, **E<sup>œ</sup>** = [**ex**(*c<sup>σ</sup>*

*νξ* <sup>=</sup> *<sup>E</sup><sup>σ</sup>*

both regions. In following discussion, the *k* vectors are normalized by ˜

HM

ky kr

The Group Velocity Picture of Metamaterial Systems 61

<sup>i</sup> H<sup>h</sup> s

H<sup>d</sup> H <sup>s</sup> <sup>d</sup> i

asymptote of HM dispersion, abnormal omnidirectional transmission occurs[17]. Although some theoretical and experimental works [16] have demonstrated that the EW really can be converted into RW by HI of the layered cylindrical HM, a full theory involving the "material dispersion"(will be explained later) has not been given so far. For meta-material systems, if without physical dispersion, some abnormal optical properties can not be clearly explained and the dynamical study of wave propagation can not be carried out[20]. Even more seriously, the causality violation because of the superluminal group velocity (*vg* > *c*) in HM is pointed out[18], which makes the observed hyper-lens effect doubtful. To solve these problems and predict new phenomena, more robust theory with stronger base is needed.

On the other hand, to compress and to stop (slow) light pulses are very essential for modern optical/photonics research and signal processing. Hence, a new mechanism, which can compress and stop (slow) light pulses and is frequency and direction insensitive, would induce wide interest in related directions.

In this Letter, we theoretically and numerically investigate the novel optical properties of flat HI[22], in which, unlike the cylindrical HI, the translational symmetry guarantees the simple physical picture for intuitive understanding, the quantitative study of the conversion between EW and RW *etc*.

A general theory of HI is constructed with physical dispersion of HM. On the HI, not only the conversion from EW to RW (CER) of hyperlens is confirmed, when RW is incident from HM to dielectric(the inverse process of hyper-lens), but also the almost total conversion from RW to EW (CRE) can occur, *i*.*e*. there is "no-transmission and no-refelection" (NTNR). More important we find that this is a new mechanism to compress and stop (slow) light pulses in wide frequency and direction range with many potential applications. Theoretically and numerically we demonstrate that the superluminal group velocity in hyperlens is artificial since the HM material dispersion is neglected in previous study[18]. At last, the feasibility to realize these functions on real structures is discussed. All theoretical results are demonstrated by finite-difference-time-domain (FDTD) simulations.

Our model is as follows. Assuming two plane waves are incident to HI from HM and isotropic dielectric, and scattered from HI, as shown in the upper-right insert of Fig.(1). The HI is in the *x*-*z* plane, while the incident surface and both HM optical axes lie in the *x*-*y* plane. The incident waves are chosen as TM wave with field components (*Ex*, *Ey*, *Hz*) and same "parallel wave-vector" *kx*. The HM is with the permittivity tensor as

$$
\hat{\mathfrak{e}}\_p = \begin{pmatrix} \mathfrak{e}\_1(\omega) & 0 \\ 0 & \mathfrak{e}\_2 \end{pmatrix} \tag{1}
$$

in its principle axes coordinate, where *�*<sup>1</sup> < 0 and *�*<sup>2</sup> > 0 are assumed. And the permittivity of isotropic dielectric material is *�*. The essential point of our model is that the negative diagonal component is dispersive *�*<sup>1</sup> = *�*1(*ω*), which is called *material dispersion* in our study. It is well known that dispersion is physically required for real meta-materials with abnormal effective constitutive coefficients, such as negative permittivity. We will see that the material dispersion will help us to obtain self-consistent explanation of abnormal optical properties of HI and to avoid causality violation.

For simplicity, the HM is nonmagnetic and Gaussian unit is employed throughout the paper. We define the angle between the HI (or *x* axis) and the positive-*�* principle axis of HM is *θ*, 4 Will-be-set-by-IN-TECH

asymptote of HM dispersion, abnormal omnidirectional transmission occurs[17]. Although some theoretical and experimental works [16] have demonstrated that the EW really can be converted into RW by HI of the layered cylindrical HM, a full theory involving the "material dispersion"(will be explained later) has not been given so far. For meta-material systems, if without physical dispersion, some abnormal optical properties can not be clearly explained and the dynamical study of wave propagation can not be carried out[20]. Even more seriously, the causality violation because of the superluminal group velocity (*vg* > *c*) in HM is pointed out[18], which makes the observed hyper-lens effect doubtful. To solve these problems and

On the other hand, to compress and to stop (slow) light pulses are very essential for modern optical/photonics research and signal processing. Hence, a new mechanism, which can compress and stop (slow) light pulses and is frequency and direction insensitive, would

In this Letter, we theoretically and numerically investigate the novel optical properties of flat HI[22], in which, unlike the cylindrical HI, the translational symmetry guarantees the simple physical picture for intuitive understanding, the quantitative study of the conversion between

A general theory of HI is constructed with physical dispersion of HM. On the HI, not only the conversion from EW to RW (CER) of hyperlens is confirmed, when RW is incident from HM to dielectric(the inverse process of hyper-lens), but also the almost total conversion from RW to EW (CRE) can occur, *i*.*e*. there is "no-transmission and no-refelection" (NTNR). More important we find that this is a new mechanism to compress and stop (slow) light pulses in wide frequency and direction range with many potential applications. Theoretically and numerically we demonstrate that the superluminal group velocity in hyperlens is artificial since the HM material dispersion is neglected in previous study[18]. At last, the feasibility to realize these functions on real structures is discussed. All theoretical results are demonstrated

Our model is as follows. Assuming two plane waves are incident to HI from HM and isotropic dielectric, and scattered from HI, as shown in the upper-right insert of Fig.(1). The HI is in the *x*-*z* plane, while the incident surface and both HM optical axes lie in the *x*-*y* plane. The incident waves are chosen as TM wave with field components (*Ex*, *Ey*, *Hz*) and same "parallel

> *�*1(*ω*) 0 0 *�*<sup>2</sup>

in its principle axes coordinate, where *�*<sup>1</sup> < 0 and *�*<sup>2</sup> > 0 are assumed. And the permittivity of isotropic dielectric material is *�*. The essential point of our model is that the negative diagonal component is dispersive *�*<sup>1</sup> = *�*1(*ω*), which is called *material dispersion* in our study. It is well known that dispersion is physically required for real meta-materials with abnormal effective constitutive coefficients, such as negative permittivity. We will see that the material dispersion will help us to obtain self-consistent explanation of abnormal optical properties of HI and to

For simplicity, the HM is nonmagnetic and Gaussian unit is employed throughout the paper. We define the angle between the HI (or *x* axis) and the positive-*�* principle axis of HM is *θ*,

(1)

*�*ˆ*<sup>p</sup>* =

predict new phenomena, more robust theory with stronger base is needed.

induce wide interest in related directions.

by finite-difference-time-domain (FDTD) simulations.

wave-vector" *kx*. The HM is with the permittivity tensor as

EW and RW *etc*.

avoid causality violation.

Fig. 1. The frequency contour of HM and isotropic dielectric material in *k* space. The inset: the schematic figure of our model.

then the most general frequency contour in *k* space of HM is:

$$\frac{(k\_x \cos \theta - k\_y \sin \theta)^2}{\varepsilon\_2} - \frac{(k\_y \cos \theta + k\_x \sin \theta)^2}{|\varepsilon\_1|} = \left(\frac{\omega}{c}\right)^2\tag{2}$$

In the HM region and the isotropic dielectric region the fields can be expressed uniformly as

**H***<sup>σ</sup>* = **ez**(*H<sup>σ</sup> iz* + *<sup>H</sup><sup>σ</sup> sz*)*eikxx*−*iω<sup>t</sup>* , **E<sup>œ</sup>** = [**ex**(*c<sup>σ</sup> ixH<sup>σ</sup> iz* + *<sup>c</sup><sup>σ</sup> sxH<sup>σ</sup> sz*) + **ey**(*c<sup>σ</sup> iyH<sup>σ</sup> iz* + *<sup>c</sup><sup>σ</sup> syH<sup>σ</sup> sz*)]*eikxx*−*iω<sup>t</sup>* where the coefficients are defined as *c<sup>σ</sup> νξ* <sup>=</sup> *<sup>E</sup><sup>σ</sup> νξ*/*H<sup>σ</sup> <sup>ν</sup><sup>z</sup>* with *ν* = *i*,*s* for the incident fields to HI or the scattered fields from HI; *ξ* = *x*, *y*; *σ* = *h*, *d* for the HM region or dielectric region. Since the translation symmetry of flat HI, the wave-vector parallel component *kx* is continuous at both regions. In following discussion, the *k* vectors are normalized by ˜ *kσ νξ* <sup>=</sup> *<sup>k</sup><sup>σ</sup> νξ*/*k*<sup>0</sup> where *k*<sup>0</sup> = *ω*/*c*.

To explore the transmission and reflection properties of HI, we define a scattering matrix associated with the incident fields and out-going fields at HI.

$$\left(H\_{\rm sz}^{\hbar}, H\_{\rm sz}^{d}\right)^{T} = \mathcal{S}\_{2 \times 2} \left(H\_{\rm iz}^{\hbar}, H\_{\rm iz}^{d}\right)^{T} \tag{3}$$

where the superscript *<sup>T</sup>* means the matrix transpose, *<sup>S</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup>(*c<sup>h</sup> ix�* <sup>+</sup> ˜ *kd y*)/(*c<sup>h</sup> sx�* + ˜ *kd <sup>y</sup>*), *S*<sup>12</sup> = (−*c<sup>h</sup> ix�* <sup>+</sup> ˜ *kd iy*)/(*c<sup>h</sup> sx�* <sup>−</sup> ˜ *kd iy*), *<sup>S</sup>*<sup>21</sup> = (*c<sup>h</sup> sx* <sup>−</sup> *<sup>c</sup><sup>h</sup> ix*)*�*/(*c<sup>h</sup> sx�* + ˜ *kd <sup>y</sup>*), and *<sup>S</sup>*<sup>22</sup> <sup>=</sup> <sup>−</sup>2˜ *kd iy*/(*c<sup>h</sup> sx�* <sup>−</sup> ˜ *kd iy*). From the standard boundary conditions, the coefficients *c<sup>h</sup> i*(*s*)*x* , *c<sup>d</sup> <sup>i</sup>*(*s*)*<sup>x</sup>* are worked out to be *ch <sup>i</sup>*(*s*)*<sup>x</sup>* = (−˜ *kh i*(*s*)*y α*<sup>2</sup> + ˜ *kxαγ*)/(|*�*1|*�*2); *<sup>c</sup><sup>d</sup> <sup>i</sup>*(*s*)*<sup>x</sup>* <sup>=</sup> <sup>−</sup>˜ *kd i*(*s*)*y* /*�*, where factors *α*, *γ* are defined as *<sup>α</sup>* = (*�*<sup>1</sup> sin2 *<sup>θ</sup>* <sup>−</sup> *�*<sup>2</sup> cos<sup>2</sup> *<sup>θ</sup>*) 1 <sup>2</sup> ; *<sup>γ</sup>* = (|*�*1<sup>|</sup> <sup>+</sup> *�*2) sin 2*θ*/2*α*, and the values of ˜ *kd <sup>i</sup>*(*s*)*<sup>y</sup>* = ± *�* <sup>−</sup> ˜ *k*2 *x* and ˜ *kh i*(*s*)*y* and ˜ *kh <sup>i</sup>*(*s*)*<sup>y</sup>* = (±(|*�*1|*�*2(˜ *k*2 *<sup>x</sup>* + *<sup>α</sup>*2))1/2 + ˜ *kxαγ*)/*α*<sup>2</sup> are uniquely determined by Eq.(2), respectively. From Eq.(3) we can easily get the reflection and transmission coefficients across the HI from upper to down, or inverse. For the case of hyperlens that the wave is incident

�0.4 �0.2 0.0 0.2 0.4

*v*gy

�0.4 �0.2 0.0 0.2 0.4

The Group Velocity Picture of Metamaterial Systems 63

Fig. 2. Two components of �*vg* of reflected field in HM. (a)Without material dispersion of *�*1,

The most general expression of the group velocity of the reflected wave (which is also the group velocity of transmitted field in HM of hyperlens case) can be obtained from Eq.(2) as:

> *kh <sup>y</sup>* + ˜ *kxβ*2)

1*�*2(*ω*)�*ω*˜ *k p*2 *<sup>x</sup>* <sup>−</sup> <sup>2</sup>*�*<sup>2</sup> 1*�*2 2

*kx* <sup>−</sup> ˜ *kh <sup>y</sup>α*2)

1*�*2(*ω*)�

if the material dispersion of HM is neglected *∂�*1/*∂ω* = *∂�*2/*∂ω* = 0, then we will obtain the superluminal group velocity as shown in Fig.2(a). When *θ* → *θ*c, the *vg* even diverges.

But with material dispersion, the *x* and *y* components of *vg* is recalculated, and we find that there is no *vg* > *c* at all cases, as shown in Fig.2(b) in which two components of *vg* versus *θ* − *θ<sup>c</sup>*

*θ*<sup>c</sup> = *π*/4. When approaching the critical angle *θ* → *θ*c, two components can be approximated

*vgx* <sup>∼</sup> <sup>1</sup> *�*� <sup>1</sup>(*ω*) (*k<sup>h</sup>*

*vgy* <sup>∼</sup> *kx �*� <sup>1</sup>(*ω*) (*k<sup>h</sup>*

What does the zero-group velocity of reflected wave mean? The analysis will give us clear

where *W* is the energy density of reflected wave, hence the energy density *W* must be infinite large at the critical angle. From Eq.(2), we can obtain that the electric field of reflected wave

**<sup>r</sup>** | is really divergent at the critical angle. The divergent field strength means that it need *infinite long time* to accumulate energy at HI for the reflected field. In other words, there is no reflected wave physically, as our intuition has told us. When the incident angle is large enough

*ry* → ∞ at the critical angle *θc*, the group velocity of the reflected wave should be zero

*<sup>y</sup>* cos *<sup>θ</sup>* + ˜

*ω*˜ *k p*2 *<sup>x</sup>* <sup>−</sup> <sup>2</sup>*�*<sup>2</sup> 1*�*2 2

<sup>2</sup> <sup>=</sup> 0, *�*<sup>2</sup> <sup>=</sup> 1, *�*1(*ω*)=(<sup>1</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup>

b

�0.4 �0.2 0.0 0.2 0.4

*v*gx

*v*gy

*kx* sin *θ*) are the "*k* components" in the

*ry*)−<sup>1</sup> (6a)

*ry*)−<sup>2</sup> (6b)

*Sr* is not zero and �

*p*

<sup>2</sup> . From Eq.(5), we find that,

*<sup>ω</sup>*<sup>2</sup> )[21] and *ω* = *ωp*/

(5a)

(5b)

<sup>√</sup>2,

*Sr* = *vgW*

ΔΘ

1

ΔΘ

*vgx* <sup>=</sup> <sup>2</sup>*c�*1*�*2(*αγ*˜

2*�*1(*ω*)�*ω*˜ *k p*2 *<sup>y</sup>* <sup>−</sup> *�*<sup>2</sup>

*vgy* <sup>=</sup> <sup>2</sup>*c�*1*�*2(*αγ*˜

*<sup>y</sup>* sin *<sup>θ</sup>*) and ˜

principal-axes coordinate of HM and *<sup>β</sup>* = (*�*<sup>2</sup> sin2 *<sup>θ</sup>* <sup>−</sup> *�*<sup>1</sup> cos<sup>2</sup> *<sup>θ</sup>*)

*ω*˜ *k p*2 *<sup>y</sup>* <sup>−</sup> *�*<sup>2</sup>

> *k p <sup>y</sup>* = (˜ *kh*

lim *θ*→*θ*c

lim *θ*→*θ*c

answer. As we have pointed out,since the reflected energy current �

*�*2

*�*2 2*�*1(*ω*)�

*kh*

**v***<sup>g</sup>* = 0 at the critical angle, as shown in Fig.2(b) too.

*kx* cos *<sup>θ</sup>* <sup>−</sup> ˜

based on Eq.(5), with the parameters *�*�

�5

a

0

*vg*c

(b)with material dispersion.

where ˜ *k p <sup>x</sup>* = (˜

as:

. Since ˜ *kh*


5

*v*gx

from the isotropic medium to the HM, *tdh* = *<sup>S</sup>*<sup>22</sup> ; *rdh* = *<sup>S</sup>*12. When ˜ *k*2 *<sup>x</sup>* > *�*, the incident and the reflected waves in the isotropic dielectric are EWs with *y*-component wave-vectors as ˜ *kd iy* = *i* ˜ *k*2 *<sup>x</sup>* <sup>−</sup> *�* <sup>=</sup> <sup>−</sup>˜ *kd sy* ≡ *iκ*. We note that, although single EW can not carry net energy current(time averaged), two EWs, *i.e.* the incident and reflected EWs, *can* carry net energy current � *Siy* in isotropic dielectric medium, since the reflected EW gains an extra-phase from complex reflecting coefficient *rdh*. The energy current � *Siy* carried by two EWs can be converted by HI into the RW energy current � *Sty* in the HM:

$$|\vec{S}\_{iy}| = \frac{\kappa}{\varepsilon} \text{Im}(r\_{dh}) = -\frac{2\mathcal{L}\_{sx}^{l}\kappa^{2}}{\mathcal{L}\_{sx}^{h} + \kappa^{2}}$$

$$= |\vec{S}\_{ty}| = -\frac{1}{2}|t|^{2}c\_{tx} \tag{4}$$

From Eq(4), the hyper-lens effect and the image-improving by CER could be quantitatively studied.

After confirming CER on HI, it is natural to wonder if CRE can occur too, or if there are other novel phenomena on HI. Next we will study the inverse process of hyper-lens, *i.e.* the RW is incident from HM and the transmitted field is in the dielectric. For such inverse processes, there is a critical condition *<sup>θ</sup>* <sup>=</sup> *<sup>θ</sup><sup>c</sup>* <sup>≡</sup> arctan *�*2/|*�*1|, which means HI (*<sup>x</sup>* axis) perpendicular to one of hyperbola-dispersion asymptotes, or in other words, the asymptote is parallel with *y* axis now, as shown by the the solid lines in Fig.(1). At this critical condition, especially when the transmitted wave is EW, we will find CRE with NTNR, compressing and stopping light pulses, etc.

Before we get into detailed derivation, for the critical case (*θ* = *θc*) we first present two seemingly conflicting conclusions of *reflected wave* from two different arguments, which will clearly show the most tricky point of HI.

The first argument is from the "intuitive way" which is based on Fig.(1). Since there is no reflection wave-vector on the dispersion curve to satisfy the *kx* continuity, we intuitively expect that there should be *no* reflected wave with omnidirectional incidence. If the incident angle is large enough ˜ *k*2 *<sup>x</sup>* > *�* so that the transmitted field is EW, and since a single EW can not carry energy current, NTNR is the only possible choice and we expect that CRE will occur on HI. But from the second argument based on Eq.(4), we will obtain a different result. Since *θ* = *θ<sup>c</sup>* is a critical case, we should be more careful and discuss in a more subtle and strict way.

We first suppose the *<sup>θ</sup>* �<sup>=</sup> *<sup>θ</sup><sup>c</sup>* as shown by dashed lines Fig.(1), so the finite ˜ *kry* of reflected field for a fixed ˜ *kx* can be found. Next we let the angle *θ* to approach *θc* continuously (which can be realized physically by choosing different direction of HI), then we find that ˜ *kry* → ∞ when *<sup>θ</sup>* <sup>→</sup> *<sup>θ</sup>*<sup>c</sup> for a fixed ˜ *kx*.

But surprisingly, when *θ* → *θ*c, the reflection coefficient *rhd*, calculated from Eq.(4) as lim*θ*→*θ<sup>c</sup> rhd* = (*�*˜ *kx* − <sup>|</sup>*�*1|*�*2(*�* <sup>−</sup> ˜ *k*2 *<sup>x</sup>*))/(*�*˜ *kx* + <sup>|</sup>*�*1|*�*2(*�* <sup>−</sup> ˜ *k*2 *<sup>x</sup>*)) is not zero, and the reflected energy current is not zero too. So the theoretical result seems against our intuition.

To explain the conflicting results, we need to calculate the group velocity inside HM with material dispersion, which will also show that the superluminal group velocity is artificial. 6 Will-be-set-by-IN-TECH

and the reflected waves in the isotropic dielectric are EWs with *y*-component wave-vectors

current(time averaged), two EWs, *i.e.* the incident and reflected EWs, *can* carry net energy

*Sty* in the HM:

<sup>=</sup> <sup>|</sup>�

*Siy* in isotropic dielectric medium, since the reflected EW gains an extra-phase from

Im(*rdh*) = <sup>−</sup> <sup>2</sup>*c<sup>h</sup>*

2 |*t*|

*Sty*<sup>|</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

From Eq(4), the hyper-lens effect and the image-improving by CER could be quantitatively

After confirming CER on HI, it is natural to wonder if CRE can occur too, or if there are other novel phenomena on HI. Next we will study the inverse process of hyper-lens, *i.e.* the RW is incident from HM and the transmitted field is in the dielectric. For such inverse processes, there is a critical condition *<sup>θ</sup>* <sup>=</sup> *<sup>θ</sup><sup>c</sup>* <sup>≡</sup> arctan *�*2/|*�*1|, which means HI (*<sup>x</sup>* axis) perpendicular to one of hyperbola-dispersion asymptotes, or in other words, the asymptote is parallel with *y* axis now, as shown by the the solid lines in Fig.(1). At this critical condition, especially when the transmitted wave is EW, we will find CRE with NTNR, compressing and stopping light

Before we get into detailed derivation, for the critical case (*θ* = *θc*) we first present two seemingly conflicting conclusions of *reflected wave* from two different arguments, which will

The first argument is from the "intuitive way" which is based on Fig.(1). Since there is no reflection wave-vector on the dispersion curve to satisfy the *kx* continuity, we intuitively expect that there should be *no* reflected wave with omnidirectional incidence. If the incident

not carry energy current, NTNR is the only possible choice and we expect that CRE will occur on HI. But from the second argument based on Eq.(4), we will obtain a different result. Since *θ* = *θ<sup>c</sup>* is a critical case, we should be more careful and discuss in a more subtle and strict way.

But surprisingly, when *θ* → *θ*c, the reflection coefficient *rhd*, calculated from Eq.(4) as

To explain the conflicting results, we need to calculate the group velocity inside HM with material dispersion, which will also show that the superluminal group velocity is artificial.

*kx* + 

energy current is not zero too. So the theoretical result seems against our intuition.

*kx* can be found. Next we let the angle *θ* to approach *θc* continuously (which can

<sup>|</sup>*�*1|*�*2(*�* <sup>−</sup> ˜

*k*2

We first suppose the *<sup>θ</sup>* �<sup>=</sup> *<sup>θ</sup><sup>c</sup>* as shown by dashed lines Fig.(1), so the finite ˜

*k*2 *<sup>x</sup>*))/(*�*˜

be realized physically by choosing different direction of HI), then we find that ˜

*sy* ≡ *iκ*. We note that, although single EW can not carry net energy

*sxκ*<sup>2</sup> *c<sup>h</sup>* <sup>2</sup> *sx* + *κ*<sup>2</sup>

*<sup>x</sup>* > *�* so that the transmitted field is EW, and since a single EW can

*k*2

*Siy* carried by two EWs can be converted

<sup>2</sup>*ctx* (4)

*<sup>x</sup>* > *�*, the incident

*kry* of reflected field

*<sup>x</sup>*)) is not zero, and the reflected

*kry* → ∞ when

from the isotropic medium to the HM, *tdh* = *<sup>S</sup>*<sup>22</sup> ; *rdh* = *<sup>S</sup>*12. When ˜

as ˜ *kd iy* = *i* ˜ *k*2

current �

studied.

pulses, etc.

for a fixed ˜

*<sup>θ</sup>* <sup>→</sup> *<sup>θ</sup>*<sup>c</sup> for a fixed ˜

lim*θ*→*θ<sup>c</sup> rhd* = (*�*˜

angle is large enough ˜

*<sup>x</sup>* <sup>−</sup> *�* <sup>=</sup> <sup>−</sup>˜

by HI into the RW energy current �

clearly show the most tricky point of HI.

*kx*.

*kx* − 

<sup>|</sup>*�*1|*�*2(*�* <sup>−</sup> ˜

*k*2

*kd*

complex reflecting coefficient *rdh*. The energy current �


Fig. 2. Two components of �*vg* of reflected field in HM. (a)Without material dispersion of *�*1, (b)with material dispersion.

The most general expression of the group velocity of the reflected wave (which is also the group velocity of transmitted field in HM of hyperlens case) can be obtained from Eq.(2) as:

$$w\_{\mathcal{S}^{\mathcal{X}}} = \frac{2c\epsilon\_1\epsilon\_2(a\gamma\vec{k}\_y^h + \vec{k}\_x\beta^2)}{\epsilon\_2^2\epsilon\_1(\omega)'\omega\vec{k}\_y^{p2} - \epsilon\_1^2\epsilon\_2(\omega)'\omega\vec{k}\_x^{p2} - 2\epsilon\_1^2\epsilon\_2^2} \tag{5a}$$

$$v\_{\mathcal{Y}\mathcal{Y}} = \frac{2c\epsilon\_1\epsilon\_2(a\gamma\tilde{k}\_{\mathcal{X}} - \tilde{k}\_y^h a^2)}{\epsilon\_2^2\epsilon\_1(\omega)'\omega\tilde{k}\_{\mathcal{Y}}^{p2} - \epsilon\_1^2\epsilon\_2(\omega)'\omega\tilde{k}\_{\mathcal{X}}^{p2} - 2\epsilon\_1^2\epsilon\_2^2} \tag{5b}$$

where ˜ *k p <sup>x</sup>* = (˜ *kx* cos *<sup>θ</sup>* <sup>−</sup> ˜ *kh <sup>y</sup>* sin *<sup>θ</sup>*) and ˜ *k p <sup>y</sup>* = (˜ *kh <sup>y</sup>* cos *<sup>θ</sup>* + ˜ *kx* sin *θ*) are the "*k* components" in the principal-axes coordinate of HM and *<sup>β</sup>* = (*�*<sup>2</sup> sin2 *<sup>θ</sup>* <sup>−</sup> *�*<sup>1</sup> cos<sup>2</sup> *<sup>θ</sup>*) 1 <sup>2</sup> . From Eq.(5), we find that, if the material dispersion of HM is neglected *∂�*1/*∂ω* = *∂�*2/*∂ω* = 0, then we will obtain the superluminal group velocity as shown in Fig.2(a). When *θ* → *θ*c, the *vg* even diverges.

But with material dispersion, the *x* and *y* components of *vg* is recalculated, and we find that there is no *vg* > *c* at all cases, as shown in Fig.2(b) in which two components of *vg* versus *θ* − *θ<sup>c</sup>* based on Eq.(5), with the parameters *�*� <sup>2</sup> <sup>=</sup> 0, *�*<sup>2</sup> <sup>=</sup> 1, *�*1(*ω*)=(<sup>1</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *p <sup>ω</sup>*<sup>2</sup> )[21] and *ω* = *ωp*/ <sup>√</sup>2, *θ*<sup>c</sup> = *π*/4. When approaching the critical angle *θ* → *θ*c, two components can be approximated as:

$$\lim\_{\theta \to \theta\_{\mathbb{C}}} v\_{\mathbb{S}^{\mathcal{X}}} \sim \frac{1}{\epsilon\_1'(\omega)} (k\_{ry}^h)^{-1} \tag{6a}$$

$$\lim\_{\theta \to \theta\_{\mathsf{C}}} v\_{\mathcal{S}y} \sim \frac{k\_{\mathsf{x}}}{\epsilon\_1'(\omega)} (k\_{ry}^{\mathsf{h}})^{-2} \tag{6b}$$

. Since ˜ *kh ry* → ∞ at the critical angle *θc*, the group velocity of the reflected wave should be zero **v***<sup>g</sup>* = 0 at the critical angle, as shown in Fig.2(b) too.

What does the zero-group velocity of reflected wave mean? The analysis will give us clear answer. As we have pointed out,since the reflected energy current � *Sr* is not zero and � *Sr* = *vgW* where *W* is the energy density of reflected wave, hence the energy density *W* must be infinite large at the critical angle. From Eq.(2), we can obtain that the electric field of reflected wave |**Eh <sup>r</sup>** | is really divergent at the critical angle. The divergent field strength means that it need *infinite long time* to accumulate energy at HI for the reflected field. In other words, there is no reflected wave physically, as our intuition has told us. When the incident angle is large enough

�0.4 �0.2 0.0 0.2 0.4

The Group Velocity Picture of Metamaterial Systems 65

*ky*<sup>a</sup><sup>2</sup><sup>Π</sup>

Fig. 4. (a) The structures of real HM: the periodic metal-dielectric layers and the periodic metal nano-wires embedded in a dielectric matrix. (b) The frequency contour of periodic

*i.e.* the HI (*x* axis) perpendicular to one of hyperbola-dispersion asymptotes. The second is that the decay (because of dissipation) of trapped field on HI is much slower than in common metallic material since the trapped field energy is mainly in the dielectric side as shown in Fig.3(b). The third is that the trapped signals are easy to take out (read) since they are on the interface. Because of these advantages, HI could be used as a removable recorder (dynamical memory) in optical/photonic signal processing, or as a wide-frequency

However, we should point out that the above theoretical and numerical studies are with the assumption of the ideal hyper-dispersion, which is still void when *kry* → ∞. In reality, such HM does't exist, so that we need to study the limit of hyper-dispersion of realizable HM. HM can be realized by many structures, i.e. one-dimensional (1D) periodic metal-dielectric binary layers [24, 25] or two-dimensional (2D) periodic metallic lines[26], as shown in Fig.4(a). For these structures, the dispersion relation can be calculated exactly. In Fig.4, the calculated frequency contour of a 1D metal-dielectric binary layers is shown, from which we can see that the effective HM medium is not available anymore when |*k*| approaches *π*/*a*. Based on this

metallic-line structure, from the modern technical limit we assume the smallest lattice constant

<sup>10</sup>−10s as in Ref [27], we obtain *vgx* <sup>∼</sup> 4.6m/s which means considerably slow light although not totally stopped, and *vgy* <sup>∼</sup> 7.07 <sup>×</sup> <sup>10</sup>−8m/s which means that the strongly-compressed

In conclusion, we have theoretically and numerically investigate the optical properties of HI. The theory with dispersion of meta-material is constructed and the hyperlens effect of CER is confirmed. At the inverse process of hyperlens, the abnormal phenomena of CRE with NTNR and a novel mechanism to compress and stop light in wide frequency range are revealed. Based the calculated group velocity, we demonstrate that the previously-pointed-out superluminal group velocity in HM is artificial since the material dispersion is neglected. FDTD simulations confirm that the HI has potential to be a removable optical/photonic

*�*� 1(*ω*)(*k<sup>h</sup>*

2, where *γ<sup>s</sup>* = *kry*/*k*<sup>0</sup> is the slowing coefficient. For the 2D

*ry*)−<sup>1</sup> ∝ 1/*γ<sup>s</sup>*

<sup>1</sup>(*ω*) = 6.9 ×

limit, we can roughly estimate the slow limit of group velocity by *vgx* <sup>∼</sup> <sup>1</sup>

as *<sup>a</sup>* <sup>=</sup> 10nm. If the incident is the micro-wave *<sup>ω</sup>* <sup>=</sup> 5.8GHz (*γ<sup>s</sup>* <sup>∼</sup> 107) and *�*�

�0.2 �0.1 0.0 0.1 0.2

(b)

(a)

*kx*a2Π

wide-angle light trapper in photovoltaic devices.

*ry*)−<sup>2</sup> ∝ 1/*γ<sup>s</sup>*

metal-dielectric layers.

and *vgy* <sup>∼</sup> *kx*

*�*� <sup>1</sup> (*ω*)(*k<sup>h</sup>*

light pulses can be easily achieved.

Fig. 3. (a)The magnetic field *Hz* distributions for a Gaussian beam incident on the interface with <sup>|</sup>˜ *kx*<sup>|</sup> <sup>&</sup>gt; <sup>√</sup>*�*. (b)The averaged field intensity versus the vertical distance to the HI. (c)Two pulses are arriving at HI at different time. (d) The fields of two pulses, which stay at the incident positions, at 14 periods after the pulse arriving.

˜ *k*2 *<sup>x</sup>* > *�*, since the energy of incident RW can not be transmitted, also can not be reflected, the only answer is that the energy is stored at the HI or CRE occurs. Thus, we can have a self-consistent explanation for our seemingly conflicting results.

To confirm our theoretical discussion at *θc*, the FDTD simulation [79] with strict physical HM dispersion (Drude mode) which satisfies Kramar-Kronig relation, is done. The parameter of HM and dielectric are *�*<sup>1</sup> <sup>=</sup> <sup>−</sup>3, *�*<sup>2</sup> <sup>=</sup> 3 , and *�<sup>l</sup>* <sup>=</sup> 1.1. For the case(˜ *kx* > *�*), as shown Fig.3(a), a light beam is incident from HM to HI in 45<sup>0</sup> angle,as we predicted, there is no reflection and no transmission, and the field energy is accumulated at HI and stopped there. More detailed observation shows that at the boundary the field energy is mainly at the dielectric side, as shown Fig.3(b).

We also has checked the group velocities of hyperlens cases and and find no violation of the causality. Actually, in FDTD simulation, if there is superluminal group velocity the program will be numerically unstable.

The dynamical study, such as with the pulse incidence, can reveal more interesting phenomena of HI. Since the group velocity along HI is zero at NTNR case as discussed, we expect that the pulse energy will accumulate on HI and stay at the incident position until it is dissipated because of absorption of HM.

The numerical experiments with incident pulses by FDTD are also done. As shown in Fig.3(c) and (d), two pulses arrive at the HI at different time, then they stop at the incident positions on HI. The pulse vertical length is compressed to almost zero, but their width keeps same so that they are still well separated in Fig.3(d). We emphasize at here that this is a novel mechanism to compress and stop (slow) light pulses with special advantages. The first advantage is that this mechanism works at very wide frequency and wide incident-angle range, which is confirmed by FDTD simulation in Fig.3 with incident of pretty short pulses. The frequency and incident-angle insensitivity is because the mechanism is from a simple geometry property, 8 Will-be-set-by-IN-TECH

Normalization of ¯ position/λ I

0 4 8 0.0 0.2 0.4 0.6 0.8 1.0

(a) (b)

0 4 8 12 16 20 24 position/λ

Fig. 3. (a)The magnetic field *Hz* distributions for a Gaussian beam incident on the interface

pulses are arriving at HI at different time. (d) The fields of two pulses, which stay at the

*kx*<sup>|</sup> <sup>&</sup>gt; <sup>√</sup>*�*. (b)The averaged field intensity versus the vertical distance to the HI. (c)Two

*<sup>x</sup>* > *�*, since the energy of incident RW can not be transmitted, also can not be reflected, the only answer is that the energy is stored at the HI or CRE occurs. Thus, we can have a

To confirm our theoretical discussion at *θc*, the FDTD simulation [79] with strict physical HM dispersion (Drude mode) which satisfies Kramar-Kronig relation, is done. The parameter of

a light beam is incident from HM to HI in 45<sup>0</sup> angle,as we predicted, there is no reflection and no transmission, and the field energy is accumulated at HI and stopped there. More detailed observation shows that at the boundary the field energy is mainly at the dielectric side, as

We also has checked the group velocities of hyperlens cases and and find no violation of the causality. Actually, in FDTD simulation, if there is superluminal group velocity the program

The dynamical study, such as with the pulse incidence, can reveal more interesting phenomena of HI. Since the group velocity along HI is zero at NTNR case as discussed, we expect that the pulse energy will accumulate on HI and stay at the incident position until it is

The numerical experiments with incident pulses by FDTD are also done. As shown in Fig.3(c) and (d), two pulses arrive at the HI at different time, then they stop at the incident positions on HI. The pulse vertical length is compressed to almost zero, but their width keeps same so that they are still well separated in Fig.3(d). We emphasize at here that this is a novel mechanism to compress and stop (slow) light pulses with special advantages. The first advantage is that this mechanism works at very wide frequency and wide incident-angle range, which is confirmed by FDTD simulation in Fig.3 with incident of pretty short pulses. The frequency and incident-angle insensitivity is because the mechanism is from a simple geometry property,

with <sup>|</sup>˜

shown Fig.3(b).

will be numerically unstable.

dissipated because of absorption of HM.

˜ *k*2 (c)

(d)

incident positions, at 14 periods after the pulse arriving.

self-consistent explanation for our seemingly conflicting results.

HM and dielectric are *�*<sup>1</sup> <sup>=</sup> <sup>−</sup>3, *�*<sup>2</sup> <sup>=</sup> 3 , and *�<sup>l</sup>* <sup>=</sup> 1.1. For the case(˜

2 4 6

**−4 −3 −2 −1 0 1 2 3 4**

*kx* > *�*), as shown Fig.3(a),

Fig. 4. (a) The structures of real HM: the periodic metal-dielectric layers and the periodic metal nano-wires embedded in a dielectric matrix. (b) The frequency contour of periodic metal-dielectric layers.

*i.e.* the HI (*x* axis) perpendicular to one of hyperbola-dispersion asymptotes. The second is that the decay (because of dissipation) of trapped field on HI is much slower than in common metallic material since the trapped field energy is mainly in the dielectric side as shown in Fig.3(b). The third is that the trapped signals are easy to take out (read) since they are on the interface. Because of these advantages, HI could be used as a removable recorder (dynamical memory) in optical/photonic signal processing, or as a wide-frequency wide-angle light trapper in photovoltaic devices.

However, we should point out that the above theoretical and numerical studies are with the assumption of the ideal hyper-dispersion, which is still void when *kry* → ∞. In reality, such HM does't exist, so that we need to study the limit of hyper-dispersion of realizable HM. HM can be realized by many structures, i.e. one-dimensional (1D) periodic metal-dielectric binary layers [24, 25] or two-dimensional (2D) periodic metallic lines[26], as shown in Fig.4(a). For these structures, the dispersion relation can be calculated exactly. In Fig.4, the calculated frequency contour of a 1D metal-dielectric binary layers is shown, from which we can see that the effective HM medium is not available anymore when |*k*| approaches *π*/*a*. Based on this limit, we can roughly estimate the slow limit of group velocity by *vgx* <sup>∼</sup> <sup>1</sup> *�*� 1(*ω*)(*k<sup>h</sup> ry*)−<sup>1</sup> ∝ 1/*γ<sup>s</sup>* and *vgy* <sup>∼</sup> *kx �*� <sup>1</sup> (*ω*)(*k<sup>h</sup> ry*)−<sup>2</sup> ∝ 1/*γ<sup>s</sup>* 2, where *γ<sup>s</sup>* = *kry*/*k*<sup>0</sup> is the slowing coefficient. For the 2D metallic-line structure, from the modern technical limit we assume the smallest lattice constant as *<sup>a</sup>* <sup>=</sup> 10nm. If the incident is the micro-wave *<sup>ω</sup>* <sup>=</sup> 5.8GHz (*γ<sup>s</sup>* <sup>∼</sup> 107) and *�*� <sup>1</sup>(*ω*) = 6.9 × <sup>10</sup>−10s as in Ref [27], we obtain *vgx* <sup>∼</sup> 4.6m/s which means considerably slow light although not totally stopped, and *vgy* <sup>∼</sup> 7.07 <sup>×</sup> <sup>10</sup>−8m/s which means that the strongly-compressed light pulses can be easily achieved.

In conclusion, we have theoretically and numerically investigate the optical properties of HI. The theory with dispersion of meta-material is constructed and the hyperlens effect of CER is confirmed. At the inverse process of hyperlens, the abnormal phenomena of CRE with NTNR and a novel mechanism to compress and stop light in wide frequency range are revealed. Based the calculated group velocity, we demonstrate that the previously-pointed-out superluminal group velocity in HM is artificial since the material dispersion is neglected. FDTD simulations confirm that the HI has potential to be a removable optical/photonic

**region I**

or *n*��. For radiative waves, since *n*� determines the real part of wavevector *k* � *n*�

Fig. 6. The schematic diagram of our model.

detected by measuring the delay time *τ* at a short distance.

based on the dynamical picture and the dissipation is critical.

wave to probe small change of *n*�

Green's function methods [42].

x 0 z

The idea is from the "dual roles" of real and imaginary parts of refractive index *n*. Supposing a medium with complex index *n* = *n*� + *in*��, our goal is to detect the very tiny change of *n*�

is easy to measure the phase change or group delay, so, it is natural to choose the radiative

*n*�� causes an extremely small decay change which is very hard to measure in the limited lab space. However, for the evanescent waves, the roles of *n*� and *n*�� are totally exchanged, i.e., *n*� dominates the decay rate, while the *n*�� introduces a phase change which is much easier to detect. Further more, we will demonstrate that *n*�� can also introduce the energy propagation for EWs whose energy velocity *ve* ∝ *n*�� can be extremely slow. Such a slow wave can be

Actually, the tenneling mechanism of EW has been widely studied[39–41]. We would like to emphasize the mechanism difference between ours and that in the previous works. In Ref[41], they are based on "two interfaces" structure (a slab). Such "two-interfaces" structure will generate both evanescent modes exp(±*κx*) and such two evanescent modes can carry energy current[9], which called as "tunneling mechanism". So, even if the material dissipation is neglected[41] , the energy propagation is still available. However, in our model, since there is only a single interface (Fig.2), obviously if without dissipation there will be no the energy current at all[42], then, no phase change and no the energy delay time. So, our mechanism is

Here we note that, because the probing light is much weaker than the external field in our model, the nonlinear effect is negligible. For a linear system, all dynamical processes can be solved numerically by sum of multi-frequency componentsˇc ˇnwhich can be obtained by

Our model is schematically shown in Fig.6, based on the total internal reflection (TIR) at the interface between a dielectric media *n*<sup>1</sup> (region *I*) and vacuum (region *I I*). When the incident angle *θ<sup>i</sup>* > *θ<sup>c</sup>* = *arcsin*(1/*n*1), the TIR will occur and the transmitted wave in the vacuum is the EW. We choose *θ<sup>i</sup>* is a little larger than *θ<sup>c</sup>* to make sure that almost all frequency components

. On the other hand, for radiative waves, tiny change of

=2

probing light

θi

The Group Velocity Picture of Metamaterial Systems 67

n <sup>2</sup> <sup>n</sup> =1+δ+in" <sup>1</sup>

**region II**

*ω*/*c* and it

L

evanescent wave

reflective light

Fig. 5. The schematic picture of vacuum polarization processes with electron-positron pair generation, with which the vacuum becomes dissipative and anisotropic. The insert is the Feynman diagram of the vacuum polarization processes.

recorder, or a wide-frequency wide-angle light trapper. At last the realizability of these phenomena on the real metallic structures is discussed. Obviously, the new mechanism works not only for electromagnetic waves, but also for acoustic or matter waves if hyperbolic dispersion is available, so that more interesting phenomena and applications are waiting for further theoretical and experimental research.

#### **3. The methods to detect vacuum polarization by evanescent modes**

Vacuum is one of the most fundamental concepts in all quantum fields[30–32] since all excitations are from the vacuum and determined by vacuum in some way. Modern vacuum concept is started from quantum electrodynamics (QED), which describes the interaction between light and matter (including vacuum), and has been continually studied both experimentally and theoretically[33–38]. According to QED, the vacuum becomes weakly anisotropic, dispersive, dissipative and even nonlinear optical medium, when external electric field is approaching the Schwinger critical value *Ec* 1018*V*/*m*. In other words, the real and imaginary parts of vacuum refractive index could deviate from unit and zero[34, 35], respectively. Physically, the deviation of the imaginary part is mainly from the electron-positron pair generation. However, the electron-positron pair generation, also generally called as *vacuum polarization* (VP) processes[34], which is schematically shown in Fig.1, has not been directly observed for over half century since very high *Ec* is beyond the contemporary technical limit. Therefore, it is natural to wonder if we can find an approach to probe VP with external field *Eext* much smaller than *Ec*.

In this work, we propose evanescent-mode-sensing methods based on new mechanism to detect the QED VP, which is based on the measuring the phase change and the energy time delay of evanescent wave (EW). We find that the required external field could be one order weaker than *Ec*, which may be realizable by contemporary experiments.

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

Fig. 5. The schematic picture of vacuum polarization processes with electron-positron pair generation, with which the vacuum becomes dissipative and anisotropic. The insert is the

recorder, or a wide-frequency wide-angle light trapper. At last the realizability of these phenomena on the real metallic structures is discussed. Obviously, the new mechanism works not only for electromagnetic waves, but also for acoustic or matter waves if hyperbolic dispersion is available, so that more interesting phenomena and applications are waiting for

Vacuum is one of the most fundamental concepts in all quantum fields[30–32] since all excitations are from the vacuum and determined by vacuum in some way. Modern vacuum concept is started from quantum electrodynamics (QED), which describes the interaction between light and matter (including vacuum), and has been continually studied both experimentally and theoretically[33–38]. According to QED, the vacuum becomes weakly anisotropic, dispersive, dissipative and even nonlinear optical medium, when external electric field is approaching the Schwinger critical value *Ec* 1018*V*/*m*. In other words, the real and imaginary parts of vacuum refractive index could deviate from unit and zero[34, 35], respectively. Physically, the deviation of the imaginary part is mainly from the electron-positron pair generation. However, the electron-positron pair generation, also generally called as *vacuum polarization* (VP) processes[34], which is schematically shown in Fig.1, has not been directly observed for over half century since very high *Ec* is beyond the contemporary technical limit. Therefore, it is natural to wonder if we can find an approach to

In this work, we propose evanescent-mode-sensing methods based on new mechanism to detect the QED VP, which is based on the measuring the phase change and the energy time delay of evanescent wave (EW). We find that the required external field could be one order

**3. The methods to detect vacuum polarization by evanescent modes**

Feynman diagram of the vacuum polarization processes.

probe VP with external field *Eext* much smaller than *Ec*.

weaker than *Ec*, which may be realizable by contemporary experiments.

further theoretical and experimental research.

Fig. 6. The schematic diagram of our model.

The idea is from the "dual roles" of real and imaginary parts of refractive index *n*. Supposing a medium with complex index *n* = *n*� + *in*��, our goal is to detect the very tiny change of *n*� or *n*��. For radiative waves, since *n*� determines the real part of wavevector *k* � *n*� *ω*/*c* and it is easy to measure the phase change or group delay, so, it is natural to choose the radiative wave to probe small change of *n*� . On the other hand, for radiative waves, tiny change of *n*�� causes an extremely small decay change which is very hard to measure in the limited lab space. However, for the evanescent waves, the roles of *n*� and *n*�� are totally exchanged, i.e., *n*� dominates the decay rate, while the *n*�� introduces a phase change which is much easier to detect. Further more, we will demonstrate that *n*�� can also introduce the energy propagation for EWs whose energy velocity *ve* ∝ *n*�� can be extremely slow. Such a slow wave can be detected by measuring the delay time *τ* at a short distance.

Actually, the tenneling mechanism of EW has been widely studied[39–41]. We would like to emphasize the mechanism difference between ours and that in the previous works. In Ref[41], they are based on "two interfaces" structure (a slab). Such "two-interfaces" structure will generate both evanescent modes exp(±*κx*) and such two evanescent modes can carry energy current[9], which called as "tunneling mechanism". So, even if the material dissipation is neglected[41] , the energy propagation is still available. However, in our model, since there is only a single interface (Fig.2), obviously if without dissipation there will be no the energy current at all[42], then, no phase change and no the energy delay time. So, our mechanism is based on the dynamical picture and the dissipation is critical.

Here we note that, because the probing light is much weaker than the external field in our model, the nonlinear effect is negligible. For a linear system, all dynamical processes can be solved numerically by sum of multi-frequency componentsˇc ˇnwhich can be obtained by Green's function methods [42].

Our model is schematically shown in Fig.6, based on the total internal reflection (TIR) at the interface between a dielectric media *n*<sup>1</sup> (region *I*) and vacuum (region *I I*). When the incident angle *θ<sup>i</sup>* > *θ<sup>c</sup>* = *arcsin*(1/*n*1), the TIR will occur and the transmitted wave in the vacuum is the EW. We choose *θ<sup>i</sup>* is a little larger than *θ<sup>c</sup>* to make sure that almost all frequency components

function method with physical dissipation and dispersion, it is found that the fluctuation will propagate on the EW from the interface to far away, as shown in Fig.7(b). So, we can measure the time delay *τ* of the fluctuation propagation on the EW to detect the VP effect. The propagation speed of irradiance fluctuation can be obtained by the energy velocity *ve* which is

The Group Velocity Picture of Metamaterial Systems 69

very weak. The physical meaning of *ve* can be understood as the "propagation" speed of

Hence, experimentally the time delay *τ* of the irradiance fluctuation at distance *L* can be

Since it is *near field* phenomenon, the detecting should be very near the interface. For the VP effect, since *n*�� is extremely small, the "propagation" speed of the irradiance fluctuation is so slow that *τ* gets to peco-second level when the distance is one tenth of the wavelength

Therefore, either the phase change Δ*φ* or the time delay *τ* are very sensitive for *n*��, and the EW is a good candidate to probe the VP effect. Here, we note that the famous Kramers-Kronig relations still fit for QED vacuum[34]. Hence, the observation of imaginary part of vacuum

Next, we will quantitatively study the VP detect by our methods. Supposing that an external homogeneous constant electric field *Eext*, which is perpendicular to the *xz* plane and smaller than the Schwinger critical electric field *Ec*, is applied to the vacuum (region *I I*) only, as shown in Fig.6, then, the optical properties of the vacuum can be described by the Euler-Heisenberg Lagrangian *Leff*[34, 37]. Physically, the imaginary part of Euler-Heisenberg Lagrangian *Leff* is related to the imaginary part of VP operator, and therefore corresponds to the *electron-positron*

Consequently, the vacuum refractive index can be deduced from the Lagrangian *Leff*[34, 37,

In our model, since the external magnetic field is supposed to be zero, thus the vacuum refractive index is determined only by the external homogeneous constant electric field *Eext*. We use *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> to refer the effective refractive index of vacuum when the electric field of probing light are parallel and perpendicular to the field *Eext*, respectively. *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> can

and *<sup>α</sup>* � 1/137 is the fine-structure constant. Therefore we have *<sup>δ</sup>* <sup>=</sup> *Re*

<sup>45</sup>*<sup>π</sup> <sup>y</sup>*<sup>2</sup> <sup>+</sup> *<sup>i</sup>* · *<sup>α</sup>*

for *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> when we solve the equations such as Eq.(7) in this letter.

<sup>4</sup>*<sup>π</sup>* <sup>∑</sup><sup>∞</sup> *n*=1 *<sup>π</sup> <sup>y</sup>*<sup>2</sup> <sup>+</sup> <sup>1</sup> *n* 1 *y* 

exp(−*nπ*/*y*),

*<sup>n</sup>*�(⊥) − 1,

exp(−*nπ*/*y*), where *y* = |*Eext*|/*Ec*,

<sup>2</sup>*Re* (*E* × *H*∗)|*<sup>z</sup>* is the averaged Poynting vector along *z*

*ve* = *χ* · *n*�� (9)

*τ* = *L*/*ve* ∝ 1/*n*��. (10)

, when the dissipation and dispersion are

<sup>2</sup>) is the local energy density of the electromagnetic wave.

defined as: *ve* <sup>=</sup> <sup>|</sup>

with *χ* � *c*/

measured:

*L* = 60*nm*.

*pair generation*.

and *<sup>n</sup>*<sup>⊥</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>7</sup>*<sup>α</sup>*

*<sup>n</sup>*�(⊥) 

*<sup>n</sup>*�� <sup>=</sup> *Im*

38].

direction, and *<sup>W</sup>* � <sup>1</sup>

*Sz*|/*W*, where

<sup>4</sup> (*�*0|*E*|

In our model, the energy velocity is obtained as:

(*n*<sup>1</sup> sin *θi*)<sup>2</sup>

*Sz* = <sup>1</sup>

(*n*<sup>1</sup> sin *<sup>θ</sup>i*)<sup>2</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*)<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>μ</sup>*0|*B*<sup>|</sup>

irradiance fluctuation of the EW, which can be measured[44].

index also confirms the dispersion of QED vacuum.

be obtained from the reference [38]:*n*� <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup>*<sup>α</sup>*

<sup>4</sup>*<sup>π</sup>* <sup>∑</sup><sup>∞</sup> *n*=1 2 <sup>3</sup>*<sup>π</sup>* <sup>+</sup> <sup>1</sup> *n* 1 *<sup>y</sup>* <sup>+</sup> <sup>1</sup> *n*2 2 *π* 

<sup>90</sup>*<sup>π</sup> <sup>y</sup>*<sup>2</sup> <sup>+</sup> *<sup>i</sup>* · *<sup>α</sup>*

Fig. 7. The irradiance of light *I* versus time *t*, (a)the incident light at the interface; (b)the transmitted EW in region *II*, where the black, the red and the green lines are for the fields at the distance from the interface *d*<sup>1</sup> = 0.1*λ*0, *d*<sup>2</sup> = 0.2*λ*<sup>0</sup> and *d*<sup>3</sup> = 0.3*λ*0, respectively, with *λ*<sup>0</sup> = 600*nm*.

are totally reflected when the incidence is the slowly-varying quasi-monochromatic wave. An interferometer or a photon detector is set at distance *L* from the interface so that the phase and intensity change can be detected.

The time-dependent Maxwell equations are given by ∇ × **E** = −*μ*(*z*)*μ*0*∂***H**/*∂t* and ∇ × **H** = *�*(*z*)*�*0*∂***E**/*∂t*, where *�*(*z*) and *μ*(*z*) are the relative permittivity and the relative permeability, respectively, and *<sup>c</sup>* = 1/√*�*0*μ*0. To obtain the concrete results, the system parameters are chosen as following, the incident angle *<sup>θ</sup><sup>i</sup>* <sup>=</sup> 0.1667*π*, the refractive index of region *I n*<sup>1</sup> <sup>=</sup> <sup>√</sup>*�*<sup>1</sup> <sup>=</sup> 2, and the vacuum refractive index of region *II n*<sup>2</sup> <sup>=</sup> <sup>√</sup>*�*2*μ*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>δ</sup>* <sup>+</sup> *in*��, where *δ* << 1 and *n*�� << 1 are the real and imaginary index deviations of vacuum, because of VP processes caused by strong external field. If the incident probing light is a plane wave, the transmitted wave in the vacuum region can be generally written in the form *E*(*x*, *z*, *t*) = *Eexp*(*ikzz* <sup>+</sup> *ik*�*r*� <sup>−</sup> *<sup>i</sup>ωt*), where *<sup>k</sup>*� <sup>=</sup> *<sup>n</sup>*<sup>1</sup> sin *<sup>θ</sup>iω*/*<sup>c</sup>* and *kz* <sup>=</sup> (*n*2*ω*/*c*)<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> � are the wave vectors parallel and perpendicular to the interface. For the EW, *kz* is described as:

$$k\_z = i\sqrt{(n\_1\sin\theta\_i)^2 - (1+\delta)^2}\frac{\omega}{c} + \frac{n''}{\sqrt{(n\_1\sin\theta\_i)^2 - (1+\delta)^2}}\frac{\omega}{c}.\tag{7}$$

The physical meaning of *kz* is very clear that the imaginary part *Im*(*kz*) = *κ<sup>z</sup>* corresponds to the exponential decay of the field, and the real part *Re*(*kz*) causes *a phase change because of VP*. The phase change at distance *z* = *L* is

$$
\Delta \phi = \text{Re}(k\_z) L \propto n^{\prime\prime} L \tag{8}
$$

which could be measured by interferometers[43].

Besides the phase change Δ*φ*, with the same model as shown in Fig.6, there is another way to detect the tiny *n*�� by *measuring the time delay of irradiance fluctuation [44] of the evanescent wave*. It is a dynamic process as following. First, we suppose that the incident light is not a plane wave anymore, but with a slow intensity fluctuation, as shown in Fig.7(a). Then, the question is "What will happen for the EW in region *II* ?" Numerically, from the strict Green's 12 Will-be-set-by-IN-TECH

x 104

Fig. 7. The irradiance of light *I* versus time *t*, (a)the incident light at the interface; (b)the transmitted EW in region *II*, where the black, the red and the green lines are for the fields at the distance from the interface *d*<sup>1</sup> = 0.1*λ*0, *d*<sup>2</sup> = 0.2*λ*<sup>0</sup> and *d*<sup>3</sup> = 0.3*λ*0, respectively, with

are totally reflected when the incidence is the slowly-varying quasi-monochromatic wave. An interferometer or a photon detector is set at distance *L* from the interface so that the phase and

The time-dependent Maxwell equations are given by ∇ × **E** = −*μ*(*z*)*μ*0*∂***H**/*∂t* and ∇ × **H** = *�*(*z*)*�*0*∂***E**/*∂t*, where *�*(*z*) and *μ*(*z*) are the relative permittivity and the relative permeability, respectively, and *<sup>c</sup>* = 1/√*�*0*μ*0. To obtain the concrete results, the system parameters are chosen as following, the incident angle *<sup>θ</sup><sup>i</sup>* <sup>=</sup> 0.1667*π*, the refractive index of region *I n*<sup>1</sup> <sup>=</sup> <sup>√</sup>*�*<sup>1</sup> <sup>=</sup> 2, and the vacuum refractive index of region *II n*<sup>2</sup> <sup>=</sup> <sup>√</sup>*�*2*μ*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>δ</sup>* <sup>+</sup> *in*��, where *δ* << 1 and *n*�� << 1 are the real and imaginary index deviations of vacuum, because of VP processes caused by strong external field. If the incident probing light is a plane wave, the transmitted wave in the vacuum region can be generally written in the form *E*(*x*, *z*, *t*) =

> *c* +

The physical meaning of *kz* is very clear that the imaginary part *Im*(*kz*) = *κ<sup>z</sup>* corresponds to the exponential decay of the field, and the real part *Re*(*kz*) causes *a phase change because of VP*.

Besides the phase change Δ*φ*, with the same model as shown in Fig.6, there is another way to detect the tiny *n*�� by *measuring the time delay of irradiance fluctuation [44] of the evanescent wave*. It is a dynamic process as following. First, we suppose that the incident light is not a plane wave anymore, but with a slow intensity fluctuation, as shown in Fig.7(a). Then, the question is "What will happen for the EW in region *II* ?" Numerically, from the strict Green's

0

0.2

0.4

I(a.u.)

0.6

0.8

1

× 10−4 (b)

4 4.5 5 5.5 6 6.5 7

t/fs

x 104

(*n*2*ω*/*c*)<sup>2</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

*ω*

*n*�� (*n*<sup>1</sup> sin *<sup>θ</sup>i*)<sup>2</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*)<sup>2</sup>

Δ*φ* = *Re*(*kz*)*L* ∝ *n*��*L* (8)

� are the wave

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

d1 d2 d3

4 4.5 5 5.5 6 6.5 7

t/fs

*Eexp*(*ikzz* <sup>+</sup> *ik*�*r*� <sup>−</sup> *<sup>i</sup>ωt*), where *<sup>k</sup>*� <sup>=</sup> *<sup>n</sup>*<sup>1</sup> sin *<sup>θ</sup>iω*/*<sup>c</sup>* and *kz* <sup>=</sup>

(*n*<sup>1</sup> sin *<sup>θ</sup>i*)<sup>2</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*)<sup>2</sup> *<sup>ω</sup>*

vectors parallel and perpendicular to the interface. For the EW, *kz* is described as:

intensity change can be detected.

*kz* = *i* 

The phase change at distance *z* = *L* is

which could be measured by interferometers[43].

*λ*<sup>0</sup> = 600*nm*.

I(a.u.)

(a)

function method with physical dissipation and dispersion, it is found that the fluctuation will propagate on the EW from the interface to far away, as shown in Fig.7(b). So, we can measure the time delay *τ* of the fluctuation propagation on the EW to detect the VP effect. The propagation speed of irradiance fluctuation can be obtained by the energy velocity *ve* which is defined as: *ve* <sup>=</sup> <sup>|</sup> *Sz*|/*W*, where *Sz* = <sup>1</sup> <sup>2</sup>*Re* (*E* × *H*∗)|*<sup>z</sup>* is the averaged Poynting vector along *z* direction, and *<sup>W</sup>* � <sup>1</sup> <sup>4</sup> (*�*0|*E*| <sup>2</sup> <sup>+</sup> *<sup>μ</sup>*0|*B*<sup>|</sup> <sup>2</sup>) is the local energy density of the electromagnetic wave. In our model, the energy velocity is obtained as:

$$
\sigma\_{\ell} = \chi \cdot \mathfrak{n}^{\prime\prime} \tag{9}
$$

with *χ* � *c*/ (*n*<sup>1</sup> sin *θi*)<sup>2</sup> (*n*<sup>1</sup> sin *<sup>θ</sup>i*)<sup>2</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*)<sup>2</sup> , when the dissipation and dispersion are very weak. The physical meaning of *ve* can be understood as the "propagation" speed of irradiance fluctuation of the EW, which can be measured[44].

Hence, experimentally the time delay *τ* of the irradiance fluctuation at distance *L* can be measured:

$$
\pi = \mathcal{L} / \upsilon\_{\mathcal{E}} \propto 1 / n'' \,. \tag{10}
$$

Since it is *near field* phenomenon, the detecting should be very near the interface. For the VP effect, since *n*�� is extremely small, the "propagation" speed of the irradiance fluctuation is so slow that *τ* gets to peco-second level when the distance is one tenth of the wavelength *L* = 60*nm*.

Therefore, either the phase change Δ*φ* or the time delay *τ* are very sensitive for *n*��, and the EW is a good candidate to probe the VP effect. Here, we note that the famous Kramers-Kronig relations still fit for QED vacuum[34]. Hence, the observation of imaginary part of vacuum index also confirms the dispersion of QED vacuum.

Next, we will quantitatively study the VP detect by our methods. Supposing that an external homogeneous constant electric field *Eext*, which is perpendicular to the *xz* plane and smaller than the Schwinger critical electric field *Ec*, is applied to the vacuum (region *I I*) only, as shown in Fig.6, then, the optical properties of the vacuum can be described by the Euler-Heisenberg Lagrangian *Leff*[34, 37]. Physically, the imaginary part of Euler-Heisenberg Lagrangian *Leff* is related to the imaginary part of VP operator, and therefore corresponds to the *electron-positron pair generation*.

Consequently, the vacuum refractive index can be deduced from the Lagrangian *Leff*[34, 37, 38].

In our model, since the external magnetic field is supposed to be zero, thus the vacuum refractive index is determined only by the external homogeneous constant electric field *Eext*. We use *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> to refer the effective refractive index of vacuum when the electric field of probing light are parallel and perpendicular to the field *Eext*, respectively. *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> can be obtained from the reference [38]:*n*� <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup>*<sup>α</sup>* <sup>45</sup>*<sup>π</sup> <sup>y</sup>*<sup>2</sup> <sup>+</sup> *<sup>i</sup>* · *<sup>α</sup>* <sup>4</sup>*<sup>π</sup>* <sup>∑</sup><sup>∞</sup> *n*=1 *<sup>π</sup> <sup>y</sup>*<sup>2</sup> <sup>+</sup> <sup>1</sup> *n* 1 *y* exp(−*nπ*/*y*), and *<sup>n</sup>*<sup>⊥</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>7</sup>*<sup>α</sup>* <sup>90</sup>*<sup>π</sup> <sup>y</sup>*<sup>2</sup> <sup>+</sup> *<sup>i</sup>* · *<sup>α</sup>* <sup>4</sup>*<sup>π</sup>* <sup>∑</sup><sup>∞</sup> *n*=1 2 <sup>3</sup>*<sup>π</sup>* <sup>+</sup> <sup>1</sup> *n* 1 *<sup>y</sup>* <sup>+</sup> <sup>1</sup> *n*2 2 *π* exp(−*nπ*/*y*), where *y* = |*Eext*|/*Ec*, and *<sup>α</sup>* � 1/137 is the fine-structure constant. Therefore we have *<sup>δ</sup>* <sup>=</sup> *Re <sup>n</sup>*�(⊥) − 1, *<sup>n</sup>*�� <sup>=</sup> *Im <sup>n</sup>*�(⊥) for *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> when we solve the equations such as Eq.(7) in this letter.

**4. The temporal coherence gain of the negative-index superlens image**

Veselago predicted that the negative-index material (NIM) has some unusual properties, such as a flat slab of the NIM could function as a lens for electromagnetic (EM) waves [1]. This research direction was further pushed by works of Pendry and others [4, 5, 46–58] who showed the lens with such NIM (i.e. *�* = *μ* = −1 + *δ*) could be a *superlens* whose image resolution can go beyond the usual diffraction limit. After that, several beyond-limit properties of NIM systems are found, such as, the sub-wavelength cavity [59] and the waveguide [60]. Some of the theoretical results are confirmed by experiments[4, 46, 49]. And these beyond-limit properties give us new physical pictures and opportunities to design devices. Recently, new numerical [50, 51] and theoretical Green's function [52] methods are used to understand the phenomena in such systems. But so far almost all studies are done with the strictly single-frequency sources, so that the coherent properties of EM waves (or photons) in the NIM systems have not been studied to the best of our knowledge. Even more seriously, there is no theory for the propagation of coherent functions in NIM systems. The importance of coherence research can not be over-estimated since the coherence is essential in the wave interference , the imaging , the signal processing and the telecommunication [61, 62]. Can we find new frontier to go beyond at the coherent properties in NIM systems? If so, can we develop a simple theoretical method to deal with the image coherence of superlens?

The Group Velocity Picture of Metamaterial Systems 71

Fig. 9. The schematic diagram of our model with ray paths(left); and the typical snapshot of

In this section, the finite difference time domain (FDTD) method is used in the two-dimensional (2D) numerical experiments to study the temporal coherence of the superlens image with random quasi-monochromatic sources. We observe the dramatic temporal-coherence gain of the superlens image even if the reflection and frequency-filtering effects are very weak. Based on the new physical picture of the signal (the fluctuation of random source) propagation in NIM, we construct a theory to obtain the image field and derive the equation of the temporal-coherence relation between the source and its image. The new mechanism of the temporal-coherent gain can be explained by the key idea that the signals on different paths have different "group" retarded time. Our theory excellently agrees

*The setup* of the 2D system is shown in Fig.9. The thickness of the infinite-long NIM slab is *d*.

*P* and the magnetic

electric field in our FDTD simulation (right).

with numerical results and the strict Green's function results.

To realize the negative *�* and negative *μ*, the electric polarization density

Fig. 8. (a)The real part of vacuum index *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> versus *Eext*; (b)The imaginary part of *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> versus the external electric field strength; (c)Δ*<sup>φ</sup>* versus with *Lp* = <sup>6</sup>*μm*; (d)*<sup>τ</sup>* versus *Eext* with *L<sup>τ</sup>* = 60*nm*. Both the results from theory and from Green's function are shown in (c) and (d).

The parameters of our model in Fig.6 are chosen as following. The wavelength of probing light is *λ*<sup>0</sup> = 600*nm*, the dielectric constant in the region *I* is *ε* = 4, and the incident angle is *θinc* = 0.1667*π* > *θc*, so that the field in vacuum is evanescent. The distance *L* for the phase detecting is *Lp* = 6*μm* = 10 × *λ*0, while for *τ* detecting is *L<sup>τ</sup>* = 60*nm* = 0.1 × *λ*0, respectively. The QED theoretical results of real and imaginary part of *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> are shown in Fig.8(a) and Fig.8(b), respectively. Bring these results into Eq.(8) and Eq.(10), the phase change Δ*φ* and the delayed time *τ* can be obtained, which are shown in Fig8(c) and Fig.8(d), respectively. Numerically, the phase change with plane wave incidence and the time delay of local amplitude maximum are calculated by Green's function method, which are also shown in Fig.8(c) and Fig.8 (d). Comparing the analytical results from Eq.(8) and Eq.(9) and numerical results, we can find that they agree with each other very well.

Next, we will analyze the possibility to observe the VP effect in experimental conditions. The recent experimental advances[45] have raised hopes that lasers may achieve fields just one or two orders of magnitude below the Schwinger critical field strength. In this case *Eext* ∼ 0.1*Ec*, from our numerical and analytical results in Fig.8, we can see the <sup>Δ</sup>*<sup>φ</sup>* can get to <sup>∼</sup> <sup>10</sup>−1*mrad* order, which are in measuring limit of contemporary interferometer [43]. Very recently, it is supposed that the electric field *E* could be effectively amplified 4 times larger by coherent constructive interference of laser beams[36]. If *Eext* can get to 0.5*Ec* by this method, not only Δ*φ* can be one order larger, but also the delay time *τ* can get to sub peco-second level and may be measured by contemporary photon detectors.

14 Will-be-set-by-IN-TECH

0

0

5

10

τ /ps

Fig. 8. (a)The real part of vacuum index *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> versus *Eext*; (b)The imaginary part of *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> versus the external electric field strength; (c)Δ*<sup>φ</sup>* versus with *Lp* = <sup>6</sup>*μm*; (d)*<sup>τ</sup>* versus *Eext* with *L<sup>τ</sup>* = 60*nm*. Both the results from theory and from Green's function are shown in

The parameters of our model in Fig.6 are chosen as following. The wavelength of probing light is *λ*<sup>0</sup> = 600*nm*, the dielectric constant in the region *I* is *ε* = 4, and the incident angle is *θinc* = 0.1667*π* > *θc*, so that the field in vacuum is evanescent. The distance *L* for the phase detecting is *Lp* = 6*μm* = 10 × *λ*0, while for *τ* detecting is *L<sup>τ</sup>* = 60*nm* = 0.1 × *λ*0, respectively. The QED theoretical results of real and imaginary part of *<sup>n</sup>*� and *<sup>n</sup>*<sup>⊥</sup> are shown in Fig.8(a) and Fig.8(b), respectively. Bring these results into Eq.(8) and Eq.(10), the phase change Δ*φ* and the delayed time *τ* can be obtained, which are shown in Fig8(c) and Fig.8(d), respectively. Numerically, the phase change with plane wave incidence and the time delay of local amplitude maximum are calculated by Green's function method, which are also shown in Fig.8(c) and Fig.8 (d). Comparing the analytical results from Eq.(8) and Eq.(9) and numerical

Next, we will analyze the possibility to observe the VP effect in experimental conditions. The recent experimental advances[45] have raised hopes that lasers may achieve fields just one or two orders of magnitude below the Schwinger critical field strength. In this case *Eext* ∼ 0.1*Ec*, from our numerical and analytical results in Fig.8, we can see the <sup>Δ</sup>*<sup>φ</sup>* can get to <sup>∼</sup> <sup>10</sup>−1*mrad* order, which are in measuring limit of contemporary interferometer [43]. Very recently, it is supposed that the electric field *E* could be effectively amplified 4 times larger by coherent constructive interference of laser beams[36]. If *Eext* can get to 0.5*Ec* by this method, not only Δ*φ* can be one order larger, but also the delay time *τ* can get to sub peco-second level and may

(c) (d)

15

1

2

Im(n)

(a) (b)

3

4 x 10−4

n|| n⊥

0 0.5 1 1.5

Analytical n|| Analytical n<sup>⊥</sup> Numerical n|| Numerical n<sup>⊥</sup>

Eext/Ec

0.4 0.5 0.6

Eext/Ec

0 0.5 1 1.5

Eext/Ec

Analytical n|| Analytical n<sup>⊥</sup> Numerical n|| Numerical n<sup>⊥</sup>

n|| n⊥

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>0</sup>

Eext/Ec

results, we can find that they agree with each other very well.

be measured by contemporary photon detectors.

1 1.0001 1.0002 1.0003 1.0004 1.0005

Δφ /mrad

(c) and (d).

Re(n)

#### **4. The temporal coherence gain of the negative-index superlens image**

Veselago predicted that the negative-index material (NIM) has some unusual properties, such as a flat slab of the NIM could function as a lens for electromagnetic (EM) waves [1]. This research direction was further pushed by works of Pendry and others [4, 5, 46–58] who showed the lens with such NIM (i.e. *�* = *μ* = −1 + *δ*) could be a *superlens* whose image resolution can go beyond the usual diffraction limit. After that, several beyond-limit properties of NIM systems are found, such as, the sub-wavelength cavity [59] and the waveguide [60]. Some of the theoretical results are confirmed by experiments[4, 46, 49]. And these beyond-limit properties give us new physical pictures and opportunities to design devices. Recently, new numerical [50, 51] and theoretical Green's function [52] methods are used to understand the phenomena in such systems. But so far almost all studies are done with the strictly single-frequency sources, so that the coherent properties of EM waves (or photons) in the NIM systems have not been studied to the best of our knowledge. Even more seriously, there is no theory for the propagation of coherent functions in NIM systems. The importance of coherence research can not be over-estimated since the coherence is essential in the wave interference , the imaging , the signal processing and the telecommunication [61, 62]. Can we find new frontier to go beyond at the coherent properties in NIM systems? If so, can we develop a simple theoretical method to deal with the image coherence of superlens?

Fig. 9. The schematic diagram of our model with ray paths(left); and the typical snapshot of electric field in our FDTD simulation (right).

In this section, the finite difference time domain (FDTD) method is used in the two-dimensional (2D) numerical experiments to study the temporal coherence of the superlens image with random quasi-monochromatic sources. We observe the dramatic temporal-coherence gain of the superlens image even if the reflection and frequency-filtering effects are very weak. Based on the new physical picture of the signal (the fluctuation of random source) propagation in NIM, we construct a theory to obtain the image field and derive the equation of the temporal-coherence relation between the source and its image. The new mechanism of the temporal-coherent gain can be explained by the key idea that the signals on different paths have different "group" retarded time. Our theory excellently agrees with numerical results and the strict Green's function results.

*The setup* of the 2D system is shown in Fig.9. The thickness of the infinite-long NIM slab is *d*. To realize the negative *�* and negative *μ*, the electric polarization density *P* and the magnetic

*paths nds* = *const* (

E

Et(a.u.)

ω(a.u.)

NIM, the GRT of a path should be *<sup>d</sup>*

0.96 0.98 1.00 1.02 1.04 ω/ω<sup>0</sup>

0 0.5 1 1.5 2 2.5 3 3.5 4

from Eq.(1) (up), and from the Green's function method (down).

x 10 t/δ <sup>4</sup> <sup>t</sup>

Fig. 10. (a)the FSs of the source(up) and the image(down) . (b)The electric field of the source(up) and its image (down) vs time from FDTD simulation (c) The image field vs time

*Physical pictures*.−To deeper understand the new mechanism of coherence gain and construct our theory, we need make two physical pictures clear. The first one is about the optical path length (OPL) *nds* which determines the wave phase and the refracted "paths" of rays in Fig.9 according to Fermat's principle (or Snell's law). Based on ray optics, the superlens and traditional lenses have same focusing mechanism, that all focusing rays have same OPL

The Group Velocity Picture of Metamaterial Systems 73

well-known that it suppresses the other important picture. Because the temporal-coherence information is in the fluctuation signals of random field, the signal propagating picture should be essential for our study. *The optical signals propagate in the group velocity vg which is always positive*. Obviously, if the path (in Fig.9) is longer (larger incident angle), the signal need a longer propagating time, which is called *group retarded time* (GRT) in this section Inside the

where *θ* is the incident angle and *vg* = *c*/3.04 is the group velocity of NIM around *ω*<sup>0</sup> [65]. The total GRT from source to image is *τ<sup>r</sup>* = *τ*0/*cos*(*θ*) where the *τ*<sup>0</sup> = *d*/*c* + *d*/*vg* is the GRT of the paraxial ray. Now, the new propagating picture for a signal through superlens is that a signal, generated at at *ts* from the source, will propagate on all focusing paths and arrive at

Et(a.u.)

0

0.2

0.4

0.6

g(1)

0.8

1

*paths nds* = 0 for superlens) from source to image [1]. But this picture is so

*cos*(*θ*)*vg* (this is confirmed by our numerical experiments),

(a) (b)

0 0.5 1 1.5 2 2.5 3 3.5 4

(c) (d)

Source FDTD Our theory Green's function

0 1000 2000 3000 4000 5000

τ/δ<sup>t</sup>

x 10 t/δ <sup>4</sup> <sup>t</sup>

moment density *M*� are phenomenologically introduced in FDTD simulation [63]. The effective permittivity and permeability of the NIM are *�r*(*ω*) = *μr*(*ω*) = 1 + *ω*<sup>2</sup> *P*/(*ω*<sup>2</sup> *<sup>a</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>γ*). In our model, *<sup>ω</sup><sup>a</sup>* <sup>=</sup> 1.884 <sup>×</sup> 1013/*s*, *<sup>γ</sup>* <sup>=</sup> *<sup>ω</sup>a*/100, *<sup>ω</sup><sup>P</sup>* <sup>=</sup> <sup>10</sup> <sup>×</sup> *<sup>ω</sup>a*. The quasi-monochromatic field is expressed as *E*(*x*, *t*) = *U*(*x*, *t*)*exp*(−*iω*0*t*), where *U*(*x*, *t*) is a slowly-varying random function, *<sup>ω</sup>*<sup>0</sup> <sup>=</sup> *<sup>π</sup>*/20*δ<sup>t</sup>* is the central frequency of our random sources and *<sup>δ</sup><sup>t</sup>* <sup>=</sup> 1.18 <sup>×</sup> <sup>10</sup>−15*<sup>s</sup>* is the smallest time-step in FDTD simulation. At *ω*0, we have *�<sup>r</sup>* = *μ<sup>r</sup>* = −1.00 − *i*0.0029. At here, we emphasize that in our FDTD simulation the smallest space-step *δ<sup>x</sup>* = *λ*0/20 (*λ*<sup>0</sup> = 2*πc*/*ω*<sup>0</sup> ) and the distance (*d*/2 = *λ*0) of the source from the lens are too large to excite strong evanescent modes of NIM [50, 51, 53]. *Actually the evanescent field in our simulation can be neglected comparing with radiating field, and what we are studying is the property dominated by the radiating field.*

The random source is composed of the randomly generated plane-wave pulses, with the average pulse length *tp* and the random starting phase and starting time. In the simulation, we record the field of the source and the image for a duration of 4 <sup>×</sup> 105*δ<sup>t</sup>* to obtain the data for analysis. For the convenience, we define *E*(*ω*) = lim*T*→<sup>∞</sup> *T* <sup>−</sup>*<sup>T</sup> <sup>E</sup>*(*t*)*exp*(−*iωt*) as the *field spectrum* (FS).

*Unusual phenomena*.−At first, the FS width of the random source is a little too large (Δ*ω<sup>s</sup>* � *ω*0/20). When we observe the image temporal-coherence gain, we also find that the FS width of the image is sharper than the source (Δ*ω<sup>i</sup>* < Δ*ωs*). It is obvious that there are the frequency-filtering effects because of the NIM dispersion, such as the frequency-dependent interface reflection and focal length. After increasing the pulse-length *tp* of the source, we reduce the source FS width to Δ*ω<sup>s</sup>* � *ω*0/100, then the reflection and focal-length difference are very small [64]. With such source, the FS widths of source and image are almost same Δ*ω<sup>i</sup>* � Δ*ωs*, as shown in Fig.10a. The difference between two widths is < 5%, which is our criterion of the *quasi*-*monochromatic* source. Even so the dramatic gain of temporal coherence is still observed. In Fig.10b, the source field (up) and the image field (down) vs time of FDTD simulation are compared. The *profiles* of them are genically similar, but the image profile is much smoother.

The normalized temporal-coherence function *g*(1)(*τ*) =< *E*∗(*t*)*E*(*t* + *τ*) > /< *E*∗(*t*)*E*(*t*) > (<> means the ensemble average) of the source (black) and the image (red) from FDTD simulation are shown in Fig. 11. The temporal coherence of the image field is obvious better than the source. From *g*(1) , the image coherent time is obtained *Tco <sup>i</sup>* <sup>=</sup> *<sup>g</sup>* (1) *<sup>i</sup>* (*τ*)*dτ* = 1268*δt*, which is about 50% longer than the source coherent time *Tco <sup>s</sup>* = 860*δt*.

Although the gain of the spatial coherence only by propagation is well-known[62], the dramatic gain of temporal coherence is generally from the high-Q cavities, contrary to our case, which have strong filtering effects. To reveal the new mechanism of the temporal coherence gain in NIM systems, we also have done more numerical experiments in which *only* the ray near a certain incident angle (shown in Fig.9), such as only paraxial rays (*θ* � 0), can pass through the superlens. Then the image field profile vs time looks very like the source field and has no gain of coherence anymore. *Therefore, the gain of temporal coherence of the superlens image is not from one ray with certain incident angle, but probably from the interference between the rays with different incident angles.* Then, what is different between the rays with different incident angles? After carefully checking the field profiles of different-incident-angle cases, we find that the profiles have different retarded time. The larger incident angle the longer retarded time.

16 Will-be-set-by-IN-TECH

moment density *M*� are phenomenologically introduced in FDTD simulation [63]. The effective

our model, *<sup>ω</sup><sup>a</sup>* <sup>=</sup> 1.884 <sup>×</sup> 1013/*s*, *<sup>γ</sup>* <sup>=</sup> *<sup>ω</sup>a*/100, *<sup>ω</sup><sup>P</sup>* <sup>=</sup> <sup>10</sup> <sup>×</sup> *<sup>ω</sup>a*. The quasi-monochromatic field is expressed as *E*(*x*, *t*) = *U*(*x*, *t*)*exp*(−*iω*0*t*), where *U*(*x*, *t*) is a slowly-varying random function, *<sup>ω</sup>*<sup>0</sup> <sup>=</sup> *<sup>π</sup>*/20*δ<sup>t</sup>* is the central frequency of our random sources and *<sup>δ</sup><sup>t</sup>* <sup>=</sup> 1.18 <sup>×</sup> <sup>10</sup>−15*<sup>s</sup>* is the smallest time-step in FDTD simulation. At *ω*0, we have *�<sup>r</sup>* = *μ<sup>r</sup>* = −1.00 − *i*0.0029. At here, we emphasize that in our FDTD simulation the smallest space-step *δ<sup>x</sup>* = *λ*0/20 (*λ*<sup>0</sup> = 2*πc*/*ω*<sup>0</sup> ) and the distance (*d*/2 = *λ*0) of the source from the lens are too large to excite strong evanescent modes of NIM [50, 51, 53]. *Actually the evanescent field in our simulation can be neglected comparing with radiating field, and what we are studying is the property dominated by the*

The random source is composed of the randomly generated plane-wave pulses, with the average pulse length *tp* and the random starting phase and starting time. In the simulation, we record the field of the source and the image for a duration of 4 <sup>×</sup> 105*δ<sup>t</sup>* to obtain the data

*Unusual phenomena*.−At first, the FS width of the random source is a little too large (Δ*ω<sup>s</sup>* � *ω*0/20). When we observe the image temporal-coherence gain, we also find that the FS width of the image is sharper than the source (Δ*ω<sup>i</sup>* < Δ*ωs*). It is obvious that there are the frequency-filtering effects because of the NIM dispersion, such as the frequency-dependent interface reflection and focal length. After increasing the pulse-length *tp* of the source, we reduce the source FS width to Δ*ω<sup>s</sup>* � *ω*0/100, then the reflection and focal-length difference are very small [64]. With such source, the FS widths of source and image are almost same Δ*ω<sup>i</sup>* � Δ*ωs*, as shown in Fig.10a. The difference between two widths is < 5%, which is our criterion of the *quasi*-*monochromatic* source. Even so the dramatic gain of temporal coherence is still observed. In Fig.10b, the source field (up) and the image field (down) vs time of FDTD simulation are compared. The *profiles* of them are genically similar, but the image profile is

The normalized temporal-coherence function *g*(1)(*τ*) =< *E*∗(*t*)*E*(*t* + *τ*) > /< *E*∗(*t*)*E*(*t*) > (<> means the ensemble average) of the source (black) and the image (red) from FDTD simulation are shown in Fig. 11. The temporal coherence of the image field is obvious better

, the image coherent time is obtained *Tco*

Although the gain of the spatial coherence only by propagation is well-known[62], the dramatic gain of temporal coherence is generally from the high-Q cavities, contrary to our case, which have strong filtering effects. To reveal the new mechanism of the temporal coherence gain in NIM systems, we also have done more numerical experiments in which *only* the ray near a certain incident angle (shown in Fig.9), such as only paraxial rays (*θ* � 0), can pass through the superlens. Then the image field profile vs time looks very like the source field and has no gain of coherence anymore. *Therefore, the gain of temporal coherence of the superlens image is not from one ray with certain incident angle, but probably from the interference between the rays with different incident angles.* Then, what is different between the rays with different incident angles? After carefully checking the field profiles of different-incident-angle cases, we find that the profiles have different retarded time. The larger incident angle the longer

*T*

*<sup>i</sup>* <sup>=</sup> *<sup>g</sup>*

*<sup>s</sup>* = 860*δt*.

(1)

*<sup>i</sup>* (*τ*)*dτ* = 1268*δt*,

*P*/(*ω*<sup>2</sup>

<sup>−</sup>*<sup>T</sup> <sup>E</sup>*(*t*)*exp*(−*iωt*) as the *field*

*<sup>a</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>γ*). In

permittivity and permeability of the NIM are *�r*(*ω*) = *μr*(*ω*) = 1 + *ω*<sup>2</sup>

for analysis. For the convenience, we define *E*(*ω*) = lim*T*→<sup>∞</sup>

which is about 50% longer than the source coherent time *Tco*

*radiating field.*

*spectrum* (FS).

much smoother.

retarded time.

than the source. From *g*(1)

*Physical pictures*.−To deeper understand the new mechanism of coherence gain and construct our theory, we need make two physical pictures clear. The first one is about the optical path length (OPL) *nds* which determines the wave phase and the refracted "paths" of rays in Fig.9 according to Fermat's principle (or Snell's law). Based on ray optics, the superlens and traditional lenses have same focusing mechanism, that all focusing rays have same OPL *paths nds* = *const* ( *paths nds* = 0 for superlens) from source to image [1]. But this picture is so well-known that it suppresses the other important picture. Because the temporal-coherence information is in the fluctuation signals of random field, the signal propagating picture should be essential for our study. *The optical signals propagate in the group velocity vg which is always positive*. Obviously, if the path (in Fig.9) is longer (larger incident angle), the signal need a longer propagating time, which is called *group retarded time* (GRT) in this section Inside the NIM, the GRT of a path should be *<sup>d</sup> cos*(*θ*)*vg* (this is confirmed by our numerical experiments), where *θ* is the incident angle and *vg* = *c*/3.04 is the group velocity of NIM around *ω*<sup>0</sup> [65]. The total GRT from source to image is *τ<sup>r</sup>* = *τ*0/*cos*(*θ*) where the *τ*<sup>0</sup> = *d*/*c* + *d*/*vg* is the GRT of the paraxial ray. Now, the new propagating picture for a signal through superlens is that a signal, generated at at *ts* from the source, will propagate on all focusing paths and arrive at

Fig. 10. (a)the FSs of the source(up) and the image(down) . (b)The electric field of the source(up) and its image (down) vs time from FDTD simulation (c) The image field vs time from Eq.(1) (up), and from the Green's function method (down).

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

<sup>1</sup> <sup>−</sup> (*τ*0/*t*)<sup>2</sup> is the response function of different incident angles,

*<sup>i</sup>* (*t*1)*hi*(*t*<sup>2</sup> + *τ*)

*<sup>i</sup>* vs superlens length *L*

*i*

*<sup>s</sup>* (*t*1)*Es*(*t*2) > is the temporal-coherence function of the source. Eq.(12)

τ / δ t

The Group Velocity Picture of Metamaterial Systems 75

Fig. 12. The coherent time versus the superlens length *L*, from the FDTD simulation(blue)

can explain the temporal-coherence gain of the image too. Even if the source field is totally temporal *incoherent Gs*(*t*<sup>2</sup> − *t*1) ∝ *δ*(*t*<sup>2</sup> − *t*1), based on Eq.(12) we can find that *Gi*(*τ*) is not a

includes the interference between paths. According to our theory, we calculate the image coherence function *g*(1) vs time (Fig.11 blue) which agree with our FDTD result (Fig.11 red)

To further confirm our theory and FDTD results, the *strict* Green's function method [52] is engaged to check our results. *We only include the radiating field (no evanescent wave) in Green's function*. The strict image field vs time is shown in Fig.10c(down), and the image temporal-coherence function *g*(1) vs time is shown in Fig.11 (blue). In Fig.11, we can see that the FDTD result (red) is almost exactly same as the strict Green's function method (green). But our theory (blue) deviates from the strict result at very large *τ* > 3000*δ<sup>t</sup>* which corresponding to very long path(or very large incident angle). This is understandable since in our theory we neglect the dispersion of NIM totally and only use *vg*(*ω*0). For the very-large-angle rays a small index difference (from the dispersion of NIM) can cause large focal-length difference. Hence the deviation is from the focus-filtering effect. When we reduce the FS width of source

*δ*-function anymore, so the image is partial temporal coherent. The product of *h*∗

to an even smaller value (*i.e.* Δ*ω<sup>s</sup>* = *ω*0/500), the deviation of our theory is smaller.

Green's function method. In Fig.(11), we plot the coherent time *Tco*

contribution to the temporal-coherence gain are missed in the short superlens.

Although our theory is only a good approximation generally, owing to the picture simplicity and clarity the theory can help us to study more complex systems qualitatively and quantitatively. The *finitely-long* 2D superlens is a good example which is hard to deal by

of the FDTD simulation (black) and of our theory (red) , respectively. They coincide with each other pretty well (the deviation reason has been discussed). The increase of the *Tco*

with the increase of *L* can be explained simply according our theory. Since the image field is

*cos<sup>θ</sup>* )*dθ*, the large-angle paths (*θ* > *θmax* and *θ* < *θmin*)and their

Radiative Filed Evanescent Filed Global Filed

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

pretty well (we will discuss the deviation later).

and from our theory (red). where *hi*(*t*)=(*τ*0/*t*)2/

and *Gs*(*t*<sup>2</sup> − *t*1) =< *E*<sup>∗</sup>

*Ei*(*t*) = <sup>1</sup>

*<sup>U</sup>*<sup>0</sup> *<sup>e</sup>*−*iω*0*<sup>t</sup> <sup>θ</sup>max*

*<sup>θ</sup>min Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*<sup>0</sup>

g(1)

image position at very different time *ts* + *τ*0/*cos*(*θ*) from different paths(this is schematically shown in Fig.9). This picture is totally different from traditional lenses, whose images don't have obvious temporal-coherence gain because their focusing rays have same OPL *and similar GRT*.

*Our theory*.−Based on these analysis, we suppose that *the superlens image field of the random quasi-monochromatic source is the sum of all signals from different paths with different GRT.* This is the key point of our theory, and then the image field can be obtained:

$$E\_l(t) = \frac{1}{\mathcal{U}\_0} e^{-i\omega\_0 t} \sum\_{paths} \mathcal{U}\_s(t - \tau\_r) = \frac{1}{\mathcal{U}\_0} e^{-i\omega\_0 t} \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \mathcal{U}\_s(t - \frac{\tau\_0}{\cos(\theta)}) d\theta \tag{11}$$

where *Us*(*t*) is the slowly-varying profile function of the source and *U*<sup>0</sup> is the normalization factor. In Fig.10c (up), we show the result of the image field based on Eq.(11), we can see it is in excellent agreement with the FDTD result in 2b(down). To show the interference effect of different paths, we assume there are only two paths (such as *A* and *B* in Fig.9). Based on Eq.(11) the image field is *Ei* = *e*−*iω*0*<sup>t</sup>* (*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup> <sup>r</sup>* ) + *Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup> <sup>r</sup>* )), then the temporal coherence of image is *G*(*τ*) =< *E*∗ *<sup>i</sup>* (*t*)*Ei*(*t* + *τ*) >=< *U*<sup>∗</sup> *<sup>s</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup> <sup>r</sup>* )*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup> <sup>r</sup>* + *τ*) + *U*<sup>∗</sup> *<sup>s</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup> <sup>r</sup>* )*Us*(*t* − *τB <sup>r</sup>* + *τ*) + *U*<sup>∗</sup> *<sup>s</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup> <sup>r</sup>* )*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup> <sup>r</sup>* + *τ*) + *U*<sup>∗</sup> *<sup>s</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup> <sup>r</sup>* )*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup> <sup>r</sup>* + *τ*) >. The first two terms are same as the source field (just a time-shift) so they don't contribute to the coherence gain. The last two terms are from interference between two paths. The third (or the forth) term could be very large at the condition *<sup>τ</sup>* � *<sup>τ</sup><sup>B</sup> <sup>r</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup> <sup>r</sup>* (or *τ<sup>A</sup> <sup>r</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup> <sup>r</sup>* ). This condition can always be satisfied between any two paths since *τ* is a continuous variable. So the interfering terms between the paths are responsible for the image temporal-coherence gain.

From Eq. (11), after the variable transformation *ts* = *t* − *τ*0/*cosθ* and some algebra, the relation of the temporal coherence between the image and the source can be obtained:

$$G\_{i}(\tau) = \left. < E\_{i}^{\*}(t)E\_{i}(t+\tau) > \right. \tag{12}$$

$$= \frac{1}{\mathbf{U}\_{0}^{2}} \int\_{-\infty}^{-\eta} dt\_{1} \int\_{-\infty}^{-\eta+\tau} dt\_{2} h\_{i}^{\*}(t\_{1})h\_{i}(t\_{2}+\tau)\mathbf{G}\_{i}(t\_{2}-t\_{1})$$

$$\stackrel{\text{uu}}{\longrightarrow}$$

$$\stackrel{\text{uu}}{\longrightarrow}$$

$$\stackrel{\text{v}}{\longrightarrow}$$

$$\stackrel{\text{v}}{\longrightarrow}$$

$$\stackrel{\text{v}}{\longrightarrow}$$

$$\stackrel{\text{au}}{\longrightarrow}$$

$$\stackrel{\text{au}}{\longrightarrow}$$

$$\stackrel{\text{au}}{\longrightarrow}$$

Fig. 11. The normalized temporal-coherence function *g*(1) vs time of the source field (black), and of the image field which obtained from the FDTD simulation (red), from Eq.(2) (blue) and from the Green's function method(green).

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

image position at very different time *ts* + *τ*0/*cos*(*θ*) from different paths(this is schematically shown in Fig.9). This picture is totally different from traditional lenses, whose images don't have obvious temporal-coherence gain because their focusing rays have same OPL *and similar*

*Our theory*.−Based on these analysis, we suppose that *the superlens image field of the random quasi-monochromatic source is the sum of all signals from different paths with different GRT.* This is

where *Us*(*t*) is the slowly-varying profile function of the source and *U*<sup>0</sup> is the normalization factor. In Fig.10c (up), we show the result of the image field based on Eq.(11), we can see it is in excellent agreement with the FDTD result in 2b(down). To show the interference effect of different paths, we assume there are only two paths (such as *A* and *B* in Fig.9). Based on

*U*<sup>0</sup>

*e*−*iω*0*<sup>t</sup>*

*<sup>r</sup>* ) + *Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup>*

*<sup>r</sup>* )*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup>*

*<sup>s</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup>*

 *<sup>π</sup>* 2 − *<sup>π</sup>* 2

*<sup>r</sup>* )*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup>*

*<sup>i</sup>* (*t*)*Ei*(*t* + *τ*) > (12)

*<sup>i</sup>* (*t*1)*hi*(*t*<sup>2</sup> + *τ*)*Gs*(*t*<sup>2</sup> − *t*1)

*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*<sup>0</sup>

cos(*θ*)

*<sup>r</sup>* )), then the temporal coherence

*<sup>r</sup>* + *τ*) >. The first two terms are

*<sup>s</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup>*

*<sup>r</sup>* )*Us*(*t* −

*<sup>r</sup>* + *τ*) + *U*<sup>∗</sup>

*<sup>r</sup>* ). This condition can always be satisfied

)*dθ* (11)

*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>r*) = <sup>1</sup>

(*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup>*

*<sup>r</sup>* (or *τ<sup>A</sup>*

 −*τ*0+*τ* −∞

*<sup>s</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup>*

same as the source field (just a time-shift) so they don't contribute to the coherence gain. The last two terms are from interference between two paths. The third (or the forth) term could be

between any two paths since *τ* is a continuous variable. So the interfering terms between the

From Eq. (11), after the variable transformation *ts* = *t* − *τ*0/*cosθ* and some algebra, the relation

*<sup>r</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup>*

*dt*2*h*<sup>∗</sup>

<sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>0</sup>

Fig. 11. The normalized temporal-coherence function *g*(1) vs time of the source field (black), and of the image field which obtained from the FDTD simulation (red), from Eq.(2) (blue)

L/2cδ<sup>t</sup>

*<sup>i</sup>* (*t*)*Ei*(*t* + *τ*) >=< *U*<sup>∗</sup>

*<sup>r</sup>* + *τ*) + *U*<sup>∗</sup>

*<sup>r</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup>*

of the temporal coherence between the image and the source can be obtained:

*dt*<sup>1</sup>

paths are responsible for the image temporal-coherence gain.

 −*τ*<sup>0</sup> −∞

the key point of our theory, and then the image field can be obtained:

*<sup>e</sup>*−*iω*0*<sup>t</sup>* ∑ *paths*

*Ei*(*t*) = <sup>1</sup>

Eq.(11) the image field is *Ei* = *e*−*iω*0*<sup>t</sup>*

*<sup>s</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>A</sup>*

very large at the condition *<sup>τ</sup>* � *<sup>τ</sup><sup>B</sup>*

of image is *G*(*τ*) =< *E*∗

*U*<sup>0</sup>

*<sup>r</sup>* )*Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>B</sup>*

*Gi*(*τ*) = < *E*<sup>∗</sup>

<sup>=</sup> <sup>1</sup> *U*<sup>2</sup> 0

T

i

co

and from the Green's function method(green).

*GRT*.

*τB*

*<sup>r</sup>* + *τ*) + *U*<sup>∗</sup>

Fig. 12. The coherent time versus the superlens length *L*, from the FDTD simulation(blue) and from our theory (red).

where *hi*(*t*)=(*τ*0/*t*)2/ <sup>1</sup> <sup>−</sup> (*τ*0/*t*)<sup>2</sup> is the response function of different incident angles, and *Gs*(*t*<sup>2</sup> − *t*1) =< *E*<sup>∗</sup> *<sup>s</sup>* (*t*1)*Es*(*t*2) > is the temporal-coherence function of the source. Eq.(12) can explain the temporal-coherence gain of the image too. Even if the source field is totally temporal *incoherent Gs*(*t*<sup>2</sup> − *t*1) ∝ *δ*(*t*<sup>2</sup> − *t*1), based on Eq.(12) we can find that *Gi*(*τ*) is not a *δ*-function anymore, so the image is partial temporal coherent. The product of *h*∗ *<sup>i</sup>* (*t*1)*hi*(*t*<sup>2</sup> + *τ*) includes the interference between paths. According to our theory, we calculate the image coherence function *g*(1) vs time (Fig.11 blue) which agree with our FDTD result (Fig.11 red) pretty well (we will discuss the deviation later).

To further confirm our theory and FDTD results, the *strict* Green's function method [52] is engaged to check our results. *We only include the radiating field (no evanescent wave) in Green's function*. The strict image field vs time is shown in Fig.10c(down), and the image temporal-coherence function *g*(1) vs time is shown in Fig.11 (blue). In Fig.11, we can see that the FDTD result (red) is almost exactly same as the strict Green's function method (green). But our theory (blue) deviates from the strict result at very large *τ* > 3000*δ<sup>t</sup>* which corresponding to very long path(or very large incident angle). This is understandable since in our theory we neglect the dispersion of NIM totally and only use *vg*(*ω*0). For the very-large-angle rays a small index difference (from the dispersion of NIM) can cause large focal-length difference. Hence the deviation is from the focus-filtering effect. When we reduce the FS width of source to an even smaller value (*i.e.* Δ*ω<sup>s</sup>* = *ω*0/500), the deviation of our theory is smaller.

Although our theory is only a good approximation generally, owing to the picture simplicity and clarity the theory can help us to study more complex systems qualitatively and quantitatively. The *finitely-long* 2D superlens is a good example which is hard to deal by Green's function method. In Fig.(11), we plot the coherent time *Tco <sup>i</sup>* vs superlens length *L* of the FDTD simulation (black) and of our theory (red) , respectively. They coincide with each other pretty well (the deviation reason has been discussed). The increase of the *Tco i* with the increase of *L* can be explained simply according our theory. Since the image field is *Ei*(*t*) = <sup>1</sup> *<sup>U</sup>*<sup>0</sup> *<sup>e</sup>*−*iω*0*<sup>t</sup> <sup>θ</sup>max <sup>θ</sup>min Us*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*<sup>0</sup> *cos<sup>θ</sup>* )*dθ*, the large-angle paths (*θ* > *θmax* and *θ* < *θmin*)and their contribution to the temporal-coherence gain are missed in the short superlens.

−1 −0.5 0 0.5 1

The Group Velocity Picture of Metamaterial Systems 77

Fig. 13. (Color online) (a) The setup of the system. And the distribution of the electric field at different moments during the process. Parameters are chosen as *Ar* = 0.4, *A<sup>θ</sup>* = 1.6, *Az* = 0.4 (in the position where *ε<sup>z</sup>* < 1) or *Az* = 1.6 (where *ε<sup>z</sup>* > 1), and *γ* = 0.012*ω*0. (a) *t* = 2.28*T*. (b) *t* = 3.60*T*. (c) *t* = 4.92*T*. (d) *t* = 7.20*T*. (e) *t* = 9.00*T*. (f) Stable state. *T* is the period of the

for the cloaking effect around the goal frequency. So the dynamical study not only gives us whole physical picture of the cloaking, but also helps us to design more effective cloaks.

In this section, the dynamical process of the electromagnetic (EM) CS is investigated by finite-difference time-domain (FDTD) numerical experiments. In our simulation, the Lorentzian dispersion relations are introduced into the permittivity and the permeability models, then the *real* dynamical process can be simulated[76–78]. Based on numerical simulation, we can follow the details of the dynamical process, such as the time-dependent scattered field, the building-up process of the cloaking effect, and the final stable cloaking state. By tuning the dispersion parameters and observing their effects on the dynamical process and the scattered field, we can find the essential elements which dominate the process. Theoretical analysis of these essential elements can help us to have a deeper physical picture

The *setup* of the system is shown in Fig. 13(a), similar as the one in Ref. [69]. *R*<sup>1</sup> and *R*<sup>2</sup> = 2*R*<sup>1</sup> are the inner and the outer cylindrical radii of the CS, respectively. A perfect electric conductor (PEC) shell is pressed against the inner surface of the CS. The CS is surrounded by the free space with *ε*<sup>0</sup> = *μ*<sup>0</sup> = 1. From the left side, an incident plane wave with working frequency *ω*<sup>0</sup> is scattered by the CS, the total field and the scattered field can be recorded inside and outside *B*1 respectively by the numerical technique[79]. So the scattering cross-section *σ* can be calculated easily. Our study is focused on the E-polarized modes, for which only the permittivity and the permeability components *εz*, *μr*, and *μθ* are needed to be considered (For H-polarized modes, considering the corresponding components *μz*, *εr*, and *εθ*, we can obtain the same numerical results in the dynamical process.). All of them are supposed to have the form 1+ *Fj*(*r*) × *fj*(*ω*), where subscript *j* could be *z*, *r*, and *θ* for *εz*, *μr*, and *μθ*, respectively. The filling factors *Fj*(*r*) are only *r*-dependent, *ω<sup>p</sup>* is the plasma frequency which set to be a constant

*aj* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>ωγ*) are the Lorentzian dispersive functions, where

**(a)**

**B1**

**z** θ **r**

beyond the phenomena and to design more effective cloaks.

*p*/(*ω*<sup>2</sup>

incident EM wave.

*ω<sup>p</sup>* = 10*ω*0, and *fj*(*ω*) = *ω*<sup>2</sup>

**electric field (Ez)**

**(d) (e) (f)**

**(b) (c)**

Obviously, Eq.(11) is suitable not only for random quasi-monochromatic source, but also for all quasi-monochromatic fields, such as the slowly-varying Gaussian pulses and slowly switching-on process mentioned by [52]. Our theory can be easily extended to 3D systems too. And owing to the fact that what we find is from the radiating field, so the temporal-coherence gain is not the near-field property. Actually, the new mechanism of the temporal-coherence gain is not limited for the *n* � −1 superlens, also applicable to other superlenses, such as the photonic crystal superlens in [46, 49, 58]. But the specialities of *n* � −1 superlens, such as almost no frequency-filtering (no frequency loss) and no reflection (no energy loss), can be used to design novel optical/photonic coherence-gain devices.

In summary, for the first time we have numerically and theoretically studied the temporal coherence of the superlens image with the quasi-monochromatic source. Numerically, we observe that the temporal coherence of the image can be improved considerably even almost without reflection and filtering effects. Based on new physical picture, we construct a theory to calculate the image field and temporal-coherence function, which excellently agree with the FDTD results and strict Green's function results. The mechanism of the temporal coherence gain is theoretically explained by the different GRT of different paths. Although the evanescent wave is very weak in this study, the coherence of evanescent wave in NIM systems is a very interesting topic which will be discussed elsewhere [66]. Other related topics, such as the spatial coherence which is very essential for the image quality of the superlens, can also be studied through the similar methods. Although our study is within the confinement of classic optics, similar investigation can be extended to the quantum optics [62], and interesting results can be expected. Obviously, the temporal-coherence gain of superlens is another evidence that the NIM phenomena are consistent with the causality [49]. We suppose that the temporal-coherence gain phenomena could be observed in micro-wave experiments[4, 46]. Therefore, this study should have important consequences in the future studies of coherence in NIM systems. The no-reflection and no-frequency-filtering coherence gain of the superlens has some potential applications in the imaging, the coherent optical communication, and the signal processing.

#### **5. The physical picture and the essential elements of the dynamical process for dispersive cloaking structures**

Recently, the theory[67, 68] has been developed based on the geometry transformation to realize a cloaking structure (CS), in which objects become invisible from outside. Then a two-dimensional (2D) cylindrical CS[69] and a nonmagnetic optical CS[70, 71] are designed. More surprisingly, the experiment[72] demonstrates that such a 2D CS really works with a "reduced" design made of split-ring resonators. These pioneers' works are really attractive and open a new window to realize the invisibility of human dream. However, so far almost all theoretical[67–71, 73–75] studies of the CS are done in the frequency domain and the geometry transformation idea is supposed to work only for a single frequency, so that the effects of the dispersion have not been intensively studied. As pointed out in Ref. [68] and the quantitatively study in our recent work [7], the dispersion is *required* for the cloaking material to avoid the divergent group velocity. For the dispersive CS, new topics, such as the *dynamical process*, can be introduced in. Dynamical study is essential for the cloaking study since without it we can not answer the questions, such as how can the field gets to its stable state?, is there any strong scattering or oscillation in the process?, and how long is the process?, etc. More important, because the real radars generally are pulsive ones, the dynamical process is critical 20 Will-be-set-by-IN-TECH

Obviously, Eq.(11) is suitable not only for random quasi-monochromatic source, but also for all quasi-monochromatic fields, such as the slowly-varying Gaussian pulses and slowly switching-on process mentioned by [52]. Our theory can be easily extended to 3D systems too. And owing to the fact that what we find is from the radiating field, so the temporal-coherence gain is not the near-field property. Actually, the new mechanism of the temporal-coherence gain is not limited for the *n* � −1 superlens, also applicable to other superlenses, such as the photonic crystal superlens in [46, 49, 58]. But the specialities of *n* � −1 superlens, such as almost no frequency-filtering (no frequency loss) and no reflection (no energy loss), can be

In summary, for the first time we have numerically and theoretically studied the temporal coherence of the superlens image with the quasi-monochromatic source. Numerically, we observe that the temporal coherence of the image can be improved considerably even almost without reflection and filtering effects. Based on new physical picture, we construct a theory to calculate the image field and temporal-coherence function, which excellently agree with the FDTD results and strict Green's function results. The mechanism of the temporal coherence gain is theoretically explained by the different GRT of different paths. Although the evanescent wave is very weak in this study, the coherence of evanescent wave in NIM systems is a very interesting topic which will be discussed elsewhere [66]. Other related topics, such as the spatial coherence which is very essential for the image quality of the superlens, can also be studied through the similar methods. Although our study is within the confinement of classic optics, similar investigation can be extended to the quantum optics [62], and interesting results can be expected. Obviously, the temporal-coherence gain of superlens is another evidence that the NIM phenomena are consistent with the causality [49]. We suppose that the temporal-coherence gain phenomena could be observed in micro-wave experiments[4, 46]. Therefore, this study should have important consequences in the future studies of coherence in NIM systems. The no-reflection and no-frequency-filtering coherence gain of the superlens has some potential applications in the imaging, the coherent optical communication, and the

**5. The physical picture and the essential elements of the dynamical process for**

Recently, the theory[67, 68] has been developed based on the geometry transformation to realize a cloaking structure (CS), in which objects become invisible from outside. Then a two-dimensional (2D) cylindrical CS[69] and a nonmagnetic optical CS[70, 71] are designed. More surprisingly, the experiment[72] demonstrates that such a 2D CS really works with a "reduced" design made of split-ring resonators. These pioneers' works are really attractive and open a new window to realize the invisibility of human dream. However, so far almost all theoretical[67–71, 73–75] studies of the CS are done in the frequency domain and the geometry transformation idea is supposed to work only for a single frequency, so that the effects of the dispersion have not been intensively studied. As pointed out in Ref. [68] and the quantitatively study in our recent work [7], the dispersion is *required* for the cloaking material to avoid the divergent group velocity. For the dispersive CS, new topics, such as the *dynamical process*, can be introduced in. Dynamical study is essential for the cloaking study since without it we can not answer the questions, such as how can the field gets to its stable state?, is there any strong scattering or oscillation in the process?, and how long is the process?, etc. More important, because the real radars generally are pulsive ones, the dynamical process is critical

used to design novel optical/photonic coherence-gain devices.

signal processing.

**dispersive cloaking structures**

Fig. 13. (Color online) (a) The setup of the system. And the distribution of the electric field at different moments during the process. Parameters are chosen as *Ar* = 0.4, *A<sup>θ</sup>* = 1.6, *Az* = 0.4 (in the position where *ε<sup>z</sup>* < 1) or *Az* = 1.6 (where *ε<sup>z</sup>* > 1), and *γ* = 0.012*ω*0. (a) *t* = 2.28*T*. (b) *t* = 3.60*T*. (c) *t* = 4.92*T*. (d) *t* = 7.20*T*. (e) *t* = 9.00*T*. (f) Stable state. *T* is the period of the incident EM wave.

for the cloaking effect around the goal frequency. So the dynamical study not only gives us whole physical picture of the cloaking, but also helps us to design more effective cloaks.

In this section, the dynamical process of the electromagnetic (EM) CS is investigated by finite-difference time-domain (FDTD) numerical experiments. In our simulation, the Lorentzian dispersion relations are introduced into the permittivity and the permeability models, then the *real* dynamical process can be simulated[76–78]. Based on numerical simulation, we can follow the details of the dynamical process, such as the time-dependent scattered field, the building-up process of the cloaking effect, and the final stable cloaking state. By tuning the dispersion parameters and observing their effects on the dynamical process and the scattered field, we can find the essential elements which dominate the process. Theoretical analysis of these essential elements can help us to have a deeper physical picture beyond the phenomena and to design more effective cloaks.

The *setup* of the system is shown in Fig. 13(a), similar as the one in Ref. [69]. *R*<sup>1</sup> and *R*<sup>2</sup> = 2*R*<sup>1</sup> are the inner and the outer cylindrical radii of the CS, respectively. A perfect electric conductor (PEC) shell is pressed against the inner surface of the CS. The CS is surrounded by the free space with *ε*<sup>0</sup> = *μ*<sup>0</sup> = 1. From the left side, an incident plane wave with working frequency *ω*<sup>0</sup> is scattered by the CS, the total field and the scattered field can be recorded inside and outside *B*1 respectively by the numerical technique[79]. So the scattering cross-section *σ* can be calculated easily. Our study is focused on the E-polarized modes, for which only the permittivity and the permeability components *εz*, *μr*, and *μθ* are needed to be considered (For H-polarized modes, considering the corresponding components *μz*, *εr*, and *εθ*, we can obtain the same numerical results in the dynamical process.). All of them are supposed to have the form 1+ *Fj*(*r*) × *fj*(*ω*), where subscript *j* could be *z*, *r*, and *θ* for *εz*, *μr*, and *μθ*, respectively. The filling factors *Fj*(*r*) are only *r*-dependent, *ω<sup>p</sup>* is the plasma frequency which set to be a constant *ω<sup>p</sup>* = 10*ω*0, and *fj*(*ω*) = *ω*<sup>2</sup> *p*/(*ω*<sup>2</sup> *aj* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>ωγ*) are the Lorentzian dispersive functions, where

**(d)**

*inct*, (13)

*inct* is the averaged energy flow

**(a) (b)**

**(c)**

The Group Velocity Picture of Metamaterial Systems 79

Fig. 15. (Color online) Direction of Poynting vectors and the intensity front (shown by red dashed curves) at moments during the dynamical process. Parameters are chosen as that of

built up step by step, at last, the field gets to the stable state shown in Fig. 13(f). Because of the dispersion, there is an obvious time delay in the cloaking effect and the strong scattered

We introduce a time-dependent scattering cross-section *σ*(*t*) to quantitatively study the

where *<sup>t</sup>* <sup>=</sup> *<sup>n</sup>* <sup>×</sup> *<sup>T</sup>*, *<sup>n</sup>* <sup>=</sup> 0, 1, 2, ..., T is the period of the incident wave, ¯*Jscat*(*t*) is the

density of incident field. To observe the dispersive effect on *σ*(*t*) during the dynamical process, at the first step, we keep *Az* and *γ* constant and change *Ar* and *Aθ*, the results are shown in Fig. 14(a). From the *σ* versus *t* curves, we can find the general properties of the dynamical process. First, there is strong scattering in the dynamical process. At the beginning, *σ* increases rapidly when the wave gets to the CS, then reaches its maximum (at about ninth period). After that, *σ* starts to decay until it gets to the stable value (of the stable cloaking state). Second, unlike other systems, there is no oscillation in the process. This property will be discussed later. Third, the time length of dynamical process, called as "relaxation time" generally, can be tuned by the dispersion. From Fig. 14(a), we can see that the main dispersive effect is on the decaying process. From case 1 to case 5, *Ar* and *A<sup>θ</sup>* become closer to *one*, so that the dispersion is stronger. We find that the stronger the dispersion, the longer the relaxation time. For comparing with the cloaking cases, we also show the *σ*(*t*) of the naked PEC shell in the case 6. From the definition of *σ*, we know that the area covered by these curve in Fig. 14(a) is proportional to the total scattered energy in the dynamical process. So the CS with *the weaker dispersion will scatter less field* (better cloaking effect) in the dynamical process. But, such

*σ*(*t*) = ¯*Jscat*(*t*)/*S*¯

Fig. 13. (a) *t* = 4.92*T*. (b) *t* = 7.20*T*. (c) *t* = 9.00*T*. (d) Stable state.

one-period-average energy flow of scattered field, and *S*¯

field is observed.

dynamical process, which is defined as

Fig. 14. (Color online) The *σ* vs *t* curves. From (a) to (d), *Az* = 0.4 where *ε<sup>z</sup>* < 1 or *Az* = 1.6 where *ε<sup>z</sup>* > 1. (a) Keep *γ* = 0.012*ω*<sup>0</sup> unchanging, choose *Ar* and *A<sup>θ</sup>* as case 1: *Ar* = 0.4, *A<sup>θ</sup>* = 1.6, case 2: *Ar* = 0.5, *A<sup>θ</sup>* = 1.5, case 3: *Ar* = 0.6, *A<sup>θ</sup>* = 1.4, case 4: *Ar* = 0.7, *A<sup>θ</sup>* = 1.3, case 5: *Ar* = 0.8, *A<sup>θ</sup>* = 1.2. Case 6: only PEC shell without CS. (b) Keep *Ar* = 0.4, *A<sup>θ</sup>* = 1.6 unchanging, choose *γ*<sup>1</sup> = 0.012*ω*0, *γ*<sup>2</sup> = 0.024*ω*0, *γ*<sup>3</sup> = 0.048*ω*0, *γ*<sup>4</sup> = 0.096*ω*<sup>0</sup> and *γ*<sup>5</sup> = 0.192*ω*0. (c) Keep *A<sup>θ</sup>* = 1.6 and *γ* = 0.012*ω*<sup>0</sup> unvaried, and change *Ar*. (d) Keep *Ar* = 0.4 and *γ* = 0.012*ω*<sup>0</sup> unvaried, and change *Aθ*.

*γ* is the "resonance width" or called as "dissipation factor," *ωaj* are the resonant frequency of "atoms" (resonant units) in metamaterials.

For the study of the dispersive CS, we suppose that the real parts of the *εz*, *μr*, and *μθ* always satisfy the geometry transformation of Ref. [69] at *ω*0:

*Re*[*μr*(*r*, *<sup>ω</sup>*0)] = (*<sup>r</sup>* <sup>−</sup> *<sup>R</sup>*1)/*r*, *Re*[*μθ* (*r*, *<sup>ω</sup>*0)] = *<sup>r</sup>*/(*<sup>r</sup>* <sup>−</sup> *<sup>R</sup>*1), and *Re*[*εz*(*r*, *<sup>ω</sup>*0)] = *<sup>R</sup>*<sup>2</sup> <sup>2</sup>(*r* − *<sup>R</sup>*1)/[(*R*<sup>2</sup> <sup>−</sup> *<sup>R</sup>*1)2*r*]. Then the filling factors *Fj*(*r*) at different *<sup>r</sup>* can be obtained: *Fr*(*r*) = {*Re*[*μr*(*r*, *ω*0)] − 1}/*Re*[ *fr*(*ω*0)], *Fθ*(*r*) = {*Re*[*μθ*(*r*, *ω*0)] − 1}/*Re*[ *fθ*(*ω*0)], and *Fz*(*r*) = {*Re*[*εz*(*r*, *ω*0)] − 1}/*Re*[ *fz*(*ω*0)].

To investigate the dispersive effect on the dynamical process, we tune the dispersion parameters *ωaj* in our numerical experiments. We use the working frequency *ω*<sup>0</sup> as the frequency unit since it is the same for all cases in this section so the ratio *Aj* = *ωaj*/*ω*<sup>0</sup> represents *ωaj*. Obviously, for the Lorentzian dispersive relation, the dispersion is stronger when *ω*<sup>0</sup> and *ωaj* are closer to each other (the working frequency is near the resonant frequency), or in other words, when *Aj* approaches *one*. Since there are singular values of real part of *ε* and *μ*, in our numerical simulation we have done some approximations,[80] such as we set the maximum and the minimum for *ε* and *μ*. Although such approximations will affect the cloaking effect of stable state,[74] we find that the influence of these approximations on the dynamical process is very small and can be neglected.

First, we show an example of evolving electronic field during the dynamical process in Fig. 13 with concrete parameters of *Ar*, *Aθ*, *Az*, and *γ*. In Fig. 13(a), the plane wave arrives at the left side of the CS, and is ready to enter the CS. From Fig. 13(b)-13(e), the cloaking effect is 22 Will-be-set-by-IN-TECH

**(a)**

**(c)**

**case1 case2 case3 case4 case5 case6**

**Ar =0.4 Ar =0.5 Ar =0.6 Ar =0.7 Ar =0.8** γ **1** γ **2** γ **3** γ **4** γ **5**

**t/T**

Fig. 14. (Color online) The *σ* vs *t* curves. From (a) to (d), *Az* = 0.4 where *ε<sup>z</sup>* < 1 or *Az* = 1.6 where *ε<sup>z</sup>* > 1. (a) Keep *γ* = 0.012*ω*<sup>0</sup> unchanging, choose *Ar* and *A<sup>θ</sup>* as case 1: *Ar* = 0.4, *A<sup>θ</sup>* = 1.6, case 2: *Ar* = 0.5, *A<sup>θ</sup>* = 1.5, case 3: *Ar* = 0.6, *A<sup>θ</sup>* = 1.4, case 4: *Ar* = 0.7, *A<sup>θ</sup>* = 1.3, case 5: *Ar* = 0.8, *A<sup>θ</sup>* = 1.2. Case 6: only PEC shell without CS. (b) Keep *Ar* = 0.4, *A<sup>θ</sup>* = 1.6

*γ* is the "resonance width" or called as "dissipation factor," *ωaj* are the resonant frequency of

For the study of the dispersive CS, we suppose that the real parts of the *εz*, *μr*, and *μθ* always

*<sup>R</sup>*1)/[(*R*<sup>2</sup> <sup>−</sup> *<sup>R</sup>*1)2*r*]. Then the filling factors *Fj*(*r*) at different *<sup>r</sup>* can be obtained: *Fr*(*r*) = {*Re*[*μr*(*r*, *ω*0)] − 1}/*Re*[ *fr*(*ω*0)], *Fθ*(*r*) = {*Re*[*μθ*(*r*, *ω*0)] − 1}/*Re*[ *fθ*(*ω*0)], and *Fz*(*r*) =

To investigate the dispersive effect on the dynamical process, we tune the dispersion parameters *ωaj* in our numerical experiments. We use the working frequency *ω*<sup>0</sup> as the frequency unit since it is the same for all cases in this section so the ratio *Aj* = *ωaj*/*ω*<sup>0</sup> represents *ωaj*. Obviously, for the Lorentzian dispersive relation, the dispersion is stronger when *ω*<sup>0</sup> and *ωaj* are closer to each other (the working frequency is near the resonant frequency), or in other words, when *Aj* approaches *one*. Since there are singular values of real part of *ε* and *μ*, in our numerical simulation we have done some approximations,[80] such as we set the maximum and the minimum for *ε* and *μ*. Although such approximations will affect the cloaking effect of stable state,[74] we find that the influence of these approximations on

First, we show an example of evolving electronic field during the dynamical process in Fig. 13 with concrete parameters of *Ar*, *Aθ*, *Az*, and *γ*. In Fig. 13(a), the plane wave arrives at the left side of the CS, and is ready to enter the CS. From Fig. 13(b)-13(e), the cloaking effect is

*Re*[*μr*(*r*, *<sup>ω</sup>*0)] = (*<sup>r</sup>* <sup>−</sup> *<sup>R</sup>*1)/*r*, *Re*[*μθ* (*r*, *<sup>ω</sup>*0)] = *<sup>r</sup>*/(*<sup>r</sup>* <sup>−</sup> *<sup>R</sup>*1), and *Re*[*εz*(*r*, *<sup>ω</sup>*0)] = *<sup>R</sup>*<sup>2</sup>

unchanging, choose *γ*<sup>1</sup> = 0.012*ω*0, *γ*<sup>2</sup> = 0.024*ω*0, *γ*<sup>3</sup> = 0.048*ω*0, *γ*<sup>4</sup> = 0.096*ω*<sup>0</sup> and *γ*<sup>5</sup> = 0.192*ω*0. (c) Keep *A<sup>θ</sup>* = 1.6 and *γ* = 0.012*ω*<sup>0</sup> unvaried, and change *Ar*. (d) Keep

**0 5 10 15 20 25 30**

**A**θ **=1.6 A**θ **=1.5 A**θ **=1.4 A**θ **=1.3 A**θ **=1.2**

**(b)**

**(d)**

<sup>2</sup>(*r* −

**0 0.5 1.0 1.5 2.0 2.5 3.0 3.5**

**0 0.5 1.0 1.5 2.0 2.5 3.0**

**0 5 10 15 20 25**

*Ar* = 0.4 and *γ* = 0.012*ω*<sup>0</sup> unvaried, and change *Aθ*.

satisfy the geometry transformation of Ref. [69] at *ω*0:

the dynamical process is very small and can be neglected.

"atoms" (resonant units) in metamaterials.

{*Re*[*εz*(*r*, *ω*0)] − 1}/*Re*[ *fz*(*ω*0)].

σ**/**λ

Fig. 15. (Color online) Direction of Poynting vectors and the intensity front (shown by red dashed curves) at moments during the dynamical process. Parameters are chosen as that of Fig. 13. (a) *t* = 4.92*T*. (b) *t* = 7.20*T*. (c) *t* = 9.00*T*. (d) Stable state.

built up step by step, at last, the field gets to the stable state shown in Fig. 13(f). Because of the dispersion, there is an obvious time delay in the cloaking effect and the strong scattered field is observed.

We introduce a time-dependent scattering cross-section *σ*(*t*) to quantitatively study the dynamical process, which is defined as

$$
\sigma(t) = \vec{f}\_{\text{scat}}(t) / \vec{S}\_{\text{inct}\prime} \tag{13}
$$

where *<sup>t</sup>* <sup>=</sup> *<sup>n</sup>* <sup>×</sup> *<sup>T</sup>*, *<sup>n</sup>* <sup>=</sup> 0, 1, 2, ..., T is the period of the incident wave, ¯*Jscat*(*t*) is the one-period-average energy flow of scattered field, and *S*¯ *inct* is the averaged energy flow density of incident field. To observe the dispersive effect on *σ*(*t*) during the dynamical process, at the first step, we keep *Az* and *γ* constant and change *Ar* and *Aθ*, the results are shown in Fig. 14(a). From the *σ* versus *t* curves, we can find the general properties of the dynamical process. First, there is strong scattering in the dynamical process. At the beginning, *σ* increases rapidly when the wave gets to the CS, then reaches its maximum (at about ninth period). After that, *σ* starts to decay until it gets to the stable value (of the stable cloaking state). Second, unlike other systems, there is no oscillation in the process. This property will be discussed later. Third, the time length of dynamical process, called as "relaxation time" generally, can be tuned by the dispersion. From Fig. 14(a), we can see that the main dispersive effect is on the decaying process. From case 1 to case 5, *Ar* and *A<sup>θ</sup>* become closer to *one*, so that the dispersion is stronger. We find that the stronger the dispersion, the longer the relaxation time. For comparing with the cloaking cases, we also show the *σ*(*t*) of the naked PEC shell in the case 6. From the definition of *σ*, we know that the area covered by these curve in Fig. 14(a) is proportional to the total scattered energy in the dynamical process. So the CS with *the weaker dispersion will scatter less field* (better cloaking effect) in the dynamical process. But, such

( √ 2*c ε <sup>z</sup>μ<sup>r</sup>* )/(2 + *<sup>ω</sup> ε z dε <sup>z</sup> <sup>d</sup><sup>ω</sup>* <sup>+</sup> *<sup>ω</sup> μr dμ<sup>r</sup>*

the velocity of light in vacuum.

how to tune *Vg<sup>θ</sup>* by modifying dispersion parameters.

of the CS, there should be an optimized trade-off.

is performed to validate our analysis.

For the anisotropic cloaking material, the *Vg<sup>θ</sup>* and *Vgr* can be expressed as: *Vg<sup>θ</sup>* = [∇**k***ω*(**k**)]*<sup>θ</sup>* =

The Group Velocity Picture of Metamaterial Systems 81

In order to illustrate our prediction, the *σ*(*t*) under different *Vg<sup>θ</sup>* and *Vgr* are investigated, respectively. First, we keep the *Vgr* unvaried by holding *Aθ*, *Az* and *γ* constant [keep *dεz*/*dω* and *dμθ*/*dω* unchanged], only modify *Ar* to change the *Vgθ*. The results are shown in Fig.14(c), when *Ar* is closer to *one*, the *Vg<sup>θ</sup>* becomes smaller (with larger *dμr*/*dω*), the relaxation time is longer and more energy scattered in the dynamical process. So the larger *Vg<sup>θ</sup>* means the better cloaking effect in the dynamical process. On the other hand, when we keep the *Vg<sup>θ</sup>* unvaried and change *Vgr* by holding *Ar*, *Az*, and *γ* constant and modifying *Aθ*, the results are shown in Fig. 14(d). We find that the relaxation time is almost unchanged with the change of *Vgr*. Obviously, *Vg<sup>θ</sup>* is the dominant element in the dynamical process. This conclusion can help us to design a better CS to defend the pulsive radars. In the expression of *Vg<sup>θ</sup>* , it is also shown

It seems that the larger *Vgθ*, the better cloaking effect in the dynamical process. However, since the *Vg* (and its components) cannot exceed *c* generally, there is a minimum limit for the relaxation time of the cloaking dynamical process. We can estimate it through dividing the mean length of the propagation rays by *Vg<sup>θ</sup>* . In our model, the mean length is *π*(*R*<sup>2</sup> + *R*1)/2, about three wavelengths. So the relaxation time can not be shorter than three periods. Figure 14 shows that our estimation is coincident with our simulation results. Actually, here we are facing a very basic conflict to make a "better" CS, which is more discussed in our other works.[7, 81] The conflict is from the fact that the *pretty strong* dispersion is required to realize a good *stable* cloaking effect at a certain frequency,[7, 68] but at this research we show that the *weaker* dispersion can realize a better cloaking effect in the *dynamical* process. At real design

Based on causality, the limitation of the electromagnetic cloak with dispersive material is investigated in this section The results show that perfect invisibility can not be achieved because of the dilemma that either the group velocity *Vg* diverges or a strong absorption is imposed on the cloaking material. It is an intrinsic conflict which originates from the demand of causality. However, the total cross section can really be reduced through the approach of coordinate transformation. A simulation of finite-difference time-domain method

Through the ages, people have dreamed to have a magic cloak whose owner can not be seen by others. For this fantastic dream, plenty of work has been done by scientists all over the world. For example, the researchers diminished the scattering or the reflection from objects by absorbing screens[82] and small, non-absorbing, compound ellipsoids[83]. More recently, based on the coordinate transformation, J. B. Pendry, et al theoretically proposed a general recipe for designing an electromagnetic cloak to hide an object from the electromagnetic(EM) wave[68]. An arbitrary object may be hidden because it remains untouched by external radiation. Meanwhile, Ulf Leonhardt described a similar method where the Helmholtz equation is transformed to produce similar effects in the geometric limit[67, 75]. Soon, Steven A. Cummer, et al simulated numerically(COMSOL) the cylindrical version of this cloak structure using ideal and nonideal (but physically realizable) electromagnetic

**6. Limitation of the electromagnetic cloak with dispersive material**

2*c ε <sup>z</sup>μθ* )/(2 + *<sup>ω</sup> ε z dε <sup>z</sup> <sup>d</sup><sup>ω</sup>* <sup>+</sup> *<sup>ω</sup> μθ dμθ*

*<sup>d</sup><sup>ω</sup>* ), where *c* is

*<sup>d</sup><sup>ω</sup>* ) and *Vgr* = [∇**k***ω*(**k**)]*<sup>r</sup>* = ( <sup>√</sup>

a general conclusion is still not enough for us to get a clear physical picture to understand the cloaking dynamical process.

Next, we check whether the absorption of the CS is important in the process. The absorption is determined by the imaginary part of *ε* and *μ*. To study this effect, we hold *Ar*, *Aθ*, and *Az* constant but modify the dispassion factor *γ*. We modify the filling factors *Fj* simultaneously, so that the real parts of *ε* and *μ* are kept unchanged at *ω*0. In such way, we can keep the dispersion strength almost unchanged, but with the imaginary parts of *ε* and *μ* changed. Results in Fig. 14(b) show that the stronger absorption only leads to larger stable value of *σ*, leaving the relaxation time nearly unchanged. Thus, we can exclude the absorption from the relevant parameter list, since it only influences the *σ*(*t*) of stable state considerably.

To obtain deeper insight of the dynamical process, we need to study the dynamical process more carefully. From Figs. 13(b)-13(e), we can see that the "field intensity" (shown by different color in the figures) propagates slower inside the CS than that in the outside vacuum. And when the inside field intensity "catches up" the outside one [in Fig. 13(f)], the field in the CS gets to the stable state and the cloaking effect is built up. In fact, this catching-up process of the field intensity can be shown more clearly by the direction of Poynting vectors during the dynamical process. From Figs. 15(a)-15(d), we show the direction of Poynting vectors in moments of Figs. 13(c)-13(f), respectively. In the Fig. 15, we see that there is the "intensity front"(shown by red dashed curve) which separates two regions of the CS. At the right side of the front, the field intensity in the CS is much weaker than the outside and the Poynting vector directions are not regular(especially near the front). But at the left-side region which is swept by the intensity front, the Poynting vectors are very regular and nearly along the "cloaking rays" which was predicted at the coordinate transformation[68]. Since the cloaking effect can be interpreted by the mimic picture that the light runs around the cloaking area through these curved cloaking rays, it is not surprising to find that the stable cloaking state is achieved when the intensity front sweeps through the whole CS and these optical rays are well constructed. The surprising thing is that *the stable cloaking state seems to be constructed locally.* We believe this property is related the original cloaking recipe,[68] which makes the cloaking material is almost impedance matched layer by layer. This also explains why there is no oscillation in the cloaking dynamical process generally. This picture also can interpret the strong scattered field in the dynamical process, since these "irregular rays" at the right-side region of the intensity front must be scattered strongly. Further, we can use this picture to analysis the dynamical process of other incident waves, such as the Gaussian beams, which are composed of different plane-wave components.

With these understanding, now we are ready to find the correlation between the relaxation time and the CS dispersion. It is well known that the field intensity (or energy) propagates at the group velocity *Vg*, which is controlled by the material dispersion. So the intensity front, which determines the dynamical process, should move in *Vg*. Thus, we can explain the results in Fig. 14, since our modification of the dispersive parameters can cause the *Vg* changed. But, because the cloaking material is the strong anisotropic material, the *Vg* at different directions could be very different. Can we predict more precisely which component dominates the relaxation time? The answer is "yes." In Fig. 15(d), we can see that the stable energy flow in the CS is nearly along the *θ* direction at most regions of the CS. Then it is reasonable for us to argue that it is the component along the *θ* direction *Vgθ*, not the component along the *r* direction *Vgr*, that dominates the relaxation time and the total scattered energy in the dynamical process.

24 Will-be-set-by-IN-TECH

a general conclusion is still not enough for us to get a clear physical picture to understand the

Next, we check whether the absorption of the CS is important in the process. The absorption is determined by the imaginary part of *ε* and *μ*. To study this effect, we hold *Ar*, *Aθ*, and *Az* constant but modify the dispassion factor *γ*. We modify the filling factors *Fj* simultaneously, so that the real parts of *ε* and *μ* are kept unchanged at *ω*0. In such way, we can keep the dispersion strength almost unchanged, but with the imaginary parts of *ε* and *μ* changed. Results in Fig. 14(b) show that the stronger absorption only leads to larger stable value of *σ*, leaving the relaxation time nearly unchanged. Thus, we can exclude the absorption from the relevant

To obtain deeper insight of the dynamical process, we need to study the dynamical process more carefully. From Figs. 13(b)-13(e), we can see that the "field intensity" (shown by different color in the figures) propagates slower inside the CS than that in the outside vacuum. And when the inside field intensity "catches up" the outside one [in Fig. 13(f)], the field in the CS gets to the stable state and the cloaking effect is built up. In fact, this catching-up process of the field intensity can be shown more clearly by the direction of Poynting vectors during the dynamical process. From Figs. 15(a)-15(d), we show the direction of Poynting vectors in moments of Figs. 13(c)-13(f), respectively. In the Fig. 15, we see that there is the "intensity front"(shown by red dashed curve) which separates two regions of the CS. At the right side of the front, the field intensity in the CS is much weaker than the outside and the Poynting vector directions are not regular(especially near the front). But at the left-side region which is swept by the intensity front, the Poynting vectors are very regular and nearly along the "cloaking rays" which was predicted at the coordinate transformation[68]. Since the cloaking effect can be interpreted by the mimic picture that the light runs around the cloaking area through these curved cloaking rays, it is not surprising to find that the stable cloaking state is achieved when the intensity front sweeps through the whole CS and these optical rays are well constructed. The surprising thing is that *the stable cloaking state seems to be constructed locally.* We believe this property is related the original cloaking recipe,[68] which makes the cloaking material is almost impedance matched layer by layer. This also explains why there is no oscillation in the cloaking dynamical process generally. This picture also can interpret the strong scattered field in the dynamical process, since these "irregular rays" at the right-side region of the intensity front must be scattered strongly. Further, we can use this picture to analysis the dynamical process of other incident waves, such as the Gaussian beams, which are composed of different

With these understanding, now we are ready to find the correlation between the relaxation time and the CS dispersion. It is well known that the field intensity (or energy) propagates at the group velocity *Vg*, which is controlled by the material dispersion. So the intensity front, which determines the dynamical process, should move in *Vg*. Thus, we can explain the results in Fig. 14, since our modification of the dispersive parameters can cause the *Vg* changed. But, because the cloaking material is the strong anisotropic material, the *Vg* at different directions could be very different. Can we predict more precisely which component dominates the relaxation time? The answer is "yes." In Fig. 15(d), we can see that the stable energy flow in the CS is nearly along the *θ* direction at most regions of the CS. Then it is reasonable for us to argue that it is the component along the *θ* direction *Vgθ*, not the component along the *r* direction *Vgr*, that dominates the relaxation time and the total scattered energy in the dynamical process.

parameter list, since it only influences the *σ*(*t*) of stable state considerably.

cloaking dynamical process.

plane-wave components.

For the anisotropic cloaking material, the *Vg<sup>θ</sup>* and *Vgr* can be expressed as: *Vg<sup>θ</sup>* = [∇**k***ω*(**k**)]*<sup>θ</sup>* = ( √ 2*c ε <sup>z</sup>μ<sup>r</sup>* )/(2 + *<sup>ω</sup> ε z dε <sup>z</sup> <sup>d</sup><sup>ω</sup>* <sup>+</sup> *<sup>ω</sup> μr dμ<sup>r</sup> <sup>d</sup><sup>ω</sup>* ) and *Vgr* = [∇**k***ω*(**k**)]*<sup>r</sup>* = ( <sup>√</sup> 2*c ε <sup>z</sup>μθ* )/(2 + *<sup>ω</sup> ε z dε <sup>z</sup> <sup>d</sup><sup>ω</sup>* <sup>+</sup> *<sup>ω</sup> μθ dμθ <sup>d</sup><sup>ω</sup>* ), where *c* is the velocity of light in vacuum.

In order to illustrate our prediction, the *σ*(*t*) under different *Vg<sup>θ</sup>* and *Vgr* are investigated, respectively. First, we keep the *Vgr* unvaried by holding *Aθ*, *Az* and *γ* constant [keep *dεz*/*dω* and *dμθ*/*dω* unchanged], only modify *Ar* to change the *Vgθ*. The results are shown in Fig.14(c), when *Ar* is closer to *one*, the *Vg<sup>θ</sup>* becomes smaller (with larger *dμr*/*dω*), the relaxation time is longer and more energy scattered in the dynamical process. So the larger *Vg<sup>θ</sup>* means the better cloaking effect in the dynamical process. On the other hand, when we keep the *Vg<sup>θ</sup>* unvaried and change *Vgr* by holding *Ar*, *Az*, and *γ* constant and modifying *Aθ*, the results are shown in Fig. 14(d). We find that the relaxation time is almost unchanged with the change of *Vgr*. Obviously, *Vg<sup>θ</sup>* is the dominant element in the dynamical process. This conclusion can help us to design a better CS to defend the pulsive radars. In the expression of *Vg<sup>θ</sup>* , it is also shown how to tune *Vg<sup>θ</sup>* by modifying dispersion parameters.

It seems that the larger *Vgθ*, the better cloaking effect in the dynamical process. However, since the *Vg* (and its components) cannot exceed *c* generally, there is a minimum limit for the relaxation time of the cloaking dynamical process. We can estimate it through dividing the mean length of the propagation rays by *Vg<sup>θ</sup>* . In our model, the mean length is *π*(*R*<sup>2</sup> + *R*1)/2, about three wavelengths. So the relaxation time can not be shorter than three periods. Figure 14 shows that our estimation is coincident with our simulation results. Actually, here we are facing a very basic conflict to make a "better" CS, which is more discussed in our other works.[7, 81] The conflict is from the fact that the *pretty strong* dispersion is required to realize a good *stable* cloaking effect at a certain frequency,[7, 68] but at this research we show that the *weaker* dispersion can realize a better cloaking effect in the *dynamical* process. At real design of the CS, there should be an optimized trade-off.

Based on causality, the limitation of the electromagnetic cloak with dispersive material is investigated in this section The results show that perfect invisibility can not be achieved because of the dilemma that either the group velocity *Vg* diverges or a strong absorption is imposed on the cloaking material. It is an intrinsic conflict which originates from the demand of causality. However, the total cross section can really be reduced through the approach of coordinate transformation. A simulation of finite-difference time-domain method is performed to validate our analysis.

#### **6. Limitation of the electromagnetic cloak with dispersive material**

Through the ages, people have dreamed to have a magic cloak whose owner can not be seen by others. For this fantastic dream, plenty of work has been done by scientists all over the world. For example, the researchers diminished the scattering or the reflection from objects by absorbing screens[82] and small, non-absorbing, compound ellipsoids[83]. More recently, based on the coordinate transformation, J. B. Pendry, et al theoretically proposed a general recipe for designing an electromagnetic cloak to hide an object from the electromagnetic(EM) wave[68]. An arbitrary object may be hidden because it remains untouched by external radiation. Meanwhile, Ulf Leonhardt described a similar method where the Helmholtz equation is transformed to produce similar effects in the geometric limit[67, 75]. Soon, Steven A. Cummer, et al simulated numerically(COMSOL) the cylindrical version of this cloak structure using ideal and nonideal (but physically realizable) electromagnetic

Fig. 17. The relation between *R*1/*r* and *α* when *Vg* = *c* for different *ω*

the dispersion relation of the anisotropic material[87] as: *k*<sup>2</sup>

(*cosα*)<sup>2</sup> *n*2 *r*�

*<sup>d</sup><sup>ω</sup>* and *mt*� = *nt*� + *ω*

We will discuss Eq.15 in two cases. The first case is with the finite *dnt*�

tangential component of *Vg*) will diverge at both peaks around *α* = 0.

<sup>+</sup> (*sinα*)<sup>2</sup> *n*2 *t*�

*dnt*� *<sup>d</sup><sup>ω</sup>* . If the transformation has the following characteristic: *f*(*r* = 0) = *R*1, *f*(*r* = *R*2) = *R*2, then when *r*� → *R*<sup>1</sup> (or *r* → 0), *nt*� will tend to zero, and the group velocity is approximated as:

*Vg* <sup>≈</sup> *<sup>c</sup>*


example, in which the transformation is *r*� = *f*(*r*)=(*R*<sup>2</sup> − *R*1)*r*/*R*<sup>2</sup> + *R*<sup>1</sup> as Ref. [68], *R*<sup>2</sup> = 2*R*1, thus *nr*� = 2 and *nt*� = 2 − 4/(*r*/*R*<sup>1</sup> + 2). The dispersion parameters are set as *mr*� = 2.5,

*<sup>d</sup><sup>ω</sup>* = 4 at working frequency. In figure 16, the curves of *Vg* vs *α* are plotted for different *R*1/*r* values. We can see that, for large *R*1/*r*(*r* → 0), the group velocity (more precisely, the

*<sup>ϕ</sup>*� , *nr*� <sup>=</sup> <sup>√</sup>*εϕ*�*μθ*� <sup>=</sup> *ni*/ *d f*(*r*)

*k*2 *<sup>t</sup>*� <sup>=</sup> *<sup>k</sup>*<sup>2</sup>

*ω dnt*� *<sup>θ</sup>*� <sup>+</sup> *<sup>k</sup>*<sup>2</sup>

Where *mr*� = *nr*� + *ω*

obtained as:

*ni* <sup>=</sup> <sup>√</sup>*εiμi*. Then we can define *kr*� <sup>=</sup> *<sup>ω</sup>*

*Vg* = *c*

*dnr*�

diverge when *sin<sup>α</sup>* <sup>→</sup> 0 for any finite *dnt*�

is very important for us to have consistent physical pictures in both spaces. At working frequency *ω*0, for a propagating mode with k-vector as {*kr*� ,*kθ*� ,*kϕ*�} inside the cloak, we have

The Group Velocity Picture of Metamaterial Systems 83

*<sup>c</sup> nr*� *cos<sup>α</sup>* and *kt*� <sup>=</sup> *<sup>ω</sup>*

/((*cosα*)<sup>2</sup> *mr*�

*dnt*� *dω*

*nr*�

*dnt*� *<sup>d</sup><sup>ω</sup>* .

*<sup>c</sup> nt*�*sinα*, the group velocity can be

*<sup>t</sup>*� <sup>=</sup> *<sup>ω</sup>*2/*c*2, where

) (14)

*<sup>d</sup><sup>ω</sup>* . Obviously, *Vg* will

(15)

*<sup>r</sup>*�/*n*<sup>2</sup> *<sup>r</sup>*� <sup>+</sup> *<sup>k</sup>*<sup>2</sup> *<sup>t</sup>*�/*n*<sup>2</sup>

*dr* , *nt*� <sup>=</sup> *<sup>n</sup>θ*� <sup>=</sup> *<sup>n</sup>ϕ*� <sup>=</sup> <sup>√</sup>*εr*�*μθ*� <sup>=</sup> *nir*/ *<sup>f</sup>*(*r*), and

+ (*sinα*)<sup>2</sup> *mt*�

*<sup>d</sup><sup>ω</sup>* . Such divergence is shown in figure 16 for a concrete

*nt*�

Fig. 16. The group velocity *Vg* versus *α* for different *R*1/*r* values

parameters[69]. Especially, Schurig, et al experimentally demonstrated such a cloak by split-ring resonators[72]. In addition, Wenshan Cai , et al proposed an electromagnetic cloak using high-order transformation to create smooth moduli at the outer interface and presented a design of a non-magnetic cloak operating at optical frequencies[70, 71]. According to the general recipe, the electromagnetic cloak is supposed to be perfect or "fully functioned" at certain frequency as long as we can get very close to the ideal design although there is a singularity in the distribution, which has been elucidated further in several literatures [84, 85]. However, in all these pioneering works, the interests are mainly focused on single-frequency EM waves, so that the effects of the dispersion, which is related with very basic physical laws, are not well studied. If the dispersion is introduced into the study, can we have a deeper insight into the cloaking physics?

In this section we will show the ideal cloaking can not be achieved because of another more basic physical limitation—the causality limitation (based on the same limitation, Chen et al obtained a constraint of the band width that limit the design of an invisibility cloak[86]). Starting from dispersion relation and combining with the demand of causality, we will demonstrate that the ideal cloaking will lead to the dilemma that either the group velocity *Vg* diverges or a strong absorption is imposed on the cloaking material. Our derivation and numerical experiments based on the finite-difference-time-domain(FDTD) methods will show that the absorption cross section will be pretty large and dominate the total cross section for a dispersive cloak, even with very small imaginary parts of permittivity and permeability.

Let's consider a more general coordinate transformation on an initial homogeneous medium with *ε<sup>i</sup>* = *μ<sup>i</sup>* in r space: *r* = *f*(*r*), *θ* = *θ*, *ϕ* = *ϕ*, following the approach in Ref. [87] and [89], we get the following radius-dependent, anisotropic relative permittivity and permeability: *ε<sup>r</sup>* = *μ<sup>r</sup>* = *εi*( *<sup>r</sup> f*(*r*))<sup>2</sup> *d f*(*r*) *dr* , *εθ* <sup>=</sup> *μθ* <sup>=</sup> *<sup>ε</sup>i*/ *d f*(*r*) *dr* and *εϕ* <sup>=</sup> *μϕ* <sup>=</sup> *<sup>ε</sup>i*/ *d f*(*r*) *dr* . We emphasize that since the transformation is directly acted on the Maxwell equations, the above equations are also suited for the imaginary parts of constitutive parameters, and all physical properties of wave propagation in *r* space should be inherited in *r* space, such as the absorption. This 26 Will-be-set-by-IN-TECH

R1 /r=0.5

R1 /r=1.5

R1 /r=3.5

R1 /r=7.5

R1 /r=15.5

R1 /r=18.5

R1 /r=31.5

R1 /r=63.5

−1.5 −1 −0.5 0 0.5 1 1.5

parameters[69]. Especially, Schurig, et al experimentally demonstrated such a cloak by split-ring resonators[72]. In addition, Wenshan Cai , et al proposed an electromagnetic cloak using high-order transformation to create smooth moduli at the outer interface and presented a design of a non-magnetic cloak operating at optical frequencies[70, 71]. According to the general recipe, the electromagnetic cloak is supposed to be perfect or "fully functioned" at certain frequency as long as we can get very close to the ideal design although there is a singularity in the distribution, which has been elucidated further in several literatures [84, 85]. However, in all these pioneering works, the interests are mainly focused on single-frequency EM waves, so that the effects of the dispersion, which is related with very basic physical laws, are not well studied. If the dispersion is introduced into the study, can we have a deeper

In this section we will show the ideal cloaking can not be achieved because of another more basic physical limitation—the causality limitation (based on the same limitation, Chen et al obtained a constraint of the band width that limit the design of an invisibility cloak[86]). Starting from dispersion relation and combining with the demand of causality, we will demonstrate that the ideal cloaking will lead to the dilemma that either the group velocity *Vg* diverges or a strong absorption is imposed on the cloaking material. Our derivation and numerical experiments based on the finite-difference-time-domain(FDTD) methods will show that the absorption cross section will be pretty large and dominate the total cross section for a dispersive cloak, even with very small imaginary parts of permittivity and permeability.

Let's consider a more general coordinate transformation on an initial homogeneous medium with *ε<sup>i</sup>* = *μ<sup>i</sup>* in r space: *r* = *f*(*r*), *θ* = *θ*, *ϕ* = *ϕ*, following the approach in Ref. [87] and [89], we get the following radius-dependent, anisotropic relative permittivity and permeability:

since the transformation is directly acted on the Maxwell equations, the above equations are also suited for the imaginary parts of constitutive parameters, and all physical properties of wave propagation in *r* space should be inherited in *r* space, such as the absorption. This

*dr* and *εϕ* <sup>=</sup> *μϕ* <sup>=</sup> *<sup>ε</sup>i*/ *d f*(*r*)

*dr* . We emphasize that

α/rad

0.2

insight into the cloaking physics?

*f*(*r*))<sup>2</sup> *d f*(*r*)

*dr* , *εθ* <sup>=</sup> *μθ* <sup>=</sup> *<sup>ε</sup>i*/ *d f*(*r*)

*ε<sup>r</sup>* = *μ<sup>r</sup>* = *εi*( *<sup>r</sup>*

Fig. 16. The group velocity *Vg* versus *α* for different *R*1/*r* values

0.6

1

Vg/c

1.4

Fig. 17. The relation between *R*1/*r* and *α* when *Vg* = *c* for different *ω dnt*� *<sup>d</sup><sup>ω</sup>* .

is very important for us to have consistent physical pictures in both spaces. At working frequency *ω*0, for a propagating mode with k-vector as {*kr*� ,*kθ*� ,*kϕ*�} inside the cloak, we have the dispersion relation of the anisotropic material[87] as: *k*<sup>2</sup> *<sup>r</sup>*�/*n*<sup>2</sup> *<sup>r</sup>*� <sup>+</sup> *<sup>k</sup>*<sup>2</sup> *<sup>t</sup>*�/*n*<sup>2</sup> *<sup>t</sup>*� <sup>=</sup> *<sup>ω</sup>*2/*c*2, where *k*2 *<sup>t</sup>*� <sup>=</sup> *<sup>k</sup>*<sup>2</sup> *<sup>θ</sup>*� <sup>+</sup> *<sup>k</sup>*<sup>2</sup> *<sup>ϕ</sup>*� , *nr*� <sup>=</sup> <sup>√</sup>*εϕ*�*μθ*� <sup>=</sup> *ni*/ *d f*(*r*) *dr* , *nt*� <sup>=</sup> *<sup>n</sup>θ*� <sup>=</sup> *<sup>n</sup>ϕ*� <sup>=</sup> <sup>√</sup>*εr*�*μθ*� <sup>=</sup> *nir*/ *<sup>f</sup>*(*r*), and *ni* <sup>=</sup> <sup>√</sup>*εiμi*. Then we can define *kr*� <sup>=</sup> *<sup>ω</sup> <sup>c</sup> nr*� *cos<sup>α</sup>* and *kt*� <sup>=</sup> *<sup>ω</sup> <sup>c</sup> nt*�*sinα*, the group velocity can be obtained as:

$$V\_{\mathcal{S}} = c\sqrt{\frac{(\cos a)^2}{n\_{r'}^2} + \frac{(\sin a)^2}{n\_{l'}^2}}/((\cos a)^2 \frac{m\_{r'}}{n\_{r'}} + (\sin a)^2 \frac{m\_{l'}}{n\_{l'}})\tag{14}$$

Where *mr*� = *nr*� + *ω dnr*� *<sup>d</sup><sup>ω</sup>* and *mt*� = *nt*� + *ω dnt*� *<sup>d</sup><sup>ω</sup>* .

If the transformation has the following characteristic: *f*(*r* = 0) = *R*1, *f*(*r* = *R*2) = *R*2, then when *r*� → *R*<sup>1</sup> (or *r* → 0), *nt*� will tend to zero, and the group velocity is approximated as:

$$V\_{\mathcal{S}} \approx \frac{c}{|\sin \mathfrak{a}| \omega \frac{d n\_{\ell'}}{d \omega}} \tag{15}$$

We will discuss Eq.15 in two cases. The first case is with the finite *dnt*� *<sup>d</sup><sup>ω</sup>* . Obviously, *Vg* will diverge when *sin<sup>α</sup>* <sup>→</sup> 0 for any finite *dnt*� *<sup>d</sup><sup>ω</sup>* . Such divergence is shown in figure 16 for a concrete example, in which the transformation is *r*� = *f*(*r*)=(*R*<sup>2</sup> − *R*1)*r*/*R*<sup>2</sup> + *R*<sup>1</sup> as Ref. [68], *R*<sup>2</sup> = 2*R*1, thus *nr*� = 2 and *nt*� = 2 − 4/(*r*/*R*<sup>1</sup> + 2). The dispersion parameters are set as *mr*� = 2.5, *ω dnt*� *<sup>d</sup><sup>ω</sup>* = 4 at working frequency. In figure 16, the curves of *Vg* vs *α* are plotted for different *R*1/*r* values. We can see that, for large *R*1/*r*(*r* → 0), the group velocity (more precisely, the tangential component of *Vg*) will diverge at both peaks around *α* = 0.

(a) (b)

structure with a PEC at radius *R*1, (b) the naked PEC with radius *R*1.

needed to avoid *Vg* > *c*.

*�*˜*z*�(*r*�

*μ*˜*r*�(*r*�

*μ*˜ *<sup>θ</sup>*�(*r*�

, *ω*)| = 1 + *Fz*�(*r*�

, *ω*) = 1 + *Fr*�(*r*�

, *ω*) = 1 + *Fθ*�(*r*�

Fig. 18. The snapshots of the electric-field distribution in the vicinity of PEC. (a) the cloaking

The Group Velocity Picture of Metamaterial Systems 85

the Lorentz co-variant transformation, but the coordinate transformation for ideal cloaking is not Lorentz co-variant. Such violation is obvious if we suppose the initial medium in the r space is not dispersive, such as the vacuum, but as we have pointed out (also mentioned in Ref. [68], the cloaking material (in *r*� space) must be dispersive to avoid the group velocity over c. Such Lorentz co-variant violation is generally true for ˛aˇrtransformation optics ˛a´s since material parameters are non-relativistic, so the causality limitation should be checked widely. Third, from Eq.14, we can find that not only the inner layers of the cloak(*r*� → *R*1) but also the other layers(*r*� > *R*1) must be dispersive. For every layer, a certain dispersive strength is

In the following, we will validate that the total cross section can be reduced drastically, and that the perfect cloaking cannot be achieved because of strong absorption by FDTD numerical experiments. Compared with other frequency-domain simulation methods, such as the finite element methods or the transfer-matrix methods, the FDTD simulation can better reflect the real physical process of cloaking. For example, we note that the FDTD calculation will be numerically unstable when the dispersion is not included in the cloak's material. For simplicity, the simulation is limited to two-dimensional cloak[69]. Without lost of generality, only TE modes are investigated in this study (TE modes have the electric field perpendicular to the two-dimensional plane of our model). Thus the constitutive parameters involved here are *�z*� , *μr*� , *μθ*� . The dispersion is introduced into our FDTD by standard Lorentz model:

)*ω*<sup>2</sup>

)*ω*<sup>2</sup>

)*ω*<sup>2</sup>

Where *ωpz*� , *ωpr*� , *ωpθ*� are plasma frequencies, *ωaz*� , *ωar*�, *ωaθ*� are atom resonated frequencies, *γz*� , *γr*� , *γθ*� are damping factors and *Fpz*� , *Fpr*� , *Fp<sup>θ</sup>*� are filling factors. In our simulation, an E-polarized time-harmonic uniform plane wave whose wavelength *λ*<sup>0</sup> in vacuum is 3.75*cm* is incident from left to right. The real parts of the constitutive parameters at *ω*<sup>0</sup> = 2*πc*/*λ*<sup>0</sup>

*<sup>r</sup>*� <sup>=</sup> *<sup>R</sup>*2(*θ*) <sup>−</sup> *<sup>R</sup>*1(*θ*)

*pz*�/(*ωaz*�(*r*�

*pr*�/(*ωar*�(*r*�

*<sup>p</sup>θ*�/(*ωaθ*�(*r*�

)

)

)

<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>ωγz*�)

<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>ωγθ*�)

*θ*� = *θ* (18)

*<sup>R</sup>*2(*θ*) *<sup>r</sup>* <sup>+</sup> *<sup>R</sup>*1(*θ*) (17)

<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>ωγr*�) (16)

Because of the causality limitation, it is well-known that the group velocity can not exceed *c* except in the "strong dispersion" frequency range (or called "resonant range"). But, if the working frequency is in the "strong-dispersion" range of the cloaking material, the absorption must be very strong and it will destroy the ideal cloaking obviously. So perfect invisibility can not be achieved for the finite *dnt*� *<sup>d</sup><sup>ω</sup>* because it will lead to superluminal velocity or strong absorption.

In addition, the curves with the criterion condition *Vg* = *c* on the plane [*R*1/*r*, *α*] are plotted for different *ω dnt*� *<sup>d</sup><sup>ω</sup>* in figure 17. The region to the left of curves is corresponding to *Vg* < *c* and the region to the right is corresponding to *Vg* > *c*. There exists a maximum *max*{*R*1/*r*} for each curve in order that *Vg* ≤ *c* can be hold for all the *α*. Especially, for the no-dispersion case *ω dnt*� *<sup>d</sup><sup>ω</sup>* = 0, we can see that *Vg* > *c* at all *R*1/*r* for large *α* values, which means the whole cloak is not physical if there is no dispersion. This "dispersion-is-required" conclusion can be generally derived from Eq.14, and it is consistent with the analysis in Ref. [68]. From figure 17, we know that the larger *ω dnt*� *<sup>d</sup><sup>ω</sup>* , the larger *max*{*R*1/*r*}. But anyway, for arbitrary finite *ω dnt*� *<sup>d</sup><sup>ω</sup>* , *max*{*R*1/*r*} can not be infinite, so that the superluminal range always exists.

The second case of Eq.15 is with divergent *dnt*� *<sup>d</sup><sup>ω</sup>* . From the previous discussion, we know that if the ideal cloak exists, the cloak must be dispersive and *dnt*� *<sup>d</sup><sup>ω</sup>* must be divergent when *<sup>r</sup>* <sup>→</sup> 0. Actually, when *<sup>r</sup>* <sup>→</sup> 0, since <sup>√</sup>*εr*� <sup>∝</sup> *<sup>r</sup>*, *dnt*� *<sup>d</sup><sup>ω</sup>* <sup>∝</sup> *<sup>d</sup>εr*� *<sup>d</sup><sup>ω</sup>* / <sup>√</sup>*εr*� is really divergent for non-zero *dεr*� *<sup>d</sup><sup>ω</sup>* . From Eq.15, we can see that now the *Vg* <sup>→</sup> 0 for a finite *<sup>d</sup>εr*� *<sup>d</sup><sup>ω</sup>* (generally true) at all *α* values except *α* = 0(or *π*), so that the group velocity difficulty seems to be overcome. But, since the causality limitation, the non-zero *<sup>d</sup>εr*� *<sup>d</sup><sup>ω</sup>* means non-zero imaginary part of permittivity (non-zero dissipation). The non-zero dissipation and the almost-zero group velocity will result in very strong absorption. This means that the energy of rays near the inner cloaking radius *R*<sup>1</sup> is almost totally absorbed by the cloaking material. As we pointed out at the beginning that the absorption in *r*� space should also appear in r space, because of the consistence between two spaces. The strong absorption in *r* space can be interpreted in the following way. From the transformation (which is also suited for imaginary part), we can find that when *r* → 0, the finite imaginary part in *r*� space corresponds to the infinite imaginary part in *r* space, which also means very strong absorption in the initial homogeneous medium. So the perfect cloaking is still impossible because of the strong absorption which is enforced by the causality limitation.

For a two-dimensional coordinate transformation: *r*� = *f*(*r*), *θ*� = *θ*, *z*� = *z*, the same conclusions of the causality limitation can be obtained through the similar analysis, although the coordinate transformation and the singularities are different from the three-dimensional case.

Next, we will discuss the physical meaning of the dilemma that either the group velocity *Vg* diverges or a strong absorption is imposed on the cloaking material. First, it is an intrinsic conflict which can not be solved by the methods, for example, "the system is imbedded in a medium"[68]. We believe that the ideal cloaking is impossible because of the causality limitation and this conclusion is consistent with the statement of previous studies[89] that the perfect invisibility is unachievable because of the wave nature of light. Second, we have to face the question: " Why the causality is violated for ideal cloaking which is based on the simple coordinate transformation?". Our answer is that the causality is only guaranteed by 28 Will-be-set-by-IN-TECH

Because of the causality limitation, it is well-known that the group velocity can not exceed *c* except in the "strong dispersion" frequency range (or called "resonant range"). But, if the working frequency is in the "strong-dispersion" range of the cloaking material, the absorption must be very strong and it will destroy the ideal cloaking obviously. So perfect invisibility

In addition, the curves with the criterion condition *Vg* = *c* on the plane [*R*1/*r*, *α*] are plotted

and the region to the right is corresponding to *Vg* > *c*. There exists a maximum *max*{*R*1/*r*} for each curve in order that *Vg* ≤ *c* can be hold for all the *α*. Especially, for the no-dispersion

*<sup>d</sup><sup>ω</sup>* . From Eq.15, we can see that now the *Vg* <sup>→</sup> 0 for a finite *<sup>d</sup>εr*�

true) at all *α* values except *α* = 0(or *π*), so that the group velocity difficulty seems to be

of permittivity (non-zero dissipation). The non-zero dissipation and the almost-zero group velocity will result in very strong absorption. This means that the energy of rays near the inner cloaking radius *R*<sup>1</sup> is almost totally absorbed by the cloaking material. As we pointed out at the beginning that the absorption in *r*� space should also appear in r space, because of the consistence between two spaces. The strong absorption in *r* space can be interpreted in the following way. From the transformation (which is also suited for imaginary part), we can find that when *r* → 0, the finite imaginary part in *r*� space corresponds to the infinite imaginary part in *r* space, which also means very strong absorption in the initial homogeneous medium. So the perfect cloaking is still impossible because of the strong absorption which is enforced

For a two-dimensional coordinate transformation: *r*� = *f*(*r*), *θ*� = *θ*, *z*� = *z*, the same conclusions of the causality limitation can be obtained through the similar analysis, although the coordinate transformation and the singularities are different from the three-dimensional

Next, we will discuss the physical meaning of the dilemma that either the group velocity *Vg* diverges or a strong absorption is imposed on the cloaking material. First, it is an intrinsic conflict which can not be solved by the methods, for example, "the system is imbedded in a medium"[68]. We believe that the ideal cloaking is impossible because of the causality limitation and this conclusion is consistent with the statement of previous studies[89] that the perfect invisibility is unachievable because of the wave nature of light. Second, we have to face the question: " Why the causality is violated for ideal cloaking which is based on the simple coordinate transformation?". Our answer is that the causality is only guaranteed by

*<sup>d</sup><sup>ω</sup>* = 0, we can see that *Vg* > *c* at all *R*1/*r* for large *α* values, which means the whole cloak is not physical if there is no dispersion. This "dispersion-is-required" conclusion can be generally derived from Eq.14, and it is consistent with the analysis in Ref. [68]. From figure 17,

*<sup>d</sup><sup>ω</sup>* in figure 17. The region to the left of curves is corresponding to *Vg* < *c*

*<sup>d</sup><sup>ω</sup>* , the larger *max*{*R*1/*r*}. But anyway, for arbitrary finite *ω*

*<sup>d</sup><sup>ω</sup>* <sup>∝</sup> *<sup>d</sup>εr*�

*<sup>d</sup><sup>ω</sup>* /

*<sup>d</sup><sup>ω</sup>* . From the previous discussion, we know

*<sup>d</sup><sup>ω</sup>* must be divergent when

<sup>√</sup>*εr*� is really divergent for

*<sup>d</sup><sup>ω</sup>* means non-zero imaginary part

*<sup>d</sup><sup>ω</sup>* (generally

*<sup>d</sup><sup>ω</sup>* because it will lead to superluminal velocity or strong

*dnt*� *<sup>d</sup><sup>ω</sup>* ,

can not be achieved for the finite *dnt*�

*dnt*�

absorption.

case *ω*

non-zero

case.

for different *ω*

*dnt*�

we know that the larger *ω*

*dεr*�

by the causality limitation.

*dnt*�

*<sup>r</sup>* <sup>→</sup> 0. Actually, when *<sup>r</sup>* <sup>→</sup> 0, since <sup>√</sup>*εr*� <sup>∝</sup> *<sup>r</sup>*, *dnt*�

overcome. But, since the causality limitation, the non-zero *<sup>d</sup>εr*�

The second case of Eq.15 is with divergent *dnt*�

*max*{*R*1/*r*} can not be infinite, so that the superluminal range always exists.

that if the ideal cloak exists, the cloak must be dispersive and *dnt*�

Fig. 18. The snapshots of the electric-field distribution in the vicinity of PEC. (a) the cloaking structure with a PEC at radius *R*1, (b) the naked PEC with radius *R*1.

the Lorentz co-variant transformation, but the coordinate transformation for ideal cloaking is not Lorentz co-variant. Such violation is obvious if we suppose the initial medium in the r space is not dispersive, such as the vacuum, but as we have pointed out (also mentioned in Ref. [68], the cloaking material (in *r*� space) must be dispersive to avoid the group velocity over c. Such Lorentz co-variant violation is generally true for ˛aˇrtransformation optics ˛a´s since material parameters are non-relativistic, so the causality limitation should be checked widely. Third, from Eq.14, we can find that not only the inner layers of the cloak(*r*� → *R*1) but also the other layers(*r*� > *R*1) must be dispersive. For every layer, a certain dispersive strength is needed to avoid *Vg* > *c*.

In the following, we will validate that the total cross section can be reduced drastically, and that the perfect cloaking cannot be achieved because of strong absorption by FDTD numerical experiments. Compared with other frequency-domain simulation methods, such as the finite element methods or the transfer-matrix methods, the FDTD simulation can better reflect the real physical process of cloaking. For example, we note that the FDTD calculation will be numerically unstable when the dispersion is not included in the cloak's material. For simplicity, the simulation is limited to two-dimensional cloak[69]. Without lost of generality, only TE modes are investigated in this study (TE modes have the electric field perpendicular to the two-dimensional plane of our model). Thus the constitutive parameters involved here are *�z*� , *μr*� , *μθ*� . The dispersion is introduced into our FDTD by standard Lorentz model:

$$\begin{aligned} \left| \mathfrak{E}\_{z'} (r', \omega) \right| &= 1 + F\_{z'} (r') \omega\_{p z'}^2 / (\omega\_{a z'} (r')^2 - \omega^2 - \mathrm{i}\omega \gamma\_{z'})\\ \left| \mathfrak{H}\_{r'} (r', \omega) = 1 + F\_{r'} (r') \omega\_{p r'}^2 / (\omega\_{a r'} (r')^2 - \omega^2 - \mathrm{i}\omega \gamma\_{r'})\\ \left| \mathfrak{H}\_{\theta'} (r', \omega) = 1 + F\_{\theta'} (r') \omega\_{p \theta'}^2 / (\omega\_{a \theta'} (r')^2 - \omega^2 - \mathrm{i}\omega \gamma\_{\theta'}) \end{aligned} \tag{16}$$

$$r' = \frac{R\_2(\theta) - R\_1(\theta)}{R\_2(\theta)}r + R\_1(\theta) \tag{17}$$

$$
\theta' = \theta \tag{18}
$$

Where *ωpz*� , *ωpr*� , *ωpθ*� are plasma frequencies, *ωaz*� , *ωar*�, *ωaθ*� are atom resonated frequencies, *γz*� , *γr*� , *γθ*� are damping factors and *Fpz*� , *Fpr*� , *Fp<sup>θ</sup>*� are filling factors. In our simulation, an E-polarized time-harmonic uniform plane wave whose wavelength *λ*<sup>0</sup> in vacuum is 3.75*cm* is incident from left to right. The real parts of the constitutive parameters at *ω*<sup>0</sup> = 2*πc*/*λ*<sup>0</sup>

metamaterial studies. From group velocity, we can find the physical origin of abnormal optical phenomena of metamaterials, such as the "no-transmission no-reflection" on the hyper-medium surface which is from a zero-group-velocity reflecting mode, and the coherence gain of superlens image which is from the different group delay on different paths. From group velocity, we can avoid some traps of violating basic physical limitation, such as the violation of causality limitation in cloaking study. These traps are very serious since the metamaterial from our imagination could exist in this world if violating basic limitations. From group velocity, we can find the key parameter of cloaking dynamical process and help us to optimizing the design of cloak design. From group velocity of evanescent wave, new detecting methods could be found, for example the detecting of QED vacuum polarization by phase change or delay time of evanescent wave. We believe that only with the well-constructed group velocity picture, the deeper understanding of the abnormal optical/photonic properties of metamaterials is possible. All these research works also show that the group velocity study of metamaterials can lead us to many new interesting topics,

The Group Velocity Picture of Metamaterial Systems 87

This work is supported by NKBRPC (Grant No. 2012CB927401), NSFC (Grant Nos. 11004212, 11174309, and 60938004), and the STCSM (Grant Nos. 11ZR1443800 and 11JC1414500).

[2] D. R. Smith, W.J.Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett.

[3] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, Appl. Phys. Lett. 78, 489

[5] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave

[6] P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 033401 (2001); Phys. Rev. E 65, 036622

[8] Zixian Liang, Peijun Yao, Xiaowei Sun and Xunya Jiang, Appl. Phys. Lett. 92, 131118

[9] Zheng Liu, Zixian Liang, Xunya Jiang, Xinhua Hu, Xin Li, and Jian Zi, Appl.Phys. Lett.

[10] Brillouin, *Wave Propagation and Group Velocity*, New York:Academic Press Inc., (1960) [11] J. H. Page, Ping Sheng, H. P. Schriemer, I.Jones, Xiaodun Jing and D. A. Weitzt,Science,

[13] R.A. Shelby, D.R. Smith, and S. Schultz, Science 92, 792(2001); D.R. Smith and D. Schurig, Phys. Rev. Lett. 90,077405 (2003);C.G. Parazzoli, R.B. Greegor, K. Li, B.E.C.

[14] J. B. Pendry, Phys. Rev. Lett. 85, 3966(2000); N. Fang, H. Lee, C. Sun, X. Zhang, Science 308, 534 (2005); T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, R.Hillenbrand, Science

Theory Tech. 47, 2075 (1999); J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).

which are still waiting for further research.

[1] V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).

[4] R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).

[7] P. Yao, Z. Liang, and X. Jiang, Appl. Phys. Lett. 92, 031111 (2008).

[12] Hailu Luo, Wei Hu and Zhongzhou Ren, Europhys. Lett. 74 1081(2006).

Koltenbah, and M. Tanielian, Phys. Rev. Lett. 90, 107401 (2003).

**8. Acknowledgement**

84, 4184 (2000).

**9. References**

(2001).

(2002).

(2008)

96, 113507 (2010)

271 634 1996

313, 1595 (2006)

satisfy the cloaking coordinate transformation[69, 91], and they are *<sup>μ</sup>r*� = *<sup>r</sup>*� −*R*<sup>1</sup> *<sup>r</sup>*� , *μθ*� <sup>=</sup> <sup>1</sup> *μr*� , *<sup>ε</sup>z*� = ( *<sup>R</sup>*<sup>2</sup> *R*2−*R*<sup>1</sup> )<sup>2</sup> *<sup>r</sup>*� −*R*<sup>1</sup> *<sup>r</sup>*� , where *R*<sup>1</sup> is 0.665*λ*0, *R*<sup>2</sup> is 1.33*λ*0. And the dispersive parameters are set as follows: if *εz*� > 1 then *ωaz*� = 1.4*ω*0, else *ωaz*� = 0.6*ω*0, *ωar*� = 0.6*ω*0, *ωaθ*� = 1.4*ω*0, *γz*� = *γr*� = *γθ*� = *ω*0/100. *ωpz*� = *ωpr*� = *ωpθ*� = 4*ω*0, *Fz*�(*r*)=(*εz*� − 1) (*ω*<sup>2</sup> *az*�−*ω*<sup>2</sup> <sup>0</sup> )<sup>2</sup>+*ω*<sup>2</sup> 0*γ*<sup>2</sup> *z*� (*ω*<sup>2</sup> *az*�−*ω*<sup>2</sup> <sup>0</sup> )*ω*<sup>2</sup> *pz*� , *Fr*�(*r*) = (*μr*� − 1) (*ω*<sup>2</sup> *ar*�−*ω*<sup>2</sup> <sup>0</sup> )<sup>2</sup>+*ω*<sup>2</sup> 0*γ*<sup>2</sup> *r*� (*ω*<sup>2</sup> *ar*�−*ω*<sup>2</sup> <sup>0</sup> )*ω*<sup>2</sup> *pr*� , *Fθ*�(*r*)=(*μθ*� − 1) (*ω*<sup>2</sup> *aθ*�−*ω*<sup>2</sup> <sup>0</sup> )<sup>2</sup>+*ω*<sup>2</sup> 0*γ*<sup>2</sup> *θ*� (*ω*<sup>2</sup> *aθ*�−*ω*<sup>2</sup> <sup>0</sup> )*ω*<sup>2</sup> *pθ*� . In fact, these parameters have many possible choices. The different groups of parameters correspond to different dynamic processes which we will discuss in another section [92].

Figure 18 shows the snapshots of the electric-field distribution in two cases: the cloak with the perfect electric conductor (PEC) at radius *R*1(left), and the naked PEC with radius *R*1(right). Obviously, the cloak is very effective. Quantitatively, we calculate the absorption cross section and the scattering cross section of the cloak at the stable state, and they are 0.67*λ*<sup>0</sup> and 0.24*λ*<sup>0</sup> respectively, while the scattering cross section of the naked PEC is 3.14*λ*0. So, with dispersive cloak, the total cross section is three times smaller, and the absorption cross section dominates as we predicted. To emphasize the huge absorption of the cloak, we use a common homogeneous isotropic media, with *ε* = *μ* = 1.1 but all other parameters are the same as the cloak, to replace the cloaking material. Then we find the absorption cross section is only 0.089*λ*<sup>0</sup> which is about one order smaller. The reason of strong absorption has been discussed before.

Now we can have a full view of cloaking recipe based on the coordinate transformation. First, the cloaking material must be dispersive, and the strong absorption can not be avoided because of the causality limitation. Thus it is not perfectly invisible. Second, the scattering cross section of the dispersive cloak could be small, so that the scattered field is weak. Although the ideal invisibility is impossible, the cloaking recipe still has a main advantage. The "strong-absorption and weak-scattering" property means that the cloak almost can not be observed except from the forward direction. so that such cloak can well defend the aˇ˛rpassive radars ˛a´s which detect the perturbation of the original field. It is well-known that at the Rayleigh scattering case, where the radius of the scatterer is much smaller than the wavelength, the absorption cross section could be larger than the scattering cross section because of the diffraction. The cloaking can be thought as giant Rayleigh scattering case, where the light rays are forced to "diffract" around the cloaked area.

In conclusion, the properties of the dispersive cloak are investigated, and the limitation of causality is revealed. Our study shows that the superluminal velocity or a strong absorption can not be overcome since the intrinsic conflict between the coordinate transformation to obtain the cloaking and the causality limitation. In addition, we validate the results using a numerical simulation which is performed in FDTD algorithm with physical parameters. The numerical experiments show that the absorption cross section is dominant and the scattering cross section can be reduced significantly. The study gives us a full view of the cloaking recipe based on the coordinate transformation, and will have further profound influence on the related topics.

#### **7. Summary**

In summary, we have investigate the metamaterial systems from group velocity(energy velocity) picture. From these topics, we demonstrate the importance of group velocity in metamaterial studies. From group velocity, we can find the physical origin of abnormal optical phenomena of metamaterials, such as the "no-transmission no-reflection" on the hyper-medium surface which is from a zero-group-velocity reflecting mode, and the coherence gain of superlens image which is from the different group delay on different paths. From group velocity, we can avoid some traps of violating basic physical limitation, such as the violation of causality limitation in cloaking study. These traps are very serious since the metamaterial from our imagination could exist in this world if violating basic limitations. From group velocity, we can find the key parameter of cloaking dynamical process and help us to optimizing the design of cloak design. From group velocity of evanescent wave, new detecting methods could be found, for example the detecting of QED vacuum polarization by phase change or delay time of evanescent wave. We believe that only with the well-constructed group velocity picture, the deeper understanding of the abnormal optical/photonic properties of metamaterials is possible. All these research works also show that the group velocity study of metamaterials can lead us to many new interesting topics, which are still waiting for further research.

#### **8. Acknowledgement**

This work is supported by NKBRPC (Grant No. 2012CB927401), NSFC (Grant Nos. 11004212, 11174309, and 60938004), and the STCSM (Grant Nos. 11ZR1443800 and 11JC1414500).

#### **9. References**

30 Will-be-set-by-IN-TECH

as follows: if *εz*� > 1 then *ωaz*� = 1.4*ω*0, else *ωaz*� = 0.6*ω*0, *ωar*� = 0.6*ω*0, *ωaθ*� = 1.4*ω*0, *γz*� =

(*ω*<sup>2</sup> *aθ*�−*ω*<sup>2</sup>

many possible choices. The different groups of parameters correspond to different dynamic

Figure 18 shows the snapshots of the electric-field distribution in two cases: the cloak with the perfect electric conductor (PEC) at radius *R*1(left), and the naked PEC with radius *R*1(right). Obviously, the cloak is very effective. Quantitatively, we calculate the absorption cross section and the scattering cross section of the cloak at the stable state, and they are 0.67*λ*<sup>0</sup> and 0.24*λ*<sup>0</sup> respectively, while the scattering cross section of the naked PEC is 3.14*λ*0. So, with dispersive cloak, the total cross section is three times smaller, and the absorption cross section dominates as we predicted. To emphasize the huge absorption of the cloak, we use a common homogeneous isotropic media, with *ε* = *μ* = 1.1 but all other parameters are the same as the cloak, to replace the cloaking material. Then we find the absorption cross section is only 0.089*λ*<sup>0</sup> which is about one order smaller. The reason of strong absorption has been discussed

Now we can have a full view of cloaking recipe based on the coordinate transformation. First, the cloaking material must be dispersive, and the strong absorption can not be avoided because of the causality limitation. Thus it is not perfectly invisible. Second, the scattering cross section of the dispersive cloak could be small, so that the scattered field is weak. Although the ideal invisibility is impossible, the cloaking recipe still has a main advantage. The "strong-absorption and weak-scattering" property means that the cloak almost can not be observed except from the forward direction. so that such cloak can well defend the aˇ˛rpassive radars ˛a´s which detect the perturbation of the original field. It is well-known that at the Rayleigh scattering case, where the radius of the scatterer is much smaller than the wavelength, the absorption cross section could be larger than the scattering cross section because of the diffraction. The cloaking can be thought as giant Rayleigh scattering case,

In conclusion, the properties of the dispersive cloak are investigated, and the limitation of causality is revealed. Our study shows that the superluminal velocity or a strong absorption can not be overcome since the intrinsic conflict between the coordinate transformation to obtain the cloaking and the causality limitation. In addition, we validate the results using a numerical simulation which is performed in FDTD algorithm with physical parameters. The numerical experiments show that the absorption cross section is dominant and the scattering cross section can be reduced significantly. The study gives us a full view of the cloaking recipe based on the coordinate transformation, and will have further profound influence on

In summary, we have investigate the metamaterial systems from group velocity(energy velocity) picture. From these topics, we demonstrate the importance of group velocity in

(*ω*<sup>2</sup> *aθ*�−*ω*<sup>2</sup> <sup>0</sup> )*ω*<sup>2</sup> *pθ*�

*<sup>r</sup>*� , where *R*<sup>1</sup> is 0.665*λ*0, *R*<sup>2</sup> is 1.33*λ*0. And the dispersive parameters are set

<sup>0</sup> )<sup>2</sup>+*ω*<sup>2</sup> 0*γ*<sup>2</sup> *θ*� −*R*<sup>1</sup>

<sup>0</sup> )<sup>2</sup>+*ω*<sup>2</sup> 0*γ*<sup>2</sup> *z*�

. In fact, these parameters have

(*ω*<sup>2</sup> *az*�−*ω*<sup>2</sup>

(*ω*<sup>2</sup> *az*�−*ω*<sup>2</sup> <sup>0</sup> )*ω*<sup>2</sup> *pz*�

*<sup>r</sup>*� , *μθ*� <sup>=</sup> <sup>1</sup>

*μr*� ,

, *Fr*�(*r*) =

satisfy the cloaking coordinate transformation[69, 91], and they are *<sup>μ</sup>r*� = *<sup>r</sup>*�

*γr*� = *γθ*� = *ω*0/100. *ωpz*� = *ωpr*� = *ωpθ*� = 4*ω*0, *Fz*�(*r*)=(*εz*� − 1)

, *Fθ*�(*r*)=(*μθ*� − 1)

where the light rays are forced to "diffract" around the cloaked area.

processes which we will discuss in another section [92].

*<sup>ε</sup>z*� = ( *<sup>R</sup>*<sup>2</sup> *R*2−*R*<sup>1</sup>

(*μr*� − 1)

before.

the related topics.

**7. Summary**

)<sup>2</sup> *<sup>r</sup>*� −*R*<sup>1</sup>

> <sup>0</sup> )<sup>2</sup>+*ω*<sup>2</sup> 0*γ*<sup>2</sup> *r*�

(*ω*<sup>2</sup> *ar*�−*ω*<sup>2</sup>

(*ω*<sup>2</sup> *ar*�−*ω*<sup>2</sup> <sup>0</sup> )*ω*<sup>2</sup> *pr*�


[44] M. Bass, Handbook of Optics, Vol. II (McGraw-Hill, Inc., New York, 1995); W. Su, J. Li

The Group Velocity Picture of Metamaterial Systems 89

[46] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis Phys. Rev.Lett. 91, 207401 (2003); E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C.M.

[47] J. B. Pendry, Phys. Rev. Lett. 91, 099701 (2003);D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).

[49] S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett. 90, 107402

[50] X. S. Rao and C. K. Ong, Phys. Rev. B 68, 113103 (2003); X. S. Rao and C. K. Ong, Phys.

[52] Lei Zhou, C. T. Chan, Appl. Phys. Lett. 86, 101104 (2005), Y. Zhang, T. M. Grzegorczyk

[56] P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, Phys. Rev.

[58] C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry Phys. Rev. B 68, 045115

[60] Ilya V. Shadrivov, Andrey A. Sukhorukov, and Yuri S. Kivshar, Phys. Rev. E 67, 057602

[61] B. E. A. Saleh, M. C. Teich, *Fundamentals of Photonics* (John Wiley & Sons, New York,

[62] L. Mandel and E. Wolf *Optical Coherence and Quantum Optics* (Cambridge University, Cambridge, 1995); Marlan O. Scully and M. Suhail Zubairy *Quantum Optics*,

[64] In our source frequency range, the index range is about <sup>−</sup><sup>1</sup> <sup>−</sup> *<sup>i</sup>*0.0029 <sup>±</sup> (0.006 <sup>+</sup> *<sup>i</sup>*10−6),

[65] The "group velocity" is not a well-defined value if the working frequency *ω*<sup>0</sup> is near the

[69] S. A. Cummer, B.-I Popa, D. Schurig, and David R. Smith, Phys. Rev. E 74, 036621 (2006). [70] W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Photonics 1, 224 (2007). [71] W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, Appl. Phys. Lett.

[72] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R.

[73] H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 063903 (2007).

[59] Nader Engheta, IEEE Antennas and Wireless Propagation Lett. 1, 10, 2002

(2003); A. C. Peacock and N. G. R. Broderick, 11, 2502 (2003).

(2003); J. B. Pendry and D. R. Smith, Phys. Rev. Lett. 90, 029703 (2003).

and N. Xu, J. Biotechnol. 105, 165 (2003). [45] T. Tajima, Eur. Phys. J. D 55, 519 (2009).

Soukoulis, Nature (London) 423, 604 (2003).

[48] G. Gomez-Santos, Phys. Rev. Lett. 90, 077401 (2003).

[51] Michael W. Feise, Yuri S. Kivshar, Phys. Lett. A 334 326 (2005)

[53] L. Chen, S. He and L. Shen, Phys. Rev. Lett. 92, 107404(2004). [54] R. W. Ziolkowski and E. Heyman, Phys. Rev. E 64, 056625 (2001).

Rev. E 68, 067601 (2003).

E 67, 025602(R) (2003).

(2003)

1991)

and J. A. Kong, PIER 35,271, (2002)

[55] S. A. Cummer, Appl. Phys. Lett. 82, 1503 (2003).

[57] R. Merlin, Appl. Phys. Lett. 84, 1290 (2004).

(Cambridge University, Cambridge, 1997)

[66] Xunya Jiang *et. al.* unpublished. [67] U. Leonhardt, Science 312, 1777 (2006).

Smith, Science 314, 977 (2006).

91, 111105 (2007).

[63] X. Jiang and C. M. Soukoulis, Phys. Rev. Lett. 85, 70 (2000)

so the focal length defference and reflection are very small.

[68] J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).

resonant frequency *ωa* of the NIM. But the GRT is still well-defined.


32 Will-be-set-by-IN-TECH

[15] J. B. Pendry, Opt. Express 11, 755 (2003); Zhaowei Liu, Hyesog Lee, Yi Xiong, Cheng Sun, and Xiang Zhang, Science 315, 1686 (2007); Igor I. Smolyaninov, Yu-Ju Hung, and

[16] Z. Jacob, L. V. Alekseyev, E. Narimanov, Opt. Express 14,8247 (2006); A. Salandrino, N. Engheta, Phys. Rev. B 74, 075103 (2006); Zubin Jacob, Leonid V. Alekseyev, and Evgenii Narimanov, J. Opt. Soc. Am. A 24,A52 (2007); J. B. Pendry, S. A. Ramakrishna, J. Phys.

[17] Xin Li,Zixian Liang,Xiaohan Liu,Xunya Jiang, and Jian Zi, Appl. Phys. Lett. 93, 171111

[19] P. W. Milonni, *Fast Light, Slow Light and Left-Handed Light* Institute of Physics, Great

[22] V. A. Podolskiy, and E. E. Narimanov, Phys. Rev. B 71, 201101 (2005); R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, J. Opt. Soc. Am. B. 23, 498 (2006); A. A.

[23] A. Taflove, *Computational Electrodynamics: The Finite-Difference Time- Domain Method*

[26] Jie Yao,Zhaowei Liu,Yongmin Liu, Yuan Wang,Cheng Sun,Guy Bartal, Angelica M.

[31] N. Ashcroft, N. Mermin, Solid State Physics (Harcourt College Publishers, New York, 1976) ; H. Bruus and K. Flensberg, Many-body quantum theory in condensed matter

[33] R. Feynman, QED: the strange theory of light and matter (Princeton university press,

[39] R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, New

[42] J. Kong, Electromagnetic Wave Theory, Electromagnetic Wave Theory

[20] As for the explanation of the static solution of the HI it'll be given in another paper.

[18] Hailu Luo, Wei Hu and Zhongzhou Ren, Europhys. Lett. 74 1081(2006).

[21] David R. Smith and Norman Kroll, Phys. Rev. Lett. 85, 2933(2000).

Govyadinov, V. A. Podolskiy, Phys. Rev. B 73,115108(2006).

[25] N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).

Stacy and Xiang Zhang, Science 321, 930 (2008).

physics (Oxford university press, Oxford, 2004).

[35] Y. Ding, A. Kaplan, Phys. Rev. Lett. 63, 2725 (1989). [36] B. King, A. Piazza and C. Keitel, Nat. photon. 4, 92 (2010).

[37] H. Gies, Phys. Rev. D 61, 085021 (2000).

(Wiley-Interscience, New York, 1990).

[24] Alessandro Salandrino and Nader Engheta, Phys. Rev. B 74,115108(2006).

[27] J. B. Pendry and A. J. Holden and W. J. Stewart, Phys. Rev. Lett. 76, 4773(1996). [28] A. N. Grigorenko, A. K. Geim, H. F. Gleeson,*et al*, Nature, 438 335(2005)

[29] G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, Opt. Lett. 32, 53 (2007). [30] D. Perkins, Introduction to high energy physics, 4th edition (Cambridge university

[32] J. Goodman, Statistical Optics (John Wiley & Sons, Inc., New York, 1985).

[34] R. Ruffini, G. Vereshchagin and S. Xue, Phys. Rep. 487, 1 (2010).

[38] J. S Heyl and L. Hernquist, J. Phys. A: Math. Gen. 30, 6485 (1997).

[40] B.Barwick, D.Flannigan, and A.H. Zewail, Nature 462, 902 (2009). [41] C. Carniglia and L. Mandel, J. Opt. Soc. Am. 61, 1035 (1971).

[43] J. Lawall and E. Kessler , Rev. Sci. Instrum. 71, 2669 (2000).

Christopher C. Davis, Science 315, 1699 (2007).

Condens. Matter 14,8463(2002).

Britain, CRC Press, 2004

(Artech House, Boston, 1995).

press, Cambridge, 2000).

Princeton, 1985).

York, 1985).

(2008).


**1. Introduction** 

**4** 

*USA* 

Elena Semouchkina

*Michigan Technological University* 

**Formation of Coherent Multi-Element** 

Employment of metamaterials in cloaking devices, which could make concealed objects invisible, is based on the capability of resonators used as metamaterial "atoms" to provide effective material parameters, ranging from any positive to any negative values, in the vicinity of the resonance frequency. Metamaterials comprising layers with variously sized resonators (Schurig et al., 2006) or layers with variable density of identical resonators (Semouchkina et al., 2010) have been proposed to obtain the desired spatial distribution of the medium parameters, for example, radial variation of the effective permittivity or permeability in the shell prescribed by the transformation optics relations for cloaking cylindrically shaped objects (Leonhardt, 2006; Pendry et al., 2006). Such approach could be justified, if resonators contribute to the value of the effective parameters by the same way as polarized atoms contribute to the value of the polarization of a dielectric medium, i.e., if the responses of individual atoms are additive so that the medium response exhibits the same dispersion as the response of an individual atom. Then the material can be described by the effective medium theory (EMT), which was adopted for metamaterails description at the onset of metamaterial studies (Pendry et al., 1999; Pendry & Smith, 2003; Smith & Pendry, 2004; Smith et al., 2006). According to the EMT, a polar dielectric material could be represented by a set of parallel dipoles switching their directions by 1800 in dependence on the phase of the external field. Resonance modes in the "atoms" of metamaterials could also be represented by equivalent dipoles and, similarly to atoms in dielectrics, these dipoles are expected to respond accordingly, i.e. to follow the phase of the external field and demonstrate a coherent response within the half wavelength of incident radiation. However, observation of the performance of the infrared invisibility cloak designed from glass resonators (Semouchkina et al., 2010) has shown that obtaining a coherent response of the cloak structure presents a serious problem, if resonators in the metamaterail are electromagnetically coupled. The problem of coupling between resonators in metamaterials,

It is well known that metamaterials, as a rule, are designed as periodic structures of closepacked resonators. When the dimensions of resonators are ten times smaller than the wavelength, the EMT is considered to be applicable, so that the waves "do not see" the atoms and propagate in the metamaterial as in a homogenized, i.e. in a uniform medium. Such homogenization is supposed to make metamaterials essentially different from

however, remains to be largely ignored in the literature.

**Resonance States in Metamaterials** 


### **Formation of Coherent Multi-Element Resonance States in Metamaterials**

Elena Semouchkina *Michigan Technological University USA* 

#### **1. Introduction**

34 Will-be-set-by-IN-TECH

90 Metamaterial

[79] A. Taflove and S. C. Hagness, *Computational Electrodynamics: The Finite-Difference*

[80] The approximation of permittivity *ε* and permeability *μ* in this section are: *Re*[*μθ*(*r*, *ω*0)]*max* = 20, *Re*[*μr*(*r*, *ω*0)]*min* = 1/20 and *Re*[*εz*(*r*, *ω*0)]*min* = 1/5.

[81] H. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, and C. T. Chan, Phys. Rev. B 76, 241104 (2007). [82] R. L. Fante, and M. T. McCormack, IEEE Trans. Antennas Propag. 30, 1443(1968). [83] M. Kerker, J. Opt. Soc. Am. 65, 376(1975). and D. R. Nature Photonics 1, 224 (2007).

[84] Hongsheng Chen, Bae-Ian Wu, Baile Zhang, and Jin Au Kong, Phys. Rev. Lett. 99, 063903

[85] Zhichao Ruan, Min Yan, Curtis W. Neff, and Min Qiu, Phys. Rev. Lett. 99, 113903 (2007).

[91] When *r*� → *R*1, *μθ*� will tend to infinite. In order that it can be realizable in our numerical

[94] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, http:

[95] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon

[96] HY Chen and CT Chan, the brief report for the detail of the causality constraint. [97] T. Koschny, M. Kafesaki, E. N. Economou and C. M. Soukoulis, Phys. Rev. Lett. 93,

[74] Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, Phys. Rev. Lett. 99, 113903 (2007).

[77] X. Jiang, W. Han, P. Yao, and W. Li, Appl. Phys. Lett. 89, 221102 (2006). [78] P. Yao, W. Li, S. Feng, and X. Jiang, Opt. Express 14, 12295 (2006).

*Time-Domain Method*, 2nd ed. (Artech House, Boston, 2000).

Graeme W. Milton, Appl. Phys. Lett. 91, 111105 (2007).

[86] Huanyang Chen, Xunya Jiang, C. T. Chan, arXiv:0707.1126v2.

[88] H. Chen and C. T. Chan, Appl. Phys. Lett. 90, 241105 (2007). [89] Ulf Leonhardt and Thomas G Philbin, New J. Phys. 8, 247(2006). [90] J. B. Pendry and D. R. Smith, Phys. Rev. Lett. 90, 029703 (2003).

simulation, we limit its maximum value to 103.

//www.arXiv:/0703059[math-ph] (2007).

Press, 1975), ch. 11, pp. 315-321

107402 (2004).

[87] D. Schurig, J. B. Pendry, and D. R. Smith, Optics Express 14, 9794(2006).

[92] Zixian Liang, Peijun Yao, Xunya Jiang, and Xiaowei Sun, unpublished. [93] G. W. Milton, M. Briane, and J. R. Willis, New J. Phys. 8, 248 (2006).

[75] U. Leonhardt, New J. Phys. 8, 118 (2006). [76] S. A. Cummer, Appl. Phys. Lett. 82, 2008 (2003).

unpublished.

(2007).

Employment of metamaterials in cloaking devices, which could make concealed objects invisible, is based on the capability of resonators used as metamaterial "atoms" to provide effective material parameters, ranging from any positive to any negative values, in the vicinity of the resonance frequency. Metamaterials comprising layers with variously sized resonators (Schurig et al., 2006) or layers with variable density of identical resonators (Semouchkina et al., 2010) have been proposed to obtain the desired spatial distribution of the medium parameters, for example, radial variation of the effective permittivity or permeability in the shell prescribed by the transformation optics relations for cloaking cylindrically shaped objects (Leonhardt, 2006; Pendry et al., 2006). Such approach could be justified, if resonators contribute to the value of the effective parameters by the same way as polarized atoms contribute to the value of the polarization of a dielectric medium, i.e., if the responses of individual atoms are additive so that the medium response exhibits the same dispersion as the response of an individual atom. Then the material can be described by the effective medium theory (EMT), which was adopted for metamaterails description at the onset of metamaterial studies (Pendry et al., 1999; Pendry & Smith, 2003; Smith & Pendry, 2004; Smith et al., 2006). According to the EMT, a polar dielectric material could be represented by a set of parallel dipoles switching their directions by 1800 in dependence on the phase of the external field. Resonance modes in the "atoms" of metamaterials could also be represented by equivalent dipoles and, similarly to atoms in dielectrics, these dipoles are expected to respond accordingly, i.e. to follow the phase of the external field and demonstrate a coherent response within the half wavelength of incident radiation. However, observation of the performance of the infrared invisibility cloak designed from glass resonators (Semouchkina et al., 2010) has shown that obtaining a coherent response of the cloak structure presents a serious problem, if resonators in the metamaterail are electromagnetically coupled. The problem of coupling between resonators in metamaterials, however, remains to be largely ignored in the literature.

It is well known that metamaterials, as a rule, are designed as periodic structures of closepacked resonators. When the dimensions of resonators are ten times smaller than the wavelength, the EMT is considered to be applicable, so that the waves "do not see" the atoms and propagate in the metamaterial as in a homogenized, i.e. in a uniform medium. Such homogenization is supposed to make metamaterials essentially different from

Formation of Coherent Multi-Element Resonance States in Metamaterials 93

Fig. 1. Cross-section of the metamaterial block depicting SRR sections (longer strips) and cut wires (shorter strips) placed normally to the figure plane (left figure); and distributions of magnetic field amplitude (upper row) and phase (lower row) in the same cross-section at four frequencies within the metamaterial transmission band. Bright spots in the upper row

This Chapter addresses the listed above underexplored metamaterial problems and describes the effects caused by coupling and splitting of resonances in two types of metamaterial multi-resonator structures, i.e. those composed of SRRs and of dielectric resonators. The results presented here demonstrate that EM responses provided by these structures cannot be reduced to the responses of their constituent single resonators as it is assumed at application of the EMT. The Chapter also shows that splitting of resonances in arrays of dielectric resonators leads to the formation of a bandgap dividing two transmission zones, one at lower, and another one at higher frequncies. While at the gap frequencies elementary resonances in arrays look sporadically flashing at various multiple frequencies, the lower frequency transmission zone is characterized by overlapping of the resonance fields and slow wave propagation. The transmission zone at higher frequencies can be characterized by a coherent response of elementary resonators within halfwavelength distances and by superluminal phase velocities of propagating waves. The obtained results bridge the gap between the properties of metamaterials and photonic

**2. Resonance splitting in metamaterial arrays composed of SRRs** 

Investigation of the resonance splitting phenomena in SRR, which is presented in the following sub-sections, has been performed on fragments of SRR arrays quite similar to the arrays used in the design of the microwave invisibility cloak (Schurig et al., 2006). This cloaking shell was formed from concentric arrays of unit cells each including one SRR. The size of the sell was of 3.33 mm x 3.33 mm in the SRR plane and 3.18 mm normal to this plane with the SRR located in the central position. The SRRs were placed on the substrate with the

correspond to the areas with highest field magnitude.

crystals employing Mie resonances.

**2.1 Specific features of SRR characterization** 

photonic crystals (Pendry & Smith, 2003). It is worth noting that the lattice parameter was not thought to be critical for the EMT application, so that the only ultimate requirement for metamaterials in this attitude is that the lattice parameter should not be equal to the half wavelength of incident radiation, which is typical for photonic crystals. Therefore, for example, the metamaterial for the microwave cloak operating at 8.5 GHz (Schurig et al., 2006), was composed of split ring resonators (SRR) with the planar dimensions of 3 mm x 3 mm packed in concentric arrays with the inter-resonators separation of 0.17 mm, which is negligible compared to the resonator size. Such close packing of SRRs causes questions about possible effects of interaction between resonators and elementary resonance splitting that could deteriorate the EMT applicability.

Our earlier studies (Semouchkina et al., 2004, 2005) indeed pointed out at strong splitting of elementary resonances in close-packed resonator arrays of conventional metamaterials consisting of metal elements. At the modeling of an extended finite metamaterial block, which consisted of 36 SRRs and 12 elongated metal strips arranged in unit cells quite similar to typical arrangements employed in conventional metamaterials, it was shown that at the frequencies corresponding to the metamaterial transmission band, elementary resonances were essentially coupled and integrated in 3D networks, an example of which is presented in Fig. 1. As a sequence of coupling and splitting, most resonators responded resonantly at multiple frequencies and formed specific accociations at different frequencies in the transmission band, making the metamaterial essentially nonuniform with the character of inhomogenities changing in dependence on frequency. No regular wave front propagation across the sample was observed, so that the energy transfer occurred rather due to the hopping mechanism similar to that thought to be inherent in the CROWs (coupled resonator optical waveguides (Yariv et al., 1999).

It is logical to suggest that phenomena similar to the mentioned above should take place in any close-packed metamaterial structures including those used in the microwave cloak described in (Schurig et al., 2006). The reason why these phenomena have not been revealed by the authors can be seen in the employment of simulation models, in which real multiresonator cloak structures were replaced by layered material structures with prescribed values of the effective permeability for each layer.

An additional reason, why coupling phenomena in metamaterials remain underexplored is seen in the typical approach to consider metamaterial properties to be identical to the properties of a single cell and to ignore interaction between cells. This approach is based on the results of unit cell simulations which could not reveal resonance splitting even at small distances between neighboring resonators defined by the dimensions of the unit cell, because of employment of plane wave excitation source at normal wave incidence and periodic boundary conditions (PBC) at the cell boundaries normal to the direction of wave propagation. Such arangement, however, could only model an infinite 2D array of unit cells and not a 3D metamaterial medium. Our results indicate that drastic changes in electromagnetic (EM) metamaterial response could be observed at stacking 2D arrays in the direction of wave propagation to form a 3D structure, because incident wave has a different phase, when encounters each 2D array. Significant changes also occur in finite 2D arrays and in arrays with some distortions of periodicity that is unavoidable in practice.

photonic crystals (Pendry & Smith, 2003). It is worth noting that the lattice parameter was not thought to be critical for the EMT application, so that the only ultimate requirement for metamaterials in this attitude is that the lattice parameter should not be equal to the half wavelength of incident radiation, which is typical for photonic crystals. Therefore, for example, the metamaterial for the microwave cloak operating at 8.5 GHz (Schurig et al., 2006), was composed of split ring resonators (SRR) with the planar dimensions of 3 mm x 3 mm packed in concentric arrays with the inter-resonators separation of 0.17 mm, which is negligible compared to the resonator size. Such close packing of SRRs causes questions about possible effects of interaction between resonators and elementary resonance splitting

Our earlier studies (Semouchkina et al., 2004, 2005) indeed pointed out at strong splitting of elementary resonances in close-packed resonator arrays of conventional metamaterials consisting of metal elements. At the modeling of an extended finite metamaterial block, which consisted of 36 SRRs and 12 elongated metal strips arranged in unit cells quite similar to typical arrangements employed in conventional metamaterials, it was shown that at the frequencies corresponding to the metamaterial transmission band, elementary resonances were essentially coupled and integrated in 3D networks, an example of which is presented in Fig. 1. As a sequence of coupling and splitting, most resonators responded resonantly at multiple frequencies and formed specific accociations at different frequencies in the transmission band, making the metamaterial essentially nonuniform with the character of inhomogenities changing in dependence on frequency. No regular wave front propagation across the sample was observed, so that the energy transfer occurred rather due to the hopping mechanism similar to that thought to be inherent in the CROWs (coupled resonator

It is logical to suggest that phenomena similar to the mentioned above should take place in any close-packed metamaterial structures including those used in the microwave cloak described in (Schurig et al., 2006). The reason why these phenomena have not been revealed by the authors can be seen in the employment of simulation models, in which real multiresonator cloak structures were replaced by layered material structures with prescribed

An additional reason, why coupling phenomena in metamaterials remain underexplored is seen in the typical approach to consider metamaterial properties to be identical to the properties of a single cell and to ignore interaction between cells. This approach is based on the results of unit cell simulations which could not reveal resonance splitting even at small distances between neighboring resonators defined by the dimensions of the unit cell, because of employment of plane wave excitation source at normal wave incidence and periodic boundary conditions (PBC) at the cell boundaries normal to the direction of wave propagation. Such arangement, however, could only model an infinite 2D array of unit cells and not a 3D metamaterial medium. Our results indicate that drastic changes in electromagnetic (EM) metamaterial response could be observed at stacking 2D arrays in the direction of wave propagation to form a 3D structure, because incident wave has a different phase, when encounters each 2D array. Significant changes also occur in finite 2D arrays and in arrays with some distortions of periodicity that is unavoidable in

that could deteriorate the EMT applicability.

optical waveguides (Yariv et al., 1999).

practice.

values of the effective permeability for each layer.

Fig. 1. Cross-section of the metamaterial block depicting SRR sections (longer strips) and cut wires (shorter strips) placed normally to the figure plane (left figure); and distributions of magnetic field amplitude (upper row) and phase (lower row) in the same cross-section at four frequencies within the metamaterial transmission band. Bright spots in the upper row correspond to the areas with highest field magnitude.

This Chapter addresses the listed above underexplored metamaterial problems and describes the effects caused by coupling and splitting of resonances in two types of metamaterial multi-resonator structures, i.e. those composed of SRRs and of dielectric resonators. The results presented here demonstrate that EM responses provided by these structures cannot be reduced to the responses of their constituent single resonators as it is assumed at application of the EMT. The Chapter also shows that splitting of resonances in arrays of dielectric resonators leads to the formation of a bandgap dividing two transmission zones, one at lower, and another one at higher frequncies. While at the gap frequencies elementary resonances in arrays look sporadically flashing at various multiple frequencies, the lower frequency transmission zone is characterized by overlapping of the resonance fields and slow wave propagation. The transmission zone at higher frequencies can be characterized by a coherent response of elementary resonators within halfwavelength distances and by superluminal phase velocities of propagating waves. The obtained results bridge the gap between the properties of metamaterials and photonic crystals employing Mie resonances.

### **2. Resonance splitting in metamaterial arrays composed of SRRs**

#### **2.1 Specific features of SRR characterization**

Investigation of the resonance splitting phenomena in SRR, which is presented in the following sub-sections, has been performed on fragments of SRR arrays quite similar to the arrays used in the design of the microwave invisibility cloak (Schurig et al., 2006). This cloaking shell was formed from concentric arrays of unit cells each including one SRR. The size of the sell was of 3.33 mm x 3.33 mm in the SRR plane and 3.18 mm normal to this plane with the SRR located in the central position. The SRRs were placed on the substrate with the

Formation of Coherent Multi-Element Resonance States in Metamaterials 95

been characterized. The values of *s* and *r* were taken very close to those characteristic for the SRRs of the 1st, 2nd, and 3rd concentric arrays of the microwave cloak described in (Schurig et al., 2006). Small corrections, however, have been introduced in order to provide fitting of the chosen *s* and *r* to the least square dependences which were built to express interrelation between these parameters of SRRs and the effective permeability found in (Schurig et al.,

mm mm GHz GHz GHz GHz

7.761

9.125

9.061

As seen from Table 1, numerical experiments with plane wave incidence and with the standard waveguide gave quite close results for *res f* , while in experiments with unit cells used in (Schurig et al., 2006) this frequency appeared to be essentially lower both for SRRs placed in air and on a substrate. The resulting S-parameter spectra in the latter simulations appeared distorted in comparison to the spectra obtained in the standard waveguide. Similar distortions of resonances are known to be observed at too close placement of metal objects or the domain boundaries to the resonator. Therefore, the simulations have been repeated for computational volumes of increased dimensions. Gradual increase of the volume caused better and better fitting of the simulation results to the data obtained in two other sets of numerical experiments. Saturation of changes and coincidence of the data was observed at the distances between resonator and domain boundaries exceeding 1.833 mm in the SRR plane and 4.183 mm in the normal direction. Apparently these dimensions characterize the "halo" of the resonance fields around the SRRs, illustrated by the field

It is obvious that the space occupied by the SRR field "halo" substantially exceeds the volume of unit cells used in (Schurig et al., 2006). As seen in Figs. 2a and 2c, the resonance magnetic and electric fields in the "halo" are not vanishing at the distances from the SRR

*free space fres*

7.673 7.657

7.745 7.19 7.76

*unit cell f res*

7.125

9.16 9.119 8.53 9.17

7.73 7.67 7.73

9.075

*waveguide f res*

Transient solver Frequency domain

8.448

9.03 9.068

7.027 7.673

8.49 9.185

solver

*halo cell f res*

2006) for unit cells with SRRs of proper dimensions.

*r*

*substrate*

yes

no

yes

no

no

Table 1. Parameters and resonance frequencies of various SRRs

yes

1 1.654 0.26

2 1.677 0.254

patterns presented in Figs. 2a and 2c.

1.716 0.244

*s*

N

Table 1

3

thickness of 381 microns and the relative permittivity of 2.33. In the simulations described in the next sub-section, the SRRs with identical parameters (Fig. 2a) were used to build arrays of unit cells similar to various fragments of the cloak described in (Schurig et al., 2006). The only difference between the investigated metamaterial fragments and those used in the publication was that fragments in our case were flat. However, as the radiuses of concentric arrays in (Schurig et al., 2006) were relatively big compared to the cell size, flat arrays with the lengths of up to six cells were a good approaximation of fractions of concentric arrays in the cloak metamaterial. The propagation vector (k-vector) of the incident wave in our studies was always directed along the array rows in the plane of the SRR location, while magnetic field was directed normally to the SRR planes. Therefore, the investigated arrays had to perform similarly to the cloak fragments located close to the diameter of the shell normal to the k-vector of the incident wave. First, the SRR responses in air and at the placement of the resonators on the dielectric substrate have been investigated. Numerical simulations were performed for three arrangements: 1) at the plane wave incidence on the object, 2) at the placement of the SRR in the waveguide of standard dimensions with perfect electric conductor (PEC) walls, and 3) at application of waveguide ports to the standard unit cell and/or to the unit cell increased to incorporate the resonance field "halos" (the following section describes how the "halo" dimensions were found).

Fig. 2. Resonance in a single SRR placed in the waveguide WR137: (a) measured and simulated transmission spectra S21; (b) arrangement for the measurements; (c) magnetic and (d) electric field patterns.

All simulations were conducted by using the commercial full-wave software package CST Microwave Studio. For the two first arrangements the Transient Solver has been used, while for the third one – the Frequency Domain Solver. Numerical experiments with waveguide ports allowed for obtaining the scattering parameter spectra convenient for controlling the resonance frequency *res f* , while in experiments with plane wave incidence, the resonance was detected by monitoring the signal of H-field probes placed at the expected maximum field locations in the resonators. Three types of SRRs that differed by the lengths of their inner slot forming strips *s* and the radii of the corners *r,* as summarized in Table 1, have

thickness of 381 microns and the relative permittivity of 2.33. In the simulations described in the next sub-section, the SRRs with identical parameters (Fig. 2a) were used to build arrays of unit cells similar to various fragments of the cloak described in (Schurig et al., 2006). The only difference between the investigated metamaterial fragments and those used in the publication was that fragments in our case were flat. However, as the radiuses of concentric arrays in (Schurig et al., 2006) were relatively big compared to the cell size, flat arrays with the lengths of up to six cells were a good approaximation of fractions of concentric arrays in the cloak metamaterial. The propagation vector (k-vector) of the incident wave in our studies was always directed along the array rows in the plane of the SRR location, while magnetic field was directed normally to the SRR planes. Therefore, the investigated arrays had to perform similarly to the cloak fragments located close to the diameter of the shell normal to the k-vector of the incident wave. First, the SRR responses in air and at the placement of the resonators on the dielectric substrate have been investigated. Numerical simulations were performed for three arrangements: 1) at the plane wave incidence on the object, 2) at the placement of the SRR in the waveguide of standard dimensions with perfect electric conductor (PEC) walls, and 3) at application of waveguide ports to the standard unit cell and/or to the unit cell increased to incorporate the resonance field "halos" (the

k H

E

(b)

(b)

simulated measured

Fig. 2. Resonance in a single SRR placed in the waveguide WR137: (a) measured and simulated transmission spectra S21; (b) arrangement for the measurements; (c) magnetic and

All simulations were conducted by using the commercial full-wave software package CST Microwave Studio. For the two first arrangements the Transient Solver has been used, while for the third one – the Frequency Domain Solver. Numerical experiments with waveguide ports allowed for obtaining the scattering parameter spectra convenient for controlling the resonance frequency *res f* , while in experiments with plane wave incidence, the resonance was detected by monitoring the signal of H-field probes placed at the expected maximum field locations in the resonators. Three types of SRRs that differed by the lengths of their inner slot forming strips *s* and the radii of the corners *r,* as summarized in Table 1, have

following section describes how the "halo" dimensions were found).

6.5 7 7.5 8 8.5 9

Frequency (GHz)

(c) (d)


S21 (dB)

(d) electric field patterns.

(d) (a)

been characterized. The values of *s* and *r* were taken very close to those characteristic for the SRRs of the 1st, 2nd, and 3rd concentric arrays of the microwave cloak described in (Schurig et al., 2006). Small corrections, however, have been introduced in order to provide fitting of the chosen *s* and *r* to the least square dependences which were built to express interrelation between these parameters of SRRs and the effective permeability found in (Schurig et al., 2006) for unit cells with SRRs of proper dimensions.


Table 1. Parameters and resonance frequencies of various SRRs

As seen from Table 1, numerical experiments with plane wave incidence and with the standard waveguide gave quite close results for *res f* , while in experiments with unit cells used in (Schurig et al., 2006) this frequency appeared to be essentially lower both for SRRs placed in air and on a substrate. The resulting S-parameter spectra in the latter simulations appeared distorted in comparison to the spectra obtained in the standard waveguide. Similar distortions of resonances are known to be observed at too close placement of metal objects or the domain boundaries to the resonator. Therefore, the simulations have been repeated for computational volumes of increased dimensions. Gradual increase of the volume caused better and better fitting of the simulation results to the data obtained in two other sets of numerical experiments. Saturation of changes and coincidence of the data was observed at the distances between resonator and domain boundaries exceeding 1.833 mm in the SRR plane and 4.183 mm in the normal direction. Apparently these dimensions characterize the "halo" of the resonance fields around the SRRs, illustrated by the field patterns presented in Figs. 2a and 2c.

It is obvious that the space occupied by the SRR field "halo" substantially exceeds the volume of unit cells used in (Schurig et al., 2006). As seen in Figs. 2a and 2c, the resonance magnetic and electric fields in the "halo" are not vanishing at the distances from the SRR

Formation of Coherent Multi-Element Resonance States in Metamaterials 97

7.1 7.3 7.5 7.7 7.9 8.1 8.3 8.5

2-SRR 3-SRR 4-SRR 5-SRR

**degrees**

**degrees**

Frequency (GHz)

1940, 2020, 15.60, 25.10, 25.20

Fig. 3. Resonances in linear arrays arranged in wave propagation direction in the waveguide WR137: (a) S21; (b) coherent response of a 5-SRR array at 7.73 GHz; and (c) assymmetric

Much stronger splitting of SRR resonances has been revealed in the SRR columns (Fig. 4) similar to those used in concentric arrays of the SRRs in (Schurig et al., 2006). Placement of such columns in the waveguide WR137 caused some concerns, since at about 1 cm height of the columns their resonance fields could interact with the waveguide top and bottom walls that could cause deterioration of the resonances. Therefore, the simulations for these columns were initially performed in free space at plane wave incidence. As such simulations did not provide an opportunity for deriving the transmission spectra, the resonance field in the columns was monitored by using H-field probes placed in geometrically identical points of the SRR cross-sections (Fig. 4a). As seen from Fig.4e the probes revealed three resonances covering more than 1 GHz of the frequency range (i.e. almost five times wider resonance band than that observed for the linear arrays of SRRs arranged along the direction of the incident wave k—vector). It is worth mentioning that each resonator in the column responded resonantly at all split frequencies while the integrated modes at these frequencies were quite different. As seen from Fig. 4b, at the lower frequency resonance sin-phase oscillations were observed in the two upper resonators, while the third resonator oscillated with a smaller magnitude and with opposite phase. At the higher frequency resonance, the two lower resonators demonstrated sin-phase oscillations while the third resonator oscillated with opposite phase (Fig. 4d). Resonance at the median frequency was supported mainly by sin-phase oscillations in the upper and the lower resonators of the column (Fig. 4c). The third resonator located in the middle of the column did not show strong

1130, 1110, 1110, 1100, 1110

S21 (dB)


(b)

(c)

response at 7.95 GHz.



0

(a)

corresponding to the boundaries of the used unit cells, therefore, the resonance formation and the resonance frequencies *res f* were affected in respective simulations of such unit cells. In comparison, field magnitudes at the boundaries of unit cells with the size equal to the "halo" size were found to be ten times smaller than those of the cells used in (Schurig et al., 2006) that explains the consistency of the data obtained for the halo-sized cells and the results of other numerical experiments. Only at building the shell of such cells, i.e. at the placement of resonators in arrays of the metamaterial at distances, when "halos" are not overlapped, interaction between resonators could be omitted from consideration. In contrast, the distance between resonators in concentric arrays in (Schurig et al., 2006) did not exceed 0.333 mm, while the distance between arrays (i.e. along the normal to the plane of resonators) was only 3.183 mm.

The data presented in Table 1 do not confirm the expectations for *res f* to be of about 8.5 GHz as reported in (Schurig et al., 2006) for both the resonance frequency of basic SRRs and the frequency of the cloak response. Resonances at frequencies close to 8.5 GHz were obtained in our experiments only for the unit cells used in (Schurig et al., 2006) at the placement of SRRs in air. It might be that the dielectric substrate was omitted in models used for simulations in (Schurig et al., 2006). The obtained discrepancy does not seem to be critical for the studies of the splitting phenomena in SRR-based metamaterials employed in (Schurig et al., 2006), although it should be taken into account at verification of the cloaking effect, since this effect should not be expected at 8.5 GHz, if the metamaterial perform in accordance to the EMT predictions.

#### **2.2 Resonances in SRR arrays representing fragments of the microwave cloak**

In order to additionally verify the results of simulations, individual SRRs and various multielement arrays of SRRs with the same dimensions as in (Schurig et al., 2006) have been fabricated, and their EM response has been measured in the waveguide WR137 operating at the single TE10 mode in the range (5-10) GHz. In experiments, the waveguide WR137 loaded with SRR arrays was connected to the PNA-L Network Analyzer N5230A to measure the transmission spectrum. In order to avoid EM interaction between the arrays and waveguide walls, resonators or arrays were supported by a styrofoam layer inserted into the center region of the waveguide. As seen from Fig. 2d, the resonance in a single resonator placed in the waveguide was observed at f=7.78 GHz and not at 8.5 GHz, i.e. quite close to the results of our simulations for the resonator with the same dimensions.

Next, linear arrays of SRRs arranged along the direction of k-vector of the incident wave have been investigated. Fig. 3a shows that the arrays of 3-5 SRRs demonstrate strong splitting of the resonance, which increases with the number of resonators in the array. It is interesting to note, that instead of multiple dips in the transmission spectrum correlated with the quantity of resonators in the array, only two well expressed drops in the transmission spectra have been observed. The first drop is accompanied by the coherent response of resonators (Fig. 3b), while at the higher frequency drop the resonators in the line respond with opposite phases in two halves of the array. The phase patterns of the wave propagating in free space inside the waveguide at both resonances show strong disturbances of the wave transmission through the waveguide (the curve between bright and dark parts of the image divides the areas corresponding to the half wavelengths of the propagating waves).

corresponding to the boundaries of the used unit cells, therefore, the resonance formation and the resonance frequencies *res f* were affected in respective simulations of such unit cells. In comparison, field magnitudes at the boundaries of unit cells with the size equal to the "halo" size were found to be ten times smaller than those of the cells used in (Schurig et al., 2006) that explains the consistency of the data obtained for the halo-sized cells and the results of other numerical experiments. Only at building the shell of such cells, i.e. at the placement of resonators in arrays of the metamaterial at distances, when "halos" are not overlapped, interaction between resonators could be omitted from consideration. In contrast, the distance between resonators in concentric arrays in (Schurig et al., 2006) did not exceed 0.333 mm, while the distance between arrays (i.e. along the normal to the plane of

The data presented in Table 1 do not confirm the expectations for *res f* to be of about 8.5 GHz as reported in (Schurig et al., 2006) for both the resonance frequency of basic SRRs and the frequency of the cloak response. Resonances at frequencies close to 8.5 GHz were obtained in our experiments only for the unit cells used in (Schurig et al., 2006) at the placement of SRRs in air. It might be that the dielectric substrate was omitted in models used for simulations in (Schurig et al., 2006). The obtained discrepancy does not seem to be critical for the studies of the splitting phenomena in SRR-based metamaterials employed in (Schurig et al., 2006), although it should be taken into account at verification of the cloaking effect, since this effect should not be expected at 8.5 GHz, if the metamaterial perform in

**2.2 Resonances in SRR arrays representing fragments of the microwave cloak** 

of our simulations for the resonator with the same dimensions.

corresponding to the half wavelengths of the propagating waves).

In order to additionally verify the results of simulations, individual SRRs and various multielement arrays of SRRs with the same dimensions as in (Schurig et al., 2006) have been fabricated, and their EM response has been measured in the waveguide WR137 operating at the single TE10 mode in the range (5-10) GHz. In experiments, the waveguide WR137 loaded with SRR arrays was connected to the PNA-L Network Analyzer N5230A to measure the transmission spectrum. In order to avoid EM interaction between the arrays and waveguide walls, resonators or arrays were supported by a styrofoam layer inserted into the center region of the waveguide. As seen from Fig. 2d, the resonance in a single resonator placed in the waveguide was observed at f=7.78 GHz and not at 8.5 GHz, i.e. quite close to the results

Next, linear arrays of SRRs arranged along the direction of k-vector of the incident wave have been investigated. Fig. 3a shows that the arrays of 3-5 SRRs demonstrate strong splitting of the resonance, which increases with the number of resonators in the array. It is interesting to note, that instead of multiple dips in the transmission spectrum correlated with the quantity of resonators in the array, only two well expressed drops in the transmission spectra have been observed. The first drop is accompanied by the coherent response of resonators (Fig. 3b), while at the higher frequency drop the resonators in the line respond with opposite phases in two halves of the array. The phase patterns of the wave propagating in free space inside the waveguide at both resonances show strong disturbances of the wave transmission through the waveguide (the curve between bright and dark parts of the image divides the areas

resonators) was only 3.183 mm.

accordance to the EMT predictions.

Fig. 3. Resonances in linear arrays arranged in wave propagation direction in the waveguide WR137: (a) S21; (b) coherent response of a 5-SRR array at 7.73 GHz; and (c) assymmetric response at 7.95 GHz.

Much stronger splitting of SRR resonances has been revealed in the SRR columns (Fig. 4) similar to those used in concentric arrays of the SRRs in (Schurig et al., 2006). Placement of such columns in the waveguide WR137 caused some concerns, since at about 1 cm height of the columns their resonance fields could interact with the waveguide top and bottom walls that could cause deterioration of the resonances. Therefore, the simulations for these columns were initially performed in free space at plane wave incidence. As such simulations did not provide an opportunity for deriving the transmission spectra, the resonance field in the columns was monitored by using H-field probes placed in geometrically identical points of the SRR cross-sections (Fig. 4a). As seen from Fig.4e the probes revealed three resonances covering more than 1 GHz of the frequency range (i.e. almost five times wider resonance band than that observed for the linear arrays of SRRs arranged along the direction of the incident wave k—vector). It is worth mentioning that each resonator in the column responded resonantly at all split frequencies while the integrated modes at these frequencies were quite different. As seen from Fig. 4b, at the lower frequency resonance sin-phase oscillations were observed in the two upper resonators, while the third resonator oscillated with a smaller magnitude and with opposite phase. At the higher frequency resonance, the two lower resonators demonstrated sin-phase oscillations while the third resonator oscillated with opposite phase (Fig. 4d). Resonance at the median frequency was supported mainly by sin-phase oscillations in the upper and the lower resonators of the column (Fig. 4c). The third resonator located in the middle of the column did not show strong

Formation of Coherent Multi-Element Resonance States in Metamaterials 99

responses with respect to the central row i.e. the resonators located in upper and lower rows responded coherently, even though neighbouring columns demonstrated opposite phases of oscillations. So, it could be said that every other column responded coherently (Fig. 6c).

90.40

90.40

1340 1340

> 4 5

13 14

15

7 8

9

6

6.5 7 7.5 8 8.5 9

6.5 7 7.5 8 8.5 9

column) for SRR pairs arranged: (a) "face-to-face" and (b) "back-to-back ".

6.95 GHz 7.7 GHz

frequency, GHz

Fig. 5. Geometry, S21 spectra, and patterns of H-field magnitude (3d column) and phase (4th

6.5 7.0 7.5 8.0 8.5 8.8 frequency, GHz

(b) (c) (d)8.15 GHz

Fig. 6. (a) The spectra of the probe signals; inset shows the SRR array and the H-field probe locations; (b)-(d) typical patterns of the resonance oscillations in the array sampled at 6.64,

Despite of the above mentioned concerns about possible interaction between multi-element arrays and waveguide walls at the measurements we have performed comparison of simulation and measurement results for the transmission spectra at the placement of the 3 x 5 arrays of SRRs in the waveguide WR137. A good matching of the results presented in Fig. 7 is obvious, although both S21 spectra demostrate a smaller quantity of the transmission

7.4 and 8.15 GHz (colour intensity shows the magnitude of resonance oscillations).

1 2 10 11

12

3


S21(dB)

*E H k*

(a)

(b)

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.0

(a)

Probe Magnitude in A/m



0


0

S21(dB)

oscillations, however, the phase of these oscillations was shifted by 1800 with respect to the oscillations of two other SRRs.

Fig. 4. (a) The model of the SRR column with H-field probes; (b)-(d) H-field resonance responses of the column at signal peak frequencies (colour intensity shows the magnitude of resonance; and (e) the spectra of probe signals.

It is interesting to note that shorted columns of 2 SRRs placed in either "face-to-face" or "back-to-back" arrangement did not demonstrate any splitting of the resonances. As seen from Fig. 5, the former arrangement provided for the unsplit resonance at almost the same frequency as the lower frequency resonance in the column of 3 SRRs, while the latter arrangement provided for the resonance at the frequency close to the frequency of upper resonance in the column of 3 SRRs. It means that the listed above resonances in a 3-SRR column are apparently related to integrated coherent resonances in couples of oppositely arranged SRRs. The coherentness is evident from the phase patterns presented in Fig. 5 on the right.

Essentially more complicated response with multiple resonances of various Q factors was demonstrated by arrays composed of several columns as, for example, the presented by the insert in Fig 6a array of 5 columns. Despite of the complexity, however, it was possible to distinguish three main groups of resonances comparable to resonances revealed in one SRR column (Fig. 4). In fact, the lower frequency group in the 3 x 5 SRR array demontsrated coherent resonances in upper rows of the array (Fig. 6b), while the higher frequency group demontsrated coherent resonances in the lower rows of the array (Fig. 6d). The group of resonances at the median frequencies could be characterized by a symmetry of DR

oscillations, however, the phase of these oscillations was shifted by 1800 with respect to the

(a) (b) (c) (d)

3

2

1

1 - 2 - 3 - 6.9 GHz 7.5 GHz 7.9 GHz

6.2 6.5 7.0 7.5 8.0 8.5 Frequency, GHz

Fig. 4. (a) The model of the SRR column with H-field probes; (b)-(d) H-field resonance responses of the column at signal peak frequencies (colour intensity shows the magnitude of

It is interesting to note that shorted columns of 2 SRRs placed in either "face-to-face" or "back-to-back" arrangement did not demonstrate any splitting of the resonances. As seen from Fig. 5, the former arrangement provided for the unsplit resonance at almost the same frequency as the lower frequency resonance in the column of 3 SRRs, while the latter arrangement provided for the resonance at the frequency close to the frequency of upper resonance in the column of 3 SRRs. It means that the listed above resonances in a 3-SRR column are apparently related to integrated coherent resonances in couples of oppositely arranged SRRs. The coherentness is evident from the phase patterns presented in Fig. 5 on

Essentially more complicated response with multiple resonances of various Q factors was demonstrated by arrays composed of several columns as, for example, the presented by the insert in Fig 6a array of 5 columns. Despite of the complexity, however, it was possible to distinguish three main groups of resonances comparable to resonances revealed in one SRR column (Fig. 4). In fact, the lower frequency group in the 3 x 5 SRR array demontsrated coherent resonances in upper rows of the array (Fig. 6b), while the higher frequency group demontsrated coherent resonances in the lower rows of the array (Fig. 6d). The group of resonances at the median frequencies could be characterized by a symmetry of DR

oscillations of two other SRRs.

Probe magnitude in A/m

resonance; and (e) the spectra of probe signals.

the right.

(e)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 responses with respect to the central row i.e. the resonators located in upper and lower rows responded coherently, even though neighbouring columns demonstrated opposite phases of oscillations. So, it could be said that every other column responded coherently (Fig. 6c).

Fig. 5. Geometry, S21 spectra, and patterns of H-field magnitude (3d column) and phase (4th column) for SRR pairs arranged: (a) "face-to-face" and (b) "back-to-back ".

Fig. 6. (a) The spectra of the probe signals; inset shows the SRR array and the H-field probe locations; (b)-(d) typical patterns of the resonance oscillations in the array sampled at 6.64, 7.4 and 8.15 GHz (colour intensity shows the magnitude of resonance oscillations).

Despite of the above mentioned concerns about possible interaction between multi-element arrays and waveguide walls at the measurements we have performed comparison of simulation and measurement results for the transmission spectra at the placement of the 3 x 5 arrays of SRRs in the waveguide WR137. A good matching of the results presented in Fig. 7 is obvious, although both S21 spectra demostrate a smaller quantity of the transmission

Formation of Coherent Multi-Element Resonance States in Metamaterials 101

5.75 6.0 6.5 7.0 7.5 8.0 8.5 9.0 frequency, GHz

Fig. 8. (a) Geometry of a three- layer SRR array with marked probe locations; (b) coherent response of SRRs at 8.35 GHz; and (c) spectra of probe signals revealing enhanced splitting

The publication (Semouchkina et al., 2004) was among the first that started the development of all-dielectric metamaterials. The data presented in Fig. 9 confirm the possibility to use dielectric cylinders of a proper size (the diameter of 6 mm and the height of 3 mm, in this case) and permittivity (dielectric constant of 37.2 in this case) to replace SRRs as magnetic components in metamaterials supporting Lorentz-type resonance necessary for obtaining the effective negative permeability of the medium at microwave frequencies. It was later shown that dielectric resonators can also replace cut metal wires used in conventional metamaterials below their plasma frequency as electric components providing for the effective negative permittivity. Therefore, it was predicted that materials composed of two types of dielectric resonators providing one for electric and another one – for magnetic response could exhibit negative index of refraction (Jylha et al., 2006; Vendik & Gashinova, 2004). In the work (Semouchkina et al., 2005) it was demonstrated that negative refraction can also be provided by dielectric arrays composed of resonators of one type at the placement of the array on a ground plane. Similar effects in arrays composed of DRs of one type with ground planes or in waveguides have been also reported in (Ueda et al., 2007,

E H

k

caused by interaction between SRRs in neighbouring layers.

(a) (b)

0.55 0.5 0.45 0.4 0.35 0.3 025 0.2 0.15 0.1 0.05 0.0

(c)

Probe Signal Magnitude in A/m

2010).

dips compared to that observed in the probe signal spectra (Fig. 7). The three groups of resonances, however, could be clearly distinguished in both spectra presented in Fig. 7.

Fig. 7. S21 spectra for the 3 x 5 array of SRRs in the waveguide WR137: dashed curve simulation results, solid curve - the results of measurements.

The responses of multi-layer structures composed of several planar arrays were found to loose the perspicuity of the splitting phenomena observed in the multicolumn arrays. As seen in Fig. 8c, which presents the probe signal spectra for a three-layer array of SRRs (Fig. 8a), it is difficult to distinguish well separated in frequency groups of resonances similar to those observed for 3 x 5 arrays. Instead, split resonances are observed in the essentially extended frequency range of more than 2 GHz. Such character of the multi-layer array response makes it quite doubtful to expect that the proposed in (Schurig et al. 2006) design of cylindrical cloak based on close-packed arrays of SRRs would provide the cloaking effect at the frequency determined from the analysis of the response of a single resonator. The reason for obtaining the invisibility effect in (Schurig et al., 2006) should be rather searched for in some specifics of split resonances, which need to be studied additionally. It is worth noting that some of split modes indeed demonstrated a possibility to support the desired coherent response of SRRs across extended areas of investigated arrays as it is shown in Fig. 8b. It is reasonable to expect that at respective frequencies the participating resonators could contribute to obtaining the desired dispersion of the effective parameters precribed by the transformation optics relations.

#### **3. Resonances in metamaterial arrays composed of dielectric resonators**

#### **3.1 Specifics of resonances in dielectric resonators and potential of all-dielectric metamaterials**

Dielectric resonators (DRs), especially those of cylindrical shape, are capable of providing EM response quite similar to that of SRRs. In fact, when the incident wave propagates normally to the axis of the cylinder with its H-field directed along this axis, the DR can support the TE01δ resonance mode, which is equivalent to the formation of a magnetic dipole along the axis of the cylinder, i.e normal to the plane, in which the cross-section of the DR is a circle (Kajfez & Guillon, 1998). The latter plane should host circular displacement currents comparable to circular currents in SRRs.

dips compared to that observed in the probe signal spectra (Fig. 7). The three groups of resonances, however, could be clearly distinguished in both spectra presented in Fig. 7.

6 6.5 7 7.5 8 8.5 9 9.5

simulated measured

Frequency (GHz)

The responses of multi-layer structures composed of several planar arrays were found to loose the perspicuity of the splitting phenomena observed in the multicolumn arrays. As seen in Fig. 8c, which presents the probe signal spectra for a three-layer array of SRRs (Fig. 8a), it is difficult to distinguish well separated in frequency groups of resonances similar to those observed for 3 x 5 arrays. Instead, split resonances are observed in the essentially extended frequency range of more than 2 GHz. Such character of the multi-layer array response makes it quite doubtful to expect that the proposed in (Schurig et al. 2006) design of cylindrical cloak based on close-packed arrays of SRRs would provide the cloaking effect at the frequency determined from the analysis of the response of a single resonator. The reason for obtaining the invisibility effect in (Schurig et al., 2006) should be rather searched for in some specifics of split resonances, which need to be studied additionally. It is worth noting that some of split modes indeed demonstrated a possibility to support the desired coherent response of SRRs across extended areas of investigated arrays as it is shown in Fig. 8b. It is reasonable to expect that at respective frequencies the participating resonators could contribute to obtaining the desired dispersion of the effective parameters precribed by the

Fig. 7. S21 spectra for the 3 x 5 array of SRRs in the waveguide WR137: dashed curve -

**3. Resonances in metamaterial arrays composed of dielectric resonators 3.1 Specifics of resonances in dielectric resonators and potential of all-dielectric** 

Dielectric resonators (DRs), especially those of cylindrical shape, are capable of providing EM response quite similar to that of SRRs. In fact, when the incident wave propagates normally to the axis of the cylinder with its H-field directed along this axis, the DR can support the TE01δ resonance mode, which is equivalent to the formation of a magnetic dipole along the axis of the cylinder, i.e normal to the plane, in which the cross-section of the DR is a circle (Kajfez & Guillon, 1998). The latter plane should host circular displacement currents

simulation results, solid curve - the results of measurements.


transformation optics relations.

comparable to circular currents in SRRs.

**metamaterials**




S21 (dB)


0

Fig. 8. (a) Geometry of a three- layer SRR array with marked probe locations; (b) coherent response of SRRs at 8.35 GHz; and (c) spectra of probe signals revealing enhanced splitting caused by interaction between SRRs in neighbouring layers.

The publication (Semouchkina et al., 2004) was among the first that started the development of all-dielectric metamaterials. The data presented in Fig. 9 confirm the possibility to use dielectric cylinders of a proper size (the diameter of 6 mm and the height of 3 mm, in this case) and permittivity (dielectric constant of 37.2 in this case) to replace SRRs as magnetic components in metamaterials supporting Lorentz-type resonance necessary for obtaining the effective negative permeability of the medium at microwave frequencies. It was later shown that dielectric resonators can also replace cut metal wires used in conventional metamaterials below their plasma frequency as electric components providing for the effective negative permittivity. Therefore, it was predicted that materials composed of two types of dielectric resonators providing one for electric and another one – for magnetic response could exhibit negative index of refraction (Jylha et al., 2006; Vendik & Gashinova, 2004). In the work (Semouchkina et al., 2005) it was demonstrated that negative refraction can also be provided by dielectric arrays composed of resonators of one type at the placement of the array on a ground plane. Similar effects in arrays composed of DRs of one type with ground planes or in waveguides have been also reported in (Ueda et al., 2007, 2010).

Formation of Coherent Multi-Element Resonance States in Metamaterials 103

**<sup>X</sup> <sup>Z</sup>**

Fig. 10. The design of the infrared invisibility cloak composed of cylindrical glass resonators arranged in concentric arrays; lower inset shows the spokes of resonators with spacers

Despite the described advances the application of dielectric metamaterials in cloaking structures has underlined once again the main problem common for all resonance metamaterials – their narrowbandness. According to initial works on cloaking it was thought that application of such metamaterials excudes other than zero bandwidth for the invisibility effects (Pendry et al., 2006). Accordingly, the described above invisibility cloak demonstrated good cloaking effect in the band less than 1.2% or 3 THz. However, it was also revealed that even at a slightly denser packing of resonators than that shown in Fig. 11, the bandwidth could be increased up to 2 times. This effect could be related to the positive consequencies of coupling, which are known since earlier applications of resonator arrays in microwave filters and other devices, where coupling was used for the bandwidth enhancement. The next section will demonstrate the complex nature of coupling related phenomena in dielectric metamaterials and outline the opportunity to use them for the partial lifting of the delay-bandwidth limitations for cloaking devices formulated in

**3.2 Specifics of coupling related phenomena in linear arrays of dielectric resonators**  The responses of linear arrays of DRs with the same parameters as those of the DR featured in Fig. 9, at their positioning along the wave propagation direction has been investigated first. In

(Semouchkina et al., 2010)

(Hashemi et al., 2010).

**Y**

Recently, a metamaterial composed of dielectric resonators has been employed to design an invisisbility cloak for the infrared range (Semouchkina et al., 2009, 2010). In difference from earlier works (Cui et al., 2009, Gaillot et al., 2008, Schurig et al., 2006) on microwave and THz cloaks, the design proposed in (Semouchkina et al., 2009, 2010) utilized identical resonators in the entire cloakling shell. Such design does not require to solve technological problems of fabricating concentric arrays of various nano-sized elements, as well as does not demand a wide band of incident light to excite differently sized resonators. In addition, employment of chalcogenide glass as a resonator material should decrease the loss related limitations for the size of concealed objects recently formulated in (Zhang et al., 2009). The estimates based on the level of the extinction coefficient at 1 micron wavelength of incident light gave more than an order less value for the loss tangent of the glass cloak than values characteristic for previously developed cloaks. The required by the transformation optics dispersion of the effective permeability in the novel cloak design was obtained by using a controlled decrease of the density of resonators from the inner layer of the cloak to the outer layer, i.e. due to radial dispersion of air fractions in the shell. In addition, the requirement to avoid both coupling between resonators and strong resonance splitting as described in previous sections has been satisfied that has led to a spoke-type arrangement of the resonators in the cloak depicted in Fig. 10. The intent to exclude overlapping of the resonance fields "halos" limited minimal angular distances between spokes and minimal gaps between concentric arrays of resonators by the "halo" size, however, it did not deteriorate the cloak performance. The effective parameters of the cloak layers have been characterized by using mixing relations incorporating responses of air fractions and individual resonators (the latter were assigned not to the physical resonator bodies, but to the "halo" volumes). This approach allowed for accurate adjustment of the permeability dispersion, so that the desired performance of the cloak with no distortions of the wave front at frequencies within the cloaking band has been provided.

Fig. 9. (a) Simulated and measured S12 spectra for a DR in WR137; (b) electric and (c) magnetic field patterns in DR cross-sections; and (c) effective permeability changes at the magnetic-type resonance in the DR (Chen et al., 2011).

Recently, a metamaterial composed of dielectric resonators has been employed to design an invisisbility cloak for the infrared range (Semouchkina et al., 2009, 2010). In difference from earlier works (Cui et al., 2009, Gaillot et al., 2008, Schurig et al., 2006) on microwave and THz cloaks, the design proposed in (Semouchkina et al., 2009, 2010) utilized identical resonators in the entire cloakling shell. Such design does not require to solve technological problems of fabricating concentric arrays of various nano-sized elements, as well as does not demand a wide band of incident light to excite differently sized resonators. In addition, employment of chalcogenide glass as a resonator material should decrease the loss related limitations for the size of concealed objects recently formulated in (Zhang et al., 2009). The estimates based on the level of the extinction coefficient at 1 micron wavelength of incident light gave more than an order less value for the loss tangent of the glass cloak than values characteristic for previously developed cloaks. The required by the transformation optics dispersion of the effective permeability in the novel cloak design was obtained by using a controlled decrease of the density of resonators from the inner layer of the cloak to the outer layer, i.e. due to radial dispersion of air fractions in the shell. In addition, the requirement to avoid both coupling between resonators and strong resonance splitting as described in previous sections has been satisfied that has led to a spoke-type arrangement of the resonators in the cloak depicted in Fig. 10. The intent to exclude overlapping of the resonance fields "halos" limited minimal angular distances between spokes and minimal gaps between concentric arrays of resonators by the "halo" size, however, it did not deteriorate the cloak performance. The effective parameters of the cloak layers have been characterized by using mixing relations incorporating responses of air fractions and individual resonators (the latter were assigned not to the physical resonator bodies, but to the "halo" volumes). This approach allowed for accurate adjustment of the permeability dispersion, so that the desired performance of the cloak with no distortions of the wave front at frequencies

9.103 4.103 2.103 6.102 0

Permeability μ

r , rel. units

7 8 9 10 frequency, GHz

V/m

A/m

within the cloaking band has been provided.

magnetic-type resonance in the DR (Chen et al., 2011).

Fig. 9. (a) Simulated and measured S12 spectra for a DR in WR137; (b) electric and (c) magnetic field patterns in DR cross-sections; and (c) effective permeability changes at the

Fig. 10. The design of the infrared invisibility cloak composed of cylindrical glass resonators arranged in concentric arrays; lower inset shows the spokes of resonators with spacers (Semouchkina et al., 2010)

Despite the described advances the application of dielectric metamaterials in cloaking structures has underlined once again the main problem common for all resonance metamaterials – their narrowbandness. According to initial works on cloaking it was thought that application of such metamaterials excudes other than zero bandwidth for the invisibility effects (Pendry et al., 2006). Accordingly, the described above invisibility cloak demonstrated good cloaking effect in the band less than 1.2% or 3 THz. However, it was also revealed that even at a slightly denser packing of resonators than that shown in Fig. 11, the bandwidth could be increased up to 2 times. This effect could be related to the positive consequencies of coupling, which are known since earlier applications of resonator arrays in microwave filters and other devices, where coupling was used for the bandwidth enhancement. The next section will demonstrate the complex nature of coupling related phenomena in dielectric metamaterials and outline the opportunity to use them for the partial lifting of the delay-bandwidth limitations for cloaking devices formulated in (Hashemi et al., 2010).

#### **3.2 Specifics of coupling related phenomena in linear arrays of dielectric resonators**

The responses of linear arrays of DRs with the same parameters as those of the DR featured in Fig. 9, at their positioning along the wave propagation direction has been investigated first. In

Formation of Coherent Multi-Element Resonance States in Metamaterials 105

not demonstrate any resonance-like field enhancement at corresponding frequencies even though they experienced deep drops of the resonance activity at some other frequencies. Most obvious resonance–like activity was seen only at frequencies corresponding to the boundaries of the transmission gap. Phase patterns in the waveguide cross-section at these frequencies reminded the patterns described above for the pair of resonators, i.e. at the low frequency boundary of the transmission gap all neighboring resonators oscillated with opposite phases, while at the higher frequency boundary all resonators in the array

Fig. 12. (a) Simulated and (b) measured spectra of S21 for linear DR arrays arranged along k-

2270 2290 2290 48.60 48.60

1330 1330 1340 1340 1340

Fig. 13. H-field phase patterns in waveguide at (a) 8.17 GHz and (b) 8.58 GHz (Chen et al., 2011)

demonstrated coherent responses (Figs. 13a and 13b).

vector at 2 mm between DR bodies (Chen et al., 2011).

(b)

(a)

difference from close packed array of SRRs, even one pair of resonators placed at the distance between their bodies of 2 mm demonstrated resonance splitting of about 0.25 GHz (Fig. 11a). Splitting decreased at increasing the distance between the resonators, however, it was still well expressed even at the separation of 10 mm. Only 16 mm separation provided for an unsplit spectrum of S21. It is worth mentioning that this distance is close to the doubled thickness of the electric field "halo" shown in Fig. 9. This confirms the suggestion that resonance coupling and splitting result from overlapping of "halos" created by neighboring resonators.

Fig. 11. (a) S21 spectra for two DRs arranged along the axis of the waveguide WR137; (b) and (c) phase patterns for H-field distributions in the longitudinal cross-section at 2 split resonances (Chen et al., 2011).

Phase patterns for magnetic field component in the waveguide cross-section presented in Fig. 11 show that at lower frequency, resonance field oscillations in two resonators proceed with opposite phases, while at higher frequency coherent resonance oscillations are observed. Since the resonance modes can be represented by magnetic dipoles, it follows that at lower frequency, magnetic dipoles formed in two resonators remain directed oppositely at any instant, while at higher frequency they always keep parallel orientation. Similar difference between split resonances at transverse interaction of resonators (side-by-side) was observed and discussed in (Liu et al., 2010), where it was concluded that dipole interaction defines the energy difference of two resonance states, and, correspondingly, the degree of their splitting in the frequency scale. In fact, parallel magnetic dipoles are expected to repel each other since the curls of their electric fields tend to cancell each other in the gap between the resonators. Just opposite, counter-directed magnetic dipoles provide for co-directed electric fields in the gap between the resonators that should attract them to each other, thus decreasing the state energy.

At increasing the quantity of resonators in the linear DR array from 2 to 5, the split spectrum of S21 demonstrated a better expressed gap of transmission in the frequency range of about 0.45 GHz (Fig. 12). Further addition of resonators to the array led to increasing the quantity of transmission drops in the gap and to its gradual flattening. However, the signals of the probes that measured the magnitude of H-field oscillations in the centers of resonators did

difference from close packed array of SRRs, even one pair of resonators placed at the distance between their bodies of 2 mm demonstrated resonance splitting of about 0.25 GHz (Fig. 11a). Splitting decreased at increasing the distance between the resonators, however, it was still well expressed even at the separation of 10 mm. Only 16 mm separation provided for an unsplit spectrum of S21. It is worth mentioning that this distance is close to the doubled thickness of the electric field "halo" shown in Fig. 9. This confirms the suggestion that resonance coupling

8 8.1 8.2 8.3 8.4 8.5 8.6 8.7

d= 2 mm d= 6 mm d= 16 mm

Frequency (GHz)

Fig. 11. (a) S21 spectra for two DRs arranged along the axis of the waveguide WR137; (b) and

Phase patterns for magnetic field component in the waveguide cross-section presented in Fig. 11 show that at lower frequency, resonance field oscillations in two resonators proceed with opposite phases, while at higher frequency coherent resonance oscillations are observed. Since the resonance modes can be represented by magnetic dipoles, it follows that at lower frequency, magnetic dipoles formed in two resonators remain directed oppositely at any instant, while at higher frequency they always keep parallel orientation. Similar difference between split resonances at transverse interaction of resonators (side-by-side) was observed and discussed in (Liu et al., 2010), where it was concluded that dipole interaction defines the energy difference of two resonance states, and, correspondingly, the degree of their splitting in the frequency scale. In fact, parallel magnetic dipoles are expected to repel each other since the curls of their electric fields tend to cancell each other in the gap between the resonators. Just opposite, counter-directed magnetic dipoles provide for co-directed electric fields in the gap between the resonators that should attract them to each other, thus

At increasing the quantity of resonators in the linear DR array from 2 to 5, the split spectrum of S21 demonstrated a better expressed gap of transmission in the frequency range of about 0.45 GHz (Fig. 12). Further addition of resonators to the array led to increasing the quantity of transmission drops in the gap and to its gradual flattening. However, the signals of the probes that measured the magnitude of H-field oscillations in the centers of resonators did

(c) phase patterns for H-field distributions in the longitudinal cross-section at 2 split

and splitting result from overlapping of "halos" created by neighboring resonators.

(b) (c)


(a)

S21 (dB)

resonances (Chen et al., 2011).

decreasing the state energy.

not demonstrate any resonance-like field enhancement at corresponding frequencies even though they experienced deep drops of the resonance activity at some other frequencies. Most obvious resonance–like activity was seen only at frequencies corresponding to the boundaries of the transmission gap. Phase patterns in the waveguide cross-section at these frequencies reminded the patterns described above for the pair of resonators, i.e. at the low frequency boundary of the transmission gap all neighboring resonators oscillated with opposite phases, while at the higher frequency boundary all resonators in the array demonstrated coherent responses (Figs. 13a and 13b).

Fig. 12. (a) Simulated and (b) measured spectra of S21 for linear DR arrays arranged along kvector at 2 mm between DR bodies (Chen et al., 2011).

Fig. 13. H-field phase patterns in waveguide at (a) 8.17 GHz and (b) 8.58 GHz (Chen et al., 2011)

Formation of Coherent Multi-Element Resonance States in Metamaterials 107

**<sup>y</sup>** (a)

frequency, GHz

Fig. 14. (a) Basic element of 3D arrays at plane wave incidence in z-and PBC in y-direction; (b) spectra of signals from probes placed in DR centres; and (c) spectra of probes placed

7.5 8.0 8.5 9.0 9.5 10.

7.5 8.0 8.5 9.0 9.5 10. frequency, GHz

(c)

(b)

*E <sup>k</sup> <sup>H</sup>*

0.055 0.05 0.04 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.05 0.0

0.3

**Z**

**x**

0.25

0.2

0.15

Probe signal magnitude in A/m

0.1

0.05

0.0

Probe signal magnitude in A/m

outside the sample to detect transmission.

It is worth noting that the revealed low frequency and high frequency resonance states in DR arrays remind bonding and anti-bonding states of electrons in molecules and crystals with such difference that instead of electron spins the type of resonance states in DR arrays is defined by respective orientations of formed magnetic dipoles. It is also important to add that a coherent response of resonators presents a special interest for all applications of metamaterials, which depend on the realization of the prescibed by the transformation optics dispersion of the effective medium parameters.

#### **3.3 EM responses of coupled 3D metamaterial arrays of dielectric resonators**

It should be pointed out that the described above simulations and experiments with placement the DR arrays in waveguides are, in fact, equivalent to modeling 3D arrays of resonators, due to the mirror reflection effect of the waveguide walls. Since the distance between resonators and the walls of the waveguide WR137 was relatively large compared to inter-resonator separation, the obtained results show that even large lattice parameters in two directions of the modelled 3D arrays (and, repectively, negligible coupling in these directions) does not provide the similarity of the array response to the response of the unit cell with a single resonator inside. However, large lattice parameters are rather unusual for metamaterials applications, therefore, the studies of dense 3D arrays that should present additional interest for practical realization of metamaterials, have also been conducted.

Numerical experiments with 3D arrays having lattice parameters between 8 mm and 12 mm (which corresponded to separations between the resonator bodies ranging from 2 mm to 6 mm, respectively) were carried out in free space. Due to employment of periodic boundary conditions (PBC) in simulations, the 3D arrays under study were infinite in one direction and finite in two other directions. The plane wave incidence was normal to the infinite direction. For simplicity, a cubic structure of the arrays was employed. The individual responses of resonators in two perpendicular to each other linear arrays of resonators were controlled by the probes placed in the centers of corresponding resonators. In additition, more probes were placed at some distances from the array on the way of propagating waves to detect transmission (see schematic in Fig. 14a). Wave propagation through the arrays was controlled by using snapshots and animation of H-field distribution patterns in XZ cross-section of the array.

As seen in Figs. 14b and 14c, the responses of array elements were characterized by multiple resonances observed even at the frequencies corresponding to the low frequecy half of the transmission gap, between 8.2 GHz and 8.63 GHz (see Fig. 14c). At frequencies below and above the transmission gap the waves passed through the array, however, the specifics of these propagations were quite different. While at lower frequencies the lengths of waves propagating in the array were essentially shorter than in free space so that the wave front movement through the array was lagging the wave front in free space (Fig. 15a), at higher frequencies, *vise versa*, the lengths of waves passing through the array were longer compared to the wavelengths in free space so that the movement of wave front through the array outrided the movement in free space (Fig. 15b). It is worth mentioning that when the frequency below the gap increased to approach its low frequency boundary, the lag in the wave movement through the array became more and more pronounced reaching its maximum at about 8.18 GHz. At frequencies above the transmission gap, the highest velocity of waves passing through the array was observed just near the gap boundary at about 8.66 GHz. At higher frequencies, wave advancing inside the array gradually decreased so that the entire wave front became flat at about 9.12 - 9.15 GHz.

It is worth noting that the revealed low frequency and high frequency resonance states in DR arrays remind bonding and anti-bonding states of electrons in molecules and crystals with such difference that instead of electron spins the type of resonance states in DR arrays is defined by respective orientations of formed magnetic dipoles. It is also important to add that a coherent response of resonators presents a special interest for all applications of metamaterials, which depend on the realization of the prescibed by the transformation

It should be pointed out that the described above simulations and experiments with placement the DR arrays in waveguides are, in fact, equivalent to modeling 3D arrays of resonators, due to the mirror reflection effect of the waveguide walls. Since the distance between resonators and the walls of the waveguide WR137 was relatively large compared to inter-resonator separation, the obtained results show that even large lattice parameters in two directions of the modelled 3D arrays (and, repectively, negligible coupling in these directions) does not provide the similarity of the array response to the response of the unit cell with a single resonator inside. However, large lattice parameters are rather unusual for metamaterials applications, therefore, the studies of dense 3D arrays that should present additional interest for practical realization of metamaterials, have also been conducted.

Numerical experiments with 3D arrays having lattice parameters between 8 mm and 12 mm (which corresponded to separations between the resonator bodies ranging from 2 mm to 6 mm, respectively) were carried out in free space. Due to employment of periodic boundary conditions (PBC) in simulations, the 3D arrays under study were infinite in one direction and finite in two other directions. The plane wave incidence was normal to the infinite direction. For simplicity, a cubic structure of the arrays was employed. The individual responses of resonators in two perpendicular to each other linear arrays of resonators were controlled by the probes placed in the centers of corresponding resonators. In additition, more probes were placed at some distances from the array on the way of propagating waves to detect transmission (see schematic in Fig. 14a). Wave propagation through the arrays was controlled by using snap-

As seen in Figs. 14b and 14c, the responses of array elements were characterized by multiple resonances observed even at the frequencies corresponding to the low frequecy half of the transmission gap, between 8.2 GHz and 8.63 GHz (see Fig. 14c). At frequencies below and above the transmission gap the waves passed through the array, however, the specifics of these propagations were quite different. While at lower frequencies the lengths of waves propagating in the array were essentially shorter than in free space so that the wave front movement through the array was lagging the wave front in free space (Fig. 15a), at higher frequencies, *vise versa*, the lengths of waves passing through the array were longer compared to the wavelengths in free space so that the movement of wave front through the array outrided the movement in free space (Fig. 15b). It is worth mentioning that when the frequency below the gap increased to approach its low frequency boundary, the lag in the wave movement through the array became more and more pronounced reaching its maximum at about 8.18 GHz. At frequencies above the transmission gap, the highest velocity of waves passing through the array was observed just near the gap boundary at about 8.66 GHz. At higher frequencies, wave advancing inside the array gradually

shots and animation of H-field distribution patterns in XZ cross-section of the array.

decreased so that the entire wave front became flat at about 9.12 - 9.15 GHz.

**3.3 EM responses of coupled 3D metamaterial arrays of dielectric resonators** 

optics dispersion of the effective medium parameters.

Fig. 14. (a) Basic element of 3D arrays at plane wave incidence in z-and PBC in y-direction; (b) spectra of signals from probes placed in DR centres; and (c) spectra of probes placed outside the sample to detect transmission.

Formation of Coherent Multi-Element Resonance States in Metamaterials 109

If the investigated array could respond as a uniform medium without any resonance splitting it could be expected that the wave movement inside the array would depend on characteristic for Lorentz-type resonators changes of the effective permeability. As known, these changes have three essential for wave movement ranges: 1) increase of the effective permeability with frequency at approaching the resonance, 2) drop of the effective permeability at the resonance frequency down to negative values and then growth back with crossing the zero level at the so-called magnetic plasma frequency, and 3) continuing increase of the effective permeability in the range of values between 0 and 1, as shown in Fig. 3c. The above changes would define the phase velocity of propagating waves in

p

ν

r r 1

μ ε<sup>=</sup>

and cause lagging of the wave movement in the array at approaching the resonance, then a transmission gap at frequencies corresponding to the negative values of permeability, and, finally, superluminal wave movement at frequencies exceeding the magnetic plasma frequency. It seems doubtful to expect similar changes of the effective permeability as those observed in the model of a uniform medium (Fig. 3c) from close-packed arrays of coupled resonators with essentially split resonances. Nevertheless, the obtained data testify in favor of some analogy. It follows that despite of the resonance splitting, coupled arrays are capable of responding in the way predicted by the above described changes of the effective permeability near the resonance. It means that coupling and splitting of elementary resonances do not prevent arrays from specific integrated contributions of multiple responses to their effective properties and from exhibiting qualitatively similar to a uniform medium changes of these properties with frequency. Since application of metamaterials in cloaking devices assumes employment of cloaking shells for speeding up waves around the concealed objects, the obtained results show that these expectations could be realized despite of complications caused by the coupling phenomena. It is worth adding that at the "superluminal" phenomena, the resonators in coupled arrays tend to respond coherently within the half wavelengths of the passing waves. Therefore, at frequencies close to the upper boundary of the transmission gap, when the strongest extention of the wavelengths by the resonating array is provided, coherent response can involve even the entire array (Fig. 15c). It should be, however, taken into account that the coherent response formation could occur at frequencies different from the resonance frequency of a single resonator and that the frequency range of the coherent response could depend on the specifics of coupling

The revealed possibility to employ arrays of coupled resonators for obtaining coherent responses within the half wavelengths of the incident wave provides the grounds for the employment of coupling phenomena in metamaterials to enhance the bandwidth of their

The performed studies of coupling and splitting of elementary resonances in arrays of metal split-ring resonators and dielectric resonators have demonstrated that ignoring coupling

operation and, in particular, for the enhancement of the cloaking effect bandwidth.

accordance with the known relation:

effects in respective arrays.

**4. Conclusion**

Fig. 15. Wave propagation in 3D DR arrays at frequencies (a) below the gap and (b, c) above the gap at: (a) 8.0 GHz; (b) 8.74 GHz; and (c) 8.64 GHz.

(a)

(b)

(c)

the gap at: (a) 8.0 GHz; (b) 8.74 GHz; and (c) 8.64 GHz.

Fig. 15. Wave propagation in 3D DR arrays at frequencies (a) below the gap and (b, c) above

If the investigated array could respond as a uniform medium without any resonance splitting it could be expected that the wave movement inside the array would depend on characteristic for Lorentz-type resonators changes of the effective permeability. As known, these changes have three essential for wave movement ranges: 1) increase of the effective permeability with frequency at approaching the resonance, 2) drop of the effective permeability at the resonance frequency down to negative values and then growth back with crossing the zero level at the so-called magnetic plasma frequency, and 3) continuing increase of the effective permeability in the range of values between 0 and 1, as shown in Fig. 3c. The above changes would define the phase velocity of propagating waves in accordance with the known relation:

$$\nu\_{\mathbb{P}} = \frac{1}{\sqrt{\mu\_r \varepsilon\_r}}$$

and cause lagging of the wave movement in the array at approaching the resonance, then a transmission gap at frequencies corresponding to the negative values of permeability, and, finally, superluminal wave movement at frequencies exceeding the magnetic plasma frequency. It seems doubtful to expect similar changes of the effective permeability as those observed in the model of a uniform medium (Fig. 3c) from close-packed arrays of coupled resonators with essentially split resonances. Nevertheless, the obtained data testify in favor of some analogy. It follows that despite of the resonance splitting, coupled arrays are capable of responding in the way predicted by the above described changes of the effective permeability near the resonance. It means that coupling and splitting of elementary resonances do not prevent arrays from specific integrated contributions of multiple responses to their effective properties and from exhibiting qualitatively similar to a uniform medium changes of these properties with frequency. Since application of metamaterials in cloaking devices assumes employment of cloaking shells for speeding up waves around the concealed objects, the obtained results show that these expectations could be realized despite of complications caused by the coupling phenomena. It is worth adding that at the "superluminal" phenomena, the resonators in coupled arrays tend to respond coherently within the half wavelengths of the passing waves. Therefore, at frequencies close to the upper boundary of the transmission gap, when the strongest extention of the wavelengths by the resonating array is provided, coherent response can involve even the entire array (Fig. 15c). It should be, however, taken into account that the coherent response formation could occur at frequencies different from the resonance frequency of a single resonator and that the frequency range of the coherent response could depend on the specifics of coupling effects in respective arrays.

The revealed possibility to employ arrays of coupled resonators for obtaining coherent responses within the half wavelengths of the incident wave provides the grounds for the employment of coupling phenomena in metamaterials to enhance the bandwidth of their operation and, in particular, for the enhancement of the cloaking effect bandwidth.

#### **4. Conclusion**

The performed studies of coupling and splitting of elementary resonances in arrays of metal split-ring resonators and dielectric resonators have demonstrated that ignoring coupling

Formation of Coherent Multi-Element Resonance States in Metamaterials 111

Jylha, L., Kolmakov, I., Maslovski, S., & Tretyakov, S. (2006). Modeling of Isotropic

Kajfez, D. & Guillon, P. (1998), *Dielectric Resonators*, Noble Publishing Corp., 2nd ed.,

Leonhardt, U. (2006). Optical Conformal Mapping. *Science*, Vol.312, No. 5781, (June 2006),

Liu, N. & Giessen, H. (2010), Coupling Effects in Optical Metamaterials. *Angewandte Chemie* 

Pendry, J. B., Holden, A., Robbins, D. J, & Stewart, W. J. (1999). Magnetism from Conductors

*Techniques,* Vol.47, No.11, (November 1999), pp. 2075–2084, ISSN: 0018-9480 Pendry, J. B. & Smith, D. R. (2003). Reversing Light: Negative Refraction. *Physics Today*,

Pendry, J. B., Schurig, D. & Smith, D. R. (2006). Controlling Electromagnetic Fields. *Science*,

Schurig, D., Mock, J. J., Justice, B. J., Cummer, S. A., Pendry, J. B., Starr, A. F. & Smith, D. R.

Semouchkina, E., Baker, A., Semouchkin, G., Randall, C. & Lanagan, M. (2004). Resonant

Semouchkina, E., Semouchkin, G., Lanagan, M. & Randall, C. A. (2005). FDTD Study of

Semouchkina, E., Werner, D. H. & Pantano, C. (2009). An Optical Cloak Composed of

Semouchkina, E., Werner, D. H., Semouchkin, G. & Pantano, C. (2010). An Infrared

Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. (2004). Metamaterials and Negative Refractive Index. *Science*, Vol.305, No.5685*,* (August 2004), pp. 788–792, ISSN 0036-8075 Smith, D. R. & Pendry, J. B. (2006). Homogenization of Metamaterails by Field Averaging.

Ueda, T., Lai, A. & Itoh, T. (2007). Demonstration of Negative Refraction in a Cutoff Parallel-

Ueda, T., Michishita, N., Akiyama, M. & and Itoh, T. (2010). Anisotropic 3-D Composite

*Physics*, Vol.99, 043102, (2006), ISSN:0021-8979

10.1002/anie.200906211, ISSN: 1521-3773

88986-409-8, Banff, Canada, July 2004

ISSN 0003-6951

pp.1280–1287, ISSN: 0018-9480

Vol.56, (December 2003), pp. 1-8, ISSN:0031-9228

Vol.312, No.5781, (June 2006), pp. 1780-1782, ISSN 0036-8075

Vol.314, No.5806, (November 2006), pp. 977–979, ISSN 0036-8075

*Techniques,* Vol.53, (April 2005), pp. 1477–1487, ISSN: 0018-9480

978-0-9551179-6-1, London, United Kingdom, September 2009

*JOSA B*, Vol.B23, No.3, (March 2006), pp.391-403, ISSN: 0740-3224

Vol.58, No.7, (July 2010), pp. 1766 –1773, ISSN: 0018-9480

Atlanta, (1998), ISBN:1884932053

pp. 1777-1779, ISSN 0036-8075

Backward-Wave Materials Composed of Resonant Spheres. *Journal of Applied* 

*Int. Ed.,* Vol.49, No.51, (December 2010), pp. 9838-9852, DOI:

and Enhanced Non-Linear Phenomena. *IEEE Trans. on Microwave Theory &* 

(2006). Metamaterial Electromagnetic Cloak at Microwave Frequencies. *Science,* 

Wave Propagation in Periodic Dielectric Structures, *Proceedings of the IASTED Int. Conf*. *ANTENNAS, RADAR AND WAVE PROPAGATION*, pp. 149–154, ISBN: 0-

Resonance Processes in Metamaterials. *IEEE Transactions on Microwave Theory &* 

Identical Chalcogenide Glass Resonators, *Proceedings of the Metamaterials'2009 - 3rd Int. Congress on Advanced Electromagnetic Materials in Microwaves and Optics*, , ISBN

Invisibility Cloak Composed of Glass. *Appl. Phys. Lett*., Vol.96, 233503, (June 2010),

Plate Waveguide Loaded with Two-Dimensional Lattice of Dielectric Resonators. *IEEE Transactions on Microwave Theory & Techniques,* Vol.55, No.6, (June 2007),

Right/Left-Handed Metamaterial Structures Using Dielectric Resonators and Conductive Mesh Plates. *IEEE Transactions on Microwave Theory & Techniques,* 

effects at metamaterial applications can lead to mistakes and malfunctioning. Even the best ideas proposed for designing various devices based on employing resonance metamaterials, can loose an opportunity to be realized properly unless coupling phenomena are taken into account. In particular, the presented data, which disclosed dramatic splitting of the resonances in close-packed arrays of SRRs have shown that these arrays could not be used for demonstrating the effects of invisibility by the cylindrical cloak of concentric resonator arrays at frequencies predicted byusing the analysis of a single SRR.

The arrays of dielectric resonators, which seem perspective due to low loss, especially for employment in cloaking devices, are also found to be a subject of coupling phenomena even at the lattice constants essentially exceeding the dimensions of resonators. However, there are reasons to express an optimism towards their applications as even at strong coupling and resonance splitting phenomena the DR arrays are able to demonstrate integrated responses qualitatively corresponding to the responses expected for the theoretical uniform media with Lorentz-type resonances. In particular, the DR arrays show an ability to support superluminal wave propagation and coherent resonance responses from multiple resonators located within the half wavelength of the propagating wave.

Similar to photonic crystals, DR arrays demonstrate the formation of a transmission gap that separates two transmission bands. However, this bandgap is not related to Bragg resonance and apparently is related to negative values of the effective permeability at elementary Lorentz-type resonances. In this attitude, metamaterial DR arrays could be compared to photonic crystals exploiting Mie resonances. The slow waves, which have been observed in DR arrays at frequencies below the transmission gap, and the superluminal waves observed at frequencies above the gap enhance the similarity with the latter photonic crystals. A detaled comparison of the properties and the physics underlying the observed phenomena in two types of artificial materials could provide a deeper understanding of their specifics.

#### **5. Acknowledgment**

This work was supported by the National Science Foundation under Grant No. 0968850. The author wishes to thank graduate students Fang Chen and Xiaohui Wang for performing some simulations and measurements.

#### **6. References**


effects at metamaterial applications can lead to mistakes and malfunctioning. Even the best ideas proposed for designing various devices based on employing resonance metamaterials, can loose an opportunity to be realized properly unless coupling phenomena are taken into account. In particular, the presented data, which disclosed dramatic splitting of the resonances in close-packed arrays of SRRs have shown that these arrays could not be used for demonstrating the effects of invisibility by the cylindrical cloak of concentric resonator

The arrays of dielectric resonators, which seem perspective due to low loss, especially for employment in cloaking devices, are also found to be a subject of coupling phenomena even at the lattice constants essentially exceeding the dimensions of resonators. However, there are reasons to express an optimism towards their applications as even at strong coupling and resonance splitting phenomena the DR arrays are able to demonstrate integrated responses qualitatively corresponding to the responses expected for the theoretical uniform media with Lorentz-type resonances. In particular, the DR arrays show an ability to support superluminal wave propagation and coherent resonance responses from multiple resonators

Similar to photonic crystals, DR arrays demonstrate the formation of a transmission gap that separates two transmission bands. However, this bandgap is not related to Bragg resonance and apparently is related to negative values of the effective permeability at elementary Lorentz-type resonances. In this attitude, metamaterial DR arrays could be compared to photonic crystals exploiting Mie resonances. The slow waves, which have been observed in DR arrays at frequencies below the transmission gap, and the superluminal waves observed at frequencies above the gap enhance the similarity with the latter photonic crystals. A detaled comparison of the properties and the physics underlying the observed phenomena in two types of artificial materials could provide a deeper understanding of their specifics.

This work was supported by the National Science Foundation under Grant No. 0968850. The author wishes to thank graduate students Fang Chen and Xiaohui Wang for performing

Chen, F., Wang, X. & Semouchkina, E. (2011). Simulation and Experimental Studies of

Cui, T. J., Smith, D. R., Liu, R. (2009), *Metamaterials: Theory, Design, and Applications*, Springer,

Gaillot, D. P., Croenne, C. & L ippens, D. (2008). An All-dielectric Route for Terahertz

Hashemi, H., Zhang, B., Joannopoulos, J. D. & Johnson, S. G. (2010). Delay-Bandwidth and

Dielectric Resonator Arrays for Designing Metamaterials, *Proceedings of IEEE International Symposium on Antennas and Propagation,* pp. 2936-2939, ISSN: 1522-

Cloaking. *Optics Express,* Vol.16, No.6, (March 2008), pp. 3986-3992, ISSN: 1094-4087

Delay-Loss Limitations for Cloaking of Large Objects. *Phys. Rev. Lett.,* Vol.104,

arrays at frequencies predicted byusing the analysis of a single SRR.

located within the half wavelength of the propagating wave.

**5. Acknowledgment**

**6. References** 

some simulations and measurements.

3965, Spokane, WA, August 2011

(June 2010), 253903, ISSN 0031-9007

1st ed., (November 2009), ISBN-10: 1441905723


**5** 

*France*

**Space Coordinate Transformation** 

Boubacar Kante, Rasta Ghasemi and Dylan Germain

André de Lustrac, Shah Nawaz Burokur, Paul-Henri Tichit,

Invisibility or cloaking is an old myth of humanity that must date probably from the time when the success in hunting or war would depend on the ability to hide as much as possible. This invisibility, after being a prolific subject for writers and filmmakers, has become almost a reality in 2006 with the first practical realization of an electromagnetic

The invariance of Maxwell's equations in the geometric transformation of coordinates has become a hot topic that year with the first proposal of a cylindrical invisibility cloak by J. B. Pendry and U. Leonhardt. The experimental fabrication and characterization of the first cloak at microwave frequencies have shown that this tool is very effective. After this realization, several applications of this transformation have been proposed for the design of concentrators, waveguides, transitions and bends, directional antennas, and even

This space transformation is therefore a powerful tool for the design of devices or components with special properties difficult to obtain from conventional materials and geometries. Theoretically, the method of coordinate transformation involves the generation of a new space derived from an original space where the solutions of Maxwell's equations

The first step is to imagine an initial space and a final space with their topological properties and link them through an analytic transformation. Most of this work is based on a continuous transformation that produces a final space with electromagnetic parameters

The challenge is then to effectively design this new space. To make the fabrication easier, simplified parameters were proposed in the early works, with the disadvantage of an impedance mismatch between the material and its environment. More recently, a transformation applied to discrete multi-layer structures has been proposed to further simplify the realization. In this chapter, we present the principles and main applications

**1. Introduction** 

invisibility cloak.

are known.

electromagnetic wormholes.

often complex, heterogeneous and anisotropic.

envisaged for this transformation.

**and Applications** 

*Institut d'Electronique Fondamentale, Univ. Paris-Sud, CNRS UMR 8622, Orsay,* 


### **Space Coordinate Transformation and Applications**

André de Lustrac, Shah Nawaz Burokur, Paul-Henri Tichit, Boubacar Kante, Rasta Ghasemi and Dylan Germain *Institut d'Electronique Fondamentale, Univ. Paris-Sud, CNRS UMR 8622, Orsay, France*

### **1. Introduction**

112 Metamaterial

Vendik, O. G. & M. S. Gashinova, M. S. (2004). Artificial Double Negative (DNG) Media

Zhang, B. L., Chen, H. S. & Wu, B.-L. (2009). Practical Limitations of an Invisibility Cloak. *Progress in Electromagnetics Research*, Vol.97, (2009), pp. 407-416, ISSN: 1070-4698

ISBN/ISSN: 1580-5399-20 9781-5805-39920, Amsterdam, October 2004 Yariv, A., Xu, Y., Lee, R. K. & Scherer, A. (1999).Coupled Resonator Optical Waveguide: a

0146-9592

Composed by Two Different Dielectric Sphere Lattices Embedded in a Dielectric Matrix, *Proceedings of the 34th European Microwave Conference*, pp. 1209-1212,

Proposal and Analysis. *Optics Letters*, Vol.24, No.11, (June 1999), pp.711-713, ISSN:

Invisibility or cloaking is an old myth of humanity that must date probably from the time when the success in hunting or war would depend on the ability to hide as much as possible. This invisibility, after being a prolific subject for writers and filmmakers, has become almost a reality in 2006 with the first practical realization of an electromagnetic invisibility cloak.

The invariance of Maxwell's equations in the geometric transformation of coordinates has become a hot topic that year with the first proposal of a cylindrical invisibility cloak by J. B. Pendry and U. Leonhardt. The experimental fabrication and characterization of the first cloak at microwave frequencies have shown that this tool is very effective. After this realization, several applications of this transformation have been proposed for the design of concentrators, waveguides, transitions and bends, directional antennas, and even electromagnetic wormholes.

This space transformation is therefore a powerful tool for the design of devices or components with special properties difficult to obtain from conventional materials and geometries. Theoretically, the method of coordinate transformation involves the generation of a new space derived from an original space where the solutions of Maxwell's equations are known.

The first step is to imagine an initial space and a final space with their topological properties and link them through an analytic transformation. Most of this work is based on a continuous transformation that produces a final space with electromagnetic parameters often complex, heterogeneous and anisotropic.

The challenge is then to effectively design this new space. To make the fabrication easier, simplified parameters were proposed in the early works, with the disadvantage of an impedance mismatch between the material and its environment. More recently, a transformation applied to discrete multi-layer structures has been proposed to further simplify the realization. In this chapter, we present the principles and main applications envisaged for this transformation.

Space Coordinate Transformation and Applications 115

In this case, we use normalized values of the electromagnetic parameters ε and µ that we

μ μ

*v*

∂

∂∂∂

*x z <sup>y</sup> <sup>F</sup> www*

<sup>0</sup> ′ ′′ <sup>=</sup> and *E E* <sup>0</sup> <sup>=</sup>

The first application of the space coordinate transformation will be the design and the

In the case of the first invisibility cloak proposed by Smith and Pendry [Pendry, 2006], the transformation of space concerned a cylindrical space of radius b which is transformed into an annular space between radii a and b (Figure 2). The initial points in space are identified by coordinates r, θ and z. Those of the transformed space are identified by r ', θ' and z' in cylindrical coordinates. Both spaces are assumed infinite in the directions z and z ', which

> θ θ

*r r a*

The three parameters obtained depend on the distance r and on the geometric parameters of the original and transformed spaces. These are the electromagnetic parameters of the

*z z*

ε μ

*uvw u u u FFF F*

∂

2 2 2

2 *uvw u u u FFF F*

2 2 2

′ = and 2

 ∂

∂

The conventionnal relations of electromagnetics are conserved in the new space :

ε ε

∂

ε

*u, v* et *w*. Generally, the new space is anisotropic.

*u*

and

In these relations, *µ'* and

**3. Cloaking** 

**3.1 Principle** 

are combined.

After transformation we obtain :

*r r*

ε μ<sup>−</sup> = = ;

*r a r*

where *Fu, Fv* et *Fw* are defined by the following equations:

*x z <sup>y</sup> <sup>F</sup> uuu*

∂

∂∂∂

*w*

*B H* μ μ

characterization of different electromagnetic invisibility cloaks [2-6].

The transformation is defined by a relatively simple set of equations (6).

θ θ

ε μ= = − ;

*b a r ra b* <sup>−</sup> ′ = + ;

=++

will define in the new space:

u(x,y, z),v(x,y, z), and w(x,y, z) (2)

′ = ; etc… (3)

 ∂

′ ′ (5)

′ = ; z'=z (6)

<sup>−</sup> = = <sup>−</sup> (7)

2

*b ra ba r*

2 2 2

*x z <sup>y</sup> <sup>F</sup> vvv*

> ∂

ε ε

*'* are tensors whose components depend on the spatial coordinates

∂

∂∂∂

=++

=++ (4)

#### **2. Space coordinate transformation**

#### **2.1 Principle**

In a letter of 1662 Pierre de Fermat established the principle that governs the geometrical optics [Tannery, 1891]. The light follows a stationary optical path between two points. Most of the time, it follows the shortest path. In some cases, it takes the longer one. The optical path is defined by the equation 1, where n is the refractive index of the space, which may depend on the spatial coordinates, and dl a small element of distance:

$$s = \bigcup ndl \tag{1}$$

If n varies with the position in space the path followed by the light can be bent instead of following a straight line. This occurs, for example over a hot road in summer when the index of the air layers above the road varies with the temperature and the height over the road. In this case we can observe a curvature of the path followed by the sunlight that gives the impression that the road is covered with water. Figure 1 shows a schematic of the path of light when the space is not distorted and when this space is distorted (Figure 1a and b).

Fig. 1. (a) Propagation of a light beam in a non distorted space. (b) propagation of the same ray of light in a distorted space. (c) isolation of a region of space by deforming the propagation of light rays around this region.

J. Pendry and U. Leonhardt noted both in their articles published in 2006, the invariance of Maxwell's equations in such a deformed space [Pendry, 2006, Leonhardt, 2006]. J. Pendry has concluded that it was possible to isolate a zone of space by bending light rays around this area (Figure 1c).

#### **2.2 Implementation**

The implementation of this transformation is relatively simple. If we consider a Cartesian space where each point is identified by three coordinates (x, y, z), we can define a new space where each point will be identified by three new coordinates (u, v, w). These three new coordinates are based on the original ones.

$$\mathbf{u}(\mathbf{x}, \mathbf{y}, \mathbf{z}), \mathbf{v}(\mathbf{x}, \mathbf{y}, \mathbf{z}), \text{and } \mathbf{w}(\mathbf{x}, \mathbf{y}, \mathbf{z})\tag{2}$$

In this case, we use normalized values of the electromagnetic parameters ε and µ that we will define in the new space:

$$
\varepsilon'\_{u} = \varepsilon\_{u} \frac{F\_{u} F\_{v} F\_{w}}{F\_{u}^{2}} \quad \text{and} \quad \mu'\_{u} = \mu\_{u} \frac{F\_{u} F\_{v} F\_{w}}{F\_{u}^{2}} \; ; \; \text{etc.} \tag{3}
$$

where *Fu, Fv* et *Fw* are defined by the following equations:

$$F\_u = \left(\frac{\partial x}{\partial u}\right)^2 + \left(\frac{\partial y}{\partial u}\right)^2 + \left(\frac{\partial z}{\partial u}\right)^2 \quad F\_v = \left(\frac{\partial x}{\partial v}\right)^2 + \left(\frac{\partial y}{\partial v}\right)^2 + \left(\frac{\partial z}{\partial v}\right)^2$$

$$\text{and } F\_w = \left(\frac{\partial x}{\partial w}\right)^2 + \left(\frac{\partial y}{\partial w}\right)^2 + \left(\frac{\partial z}{\partial w}\right)^2\tag{4}$$

The conventionnal relations of electromagnetics are conserved in the new space :

$$
\vec{B}' = \mu\_0 \mu' \vec{H}' \quad \text{and} \qquad \vec{E} = \varepsilon\_0 \varepsilon' \vec{E}' \tag{5}
$$

In these relations, *µ'* and ε*'* are tensors whose components depend on the spatial coordinates *u, v* et *w*. Generally, the new space is anisotropic.

#### **3. Cloaking**

114 Metamaterial

In a letter of 1662 Pierre de Fermat established the principle that governs the geometrical optics [Tannery, 1891]. The light follows a stationary optical path between two points. Most of the time, it follows the shortest path. In some cases, it takes the longer one. The optical path is defined by the equation 1, where n is the refractive index of the space, which may

If n varies with the position in space the path followed by the light can be bent instead of following a straight line. This occurs, for example over a hot road in summer when the index of the air layers above the road varies with the temperature and the height over the road. In this case we can observe a curvature of the path followed by the sunlight that gives the impression that the road is covered with water. Figure 1 shows a schematic of the path of light when the space is not distorted and when this space is distorted (Figure

*s ndl* <sup>=</sup> (1)

depend on the spatial coordinates, and dl a small element of distance:

(a) (b) (c)

propagation of light rays around this region.

coordinates are based on the original ones.

this area (Figure 1c).

**2.2 Implementation** 

ray of light in a distorted space. (c) isolation of a region of space by deforming the

Fig. 1. (a) Propagation of a light beam in a non distorted space. (b) propagation of the same

J. Pendry and U. Leonhardt noted both in their articles published in 2006, the invariance of Maxwell's equations in such a deformed space [Pendry, 2006, Leonhardt, 2006]. J. Pendry has concluded that it was possible to isolate a zone of space by bending light rays around

The implementation of this transformation is relatively simple. If we consider a Cartesian space where each point is identified by three coordinates (x, y, z), we can define a new space where each point will be identified by three new coordinates (u, v, w). These three new

**2. Space coordinate transformation** 

**2.1 Principle** 

1a and b).

The first application of the space coordinate transformation will be the design and the characterization of different electromagnetic invisibility cloaks [2-6].

#### **3.1 Principle**

In the case of the first invisibility cloak proposed by Smith and Pendry [Pendry, 2006], the transformation of space concerned a cylindrical space of radius b which is transformed into an annular space between radii a and b (Figure 2). The initial points in space are identified by coordinates r, θ and z. Those of the transformed space are identified by r ', θ' and z' in cylindrical coordinates. Both spaces are assumed infinite in the directions z and z ', which are combined.

The transformation is defined by a relatively simple set of equations (6).

$$r' = \frac{b-a}{b}r + a \; ; \; \theta' = \theta \; ; \; z' = z \tag{6}$$

After transformation we obtain :

$$
\varepsilon\_r = \mu\_r = \frac{r-a}{r} \; ; \; \varepsilon\_\theta = \mu\_\theta = \frac{r}{r-a} \; ; \; \varepsilon\_z = \mu\_z = \left(\frac{b}{b-a}\right)^2 \frac{r-a}{r} \tag{7}
$$

The three parameters obtained depend on the distance r and on the geometric parameters of the original and transformed spaces. These are the electromagnetic parameters of the

Space Coordinate Transformation and Applications 117

(a) (b)

(c) Fig. 3. (a) Elementary cell of the metamaterial. (b) values of the parameters. (c) Photography

Figures 4b and 4d are quite similar and clearly show the influence of electromagnetic radiation on the cloak with in particular the cloaking of the central metallic cylinder by the electromagnetic energy. Both figures also illustrate the limits of the exercise as the use of a reduced set of parameters causes reflection of part of the incident energy, and therefore an

of the prototype. The insert shows the variations of μr, μθ and εz.

imperfect reconstruction of the wave after the cloak.

transformed space. Note that the permeability and permittivity have here the same expressions. This guarantees the matching of the wave impedance of the transformed medium with the initial medium.

Fig. 2. Initial circular space of radius b, transformed to a ring between a and b.

#### **3.2 Realization of the first microwave cloak**

The practical realization of the material forming the transformed space is a difficult task since the three parameters vary simultaneously [Schurig, 2006]. To simplify this material, one solution is to choose a polarization, namely the transverse magnetic (TM) polarization, with the electric field E parallel to the axis z. In this case, only the parameters , et *z r* θ ε μ μ are important. In this case, a simple set of parameters is possible provided it meets the propagation equation. The next set is one of the possibilities:

$$
\varepsilon\_z = \left(\frac{b}{b-a}\right)^2; \ \mu\_r = \left(\frac{r-a}{r}\right)^2; \ \mu\_\theta = 1\tag{8}
$$

In these three parameters, two are fixed and one varies; the radial permeability ur. This material can be realized by the metamaterial concept [Soukoulis, 2011]. Figure 3a shows the basic pattern of the material and the geometric values used with the corresponding values of ur. The permittivity εz is realized using a conventional dielectric.

Figure 3c shows a part of the realized circuit with, in the insert, the evolution of the three parameters μr, μθ and εz. The shape of the elementary patterns depends on the layer of material so that the permeability μr also varies and follows the red curve in the figure inset. Figure 4(a) and 4(b) shows the simulation of the cloak with respectively the theoretical parameters of equation 13 and the reduced parameters of equation 14. Figure 4(c) presents the measured electric field cartography of the central bare metallic cylinder where we can observe strong reflections and shadows compared to the case of very low reflections when the cloak is applied around the cylinder (Figure 4(d)).

transformed space. Note that the permeability and permittivity have here the same expressions. This guarantees the matching of the wave impedance of the transformed

Fig. 2. Initial circular space of radius b, transformed to a ring between a and b.

meets the propagation equation. The next set is one of the possibilities:

 <sup>=</sup> <sup>−</sup> ;

*b b a*

*z*

ur. The permittivity εz is realized using a conventional dielectric.

the cloak is applied around the cylinder (Figure 4(d)).

ε

2

*r*

In these three parameters, two are fixed and one varies; the radial permeability ur. This material can be realized by the metamaterial concept [Soukoulis, 2011]. Figure 3a shows the basic pattern of the material and the geometric values used with the corresponding values of

Figure 3c shows a part of the realized circuit with, in the insert, the evolution of the three parameters μr, μθ and εz. The shape of the elementary patterns depends on the layer of material so that the permeability μr also varies and follows the red curve in the figure inset. Figure 4(a) and 4(b) shows the simulation of the cloak with respectively the theoretical parameters of equation 13 and the reduced parameters of equation 14. Figure 4(c) presents the measured electric field cartography of the central bare metallic cylinder where we can observe strong reflections and shadows compared to the case of very low reflections when

μ

The practical realization of the material forming the transformed space is a difficult task since the three parameters vary simultaneously [Schurig, 2006]. To simplify this material, one solution is to choose a polarization, namely the transverse magnetic (TM) polarization, with the electric field E parallel to the axis z. In this case, only the parameters

are important. In this case, a simple set of parameters is possible provided it

*r a r*

<sup>−</sup> <sup>=</sup>

2

 ; 1 μθ

= (8)

**3.2 Realization of the first microwave cloak** 

, et *z r*

ε μ μ

θ

medium with the initial medium.


(c)

Fig. 3. (a) Elementary cell of the metamaterial. (b) values of the parameters. (c) Photography of the prototype. The insert shows the variations of μr, μθ and εz.

Figures 4b and 4d are quite similar and clearly show the influence of electromagnetic radiation on the cloak with in particular the cloaking of the central metallic cylinder by the electromagnetic energy. Both figures also illustrate the limits of the exercise as the use of a reduced set of parameters causes reflection of part of the incident energy, and therefore an imperfect reconstruction of the wave after the cloak.

Space Coordinate Transformation and Applications 119

Fig. 5. (a) Spiral coil used by S. Tretyakov in his cloak. (b) schematic view of the cloak.

This reduced set of parameters holds for a polarization perpendicular to the cylinder axis and satisfies the dispersion relation, but not the equality of the wave impedance between the vacuum and the cloak. A non-zero reflection is then predictable. In our implementation, the radial permittivity profile is designed using the electric response of SRRs by locally changing the dimension of resonators, actually only the SRRs gap size. The typical unit cell and the effective parameters of discrete SRRs at the design frequency (11 GHz) are presented in Fig. 6. While SRRs (embedded in the host medium) are used to achieve the radial variation of the permittivity function, the azimuthal permittivity is mainly implemented by the permittivity of the host medium itself since SRRs have no electric response in this direction. The realized cloak is composed of 15700 elementary SRRs. The cylindrical shell is divided in 20 annular regions of equal thickness (lr=4.5 mm) with a linear radial variation of the permittivity from 0 to 1 (from the inner to the outer boundary of the cloak) and 157 stripes separated by an angle of about 2.3° (Fig.7 (a)). The inner and outer radius of the cloak are a=6 cm, b=15 cm and the cloak height is 2.25 cm corresponding to 5×lz i.e. 5 SRR layers. For this set of parameters the reflection coefficient Rp is very weak, equal to 0.0625. The SRRs within a given annular region are identical and designed to have the proper local radial permittivity. The host medium, a commercially available resin is an important design component (closely linked to εθ). Its permittivity has been measured and found to be equal to εresin= 2.75. The SRRs have been printed on a dielectric substrate (as seen in the picture of Fig.1 (c)) with a permittivity close to the resin's one. We chose RO3003 with εsubstrate= 3±0.04 and a dielectric loss tangent at 11 GHz of about 0.0013. The 157 stripes were arranged in a moulded water-tight polymeric matrix designed accordingly (Fig. 7 (a)). In contrast with previously reported structure, the measurements are performed in free space and not in a waveguide configuration(Fig. 7b). A loop antenna, consisting of a circular coil made of the inner conductor of a SMA cable has been designed to map the magnetic field (Hz). The magnetic field is output from the X-band horn antenna. Both antennas are connected to an Agilent 8722ES Vectorial Network Analyzer. The loop antenna position can be controlled via an automated Labview program over a surface of 40 cm\*40 cm and getting for each spatial position of the loop antenna the complex (magnitude and phase) scattering parameters. The experimental setup can be seen on Fig. 7(b). Since the resin fills the

(a) (b)

Fig. 4. (a) Simulation of the theoretical cloak. (b) simulation of the cloak with reduced parameters. (c) simulation of the central metallic cylinder. (d) E field measurement.

#### **3.3 Others realizations**

The experimental verification of this first cloak has excited the imagination of researchers who have tried to extend to other areas in optics but also in acoustics.

#### **3.3.1 Cloak insensitive to the polarization**

In the field of electromagnetics, few achievements have been proposed and tested experimentally. But there are some exceptions [Guven, 2008, Kante, 2008, Kante, 2009]. In reference [Guven, 2008], S. Tretyakov and his team tried to design a cloak based on spiral type resonators and the principle of homogenization. These coils act as a combination of an electric and a magnetic dipole. They can therefore meet the criteria to realize the material of the cloak. Fig. 5a shows the unit spiral resonator cell used by S. Tretyakov and Fig. 5b illustrates the distribution of the resonators in the cloak. The realization of the cloak is simpler than that of Smith and is supposed to work for both TM and TE polarizations of the incident wave with the disadvantage of its large size compared to the cloaked object.

#### **3.3.2 Cloak based on the electric resonance of the SRR**

In the references [Kante, 2008, Kante, 2009] the principle is completely different. Instead of using the magnetic resonance of the resonators of Pendry, the authors use the electrical resonance of these resonators. Smith's cloak works for a TM polarization (E-field vertical and H-field in the plane of the cloak). The new cloak works with a TE polarized wave (Hfield vertical and E-field in the plane of the cape). For this non-magnetic cloak, the set of parameters is as follows:

$$
\mu\_z = 1, \quad \varepsilon\_\theta = (\frac{b}{b-a})^2, \quad \varepsilon\_r = \left(\frac{b}{b-a}\right)^2 \left(\frac{r-a}{r}\right)^2 \tag{9}
$$

Fig. 4. (a) Simulation of the theoretical cloak. (b) simulation of the cloak with reduced parameters. (c) simulation of the central metallic cylinder. (d) E field measurement.

who have tried to extend to other areas in optics but also in acoustics.

**3.3.2 Cloak based on the electric resonance of the SRR** 

1 μ

*<sup>z</sup>* <sup>=</sup> , <sup>2</sup> ( ) *<sup>b</sup>*

θ ε

*b a*

<sup>=</sup> <sup>−</sup> ,

The experimental verification of this first cloak has excited the imagination of researchers

In the field of electromagnetics, few achievements have been proposed and tested experimentally. But there are some exceptions [Guven, 2008, Kante, 2008, Kante, 2009]. In reference [Guven, 2008], S. Tretyakov and his team tried to design a cloak based on spiral type resonators and the principle of homogenization. These coils act as a combination of an electric and a magnetic dipole. They can therefore meet the criteria to realize the material of the cloak. Fig. 5a shows the unit spiral resonator cell used by S. Tretyakov and Fig. 5b illustrates the distribution of the resonators in the cloak. The realization of the cloak is simpler than that of Smith and is supposed to work for both TM and TE polarizations of the

incident wave with the disadvantage of its large size compared to the cloaked object.

In the references [Kante, 2008, Kante, 2009] the principle is completely different. Instead of using the magnetic resonance of the resonators of Pendry, the authors use the electrical resonance of these resonators. Smith's cloak works for a TM polarization (E-field vertical and H-field in the plane of the cloak). The new cloak works with a TE polarized wave (Hfield vertical and E-field in the plane of the cape). For this non-magnetic cloak, the set of

*r*

ε

2 2

<sup>−</sup> <sup>=</sup> <sup>−</sup> (9)

*b ra ba r*

**3.3 Others realizations** 

parameters is as follows:

**3.3.1 Cloak insensitive to the polarization** 

Fig. 5. (a) Spiral coil used by S. Tretyakov in his cloak. (b) schematic view of the cloak.

This reduced set of parameters holds for a polarization perpendicular to the cylinder axis and satisfies the dispersion relation, but not the equality of the wave impedance between the vacuum and the cloak. A non-zero reflection is then predictable. In our implementation, the radial permittivity profile is designed using the electric response of SRRs by locally changing the dimension of resonators, actually only the SRRs gap size. The typical unit cell and the effective parameters of discrete SRRs at the design frequency (11 GHz) are presented in Fig. 6. While SRRs (embedded in the host medium) are used to achieve the radial variation of the permittivity function, the azimuthal permittivity is mainly implemented by the permittivity of the host medium itself since SRRs have no electric response in this direction. The realized cloak is composed of 15700 elementary SRRs. The cylindrical shell is divided in 20 annular regions of equal thickness (lr=4.5 mm) with a linear radial variation of the permittivity from 0 to 1 (from the inner to the outer boundary of the cloak) and 157 stripes separated by an angle of about 2.3° (Fig.7 (a)). The inner and outer radius of the cloak are a=6 cm, b=15 cm and the cloak height is 2.25 cm corresponding to 5×lz i.e. 5 SRR layers. For this set of parameters the reflection coefficient Rp is very weak, equal to 0.0625. The SRRs within a given annular region are identical and designed to have the proper local radial permittivity. The host medium, a commercially available resin is an important design component (closely linked to εθ). Its permittivity has been measured and found to be equal to εresin= 2.75. The SRRs have been printed on a dielectric substrate (as seen in the picture of Fig.1 (c)) with a permittivity close to the resin's one. We chose RO3003 with εsubstrate= 3±0.04 and a dielectric loss tangent at 11 GHz of about 0.0013. The 157 stripes were arranged in a moulded water-tight polymeric matrix designed accordingly (Fig. 7 (a)).

In contrast with previously reported structure, the measurements are performed in free space and not in a waveguide configuration(Fig. 7b). A loop antenna, consisting of a circular coil made of the inner conductor of a SMA cable has been designed to map the magnetic field (Hz). The magnetic field is output from the X-band horn antenna. Both antennas are connected to an Agilent 8722ES Vectorial Network Analyzer. The loop antenna position can be controlled via an automated Labview program over a surface of 40 cm\*40 cm and getting for each spatial position of the loop antenna the complex (magnitude and phase) scattering parameters. The experimental setup can be seen on Fig. 7(b). Since the resin fills the

Space Coordinate Transformation and Applications 121

incident and reflected beams. Fig. 8(d) shows that in the presence of the cloak, the shadowing effect of the metallic cylinder is suppressed and the wave fronts are maintained thus demonstrating the cloaking effect. For comparison, simulation result using commercial finite element code (Comsol Multiphysics) for a cloak with the reduced parameters of equations [Kante, 2009]. is reported in Fig. 8(c). The fact that a non-zero field is detected in the central region of Fig. 8(b) and 8(d) results from radiation leakage below the metallic cylinder in our measurements. More importantly, the bending and redirection of quasicylindrical wave fronts inside the cloak can be nicely observed as a change in the radius of

Fig. 8. Real part of the measured magnetic field output from the horn antenna in free space (a) with the metallic cylinder alone (b) and with the cloak surrounding the metallic cylinder (d). Finite element simulation exciting the cloak by the appropriate optical excitation (Comsol Multiphysics) with the reduced set of parameters presented in equations (1) is reported for comparison (c). In all cases, the 11 GHz wave travels from bottom to top.

In the reference [Cai, 2008], V. Shalaev proposes two possible achievements of cloaks in the

infrared and visible domains with a TE and TM polarization (figure 9).

the horn antenna waves fronts in Fig. 8(d).

**3.4 Optical cloaks** 

**3.4.1 Propositions of V. Shalaev** 

Fig. 6. (a) Unit cell. (b) The dimensions of a typical square SRR are: L=3.6 mm, w=0.3 mm and copper thickness t=35 µm. The SRRs gap g and lθi, are the only varying parameters. lθ<sup>i</sup> linearly decreases from the outer to the inner boundary of the cloak.

Fig. 7. (a) Realized metamaterial cloaking device (b) Picture of a portion of the experimental setup with the loop antenna mapping the magnetic field at the bottom surface of the cloak.

structure, it is difficult to access the internal field. Instead, the bottom surface of the cloak (see Fig. 8d) has been scanned taking profit of the continuity of field at this boundary in quasi-contact mode. The first measurement maps the magnetic field of the free space radiation from the horn antenna (Fig. 8(a)). The second and third measurements use a metallic cylinder alone (diameter 12 cm) (Fig. 8(b)) and surrounded with the cloak (outer diameter 30 cm) (Fig. 8(d)). The results are presented in Fig. 8 (real part of the complex transmission). The quasi-cylindrical wave output from the horn antenna is nicely resolved in our measurement (Fig. 8(a)). In presence of the metallic cylinder, the scattering and shadowing effects can be clearly observed in Fig. 8(b) as well as interferences between the

Fig. 6. (a) Unit cell. (b) The dimensions of a typical square SRR are: L=3.6 mm, w=0.3 mm and copper thickness t=35 µm. The SRRs gap g and lθi, are the only varying parameters. lθ<sup>i</sup>

**(a) (b)** 

Fig. 7. (a) Realized metamaterial cloaking device (b) Picture of a portion of the experimental setup with the loop antenna mapping the magnetic field at the bottom surface of the cloak.

structure, it is difficult to access the internal field. Instead, the bottom surface of the cloak (see Fig. 8d) has been scanned taking profit of the continuity of field at this boundary in quasi-contact mode. The first measurement maps the magnetic field of the free space radiation from the horn antenna (Fig. 8(a)). The second and third measurements use a metallic cylinder alone (diameter 12 cm) (Fig. 8(b)) and surrounded with the cloak (outer diameter 30 cm) (Fig. 8(d)). The results are presented in Fig. 8 (real part of the complex transmission). The quasi-cylindrical wave output from the horn antenna is nicely resolved in our measurement (Fig. 8(a)). In presence of the metallic cylinder, the scattering and shadowing effects can be clearly observed in Fig. 8(b) as well as interferences between the

linearly decreases from the outer to the inner boundary of the cloak.

incident and reflected beams. Fig. 8(d) shows that in the presence of the cloak, the shadowing effect of the metallic cylinder is suppressed and the wave fronts are maintained thus demonstrating the cloaking effect. For comparison, simulation result using commercial finite element code (Comsol Multiphysics) for a cloak with the reduced parameters of equations [Kante, 2009]. is reported in Fig. 8(c). The fact that a non-zero field is detected in the central region of Fig. 8(b) and 8(d) results from radiation leakage below the metallic cylinder in our measurements. More importantly, the bending and redirection of quasicylindrical wave fronts inside the cloak can be nicely observed as a change in the radius of the horn antenna waves fronts in Fig. 8(d).

Fig. 8. Real part of the measured magnetic field output from the horn antenna in free space (a) with the metallic cylinder alone (b) and with the cloak surrounding the metallic cylinder (d). Finite element simulation exciting the cloak by the appropriate optical excitation (Comsol Multiphysics) with the reduced set of parameters presented in equations (1) is reported for comparison (c). In all cases, the 11 GHz wave travels from bottom to top.

#### **3.4 Optical cloaks**

#### **3.4.1 Propositions of V. Shalaev**

In the reference [Cai, 2008], V. Shalaev proposes two possible achievements of cloaks in the infrared and visible domains with a TE and TM polarization (figure 9).

Space Coordinate Transformation and Applications 123

Indeed, the magnetic resonators of Pendry and the nanowires are equivalent for a TE polarized incident wave, where the electric field is parallel to the long side of the resonator or to the wire. The invisibility cloak that was made in the microwave domain and that operates at 11GHz can be implemented in optics by replacing the resonators used in microwave by gold nanowires (Fig. 10). This figure shows a detail of the cape made using the magnetic resonators of Pendry (Fig. 10 (a)), the simulation of the cloak at 1.5 microns (Fig. 10 (b)), a gold nanowire for a TE polarization of the incident wave (Fig. 10 (c)), the equivalent parameters of the nanowire around its first resonance frequency (Fig. 10 (d)), and a schematic view of the cape in which the resonators are replaced by gold

The principles used in the cylindrical cloaks described above can be generalized to a variety of different shapes. The figure below is taken from reference [Nicolet, 2008] where a Fourier expansion is used to access to convex shapes. Reference [Rahm, 2008] proposes a square cloak (Figure 12a), which has been generalized to a polygonal cloak in [Tichit, 2008] (Figure

The major disadvantages of the first invisibility cloak were the narrow frequency-operating band and the extreme values of electromagnetic parameters needed. Various attempts have then been proposed to achieve broadband cloaks, or to design cloaks with more realistic parameter values. The most amazing proposition was suggested by U. Leonhardt who proposed to benefit from a non-Euclidean geometry to achieve the broadband [Leonhardt, 2009]. A. V. Kildishev also proposed an approximate solution to achieve a broadband cloak [Kildishev, 2008]. Following the preceding reference, we show the transition from a twodimensional "conventional" cloak (Figure 13a) to a non-Euclidean one, namely a sphere replacing a single circle (Figure 13b). In the broadband cloak proposed by Shalaev, the principle is simple: light follows a different path depending on the operating frequency

Recently other concepts of broadband invisibility cloaks based on the use of broadband nonresonant metamaterials have been proposed [Qiu, 2009, Feng, 2011] (figure 14). These recent works are based on the use of materials made of broadband dielectric multilayer where the electromagnetic parameters are extracted using the relations of Wiener on one-dimensional multi-layer materials. Figure 14 shows an example of a dielectric multi-layer structure used in a cylindrical invisibility cloak. Each concentric layer is constituted by a sub-layer of permittivity εA and a sub-layer of permittivity εB and η is the thicknesses ratio of the two

layers. The effective permittivity parameters of the layer are given by:

nanowires.

**3.5 Cloak with arbitrary shapes** 

**3.6 Broadband cloak** 

(Figure 13c).

12b), then to an elliptical one (Figure 12c).

Fig. 9. Proposition of an invisibility cloak for a TE polarization (a) and a TM one (b).

The first one corresponds to a TE polarization. For this polarization, the main parameters are:

$$\varepsilon\_r = \left(\bigvee\_r'\right)^2 \; : \; \varepsilon\_\theta = \left(\bigvee\_r \mathbf{g}(r') \bigvee\_\theta \mathbf{g}(r)\right)^{-2} \; : \; \mu\_z = 1 \tag{10}$$

Instead for the TM polarization the main parameters are:

$$\mu\_r = \left(\stackrel{r}{\vee}\_r\right)^2 \left(\stackrel{\partial}{\otimes} \mathbf{g}\left(r'\right)\Big|\_{\partial}\right)^2 : \mu\_\theta = 1 \colon \mathcal{e}\_z = \left(\stackrel{\partial}{\otimes} \mathbf{g}\left(r'\right)\Big|\_{\partial}\right)^{-2} \tag{11}$$

In these expressions g(r) is the relation between the intial space and the final one. For example in the case of a cylindrical cloak :

$$\mathbf{r} = \mathbf{g}(\mathbf{r}') = (1 - \mathbf{a} \;/\; b)\mathbf{r}' + \mathbf{a} \tag{12}$$

In the case of a TM polarization, where ε must vary between the inside and outside of the cloak, Shalaev proposes to use an effective permittivity given by Wiener relations, varying between the permittivity of a metal (in this case silver or silicon carbide) and that of a dielectric, which can vary with the wavelength of work (like silica or barium fluoride) (Figure 9(a)).

For the TE polarization, where the permeability must vary, the material used can be silicon carbide with the shape of rods embedded in air, which seems to be a bit unrealistic at visible wavelengths. However no practical realization has been proposed.

#### **3.4.2 Other optical cloak for TE polarization**

In reference [Kante, 2008] another proposal is to use the electromagnetic properties of metal cut wires. This proposal concerns only the TE polarization, and is directly correlated to the proposed cloak in the microwave region [10] by replacing the resonators of Pendry by gold nanowires.

Fig. 9. Proposition of an invisibility cloak for a TE polarization (a) and a TM one (b).

( )<sup>2</sup> *<sup>r</sup> <sup>r</sup> r*

′ <sup>=</sup> : ( )

<sup>2</sup> <sup>2</sup>

∂

ε

θ

( )

: <sup>1</sup>

ε

*r*

example in the case of a cylindrical cloak :

**3.4.2 Other optical cloak for TE polarization** 

μ

(Figure 9(a)).

nanowires.

Instead for the TM polarization the main parameters are:

( ) ( )

′ ′ <sup>=</sup>

wavelengths. However no practical realization has been proposed.

*g r r r g r* ∂

The first one corresponds to a TE polarization. For this polarization, the main parameters are:

*g r*

μθ

In these expressions g(r) is the relation between the intial space and the final one. For

In the case of a TM polarization, where ε must vary between the inside and outside of the cloak, Shalaev proposes to use an effective permittivity given by Wiener relations, varying between the permittivity of a metal (in this case silver or silicon carbide) and that of a dielectric, which can vary with the wavelength of work (like silica or barium fluoride)

For the TE polarization, where the permeability must vary, the material used can be silicon carbide with the shape of rods embedded in air, which seems to be a bit unrealistic at visible

In reference [Kante, 2008] another proposal is to use the electromagnetic properties of metal cut wires. This proposal concerns only the TE polarization, and is directly correlated to the proposed cloak in the microwave region [10] by replacing the resonators of Pendry by gold

∂

( )

: 1

<sup>=</sup> : ( )

*z*

ε

*g r*

∂

<sup>−</sup> ′ <sup>=</sup>

2

μ

*g r*

r = ′ = − ′+ g(r ) (1 a / b)r a (12)

∂

*<sup>z</sup>* = (10)

( )

*g r*

∂

<sup>−</sup> ′ <sup>=</sup>

2

(11)

Indeed, the magnetic resonators of Pendry and the nanowires are equivalent for a TE polarized incident wave, where the electric field is parallel to the long side of the resonator or to the wire. The invisibility cloak that was made in the microwave domain and that operates at 11GHz can be implemented in optics by replacing the resonators used in microwave by gold nanowires (Fig. 10). This figure shows a detail of the cape made using the magnetic resonators of Pendry (Fig. 10 (a)), the simulation of the cloak at 1.5 microns (Fig. 10 (b)), a gold nanowire for a TE polarization of the incident wave (Fig. 10 (c)), the equivalent parameters of the nanowire around its first resonance frequency (Fig. 10 (d)), and a schematic view of the cape in which the resonators are replaced by gold nanowires.

#### **3.5 Cloak with arbitrary shapes**

The principles used in the cylindrical cloaks described above can be generalized to a variety of different shapes. The figure below is taken from reference [Nicolet, 2008] where a Fourier expansion is used to access to convex shapes. Reference [Rahm, 2008] proposes a square cloak (Figure 12a), which has been generalized to a polygonal cloak in [Tichit, 2008] (Figure 12b), then to an elliptical one (Figure 12c).

#### **3.6 Broadband cloak**

The major disadvantages of the first invisibility cloak were the narrow frequency-operating band and the extreme values of electromagnetic parameters needed. Various attempts have then been proposed to achieve broadband cloaks, or to design cloaks with more realistic parameter values. The most amazing proposition was suggested by U. Leonhardt who proposed to benefit from a non-Euclidean geometry to achieve the broadband [Leonhardt, 2009]. A. V. Kildishev also proposed an approximate solution to achieve a broadband cloak [Kildishev, 2008]. Following the preceding reference, we show the transition from a twodimensional "conventional" cloak (Figure 13a) to a non-Euclidean one, namely a sphere replacing a single circle (Figure 13b). In the broadband cloak proposed by Shalaev, the principle is simple: light follows a different path depending on the operating frequency (Figure 13c).

Recently other concepts of broadband invisibility cloaks based on the use of broadband nonresonant metamaterials have been proposed [Qiu, 2009, Feng, 2011] (figure 14). These recent works are based on the use of materials made of broadband dielectric multilayer where the electromagnetic parameters are extracted using the relations of Wiener on one-dimensional multi-layer materials. Figure 14 shows an example of a dielectric multi-layer structure used in a cylindrical invisibility cloak. Each concentric layer is constituted by a sub-layer of permittivity εA and a sub-layer of permittivity εB and η is the thicknesses ratio of the two layers. The effective permittivity parameters of the layer are given by:

$$
\varepsilon\_{\vartheta} = \frac{\varepsilon\_{\Lambda} + \eta \varepsilon\_{\text{B}}}{1 + \eta}, \quad \frac{1}{\varepsilon\_{r}} = \frac{1}{1 + \eta} \left( \frac{1}{\varepsilon\_{\Lambda}} + \frac{\eta}{\varepsilon\_{\text{B}}} \right),
$$

Space Coordinate Transformation and Applications 125

Fig. 11. Electric field radiated by a point source illuminating a cloak whose the shape is

(c)

Fig. 12. (a) Squared invisibility cloak [Rahm, 2008]. (b) polygonal cloak [Tichit, 2008]. (c)

(b)

source

cloak

obtained by Fourier expansion of a simpler one.

(a)

elliptical cloak [Tichit, 2008].

Fig. 10. (a) Cloak made of resonators of Pendry for the TE polarization of the incident field. (b) Simulation of this cloak at 1.5µm. (c) gold nanowire in TE polarization. (d) Variation of the effective permittivity and permeability of the wire as a function of frequency for a nanowire with 300nm length and a width and height of 50 nm on silicon. (e) Portion of the infra-red cloak made by a juxtaposition of gold nanowires on silicon.

(b)

(e)

(a)

(c)

(d)

infra-red cloak made by a juxtaposition of gold nanowires on silicon.

Fig. 10. (a) Cloak made of resonators of Pendry for the TE polarization of the incident field. (b) Simulation of this cloak at 1.5µm. (c) gold nanowire in TE polarization. (d) Variation of the effective permittivity and permeability of the wire as a function of frequency for a nanowire with 300nm length and a width and height of 50 nm on silicon. (e) Portion of the

Fig. 11. Electric field radiated by a point source illuminating a cloak whose the shape is obtained by Fourier expansion of a simpler one.

Fig. 12. (a) Squared invisibility cloak [Rahm, 2008]. (b) polygonal cloak [Tichit, 2008]. (c) elliptical cloak [Tichit, 2008].

Space Coordinate Transformation and Applications 127

main difficulty in this kind of design is the realization of permittivity lower than 1. This 2D

The transposition of the concept of electromagnetic cloak in acoustics has been proposed by several research laboratories in 2007 and 2008 [Torrent, 2007, Chen, 2007, Fahrat, 2008, Cummer, 2008], and particularly by the Fresnel Institute at University of Marseille. The variables considered here are the scalar pressure *p*, the fluid velocity, the density *ρ*0, the tensor density and modulus of the fluid density *λ*. As for the electromagnetic cloak, we have a variation of the above parameters in spherical coordinates as in the set of equations (13)

are the components in the plane of the relative bulk density

This example shows the versatility of the concept that can be applied to all media where a

(a)

(b)

ρ,

relative to

ρ0.

approach was also generalized to a 3D cloak [Qiu, 2009].

**3.7 Acoustic cloak** 

wave can propagate.

where ρ*<sup>r</sup>* and ρφ

Fig. 13. (a) Classical cylindrical cloak. (b) Cloak in a non-Euclidian space. (c) Principle of the broadband cloak proposed by Shalaev : the light path changes following the frequency.

Fig. 14. (a) TM plane wave incident on a PEC cylinder surrounded by concentric multilayers. The inner and outer radii of the shell are a and b, respectively. (b) The total magnetic field distribution for an optimized six-layer cloak.

A TM plane wave is incident on a PEC cylinder surrounded by concentric multilayers as shown in Fig. 14a. The inner and outer radii of the shell are *a* and *b*, respectively. The total magnetic field distribution for an optimized six-layer cloak is presented in Fig. 14b. The main difficulty in this kind of design is the realization of permittivity lower than 1. This 2D approach was also generalized to a 3D cloak [Qiu, 2009].

#### **3.7 Acoustic cloak**

126 Metamaterial

(a) (b)

(c) Fig. 13. (a) Classical cylindrical cloak. (b) Cloak in a non-Euclidian space. (c) Principle of the broadband cloak proposed by Shalaev : the light path changes following the frequency.

Fig. 14. (a) TM plane wave incident on a PEC cylinder surrounded by concentric multilayers. The inner and outer radii of the shell are a and b, respectively. (b) The total magnetic field

A TM plane wave is incident on a PEC cylinder surrounded by concentric multilayers as shown in Fig. 14a. The inner and outer radii of the shell are *a* and *b*, respectively. The total magnetic field distribution for an optimized six-layer cloak is presented in Fig. 14b. The

(a) (b)

distribution for an optimized six-layer cloak.

The transposition of the concept of electromagnetic cloak in acoustics has been proposed by several research laboratories in 2007 and 2008 [Torrent, 2007, Chen, 2007, Fahrat, 2008, Cummer, 2008], and particularly by the Fresnel Institute at University of Marseille. The variables considered here are the scalar pressure *p*, the fluid velocity, the density *ρ*0, the tensor density and modulus of the fluid density *λ*. As for the electromagnetic cloak, we have a variation of the above parameters in spherical coordinates as in the set of equations (13) where ρ*<sup>r</sup>* and ρφ are the components in the plane of the relative bulk density ρ, relative to ρ0. This example shows the versatility of the concept that can be applied to all media where a wave can propagate.

Space Coordinate Transformation and Applications 129

*xyz <sup>J</sup> <sup>y</sup> <sup>x</sup> y yy <sup>z</sup>*

′′′ ′′′ = = ′′′

In the initial space, an antenna emits a certain type of radiation. This radiation is then modified by the transformation of the space in which it propagates. Several examples have been proposed in recent papers [Kong,2007, Tichit, 2009, Tichit, 2011, Rui, 2011, Cui 2011].

An example of a directional antenna is given below [Kong, 2007], where a parabolic space is transformed into a rectangular one (Figure 16a). In this example we are in TM polarization.

(a)

(b) Fig. 16. (a) Initial parabolic space and transformed rectangular space. (b) Variation of the

Figure 16b shows the variations of the calculated electromagnetic parameters. Figure 17 shows the radiation of a horn in the transformed space. In this example, the benefit of the transformation of space is not real in the sense that the obtained antenna has almost the same size as the original antenna. Moreover, in this reference, the antenna has not been

electromagnetic parameters µx, µy et εz of the transformed space.

∂∂

∂ ∂∂

∂ ∂∂

/// , , / / / , , ///

*xyz zxz <sup>y</sup> z z*

*x xx y x z*

 ∂∂

 ∂∂

′′′

∂∂∂∂

 ∂

 ∂

(16)

( ) ( )

∂

**4.1.1 1st example: Parabolic transformed antenna** 

∂

Fig. 15. (a) The principle of acoustic cloak proposed in [Cummer, 2008]. (b) Real part of the fluid pressure around the object. (c) The cloak proposed in [Fahrat, 2009].

$$\begin{aligned} \rho\_{\varphi} &= \rho\_{\theta} = \frac{b-a}{b} \\ \rho\_r &= \frac{b-a}{b} \frac{r^2}{\left(r-a\right)^2} \\ \mathcal{A} &= \frac{\left(b-a\right)^3}{b^3} \frac{r^2}{\left(r-a\right)^2} \end{aligned} \tag{13}$$

#### **4. Others applications of the space coordinate transformation**

#### **4.1 Antennas**

The space coordinate transformation was proposed initially by J. Pendry and U. Leonhardt to design invisibility cloaks. This transformation of space was then used to design new devices including microwave antennas. The principle of these antennas is as follows: we define an initial space in which there is an emitter and a transformed space connected by a geometric transformation to the original space. The transformed space is realized to control the field emitted outside by the antenna. The new coordinates of the transformed space x ', y' and z 'are expressed in terms of x, y, z of the initial space

$$\mathbf{x'=x'(x,y,z), y'=y'(x,y,z), z'=z'(x,y,z)} \tag{14}$$

Then we calculate the electromagnetic parameters of the transformed space

$$\overbrace{\varepsilon'=\{\varepsilon\}}^{T} \left(\overbrace{\det^{\top}}^{\top}\right)^{-1} \overbrace{\mu'=\{\mu\}}^{T} \left(\overbrace{\det^{\top}}^{\top}\right)^{-1} \tag{15}$$

where ε is the permittivity tensor, μ the permeability tensor and *J* the Jacobian matrix defined by

(c) Fig. 15. (a) The principle of acoustic cloak proposed in [Cummer, 2008]. (b) Real part of the

> 2 2

(13)

3 2 3 2

( )

x'=x'(x,y, z), y'=y'(x,y, z), z'=z '(x,y, z) (14)

*JJ J* <sup>−</sup> ′ = (15)

the permeability tensor and *J* the Jacobian matrix

*T*

( )

<sup>−</sup> = =

<sup>−</sup> <sup>=</sup> <sup>−</sup>

<sup>−</sup> <sup>=</sup> <sup>−</sup>

The space coordinate transformation was proposed initially by J. Pendry and U. Leonhardt to design invisibility cloaks. This transformation of space was then used to design new devices including microwave antennas. The principle of these antennas is as follows: we define an initial space in which there is an emitter and a transformed space connected by a geometric transformation to the original space. The transformed space is realized to control the field emitted outside by the antenna. The new coordinates of the transformed space x ',

( )

*b a b ba r b r a ba r b ra*

fluid pressure around the object. (c) The cloak proposed in [Fahrat, 2009].

*r*

ρ

λ

**4. Others applications of the space coordinate transformation** 

Then we calculate the electromagnetic parameters of the transformed space

μ

*T*

<sup>1</sup> (det )

*JJ J* <sup>−</sup> ′ = <sup>1</sup> (det )

μ μ

y' and z 'are expressed in terms of x, y, z of the initial space

ε ε

is the permittivity tensor,

**4.1 Antennas** 

where ε

defined by

ρ ρ ϕ θ

$$\overline{J} = \frac{\partial \left( \mathbf{x'}, y', z' \right)}{\partial \left( \mathbf{x}, y, z \right)} = \begin{pmatrix} \partial \mathbf{x'} / \partial \mathbf{x} & \partial \mathbf{x'} / \partial y & \partial \mathbf{x'} / \partial z \\ \partial y' / \partial \mathbf{x} & \partial y' / \partial y & \partial y' / \partial z \\ \partial z' / \partial \mathbf{x} & \partial z' / \partial y & \partial z' / \partial z \end{pmatrix} \tag{16}$$

In the initial space, an antenna emits a certain type of radiation. This radiation is then modified by the transformation of the space in which it propagates. Several examples have been proposed in recent papers [Kong,2007, Tichit, 2009, Tichit, 2011, Rui, 2011, Cui 2011].

#### **4.1.1 1st example: Parabolic transformed antenna**

An example of a directional antenna is given below [Kong, 2007], where a parabolic space is transformed into a rectangular one (Figure 16a). In this example we are in TM polarization.

Fig. 16. (a) Initial parabolic space and transformed rectangular space. (b) Variation of the electromagnetic parameters µx, µy et εz of the transformed space.

Figure 16b shows the variations of the calculated electromagnetic parameters. Figure 17 shows the radiation of a horn in the transformed space. In this example, the benefit of the transformation of space is not real in the sense that the obtained antenna has almost the same size as the original antenna. Moreover, in this reference, the antenna has not been

monopole

ε

with

et (c) *ε*zz.

and the air.

εε

= ′ ′

ε

Space Coordinate Transformation and Applications 131

Fig. 18. Initial cylindrical space with the radiating monopole (left) and the transformed

Plane source

achieve the space transformation. The expressions of these parameters are as follows:

0

xx

*x y*

*xy xy*

where d, e and L are the geometrical dimensions of the initial and transformed spaces.

′′ ′′ ′ =μ = <sup>1</sup> (,) (,) (,) *yy yy*

Fig. 19. Variations of the electromagnetic parameters of the transformed space: (a) *ε*xx, (b) *ε*yy ,

The expressions of the electromagnetic parameters vary continuously, and remain limited to reasonable values. Figure 20 shows the calculated magnetic field at 5, 10 and 40 GHz. The directivity of the antenna increases as the frequency rises. The dimensions of the antenna are shown in Figure 20a. One can observe that there is no reflection between the metamaterial

ε

 ε

*e* π ε*xx,*ε

(,) 0 0

 <sup>μ</sup> ′ ′ μ = μ μ ′ ′ μ ′ ′

*yy*

0 (,) 0 0 0 (,)

*xx*

ε′′ ′′ =μ = ′ ′ (18)

*x y*

*x y*

*zz*

*x y*

*yy , and μzz* needed to

0

(17)

space (right) with the transformed plane source.

(x ,y ) 0 0 xx

′ ′

′ ′

0 (x ,y ) 0 yy 0 0 (x ,y ) zz

ε

(,) (,) *xx xx xy xy x*

Figure 19 shows the variations of the electromagnetic parameters

realized and the far-field radiation patterns have not been presented. However, space coordinate transformation can be used to design antennas more compact than conventional ones. This is the case of the antenna proposed theoretically in reference 23 and experimentally measured in [Tichit, 2009].

Fig. 17. (a) Horn antenna emitting in the parabolic space. (b) horn antenna emitting in the transformed space. In both cases the near fields are almost equivalent.

#### **4.1.2 2nd example: Directive antenna**

The 2nd example concerns the transformation of an isotropic antenna into a directive one [Tichit, 2009, Tichit, 2011]. This isotropic antenna is taken as an infinite radiating wire. The initial space is then supposed to be the cylindrical space surrounding the wire. The transformed space is a rectangular one as illustrated in Figure 18. After the transformation, the radiating wire in the cylindrical space is then comparable to a plane source radiating in the rectangular space.

Fig. 18. Initial cylindrical space with the radiating monopole (left) and the transformed space (right) with the transformed plane source.

Figure 19 shows the variations of the electromagnetic parameters ε*xx,* ε*yy , and μzz* needed to achieve the space transformation. The expressions of these parameters are as follows:

$$\mathbf{E} = \begin{pmatrix} \varepsilon\_{\text{xx}} \langle \mathbf{x}', \mathbf{y}' \rangle & 0 & 0\\ 0 & \varepsilon\_{\text{yy}} \langle \mathbf{x}', \mathbf{y}' \rangle & 0\\ 0 & 0 & \varepsilon\_{\text{zz}} \langle \mathbf{x}', \mathbf{y}' \rangle \end{pmatrix} \varepsilon\_0 \quad \text{\(\mu = \begin{pmatrix} \mu\_{\text{xx}} \langle \mathbf{x}', \mathbf{y}' \rangle & 0 & 0\\ 0 & \mu\_{\text{yy}} \langle \mathbf{x}', \mathbf{y}' \rangle & 0\\ 0 & 0 & \mu\_{\text{zz}} \langle \mathbf{x}', \mathbf{y}' \rangle \end{pmatrix} \mu\_0 \text{ (17)}$$

with

130 Metamaterial

realized and the far-field radiation patterns have not been presented. However, space coordinate transformation can be used to design antennas more compact than conventional ones. This is the case of the antenna proposed theoretically in reference 23 and

(a)

Fig. 17. (a) Horn antenna emitting in the parabolic space. (b) horn antenna emitting in the

The 2nd example concerns the transformation of an isotropic antenna into a directive one [Tichit, 2009, Tichit, 2011]. This isotropic antenna is taken as an infinite radiating wire. The initial space is then supposed to be the cylindrical space surrounding the wire. The transformed space is a rectangular one as illustrated in Figure 18. After the transformation, the radiating wire in the cylindrical space is then comparable to a plane source radiating in

transformed space. In both cases the near fields are almost equivalent.

**4.1.2 2nd example: Directive antenna** 

the rectangular space.

(b)

experimentally measured in [Tichit, 2009].

$$
\varepsilon\_{xx}(\mathbf{x}', \mathbf{y}') = \mu\_{xx}(\mathbf{x}', \mathbf{y}') = \frac{\pi}{c} \mathbf{x}' \qquad \varepsilon\_{yy}(\mathbf{x}', \mathbf{y}') = \mu\_{yy}(\mathbf{x}', \mathbf{y}') = \frac{1}{\varepsilon\_{xx}(\mathbf{x}', \mathbf{y}')} \tag{18}
$$

where d, e and L are the geometrical dimensions of the initial and transformed spaces.

Fig. 19. Variations of the electromagnetic parameters of the transformed space: (a) *ε*xx, (b) *ε*yy , et (c) *ε*zz.

The expressions of the electromagnetic parameters vary continuously, and remain limited to reasonable values. Figure 20 shows the calculated magnetic field at 5, 10 and 40 GHz. The directivity of the antenna increases as the frequency rises. The dimensions of the antenna are shown in Figure 20a. One can observe that there is no reflection between the metamaterial and the air.

Space Coordinate Transformation and Applications 133

b

The previous antenna made use of resonant metamaterials. Then its bandwidth is inherently narrow. An interesting broadband operation proposal was recently presented by Y. Hao [Rui, 2011]. A broadband Fresnel lens can be realized using a multilayer dielectric structure. The permittivities of the different layers are calculated using space coordinate transformation. Figure 22 shows an example of such antenna. The performances are presented in part II of the figure. We can note the broadband characteristics of the antenna

I

Fig. 21. (a) Normalized measured radiation pattern of the antenna. (b) detail of the

metamaterial and of the realized antenna.

in IId.

**4.1.3 3rd example: Broadband Fresnel antenna** 

Fig. 20. Magnetic field cartography for a TM wave polarization calculated at (a) 5, (b) 10 and (c) 40 GHz.

The practical realization of this antenna, however, requires a simplification of these parameters. One solution proposed recently is to use a discrete variation of these parameters. The simplification is performed with a conservation of the propagation equation. The following set of parameters is then obtained:

$$
\varepsilon\_{yy} = \mu\_{zz} = 1 \quad \text{:} \quad \varepsilon\_{xx} = \left(\frac{\pi x}{e}\right)^2 \tag{19}
$$

The material needed must have a variable permittivity in the direction of propagation Ox. The other parameters remain constant. Figure 21a shows a detail of the material used to make the variable permittivity, and the fabricated antenna prototype for an operation near 10 GHz. Figure 21b shows the performances of this antenna. It can be observed that the radiation pattern of the antenna is not affected by the simplification and the discretization of the material.

Fig. 20. Magnetic field cartography for a TM wave polarization calculated at (a) 5, (b) 10 and

The practical realization of this antenna, however, requires a simplification of these parameters. One solution proposed recently is to use a discrete variation of these parameters. The simplification is performed with a conservation of the propagation

*xx*

ε<sup>=</sup>

The material needed must have a variable permittivity in the direction of propagation Ox. The other parameters remain constant. Figure 21a shows a detail of the material used to make the variable permittivity, and the fabricated antenna prototype for an operation near 10 GHz. Figure 21b shows the performances of this antenna. It can be observed that the radiation pattern of the antenna is not affected by the simplification and the discretization of

(a)

2

(19)

*x e* π

1 *yy zz*

= μ = :

ε

equation. The following set of parameters is then obtained:

(c) 40 GHz.

the material.

Fig. 21. (a) Normalized measured radiation pattern of the antenna. (b) detail of the metamaterial and of the realized antenna.

#### **4.1.3 3rd example: Broadband Fresnel antenna**

The previous antenna made use of resonant metamaterials. Then its bandwidth is inherently narrow. An interesting broadband operation proposal was recently presented by Y. Hao [Rui, 2011]. A broadband Fresnel lens can be realized using a multilayer dielectric structure. The permittivities of the different layers are calculated using space coordinate transformation. Figure 22 shows an example of such antenna. The performances are presented in part II of the figure. We can note the broadband characteristics of the antenna in IId.

Space Coordinate Transformation and Applications 135

Fig. 23. (a) The 3D flat-lens antenna made of gradient index metamaterial. The aperture size is 9.6 cm. (b) The measured far-field radiation patterns of the 3D metamaterial flat lens

Other devices were recently proposed in the domain of the optical waveguiding devices [Ghasemi, 2010, Liu, 2008]. The proposal in [Ghasemi, 2010] tends to answer to a main drawback of use of metallic metamaterials at optical frequencies which is their high losses. A promising approach consists in creating hybrid photonic structures in which metallic parts are coupled with dielectric (and almost lossless) waveguides. In this configuration, useful functionalities are obtained by allowing just enough light to interact with the metallic parts of the system. The remaining part of the energy propagates in the dielectric waveguide, thereby considerably mitigating the losses. Figure 25 shows a mode adapter designed using this approach. The mode adapter allows the transition of the energy flow from a large SOI ridge waveguide to a narrower one. The taper has been achieved using the method of transformation optics. Although the authors simply considered a 2D transformation, they show that this structure can effectively act upon the three dimensional

antenna in the X band.

flow of light guided by the SOI structure.

Fig. 22. I/ Schematic showing of the transformed zone plate lens antenna. (a) 2D hyperbolic lens with nearly orthogonal mapping. (b) 2D flat lens with the permittivity map consisting of 110×20 blocks. (c) 2D flat lens with the permittivity map consisting of 22×4 blocks. (d) 3D transformed zone plate lens antenna. II/ The radiation patterns of the conventional 3D hyperbolic lens, 3D phase-correcting Fresnel lens and 3D transformed zone plate lens at (a) 20 GHz, (b) 30 GHz, (c) 40 GHz. (d) The comparison of the bandwidth performance of 3D phase-correcting Fresnel lens and 3D transformed zone plate lens from 20 GHz to 45 GHz

#### **4.1.4 4th example: Three-dimensional metamaterial lens antennas**

The proposal of T.J. Cui to realize a lens using variable index material so as to focalize the beam of a waveguide is also an interesting application of transformation optics [Cui, 2011]]. Figure 23 shows a photo of the realized prototype and the performances of this lens in X band. The antenna presents two main advantages: the broadband behavior of the dielectric and the easiness of the realization (at microwave frequencies). Indeed the index gradient is realized with an array of variable size closed square rings printed on a dielectric substrate. Remains the classical drawback of the impossibility to realize an index lower than 1.

#### **4.2 Circuits**

M. Rahm proposed in [Rahm, 2008] a general method to achieve an invisibility cloak. But he also proposed the implementation of energy concentrator. Figure 24a shows a simulation of such a device. In [Lin, 2008], L. Lin proposed a number of applications such as a phase transformer to transform a cylindrical wavefront to a plane wavefront (Figure 24b) or a power divider (Figure 24c). In [Huangfu, 2008], J. Huangfu proposed a method to achieve wave guiding without reflection at 90° bends (Figure 24d).

II Fig. 22. I/ Schematic showing of the transformed zone plate lens antenna. (a) 2D hyperbolic lens with nearly orthogonal mapping. (b) 2D flat lens with the permittivity map consisting of 110×20 blocks. (c) 2D flat lens with the permittivity map consisting of 22×4 blocks. (d) 3D transformed zone plate lens antenna. II/ The radiation patterns of the conventional 3D hyperbolic lens, 3D phase-correcting Fresnel lens and 3D transformed zone plate lens at (a) 20 GHz, (b) 30 GHz, (c) 40 GHz. (d) The comparison of the bandwidth performance of 3D phase-correcting Fresnel lens and 3D transformed zone plate lens from 20 GHz to 45 GHz

The proposal of T.J. Cui to realize a lens using variable index material so as to focalize the beam of a waveguide is also an interesting application of transformation optics [Cui, 2011]]. Figure 23 shows a photo of the realized prototype and the performances of this lens in X band. The antenna presents two main advantages: the broadband behavior of the dielectric and the easiness of the realization (at microwave frequencies). Indeed the index gradient is realized with an array of variable size closed square rings printed on a dielectric substrate.

M. Rahm proposed in [Rahm, 2008] a general method to achieve an invisibility cloak. But he also proposed the implementation of energy concentrator. Figure 24a shows a simulation of such a device. In [Lin, 2008], L. Lin proposed a number of applications such as a phase transformer to transform a cylindrical wavefront to a plane wavefront (Figure 24b) or a power divider (Figure 24c). In [Huangfu, 2008], J. Huangfu proposed a method to achieve

Remains the classical drawback of the impossibility to realize an index lower than 1.

**4.1.4 4th example: Three-dimensional metamaterial lens antennas** 

wave guiding without reflection at 90° bends (Figure 24d).

**4.2 Circuits** 

Fig. 23. (a) The 3D flat-lens antenna made of gradient index metamaterial. The aperture size is 9.6 cm. (b) The measured far-field radiation patterns of the 3D metamaterial flat lens antenna in the X band.

Other devices were recently proposed in the domain of the optical waveguiding devices [Ghasemi, 2010, Liu, 2008]. The proposal in [Ghasemi, 2010] tends to answer to a main drawback of use of metallic metamaterials at optical frequencies which is their high losses. A promising approach consists in creating hybrid photonic structures in which metallic parts are coupled with dielectric (and almost lossless) waveguides. In this configuration, useful functionalities are obtained by allowing just enough light to interact with the metallic parts of the system. The remaining part of the energy propagates in the dielectric waveguide, thereby considerably mitigating the losses. Figure 25 shows a mode adapter designed using this approach. The mode adapter allows the transition of the energy flow from a large SOI ridge waveguide to a narrower one. The taper has been achieved using the method of transformation optics. Although the authors simply considered a 2D transformation, they show that this structure can effectively act upon the three dimensional flow of light guided by the SOI structure.

Space Coordinate Transformation and Applications 137

x position (µm) 0 2.5 5

(d) Fig. 25. (a) Geometry of the mode adapter considered in this study; (b) cross-sectional view

Transition from the large to the narrow waveguide using a mode adapter. The y-component

The same principle can be applied in infrared and visible domains [Gabrielli, 2009, Greenleaf, 2007, Cheng, 2009]. In reference [Gabrielli, 2009], the authors present the realization and the characterization of a carpet cloak operating in the optical domain. Figure

of the input SOI ridge waveguide; (c) cross-sectional view of the mode adapter. (d)

of the electric field is shown in the x-y plane located halfway through the Si slab.

y position (µm)

**4.3.2 Optical carpet cloak** 


27 shows a view of the realized carpet on silicon.

0

1.5

Fig. 24. (a) Energy concentrator proposed in the reference 12. (b) Phase Transformer between 2 regions. (c) Power divider. (d) 90° waveguide bend without loss.

#### **4.3 Broadband carpet cloak**

#### **4.3.1 Microwave broadband carpet cloak**

D. R. Smith has recently proposed a broadband cloak that can be adjusted to any object placed on the ground [Valentine, 2009]. This cloak allows to reconstruct the reflection of light incident on an object in order to make as if the object was not present. The object must however have dimensions small compared to the dimensions of the cloak. Figure 26a illustrates the operating principle of the cloak. Figure 26b gives a picture of its implementation and shows the pattern of the material permittivity variable used. The idea is to change the optical path followed by the reflected beam. The carpet cloak reconstructs the reflected beam as it is when no object is placed on the ground. This is clearly shown in Figure 26a: in I the ground reflects an incident beam without obstacle, in II the beam is reflected in the presence of an obstacle, in III the reflected beam consists of parallel rays reconstructed by the cloak covering the obstacle.

(a) (b)

(c) (d) Fig. 24. (a) Energy concentrator proposed in the reference 12. (b) Phase Transformer between

D. R. Smith has recently proposed a broadband cloak that can be adjusted to any object placed on the ground [Valentine, 2009]. This cloak allows to reconstruct the reflection of light incident on an object in order to make as if the object was not present. The object must however have dimensions small compared to the dimensions of the cloak. Figure 26a illustrates the operating principle of the cloak. Figure 26b gives a picture of its implementation and shows the pattern of the material permittivity variable used. The idea is to change the optical path followed by the reflected beam. The carpet cloak reconstructs the reflected beam as it is when no object is placed on the ground. This is clearly shown in Figure 26a: in I the ground reflects an incident beam without obstacle, in II the beam is reflected in the presence of an obstacle, in III the reflected beam consists of parallel rays

2 regions. (c) Power divider. (d) 90° waveguide bend without loss.

**4.3 Broadband carpet cloak** 

**4.3.1 Microwave broadband carpet cloak** 

reconstructed by the cloak covering the obstacle.

(d)

Fig. 25. (a) Geometry of the mode adapter considered in this study; (b) cross-sectional view of the input SOI ridge waveguide; (c) cross-sectional view of the mode adapter. (d) Transition from the large to the narrow waveguide using a mode adapter. The y-component of the electric field is shown in the x-y plane located halfway through the Si slab.

#### **4.3.2 Optical carpet cloak**

The same principle can be applied in infrared and visible domains [Gabrielli, 2009, Greenleaf, 2007, Cheng, 2009]. In reference [Gabrielli, 2009], the authors present the realization and the characterization of a carpet cloak operating in the optical domain. Figure 27 shows a view of the realized carpet on silicon.

Space Coordinate Transformation and Applications 139

Fig. 28. Optical carpet cloaking at a wavelength of 1,540 nm: The results for a Gaussian beam reflected from a flat surface (a), a curved (without a cloak) surface (b) and the same curved

One of the most amazing applications has been proposed by A. Greenleaf [Greenleaf, 2007]. He imagined to create a wormhole using electromagnetic invisibility cloak able to link two remote areas of space and ensures the propagation of an electromagnetic wave between both regions invisible from the outside. Figure 29 shows a schematic illustration of the wormhole where its exterior deflects the incident electromagnetic waves and a section of the

Fig. 29. (a) A schematic illustration of the wormhole whose exterior cloak deflects the incident electromagnetic waves. (b) a section of the wormhole showing a wave propagating

**4.4 Electromagnetic wormhole and other cosmological objects** 

reflecting surface with a cloak (c).

inside.

wormhole showing a wave propagating inside.

(a)

(b)

Fig. 26. Principle of the carpet cloak: in I the ground reflects an incident beam without obstacle, in II the beam is reflected in the presence of an obstacle, in III the reflected beam consists of parallel rays reconstructed by the cloak covering the obstacle. (b) View of the realized carpet cloak and the metamaterial unit cell with variable permittivity used in the carpet.

Fig. 27. Principle (a) and realization (b) of an optical carpet cloak on silicon.

Figure 28 shows the carpet cloak operating at a wavelength of 1,540 nm, for an incident Gaussian beam reflected from a curved reflecting surface. Similar reflection characteristics can be observed when compared to a reflection on a flat surface.

(a)

(b)

Fig. 26. Principle of the carpet cloak: in I the ground reflects an incident beam without obstacle, in II the beam is reflected in the presence of an obstacle, in III the reflected beam consists of parallel rays reconstructed by the cloak covering the obstacle. (b) View of the realized carpet cloak and the metamaterial unit cell with variable permittivity used in the

Fig. 27. Principle (a) and realization (b) of an optical carpet cloak on silicon.

can be observed when compared to a reflection on a flat surface.

Figure 28 shows the carpet cloak operating at a wavelength of 1,540 nm, for an incident Gaussian beam reflected from a curved reflecting surface. Similar reflection characteristics

carpet.

Fig. 28. Optical carpet cloaking at a wavelength of 1,540 nm: The results for a Gaussian beam reflected from a flat surface (a), a curved (without a cloak) surface (b) and the same curved reflecting surface with a cloak (c).

#### **4.4 Electromagnetic wormhole and other cosmological objects**

One of the most amazing applications has been proposed by A. Greenleaf [Greenleaf, 2007]. He imagined to create a wormhole using electromagnetic invisibility cloak able to link two remote areas of space and ensures the propagation of an electromagnetic wave between both regions invisible from the outside. Figure 29 shows a schematic illustration of the wormhole where its exterior deflects the incident electromagnetic waves and a section of the wormhole showing a wave propagating inside.

Fig. 29. (a) A schematic illustration of the wormhole whose exterior cloak deflects the incident electromagnetic waves. (b) a section of the wormhole showing a wave propagating inside.

Space Coordinate Transformation and Applications 141

the rare realizations proposed in literature, these parameters have been simplified, and often the impedance matching has been sacrificed to obtain a feasible material. The metamaterial based design encountered in this case the problem of reflection losses, and are already comparable to existing solutions that have proved their performances, for example in the field of antennas. Another difficulty is the narrow bandwidth of the metamaterials used, in particular those based on resonant structures of the type of split ring resonators of Pendry. In reality this is not really a problem, because broadband metamaterials can be realized from

Tannery P., Henry C. et de Waard C., *Œuvres de Pierre de Fermat*, Gauthier-Villars et Cie,

Pendry, J. B., Schurig, D., Smith, D. R., (2006), Controlling Electromagnetic Fields, *Science,* 

Cai W., Chettiar U. K., Kildishev A. V., and Shalaev V. M.. (2008), Designs for optical cloaking with high-order transformations, *Optics Express,* Vol. 16, No. 8, pp.5444-5452. Schurig, D., J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith,

Soukoulis C.M., Wegener M., (2011), Past achievements and future challenges in the

Guven K., Saenz E., Gonzalo R., Ozbay E. and Tretyakov S., (2008), Electromagnetic cloaking with canonical spiral inclusions, *New Journal of Physics*, Vol.10, 115037 Kanté B., de Lustrac A., Lourtioz J.-M., Burokur S. N., (2008), Infrared cloaking based on the

Kanté B., Germain D., and de Lustrac A., (2009), Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies » *Physical Review B*, Vol. 80, 201104(R). Nicolet A., Zolla F., Guenneau S., (2008), Electromagnetic analysis of cylindrical cloaks of an

Rahm M., Schurig D., Roberts D. A., Cummer S.A., Smith D. R., Pendry J. B., (2008), Design

Tichit P-H., Kante B., de Lustrac A., (2008) Design of polygonal or elliptical invisibility cloak,

Leonhardt U., Tyc T., (2009), Broadband Invisibility by Non-Euclidean Cloaking, *Science,* 

Kildishev A. V., Cai W., Chettiar U. K., Shalaev V.M., (2008), Transformation optics: approaching broadband electromagnetic cloaking, *New Journal of Physics*, Vol.10, 115029. Qiu C.-W., Hu L., Xu X., Feng Y., (2009), Spherical cloaking with homogeneous isotropic

Feng Y. J., Xu X. F., Yu Z. Z., (2011), Practical realization of transformation-optics designed

invisibility cloak through layered structures, *Antennas and Propagation (EUCAP), Proceedings of the 5th European Conference*, pp. 3456 – 3460, ISBN: 978-1-4577-0250-1,

arbitrary cross section, *Optics Letters,* Vol.33, No. 14, pp.1584-1586.

Nato Workshop Meta 08, Marrakesh, may 07-10 2008.

multilayered structures, *Physical Review E,* Vol.79, 047602.

(2006), Metamaterial Electromagnetic Cloak at Microwave Frequencies, *Science,* 

development of three-dimensional photonic metamaterials, *Nature Photonics*, Vol.5,

electric response of split ring resonators, *Optics Express*, Vol.16, N° 12, pp.9191-9198,

of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations, *Photonics and Nanostructures. Fundamental* 

composite metamaterials [Djermoun, 2007], or by using all dielectric structures.

Leonhardt, U., (2006), Optical Conformal Mapping, *Science,* Vol.312, pp.1777-1780.

**6. References** 

Paris, 1891-1922.

Vol. 312, pp. 1780-1782*.* 

Vol.314, pp. 977-980

*and Applications,* Vol.6, 87.

Vol.323, pp.110-112;

Rome, 11-15 April 2011.

pp.523-530

Fig. 30. (a) Distributions of electric fields |Ez| for the designed black hole at the frequency of 18 GHz: The full-wave simulation result under the on-center incidence of a Gaussian beam. (b) The full-wave simulation result under the off-center incidence of a Gaussian beam. (c) Photograph of the fabricated artificial black hole based on metamaterials, which is composed of 60 concentric layers, with ELC structures in the core layers and I-shaped structures in the shell layers.

#### **5. Conclusion and outlooks**

The potential applications of the space coordinate transformation seem to be very various. The examples presented in this chapter show their usefulness, even if they are still far from industrial achievements. Also it appears that these applications can be transposed to any frequency. The conventional metamaterials used in the microwave region are metaldielectric structures. However metals have different present high losses at infrared and optical frequencies. Therefore the applicability will differ greatly between the microwave domain on one hand, and optical frequencies on the other. In the optical domain, the problem is mainly the achievement of materials with metallic patterns having sizes of about one-tenth of the wavelength (a few hundred nanometers) and the control of their geometry [Soukoulis, 2011]. The other problem is the losses of the metallic metamaterials at optical wavelengths. Innovative approaches have recently been proposed to solve partially this problem. But the problem is not completely solved.

In the microwave field, the achievements seem easier as they involve usually inexpensive materials, and metallic parts of the metamaterials present low losses at these frequencies. But they depend strongly on the complexity of the electromagnetic parameters to achieve. In the rare realizations proposed in literature, these parameters have been simplified, and often the impedance matching has been sacrificed to obtain a feasible material. The metamaterial based design encountered in this case the problem of reflection losses, and are already comparable to existing solutions that have proved their performances, for example in the field of antennas. Another difficulty is the narrow bandwidth of the metamaterials used, in particular those based on resonant structures of the type of split ring resonators of Pendry. In reality this is not really a problem, because broadband metamaterials can be realized from composite metamaterials [Djermoun, 2007], or by using all dielectric structures.

#### **6. References**

140 Metamaterial

(a) (b)

Fig. 30. (a) Distributions of electric fields |Ez| for the designed black hole at the frequency of 18 GHz: The full-wave simulation result under the on-center incidence of a Gaussian beam. (b) The full-wave simulation result under the off-center incidence of a Gaussian beam. (c) Photograph of the fabricated artificial black hole based on metamaterials, which is composed of 60 concentric layers, with ELC structures in the core layers and I-shaped

The potential applications of the space coordinate transformation seem to be very various. The examples presented in this chapter show their usefulness, even if they are still far from industrial achievements. Also it appears that these applications can be transposed to any frequency. The conventional metamaterials used in the microwave region are metaldielectric structures. However metals have different present high losses at infrared and optical frequencies. Therefore the applicability will differ greatly between the microwave domain on one hand, and optical frequencies on the other. In the optical domain, the problem is mainly the achievement of materials with metallic patterns having sizes of about one-tenth of the wavelength (a few hundred nanometers) and the control of their geometry [Soukoulis, 2011]. The other problem is the losses of the metallic metamaterials at optical wavelengths. Innovative approaches have recently been proposed to solve partially this

In the microwave field, the achievements seem easier as they involve usually inexpensive materials, and metallic parts of the metamaterials present low losses at these frequencies. But they depend strongly on the complexity of the electromagnetic parameters to achieve. In

structures in the shell layers.

**5. Conclusion and outlooks** 

problem. But the problem is not completely solved.

(c)


**6** 

*1Saudi Arabia 2,3China* 

**Effective Medium Theories and Symmetry** 

*3Department of Physics and William Mong Institute of Nano Science and Technology* 

Recently, metamaterials have attracted a great deal of attention due to their unusual properties not seen in naturally occurring materials, such as negative refraction (Lezec et al., 2007; Pendry, 2000; Veselago, 1968), superlensing (Grbic & Eleftheriades, 2004) and cloaking (Leonhardt 2006; Pendry et al., 2006), etc. These unusual properties are derived from the resonant structures in their artificial building blocks. The resonant structures interact with the wave, but their small size prevents them from being "seen" individually by the wave with wavelength inside the background much larger than the size of the structures (Pendry & Smith, 2006). Thus, the properties of a metamaterial can be described with homogenized parameters or effective medium parameters. The theory that links the microscopic resonant structures to their effective medium parameters is called the effective medium theory (EMT). For example, the left-handed metamaterial consisting of a periodic array of split ring resonators and conducting thin wires has been successfully demonstrated that, in a frequency regime, it behaves like a homogeneous medium exhibiting negative effective permittivity,

μ

consequence of double negativity in permittivity and permeability. This example shows how effective medium parameters of particular resonant structures can be used to describe unusual properties of a metamaterial. In turn, a valid and accurate EMT provides an efficient and systematic tool to design and engineer the resonant structures according to certain desired metamaterial properties. During the development of metamaterials, there has always been a continuous effort to find an appropriate EMT for metamaterials such that various novel phenomena can have a theoretical explanation. One can obtain the effective parameters from some phenomenological results, such as transmission and reflections (Baker-Jarvis et al., 1990; Smith et al., 2002), and wave propagation (Andryieuski et al., 2009), but only a theory can give a clear understanding on the physical origin of effecitve

ε μ

**1. Introduction** 

*eff* ε

, and negative effective permeability,

Since the refractive index is defined as *neff eff eff* =

**Properties of Elastic Metamaterials** 

*1Division of Mathematical and Computer Sciences and Engineering,* 

Ying Wu1, Yun Lai2 and Zhao-Qing Zhang3

*King Abdullah University of Science and Technology* 

*Hong Kong University of Science and Technology* 

*eff* , simultaneously (Pendry et al., 1999, 1996).

, negative refraction is a

*2Department of Physics, Soochow University* 


### **Effective Medium Theories and Symmetry Properties of Elastic Metamaterials**

Ying Wu1, Yun Lai2 and Zhao-Qing Zhang3

*1Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology 2Department of Physics, Soochow University 3Department of Physics and William Mong Institute of Nano Science and Technology Hong Kong University of Science and Technology 1Saudi Arabia 2,3China* 

#### **1. Introduction**

142 Metamaterial

Torrent D. and Sanchez-Dehesa J., (2009), Acoustic metamaterials for new two-dimensional

Chen H., Chan C. T., (2007), Acoustic cloaking in three dimensions using acoustic

Farhat M. Enoch S., Guenneau S., Movchan A. B., (2008), Broadband cylindrical acoustic cloak for linear surface waves in a fluid, *Physical Review Letters* Vol.101, 134501 Cummer S. A., Popa B. I., Schurig D., Smith D. R., Pendry J., Rahm M., Starr A., (2008),

Kong F., Wu B., Kong J. A., Huangfu J. H., and Xi S., (2007), Planar focusing antenna design by using coordinate transformation technology, *Applied Physics Letters,* Vol.91, 253509. Tichit P.-H., Burokur S. N., de Lustrac A., (2009), Ultra-directive antenna via transformation

Tichit P.-H., Burokur S. N., de Lustrac A., (2011), Design and experimental demonstration of a

Rui Y., Wenxuan T., Hao Y., (2011), Broadband Dielectric Zone Plate Antenna from Transformation Electromagnetics, *Optics Express,* Vol.19, n°13, pp.12348-12356. Cui T. J., Zhou X. Y., Ma H. F., (2011) Three-Dimensional Metamaterial Lens Antennas,

Lin L., Wang W., Cui J., Du C., Luo X., (2008), Design of electromagnetic refractor and phase

Huangfu J., Xi S., Kong F., Zhang J., Chen H., Wang D., Wu B.J., Ran L., Kong J.A., (2008)

Tichit P.H., Burokur S.N., de Lustrac A., (2010), Waveguide taper engineering using coordinate transformation technology, *Optics Express*, Vol.18, n°2, pp. 767-772. Ghasemi R., Tichit P.H., Degiron A., Lupu A. de Lustrac A., (2010), Efficient control of a 3D

Liu R., Ji C., Mock J.J., Chin J.Y., Cui T.J., Smith D.R., (2009), Broadband Ground-Plane

Xu X., Feng Y., Hao Y., Zhao J., Jiang T., (2009), Infrared carpet cloak designed with uniform

Valentine J., Li J., Zentgraf T., Bartal G., Zhang X., (2009), An optical cloak made of

Gabrielli L. H., Cardenas J., Poitras C.B., Lipson M., (2009), Silicon nanostructure cloak

Greenleaf A., Kurylev Y., Lassas M., and Uhlmann G., (2007), Electromagnetic Wormholes

Cheng, Q. & Cui, T. J. (2009), An omnidirectional electromagnetic absorber made of

Djermoun A., de Lustrac A., Lourtioz J.M., (2007), A wide band left handed material with

and Virtual Magnetic Monopoles from Metamaterials, *Physical Review Letters*,

high transmission, *Photonics and Nanostructures - Fundamentals and Applications*,

silicon grating structure, *Applied Physics Letters,* Vol.95, pp 184102.

operating at optical frequencies, *Nature Photonics* Vol.3, pp.461-463.

3301 - 3303 , ISBN: 978-1-4577-0250-1, Rome, 11-15 April 2011.

high-directive emission with transformation optics, *Physical Review B,* Vol.83, 155108.

*Antennas and Propagation (EUCAP), Proceedings of the 5th European Conference,* pp.

transformer using coordinate transformation theory, *Optics Express,* Vol.16, n°10,

Application of coordinate transformation in bent waveguides*, Journal of Applied* 

optical mode using a thin sheet of transformation optical medium, *Optics Express*,

Scattering Theory Derivation of a 3D Acoustic Cloaking Shell*, Physical Review* 

sonic devices, *New Journal of Physics*, Vol.9, 323 (2007).

metamaterials, *Applied Physics Letters,* Vol.91, 183518.

optics, *Journal of Applied Phys*ics, Vol.105, 104912

*Letters*, Vol.100, 024301.

pp.6815-6821.

Vol.99, 183901.

Vol.5, n1, pp. 21-28.

*Phys*ics Vol.104, 014502.

Vol.18, n 19, pp. 20305-20312.

Cloak, *Science*, Vol.323, pp.366-369.

dielectrics, *Nature Materials*, Vol.8, pp.568-571.

metamaterials, *New Journal of Physics*, Vol.12, 063006

Recently, metamaterials have attracted a great deal of attention due to their unusual properties not seen in naturally occurring materials, such as negative refraction (Lezec et al., 2007; Pendry, 2000; Veselago, 1968), superlensing (Grbic & Eleftheriades, 2004) and cloaking (Leonhardt 2006; Pendry et al., 2006), etc. These unusual properties are derived from the resonant structures in their artificial building blocks. The resonant structures interact with the wave, but their small size prevents them from being "seen" individually by the wave with wavelength inside the background much larger than the size of the structures (Pendry & Smith, 2006). Thus, the properties of a metamaterial can be described with homogenized parameters or effective medium parameters. The theory that links the microscopic resonant structures to their effective medium parameters is called the effective medium theory (EMT). For example, the left-handed metamaterial consisting of a periodic array of split ring resonators and conducting thin wires has been successfully demonstrated that, in a frequency regime, it behaves like a homogeneous medium exhibiting negative effective permittivity, *eff* ε , and negative effective permeability, μ*eff* , simultaneously (Pendry et al., 1999, 1996). Since the refractive index is defined as *neff eff eff* = ε μ , negative refraction is a consequence of double negativity in permittivity and permeability. This example shows how effective medium parameters of particular resonant structures can be used to describe unusual properties of a metamaterial. In turn, a valid and accurate EMT provides an efficient and systematic tool to design and engineer the resonant structures according to certain desired metamaterial properties. During the development of metamaterials, there has always been a continuous effort to find an appropriate EMT for metamaterials such that various novel phenomena can have a theoretical explanation. One can obtain the effective parameters from some phenomenological results, such as transmission and reflections (Baker-Jarvis et al., 1990; Smith et al., 2002), and wave propagation (Andryieuski et al., 2009), but only a theory can give a clear understanding on the physical origin of effecitve

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 145

While the field of EM metamaterial has developed rapidly during the past decade, one of its counterparts, denoted as acoustic metamaterial has also seen fast growth (Ding et al., 2007; Fang et al., 2006; Lee et al., 2010; Yang et al., 2008). The acoustic metamaterial is designed to manipulate acoustic waves (Chen et al., 2010; Lu et al., 2009). Analogous to EM metamaterials, there are two material parameters that describe the wave propagation, which

metamaterials are all fluids, the governing equation can be mapped into 2D EM equations so that the EMT for 2D acoustic metamaterials is the same as that for 2D EM metamaterials. If the scatterers in acoustic metamaterials are solid, the shear modulus of the scatterers can be ignored when the longitudinal velocity contrast between the scatterer and the host is high (Kafesaki & Economou, 1999). In this case, the scattering property is basically the same as the EM cases. The mapping from the EM waves to acoustic waves facilitates the development of EMTs which have also been extensively studied by using various types of methods. MST (Mei et al., 2006; Torrent et al., 2006) and the CPA (Kafesaki et al., 2000; Li & Chan, 2004) represent two classes of them. Exciting news also came from the experimental realizations of acoustic metamaterials, such as acoustic negative refraction (S. Zhang et al.,

The term *elastic* metamaterial refers to those metamaterials which are able to sustain not only longitudinal but also shear waves in their effective media. It is well-known that the EMT for an elastic composite in the quasi-static limit is anisotropic in general. The only exception is the hexagonal lattice in two dimensions (Landau & Lifshitz, 1986; Royer & Dieulesaint, 1999; Wu & Z. Zhang 2009). Even for this case, the EMT involves the

EMTs which involve only two effective parameters. One more effective medium parameter greatly enriches the types and physics of wave propagation (Chen et al., 2008; Wu et al., 2007), such as mode conversion between longitudinal and transverse waves (Wu et al., 2011); however, it also adds complexity to the EMT. Recently, various EMTs for elastic metamaterials have been proposed, such as those ones based on the plane-wave-expansion

In this chapter, the symmetry property of an elastic metamaterial is examined based on the MST. It is shown that the elastic metamaterial preserves the quasi-static symmetry properties. For isotropic elastic metamaterials, CPA provides a simple and accurate EMT in the long wavelength limit, which links the scattering properties of the scatterer and the effective medium parameters (Wu et al, 2007). Those formulae can also be derived by using the MST method (Wu & Z. Zhang, 2009). For anisotropic elastic metamaterial, the EMT involves three or more effective elastic moduli (Landau & Lifshitz, 1986). In this case, CPA fails. To tackle this problem, a method based on MST in conjunction with the Christoffel's equation (Royer & Dieulesaint, 1999) has been proposed and the expressions for effective elastic moduli have been obtained (Wu & Z. Zhang, 2009). The combinations of anisotropy

. This is in contrast to the cases of previously mentioned EM and acoustic

. In two dimensions (2D), if the constituents of the

ρ

, bulk modulus,

κ, and

ρ

**1.2 Effective medium theories for acoustic metamaterials** 

, and mass density,

2009) and acoustic cloaking(S. Zhang et al., 2011).

**1.3 Effective medium theories for elastic metamaterials** 

determination of three effective parameters, i.e., mass density,

method (Krokhin et al., 2003), and integration of fields (Zhou & Hu, 2009).

are bulk modulus,

shear modulus,

μ

κ

parameters. Therefore, EMTs with physical insights can be regarded as theoretical foundations of metamaterials.

The development of EMT has seen a long history accompanied by various approaches. A famous one for electromagnetic (EM) waves is the Maxwell-Garnett theory (Sheng, 2006), which is valid in the quasi-static limit, aka the zero frequency limit. The quasi-static limit requires the wavelengths inside the scatterer, the host, and the effective medium to all be very large compared to the size of the building block (Lamb et al., 1980). However, for metamaterials, the wavelength inside the scatterer could be smaller than the size of the building block and thus lead to resonances at low frequencies. This results in the failure of the widely used quasi-static EMT. Nevertheless, as long as the wavelength in the effective medium is still large compared to the size of the building block, there exists an effective medium description as the wave still cannot probe the fine structures of the building blocks. In this context, another limit is introduced, which is the long wavelength limit. Compared with the quasi-static limit, the long wavelength limit does not have restrictions on the wavelength inside the scatterer, while the wavelengths inside the host and the effective medium should still be large (Lamb et al., 1980). In the study of metamaterials, one aspect is to develop EMTs that are valid in the long wavelength limit. In this chapter, we will focus on the recent developments of EMTs for elastic metamaterials.

#### **1.1 Effective medium theories for electromagnetic metamaterials**

Though our focus is on elastic metamaterials, it is necessary to briefly review the EMTs for EM metamaterials to offer a systematic picture of the EMTs. Ever since the birth of EM metamaterials, EMTs have played an integral role in designing metamaterials and explaining their unusual properties. The EMTs for EM metamaterials can be broken down into one of several classes. In one class, the effective parameters are obtained from the average of the computed eigenfields in the unit cell (Chern & Chen, 2009; Chui & L. Hu, 2002; Pendry et al., 1999; Smith & Pendry, 2006). This method gives inherently nonlocal parameters, i.e. parameters that depend on not only frequency but also the Bloch wave vector. For metamaterials with a good effective medium approximation, the nonlocality may be ignored. This method is especially helpful for use with metamaterials with complicated unit structures, such as split rings. Another class of EMTs is called the coherent potential approximation (CPA) method. In this method the effective medium is taken as the background embedded with the scatterer in the unit cell and by implying zero scattering, some elegent formulas of the effective parameters have been obtained (X. Hu et al., 2006; Jin et al., 2009; Wu et al., 2006). This method currently only works for scatterers with isotropic geometry, but it is very accurate in the long wavelength limit even at relatively high frequencies. Interestingly, the obtained effective parameters do not have any imaginary parts if the system does not have any absorption. Similar formulas can also be obtained from the multiple-scattering theory (MST) (Chui & Lin, 2008; Wu & Z. Zhang, 2009). The MST, which will be introduced in Section 2, is capable of producing the dispersion relations of a periodic structure. From dispersion relations, the effective wave speed can be easily calculated while the impedance still remains unknown. Recently, other methods have appeared, such as the quasimode method (Sun et al., 2009) and the first-principles method (Andrea, 2011).

#### **1.2 Effective medium theories for acoustic metamaterials**

144 Metamaterial

parameters. Therefore, EMTs with physical insights can be regarded as theoretical

The development of EMT has seen a long history accompanied by various approaches. A famous one for electromagnetic (EM) waves is the Maxwell-Garnett theory (Sheng, 2006), which is valid in the quasi-static limit, aka the zero frequency limit. The quasi-static limit requires the wavelengths inside the scatterer, the host, and the effective medium to all be very large compared to the size of the building block (Lamb et al., 1980). However, for metamaterials, the wavelength inside the scatterer could be smaller than the size of the building block and thus lead to resonances at low frequencies. This results in the failure of the widely used quasi-static EMT. Nevertheless, as long as the wavelength in the effective medium is still large compared to the size of the building block, there exists an effective medium description as the wave still cannot probe the fine structures of the building blocks. In this context, another limit is introduced, which is the long wavelength limit. Compared with the quasi-static limit, the long wavelength limit does not have restrictions on the wavelength inside the scatterer, while the wavelengths inside the host and the effective medium should still be large (Lamb et al., 1980). In the study of metamaterials, one aspect is to develop EMTs that are valid in the long wavelength limit. In this chapter, we will focus

Though our focus is on elastic metamaterials, it is necessary to briefly review the EMTs for EM metamaterials to offer a systematic picture of the EMTs. Ever since the birth of EM metamaterials, EMTs have played an integral role in designing metamaterials and explaining their unusual properties. The EMTs for EM metamaterials can be broken down into one of several classes. In one class, the effective parameters are obtained from the average of the computed eigenfields in the unit cell (Chern & Chen, 2009; Chui & L. Hu, 2002; Pendry et al., 1999; Smith & Pendry, 2006). This method gives inherently nonlocal parameters, i.e. parameters that depend on not only frequency but also the Bloch wave vector. For metamaterials with a good effective medium approximation, the nonlocality may be ignored. This method is especially helpful for use with metamaterials with complicated unit structures, such as split rings. Another class of EMTs is called the coherent potential approximation (CPA) method. In this method the effective medium is taken as the background embedded with the scatterer in the unit cell and by implying zero scattering, some elegent formulas of the effective parameters have been obtained (X. Hu et al., 2006; Jin et al., 2009; Wu et al., 2006). This method currently only works for scatterers with isotropic geometry, but it is very accurate in the long wavelength limit even at relatively high frequencies. Interestingly, the obtained effective parameters do not have any imaginary parts if the system does not have any absorption. Similar formulas can also be obtained from the multiple-scattering theory (MST) (Chui & Lin, 2008; Wu & Z. Zhang, 2009). The MST, which will be introduced in Section 2, is capable of producing the dispersion relations of a periodic structure. From dispersion relations, the effective wave speed can be easily calculated while the impedance still remains unknown. Recently, other methods have appeared, such as the quasimode method (Sun et al., 2009) and the first-principles method

on the recent developments of EMTs for elastic metamaterials.

**1.1 Effective medium theories for electromagnetic metamaterials** 

foundations of metamaterials.

(Andrea, 2011).

While the field of EM metamaterial has developed rapidly during the past decade, one of its counterparts, denoted as acoustic metamaterial has also seen fast growth (Ding et al., 2007; Fang et al., 2006; Lee et al., 2010; Yang et al., 2008). The acoustic metamaterial is designed to manipulate acoustic waves (Chen et al., 2010; Lu et al., 2009). Analogous to EM metamaterials, there are two material parameters that describe the wave propagation, which are bulk modulus, κ , and mass density, ρ . In two dimensions (2D), if the constituents of the metamaterials are all fluids, the governing equation can be mapped into 2D EM equations so that the EMT for 2D acoustic metamaterials is the same as that for 2D EM metamaterials. If the scatterers in acoustic metamaterials are solid, the shear modulus of the scatterers can be ignored when the longitudinal velocity contrast between the scatterer and the host is high (Kafesaki & Economou, 1999). In this case, the scattering property is basically the same as the EM cases. The mapping from the EM waves to acoustic waves facilitates the development of EMTs which have also been extensively studied by using various types of methods. MST (Mei et al., 2006; Torrent et al., 2006) and the CPA (Kafesaki et al., 2000; Li & Chan, 2004) represent two classes of them. Exciting news also came from the experimental realizations of acoustic metamaterials, such as acoustic negative refraction (S. Zhang et al., 2009) and acoustic cloaking(S. Zhang et al., 2011).

#### **1.3 Effective medium theories for elastic metamaterials**

The term *elastic* metamaterial refers to those metamaterials which are able to sustain not only longitudinal but also shear waves in their effective media. It is well-known that the EMT for an elastic composite in the quasi-static limit is anisotropic in general. The only exception is the hexagonal lattice in two dimensions (Landau & Lifshitz, 1986; Royer & Dieulesaint, 1999; Wu & Z. Zhang 2009). Even for this case, the EMT involves the determination of three effective parameters, i.e., mass density, ρ , bulk modulus, κ , and shear modulus, μ . This is in contrast to the cases of previously mentioned EM and acoustic EMTs which involve only two effective parameters. One more effective medium parameter greatly enriches the types and physics of wave propagation (Chen et al., 2008; Wu et al., 2007), such as mode conversion between longitudinal and transverse waves (Wu et al., 2011); however, it also adds complexity to the EMT. Recently, various EMTs for elastic metamaterials have been proposed, such as those ones based on the plane-wave-expansion method (Krokhin et al., 2003), and integration of fields (Zhou & Hu, 2009).

In this chapter, the symmetry property of an elastic metamaterial is examined based on the MST. It is shown that the elastic metamaterial preserves the quasi-static symmetry properties. For isotropic elastic metamaterials, CPA provides a simple and accurate EMT in the long wavelength limit, which links the scattering properties of the scatterer and the effective medium parameters (Wu et al, 2007). Those formulae can also be derived by using the MST method (Wu & Z. Zhang, 2009). For anisotropic elastic metamaterial, the EMT involves three or more effective elastic moduli (Landau & Lifshitz, 1986). In this case, CPA fails. To tackle this problem, a method based on MST in conjunction with the Christoffel's equation (Royer & Dieulesaint, 1999) has been proposed and the expressions for effective elastic moduli have been obtained (Wu & Z. Zhang, 2009). The combinations of anisotropy

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 147

where *J x <sup>m</sup>*( ) and (1) *H x <sup>m</sup>* ( ) are Bessel functions and Hankel functions of the first kind,

coordinates originating at the center of the scatterer. The longitudinal and transverse waves in the matrix are coupled by the scatterings of the scatterers, inside which the displacement

> ( ) ( ) ( ) <sup>ˆ</sup> ( ) *p p p p im im p p lm m ls p tm m ts p*

vectors inside the scatterer, respectively. The coefficients of those Bessel and Hankel functions can be determined by considering the elastic boundary conditions which are the continuities of the radial and tangential component of the displacement field, i.e., *ur* and

> σ *r*θ

β

′ ′

α

<sup>=</sup> ( , ) ,

*lma* and *<sup>p</sup>*

*atm* through:

are elastic Mie-like scattering coefficients for isotropic

= ∇ + ∇×

, (4)

*u r c J k r e c zJ k r e* θ

> σ*rr* and

> > αβ

scatterers and are functions of *ls k* , *ts k* , *<sup>l</sup>*<sup>0</sup> *k* , *<sup>t</sup>*<sup>0</sup> *k* and *sr* . The explicit expressions for *Dm*

For a collection of scatterers, the MST takes full account of the multiple scatterings between any two scatterers (Liu et al., 2000a; Mei et al., 2003). The wave incident on the scatterer *p* is contributed by two parts: one is the external incident waves from outside the system, and the other part is the scattered waves coming from all the other scatterers inside the system.

( )

( ) ˆ () ,

and *p*, respectively. For simplicity, the center of scatterer *p* is chosen as the origin and the position of scatterer *q* is denoted by R = (R , qp qp qp <sup>Θ</sup> ) . Thus, *q p qp rrR* = − . The relation

0 0

( )

*q q im im lm m l q tm m t q*

refer to the same spatial point measured from the positions of scatterers *<sup>q</sup>*

'' 0 0 '' ''

θ

+ ∇ + ∇×

(1) '' (1) ''

in the polar coordinates originating at the center of scatterer *q*.

 θ

*q q*

*b H k r e b zH k r e* (6)

 θ

*lmb* and *<sup>p</sup> btm* to *<sup>p</sup>*

, *p p m mm m ltm b ta*

= ′

β

α

αβ

Thus, the total incident waves on the scatterer *p* are expressed as:

''

is depicted in Fig. 1.

''

≠

*q pm*

*m*

denote *qr*

and *Rqp*

0 0

*u r a J k r e a zJ k r e*

( ) ( ) ˆ ( )

*p p*

*inc p p im im p p lm m lp tm m tp*

θ

= ∇ + ∇×

**2.2 Periodic structures and multiple-scattering** 

represent the longitudinal and transverse wave

θ

 θ

are the longitudinal and transverse wave

, at the interface. These continuities on

= *l t* (5)

αβcan

are the polar

respectively.

is given by:

*u*θ

0 00 0 ( ) *<sup>l</sup> k* = + ω ρ

κ

where ( ) *ls s s s k* = + ω ρ κ

where *t D mm m mm*

be found in (Wu et al., 2007).

αβ

where (, ) *q q r* θ

and *pr*

, *qr*

Here *qr*

between *pr*

μ

 and *<sup>t</sup>*0 00 *k* = ω ρ μ

*m*

μ

, and the continuities of the stresses,

αβ

 ′ ′ = δ. *Dm*

the surface of a cylinder relate *<sup>p</sup>*

vectors in the matrix, respectively. *ω* is the angular frequency. ( ) , *<sup>p</sup> p p r r* =

 and *ts s s k* = ω ρ μ

and negativities in various effective moduli can give rise to many types of novel wave propagation behaviors that are unseen in normal solids (Lai et al., 2011).

Within the scope of this chapter, all the EMTs mentioned above are limited to linear elastodynamics and do not consider the micro-structure introduced local rotation (Milton & Willis, 2007).

#### **2. Scattering properies of elastic metamaterials**

In order to illustrate EMTs and symmetry properties of elastic metamaterials, we start with a simple case where the resonant scatterer in a building block is homogeneous. More complicated scatterers will be discussed in Section 5. The elastic metamaterial considered here is composed of cylindrical inclusions of radius *sr* with mass density ρ*<sup>s</sup>* , shear modulus μ*<sup>s</sup>* , and bulk modulus κ *<sup>s</sup>* , embedded in an isotropic matrix, whose material parameters are denoted by ( ρ0 , μ0 , κ <sup>0</sup> ). In two dimensions, the bulk modulus, *κ*, is related to the shear modulus through the relation κλμ = + , where λ represents the Lamé constant (Royer & Dieulesaint, 1999). Due to the translational symmetry along the cylinder's axis, denoted as *z*-axis, the elastic modes in the system can be decoupled into a scalar part, which is also called shear horizontal mode with vibrations along the *z*-axis, and a vector part, i.e.,*xy*-mode with vibrations in the *x-y* plane. *xy*-mode is a mixed polarization of quasi-longitudinal and quasi-shear vertical modes. Since the shear horizontal mode satisfies the scalar wave equation with the same mathematical structure as those for acoustic (Krokhin et al., 2003) and 2D EM cases, this part is skipped. Rather, the focus is on the more complicated case of the *xy*-mode whose wave equation is given by:

$$\rho\left(\vec{r}\right)\frac{\partial^2 u\_i(\vec{r})}{\partial t^2} = \nabla \cdot \left(\mu(\vec{r})\nabla u\_i(\vec{r})\right) + \nabla \cdot \left(\mu(\vec{r})\frac{\partial \vec{u}(\vec{r})}{\partial x\_i}\right) + \frac{\partial}{\partial x\_i} \left[\mathcal{A}(\vec{r})\nabla \cdot \vec{u}(\vec{r})\right],\tag{1}$$

where *u* is the displacement field. In general, *<sup>u</sup>* can be decoupled into a longitudinal part and a transverse part, i.e. ( ) ˆ *u e* =∇ +∇× φ φ *l tz* , where φ*l* and φ*<sup>t</sup>* are the longitudinal and transverse gauge potentials, respectively.

#### **2.1 Single-scattering, the scattering coefficients**

If there is only one scatterer, the solutions to φ*l* and φ*<sup>t</sup>* can be expanded by using Bessel and Hankel functions. The wave incident on a single scatterer, *p* , is:

$$\bar{u}\_p^{inc}(\bar{r}\_p) = \sum\_m \left( a\_{lm}^p \nabla \left[ J\_m(k\_{l0}r\_p) e^{im\theta\_p} \right] + a\_{lm}^p \nabla \times \left[ \hat{z} J\_m(k\_{t0}r\_p) e^{im\theta\_p} \right] \right), \tag{2}$$

and the wave scattered by the same scatterer is

$$\bar{u}\_p^{\text{sca}}(\bar{r}\_p) = \sum\_m \left( b\_{lm}^p \nabla \left[ H\_m^{(1)}(k\_{l0}r\_p) e^{im\theta\_p} \right] + b\_{lm}^p \nabla \times \left[ \hat{z} H\_m^{(1)}(k\_{l0}r\_p) e^{im\theta\_p} \right] \right), \tag{3}$$

and negativities in various effective moduli can give rise to many types of novel wave

Within the scope of this chapter, all the EMTs mentioned above are limited to linear elastodynamics and do not consider the micro-structure introduced local rotation (Milton &

In order to illustrate EMTs and symmetry properties of elastic metamaterials, we start with a simple case where the resonant scatterer in a building block is homogeneous. More complicated scatterers will be discussed in Section 5. The elastic metamaterial considered

Lamé constant (Royer & Dieulesaint, 1999). Due to the translational symmetry along the cylinder's axis, denoted as *z*-axis, the elastic modes in the system can be decoupled into a scalar part, which is also called shear horizontal mode with vibrations along the *z*-axis, and a vector part, i.e.,*xy*-mode with vibrations in the *x-y* plane. *xy*-mode is a mixed polarization of quasi-longitudinal and quasi-shear vertical modes. Since the shear horizontal mode satisfies the scalar wave equation with the same mathematical structure as those for acoustic (Krokhin et al., 2003) and 2D EM cases, this part is skipped. Rather, the focus is on the more complicated case of the *xy*-mode whose wave equation is given

( ) ( ) ( ) () () ( ) ( ) () ()

*r r ur r r ur t x x* ρμμλ

> φ*l* and φ

( ) 0 0 ( ) ( ) <sup>ˆ</sup> () , *p p inc p p im im p p lm m lp tm m tp*

( ) (1) (1) 0 0 ( ) ( ) <sup>ˆ</sup> ( ) *p p sca p p im im p p lm m lp tm m tp*

= ∇ + ∇×

= ∇ + ∇×

*u r a J k r e a zJ k r e* θ

*u r b H k r e b zH k r e* θ

*i*

 φ *l tz* , where

φ

Hankel functions. The wave incident on a single scatterer, *p* , is:

*u r u r*

<sup>2</sup> , *<sup>i</sup>*

(1)

φ*l* and φ

(2)

, (3)

∂ ∂ <sup>∂</sup> =∇⋅ ∇ +∇⋅ <sup>+</sup> ∇ ⋅ <sup>∂</sup> ∂ ∂

*<sup>s</sup>* , embedded in an isotropic matrix, whose material

κλμ

*i i*

can be decoupled into a longitudinal part

*<sup>t</sup>* can be expanded by using Bessel and

 θ

> θ

*<sup>t</sup>* are the longitudinal and

<sup>0</sup> ). In two dimensions, the bulk modulus, *κ*, is

= + , where

ρ

represents the

λ

*<sup>s</sup>* , shear

here is composed of cylindrical inclusions of radius *sr* with mass density

κ

ρ0 , μ0 , κ

related to the shear modulus through the relation

is the displacement field. In general, *<sup>u</sup>*

and a transverse part, i.e. ( ) ˆ *u e* =∇ +∇×

If there is only one scatterer, the solutions to

**2.1 Single-scattering, the scattering coefficients** 

*m*

and the wave scattered by the same scatterer is

*m*

transverse gauge potentials, respectively.

propagation behaviors that are unseen in normal solids (Lai et al., 2011).

**2. Scattering properies of elastic metamaterials** 

*<sup>s</sup>* , and bulk modulus

Willis, 2007).

modulus

by:

where *u*

μ

parameters are denoted by (

2

where *J x <sup>m</sup>*( ) and (1) *H x <sup>m</sup>* ( ) are Bessel functions and Hankel functions of the first kind, respectively.

0 00 0 ( ) *<sup>l</sup> k* = + ω ρ κ μ and *<sup>t</sup>*0 00 *k* = ω ρ μ represent the longitudinal and transverse wave vectors in the matrix, respectively. *ω* is the angular frequency. ( ) , *<sup>p</sup> p p r r* = θ are the polar coordinates originating at the center of the scatterer. The longitudinal and transverse waves in the matrix are coupled by the scatterings of the scatterers, inside which the displacement is given by:

$$\overline{u}\_p(\overline{r}\_p) = \sum\_m \left( c\_{lm}^p \nabla \left[ f\_m(k\_{ls} r\_p) e^{im\theta\_p} \right] + c\_{lm}^p \nabla \times \left[ \hat{z} f\_m(k\_{ls} r\_p) e^{im\theta\_p} \right] \right), \tag{4}$$

where ( ) *ls s s s k* = + ω ρ κ μ and *ts s s k* = ω ρ μ are the longitudinal and transverse wave vectors inside the scatterer, respectively. The coefficients of those Bessel and Hankel functions can be determined by considering the elastic boundary conditions which are the continuities of the radial and tangential component of the displacement field, i.e., *ur* and *u*θ , and the continuities of the stresses, σ *rr* and σ *r*θ , at the interface. These continuities on the surface of a cylinder relate *<sup>p</sup> lmb* and *<sup>p</sup> btm* to *<sup>p</sup> lma* and *<sup>p</sup> atm* through:

$$b\_{\alpha m}^{p} = \sum\_{\beta=1,\,t} \sum\_{m'} t\_{\alpha \beta mm'} a\_{\beta m'}^{p} \quad (\alpha = 1,\,t) \; , \tag{5}$$

where *t D mm m mm* αβ αβ ′ ′ = δ . *Dm* αβ are elastic Mie-like scattering coefficients for isotropic scatterers and are functions of *ls k* , *ts k* , *<sup>l</sup>*<sup>0</sup> *k* , *<sup>t</sup>*<sup>0</sup> *k* and *sr* . The explicit expressions for *Dm* αβ can be found in (Wu et al., 2007).

#### **2.2 Periodic structures and multiple-scattering**

For a collection of scatterers, the MST takes full account of the multiple scatterings between any two scatterers (Liu et al., 2000a; Mei et al., 2003). The wave incident on the scatterer *p* is contributed by two parts: one is the external incident waves from outside the system, and the other part is the scattered waves coming from all the other scatterers inside the system. Thus, the total incident waves on the scatterer *p* are expressed as:

$$\begin{split} \boldsymbol{u}\_{p}^{\text{inc}}(\overline{\boldsymbol{r}}\_{p}) &= \sum\_{m} \Big( \boldsymbol{a}\_{lm}^{p0} \nabla \Big[ \boldsymbol{J}\_{m}(\boldsymbol{k}\_{l0} \boldsymbol{r}\_{p}) e^{i m \boldsymbol{\theta}\_{p}} \Big] + \boldsymbol{a}\_{lm}^{p0} \nabla \times \Big[ \hat{\boldsymbol{z}} \boldsymbol{J}\_{m}(\boldsymbol{k}\_{l0} \boldsymbol{r}\_{p}) e^{i m \boldsymbol{\theta}\_{p}} \Big] \Big) \\ &+ \sum\_{q \neq p} \sum\_{m^{\circ}} \Big( \boldsymbol{b}\_{lm^{\circ}}^{q} \nabla \Big[ \boldsymbol{H}\_{m}^{(1)}(\boldsymbol{k}\_{l0} \boldsymbol{r}\_{q}) e^{i m^{\circ} \boldsymbol{\theta}\_{q}} \Big] + \boldsymbol{b}\_{lm}^{q} \nabla \times \Big[ \hat{\boldsymbol{z}} \boldsymbol{H}\_{m}^{(1)}(\boldsymbol{k}\_{l0} \boldsymbol{r}\_{q}) e^{i m^{\circ} \boldsymbol{\theta}\_{q}} \Big] \Big), \end{split} \tag{6}$$

where (, ) *q q r* θ denote *qr* in the polar coordinates originating at the center of scatterer *q*. Here *qr* and *pr* refer to the same spatial point measured from the positions of scatterers *<sup>q</sup>* and *p*, respectively. For simplicity, the center of scatterer *p* is chosen as the origin and the position of scatterer *q* is denoted by R = (R , qp qp qp <sup>Θ</sup> ) . Thus, *q p qp rrR* = − . The relation between *pr* , *qr* and *Rqp* is depicted in Fig. 1.

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 149

β

'

det ( ) ' *mm* , ' *mm* 0 ,

*t S mm*

( ) ', ' *<sup>q</sup> iK R*

represents lattice sums. The solution of Eq. (12) offers the dependence of frequency,

0 1 (1)

Ω

<sup>4</sup> ( ) <sup>2</sup> (,) - () ( ) ( )

*i k J Qa <sup>i</sup> S n e H ka*

+

β

*n h h h*

*S mm g e*

β

 − = 

β

−− = (12)

, which is known as the dispersion relation. To solve Eq. (12) for

1 0 ,0 2 2

β

, ( ) *<sup>n</sup>* <sup>≥</sup> 0 (13)

is the volume of the unit cell and h h (Q , )

ω*K*

and () ( ) ( ) (1) n n H x 2 n 1! x <sup>n</sup> ≅− −

2 2

β

 β

is the reciprocal lattice vector.

αβ

π

 π δ

*n h in <sup>n</sup>*

. In the framework of MST, the dispersion relation can be calculated

− and the scattering matrix *mm*' *t*

ω, on

ϕ

,does not depend on the

. This simplifies the

for *l s* <sup>0</sup> *x kr* = ,

stands

*t ge b* (11)

δ<sup>⋅</sup>

0 *<sup>q</sup> iK R <sup>p</sup>*

'

 δ

*m m*

β

*q p*

 <sup>⋅</sup> ≠ − =

dispersion relations, one needs to evaluate the lattice sums first, which can be accomplished through several techniques (Chin et al., 1994; Mei et al., 2003). The expression of

1 0 0 0

Since the MST is capable of producing accurate dispersion relations for all frequencies, it is a good approach in the study of the symmetry properties of an elastic metamaterial. The

numerically by solving Eq. (12). In the long wavelength limit, both *Ka* and 0*k a* are all much smaller than unity and appropriate approximations can be made in the Bessel and Hankel

π

*t s* <sup>0</sup> *k r* , *l*<sup>0</sup> *k a* , *t*<sup>0</sup> *k a* and *Ka* in Eq. (12), it is easy to find that the leading terms are those with *m* ≤ 2 , where *m* corresponds to the order of Bessel and Hankel functions and is called the angular quantum number. Thus, we only need to consider the terms with 0 4 ≤ ≤ *n* in

> 1 0 () ( ) *<sup>h</sup> in nh h h*

*J Qae Q k Q* ϕ

<sup>−</sup>

*J ka Qk Q k a*

= + <sup>−</sup>

*h*

ϕ

*mm m m mm m*

αβ

> αβ′

β

= ≠ ′

*m*

′

*lt m m q p*

′

β

, '− is given by (Wu and Z. Zhang, 2009):

β

β

for the vector Q KK h h = + in polar coordinates, and Kh

**3. Symmetry properties of elastic metamaterials** 

elastic metamaterial is isotropic if its dispersion relation, ( )

β

*h*

secular equation and benefits the derivation of analytic expression of EMT.

<sup>0</sup> *Hx x* ≅ 2ln

+

1

+

\*

+

 β

*n*

(, ) (,)

where a denotes the lattice constant,

Ω

*S nSn*

− =−

functions in the lattice sum *S mm* (, ' )

Taking () 2 ! ( ) *n n nJx x n* <sup>≅</sup> , ( ) (1)

Eq. (13), in which the summation

Eq. (11) has nontrivial solutions if and only if,

where

*S mm* ( ) β

the Block wave vector, *K*

β

β

direction of *K*

,

Fig. 1. The spacial relation of *pr* , *qr* and *Rqp* .

With the help of Graf's addition theorem (Abramowitz & Stegun, 1972), the Hankel functions depicting the scattered wave coming from the scatterer *q* can be changed into Bessel functions describing the wave incident on the scatterer *p*, which is:

$$H\_m^{\{1\}}(k\_{\alpha 0}r\_q)e^{im^{\alpha}\theta\_q} = \sum\_m \mathbf{g}\_{mm^m}^{\alpha} \mathbf{l}\_m(k\_{\alpha 0}r\_p)e^{im\theta\_p} \quad (\alpha = \mathbf{l}, \mathbf{t})\,\tag{7}$$

where

$$\mathbf{g}\_{mm^{\prime\prime}}^{\alpha} = H\_{m-m^{\prime\prime}}^{(1)}(k\_{\alpha 0} R\_{q\overline{p}})e^{i(m^{\prime\prime} - m)\Theta\_{q\overline{p}}} \quad \text{( $\alpha = l\_{\prime}$   $t$ )}\tag{8}$$

Substituting Eqs. (7) and (8) into (6), we obtain:

$$\begin{split} \bar{u}\_p^{inc}(\bar{r}\_p) &= \sum\_m \left( a\_{lm}^{p0} + \sum\_{q \neq p} \sum\_m b\_{lm}^q g\_{mm^\*}^{l} \right) \nabla \left[ J\_m(k\_{l0} r\_p) e^{im\theta\_p} \right] \\ &+ \sum\_m \left( a\_{lm}^{p0} + \sum\_{q \neq p} \sum\_m b\_{lm}^q g\_{mm^\*}^{l} \right) \nabla \times \left[ \hat{z} f\_m(k\_{t0} r\_p) e^{im\theta\_p} \right] \end{split} \tag{9}$$

Eq. (2) together with Eqs. (5) and (9) leads to the following self-consistent equation:

$$b\_{\mathcal{O}mn}^{p} = \sum\_{\mathcal{J}=\mathbf{l},\mathbf{t}} \sum\_{m'} t\_{\mathcal{alpha}\mathcal{B}mm'} \left( a\_{\mathcal{J}m'}^{p\mathcal{O}} + \sum\_{q \neq p} \sum\_{m''} g\_{m'm''}^{\mathcal{J}} b\_{\mathcal{J}m''}^{q} \right) \tag{10}$$

This set of self-consistent equations can be written into the standard form of linear equations *Ax B* = , and the multiple-scattering problem is numerically solved.

If the scatterers are arranged in a periodic array, the Bloch theorem, qp q p iK R b be α α m '' m '' <sup>⋅</sup> = ( ) α = l,t , can be applied, where *K* is the Bloch wave vector. For an eigenvalue problem, the external incident field is dropped off so that Eq. (10) is converted into:

$$\sum\_{l,\bar{\mathcal{B}}=\mathrm{l},\mathrm{t}} \sum\_{m} \left( \sum\_{m'} t\_{\alpha \bar{\mathcal{B}}^{mm'}} \sum\_{q \neq p} \mathcal{g}\_{m'm}^{\mathcal{B}} e^{i\bar{\mathcal{K}} \cdot \bar{\mathcal{R}}\_q} - \mathcal{S}\_{mm'} \right) b\_{\beta m}^p = 0 \tag{11}$$

 Eq. (11) has nontrivial solutions if and only if,

$$\det \left| \sum\_{m'} t\_{\alpha \beta \mu m'} S \left( \beta \,\, m' \text{-} m \right) - \delta\_{mm'} \right| = 0 \,, \tag{12}$$

where

148 Metamaterial

Fig. 1. The spacial relation of *pr*

where

α

into:

 α m '' m '' <sup>⋅</sup> =  , *qr*

*p* 

(1) '' ''

> ''''

α

0

α

( ) α

β

*Ax B* = , and the multiple-scattering problem is numerically solved.

= l,t , can be applied, where *K*

Substituting Eqs. (7) and (8) into (6), we obtain:

α

and *Rqp*

Θ*qp*

*rp* 

θ*p*

Bessel functions describing the wave incident on the scatterer *p*, which is:

*mm m m qp g H kR e*

≠

*m q pm*

*m q pm*

0

≠

αβ .

 <sup>0</sup> '' 0 ( ) ( ) *q p im im m q mm m p m H k re g J k re* θ

α

(1) ( '' )

α

''

*u r a b g J kr e*

( ) ( )

*inc p q l im p p lm lm mm m l p*

''

Eq. (2) together with Eqs. (5) and (9) leads to the following self-consistent equation:

<sup>0</sup> ( ) *qp im m*

=+ ∇

''

0 '

= +

β

This set of self-consistent equations can be written into the standard form of linear equations

If the scatterers are arranged in a periodic array, the Bloch theorem, qp q p iK R b be

eigenvalue problem, the external incident field is dropped off so that Eq. (10) is converted

, '' *p p q m mm m mm m ltm q pm b t a gb*

′ = ≠ ′

ˆ ()

''

+ + ∇ ×

*a b g zJ k r e*

*p q t im tm tm mm mtp*

 α<sup>=</sup> ( , ) ,

With the help of Graf's addition theorem (Abramowitz & Stegun, 1972), the Hankel functions depicting the scattered wave coming from the scatterer *q* can be changed into

*Rqp* 

 θ

*q* 

θ*q*

*rq* 

α

Θ

− <sup>−</sup> = ( ,)

'' 0

'' 0

' '' ''

β

> β

(10)

α

θ

*p*

θ

*p*

is the Bloch wave vector. For an

= *l t* (7)

= *l t* (8)

(9)

$$S\left(\mathcal{J}, m^\circ - m\right) = \sum\_{q \neq p} g\_{m^\circ m}^{\mathcal{J}} e^{i\vec{K} \cdot \vec{R}\_q}$$

 represents lattice sums. The solution of Eq. (12) offers the dependence of frequency,ω , on the Block wave vector, *K* , which is known as the dispersion relation. To solve Eq. (12) for dispersion relations, one needs to evaluate the lattice sums first, which can be accomplished through several techniques (Chin et al., 1994; Mei et al., 2003). The expression of *S mm* ( ) β, '− is given by (Wu and Z. Zhang, 2009):

$$\begin{aligned} S(\beta, n) &= \frac{4i^{n+1}k\_{\beta 0}}{\Omega l\_{n+1}(k\_{\beta 0}a)} \sum\_{h} \frac{J\_{n+1}(Q\_{h}a)}{Q\_{h}(k\_{\beta 0}^{2} - Q\_{h}^{2})} e^{in\phi\_{h}} \cdot \left( H\_{1}^{(1)}(k\_{\beta 0}a) + \frac{2i}{\pi k\_{\beta 0}a} \right) \delta\_{n,0} \\ S(\beta, -n) &= -S^{\circ}(\beta, n) \end{aligned} \tag{13}$$

where a denotes the lattice constant, Ω is the volume of the unit cell and h h (Q , ) ϕ stands for the vector Q KK h h = + in polar coordinates, and Kh is the reciprocal lattice vector.

#### **3. Symmetry properties of elastic metamaterials**

Since the MST is capable of producing accurate dispersion relations for all frequencies, it is a good approach in the study of the symmetry properties of an elastic metamaterial. The elastic metamaterial is isotropic if its dispersion relation, ( ) ω *K* ,does not depend on the direction of *K* . In the framework of MST, the dispersion relation can be calculated numerically by solving Eq. (12). In the long wavelength limit, both *Ka* and 0*k a* are all much smaller than unity and appropriate approximations can be made in the Bessel and Hankel functions in the lattice sum *S mm* (, ' ) β − and the scattering matrix *mm*' *t*αβ . This simplifies the secular equation and benefits the derivation of analytic expression of EMT.

Taking () 2 ! ( ) *n n nJx x n* <sup>≅</sup> , ( ) (1) <sup>0</sup> *Hx x* ≅ 2ln π and () ( ) ( ) (1) n n H x 2 n 1! x <sup>n</sup> ≅− − π for *l s* <sup>0</sup> *x kr* = , *t s* <sup>0</sup> *k r* , *l*<sup>0</sup> *k a* , *t*<sup>0</sup> *k a* and *Ka* in Eq. (12), it is easy to find that the leading terms are those with *m* ≤ 2 , where *m* corresponds to the order of Bessel and Hankel functions and is called the angular quantum number. Thus, we only need to consider the terms with 0 4 ≤ ≤ *n* in Eq. (13), in which the summation

$$\sum\_{h} I\_{n+1}(Q\_h a) e^{i n \phi\_h} \left\langle \left[ Q\_h (k\_{\beta 0}^2 - Q\_h^2) \right] \right\rangle$$

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 151

<sup>3</sup> 4D 2 ia

+ ++ = − + −

κ

<sup>2</sup> F D , <sup>3</sup> 4D i a

( )

( ) ( )( )

(,) ,

wavelength approximation. It is obviously seen that the roots ( )<sup>2</sup> tri K1 and ( )<sup>2</sup> tri K2 given in

relation for an elastic metamaterial with a hexagonal structure are isotropic near the Γ point

For the case of a square lattice, the lattice sum is almost the same as that of the hexagonal lattice case except for the 0 *n* ≠ case. The reciprocal lattice vector of a square lattice is

( ) <sup>2</sup> ˆ ˆ ; , *K hi h j h h h i j ij <sup>a</sup>*

=+ ∈

cancel to zero when 1 3 ≤ ≤ *n* , and equals to 4 when 4 *n* = respectively. This indicates that the second term of Eq. (14) only vanishes when 1 3 ≤ ≤ *n* . Thus, in the long wavelength

( )

( )

<sup>4</sup> ( ,4) *K K i i iK <sup>S</sup> e e*

 ≅ + <sup>−</sup>

4

24 2 2 0 0

*ak k K*

β β

*n n in*

1 2 22 0 0 <sup>4</sup> (,) *<sup>K</sup>*

<sup>+</sup> <sup>−</sup> <sup>≅</sup>

*n i K S n <sup>e</sup> ak k K*

β β

π

For an arbitrarily chosen reciprocal lattice vector 1 1 (, ) *Kh Kh*

β

β

 , i.e., φ

3 3 4 42

*D ia D i a*

() () ( )

*ll ll ll*

ll 2 2 20 0 0 0 0 0 ll

> ll 2 2 0 20 0 0 0 0

00 0 0 20 0 0 0 0 0

 κ μ κ

2 2

0 0 0 20 0 0 0 20 0 0 0 0

 ωρ

*D D ia D i a*

+ + − + − 

3 3 <sup>32</sup> <sup>4</sup>

is the Mie-like scattering coefficient ll D after taking the long m

+ + + ++ = −

μ

2

 ωμρ

2 2 2 2

Ζ(19)

φ

, *N* = 1,2,3 , which makes the summation <sup>3</sup> inN 2

φ

4 4

 φ

φ

β

 γ−

2 2 2 2 2

 μ

2 2

 κ κ μ

ω μ ρ

(18)

 ωμρ

*<sup>K</sup>* . This implies that all the dispersion

, there always exist the other

N 0 <sup>e</sup><sup>−</sup>

<sup>−</sup> ( ) 0 3 ≤ ≤ *<sup>n</sup>* (20)

<sup>=</sup>

π

(21)

 ωμρ

( ) ( )( )

μ

 κ κ μ

 ωρ

κ

*ll ll*

μ

 μ κ μ

2 2

32 0

where *ll Dm*

expressed by:

three at

for 0 3 ≤ ≤ *n* , and

for 4 *n* = , where

*FD D*

*ll ll*

κ μ

in the long wavelength limit.

**3.2 Anisotropic dispersions** 

1 1 ( , /2) *K N h Kh* φ+

limit, the lattice sum can be written, as

π

κ μ

Eq. (17) do not depend on the direction of *K*

can be separated into two terms: the first is 0 *Kh* <sup>=</sup> or *Q K <sup>h</sup>* <sup>≅</sup> , the second is the sum of all other terms with 0 *Kh* <sup>≠</sup> or *Q K h h* <sup>≅</sup> . Then, the lattice sum takes the following expression:

$$\begin{split} S(\mathcal{B},n) &\equiv \frac{4i^{n+1}}{\Omega} \left| \frac{K^n}{k\_{\mathcal{P}0}^n (k\_{\mathcal{P}0}^2 - K^2)} e^{-in\phi\_k} + \frac{k\_{\mathcal{P}0} 2^{n+1} (n+1)!}{\left(k\_{\mathcal{P}0} a\right)^{n+1}} \sum\_{h \in \mathcal{K}\_h \neq \mathcal{0}} \frac{J\_{n+1} \{K\_h a\}}{K\_h \{k\_{\mathcal{P}0}^2 - K\_h^2\}} e^{-in\phi\_h} \right|, \newline (0 \le n \le 4) \\ S(\mathcal{B},-n) &= -\mathcal{S}^\* \left(\mathcal{B},n\right), \end{split} \tag{14}$$

where ( ) , *Kh Kh* φ denotes the polar coordinates of *Kh* , the reciprocal lattice vector of the lattice. The summation of all nonzero reciprocal lattice vectors in the second term in the bracket reveals the dependence of lattice sum on the lattice structure, which influences the symmetry properties of the dispersion relations.

#### **3.1 Isotropic dispersions**

For a 2D hexagonal lattice with a lattice constant *a* , the reciprocal lattice follows:

$$\vec{K}\_{h} = \frac{4\pi}{\sqrt{3}a} \left( h\_i \hat{i} + h\_j \left( \frac{1}{2} \hat{i} + \frac{\sqrt{3}}{2} \hat{j} \right) \right); \ h\_i, h\_j \in Z \tag{15}$$

Here, ˆ *i* and ˆ *j* represent the unit vectors along the *x*- and *y*-axes in the reciprocal space. When 0 *n* ≠ , due to the symmetry of a hexagonal lattice, the summation in the second term of Eq. (14) is zero, which can be proved in the following way. For an arbitrarily chosen reciprocal lattice vector, 1 1 (, ) *Kh Kh* φ , there always exist five other reciprocal lattice vectors at h1 Kh1 (K , N /3) φ + π , N 1,2,3,4,5 = such that <sup>5</sup> inN 3 N 0 e 0 <sup>−</sup> π <sup>=</sup> <sup>=</sup> . Thus, the summation in Eq. (14) vanishes after summing over all the non-zero *Kh* and only the first term of Eq. (14) survives. When 0 *n* = , the second term in Eq. (14) no longer sums to zero as 1 *Kh in e* − φ = . However, this term can be ignored in the long wavelength limit because compared to the first term in Eq. (14) which is on the order of <sup>2</sup> ω<sup>−</sup> , it is on the order of <sup>0</sup> ω . Thus, Eq. (14) is further reduced to:

$$S(\beta, n) \equiv \frac{8i^{n+1}K^n}{\sqrt{3}a^2k\_{\beta 0}^n \left(k\_{\beta 0}^2 - K^2\right)} e^{-in\phi\_k} \text{, } 0 \le n \le 4 \tag{16}$$

Substituting Eq. (16) into Eq.(12), we find the following two roots:

$$\left(\mathbf{K}\_{1}^{\text{tr}}\right)^{2} = \mathbf{F}\_{1}\left(\tilde{\mathbf{D}}\_{1}^{\text{ll}}\right)\mathbf{F}\_{2}\left(\tilde{\mathbf{D}}\_{2}^{\text{ll}}\right)\text{ and }\left(\mathbf{K}\_{2}^{\text{tr}}\right)^{2} = \mathbf{F}\_{1}\left(\tilde{\mathbf{D}}\_{1}^{\text{ll}}\right)\mathbf{F}\_{3}\left(\tilde{\mathbf{D}}\_{2}^{\text{ll}},\tilde{\mathbf{D}}\_{0}^{\text{ll}}\right),\tag{17}$$

with

$${}\_{1}F\_{1}\left(\tilde{D}\_{1}^{ll}\right) = -\frac{16i\tilde{D}\_{1}^{ll}\left(\kappa\_{0} + \mu\_{0}\right) - \sqrt{3}a^{2}a\rho^{2}\rho\_{0}}{\sqrt{3}a^{2}}\ \ .$$

$$\mathrm{F}\_{2}\left(\tilde{\mathrm{D}}\_{2}^{\mathrm{ul}}\right) = -\frac{4\tilde{\mathrm{D}}\_{2}^{\mathrm{ul}}\left(\kappa\_{0} + \mu\_{0}\right)\left(\kappa\_{0} + 2\mu\_{0}\right) + \mathrm{i}\frac{\sqrt{3}}{2}\mathrm{a}^{2}\phi^{2}\mu\_{0}\rho\_{0}}{\mu\_{0}\left(4\tilde{\mathrm{D}}\_{2}^{\mathrm{ul}}\kappa\_{0}\left(\kappa\_{0} + \mu\_{0}\right) - \mathrm{i}\frac{\sqrt{3}}{2}\mathrm{a}^{2}\phi^{2}\mu\_{0}\rho\_{0}\right)},\tag{18}$$

$$F\_3(\tilde{D}\_2^{\text{II}}, \tilde{D}\_0^{\text{II}}) = -\frac{\left(4\tilde{D}\_0^{\text{II}}(\kappa\_0 + \mu\_0) + i\frac{\sqrt{3}}{2}a^2a^2\rho\_0\right)\left(4\tilde{D}\_2^{\text{II}}(\kappa\_0 + \mu\_0)(\kappa\_0 + 2\mu\_0) + i\frac{\sqrt{3}}{2}a^2a^2\mu\_0\rho\_0\right)}{\left(\kappa\_0 + \mu\_0\right)\left(32\tilde{D}\_0^{\text{II}}\tilde{D}\_2^{\text{II}}\mu\_0\left(\kappa\_0 + \mu\_0\right)^2 - i\frac{\sqrt{3}}{2}a^2a^2\rho\_0\left(4\tilde{D}\_2^{\text{II}}\kappa\_0\left(\kappa\_0 + \mu\_0\right) - i\frac{\sqrt{3}}{2}a^2a^2\mu\_0\rho\_0\right)\right)}\tilde{D}\_1\left(\kappa\_0 + \mu\_0\right)\left(\frac{\sqrt{3}}{2}a^2a^2\mu\_0\rho\_0\right)^{1/2}$$

where *ll Dm* is the Mie-like scattering coefficient ll D after taking the long m wavelength approximation. It is obviously seen that the roots ( )<sup>2</sup> tri K1 and ( )<sup>2</sup> tri K2 given in Eq. (17) do not depend on the direction of *K* , i.e., φ*<sup>K</sup>* . This implies that all the dispersion relation for an elastic metamaterial with a hexagonal structure are isotropic near the Γ point in the long wavelength limit.

#### **3.2 Anisotropic dispersions**

150 Metamaterial

( )

<sup>+</sup> <sup>≅</sup> <sup>+</sup> ≤ ≤ − −

2 2 1 2 2 0 0 ( 0) 0 0

lattice. The summation of all nonzero reciprocal lattice vectors in the second term in the bracket reveals the dependence of lattice sum on the lattice structure, which influences the

> 4 13 ˆ ˆˆ ; , <sup>3</sup> 2 2 *K hi h i j h h h ij i j <sup>a</sup>*

When 0 *n* ≠ , due to the symmetry of a hexagonal lattice, the summation in the second term of Eq. (14) is zero, which can be proved in the following way. For an arbitrarily chosen

(14) survives. When 0 *n* = , the second term in Eq. (14) no longer sums to zero as 1 *Kh in e*

However, this term can be ignored in the long wavelength limit because compared to the

ω

( )

<sup>8</sup> (,) , 0 4

<sup>+</sup> <sup>−</sup> <sup>≅</sup> ≤ ≤ −

*n n in*

β β

*n*

3

Substituting Eq. (16) into Eq.(12), we find the following two roots:

1 2 22 0 0

*i K S n e n*

( ) ( ) 2 2

+ − = −

κ μ

*ll ll iD a*

1 1 2

*j* represent the unit vectors along the *x*- and *y*-axes in the reciprocal space.

N 0 e 0 <sup>−</sup> π

φ

( ) ( )( ) ( ) ( )( ) 2 2 tri ll ll tri ll ll ll K F D F D and K F D F D ,D , 1 1 12 2 = = 2 1 13 2 0 (17)

 ωρ

10 0 0

<sup>16</sup> <sup>3</sup> , <sup>3</sup>

*a*

*K*

 = ++ ∈ 

<sup>4</sup> 2 1! ( ) (,) , (0 4) ( ) ( )

0 1

− + − <sup>+</sup> <sup>≠</sup>

*in n h in*

*K Kh h*

β

Ζ

<sup>=</sup> <sup>=</sup> . Thus, the summation in Eq.

and only the first term of Eq.

ω

, there always exist five other reciprocal lattice vectors at

<sup>−</sup> , it is on the order of <sup>0</sup>

*ak k K* (16)

other terms with 0 *Kh* <sup>≠</sup> or *Q K h h* <sup>≅</sup> . Then, the lattice sum takes the following expression:

( )

β

β

For a 2D hexagonal lattice with a lattice constant *a* , the reciprocal lattice follows:

*n n h K h h i K k n J Ka S n <sup>e</sup> e n kk K k a Kk K*

, the second is the sum of all

(15)

− φ= .

. Thus, Eq. (14) is

(14)

φ

, the reciprocal lattice vector of the

can be separated into two terms: the first is 0 *Kh* <sup>=</sup> or *Q K <sup>h</sup>* <sup>≅</sup>

1 1

+ +

φ

denotes the polar coordinates of *Kh*

π

φ

(14) vanishes after summing over all the non-zero *Kh*

first term in Eq. (14) which is on the order of <sup>2</sup>

β

*F D*

, N 1,2,3,4,5 = such that <sup>5</sup> inN 3

*n n n*

symmetry properties of the dispersion relations.

\*

 β β β

( , ) ( , ) ,

**3.1 Isotropic dispersions** 

*i* and ˆ

h1 Kh1 (K , N /3) φ+

further reduced to:

reciprocal lattice vector, 1 1 (, ) *Kh Kh*

π

Ω

*S nSn*

− =−

where ( ) , *Kh Kh* φ

β

β

Here, ˆ

with

For the case of a square lattice, the lattice sum is almost the same as that of the hexagonal lattice case except for the 0 *n* ≠ case. The reciprocal lattice vector of a square lattice is expressed by:

$$\vec{K}\_{li} = \frac{2\pi}{a} \left( h\_i \hat{\mathbf{i}} + h\_j \hat{\mathbf{j}} \right); \ h\_i, h\_j \in Z \tag{19}$$

For an arbitrarily chosen reciprocal lattice vector 1 1 (, ) *Kh Kh* φ , there always exist the other three at

$$\text{(}\text{(}\text{K}\_{\text{h1}},\text{\(}\text{\(}\text{N}\text{\(}\text{?}\text{ }\text{}\text{}\text{)}\text{, N}=1,\text{2,3, which makes the summation }\sum\_{\text{N}=0}^{5}\text{e}^{-\text{inN}\text{st}\text{f2}}$$

cancel to zero when 1 3 ≤ ≤ *n* , and equals to 4 when 4 *n* = respectively. This indicates that the second term of Eq. (14) only vanishes when 1 3 ≤ ≤ *n* . Thus, in the long wavelength limit, the lattice sum can be written, as

$$S(\beta, n) \equiv \frac{4i^{n+1}K^n}{a^2 k\_{\beta 0}^n \left(k\_{\beta 0}^2 - K^2\right)} e^{-in\phi\_k} \tag{20}$$

for 0 3 ≤ ≤ *n* , and

$$S(\beta, 4) \equiv \left(\frac{4iK^4}{a^2 k\_{\beta 0}^4 \left(k\_{\beta 0}^2 - K^2\right)} + \mathcal{Y}\_{\beta} e^{i4\phi\_k}\right) e^{-i4\phi\_k} \tag{21}$$

for 4 *n* = , where

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 153

αβ = ( , ,) α β

In the long wavelength limit, where 0 0 1 *<sup>l</sup> k r* << , 0 0 1 *<sup>t</sup> k r* << , 0 1 *le k r* << and 0 1 *te k r* << , Eq. (23)can be simplified into the following effective medium equations for the elastic

0 0

0 1

( ) <sup>8</sup> *ll <sup>e</sup>*

0 0 0

0 0 2

00 0 0 0 0 ( ) 4

The dispersion relation can be reproduced from effective medium parameters. Comparing

the dispersion relation is isotropic for a hexagonal lattice and its effective medium

Eqs. (24)-(26) require the wavelengths in both the host and the effective medium to be much larger than the size of the unit cell, but they do not impose any restriction on the wavelengths inside the scatterer. If the condition of 1 *ls s k r* << , 1 *ts s k r* << is further

> 0 0 0 0 ( )( ) , ( )( ) *e s e s p*

 κκ

 μκ

 ρρ

> κμ

μ μ

*e s*

 κ

> π

0 0 ( ) ( ) , *e s*

− − <sup>=</sup> + + + + 0 0 00 0 0 00 0 0 () () ( ( 2 )) ( ( 2 )) *e s*

The 3D version was reported by Berryman decades ago (Berryman, 1980). It should be pointed out that elastic EMT cannot recover the acoustic EMT by setting all the shear moduli to be zero, because of the different boundary conditions of elastic and acoustic waves.

Figure 3 shows the equifrequency surface (EFS) of a hexagonal array of silicone rubber cylinders with radii of 0.2 *a* embedded in an epoxy host. An EFS is a collection of all states

space that have the same frequency. The metamaterial is isotropic if its EFS is a

ω

 μ μ

<sup>−</sup> <sup>=</sup> + +

π

*<sup>e</sup>* are independently determined by *ll Dm*

( )4 ( )

<sup>−</sup> <sup>=</sup> <sup>+</sup>

0 0 0

*ll <sup>e</sup> e l*

π

2 2

2 2

*D irk*

*ll <sup>e</sup>*

*l*

*e l*

π

2 2

ρ μ

. The equivalence provides a strong evidence that

− − <sup>=</sup> + + (27)

−= − *p* (28)

*p* (29)

*a c* , is used, where *t*<sup>0</sup> *<sup>c</sup>* is the

 μ μ

*D irk*

*D irk*

= *l t* (23)

, (24)

(26)

*e e* coincides with ( )<sup>2</sup>

of the embedded

1 *tri K* ω

<sup>−</sup> = − , (25)

*DeD m m* ( ) αβ

> κ κ

μ κ

ρ ρ

μμ

 κ

ω

considered, the quasi-static limit is reached and Eqs. (24)-(26) becomes:

κκ

μκ

ρρ

 μ μ

κμ

μ

cylinders alone with angular quantum numbers *m* = 0, 1 and 2.

Eqs. (24)-(26) to Eqs. (17) and (18), it is easy to show that

is identical to ( )<sup>2</sup> tri K2

properties can be evaluated from the EMT derived from CPA.

μ μ

circle. Here the dimensionless frequency, 0 ( ) (2 ) *<sup>t</sup> f* =

 κ

κμ

ρ

( ( 2))

 μ

metamaterial:

Obviously,

and ( ) ρ *ee e* κ + μ

in the *K*  κ *<sup>e</sup>* , ρ*<sup>e</sup>* and

$$\gamma\_{\beta} = \frac{16 \cdot 2^5 \cdot 5! i}{a^6 k\_{\beta 0}^4} \left| \sum\_{h\_j = 0}^{N} \sum\_{h\_i = 1}^{N} \frac{J\_5 \left( 2 \pi \sqrt{h\_i^2 + h\_j^2} \right) e^{i4 \arctan(h\_j / h\_i)}}{2 \pi \sqrt{h\_i^2 + h\_j^2} \left( k\_{\beta 0}^2 - (\frac{2 \pi}{a})^2 (h\_i^2 + h\_j^2) \right)} \right| \tag{22}$$

Due to the non-zero β γ term in Eq. (22), the determinant in Eq. (12) is φ*<sup>K</sup>* -dependent, which gives rise to anisotropic dispersion relations. The explicit expressions for 1 *squ <sup>K</sup>* and 2 *squ K* are very complicated and will be futher discussed in the next Section.

#### **4. Effective medium theory for elastic metamaterials**

The MST method is capable of producing the dispersion relations of an elastic metamaterial so that the effective wave speed for elastic waves in the metamaterial can be obtained accordingly. However, it is not able to provide an effective description for each parameters. Knowing the effective parameters will provide a clear theoretical explanation of the unusual phenomenon of a metamaterial and greatly benefit the design of new metamaterials. This Section is devoted to the derivation of EMTs.

#### **4.1 Isotropic media: Coherent potential approximation approach**

If the elastic metamaterial is isotropic, i.e., cylinders arranged in a hexogonal lattice, the EMT can be derived by considering the scattering of elastic waves by a coated cylinder embedded in the effective medium with effective parameters ( κ *<sup>e</sup>* , μ*e* , ρ *<sup>e</sup>* ), which is shown in Figure 2 (Wu et al., 2007). The coated cylinder consists of the scatterer surrounded by a layer of the matrix. The inner and outer radii, which are denoted by *sr* and 0*r* , respectively, satisfy 2 2 *<sup>s</sup>* / <sup>0</sup> *rr p* = , where *p* is the filling ratio of the scatterer. The effective parameters κ *<sup>e</sup>* , μ*e* and ρ *<sup>e</sup>* are determined by the condition that the total scattering of the coated cylinder vanishes which is so-called CPA. This condition together with the boundary conditions on the surface of the coated cylinder at = <sup>0</sup> *r r* , provides another two relations: = + () () *ll lt lm m lm m tm b D ea D ea* and = + () () *tl tt tm m lm m tm b D ea D ea* , where αβ ( ) *D e <sup>m</sup>* ( , ,) α β = *l t* can be obtained by replacing λ*s* ρ *s* μ *<sup>s</sup>* , *ls <sup>k</sup>* , *ts <sup>k</sup>* and *<sup>s</sup> <sup>r</sup>* in αβ *Dm* ( , ,) α β = *l t* mentioned in Section 2.1 with λ *<sup>e</sup>* , ρ *<sup>e</sup>* , μ *<sup>e</sup>* , *le <sup>k</sup>* , *te <sup>k</sup>* and 0 *<sup>r</sup>* , respectively, where *le <sup>k</sup>* ( *te <sup>k</sup>* ) are the longitudinal (transverse) wave vectors in the effective medium. These relations together with the relations between α*<sup>m</sup> b* and α*<sup>m</sup> a* (α = *l t*, ) shown in Section 2.1 give the following effective medium condition:

Fig. 2. Micro-sctructure of the effective medium. (Wu et al.,2007)

π

2 2 4arctan( / ) <sup>5</sup> <sup>5</sup>

= =

6 4 2 2 2 22 2 <sup>0</sup> 0 1 <sup>0</sup>

<sup>+</sup> ⋅ ⋅ <sup>=</sup> +− +

*a k h hk h h*

term in Eq. (22), the determinant in Eq. (12) is

The MST method is capable of producing the dispersion relations of an elastic metamaterial so that the effective wave speed for elastic waves in the metamaterial can be obtained accordingly. However, it is not able to provide an effective description for each parameters. Knowing the effective parameters will provide a clear theoretical explanation of the unusual phenomenon of a metamaterial and greatly benefit the design of new metamaterials. This

If the elastic metamaterial is isotropic, i.e., cylinders arranged in a hexogonal lattice, the EMT can be derived by considering the scattering of elastic waves by a coated cylinder

Figure 2 (Wu et al., 2007). The coated cylinder consists of the scatterer surrounded by a layer of the matrix. The inner and outer radii, which are denoted by *sr* and 0*r* , respectively,

vanishes which is so-called CPA. This condition together with the boundary conditions on the surface of the coated cylinder at = <sup>0</sup> *r r* , provides another two relations:

(transverse) wave vectors in the effective medium. These relations together with the

κ0, μ0, ρ<sup>0</sup>

rs κs, μs, ρ<sup>s</sup>

r0

*<sup>s</sup>* , *ls <sup>k</sup>* , *ts <sup>k</sup>* and *<sup>s</sup> <sup>r</sup>* in

*tm m lm m tm b D ea D ea* , where

*<sup>s</sup>* / <sup>0</sup> *rr p* = , where *p* is the filling ratio of the scatterer. The effective parameters

*<sup>e</sup>* are determined by the condition that the total scattering of the coated cylinder

αβ

*<sup>e</sup>* , *le <sup>k</sup>* , *te <sup>k</sup>* and 0 *<sup>r</sup>* , respectively, where *le <sup>k</sup>* ( *te <sup>k</sup>* ) are the longitudinal

 *Dm* ( , ,) α β

= *l t*, ) shown in Section 2.1 give the following effective

*N N i j*

*J h he <sup>i</sup>*

π

*j i*

gives rise to anisotropic dispersion relations. The explicit expressions for 1

<sup>2</sup> 16 2 5!

very complicated and will be futher discussed in the next Section.

**4.1 Isotropic media: Coherent potential approximation approach** 

embedded in the effective medium with effective parameters (

*lm m lm m tm b D ea D ea* and = + () () *tl tt*

α*<sup>m</sup> a* (α

Fig. 2. Micro-sctructure of the effective medium. (Wu et al.,2007)

κe, μe, ρ<sup>e</sup>

λ*s* ρ *s* μ

α*<sup>m</sup> b* and

**4. Effective medium theory for elastic metamaterials** 

β

β γ

Section is devoted to the derivation of EMTs.

γ

Due to the non-zero

satisfy 2 2

λ *<sup>e</sup>* , ρ *<sup>e</sup>* , μ

relations between

medium condition:

= + () () *ll lt*

obtained by replacing

μ*e* and ρ

with

β ( )

*h h i j i j*

β

<sup>2</sup> <sup>2</sup> ( )( )

π

*a*

κ *<sup>e</sup>* , μ*e* , ρ

αβ

 ( ) *D e <sup>m</sup>* ( , ,) α β

= *l t* mentioned in Section 2.1

*j i*

φ

(22)

*<sup>K</sup>* -dependent, which

*squ K* are

*squ <sup>K</sup>* and 2

*<sup>e</sup>* ), which is shown in

= *l t* can be

κ*<sup>e</sup>* ,

*i hh*

$$D\_{m}^{\alpha\beta} \left( e \right) = D\_{m}^{\alpha\beta} \text{ (}\alpha \text{ } \beta = \text{l} \text{ } \text{t} \text{)}\tag{23}$$

In the long wavelength limit, where 0 0 1 *<sup>l</sup> k r* << , 0 0 1 *<sup>t</sup> k r* << , 0 1 *le k r* << and 0 1 *te k r* << , Eq. (23)can be simplified into the following effective medium equations for the elastic metamaterial:

$$\frac{\left(\kappa\_0 - \kappa\_e\right)}{\left(\mu\_0 + \kappa\_e\right)} = \frac{4\tilde{D}\_0^{ll}}{i\pi r\_0^2 k\_{l0}^2} \,\text{\,\,\,\tag{24}$$

$$\frac{(\rho\_0 - \rho\_\varepsilon)}{\rho\_0} = -\frac{8\tilde{D}\_1^{ll}}{i\pi r\_0^2 k\_{l0}^2} \,\,\,\,\tag{25}$$

$$\frac{\mu\_0(\mu\_0 - \mu\_e)}{(\kappa\_0 \mu\_0 + (\kappa\_0 + 2\mu\_0)\mu\_e)} = \frac{4\tilde{D}\_2^{ll}}{i\pi r\_0^2 k\_{l0}^2} \tag{26}$$

Obviously, κ *<sup>e</sup>* , ρ *<sup>e</sup>* and μ*<sup>e</sup>* are independently determined by *ll Dm* of the embedded cylinders alone with angular quantum numbers *m* = 0, 1 and 2.

The dispersion relation can be reproduced from effective medium parameters. Comparing Eqs. (24)-(26) to Eqs. (17) and (18), it is easy to show that ρ μ *e e* coincides with ( )<sup>2</sup> 1 *tri K* ω and ( ) ρ *ee e* κ + μ is identical to ( )<sup>2</sup> tri K2 ω . The equivalence provides a strong evidence that the dispersion relation is isotropic for a hexagonal lattice and its effective medium properties can be evaluated from the EMT derived from CPA.

Eqs. (24)-(26) require the wavelengths in both the host and the effective medium to be much larger than the size of the unit cell, but they do not impose any restriction on the wavelengths inside the scatterer. If the condition of 1 *ls s k r* << , 1 *ts s k r* << is further considered, the quasi-static limit is reached and Eqs. (24)-(26) becomes:

$$\frac{\left(\kappa\_0 - \kappa\_\varepsilon\right)}{\left(\mu\_0 + \kappa\_\varepsilon\right)} = p \frac{\left(\kappa\_0 - \kappa\_s\right)}{\left(\mu\_0 + \kappa\_s\right)}\ ,\tag{27}$$

0 0 ( ) ( ) , *e s* ρρ ρρ−= − *p* (28)

$$\frac{(\mu\_0 - \mu\_e)}{(\kappa\_0 \mu\_0 + (\kappa\_0 + 2\mu\_0)\mu\_e)} = p \frac{(\mu\_0 - \mu\_s)}{(\kappa\_0 \mu\_0 + (\kappa\_0 + 2\mu\_0)\mu\_s)}\tag{29}$$

The 3D version was reported by Berryman decades ago (Berryman, 1980). It should be pointed out that elastic EMT cannot recover the acoustic EMT by setting all the shear moduli to be zero, because of the different boundary conditions of elastic and acoustic waves.

Figure 3 shows the equifrequency surface (EFS) of a hexagonal array of silicone rubber cylinders with radii of 0.2 *a* embedded in an epoxy host. An EFS is a collection of all states in the *K* space that have the same frequency. The metamaterial is isotropic if its EFS is a circle. Here the dimensionless frequency, 0 ( ) (2 ) *<sup>t</sup> f* = ω π*a c* , is used, where *t*<sup>0</sup> *<sup>c</sup>* is the

where φ

and (28) for isotropic media,

where ( )( ) <sup>2</sup>

 μ

 γ

 δμ

Here ( ) 4 4 22 *ll tt* 0 0

 = − *ik k a* γ

δ

2

2

β γ

between two extrema, ( )

. If both

of isotropic EMT given by Eqs. (24)-(26).

<sup>2</sup> (/ ) ω

amplitude of the oscillation,

1

ω

2

recover when 0

Similarly, <sup>2</sup>

Δ2 ρ

( )

and <sup>2</sup> <sup>2</sup> (/ ) ω

 *ee e* + + μ

κ

*K* ω

Δ

δ

expression:

Eq. (30) gives:

1,2 0 <sup>0</sup> <sup>0</sup> 8

=− − *e e*

π

*e*

ρ

*e*

 = (or 0 δ

and Eqs. (35) and (36) can be reduced to <sup>2</sup>

μ

Δ1 and

*K* , can be well approximated by

*e e* + Δ2 ρ

> Δ Δ

ρ

= − <sup>+</sup>

 δμ

ω

κ ρ

Since the origin of anisotropy comes from the term 0

μκ

 μ

 and *l t*, γ

responsible for the anisotropy. In the long wavelength limit,

 μ

( )

<sup>2</sup> <sup>1920</sup>

= =

<sup>5</sup> 0 0

*j i*

φ

φ

<sup>1</sup> sin 2 cos 2 2 *K K*

= +− ++ −

= ++ ++ −

ω*K* =

longitudinal bands (Royer & Dieulesaint, 1999). Eq. (35) shows that <sup>2</sup>

Δ

which are the square of two known wave speeds. For the case of anisotropic dispersions, Eqs. (35) and (36) give the dispersion relations for the quasi-transverse and quasi-

> and ( ) μ

*K* oscillates between its two extrema, ( )

μ ρ

Figure 4(a) is the same as Figure 3(a), but the rubber cylinders are arranged in a square array. The inner ring represents the quasi-longitudinal branch with distinct anisotropy and the outer one is the quasi-transverse branch with weak anisotropy. The corresponding *Ka*

*h h*

5 5 00 0 0 1 2 2

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 155

effective mass density derived from CPA. It is convenient to express the three effective

<sup>11</sup> <sup>1</sup> , *C* =++ κ *e e* μ

<sup>12</sup> <sup>1</sup> , *C* =−− κ *e e* μ

> 44 2 , *C* = + μ*e* Δ

ρ,with − and + for

Δ

Δ

is given by Eq.(22). It is clear that

( )

*J h he*

<sup>+</sup>

<sup>+</sup>

π

11 44 12 44 11 44

11 44 12 44 11 44

*C C C C C C K* (36)

<sup>1</sup> (/ ) /

*e e* + Δ1 ρ

2 are much smaller than

<sup>1</sup> sin 2 cos 2 , <sup>2</sup> *K K*

*C C C C C C*

*N N i j*

( )

( )( ) ( )( )

( )( ) ( )( )

β γ

μ ρ

= ). In this case, 11 12 44 *CC C* = + 2 (Royer & Dieulesaint, 1999)

 at 0 φ

1 2 <sup>−</sup> , is small, the angle averaged dispersions, <sup>2</sup>

 *e e* and ( ) κ *eee* + μ ρ

2 2 2 2

2 2 2 2

*h h*

*i j*

κ*<sup>e</sup>* and

and the *x*-axis, and

(31)

(32)

(33)

Δ

Δ1 and

Δ1 and

2 2 4arctan( / )

 φ

> φ

*e e* and <sup>2</sup>

*<sup>K</sup>* = and

μ*e* and

κ

ω

≠ , the isotropy is expected to

<sup>2</sup> ( / ) ( )/

<sup>1</sup> (/ ) ω

 *ee e* + + μ

> κ *e e* + μ

Δ1 ρ

, which are the results

π

 κ *K* = +*ee e* μ ρ,

*K* oscillates

4 , respectively.

and

, and the

<sup>1</sup> (/ ) ω*K*

*i hh*

δ

*j i*

μ

ρ*<sup>e</sup>* is the

<sup>2</sup> , respectively.

(34)

(35)

has the following

Δ2 are

*<sup>e</sup>* shown in Eqs. (26)

*<sup>K</sup>* denotes the angle between the Bloch wave vector *K*

moduli in Eq. (32) in terms of the two effective parameters

transverse wave speed inside the host. The silicone rubber's material parameters are 3 3 ρ = × 1.3 10 / *kg m* , 5 2 λ = ×6 10 / *N m* and 4 2 μ = ×4 10 / *N m* , which means the wave speeds inside the rubber are: 22.87 *m/s* for longitudinal waves and 5.54 *m/s* for transverse waves. The corresponding parameters in the epoxy host are 3 3 ρ = × 1.18 10 / *kg m* , 9 2 λ = × 4.43 10 / *N m* and 9 2 μ = × 1.59 10 / *N m* , which indicates the wave speeds are 2539.52m /s ( 1160.80m /s ) for longitudinal (transverse) waves (Liu et al., 2000b). Apparently, slow wave speeds imply that wavelengths inside the silicone rubber cylinder may be comparable to or even much smaller than the size of the cylinder at low frequencies. Thus, Mie-like resonances may occur, which serve as the built-in resonances required for metamaterials. Here the frequency *f* is chosen to be 0.03 where both 1.9 *ls s k r* and 7.9 *ts s k r* are larger than unity indicating it is not in the quasi-static limit. Figure 3(a) shows the corresponding EFS, which exhibits two circular rings, with the inner one denoting the quasi-longitudinal branch and the outer one representing the quasi-transverse branch. The corresponding *Ka* as a function of φK is plotted in Fig. 3(b) by open circles. These circles form two horizontal lines, indicating dispersions are isotropic, i.e., effective wave speeds do not vary with directions. Also plotted in Fig. 3(b) are the results of EMT calculated from Eq. (17) or Eqs. (24)-(26), depicted by two solid lines. The complete overlaps between solid lines and circles give a numerical support to the correctness of the EMT in the long wavelength limit.

Fig. 3. (a) The equifrequency surface for a hexagonal array. (b) . *Ka* . as a function of φ*<sup>K</sup>* . (Wu & Z. Zhang, 2009)

#### **4.2 Anisotropic media: Christoffel's equation**

If the elastic metamaterial is anisotropic, such as cylindrical scatterers arranged in a square lattice, the CPA fails as it only deals with isotropic cases. In this case, the result of MST, i.e., Eq. (14), can give an anisotropic EMT in the form of Christoffel's equation.

Taking the long wavelength limit approximation on Eq. (12) and plug in Eqs. (20) and (22), the expressions for 1 *squ <sup>K</sup>* and 2 *squ K* can be written as the solutions of the following Christoffel's equation for an anisotropic medium (Royer & Dieulesaint, 1999)

$$\det \begin{vmatrix} \cos^2 \phi\_\mathbf{k} \mathbb{C}\_{11} \rho\_\varepsilon^{-1} + \sin^2 \phi\_\mathbf{k} \mathbb{C}\_{44} \rho\_\varepsilon^{-1} - \left(a\rho'\mathbf{k}\right)^2 & \cos \phi\_\mathbf{k} \sin \phi\_\mathbf{k} \left(\mathbb{C}\_{12} + \mathbb{C}\_{44}\right) \rho\_\varepsilon^{-1} \\\ \cos \phi\_\mathbf{k} \sin \phi\_\mathbf{k} \left(\mathbb{C}\_{12} + \mathbb{C}\_{44}\right) \rho\_\varepsilon^{-1} & \cos^2 \phi\_\mathbf{k} \mathbb{C}\_{44} \rho\_\varepsilon^{-1} + \sin^2 \phi\_\mathbf{k} \mathbb{C}\_{11} \rho\_\varepsilon^{-1} - \left(a\rho'\mathbf{k}\right)^2 \end{vmatrix} = 0 \tag{30}$$

where φ*<sup>K</sup>* denotes the angle between the Bloch wave vector *K* and the *x*-axis, and ρ *<sup>e</sup>* is the effective mass density derived from CPA. It is convenient to express the three effective moduli in Eq. (32) in terms of the two effective parameters κ *<sup>e</sup>* and μ*<sup>e</sup>* shown in Eqs. (26) and (28) for isotropic media,

$$\mathbf{C}\_{11} = \mathfrak{x}\_{\boldsymbol{e}} + \mu\_{\boldsymbol{e}} + \Delta\_{\mathbf{1}\prime} \tag{31}$$

$$\mathbf{C}\_{12} = \mathbf{x}\_{\boldsymbol{\varepsilon}} - \boldsymbol{\mu}\_{\boldsymbol{\varepsilon}} - \boldsymbol{\Delta}\_{\mathbf{1}} \tag{32}$$

$$C\_{44} = \mu\_{\varepsilon} + \Delta\_{2},\tag{33}$$

where ( )( ) <sup>2</sup> 1,2 0 <sup>0</sup> <sup>0</sup> 8 Δ δμ μ δμ μ =− − *e e* ρ ,with − and + for Δ1 and Δ<sup>2</sup> , respectively. Here ( ) 4 4 22 *ll tt* 0 0 δ = − *ik k a* γ γ ω and *l t*, γ is given by Eq.(22). It is clear that Δ1 and Δ2 are responsible for the anisotropy. In the long wavelength limit, δ has the following expression:

$$\mathcal{S} = -\frac{1920}{\pi^5} \frac{\kappa\_0 \rho\_0}{\mu\_0 (\kappa\_0 + \mu\_0)} \left( \sum\_{h\_j = 0}^N \sum\_{h\_i = 1}^N \frac{J\_5 \left( 2\pi \sqrt{h\_i^2 + h\_j^2} \right) e^{i4\arctan(h\_j/h\_i)}}{\left( \sqrt{h\_i^2 + h\_j^2} \right)^5} \right) \tag{34}$$

Eq. (30) gives:

154 Metamaterial

transverse wave speed inside the host. The silicone rubber's material parameters are

inside the rubber are: 22.87 *m/s* for longitudinal waves and 5.54 *m/s* for transverse waves. The

chosen to be 0.03 where both 1.9 *ls s k r* and 7.9 *ts s k r* are larger than unity indicating it is not in the quasi-static limit. Figure 3(a) shows the corresponding EFS, which exhibits two circular rings, with the inner one denoting the quasi-longitudinal branch and the outer one

in Fig. 3(b) by open circles. These circles form two horizontal lines, indicating dispersions are isotropic, i.e., effective wave speeds do not vary with directions. Also plotted in Fig. 3(b) are the results of EMT calculated from Eq. (17) or Eqs. (24)-(26), depicted by two solid lines. The complete overlaps between solid lines and circles give a numerical support to the correctness

(b)

0.06 0.08 0.10 0.12 0.14 0.16 0.18

If the elastic metamaterial is anisotropic, such as cylindrical scatterers arranged in a square lattice, the CPA fails as it only deals with isotropic cases. In this case, the result of MST, i.e.,

Taking the long wavelength limit approximation on Eq. (12) and plug in Eqs. (20) and (22),

2 12 1 2 1 11 44 12 44

*C CK C C*

*Ke Ke KK e KK e Ke Ke*

det 0

Ka

Fig. 3. (a) The equifrequency surface for a hexagonal array. (b) . *Ka* . as a function of

Eq. (14), can give an anisotropic EMT in the form of Christoffel's equation.

Christoffel's equation for an anisotropic medium (Royer & Dieulesaint, 1999)

cos sin cos sin

 ω

cos sin cos sin

 ρ ρ

 = × 1.59 10 / *N m* , which indicates the wave speeds are 2539.52m /s ( 1160.80m /s ) for longitudinal (transverse) waves (Liu et al., 2000b). Apparently, slow wave speeds imply that wavelengths inside the silicone rubber cylinder may be comparable to or even much smaller than the size of the cylinder at low frequencies. Thus, Mie-like resonances may occur, which serve as the built-in resonances required for metamaterials. Here the frequency *f*

= ×4 10 / *N m* , which means the wave speeds

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

φ k

*squ K* can be written as the solutions of the following

 φρ

1 2 12 1 2

φφ

( ) ( )

− −− + − <sup>+</sup> <sup>=</sup> + + −

( ) ( )

*C C C CK*

12 44 44 11

− − −

 φρ

 = × 1.18 10 / *kg m* , 9 2 λ

= × 4.43 10 / *N m*

φ

 MST EMT

> ρ

> > ω

(30)

φ*<sup>K</sup>* . is

K is plotted

 = ×6 10 / *N m* and 4 2 μ

representing the quasi-transverse branch. The corresponding *Ka* as a function of

corresponding parameters in the epoxy host are 3 3

3 3

and 9 2

 = × 1.3 10 / *kg m* , 5 2 λ

of the EMT in the long wavelength limit.

(a)


0.0

KY

(Wu & Z. Zhang, 2009)

the expressions for 1

φρ

φφ

a

0.1

0.2


KX a

*squ <sup>K</sup>* and 2

 φρ

**4.2 Anisotropic media: Christoffel's equation** 

ρ

μ

$$\left(\frac{\alpha}{K\_1}\right)^2 = \frac{1}{2\rho\_\varepsilon} \left(\mathbf{C}\_{11} + \mathbf{C}\_{44} - \sqrt{\sin^2\left(2\phi\_\mathbf{K}\right)\left(\mathbf{C}\_{12} + \mathbf{C}\_{44}\right)^2 + \cos^2\left(2\phi\_\mathbf{K}\right)\left(\mathbf{C}\_{11} - \mathbf{C}\_{44}\right)^2}\right),\tag{35}$$

$$\left(\frac{\partial \rho}{\partial \xi\_2}\right)^2 = \frac{1}{2\rho\_\varepsilon} \left(\mathbb{C}\_{11} + \mathbb{C}\_{44} + \sqrt{\sin^2 \left(2\phi\_\mathbb{K}\right) \left(\mathbb{C}\_{12} + \mathbb{C}\_{44}\right)^2 + \cos^2 \left(2\phi\_\mathbb{K}\right) \left(\mathbb{C}\_{11} - \mathbb{C}\_{44}\right)^2}\right) \tag{36}$$

Since the origin of anisotropy comes from the term 0 β γ ≠ , the isotropy is expected to recover when 0 β γ = (or 0 δ = ). In this case, 11 12 44 *CC C* = + 2 (Royer & Dieulesaint, 1999) and Eqs. (35) and (36) can be reduced to <sup>2</sup> <sup>1</sup> (/ ) / ω *K* = μ ρ *e e* and <sup>2</sup> <sup>2</sup> ( / ) ( )/ ω κ *K* = +*ee e* μ ρ , which are the square of two known wave speeds. For the case of anisotropic dispersions, Eqs. (35) and (36) give the dispersion relations for the quasi-transverse and quasilongitudinal bands (Royer & Dieulesaint, 1999). Eq. (35) shows that <sup>2</sup> <sup>1</sup> (/ ) ω *K* oscillates between two extrema, ( ) μ*e e* + Δ2 ρ and ( ) μ*e e* + Δ1 ρ at 0 φ*<sup>K</sup>* = and π 4 , respectively. Similarly, <sup>2</sup> <sup>2</sup> (/ ) ω *K* oscillates between its two extrema, ( ) κ *ee e* + + μ Δ1 ρ and ( ) κ *ee e* + + μ Δ2 ρ . If both Δ1 and Δ2 are much smaller than μ*e* and κ *e e* + μ , and the amplitude of the oscillation, Δ Δ 1 2 <sup>−</sup> , is small, the angle averaged dispersions, <sup>2</sup> <sup>1</sup> (/ ) ω *K* and <sup>2</sup> <sup>2</sup> (/ ) ω *K* , can be well approximated by μ ρ *e e* and ( ) κ *eee* + μ ρ , which are the results of isotropic EMT given by Eqs. (24)-(26).

Figure 4(a) is the same as Figure 3(a), but the rubber cylinders are arranged in a square array. The inner ring represents the quasi-longitudinal branch with distinct anisotropy and the outer one is the quasi-transverse branch with weak anisotropy. The corresponding *Ka*

(or κ *e e* + μ

combinations of signs in

κ *e e* + μ , ρ*<sup>e</sup>* and

expected. Examples will be shown in a later section.

**5.1 Isotropic elastic metamaterials** 

implied by the EMT. The negative

the quadrupolar ( <sup>2</sup>

κ

negative

mass

the effective bulk modulus and shear modulus

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 157

1 lists eight possible types of wave propagation in 2D elastic metamaterials with different

(24)-(26) do not satisfy the well-known bounds (Hashin & Shtrikman, 1963; Torquato, 1991) on the effective elastic moduli as these bounds are derived in the quasi-static limit. For anisotropic metamaterials, there exists at least one more effective elastic modulus which can also turn negative. Thus, many more interesting novel wave transport behaviors would be

The simplest isotropic elastic metamaterial which is comprised of silicone rubber cylinders embedded in an epoxy host was exhibited in the last section. If the rubber cylinders' radii are chosen to be 0.3 *a* , it has been shown that various types of resonances were produced (Wu et al., 2007). The displacement fields for three typical resonances are plotted in Fig. 5, which clearly shows in (a), (b) and (c) the dipolar, quadrupolar and monopolar resonances. These resonances are linked to the effective medium parameters in the following manner

parameters by enhancing these resonances. For instance, the dipolar resonance shown in Fig. 5(a) exhibits the collective motion of the core part of rubber. This mode can be regarded as a simple "mass-spring" harmonic oscillator, with the central part serving as a "mass" and the boundary layer of the rubber serving as "spring". Replacing the inner region of rubber with another heavier cylinder, e.g. lead, will enhance the field oscillation of the silicone

by Liu *et al* and was named as "locally resonant sonic materials" (Liu et al., 2000b), which were comprised of a cubic array of rubber-coated lead spheres embedded in epoxy. A large low-frequency band gap for both longitudinal and transverse waves, induced by negative

Fig. 5. Displacement fields for different resonances. The arrows represent the direction and

the brightness denotes the amplitude, with white indicates larger. (Wu et al., 2007)

ρ

*ll <sup>D</sup>* ) and monopolar ( <sup>0</sup>

rubber, which, in turn, will widen the resonant region of

μ

) induces negative refractive index for the transverse (longitudinal) waves. Table

κ*<sup>e</sup>* and

μ

*<sup>e</sup>* arises from a dipolar resonance ( <sup>1</sup>

ρ

*<sup>e</sup>* , respectively. We can enlarge the negative regions of effective medium

*ll D* ) resonances give rise to negative

ρ*<sup>e</sup>* and

*ll D* ), whereas

μ*<sup>e</sup>* and

*<sup>e</sup>* . This design was first proposed

*<sup>e</sup>* determined according to Eqs.

*<sup>e</sup>* . Like acoustic metamaterials (Li & Chan, 2004),

μ*e*

transverse waves (Liu et al., 2000b). On the other hand, a simultaneous negative

as a function of φ*<sup>K</sup>* is plotted in Fig. 4(b) in open circles, which form two oscillating curves induced by the β γ term in Eq. (21). In this case, 1 Δ = 0.0158 and 2 Δ = −0.0179 (in the unit of μ<sup>0</sup> ), which are very small compared to κ *<sup>e</sup>* (2.241) and μ*e* (0.733). <sup>2</sup> <sup>1</sup> (/ ) ω *K* and <sup>2</sup> <sup>2</sup> (/ ) ω *K* should reach their maximum and minimum at 0 φ*<sup>K</sup>* = ,respectively. This implies *K*1 ( *K*<sup>2</sup> ) is at its minimum (maximum). Also, *K*1 arrives at its maximum at / 4 φ*<sup>K</sup>* = π , where *K*2 takes its minimal value. These are clearly illustrated Fig. 4(b). If we use the ratio ( ) max min / ,( 1,2) *ii i i d K K Ki* =− = to characterize the amount of anisotropy (Ni and Cheng, 2005), where max *Ki* , min *Ki* and *Ki* are the maximum, minimum and average of *Ki* , the corresponding quantities are 1 *d* = 5.39% and 1 *K a* = 0.1603 for the transverse waves and <sup>2</sup> *d* = 1.20% and 2 *K a* = 0.0793 for the longitudinal waves. The averaged values of *K ai* coincide with the results calculated from the isotropic effective medium, i.e., Eq. (24)-(26), which give 0.1599 *K at* = and 0.0794 *K al* = as shown in Fig. 4 (b) in two horizontal solid lines. Fig. 4(b) demonstrates that the isotropic EMT can well predict the angle-averaged value of *K ai* in the case of anisotropy.

Fig. 4. The same as Figure 3 but the lattice is a square lattice. (Wu & Z. Zhang, 2009)

#### **5. Design of elastic metamaterials**

The purpose of deriving EMTs is to reveal the relationship between the resonances of the microstructures and the effective parameters and to provide a guide in the design of new metamaterials with novel properties. Even for isotropic metamaterials, the negativities in three effective parameters as well as their combinations can give rise to various interesting properties unseen in natural materials. For instance, since the effective phase velocities in 2D elastic metamaterials are 1 *le e e e c* = + κ μ ρ and 1 *te e e c* = μ ρ for longitudinal and transverse waves, respectively, a single negative ρ *<sup>e</sup>* in a frequency regime leads to imaginary *le c* and *te c* , which implies the existence of a band gap for both longitudinal and


Table 1. Various wave propagation properties. Positive (negative) *n* indicates positive (negative) propagating bands. *l* and *t* represent longitudinal and transverse waves, respectively.

transverse waves (Liu et al., 2000b). On the other hand, a simultaneous negative ρ *<sup>e</sup>* and μ*e* (or κ *e e* + μ ) induces negative refractive index for the transverse (longitudinal) waves. Table 1 lists eight possible types of wave propagation in 2D elastic metamaterials with different combinations of signs in κ *e e* + μ , ρ *<sup>e</sup>* and μ*<sup>e</sup>* . Like acoustic metamaterials (Li & Chan, 2004), the effective bulk modulus and shear modulus κ *<sup>e</sup>* and μ*<sup>e</sup>* determined according to Eqs. (24)-(26) do not satisfy the well-known bounds (Hashin & Shtrikman, 1963; Torquato, 1991) on the effective elastic moduli as these bounds are derived in the quasi-static limit. For anisotropic metamaterials, there exists at least one more effective elastic modulus which can also turn negative. Thus, many more interesting novel wave transport behaviors would be expected. Examples will be shown in a later section.

#### **5.1 Isotropic elastic metamaterials**

156 Metamaterial

κ

Δ

*<sup>e</sup>* (2.241) and

φ

its minimal value. These are clearly illustrated Fig. 4(b). If we use the ratio ( ) max min / ,( 1,2) *ii i i d K K Ki* =− = to characterize the amount of anisotropy (Ni and Cheng, 2005), where max *Ki* , min *Ki* and *Ki* are the maximum, minimum and average of *Ki* , the corresponding quantities are 1 *d* = 5.39% and 1 *K a* = 0.1603 for the transverse waves and <sup>2</sup> *d* = 1.20% and 2 *K a* = 0.0793 for the longitudinal waves. The averaged values of *K ai* coincide with the results calculated from the isotropic effective medium, i.e., Eq. (24)-(26), which give 0.1599 *K at* = and 0.0794 *K al* = as shown in Fig. 4 (b) in two horizontal solid lines. Fig. 4(b) demonstrates that the isotropic EMT can well predict the angle-averaged

> 0.06 0.08 0.10 0.12 0.14 0.16

The purpose of deriving EMTs is to reveal the relationship between the resonances of the microstructures and the effective parameters and to provide a guide in the design of new metamaterials with novel properties. Even for isotropic metamaterials, the negativities in three effective parameters as well as their combinations can give rise to various interesting properties unseen in natural materials. For instance, since the effective phase velocities in 2D

> ρ

imaginary *le c* and *te c* , which implies the existence of a band gap for both longitudinal and

0

0 μ*e* <

Table 1. Various wave propagation properties. Positive (negative) *n* indicates positive (negative) propagating bands. *l* and *t* represent longitudinal and transverse waves,

<sup>&</sup>gt; 0 *nl* > ; 0 *nt* > 0 *nl* > ; *t*:gap 0 *nt* > ; *l*:gap *l*&*t*: gap

<sup>&</sup>lt; *l*&*t*: gap 0 *nt* < ; *l*:gap 0 *nl* < ; *t*:gap 0 *nl* < ; 0 *nt* <

Ka

Fig. 4. The same as Figure 3 but the lattice is a square lattice. (Wu & Z. Zhang, 2009)

(b)

term in Eq. (21). In this case, 1

at its minimum (maximum). Also, *K*1 arrives at its maximum at / 4

*<sup>K</sup>* is plotted in Fig. 4(b) in open circles, which form two oscillating curves

= 0.0158 and 2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

 and 1 *te e e c* = μ

> κ *e e* + < μ

ρ

 ρ

0

0 μ*e* >

*<sup>e</sup>* in a frequency regime leads to

for longitudinal and

κ *e e* + < μ

0

0 μ*e* <

φ K

μ

Δ

φ*<sup>K</sup>* = π

*e* (0.733). <sup>2</sup> <sup>1</sup> (/ ) ω

*<sup>K</sup>* = ,respectively. This implies *K*1 ( *K*<sup>2</sup> ) is

 MST EMT

= −0.0179 (in the unit of

*K* and <sup>2</sup>

, where *K*2 takes

<sup>2</sup> (/ ) ω*K*

as a function of

induced by the

μ

φ

β γ

<sup>0</sup> ), which are very small compared to

value of *K ai* in the case of anisotropy.


**5. Design of elastic metamaterials** 

elastic metamaterials are 1 *le e e e c* = +

transverse waves, respectively, a single negative

0

0 μ*e* >

κ *e e* + > μ

0 *e* ρ

0 *e* ρ

respectively.

0.0

KY

a

0.1

0.2

should reach their maximum and minimum at 0


KX a

> κ

> > κ *e e* + > μ

μ

0.18 (a)

The simplest isotropic elastic metamaterial which is comprised of silicone rubber cylinders embedded in an epoxy host was exhibited in the last section. If the rubber cylinders' radii are chosen to be 0.3 *a* , it has been shown that various types of resonances were produced (Wu et al., 2007). The displacement fields for three typical resonances are plotted in Fig. 5, which clearly shows in (a), (b) and (c) the dipolar, quadrupolar and monopolar resonances. These resonances are linked to the effective medium parameters in the following manner implied by the EMT. The negative ρ *<sup>e</sup>* arises from a dipolar resonance ( <sup>1</sup> *ll D* ), whereas the quadrupolar ( <sup>2</sup> *ll <sup>D</sup>* ) and monopolar ( <sup>0</sup> *ll D* ) resonances give rise to negative μ*<sup>e</sup>* and negative κ *<sup>e</sup>* , respectively. We can enlarge the negative regions of effective medium parameters by enhancing these resonances. For instance, the dipolar resonance shown in Fig. 5(a) exhibits the collective motion of the core part of rubber. This mode can be regarded as a simple "mass-spring" harmonic oscillator, with the central part serving as a "mass" and the boundary layer of the rubber serving as "spring". Replacing the inner region of rubber with another heavier cylinder, e.g. lead, will enhance the field oscillation of the silicone rubber, which, in turn, will widen the resonant region of ρ *<sup>e</sup>* . This design was first proposed by Liu *et al* and was named as "locally resonant sonic materials" (Liu et al., 2000b), which were comprised of a cubic array of rubber-coated lead spheres embedded in epoxy. A large low-frequency band gap for both longitudinal and transverse waves, induced by negative mass

Fig. 5. Displacement fields for different resonances. The arrows represent the direction and the brightness denotes the amplitude, with white indicates larger. (Wu et al., 2007)

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 159

The structure of the hollow silicone rubber cylinder in epoxy does provide a frequency region of negative band for shear waves, but the bandwidth is too small to be of any practical use. Enlarging the quadrupolar resonance is a challenge task that has yet to be accomplished. Since replacing the inner part of the rubber by an easier deformed material is the direction, another common material, water, becomes a candidate. The material

> = × 1.0 10 kg m and 9 2 λ

0.256 0.258 0.260 0.262 0.264

Fig. 7. Effective medium parameters for rubber-coated water cylinders embedded in epoxy

confined in the cylinders locally. To broaden the resonance, one strategy that could be adopted is to make the energy "spread" out of the cylinders so that resonances in different cylinders become "coupled" to each other. This can be realized by reducing the impedance

Figure 8 shows the effective medium parameters for a rubber-coated water cylinder embedded in a foam host in a hexagonal lattice. The foam is polyethylene foam (HD115)

(Zhou & Hu, 2009). The light foam makes the host more matched to the rubber than epoxy

 = 115 / *kg m* , 6 2 λ

μ

ρ

<sup>ω</sup>a/(2πct0)

resonance is very sharp, indicating that the energy is

 = × 6.0 10 / *N m* and 6 2 μ

= × 3.0 10 / *N m*

ρe /ρ0 κe /κ0 μe /μ0

effective medium parameters for the metamaterial with air core being replaced by water, which exhibits an improvement in the absolute bandwidth for negative shear modulus. However, the bandwidth to mid-frequency ratio (0.00145) is comparable to the previous air core case (0.00183). Moreover, the negative shear band disappears as the region for negative mass density does not overlap with that for negative shear modulus. This example demonstrates that simple replacement of air by water does not improve the negative shear band. Nevertheless, the replacement does enhance the dipolar resonance greatly in the very low frequency regime (which is not plotted here). This fact suggests that water core is a better candidate than air in the context of realizing negative mass density. The difficulty lies in increasing the negative region for shear modulus and adjusting it to overlap with that for the negative mass. This requires optimizing the inner and outer radii of the rubber cylinder. However, it can be shown that only altering the geometry parameters will not make a

= × 2.22 10 *N m* . Figure 7 shows the

parameters of water are: 3 3

significant change.

in a hexogonal lattice.

Both Figs. 6(b) and 7 show that the

mismatch between the rubber and the host.

whose material parameters are <sup>3</sup>

ρ



0.0

0.5

effective parameters

1.0

1.5

2.0

density, was found. Figure 5(c) exhibits the field pattern of a monopolar resonance, where the shape of the silicone rubber cylinder remains as a circle, with its cross-sectional area oscillating in time. This suggests that by making the inner core more easily compressed, we would enhance the monopolar resonance. This notion was supported by using air bubbles in water to achieve a large frequency region of negative bulk modulus (Ding et al., 2007). Figure 5(b) shows the relative motion of the rubber. This suggests that by making the core areas easier to deform, we would enhance the quadrupolar resonance so as to enlarge the negative region for shear modulus. An intuitive design is to make the rubber cylinder hollow. A metamaterial based on this design is made of rubber-coated air cylinders embedded in epoxy. The material parameters of air are given by <sup>3</sup> ρ = 1.23 / *kg m* and 10 2 λ = × 1.42 10 / *N m* . The effective medium parameters are evaluated by a generalized EMT, which uses the standard transfer-matrix method to obtain the quantities *ll Dm* if the scatterers are layered cylinders.

Fig. 6. Band structure and effective medium parameters for an triangular array of hollow rubber cylinders embedded in epoxy. (Wu et al., 2007)

Figure 6(a) shows the band structure of hollow rubber cylinders embedded in epoxy in a hexagonal lattice. The inner and outer radii of hollow rubber cylinders are given as 0.87 *air s r r* = and 0.3 *sr a* = , which are carefully chosen so that a negative band for shear wave can be realized. The accurate MST results are plotted in open circles and the EMT predictions are featured by curves, with solid representing the longitudinal branch and dashed corresponding to the transverse branch. In the region of 0.12240 0.12253 < < *f* , negative-*n* bands of both longitudinal and transverse waves are found, which implies that ρ *<sup>e</sup>* , κ *e e* + μ and μ*e* are all negative. These negative values are induced only by the resonances of ρ *<sup>e</sup>* and μ*<sup>e</sup>* as shown in Fig. 6(b), in which the individual effective medium parameters are plotted. In another region of 0.12340 0.12356 < < *f* , negative-*n* band purely for longitudinal waves is found, which implies that ρ *<sup>e</sup>* and κ *e e* + μ are both negative. These negative values arise from resonances of both ρ *<sup>e</sup>* and κ *<sup>e</sup>* . Figure 6(a) demonstrates that the isotropic EMT is still a good approximation even for complex scatterers with layered structures. The small discrepancies between the band-structure calculation and the effective medium prediction for the longitudinal branches shown in Fig. 6(a) is due to the less accurate approximation of the Hankel functions when the values of *l*0 0 *k r* and *t*0 0 *k r* are not much less than 1.

density, was found. Figure 5(c) exhibits the field pattern of a monopolar resonance, where the shape of the silicone rubber cylinder remains as a circle, with its cross-sectional area oscillating in time. This suggests that by making the inner core more easily compressed, we would enhance the monopolar resonance. This notion was supported by using air bubbles in water to achieve a large frequency region of negative bulk modulus (Ding et al., 2007). Figure 5(b) shows the relative motion of the rubber. This suggests that by making the core areas easier to deform, we would enhance the quadrupolar resonance so as to enlarge the negative region for shear modulus. An intuitive design is to make the rubber cylinder hollow. A metamaterial based on this design is made of rubber-coated air cylinders embedded in epoxy. The material parameters of air are given by <sup>3</sup>

= × 1.42 10 / *N m* . The effective medium parameters are evaluated by a generalized

(b)

0.0 0.1 0.2 0.3 0.4 -1 0 1 2

Fig. 6. Band structure and effective medium parameters for an triangular array of hollow

Figure 6(a) shows the band structure of hollow rubber cylinders embedded in epoxy in a hexagonal lattice. The inner and outer radii of hollow rubber cylinders are given as 0.87 *air s r r* = and 0.3 *sr a* = , which are carefully chosen so that a negative band for shear wave can be realized. The accurate MST results are plotted in open circles and the EMT predictions are featured by curves, with solid representing the longitudinal branch and dashed

*e* are all negative. These negative values are induced only by the resonances of

*<sup>e</sup>* as shown in Fig. 6(b), in which the individual effective medium parameters are plotted. In

approximation even for complex scatterers with layered structures. The small discrepancies between the band-structure calculation and the effective medium prediction for the longitudinal branches shown in Fig. 6(a) is due to the less accurate approximation of the

K (π/a)

corresponding to the transverse branch. In the region of 0.12240 0.12253 < < *f*

bands of both longitudinal and transverse waves are found, which implies that

κ *e e* + μ

Hankel functions when the values of *l*0 0 *k r* and *t*0 0 *k r* are not much less than 1.

Effective Parameters

, negative-*n* band purely for longitudinal waves is

*<sup>e</sup>* . Figure 6(a) demonstrates that the isotropic EMT is still a good

are both negative. These negative values arise from

ρe /ρ0 μe /μ0 κe /κ0

EMT, which uses the standard transfer-matrix method to obtain the quantities *ll Dm*

10 2

scatterers are layered cylinders.

0.1220 0.1222 0.1224 0.1226 0.1228 0.1230 0.1232 0.1234 0.1236

rubber cylinders embedded in epoxy. (Wu et al., 2007)

ρ*<sup>e</sup>* and

κ

(a)

ωa/(2πct0)

another region of 0.12340 0.12356 < < *f*

ρ*<sup>e</sup>* and

found, which implies that

resonances of both

λ

and μ

μ

ρ

= 1.23 / *kg m* and

, negative-*n*

ρ *<sup>e</sup>* , κ *e e* + μ

> ρ*<sup>e</sup>* and

if the

The structure of the hollow silicone rubber cylinder in epoxy does provide a frequency region of negative band for shear waves, but the bandwidth is too small to be of any practical use. Enlarging the quadrupolar resonance is a challenge task that has yet to be accomplished. Since replacing the inner part of the rubber by an easier deformed material is the direction, another common material, water, becomes a candidate. The material parameters of water are: 3 3 ρ = × 1.0 10 kg m and 9 2 λ = × 2.22 10 *N m* . Figure 7 shows the effective medium parameters for the metamaterial with air core being replaced by water, which exhibits an improvement in the absolute bandwidth for negative shear modulus. However, the bandwidth to mid-frequency ratio (0.00145) is comparable to the previous air core case (0.00183). Moreover, the negative shear band disappears as the region for negative mass density does not overlap with that for negative shear modulus. This example demonstrates that simple replacement of air by water does not improve the negative shear band. Nevertheless, the replacement does enhance the dipolar resonance greatly in the very low frequency regime (which is not plotted here). This fact suggests that water core is a better candidate than air in the context of realizing negative mass density. The difficulty lies in increasing the negative region for shear modulus and adjusting it to overlap with that for the negative mass. This requires optimizing the inner and outer radii of the rubber cylinder. However, it can be shown that only altering the geometry parameters will not make a significant change.

Fig. 7. Effective medium parameters for rubber-coated water cylinders embedded in epoxy in a hexogonal lattice.

Both Figs. 6(b) and 7 show that the μ resonance is very sharp, indicating that the energy is confined in the cylinders locally. To broaden the resonance, one strategy that could be adopted is to make the energy "spread" out of the cylinders so that resonances in different cylinders become "coupled" to each other. This can be realized by reducing the impedance mismatch between the rubber and the host.

Figure 8 shows the effective medium parameters for a rubber-coated water cylinder embedded in a foam host in a hexagonal lattice. The foam is polyethylene foam (HD115) whose material parameters are <sup>3</sup> ρ = 115 / *kg m* , 6 2 λ = × 6.0 10 / *N m* and 6 2 μ = × 3.0 10 / *N m* (Zhou & Hu, 2009). The light foam makes the host more matched to the rubber than epoxy

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 161

The rubber-coated water cylinder embedded in foam provides the possiblity of realizing double negative shear bands. The isotropic scatterers and the hexagonal lattice structure result in a simple isotropic effective medium description of the metamaterial, which makes the design of the elastic metamaterial easier. If there is no restriction on the symmetery properties of the scatterer, there would be much more choices at the cost of many more

Fig. 10. A schematic figure of the physical model and the practical design. (Lai et al., 2011)

Since Fig. 5(b) exhibits a four-fold symmetry of a quadrupolar resonance, inserting heavier objects into the rubber in a way that is in accordance with the field pattern would help enhance the resonance. Figure 10(a) is a schematic figure of the physical model of the unit cell, which shows four masses connected to their center and the host (Lai et al., 2011). Such a structure is favorable of enhancing the dipolar resonance by the collective motion of the four masses, and the quadrupolar and monopolar resonances by relative motions of the masses. A practical realization of the model is illustrated in Fig. 10(b). The scatterers are composed of four steel rods surrounding a hard silicone rubber cylinder embedded in a soft silicone rubber cylinder. The matrix material is still foam. The lattice structure is a square with lattice constant of 10*cm* ; the radii of the soft and hard silicone rubber rods are 4*cm* and 1*cm* , respectively; the rectangular steel rods are 1.6 2.4 *cm cm* × in size, located at a distance of 2.4*cm* from the center. The material parameters for the foam and soft silicone rubber remain the same as the ones used in the design of rubber-coated water cylinder. The hard siliconerubber and the steel have parameters of: 3 3

= × 1.415 10 / *kg m* , 9 2

Γ*M* and

Γ

 = × 1.11 10 / *N m* and 10 2 μ

rectangular steel rods serve as the four masses and the soft silicone rubber rods serve as the springs. The insertion of the hard silicone rubber is for the purpose of adjusting the spring

The band structure of such metamaterial was calculated by using a finite element solver (COMSOL Multiphysics) and is shown in Fig. 11(a). There are two negative bands (red and blue dots), where the lower one (red dots) has a bandwidth about 18 Hz and the higher one

These two bands are separated by a small complete gap (178 Hz~198 Hz). The two

ρ

= × 8.28 10 / *N m* for steel. The four

*X* direction respectively.

point are plotted in Figs. 11(b)

= × 1.78 10 / *N m* for hard silicone rubber and

Γ

**5.2 Anisotropic elastic metamaterials and boundary effective medium theory** 

complicated microstructures of the scatterers.

= × 1.27 10 / *N m* and 6 2

(blue dots) has band widths of 18Hz and 10 Hz along

eigenstates in the lower and upper negative bands at the

 = × 7.9 10 / *kg m* , 11 2 λ

3 3

constants between the masses.

μ

λ

ρ

which will benefit the enhancement of the resonance for shear modulus. It also makes the water-coated rubber core relatively heavier so that the resonance for mass density is also enhanced. By adjusting the geometric parameters to be 0.24*a* and 0.32*a* for inner and outer radii, respectively, a large frequency region, marked by "A" and "B" in Fig. 8, for both negative shear modulus and negative mass density is obtained. The bandwidth to midfrequency ratio reaches 0.258, which is two orders of magnitude greater than the rubbercoated air cylinders in epoxy. The corresponding band structures as well as the transmission coefficeints of a slab numerically calcuated by MST are plotted in Fig. 9(a), which clearly shows a large negative band for transverse waves denoted by red dots and a narrow negative band for longitudinal waves denoted by blue dots. The polarization of these negative bands is determined through the transmission as shown in Fig. 9(b).

Fig. 8. Effective medium parameters for rubber-coated water cylinders embedded in foam in a hexogonal lattice.

Fig. 9. (a) The dispersion along Γ *K* direction for the same system as Fig. 8. (b) Transmission coefficients for longtidinal (solid blue) wave incident and transverse (dashed red) wave incident on a slab of witdth 6a and length 50a. The incident wave is along Γ*K* direction. (Wu et al., 2011)

which will benefit the enhancement of the resonance for shear modulus. It also makes the water-coated rubber core relatively heavier so that the resonance for mass density is also enhanced. By adjusting the geometric parameters to be 0.24*a* and 0.32*a* for inner and outer radii, respectively, a large frequency region, marked by "A" and "B" in Fig. 8, for both negative shear modulus and negative mass density is obtained. The bandwidth to midfrequency ratio reaches 0.258, which is two orders of magnitude greater than the rubbercoated air cylinders in epoxy. The corresponding band structures as well as the transmission coefficeints of a slab numerically calcuated by MST are plotted in Fig. 9(a), which clearly shows a large negative band for transverse waves denoted by red dots and a narrow negative band for longitudinal waves denoted by blue dots. The polarization of these

0.16 0.18 0.20 0.22 0.24

A B

ωα/(2πct0)

0.0 0.2 0.4 0.6 0.8 1.0

Transmission

 l t

*K* direction for the same system as Fig. 8.

(b)

Fig. 8. Effective medium parameters for rubber-coated water cylinders embedded in foam in

negative bands is determined through the transmission as shown in Fig. 9(b).

ρe /ρ0 κe /κ0 μe /μ0


0.00

Γ K

(b) Transmission coefficients for longtidinal (solid blue) wave incident and transverse (dashed red) wave incident on a slab of witdth 6a and length 50a. The incident wave is along

(a)

Γ

0.05

0.10

ωa/ct0

Fig. 9. (a) The dispersion along

*K* direction. (Wu et al., 2011)

Γ

0.15

0.20

0

Effective medium parameters

a hexogonal lattice.

2

4

#### **5.2 Anisotropic elastic metamaterials and boundary effective medium theory**

The rubber-coated water cylinder embedded in foam provides the possiblity of realizing double negative shear bands. The isotropic scatterers and the hexagonal lattice structure result in a simple isotropic effective medium description of the metamaterial, which makes the design of the elastic metamaterial easier. If there is no restriction on the symmetery properties of the scatterer, there would be much more choices at the cost of many more complicated microstructures of the scatterers.

Fig. 10. A schematic figure of the physical model and the practical design. (Lai et al., 2011)

Since Fig. 5(b) exhibits a four-fold symmetry of a quadrupolar resonance, inserting heavier objects into the rubber in a way that is in accordance with the field pattern would help enhance the resonance. Figure 10(a) is a schematic figure of the physical model of the unit cell, which shows four masses connected to their center and the host (Lai et al., 2011). Such a structure is favorable of enhancing the dipolar resonance by the collective motion of the four masses, and the quadrupolar and monopolar resonances by relative motions of the masses. A practical realization of the model is illustrated in Fig. 10(b). The scatterers are composed of four steel rods surrounding a hard silicone rubber cylinder embedded in a soft silicone rubber cylinder. The matrix material is still foam. The lattice structure is a square with lattice constant of 10*cm* ; the radii of the soft and hard silicone rubber rods are 4*cm* and 1*cm* , respectively; the rectangular steel rods are 1.6 2.4 *cm cm* × in size, located at a distance of 2.4*cm* from the center. The material parameters for the foam and soft silicone rubber remain the same as the ones used in the design of rubber-coated water cylinder. The hard siliconerubber and the steel have parameters of: 3 3 ρ = × 1.415 10 / *kg m* , 9 2 λ = × 1.27 10 / *N m* and 6 2 μ = × 1.78 10 / *N m* for hard silicone rubber and 3 3 ρ = × 7.9 10 / *kg m* , 11 2 λ = × 1.11 10 / *N m* and 10 2 μ = × 8.28 10 / *N m* for steel. The four rectangular steel rods serve as the four masses and the soft silicone rubber rods serve as the springs. The insertion of the hard silicone rubber is for the purpose of adjusting the spring constants between the masses.

The band structure of such metamaterial was calculated by using a finite element solver (COMSOL Multiphysics) and is shown in Fig. 11(a). There are two negative bands (red and blue dots), where the lower one (red dots) has a bandwidth about 18 Hz and the higher one (blue dots) has band widths of 18Hz and 10 Hz along Γ *M* and Γ *X* direction respectively. These two bands are separated by a small complete gap (178 Hz~198 Hz). The two eigenstates in the lower and upper negative bands at the Γpoint are plotted in Figs. 11(b)

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 163

<sup>0</sup> <sup>0</sup> ; 2 2 *xy xy xy xy e e <sup>y</sup> y a <sup>x</sup> x a xy yx T dx T dx T dy T dy*

*a a* = = = = + +

2 2

====

= = = =

*a a u dx u dx u dy u dy*

2

*a*

Though the above equations are presented for calculations along the ГХ direction, the corresponding formula for ГM direction can be similarly transcribed. Due to the obvious link between the bulk (shear) modulus and monopolar (quadrupolar) resonance,

Knowing the effective moduli, the corresponding dispersion relations can be calculated by using Christoffel's equation. Along the ГХ direction, compressional wave and shear wave

supports a longitudinal (transverse) wave along ГХ (ГM) direction, whereas the upper negative band only allows longitudinal wave in both ГХ and ГM directions. The corresponding results obtained from the EMT are also plotted in Fig. 11(a) by crosses. Excellent agreements between the finite element results and the EMT prediction are found.

Like their EM and acoustic counterparts, elastic metamaterials have shown many intriguing wave transport properties. For example, the total mode conversion and the super-anisotropy are two of them. The total mode conversion can completely convert the incident transverse (longitudinal) wave into a refracted longitudinal (transverse) wave. It is an analogue of the Brewster angle in the EM case (for example, Jackson, 1999), but in a much more stringent and complex manner. It only occurs on the interface between a normal solid and an elastic metamaterial with negative refractive index (Wu et al. 2011). The super-anisotropic behavior has been demonstrated in Section 5.2. Also shown there is the property of sustaining only a longitudinal wave at certain frequencies, which is so-called "fluid-like" solids, blurring the

ρ

. The effective medium results show that the lower negative band

and *C*<sup>44</sup>

direction the compressional and shear wave velocities are (*CC C* 11 12 44 + + 2 2 )( )

−+−

2

*xxyy <sup>e</sup> ya y xa x*

relevant eigenstates are plotted in Figs. 11(d) and 11(e). In the lower negative band,

;

*u dy u dy u dy u dy*

− −

*y y x x e e ya y xa x xx yy*

0 0

(40)

 eff 11 12 = + ( ) CC2 and ( ) 11 12 2 μ

*eff* is negative and diverges at the Г point, which is induced by the

κ*eff* and

*eff* is negative and diverges at the Г point, which is in

0 0

(39)

*eff* = − *C C* as effective

μ

*eff* evaluated from the

κ*eff* is

*eff* is positive

ρ

and

μ

, respectively; whereas along the ГМ

*T T*

*S S*

= =

κ

accordance with the quadrupolar resonance. In the higher negative band,

ρ

**6. Some intriguing properties of elastic metamaterials** 

distinction between solids and fluids (Lai et al., 2011).

elastic bulk modulus and shear modulus. The results for

μ

*xy*

=

*S*

it is more convenient to introduce

κ

velocities are given by by *C*<sup>11</sup>

ρ

positive and finite, while

and finite, while

monopolar resonance.

( )( ) 11 12 *C C*− 2

and

= =

and 11(c), respectively. The eigenstate in Fig. 11(b) is clearly a quadrupolar resonance, whereas the eigenstate in Fig. 11(c) is a monopolar resonance.

The negative bands can also be understood from an effective medium point of view. Since the scatterer involves a four-fold symmetry, the previously derived formula based on MST for isotropic inclusions does not apply and an EMT based on boundary integration is

Fig. 11. (a) Band structure of the multi-mass metamaterial. (b) and (c) Displacement field of eigenstates. The color represents the amplitude of displacement (blue/red for small/large values) and the arrows show the displacement vectors directly. (d) and (e) effective medium parameters calculated by a boundary EMT. (Lai et al., 2011)

developed. Though the scatterer is anisotropic, the dispersions and the associated modes can still be obtained from Christoffel's equation, i.e., Eqs. (35) and (36), with three independent effective moduli, *C*<sup>11</sup> , *C*12 and *C*<sup>44</sup> , and a mass density, ρ . The task is to determine the values of these parameters. The mass density is determined by Newton's law, 2 2 / *ee e* ρ ω = −*F ua x x* , where both the effective force *<sup>e</sup> Fx* on the unit cell and its effective displacement *<sup>e</sup> ux* may be obtained from surface integration of the stresses (along the *<sup>x</sup>* direction) and the displacements over the unit cell, i.e.,

$$\left. \left\{ F\_{\mathbf{x}}^{e} = \int T\_{\mathbf{x}\mathbf{x}} dy \right\vert\_{\mathbf{x}=a} - \left\{ T\_{\mathbf{x}\mathbf{x}} dy \right\vert\_{\mathbf{x}=0} + \left\{ T\_{\mathbf{x}\mathbf{y}} dx \right\vert\_{\mathbf{y}=a} - \left\{ T\_{\mathbf{x}\mathbf{y}} dx \right\vert\_{\mathbf{y}=0} \tag{37}$$

and

$$
\mu\_{\chi}^{\ell} = \frac{\left. \int \mu\_{\chi} dy \right|\_{\chi=0} + \left. \int \mu\_{\chi} dy \right|\_{\chi=a}}{2a} \tag{38}
$$

The stresses and displacements can be obtained from the COMSOL calculation. Similarly, the effective moduli are evaluated from the effective stress and strain relations: 11 12 *e ee ee T CS CS xx xx yy* = + , 12 11 *e ee ee T CS CS yy xx yy* = + , and 44 <sup>2</sup> *e ee T CS xy xy* <sup>=</sup> , where both the effective stresses and the effective strains are evaluated on the unit cell boundary as follows:

$$T\_{\chi\chi}^{e} = \frac{\int T\_{\chi\chi} dy \Big|\_{\chi=0} + \int T\_{\chi\chi} dy \Big|\_{\chi=a}}{2a};\\T\_{yy}^{e} = \frac{\int T\_{yy} dy \Big|\_{y=0} + \int T\_{yy} dy \Big|\_{y=a}}{2a};$$

$$\left.T\_{xy}^{\epsilon}\right|\_{xy} = \frac{\left.\int T\_{xy}d\mathbf{x}\right|\_{y=0} + \left.\int T\_{xy}d\mathbf{x}\right|\_{y=a}}{2a};\\T\_{yx}^{\epsilon} = \frac{\left.\int T\_{xy}d\mathbf{y}\right|\_{\mathbf{x}=0} + \left.\int T\_{xy}d\mathbf{y}\right|\_{\mathbf{x}=a}}{2a} \tag{39}$$

and

162 Metamaterial

and 11(c), respectively. The eigenstate in Fig. 11(b) is clearly a quadrupolar resonance,

The negative bands can also be understood from an effective medium point of view. Since the scatterer involves a four-fold symmetry, the previously derived formula based on MST

Fig. 11. (a) Band structure of the multi-mass metamaterial. (b) and (c) Displacement field of eigenstates. The color represents the amplitude of displacement (blue/red for small/large values) and the arrows show the displacement vectors directly. (d) and (e) effective medium

developed. Though the scatterer is anisotropic, the dispersions and the associated modes can still be obtained from Christoffel's equation, i.e., Eqs. (35) and (36), with three

determine the values of these parameters. The mass density is determined by Newton's law,

*x xx xx xy xy xa x ya y*

0 2 *x x <sup>e</sup> <sup>x</sup> x a <sup>x</sup> u dy u dy*

The stresses and displacements can be obtained from the COMSOL calculation. Similarly, the effective moduli are evaluated from the effective stress and strain relations: 11 12 *e ee ee T CS CS xx xx yy* = + , 12 11 *e ee ee T CS CS yy xx yy* = + , and 44 <sup>2</sup> *e ee T CS xy xy* <sup>=</sup> , where both the effective

> *yy yy xx xx e e y y <sup>a</sup> <sup>x</sup> x a xx yy T dy T dy T dy T dy*

*a a*

stresses and the effective strains are evaluated on the unit cell boundary as follows:

= =

*a* = = <sup>+</sup>

<sup>0</sup> <sup>0</sup> ; ; 2 2

= = = = <sup>+</sup> <sup>+</sup>

 = −*F ua x x* , where both the effective force *<sup>e</sup> Fx* on the unit cell and its effective displacement *<sup>e</sup> ux* may be obtained from surface integration of the stresses (along the *<sup>x</sup>*

0 0

<sup>=</sup> (38)

*F T dy T dy T dx T dx* ==== =−+− (37)

ρ

. The task is to

for isotropic inclusions does not apply and an EMT based on boundary integration is

whereas the eigenstate in Fig. 11(c) is a monopolar resonance.

parameters calculated by a boundary EMT. (Lai et al., 2011)

direction) and the displacements over the unit cell, i.e.,

*e*

2 2 / *ee e*

 ω

ρ

and

independent effective moduli, *C*<sup>11</sup> , *C*12 and *C*<sup>44</sup> , and a mass density,

*u*

*T T*

$$\begin{aligned} S\_{\text{xx}}^{\epsilon} &= \frac{\int \boldsymbol{\mu}\_{\text{x}} d\boldsymbol{y} \Big|\_{\mathbf{x}=a} - \int \boldsymbol{\mu}\_{\text{x}} d\boldsymbol{y} \Big|\_{\mathbf{x}=0}}{a^2}; S\_{yy}^{\epsilon} = \frac{\int \boldsymbol{\mu}\_{\text{y}} d\boldsymbol{y} \Big|\_{\mathbf{y}=a} - \int \boldsymbol{\mu}\_{\text{y}} d\boldsymbol{y} \Big|\_{\mathbf{y}=0}}{a^2} \\ S\_{xy}^{\epsilon} &= \frac{\int \boldsymbol{\mu}\_{\text{x}} d\mathbf{x} \Big|\_{\mathbf{y}=a} - \int \boldsymbol{\mu}\_{\text{x}} d\mathbf{x} \Big|\_{\mathbf{y}=0} + \int \boldsymbol{\mu}\_{\text{y}} d\mathbf{y} \Big|\_{\mathbf{x}=a} - \int \boldsymbol{\mu}\_{\text{y}} d\mathbf{y} \Big|\_{\mathbf{x}=0}}{2a^2} \end{aligned} \tag{40}$$

Though the above equations are presented for calculations along the ГХ direction, the corresponding formula for ГM direction can be similarly transcribed. Due to the obvious link between the bulk (shear) modulus and monopolar (quadrupolar) resonance, it is more convenient to introduce κ eff 11 12 = + ( ) CC2 and ( ) 11 12 2 μ*eff* = − *C C* as effective elastic bulk modulus and shear modulus. The results for κ *eff* and μ*eff* evaluated from the relevant eigenstates are plotted in Figs. 11(d) and 11(e). In the lower negative band, κ *eff* is positive and finite, while μ*eff* is negative and diverges at the Г point, which is in accordance with the quadrupolar resonance. In the higher negative band, μ*eff* is positive and finite, while κ *eff* is negative and diverges at the Г point, which is induced by the monopolar resonance.

Knowing the effective moduli, the corresponding dispersion relations can be calculated by using Christoffel's equation. Along the ГХ direction, compressional wave and shear wave velocities are given by by *C*<sup>11</sup> ρ and *C*<sup>44</sup> ρ , respectively; whereas along the ГМ direction the compressional and shear wave velocities are (*CC C* 11 12 44 + + 2 2 )( ) ρ and ( )( ) 11 12 *C C*− 2ρ . The effective medium results show that the lower negative band supports a longitudinal (transverse) wave along ГХ (ГM) direction, whereas the upper negative band only allows longitudinal wave in both ГХ and ГM directions. The corresponding results obtained from the EMT are also plotted in Fig. 11(a) by crosses. Excellent agreements between the finite element results and the EMT prediction are found.

#### **6. Some intriguing properties of elastic metamaterials**

Like their EM and acoustic counterparts, elastic metamaterials have shown many intriguing wave transport properties. For example, the total mode conversion and the super-anisotropy are two of them. The total mode conversion can completely convert the incident transverse (longitudinal) wave into a refracted longitudinal (transverse) wave. It is an analogue of the Brewster angle in the EM case (for example, Jackson, 1999), but in a much more stringent and complex manner. It only occurs on the interface between a normal solid and an elastic metamaterial with negative refractive index (Wu et al. 2011). The super-anisotropic behavior has been demonstrated in Section 5.2. Also shown there is the property of sustaining only a longitudinal wave at certain frequencies, which is so-called "fluid-like" solids, blurring the distinction between solids and fluids (Lai et al., 2011).

Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 165

Chen, H.; Fung, K.; Ma, H. & Chan, C. (2008). Polarization Gaps and Negative Group

Chin, S.; Nicorovici, N. & McPhedran, R. (1994). Green's Function and Lattice Sums for

Chui, S. & Hu, L. (2002). Theoretical Investigation on the Possibility of Preparing Left-

Chui, S. & Lin, Z. (2008). Long-Wavelength Behavior of Two-Dimensional Photonic Crystals. *Physical Review E,* Vol.78, No.6, (December 2008), pp.065601 ISSN 1539-3755 Ding, Y.; Liu, Z.; Qiu, C. & Shi, J. (2007). Metamaterial with Simultaneously Negative Bulk

Fang, N.; Xi, D.; Xu, J.; Ambati, M.; Srituravanich, W.; Sun, C. & Zhang, X. (2006). Ultrasonic

Grbic, A. & Eleftheriades, G. (2004). Overcoming the Diffraction Limit with a Planar Left-

Hashin, Z. & Shtrikman, S. (1963). A Variational Approach to The Theory of The Elastic

Hu, X.; Chan, C.; Zi, J.; Li, M. & Ho, K. (2006). Diamagnetic Response of Metallic Photonic

Jackson, J.; (1999). *Classical Electrodynamics* (Third edition), John Wiley,ISBN 0-471-30932-X,

Jin, J.; Liu, S.; Lin, Z. & Chui, S. (2009). Effective-Medium Theory for Anisotropic Magnetic

Kafesaki, M. & Economou, E. (1999). Multiple-scattering Theory for Three-dimensional

Kafesaki, M.; Penciu, R. & Economou, E. (2000). Air Bubbles in Water: A Strongly Multiple

Krokhin, A.; Arriaga, J. & Gumen, L. (2003). Speed of Sound in Periodic Elastic

Lai, Y.; Wu, Y.; Sheng, P. & Zhang, Z. (2011). Hybrid Elastic Solids, *Nature Materials,*Vol.10,

Vol.49, No.5, (May 1994), pp. 4590-4602, ISSN 1063-651X

Vol.65, No.14, (March 2002), pp. 144407, ISSN 0163-1829

pp. 224304, ISSN 0163-1829

093904, ISSN 0031-9007

New York

0163-1829

ISSN 0031-9007

pp. 452-456, ISSN 1476-1122

2004),pp. 117403 ISSN 0031-9007

Vol.11, No.2, pp. 127-140, ISSN: 0022-5096

(June 2006), pp. 223901, ISSN 0031-9007

pp. 11993-12001, ISSN 0163-1829

2000), pp. 6050-6053, ISSN 0031-9007

No.8,(August 2011), pp. 620-624, ISSN 1476-1122

Velocity in Chiral Phononic Crystals, *Physical Review B,* Vol.77, No.22, (June 2008),

Electromagnetic Scattering by a Square Array of Cylinders. *Physical Review E,* 

Handed Materials in Metallic Magnetic Granular Composites, *Physical Review B,*

Modulus and Mass Density, *Physical Review Letters,* Vol.99, No.9, (August 2007), pp.

Metamaterials with Negative Modulus, *Nature Materials,* Vol.5, No.6, (June 2006),

Handed Transmission-Line Lens, *Physical Review Letters*, Vol.92, No.11, (March

Behaviour of Multiphase Materials, *Journal of the Mechanics and Physics of Solids*,

Crystals at Infrared and Visible Frequencies, *Physical Review Letters,* Vol.96, No.22,

Metamaterials, *Physical Review B,* Vol.80, No.11, (September 2009), pp.115101, ISSN

Periodic Acoustic Composites, *Physical Review B,* Vol.60, No.17, (November 1999),

Scattering Medium for Acoustic Waves, *Physical Review Letters,* Vol.84, No.26, (June

Composites, *Physical Review Letters,* Vol.91, No.26, (December 2003), pp. 264302,

#### **7. Conclusion**

In this chapter, the effective medium properties of 2D elastic metamaterials have been reviewed. Unlike EM or acoustic metamaterials, the elastic metamaterial is in general anisotropic unless the lattice structure is a hexagon with isotropic scatterers. For the isotropic elastic metamaterial, the EMT may be derived from CPA. For the anisotropic metamaterial, the EMT may be obtained from the MST in conjunction with Christoffel's equation, or from the integration of eigenfields on the boundaries. EMT could greatly facilitate the design of new elastic metamaterials, such as rubber-coated water cylinder embedded in foam which gives rise to large negative bands for shear waves and a multimass locally resonant structure which results in both negative bands for longitudinal waves and super-anisotropic negative bands.

Elastic metamaterial opens a new research area. The experimental realization would be much more challenging and exciting. The generalization of EMT as well as the symmetry property to more complex lattice structures, such as rectangular lattices, would also be of interest as it will introduce even stronger anisotropy. Meanwhile, finding an EMT that can also treat the rotational modes is a challenging task. Such modes are normally excited at lower frequencies and form flat bands in the band structures.

#### **8. Acknowledgments**

The relevant work was supported by Hong Kong RGC Grant No. 605008, and start-up packages from KAUST and Soochow University.

#### **9. References**


In this chapter, the effective medium properties of 2D elastic metamaterials have been reviewed. Unlike EM or acoustic metamaterials, the elastic metamaterial is in general anisotropic unless the lattice structure is a hexagon with isotropic scatterers. For the isotropic elastic metamaterial, the EMT may be derived from CPA. For the anisotropic metamaterial, the EMT may be obtained from the MST in conjunction with Christoffel's equation, or from the integration of eigenfields on the boundaries. EMT could greatly facilitate the design of new elastic metamaterials, such as rubber-coated water cylinder embedded in foam which gives rise to large negative bands for shear waves and a multimass locally resonant structure which results in both negative bands for longitudinal waves

Elastic metamaterial opens a new research area. The experimental realization would be much more challenging and exciting. The generalization of EMT as well as the symmetry property to more complex lattice structures, such as rectangular lattices, would also be of interest as it will introduce even stronger anisotropy. Meanwhile, finding an EMT that can also treat the rotational modes is a challenging task. Such modes are normally excited at

The relevant work was supported by Hong Kong RGC Grant No. 605008, and start-up

Abramowitz, M. & Stegun, I. (Eds.). (1972). *Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables*, Dover, ISBN 0486612724 , New York Andrea, A. (2011). First-Principle Homogenization Theory for Periodic Metamaterials, *Physical Review B*, Vol.84, No.7, (Aug 2011), pp. 075153, ISSN 1098-0121 Andryieuski, A.; Malureanu, R. & Lavrinenko, A. (2009). Wave Propagation Retrieval

*Physical Review B,* Vol.80, No.19, (Nov 2009), pp.193101, ISSN 1098-0121 Baker-Jarvis, J.; Vanzura, E. & Kissick, W. (1990). Improved Technique for Determining

Berryman, J. (1980). Long-wavelength Propagation in Composite Elastic Media. 1. Spherical

Chern, R. & Chen, Y. (2009). Effective Parameters for Photonic Crystals with Large Dielectric

Chen, H.; Chan, C. & Sheng, P. (2010). Transformation Optics and Metamaterials, *Nature* 

*Materials,*Vol.9, No.5, (May 2010), pp.387-396, ISSN 1476-1122

Method for Metamaterials: Unambiguous Restoration of Effective Parameters,

Complex Permittivity with the Transmission/Reflection Method, *IEEE Transactions on microwave theory and techniques*, Vol.38, No.8, (August 1990), pp. 1096-1103, ISSN

Inclusions, *Journal of the Acoustical Society of America,* Vol.68, No.6, pp. 1809-1819,

Contrast, *Physical Review B,* Vol.80, No.7, (August 2009), pp. 075118, ISSN 0163-1829

**7. Conclusion** 

and super-anisotropic negative bands.

**8. Acknowledgments** 

0018-9480

ISSN 0001-4966

**9. References** 

lower frequencies and form flat bands in the band structures.

packages from KAUST and Soochow University.


Effective Medium Theories and Symmetry Properties of Elastic Metamaterials 167

Pendry, J.; Schurig, D. & Smith, D. (2006). Controlling Electromagnetic Fields, *Science*,

Pendry, J. & Smith, D. (2006). The Quest for the Superlens, *Scientific America,* Vol.295, No.1,

Royer, D. & Dieulesaint, E. (1999). *Elastic Waves in Solids I: Free and Guided Propagation* (First

Sheng, P. (2006). *Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena* (Second edition), Springer, ISBN-10 3-540-29155-5, Berlin Heidelberg Smith, D. & Pendry, J. (2006). Homogenization of Metamaterials by Field Averaging(Invited

Smith, D.; Schultz, S.; Markos, P. & Soukoulis, C. (2002). Determination of Effective

Sun, S.; Shui, S. & Zhou, L. (2009). Effective-Medium Properties of Metamaterials: A

Torquato, S. (1991). Random Heterogeneous Media: Microstructure and Improved Bounds

Torrent, D.; Hakansson, A.; Cervera, F. & Sanchez-Dehesa, J. (2006). Homogenization of

Veselago, V. (1968). The Electrodynamics of Substances with Simultaneously Negative

Wu, Y.; Li, J.; Zhang, Z. & Chan C. (2006). Effective Medium Theory for Magnetodielectric

Wu, Y.; Lai, Y. & Zhang, Z. (2007). Effective Medium Theory for Elastic Metamaterials in

Wu, Y.; Lai, Y. & Zhang, Z. (2011). Elastic Metamaterials with Simultaneously Negative

Wu, Y. & Zhang, Z. (2009). Dispersion Relations and Their Symmetry Properties of

Yang, Z.; Mei, J.; Yang, M.; Chan, N. & Sheng, P. (2008) Membrane-Type Acoustic

Zhang, S.; Xia, C. & Fang, N. (2011). Broadband Acoustic Cloak for Ultrasound Waves,

Values of ε and μ, *Soviet Physics Uspekhi,* Vol.10, No.4, pp. 509

Paper), *Journal of the Optical Society of America B-Optical Physics,*Vol.23, No.2 (March

Permittivity and Permeability of Metamaterials from Reflection and Transmission Coefficients, *Physical Review B,* Vol.65, No.19, (May 2002),

Quasimode Theory, *Physical Review E*, Vol.79, No.6, (June 2009), pp.066604, ISSN

on Effective Properties, *Applied Mechanics Reviews,* Vol.44, No.2, (Febrary 1991) pp.

Two-Dimensional Clusters of Rigid Rods in Air, *Physical Review Letters,* Vol.96,

Composites: Beyond the Long-Wavelength Limit, *Physical Review B*, Vol.74, No.8,

Two Dimensions. *Physical Review B,* Vol.76, No.20, (Novemver 2007), pp.025313,

Effective Shear Modulus and Mass Density, *Physical Review Letters,* Vol.107, No.10,

Electromagnetic and Elastic Metamaterials in Two Dimensions, *Physical Review B*,

Metamaterial with Negative Dynamic Mass, *Physical ReviewLetters,*Vol.101, No.20,

*Physical Review Letters,* Vol.106, No.2, (January 2011), pp. 024301, ISSN 0031-

Vol.312, No. 5781, (June 2006) pp.1780-1782, ISSN 0036-8075

(July 2006), pp. 60-67, ISSN 0036-8733

2006), pp.391-403, ISSN 0740-3224

pp.195104, ISSN 1098-012

37-76, ISSN 0003-6900

ISSN 1098-0121

9007

1063-651X

edition), Springer, ISBN 3540659323, Berlin

No.20, (May 2006), pp. 204302, ISSN 0031-9007

(August 2006), pp.085111, ISSN 1098-0121

(September 2011), pp. 105506, ISSN 0031-9007

(November 2008),pp.204301, ISSN 0031-9007

Vol.79, No.19, (May 2009), pp.195111, ISSN 1098-0121


Lamb, W.; Wood, D. & Ashcroft, N. (1980). Long-Wavelength Electromagnetic Propagation

Landau, L. & Lifshitz, E. (1986). *Theory of Elasticity* (Third edition), Butterworth-Heinemann,

Lee, S.; Park, C.; Seo, Y.; Wang, Z. & Kim, C. (2010). Composite Acoustic Medium with

Leonhardt, U. (2006). Optical Conformal Mapping, *Science*, Vol.312, No.5781, (June 2006),

Lezec, H.; Dionne, J. & Atwater, H. (2007). Negative Refraction at Visible Frequencies,

Li, J. & Chan C. (2004). Double-negative Acoustic Metamaterial, *Physical Review E,* Vol.70,

Liu, Z.; Chan, C.; Sheng, P.; Goertzen, A. & Page, J. (2000). Elastic Wave Scattering by

Liu, Z.; Zhang, X.; Mao, Y.; Zhu, Y.; Yang, Z.; Chan, C. & Sheng, P. (2000) Locally Resonant

Lu, M.; Feng, L. & Chen, Y. (2009). Phononic Crystals and Acoustic Metamaterials, *Materials* 

Mei, J.; Liu, Z.; Wen, W. & Sheng, P. (2006). Effective Mass Density of Fluid-solid

Mei, J.; Liu, Z.; Shi, J. & Tian, D. (2003). Theory for Elastic Wave Scattering by a Two-

Milton, G. & Willis, J. (2007). On Modification of Newton's Second Law and Linear

Ni, Q. & Cheng, J. (2005). Anisotropy of Effective Velocity for Elastic Wave Propagation in

Pendry, J. (2000). Negative Refraction Makes a Perfect Lens, *Physical Review Letters,* Vol.85,

Pendry, J.; Holden, A.; Robbins, D. & Stewart, W. (1999). Magnetism from Conductors and

*Techniques,* Vol.47, No.11, (November 1999), pp. 2075-2084, ISSN 0018-9480 Pendry, J.; Holden, A.; Stewart, W. & Youngs, I. (1996). Extremely Low Frequency Plasmons

*Today,* Vol.12, No.12, (December 2009), pp. 34-42, ISSN: 1369-7021

Periodic Structures of Spherical Objects: Theory and Experiment, *Physical Review B,*

Sonic Materials, *Science,* Vol.289, No.5485, (September 2000), pp. 1734-1736, ISSN

Composites, *Physical Review Letters,* Vol.96, No.2, (January 2006), pp. 024301, ISSN

Dimensional Periodical Array of Cylinders: An Ideal Approach for Band-Structure Calculations, *Physical Review B,* Vol.67,No.24, (June 2003), pp. 245107, ISSN 1098-

Continuum Elastodynamics,*Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences,* Vol.463,No.2079,(March 2007), pp.855-880,

Two-Dimensional Phononic Crystals at Low Frequencies, *Physical Review B,* Vol.72,

Enhanced Nonlinear Phenomena, *IEEE Transactions on Microwave Theory and* 

in Metallic Mesostructures, *Physical Review Letters,* Vol.76, No.25, (June 1996), pp.

*Science*, Vol.361, No.5823, (April 2007), pp. 430-432, ISSN 0036-8075

2266, ISSN 0163-1829

0036-8075

0031-9007

ISSN1364-5021

4773-4776, ISSN 0031-9007

0121

ISBN 0 7506 2633 X, Oxford

pp.1777-1780, ISSN 0036-8075

No.5, (February 2010), pp. 054301, ISSN 0031-9007

No.5,(November 2004),pp.055602, ISSN 1063-651X

No.1, (July 2005), pp. 014305, ISSN 0163-1829

No.18, (October 2000), pp. 3966-3969, ISSN 0031-9007

Vol.62, No.4, (July 2000), pp. 2446-2457, ISSN 1098-0121

in Heterogeneous Media, *Physical Review B,* Vol.21, No.6, (March 1980), pp.2248-

Simultaneously Negative Density and Modulus, *Physical Review Letters,* Vol.104,


**Section 2** 

**The Performance of Metamaterials** 

