**Applications of Nano-Scale Plasmonic Structures in Design of Stub Filters — A Step Towards Realization of Plasmonic Switches**

Hassan Kaatuzian and Ahmad Naseri Taheri

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59877

#### **1. Introduction**

As electronic devices and circuits were shrinking to the nano-scale chips, some drawbacks hindered the reaching to the speed higher than tens of Giga Hertz, such as the higher power consumption, delay, and interference of the signals. However, the need for transmitting the huge amount of data over communication networks urged everyone in this area of technology to find an alternative for pure electronic devices. Despite rapidly developing the photonics technology, in practice the implementation of the photonic counterparts of the electronic devices encountered many problems such as inability to become integrated. The size of photonic devices halted on the half the wavelength of the operating signal due to the Diffrac‐ tion limit of light.

In recent years, Surface Plasmon Polaritons (SPPs) have been considered as one of the most promising ways to overcome the diffraction limits of photonic devices. The field of manipu‐ lating the SPPs as the carrier of the signal is known as Plasmonics. In plasmonics, the advan‐ tages of electronics such as nano-scale component design are joint together with the benefits of photonics. Therefore, plasmonics shapes a key part of the fascinating field of nanophotonics, which discovers how electromagnetic fields can be confined over dimensions on the order of or smaller than the wavelength. It is constructed based on interaction of electromagnetic radiation and conduction electrons at metallic interfaces or in small metallic nanostructures, leading to enhance the confinement of optical field in a sub-wavelength dimension.

The strong interaction between microscopic metal particles and light has been utilized for thousands of years, the Lycurgus Roman Cup dating from 4'th century A.D. [1], brightly colored stained-glass window panels by annealing metallic salts in transparent glass [1]. The practical instructions for this ancient "nanofabrication" technique survive from as early as the

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8th century A.D. [1]. From the beginning of the 20th century many phenomena related to the unusual optical properties of particles and spheres were explained; the blue color of the sky in terms of a simple derivation of the scattering power of spheres by Rayleigh [2], the bright color of metal glass by Garnett [3], presentation of a general formulation for the scattering of light from spherical surface by Gustav Mie [4]. In 1957, Ritchie [5] described surface plasmons in thin films in terms of electron energy loss spectroscopy. Otto [6], Kretschmann and Raether [7] explained the procedures for exciting surface plasmons on thin films optically in 1968. In 1974, the enhancement of Raman scattering from molecules influenced by the enhanced local fields at a rough metal surface was first observed by Fleischmann et al. [8].

In recent years, the applications of the surface plasmons are extended in optical integrated circuits and create a rapid developing field of photonics known as Plasmonics. Many re‐ searches are being carried out on developing plasmonic structures. Plasmonic switches and modulators are the most important components used for light routing and switching in the rapidly developing area of broadband optical communications and designing ultra-high speed switching devices, where sub-femtosecond interaction of light with matter is our main design concern [9]. Numerous plasmonic devices and components are introduced and huge of research are published. By exciting of the surface plasmons at the boundary of dielectric and metallic materials, an optical field can be confine into a subwavelength dimension [10]. A plasmonic circuit transforms the light into surface plasmons at the same frequency but reduced dimensions, in addition to guidance that is more flexible and highest speed.

In the next section of this chapter, Metal-Optics Electromagnetism is discussed. In section 3, wave analysis is used for explaining the propagation of surface plasmon polaritons. Section 4 is dedicated to investigate the various schemes of plasmonic guiding and their characteristics. In section 5, the modeling methods of plasmonic waveguides are introduced, and in section 6, a case study of plasmonic application in designing a switch is presented.

#### **2. Metal-optics electromagnetism**

Surface plasmons polaritons (SPPs) are surface electromagnetic modes that propagate at the interface of a dielectric with real electric permittivity ε1 and a metal with permittivity ε2(ω)<0 as shown in figure 1.

**Figure 1.** Electric and magnetic field distributions and charge oscillations at the dielectric/metal interface. SPPs propa‐ gate along the x-direction.

The complex permittivity or complex dielectric function can be expressed as [11]:

$$
\hat{\mathbf{z}} = \hat{\mathbf{z}}^{\cdot} + i\hat{\mathbf{z}}^{\cdot} \tag{1}
$$

where *ε* ' is the real part and *ε* '' is th imaginary part of the permitivitty. In addition, the response of the material to the incoming optical field is expressed as the complex refractive index as [12]:

$$N = n + ik \tag{2}$$

where *n* and *k* are the real and imaginary part of the refractive index. The equations (1) and (2) are related to each other by:

$$
\vec{\omega}' = \mathfrak{n}^2 - \mathfrak{k}^2 \tag{3}
$$

$$
\hat{\boldsymbol{\varepsilon}}^{\cdot} = \mathbf{2}nk\tag{4}
$$

These parameters are known as optical constants of the material however, in many of the materials they change with the frequency of the incident optical field. Especially in metals, the dielectric parameters are strongly dependent to the optical frequency. In order to use the permittivity of metals in the calculations of the spectrometry some mathematical models are introduced, such as Lorentz model, Lorentz-Drude model andExtended Drude model.

#### **2.1. Lorentz model**

8th century A.D. [1]. From the beginning of the 20th century many phenomena related to the unusual optical properties of particles and spheres were explained; the blue color of the sky in terms of a simple derivation of the scattering power of spheres by Rayleigh [2], the bright color of metal glass by Garnett [3], presentation of a general formulation for the scattering of light from spherical surface by Gustav Mie [4]. In 1957, Ritchie [5] described surface plasmons in thin films in terms of electron energy loss spectroscopy. Otto [6], Kretschmann and Raether [7] explained the procedures for exciting surface plasmons on thin films optically in 1968. In 1974, the enhancement of Raman scattering from molecules influenced by the enhanced local

In recent years, the applications of the surface plasmons are extended in optical integrated circuits and create a rapid developing field of photonics known as Plasmonics. Many re‐ searches are being carried out on developing plasmonic structures. Plasmonic switches and modulators are the most important components used for light routing and switching in the rapidly developing area of broadband optical communications and designing ultra-high speed switching devices, where sub-femtosecond interaction of light with matter is our main design concern [9]. Numerous plasmonic devices and components are introduced and huge of research are published. By exciting of the surface plasmons at the boundary of dielectric and metallic materials, an optical field can be confine into a subwavelength dimension [10]. A plasmonic circuit transforms the light into surface plasmons at the same frequency but reduced

In the next section of this chapter, Metal-Optics Electromagnetism is discussed. In section 3, wave analysis is used for explaining the propagation of surface plasmon polaritons. Section 4 is dedicated to investigate the various schemes of plasmonic guiding and their characteristics. In section 5, the modeling methods of plasmonic waveguides are introduced, and in section 6,

Surface plasmons polaritons (SPPs) are surface electromagnetic modes that propagate at the interface of a dielectric with real electric permittivity ε1 and a metal with permittivity ε2(ω)<0

**Figure 1.** Electric and magnetic field distributions and charge oscillations at the dielectric/metal interface. SPPs propa‐

fields at a rough metal surface was first observed by Fleischmann et al. [8].

dimensions, in addition to guidance that is more flexible and highest speed.

a case study of plasmonic application in designing a switch is presented.

**2. Metal-optics electromagnetism**

as shown in figure 1.

94 Photonic Crystals

gate along the x-direction.

In plasmonic, the optical properties of material can be expressed as a classical physical model of the microscopic structure [3], in which the charge carriers are assumed as damped harmonic oscillators subjected to incident electromagnetic field as driving forces. In the Lorentz model a charge carrier is supposed to have a mass of **m**, charge **e** and displacement from equilibrium **x**. The force on this particle as a linear spring force is *F* = *K*.*x*, the velocity dependent damping is *F* =*b*.*x*˙ and the driving force by any incident light is *F* =*eE*. Therfore, the equation of motion is [12, 13]:

$$a\ddot{\mathbf{x}} + b\dot{\mathbf{x}} + K\mathbf{x} = e\mathbf{E} \tag{5}$$

normalizing the above equation by mass:

$$
\ddot{\mathbf{x}} + \mathbf{y} + \alpha\_0^2 \mathbf{x} = \frac{e}{m} \mathbf{E} \tag{6}
$$

where *ω*<sup>0</sup> <sup>2</sup> = *K* / *m* and *γ* =*b* / *m*. By solving the equation (6) as a time-harmonic and substitution of *x*˙ ↔ −*iωx* and *x* ¨ ↔ −*ω* <sup>2</sup> *x* :

$$\chi = \frac{\left(\frac{e}{m}\right)E}{o\rho\_0^2 - \alpha^2 - i\rho\alpha} \tag{7}$$

Supposing N particles per volume V and dipole *p* =*ex* for each particle, the polarization is *<sup>P</sup>* =( *<sup>N</sup> <sup>V</sup>* )*p*. Therfore, equation (7) can be expressed:

$$\mathbf{P} = \frac{\alpha\_p^2}{\alpha\_0^2 - \alpha^2 - i\gamma\alpha} \varepsilon\_0 \mathbf{E} \tag{8}$$

where *ωp* is the plasma frequency and is defined as:

$$
\rho \alpha\_p^2 = \frac{\left(\frac{N}{V}\right)\varepsilon^2}{m\varepsilon\_0} \tag{9}
$$

From the Maxwell's equation, we have:

$$\mathbf{z} = \mathbf{1} + \mathbf{y} \tag{10}$$

$$\mathbf{P} = \varepsilon\_0 \chi \mathbf{E} \tag{11}$$

Therefore, from the equations (8), (10) and (11) the Lorentz model for the dielectric function is derived:

$$\varepsilon\_{Lorentz} \left( \alpha \right) = 1 + \frac{\alpha\_p^2}{\alpha\_0^2 - \alpha^2 - i\gamma\alpha} \tag{12}$$

#### **2.2. Multi-oscillator Drude-Lorentz model**

Here, the model is the superposition of j+1 individual oscillators [12]:

$$\mathcal{L}\_{Drade\text{-}Lorentz}\left(\alpha\right) = 1 - \frac{f\_0 \alpha\_{p,0}^2}{\alpha^2 + \text{i}\gamma\_0 \alpha} + \sum\_{j=1}^{l\_{\text{max}}} \frac{f\_j \alpha\_{p,j}^2}{\alpha\_j^2 - \alpha^2 - \text{i}\gamma\_j \alpha} \tag{13}$$

where *f <sup>j</sup>* ' s are the oscillator strengths, *ωp*, *<sup>j</sup>* ' s are the plasma frequencies, *γ<sup>j</sup>* ' s are damping rates and *ω <sup>j</sup>* ' *s* are the oscillator frequencies. The second term of equation (13) is the Drude term and represents the free electrons, which are not under the restoring force. Using 5 terms of the Lorentz model constructs an excellent approximation of optical properties of metals and many semiconductors in the visible spectrum [12].

### **3. Wave approach for the propagation of the surface plasmon polaritons**

Surface plasmon polaritons are electromagnetic waves propagating at the interface between a dielectric and a metal evanescently bounded in the perpendicular direction [4]. These electro‐ magnetic surface waves are created from the coupling of the optical fields to oscillations of the metal's electron plasma. Based on the dispersion relation and the spatial field distribution, the surface plasmons are described quantitatively.

#### **3.1. Propagating the SPPs at a single metal-insulator interface**

#### *3.1.1. Deriving the dispersion relation*

2 2 0

2 2 2 0

*p i*

w

0

2

*p*

e

w

w w gw

*x* w w gw*i*

*<sup>V</sup>* )*p*. Therfore, equation (7) can be expressed:

where *ωp* is the plasma frequency and is defined as:

From the Maxwell's equation, we have:

*<sup>P</sup>* =( *<sup>N</sup>*

96 Photonic Crystals

derived:

where *f <sup>j</sup>* '

and *ω <sup>j</sup>* '

*e m*

*E*

Supposing N particles per volume V and dipole *p* =*ex* for each particle, the polarization is

e

2

0

e

 c

Therefore, from the equations (8), (10) and (11) the Lorentz model for the dielectric function is

2 2 0 1 *<sup>p</sup> Lorentz i*

w

<sup>0</sup> *P E* = e c

( ) <sup>2</sup>

w w gw

( ) 2 2


w

=- +

'

0 ,0 , 2 2 2 0 1

=

*s* are the oscillator frequencies. The second term of equation (13) is the Drude term

*i i*

 w w gw

*j j j*

s are the plasma frequencies, *γ<sup>j</sup>*

 w

*max j p j pj*

*f f*

e

**2.2. Multi-oscillator Drude-Lorentz model**

e

*Drude Lorentz*

s are the oscillator strengths, *ωp*, *<sup>j</sup>*

 w

Here, the model is the superposition of j+1 individual oscillators [12]:

1

w gw

 w *N e V m*

æ ö ç ÷

<sup>=</sup> - - *P E* (8)

è ø <sup>=</sup> (9)

= +1 (10)

= + - - (12)

+ -- å (13)

'

s are damping rates

(11)

(7)

æ ö ç ÷ è ø <sup>=</sup> - -

> By solving the Maxwell's equation at a single metal-insulator interface, the wave equation is produced, in which for the transverse magnetic (TM or p) modes is [14]:

$$\frac{\partial^2 H\_y}{\partial \mathbf{z}^2} + \left(k\_0^2 \boldsymbol{\varepsilon} - \boldsymbol{\beta}^2\right) H\_y = 0 \tag{14}$$

and for transverse electric (TE or s) modes is:

$$\frac{\partial^2 E\_y}{\partial \mathbf{z}^2} + \left(k\_0^2 \mathbf{z} - \boldsymbol{\beta}^2\right) E\_y = 0 \tag{15}$$

where *k*<sup>0</sup> =*ω* / *c*0 is the wave vector of the propagating wave in vacuum. Now, we should consider a simple flat boundary between an insulator (z>0) with a real positive permittivity *ε*<sup>2</sup> and a metal (z<0) with a complex permittivity *ε*1(*ω*) (for metals *Re*{*ε*<sup>1</sup> (*ω*)} <0). Supposing the condition of propagating wave bounded to the interface with evanescent falloff in z-direction, the solutions for TM waves in z > 0 is [14]:

$$H\_y\left(z\right) = A\_2 e^{i\theta x} e^{-k\_2 z} \tag{16}$$

$$E\_x\left(z\right) = \mathrm{i}A\_2 \frac{1}{\alpha \varepsilon\_0 \varepsilon\_2} k\_2 e^{i\beta \varepsilon} e^{-k\_2 z} \tag{17}$$

$$E\_z\left(z\right) = -A\_1 \frac{\beta}{\alpha \varepsilon\_0 \varepsilon\_2} k\_2 e^{i\beta \varepsilon} e^{-k\_2 z} \tag{18}$$

and for z < 0 is:

$$H\_y\left(z\right) = A\_\mathbf{i} e^{i\boldsymbol{\beta}\cdot\mathbf{x}} e^{k\_\mathbf{i}z} \tag{19}$$

$$E\_x\left(z\right) = -iA\_1 \frac{1}{\alpha \varepsilon\_0 \varepsilon\_1} k\_1 e^{i\beta x} e^{k\_1 z} \tag{20}$$

$$E\_z\left(z\right) = -A\_1 \frac{\beta}{\alpha \varepsilon\_0 \varepsilon\_1} k\_2 e^{i\beta \varepsilon} e^{k\_1 z} \tag{21}$$

where *ki* ≡*kz*,*<sup>i</sup>* is the component of the wave vector at z-direction. The reciprocal value of the *ki* , *z* ^ =1 / <sup>|</sup>*kz* |, is evanescent decay length of the fields in the perpendicular of the propagation [14]. The continuity condition of *Hy* and *εEz* at the interface implies that *A*<sup>1</sup> = *A*2 and

$$\frac{k\_2}{k\_1} = -\frac{\varepsilon\_2}{\varepsilon\_1} \tag{22}$$

Because at the interface of insulator and metal *Re*{*ε*2} >0 and *Re*{*ε*1} <0 the propagating waves confines at the surface. Based on equation (14) for TM waves at metal and insulator [14],

$$k\_1 = \beta^2 - k\_0^2 \varepsilon\_1 \tag{23}$$

$$k\_2 = \mathcal{J}^2 - k\_0^2 \varepsilon\_2 \tag{24}$$

Therefore, by combining the equations (22), (23) and (24), the dispersion relation of the SPPs propagating at the metal-insulator yields

$$\mathcal{B} = k\_0 \sqrt{\frac{\varepsilon\_1 \varepsilon\_2}{\varepsilon\_1 + \varepsilon\_2}}\tag{25}$$

#### *3.1.2. Forbidden modes for surface plasmons*

Based on equation (15), the field components for TE modes are

Applications of Nano-Scale Plasmonic Structures in Design of Stub Filters — A Step Towards Realization… http://dx.doi.org/10.5772/59877 99

$$E\_y\left(z\right) = A\_2 e^{i\beta x} e^{-k\_2 z} \tag{26}$$

$$i\,H\_{\times}\left(\mathbf{z}\right) = -iA\_2 \frac{1}{\alpha\mu\_0} k\_2 e^{i\beta\varepsilon} e^{-k\_2 z} \tag{27}$$

$$H\_z\left(z\right) = A\_2 \frac{\beta}{\alpha \mu\_0} k\_2 e^{i\beta \ge} e^{-k\_2 z} \tag{28}$$

for z > 0, and

( ) <sup>2</sup> 1 2 0 2

> ( ) <sup>1</sup> 1 *i x k z H z Ae e <sup>y</sup>* b

( ) <sup>1</sup> 1 1 0 1

we e

( ) <sup>1</sup> 1 2 0 1

where *ki* ≡*kz*,*<sup>i</sup>* is the component of the wave vector at z-direction. The reciprocal value of the *ki*

^ =1 / <sup>|</sup>*kz* |, is evanescent decay length of the fields in the perpendicular of the propagation [14].

we e

2 2 1 1

e

e

Because at the interface of insulator and metal *Re*{*ε*2} >0 and *Re*{*ε*1} <0 the propagating waves confines at the surface. Based on equation (14) for TM waves at metal and insulator [14],

> 2 2 <sup>1</sup> 0 1 *k k* = b

> 2 2 <sup>2</sup> 0 2 *k k* = b

 e

 e

Therefore, by combining the equations (22), (23) and (24), the dispersion relation of the SPPs

1 2

e e

e e

1 2

0

*k*

b

Based on equation (15), the field components for TE modes are

propagating at the metal-insulator yields

*3.1.2. Forbidden modes for surface plasmons*

*<sup>z</sup> E z A ke e* b

The continuity condition of *Hy* and *εEz* at the interface implies that *A*<sup>1</sup> = *A*2 and

*k k*

*<sup>x</sup> E z iA k e e*

we e

*<sup>z</sup> E z A ke e* b

and for z < 0 is:

98 Photonic Crystals

*z*

*i x k z*


= (19)

= - (20)

= - (21)

= - (22)

(23)

(24)

<sup>=</sup> <sup>+</sup> (25)

,

b

1 *i x k z*

b

*i x k z*

b

$$E\_y\left(z\right) = A\_\text{r}e^{i\beta\chi}e^{k\_1z} \tag{29}$$

$$iH\_{\chi} \left( z \right) = iA\_1 \frac{1}{\alpha \mu\_0} k\_1 e^{i\beta \chi} e^{k\_1 z} \tag{30}$$

$$H\_z\left(\mathbf{z}\right) = A\_1 \frac{\beta}{\alpha \mu\_0} e^{i\beta \mathbf{x}} e^{k\_z z} \tag{31}$$

In calculations of the equations (26) to (31), the following equations are used:

$$H\_x = i \frac{1}{\alpha \mu\_0} \frac{\partial E\_y}{\partial z} \tag{32}$$

$$H\_z = \frac{\beta}{o\mu\_0} E\_y \tag{33}$$

From the continuity of *Ey* and *Hx* at the interface

$$A\_1 \left(k\_1 + k\_2\right) = 0\tag{34}$$

and because of the confinement condition requires *Re*{*k*1} >0 and *Re*{*k*2} >0, hence *A*<sup>1</sup> =0 and *A*<sup>2</sup> =0 ; therefore, there are no TE modes at the surface. That means, the surface plasmon polaritons propagate only in TM polarization [14].

#### *3.1.3. Propagation Length of SPPs*

An important parameter in comparing plasmonic waveguide structures is the **propagation length** of a supported mode. As a surface plasmon propagates along the surface, it quickly loses its energy to the metal due to absorption. The intensity of the surface plasmon decays with the square of the electric field, so at a distance x, the intensity has decreased by a factor of exp(2*Im*(*βz*)*z*) [15]. The propagation length is defined as the distance for the surface plasmon to decay by a factor of 1/e. This condition is satisfied at a length

$$L\_{\rm SPP} = \frac{1}{2\operatorname{Im}\left(\mathcal{J}\_z\right)}\tag{35}$$

The typical value for propagation length in plasmonic metal-insulator waveguides is up to 100µm in visible regime [14].

#### **4. Plasmonic waveguide schemes**

The fundamental element in integrated plasmonic circuitry, as in photonics, is the waveguide. Because the necessity of existence of metal layer and various structural possibilities of using metal layer, many of schemes are developed in the recent years [11]. Each of the schemes has some benefits and drawbacks, in which make a trade-off between their characteristics. Two main characteristics of a waveguide are the confinement area of the mode and the propagation length. This features of the plasmonic waveguides are in opposite and there are a trade-off between them.

**Figure 2.** Optical field distribution at cross section of insulator-metal waveguide

7

In this section, some of the well-known types of plasmonic waveguides structures will be briefly introduced. In each case, the advantage(s) and weakness(s) of the waveguide are investigated.

#### **4.1. Insulator-Metal (IM) waveguide**

*3.1.3. Propagation Length of SPPs*

100 Photonic Crystals

100µm in visible regime [14].

between them.

**4. Plasmonic waveguide schemes**

An important parameter in comparing plasmonic waveguide structures is the **propagation length** of a supported mode. As a surface plasmon propagates along the surface, it quickly loses its energy to the metal due to absorption. The intensity of the surface plasmon decays with the square of the electric field, so at a distance x, the intensity has decreased by a factor of exp(2*Im*(*βz*)*z*) [15]. The propagation length is defined as the distance for the surface plasmon

> ( ) 1

The typical value for propagation length in plasmonic metal-insulator waveguides is up to

The fundamental element in integrated plasmonic circuitry, as in photonics, is the waveguide. Because the necessity of existence of metal layer and various structural possibilities of using metal layer, many of schemes are developed in the recent years [11]. Each of the schemes has some benefits and drawbacks, in which make a trade-off between their characteristics. Two main characteristics of a waveguide are the confinement area of the mode and the propagation length. This features of the plasmonic waveguides are in opposite and there are a trade-off

*Im* b

*z*

<sup>=</sup> (35)

2 *SPP*

*L*

to decay by a factor of 1/e. This condition is satisfied at a length

**Figure 2.** Optical field distribution at cross section of insulator-metal waveguide

The simplest structure for guiding the plasmon polaritons is the Insulator-Metal (IM) wave‐ guide (figure 2). This scheme consists a thin film metal coated on a simple insulator strip waveguide [11]. The silicon core of the waveguide has dimension about 300nm×300nm and the cladding of the waveguide is the air. The metal cap has a 50nm thickness and can be gold or silver. The structure placed on a SiO2 substrate. The propagation length of this fundamental scheme is 2µm at 1550nm [11]. In this structure, a relative good balance exists between propagation length and confinement. In addition, the material used in fabrication allows feasibility of integrated plasmonic circuitry [11]. 1 2 **Fig. 2.** Optical field distribution at cross section of insulator-metal waveguide 3 In this section, some of the well-known types of plasmonic waveguides structures will be 4 briefly introduced. In each case, the advantage(s) and weakness(s) of the waveguide are 5 investigated. 6 **4.1. Insulator-Metal (IM) waveguide**  7 The simplest structure for guiding the plasmon polaritons is the Insulator-Metal (IM)

8 waveguide (figure 2). This scheme consists a thin film metal coated on a simple insulator

#### **4.2. Dielectric-loaded SPP waveguide** 9 strip waveguide [11]. The silicon core of the waveguide has dimension about 300nm×300nm 10 and the cladding of the waveguide is the air. The metal cap has a 50nm thickness and can be

The dielectric-loaded SPP waveguides (DLSPPWs) are formed by placing a polymethylme‐ thacrylate (PMMA) ridge with a cross section of 500 nm in width by 600nm in height as the dielectric material on top of a 65nm thick and 3µm wide gold strip [16] (figure 3a). This scheme provides a full confinement in the plane perpendicular to the propagation direction. This geometry is one of the most popular for plasmonic waveguides because PMMA can work as both resist and the dielectric core for the DLSPPW [16]. There is a convenient and simple process to fabricate plasmonic devices based on DLSPPW using deep-ultraviolet (UV) lithography for those devices with larger dimensions working at telecom wavelengths or standard E-beam lithography (EBL) for those working in the near-infrared. In addition, due to the physical dimension and therefore the mode size increase, this structure has slightly better propagation length for SPP's going from 5µm to 25µm. 11 gold or silver. The structure placed on a SiO2 substrate. The propagation length of this 12 fundamental scheme is 2m at 1550nm [11]. In this structure, a relative good balance exists 13 between propagation length and confinement. In addition, the material used in fabrication 14 allows feasibility of integrated plasmonic circuitry [11]. 15 **4.2. Dielectric-loaded SPP waveguide**  16 The dielectric-loaded SPP waveguides (DLSPPWs) are formed by placing a 17 polymethylmethacrylate (PMMA) ridge with a cross section of 500 nm in width by 600nm in 18 height as the dielectric material on top of a 65nm thick and 3μm wide gold strip [16] (figure 19 3a). This scheme provides a full connement in the plane perpendicular to the propagation 20 direction. This geometry is one of the most popular for plasmonic waveguides because 21 PMMA can work as both resist and the dielectric core for the DLSPPW [16]. There is a 22 convenient and simple process to fabricate plasmonic devices based on DLSPPW using 23 deep-ultraviolet (UV) lithography for those devices with larger dimensions working at

24 telecom wavelengths or standard E-beam lithography (EBL) for those working in the near-25 infrared. In addition, due to the physical dimension and therefore the mode size increase,

**Figure 3.** (a) Schematic representation of the DLSPPW cross section [16]. (b) Mode distribution of DLSPPW of optimal configuration.

Compared with other waveguide schemes, DLSPPWs are well-matched with different dielectrics and have a rather good trade-off between mode confinement and propagation distance [16]. These characteristics make them suitable for realization of dynamic components by utilizing of material (e.g., thermo- and electro-optic) effects, while strong mode confinement and long propagation distances are required for realization of compact and complex plasmonic circuits [16]. 1 this structure has slightly better propagation length for SPP's going from 5µm to 25µm. 2 **Fig. 3.** (a) Schematic representation of the DLSPPW cross section [16]. (b) Mode distribution 3 of DLSPPW of optimal conguration. 4 Compared with other waveguide schemes, DLSPPWs are well-matched with different 5 dielectrics and have a rather good trade-off between mode connement and propagation

6 distance [16]. These characteristics make them suitable for realization of dynamic

8 Photonic Crystals

#### *4.2.1. Long-Range Dielectric-Loaded Surface Plasmon Polaritons waveguides (LR-DLSPPs)* 7 components by utilizing of material (e.g., thermo- and electro-optic) effects, while strong 8 mode connement and long propagation distances are required for realization of compact

17 connement to the dielectric ridge and underlying dielectric lm [17].

The long-range plasmonic waveguide is, unlike previous studies of DLSPPWs, based on a metal film with finite width [17], i.e., a metal strip, and can thus be considered a hybrid plasmonic waveguide (discussed later). The LR-DLSPPW configuration consists of a dielectric ridge placed on top of a thin metal strip, which is supported by a dielectric film (figure 4a). The entire structure is supported by a low-index substrate that ensures mode confinement to the dielectric ridge and underlying dielectric film [17]. 9 and complex plasmonic circuits [16]. 10 **4.2.1. Long-range dielectric-loaded surface plasmon polaritons waveguides (LR-**11 **DLSPPs)** 12 The long-range plasmonic waveguide is, unlike previous studies of DLSPPWs, based on a 13 metal lm with nite width [17], i.e., a metal strip, and can thus be considered a hybrid 14 plasmonic waveguide (discussed later). The LR-DLSPPW conguration consists of a 15 dielectric ridge placed on top of a thin metal strip, which is supported by a dielectric lm 16 (figure 4a). The entire structure is supported by a low-index substrate that ensures mode

19 stripe deposited on an underlying dielectric layer supported by a low-index glass substrate, 20 (b) optical field distribution in the cross-section of the LR-DLSPPW [17]. 21 Because of varying the thickness of the PMMA layer and hence, balancing the mode field on **Figure 4.** (a) Layout of the LR-DLSPPW structure, with a dielectric ridge on top of a thin metal stripe deposited on an underlying dielectric layer supported by a low-index glass substrate, (b) optical field distribution in the cross-section of the LR-DLSPPW [17].

18 **Fig. 4.** (a) Layout of the LR-DLSPPW structure, with a dielectric ridge on top of a thin metal

22 either side of the metal strip (figure 4b), this structure minimizes the losses and increase the 23 propagation length to L = 3100m, while retaining strong mode connement [17]. 24 **4.3. Hybrid Plasmonic Waveguide**  25 As a solution to the issue of propagation loss of SPPs, a new waveguide known as hybrid Because of varying the thickness of the PMMA layer and hence, balancing the mode field on either side of the metal strip (figure 4b), this structure minimizes the losses and increase the propagation length to L = 3100µm, while retaining strong mode confinement [17].

#### 26 plasmonic waveguide (HPWG) is presented [18]. Figure 5a shows the cross section of a two 27 dimensional HPWG. It consists of a high index region (silicon) disjointed from a silver 28 surface by a low index layer (SiO2). The close area of the silver-silica interface and the silicon **4.3. Hybrid plasmonic waveguide**

29 slab results in coupling of the SPP mode and dielectric waveguide mode supported by these 30 two structures. Figure 5b shows the resulting hybrid mode. The guide also supports a As a solution to the issue of propagation loss of SPPs, a new waveguide known as hybrid plasmonic waveguide (HPWG) is presented [18]. Figure 5a shows the cross section of a two dimensional HPWG. It consists of a high index region (silicon) disjointed from a silver surface by a low index layer (SiO2). The close area of the silver-silica interface and the silicon slab results in coupling of the SPP mode and dielectric waveguide mode supported by these two structures. Figure 5b shows the resulting hybrid mode. The guide also supports a conventional TE mode, which is shown in figure 5c. The mode sizes for both TE and TM modes are com‐ parable in this case and are very similar to mode size achievable in case of silicon waveguide. HPWG offers a number of benefits [18]: it has an improved compromise between loss and confinement compared to purely plasmonic waveguides, and is compatible with silicon on insulator technology. The power of the TE and TM modes in the HPWG are concentrated in two different layers, therefore their guiding characteristics can be controlled in different manners by altering the material properties, and waveguide dimensions of the layers [18].

**Figure 5.** (a) Schematic of a hybrid waveguide. (b) and (c) power density profile for the TM and TE modes respectively for waveguide dimensions are w = 350 nm, t = 200 nm, h = 150nm, d = 150 nm and T = 2µm. (d) power density profile for the TM mode for w = 350 nm, t = 200 nm, h = 150 nm, d = 45 nm and T = 2µm. Wavelength of light is 1550 nm [18].

#### **4.4. Metal-insulator-metal waveguide**

distance [16]. These characteristics make them suitable for realization of dynamic components by utilizing of material (e.g., thermo- and electro-optic) effects, while strong mode confinement and long propagation distances are required for realization of compact and complex plasmonic

4 Compared with other waveguide schemes, DLSPPWs are well-matched with different 5 dielectrics and have a rather good trade-off between mode connement and propagation 6 distance [16]. These characteristics make them suitable for realization of dynamic 7 components by utilizing of material (e.g., thermo- and electro-optic) effects, while strong 8 mode connement and long propagation distances are required for realization of compact

1 this structure has slightly better propagation length for SPP's going from 5µm to 25µm. 2 **Fig. 3.** (a) Schematic representation of the DLSPPW cross section [16]. (b) Mode distribution

8 Photonic Crystals

The long-range plasmonic waveguide is, unlike previous studies of DLSPPWs, based on a metal film with finite width [17], i.e., a metal strip, and can thus be considered a hybrid plasmonic waveguide (discussed later). The LR-DLSPPW configuration consists of a dielectric ridge placed on top of a thin metal strip, which is supported by a dielectric film (figure 4a). The entire structure is supported by a low-index substrate that ensures mode confinement to

10 **4.2.1. Long-range dielectric-loaded surface plasmon polaritons waveguides (LR-**

12 The long-range plasmonic waveguide is, unlike previous studies of DLSPPWs, based on a 13 metal lm with nite width [17], i.e., a metal strip, and can thus be considered a hybrid 14 plasmonic waveguide (discussed later). The LR-DLSPPW conguration consists of a 15 dielectric ridge placed on top of a thin metal strip, which is supported by a dielectric lm 16 (figure 4a). The entire structure is supported by a low-index substrate that ensures mode

18 **Fig. 4.** (a) Layout of the LR-DLSPPW structure, with a dielectric ridge on top of a thin metal 19 stripe deposited on an underlying dielectric layer supported by a low-index glass substrate,

**Figure 4.** (a) Layout of the LR-DLSPPW structure, with a dielectric ridge on top of a thin metal stripe deposited on an underlying dielectric layer supported by a low-index glass substrate, (b) optical field distribution in the cross-section of

21 Because of varying the thickness of the PMMA layer and hence, balancing the mode field on 22 either side of the metal strip (figure 4b), this structure minimizes the losses and increase the 23 propagation length to L = 3100m, while retaining strong mode connement [17].

Because of varying the thickness of the PMMA layer and hence, balancing the mode field on either side of the metal strip (figure 4b), this structure minimizes the losses and increase the

25 As a solution to the issue of propagation loss of SPPs, a new waveguide known as hybrid 26 plasmonic waveguide (HPWG) is presented [18]. Figure 5a shows the cross section of a two 27 dimensional HPWG. It consists of a high index region (silicon) disjointed from a silver 28 surface by a low index layer (SiO2). The close area of the silver-silica interface and the silicon 29 slab results in coupling of the SPP mode and dielectric waveguide mode supported by these 30 two structures. Figure 5b shows the resulting hybrid mode. The guide also supports a

As a solution to the issue of propagation loss of SPPs, a new waveguide known as hybrid plasmonic waveguide (HPWG) is presented [18]. Figure 5a shows the cross section of a two dimensional HPWG. It consists of a high index region (silicon) disjointed from a silver surface by a low index layer (SiO2). The close area of the silver-silica interface and the silicon slab results in coupling of the SPP mode and dielectric waveguide mode supported by these two structures. Figure 5b shows the resulting hybrid mode. The guide also supports a conventional TE mode, which is shown in figure 5c. The mode sizes for both TE and TM modes are com‐ parable in this case and are very similar to mode size achievable in case of silicon waveguide. HPWG offers a number of benefits [18]: it has an improved compromise between loss and confinement compared to purely plasmonic waveguides, and is compatible with silicon on insulator technology. The power of the TE and TM modes in the HPWG are concentrated in two different layers, therefore their guiding characteristics can be controlled in different manners by altering the material properties, and waveguide dimensions of the layers [18].

propagation length to L = 3100µm, while retaining strong mode confinement [17].

*4.2.1. Long-Range Dielectric-Loaded Surface Plasmon Polaritons waveguides (LR-DLSPPs)*

17 connement to the dielectric ridge and underlying dielectric lm [17].

(b) (a)

20 (b) optical field distribution in the cross-section of the LR-DLSPPW [17].

the dielectric ridge and underlying dielectric film [17].

24 **4.3. Hybrid Plasmonic Waveguide** 

**4.3. Hybrid plasmonic waveguide**

3 of DLSPPW of optimal conguration.

9 and complex plasmonic circuits [16].

11 **DLSPPs)**

circuits [16].

102 Photonic Crystals

the LR-DLSPPW [17].

In contrast to the hybrid waveguides, metal-insulator-metal (MIM) structure has a very good modal confinement in cost of a short propagation length. A MIM waveguide is a dielectric slot sandwiched as core between two layers of metal as cladding. The coupling of two SPP's from two metal-dielectric boundaries is the only aspect in common. In the MIM structure, these two SPP's coupled into the central dielectric slot and thus gives rise to a huge field concentration. However, due to the close proximity of the mode with both metal layers, the losses are extremely large. A typical value for propagation length is in the few microns as in the IM case. The most noticeable advantage of the MIM mode is its sub-wavelength size.

**Figure 6.** Left panel: Schematic of an MIM waveguide with dielectric layer of thickness h and permittivity ε<sup>1</sup> sand‐ wiched between two metallic layers of permittivity ε2. Right panel: Density plots of longitudinal (Ex) and transverse (Ey) electric fields corresponding to the fundamental anti-symmetric SPP mode [19].

In addition, in the MIM waveguides one could make any sharp geometries or bending regardless of wavelength of the incident optical field [20]. MIM subwavelength plasmonic waveguide bends and splitters have low loss over a wide frequency range [20]. While metals are naturally lossy, the bounded SP modes of a single insulator-metal interface can propagate over several microns under optical incident fields [21]. In such a geometry, the field skin depth increases exponentially with wavelength in the insulator but is almost constant (~25nm) in the metal for visible and near-infrared excitation frequencies. Not unlike conventional wave‐ guides, these metal-insulator-metal (MIM) structures guide light via the refractive index differential between the core and cladding [21]. However, unlike dielectric slot waveguides, both plasmonic and conventional waveguiding modes can be accessed, depending on transverse core dimensions. MIM waveguides may thus allow optical mode volumes to be reduced to subwavelength scales—with minimal field decay out of the waveguide physical cross section— even for frequencies far from the plasmon resonance.

#### *4.4.1. Dispersion relation for MIM waveguides*

The modal analysis of planar multilayer structures can be solved via the vector wave equation under constraint of tangential **E** and normal **D** field continuity [21]. For unpolarized waves in a three-layer symmetric structure, the electromagnetic fields take the form [21]:

$$E\left(\mathbf{x}, z, t\right) = \left(E\_x \hat{\mathbf{x}} + E\_y \hat{\mathbf{y}} + E\_z \hat{\mathbf{z}}\right) e^{i\left(k\_x - at\right)}\tag{36}$$

$$B\left(\mathbf{x}, z, t\right) = \left(B\_x \hat{\mathbf{x}} + B\_y \hat{\mathbf{y}} + B\_z \hat{\mathbf{z}}\right) e^{i\left(k\_x - at\right)}\,,\tag{37}$$

with *Ey*, *Bx* and *Bz* identically zero for transverse magnetic (TM) polarization and *Ex*, *Ez* and *By* identically zero for transverse electric (TE) polarization. Inside the waveguide, the field components may be written as [21]:

$$\begin{aligned} E\_x^{in} &= e^{-ik\_{z1}z} \pm e^{ik\_{z1}z} \\ E\_y^{in} &= 0, \end{aligned} $$
 
$$\begin{aligned} E\_z^{in} &= \left(\frac{k\_x}{k\_{z1}}\right) \left(e^{-ik\_{z1}z} \mp e^{ik\_{z1}z}\right), \\ B\_x^{in} &= 0, \\ B\_y^{in} &= \left(\frac{-\alpha\rho\mathcal{E}\_1}{ck\_x}\right) \left(e^{-ik\_{z1}z} \mp e^{ik\_{z1}z}\right), \\ B\_z^{in} &= 0, \end{aligned} \tag{38}$$

for the TM polarization and as:

Applications of Nano-Scale Plasmonic Structures in Design of Stub Filters — A Step Towards Realization… http://dx.doi.org/10.5772/59877 105

$$\begin{aligned} E\_x^{in} &= \mathbf{0}\_\prime \\ E\_y^{in} &= e^{-ik\_{11}z} \pm e^{ik\_{11}z} \,, \\ B\_z^{in} &= \mathbf{0}\_\prime \end{aligned}$$

$$\begin{aligned} B\_x^{in} &= \left(\frac{-k\_{z1}c}{co}\right) \left(e^{-ik\_{11}z} \mp e^{ik\_{11}z}\right) \,, \\ B\_y^{in} &= \mathbf{0}\_\prime \end{aligned} \tag{39}$$

$$B\_z^{in} = \left(\frac{k\_{x}c}{co}\right) \left(e^{-ik\_{11}z} \pm e^{ik\_{11}z}\right) \,,$$

for the TE polarization. Outside the waveguide, the components are given [21]:

$$\begin{aligned} E\_x^{out} &= \left( e^{-ik\_{z1}d/2} \pm e^{ik\_{z1}d/2} \right), \\ E\_y^{out} &= 0, \end{aligned} \tag{40}$$

$$\begin{aligned} E\_z^{out} &= \left( \frac{\varepsilon\_1 k\_x}{\varepsilon\_2 k\_{z1}} \right) \left( e^{-ik\_{z1}d/2} \mp e^{ik\_{z1}d/2} \right) e^{ik\_{z1}\left(z - \frac{d}{2}\right)} \\ B\_x^{in} &= 0, \end{aligned} \tag{40}$$

$$\begin{aligned} B\_y^{out} &= \left( \frac{-\alpha \varepsilon\_1}{c k\_x} \right) \left( e^{-ik\_{z1}d/2} \mp e^{ik\_{z1}d/2} \right) e^{ik\_{z2}\left(z - \frac{d}{2}\right)}, \\ B\_z^{out} &= 0, \end{aligned}$$

for TM polarization and as:

waveguide bends and splitters have low loss over a wide frequency range [20]. While metals are naturally lossy, the bounded SP modes of a single insulator-metal interface can propagate over several microns under optical incident fields [21]. In such a geometry, the field skin depth increases exponentially with wavelength in the insulator but is almost constant (~25nm) in the metal for visible and near-infrared excitation frequencies. Not unlike conventional wave‐ guides, these metal-insulator-metal (MIM) structures guide light via the refractive index differential between the core and cladding [21]. However, unlike dielectric slot waveguides, both plasmonic and conventional waveguiding modes can be accessed, depending on transverse core dimensions. MIM waveguides may thus allow optical mode volumes to be reduced to subwavelength scales—with minimal field decay out of the waveguide physical

The modal analysis of planar multilayer structures can be solved via the vector wave equation under constraint of tangential **E** and normal **D** field continuity [21]. For unpolarized waves in

> ( ) ( ) ( ) , , ˆˆˆ *<sup>x</sup> ik t xyz E xzt Ex Ey Ez e* -

( ) ( ) ( ) , , ˆ ˆ ˆ , *<sup>x</sup> ik t xyz B xzt Bx By Bz e* -

with *Ey*, *Bx* and *Bz* identically zero for transverse magnetic (TM) polarization and *Ex*, *Ez* and *By* identically zero for transverse electric (TE) polarization. Inside the waveguide, the field

1 1

*z z*

*in ik z ik z*


=

*in y in x ik z ik z*

*Ee e E*

= ± =


0,

0,


0,

=

( )

1

*<sup>k</sup> E ee k B*

*z in x in ik z ik z*

æ ö = ç ÷ ç ÷ è ø

*x*

*z*

*y*

1

*B ee ck B*

*x in z*

we

æ ö - <sup>=</sup> ç ÷ è ø 1 1

*z z*

m

,

,

,

( )

1 1

*z z*

m

w

w

= ++ (36)

= ++ (37)

(38)

a three-layer symmetric structure, the electromagnetic fields take the form [21]:

cross section— even for frequencies far from the plasmon resonance.

*4.4.1. Dispersion relation for MIM waveguides*

104 Photonic Crystals

components may be written as [21]:

for the TM polarization and as:

$$E\_x^{out} = \mathbf{0},$$

$$E\_y^{out} = \left(e^{-ik\_{z1}d/2} \pm e^{ik\_{z1}d/2}\right)e^{-k\_{z2}\left(z - \frac{d}{2}\right)}$$

$$E\_z^{out} = \mathbf{0},$$

$$B\_x^{out} = \left(\frac{-k\_{z1}c}{co}\right)\left(e^{-ik\_{z1}d/2} \mp e^{ik\_{z1}d/2}\right)e^{ik\_{z2}\left(z - \frac{d}{2}\right)},\tag{41}$$

$$B\_y^{out} = \mathbf{0},$$

$$B\_z^{out} = \left(\frac{k\_{z1}c}{co}\right)\left(e^{-ik\_{z1}d/2} \pm e^{ik\_{z1}d/2}\right)e^{ik\_{z2}\left(z - \frac{d}{2}\right)}.$$

for TE polarization. The in-plane wave vector *kx* is defined by the dispersion relations [21]:

$$L + \begin{cases} \varepsilon\_1 k\_{z2} + \varepsilon\_2 k\_{z1} \tanh\left(\frac{-ik\_{z1}d}{2}\right) = 0 & TM \\\\ k\_{z2} + k\_{z1} \tanh\left(\frac{-ik\_{z1}d}{2}\right) = 0 & TE \end{cases} \tag{42}$$

$$L-:\left\{\begin{aligned} &\varepsilon\_{1}k\_{z2}+\varepsilon\_{2}k\_{z1}\coth\left(\frac{-ik\_{z1}d}{2}\right)=0 \ T M\\ &k\_{z2}+k\_{z1}\coth\left(\frac{-ik\_{z1}d}{2}\right)=0 \ T E \end{aligned}\right.\tag{43}$$

where *kz*2 is the momentum conservation:

$$k\_{z1,2}^2 = \varepsilon\_{1,2} \left(\frac{\alpha}{c}\right)^2 - k\_x^2 \tag{44}$$

and *L* + and *L* − denote the antisymmetric and symmetric tangential electric field with re‐ spect to the waveguide medium, respectively.

**Figure 7.** Geometry and characteristic tangential (x) electric field profiles for MIM slot waveguides; (a) Field antisym‐ metric mode, (b) Field symmetric mode [21].

In above analysis the structure is centered at z=0 with core thickness **d** and wave propagates along the positive x direction (figure 7). The core (cladding) is composed of material with complex dielectric constant *ε*<sup>1</sup> (*ε*2). Since continuity of Ey forbids charge accumulation at the interface, TE surface plasmon waves do not generally exist in planar insulator-metal structures [21]. If SPPs are excited at an insulator-metal interface, electrons in the metal make a surface polarization that leads to a localized electric field. In insulator-metal-insulator (IMI) structures, electrons of the metallic core screen the charge configuration at each interface and maintain a near-zero (or minimal) field within the waveguide. Therefore, the surface polarizations on both side of the metal film persist in phase and there is no cutoff frequency for any transverse waveguide dimension [21]. In contrast, screening does not occur within the insulator core of MIM waveguides. At each insulator-metal interface, surface polarizations of each side of interface are independent, and therefore SPPs oscillations need not be energy- or wave-vectormatched to each other. As a result, for certain MIM dielectric core thicknesses, interface SPPs may not remain in phase but will exhibit a beating frequency; as transverse core dimensions are increased, "bands" of allowed energies or wave vectors and "gaps" of forbidden energies will be observed [21].

The effective refractive index neff of an MIM waveguide is complex. Its real part controls the guided wavelength λMIM and its imaginary part limits the propagation length LSPP of SPPs through the relation [19]:

$$m\_{\rm eff} = \frac{\beta}{k} = \frac{\lambda}{\lambda\_{\rm MIDM}} + i \frac{\lambda}{4\pi\lambda\_{\rm SPP}}\tag{45}$$

where *k* =2*π* / *λ*, *λ* =2*πc*<sup>0</sup> / *ω* and c0 is the speed of light in vacuum. One can calculate *β* using the following dispersion relation for the TM-SPP modes [19]:

$$\tanh\left(\frac{ik\_1d}{2}\right) = \left(\frac{\varepsilon\_2}{\varepsilon\_1}\frac{k\_1}{k\_2}\right)^{\pm 1} \tag{46}$$

where the signs ± correspond to symmetric and antisymmetric modes. Based on Equation (46), for *h* ≪*λ*, an MIM waveguide supports only a single antisymmetric mode that is similar to the fundamental TEM mode of a parallel-plate waveguide with perfect-electric-conductor (PEC) boundaries: The symmetric mode stops to exist because it experiences a cut-off in the reciprocal space [19]. Thus, in the deep subwavelength regime, an MIM waveguide operates as a single-mode plasmonic waveguide. Since this regime is the most interesting from the standpoint of nanophotonics applications, we assume in this chapter that the condition *h* ≪*λ* is satisfied and focus on a single-mode plasmonic waveguide [19].

#### **5. Modeling of plasmonic MIM waveguides**

#### **5.1. Challenges in modeling of plasmonics**

1

(42)

(43)

(44)

*z*

1

1

*z*

1

*z*

coth 0 2

2 2 2 *z x* 1,2 1,2 *k k c* w

and *L* + and *L* − denote the antisymmetric and symmetric tangential electric field with re‐

**Figure 7.** Geometry and characteristic tangential (x) electric field profiles for MIM slot waveguides; (a) Field antisym‐

In above analysis the structure is centered at z=0 with core thickness **d** and wave propagates along the positive x direction (figure 7). The core (cladding) is composed of material with complex dielectric constant *ε*<sup>1</sup> (*ε*2). Since continuity of Ey forbids charge accumulation at the interface, TE surface plasmon waves do not generally exist in planar insulator-metal structures [21]. If SPPs are excited at an insulator-metal interface, electrons in the metal make a surface polarization that leads to a localized electric field. In insulator-metal-insulator (IMI) structures, electrons of the metallic core screen the charge configuration at each interface and maintain a

*ik d k k TM*

*ik d k k TE*

coth 0

*z*

tanh 0 2

*ik d k k TM*

*ik d k k TE*

tanh 0

12 21

e

e

*L*

106 Photonic Crystals

*L*

where *kz*2 is the momentum conservation:

spect to the waveguide medium, respectively.

metric mode, (b) Field symmetric mode [21].

*z z*

 e

2 :

<sup>ì</sup> æ ö - <sup>ï</sup> + = ç ÷ <sup>ï</sup> è ø <sup>+</sup> <sup>í</sup>

<sup>ï</sup> æ ö - + = ç ÷ <sup>ï</sup> î è ø

2 1

*z z*

12 21

*z z*

 e

2 :

<sup>ì</sup> æ ö - <sup>ï</sup> + = ç ÷ <sup>ï</sup> è ø - <sup>í</sup>

<sup>ï</sup> æ ö - + = ç ÷ <sup>ï</sup> î è ø

e

æ ö = - ç ÷ è ø

2 1

*z z*

Numerical simulation and modeling of plasmonic devices involves several challenges specific to plasmonics [22]. The permittivity of metals at optical wavelengths is complex, i.e. *ε*(*ω*)=*ε* ' (*ω*) + *iε* '' (*ω*), and is a complicated function of frequency [22]. Thus, several simulation techniques which are limited to lossless, non-dispersive materials are not applicable to plasmonic devices [22]. In addition, the dispersion properties of metals have to be approxi‐ mated by suitable analytical expressions, such as Drude model and Drude-Lorentz model. Furthermore, in surface plasmons propagating along the interface of a metal and an insulator, the field is confined at the interface, and decays exponentially perpendicular to the direction of propagation. Consequently, for numerical methods based on discretization of the fields, a very fine mesh resolution is required at the metal-dielectric interface [22]. Generally, simula‐ tion of plasmonic devices necessitates much finer grid resolution than modeling of low- or high-index-contrast dielectric devices, because of the high localization of the field at metaldielectric interfaces of plasmonic components. The required grid size depends on the shape and feature size of the modeled plasmonic device, the metallic material used and the operating frequency.

#### **5.2. General simulation methods for plasmonic components**

Due to the wave nature of the surface plasmon polaritons, most of the simulation methods of plasmonic devices and circuits are the same as the radio frequency approaches; therefore, the most common methods in simulation and modeling of the plasmonics are Finite-Difference Time Domain (FDTD) [23], Finite Element Method (FEM) [24], Transmission Line Method (TLM) [19] and Coupled Mode Theory (CMT) [25]. Each of the methods has its advantages over a specific plasmonic structure, e.g. in order to simulation of a 3D hybrid structure of a ring resonator FDTD or FEM are more convenient and for modeling of a metal-insulator-metal stub filter the accurate and fast approach is TLM. Transient response of a structure may be modelled by simulating electromagnetic pulses in time domain, by methods such as FDTD. The discussion for investigation of the best way to simulate plasmonic waveguides is out of the scope of this chapter; however, some of the most widely used methods are briefly ex‐ plained.

#### *5.2.1. Finite-Difference Time Domain (FDTD)*

Finite Difference Time Domain is a common technique in modelling electromagnetics prob‐ lems [15]. It is considered easy to understand and implement [26]. It is a time-domain method, so depending on the excitation type used, it could cover a wide range of frequencies in a single run, and hence it is usually the method of choice for wideband systems [15]. This method was introduced first by Kane Yee. FDTD does not require the existence of Green's function and directly approximates the differential operators in Maxwell's equations on a grid discretized in time and space. FDTD is an explicit finite difference approach, i.e. no matrix equation is established, stored and solved. The field values at the next time step are given entirely in terms of the field at the current and the previous time steps [15].

FDTD discretizes into unit cells called "Yee cells" [27, 28]. In such Yee cells, Electric Fields are represented by the edges of a cube, where the faces of the cubic cell denote the magnetic field. Given the offset (in space) of the magnetic fields from the electric fields, the values of the field with respect to time are also offset. Time is distributed into small steps, which are correspond‐ ing to the amount of time needed for the fields to travel from one Yee cell to the next or less. FDTD method solves the Maxwell's equations using the relationship between the partial time and space derivatives [15]. Yee's algorithm solves both E and H in time and space using the Maxwell's curl equations [26]. The biggest advantage of FDTD is its simplicity of use and fast application.

#### *5.2.2. Finite Element Method (FEM)*

the field is confined at the interface, and decays exponentially perpendicular to the direction of propagation. Consequently, for numerical methods based on discretization of the fields, a very fine mesh resolution is required at the metal-dielectric interface [22]. Generally, simula‐ tion of plasmonic devices necessitates much finer grid resolution than modeling of low- or high-index-contrast dielectric devices, because of the high localization of the field at metaldielectric interfaces of plasmonic components. The required grid size depends on the shape and feature size of the modeled plasmonic device, the metallic material used and the operating

Due to the wave nature of the surface plasmon polaritons, most of the simulation methods of plasmonic devices and circuits are the same as the radio frequency approaches; therefore, the most common methods in simulation and modeling of the plasmonics are Finite-Difference Time Domain (FDTD) [23], Finite Element Method (FEM) [24], Transmission Line Method (TLM) [19] and Coupled Mode Theory (CMT) [25]. Each of the methods has its advantages over a specific plasmonic structure, e.g. in order to simulation of a 3D hybrid structure of a ring resonator FDTD or FEM are more convenient and for modeling of a metal-insulator-metal stub filter the accurate and fast approach is TLM. Transient response of a structure may be modelled by simulating electromagnetic pulses in time domain, by methods such as FDTD. The discussion for investigation of the best way to simulate plasmonic waveguides is out of the scope of this chapter; however, some of the most widely used methods are briefly ex‐

Finite Difference Time Domain is a common technique in modelling electromagnetics prob‐ lems [15]. It is considered easy to understand and implement [26]. It is a time-domain method, so depending on the excitation type used, it could cover a wide range of frequencies in a single run, and hence it is usually the method of choice for wideband systems [15]. This method was introduced first by Kane Yee. FDTD does not require the existence of Green's function and directly approximates the differential operators in Maxwell's equations on a grid discretized in time and space. FDTD is an explicit finite difference approach, i.e. no matrix equation is established, stored and solved. The field values at the next time step are given entirely in terms

FDTD discretizes into unit cells called "Yee cells" [27, 28]. In such Yee cells, Electric Fields are represented by the edges of a cube, where the faces of the cubic cell denote the magnetic field. Given the offset (in space) of the magnetic fields from the electric fields, the values of the field with respect to time are also offset. Time is distributed into small steps, which are correspond‐ ing to the amount of time needed for the fields to travel from one Yee cell to the next or less. FDTD method solves the Maxwell's equations using the relationship between the partial time and space derivatives [15]. Yee's algorithm solves both E and H in time and space using the Maxwell's curl equations [26]. The biggest advantage of FDTD is its simplicity of use and fast

**5.2. General simulation methods for plasmonic components**

frequency.

108 Photonic Crystals

plained.

application.

*5.2.1. Finite-Difference Time Domain (FDTD)*

of the field at the current and the previous time steps [15].

The finite element method (FEM) is a numerical technique, which approximates the solutions of differential equations [15]. In electromagnetics, FEM generally approaches the solution to the Maxwell's equations in the frequency domain, therefore is usually used with timeharmonic conditions. Furthermore, it is capable of time domain simulations. FEM can be derived using two methods [15]; First, the variational method, which finds a variational functional whose minimum/maximum/stationary point corresponds to the solution of the PDE subject to certain boundary conditions (a brief formulation is given in section 2.7.1). Second method is called the "weak formulation" in the literatures. It works by introducing a weighted residual error to one of the differentials in the PDE form of Maxwell's equations [29] and equating the sum of the error to zero. If the weighting functions are Dirac delta functions, the resulting procedure is similar to finite difference method. If the weighting functions are the basis functions, then the method is called the Galerkin's method [15].

Several commercial software are presented in which simulate the electromagnetic problems based on FEM, such as CST Microwave Studio, COMSOL Multiphysics and ANSYS.

#### *5.2.3. Transmission Line Modelling (TLM) method*

Transmission Line modelling is one of the time domain techniques that can solve the electro‐ magnetic problems [15]. The concept of impedance and understanding the effects of wave‐ guide discontinuities in terms of lumped circuit elements were crucial in this respect [22]. Although the properties of metals are quite different at optical wavelengths compared to the microwave, designs that are qualitatively similar to their low frequency counterparts have been demonstrated at optical frequencies.

In this method, a unit cell is formed by conceptually filling space with a network of transmis‐ sion-lines in such a way that the voltage and current give information on the electric and magnetic fields [15]. A node represents the intersection point of the transmission-lines. At each time step, voltage pulses come to the node from each of the transmission-lines. These pulses are then scattered to produce a new set of pulses that become incident on adjacent nodes at the next time step. The relationship between the incident pulses and the scattered pulses is determined by the scattering matrix, which is set to be consistent with Maxwell's equations. Additional elements, such as transmission-line stubs, can be added to the node so that different material properties can be represented [15].

#### **6. A case study: Electro-plasmonic switch based on MIM stub filter**

#### **6.1. Stub filters in MIM plasmonic structures**

A stub is one of the key elements in microwave engineering and is employed in various microwave devices to reduce their size [30]. Some research groups have proposed a wave‐ length filter by using a stub structure in a photonic crystal waveguide [30, 31]. Such a structure may be employed in a plasmon waveguide to perform as a wavelength selective filter.

Running Title

3 have been demonstrated at optical frequencies.

14 **6.1. Stub filters in MIM plasmonic structures** 

1 [22]. Although the properties of metals are quite different at optical wavelengths compared 2 to the microwave, designs that are qualitatively similar to their low frequency counterparts

4 In this method, a unit cell is formed by conceptually lling space with a network of 5 transmission-lines in such a way that the voltage and current give information on the 6 electric and magnetic elds [15]. A node represents the intersection point of the 7 transmission-lines. At each time step, voltage pulses come to the node from each of the 8 transmission-lines. These pulses are then scattered to produce a new set of pulses that 9 become incident on adjacent nodes at the next time step. The relationship between the 10 incident pulses and the scattered pulses is determined by the scattering matrix, which is set 11 to be consistent with Maxwell's equations. Additional elements, such as transmission-line 12 stubs, can be added to the node so that different material properties can be represented [15].

16 microwave devices to reduce their size [30]. Some research groups have proposed a 17 wavelength filter by using a stub structure in a photonic crystal waveguide [30, 31]. Such a 18 structure may be employed in a plasmon waveguide to perform as a wavelength selective

13 **6. A case study: electro-plasmonic switch based on MIM stub filter** 

15

**Figure 8.** (a) three-dimension and (b) two-dimensional schematic depiction of the proposed Electro-Plasmonic switch [34].

Stub filters, also could be employed in order to make plasmonic switches [32-34]. MIM plasmonic waveguide structure helps to shrink the size of a stub filter to nanoscale and the stub structures cause to reduce the threshold voltage of the switch.

#### **6.2. MIM Stub filter as a plasmonic switch**

Figures 8(a) and 8(b) show the three and two-dimensional schematic of the proposed electroplasmonic switch based on the MIM structure, respectively [34]. This switch is essentially an MIM stub filter, which comprises of a main waveguide and one or more stub(s) vertically connected to it. Several studies investigated the properties of the stub filters and many researches used their filtering characteristics [19, 35-36].

The width of the waveguide and stubs, the length of the stubs and the displacement between them, and the refractive index of the core adjust the properties of the stub filter [19]. Any change in these factors will change the transfer function of the filter; therefore, one can control the transmittance valley or peak of the filter with a change in above parameters. Thus, through altering the refractive index of the core by applying an electric field to an EO material one could have an EO switch. In an MIM structure, the metal cladding boosts the electro optic effect in the core; subsequently, it decreases the threshold voltage of the switch.

Here, it is supposed that the entire of the switch filled with an EO material known as 4 dimethyl-amino-Nmethyl-4-stilbazolium tosylate (DAST) as the core, with a linear refractive index of nd = 2.2 and a large EO coefficient (dn/dE = 3.41 nm/V) [37].

Due to the low losses for the surface plasmons propagation, here silver is selected as the metal cladding of the waveguide [38]. In all simulations, the Drude-Lorentz model of the silver is utilized in order to obtain accurate results. A seven-pole Drude–Lorentz model is used in the wavelength range from 0.4µm to 2µm [19]:

$$\varepsilon\_{\rm u} \left( \alpha \right) = 1 - \frac{\alpha \overline{\boldsymbol{\sigma}}\_p^2}{\alpha \left( \alpha + i \boldsymbol{\gamma} \right)} + \sum\_{n=1}^5 \frac{f\_n \boldsymbol{\alpha}\_n^2}{\alpha\_n^2 - \alpha^2 - i \alpha \boldsymbol{\gamma}\_n} \tag{47}$$

where, *ω<sup>p</sup>* =2002.6 *THz* is the bulk plasma frequency of silver and *γ* =11.61 *THz* is a damping constant [19]. Table 1 listed the other parameters. In addition, we used an analytical expression to calculate of the effective index of the fundamental TM0 mode in the entire of the device [39]:

$$m\_{eff} = \sqrt{\varepsilon\_d} \left( 1 + \frac{\lambda}{\pi \varepsilon w \sqrt{-\varepsilon\_m}} \sqrt{1 + \frac{\varepsilon\_d}{-\varepsilon\_m}} \right)^{\frac{1}{2}} \tag{48}$$

where, *w* is the width of the waveguide and *ε<sup>d</sup>* =*nd* 2. The real and imaginary parts of the effective index for *w* =50nm over the wavelength range from 0.4 to 2µm is depicted in figure 9.


**Table 1.** Parameters of the Drude–Lorentz Model for Silver [19].

Running Title

3 have been demonstrated at optical frequencies.

14 **6.1. Stub filters in MIM plasmonic structures** 

*L*

**6.2. MIM Stub filter as a plasmonic switch**

researches used their filtering characteristics [19, 35-36].

(b)

(a)

110 Photonic Crystals

*stub*

*Wstub*

*Wwg*

stub structures cause to reduce the threshold voltage of the switch.

the core; subsequently, it decreases the threshold voltage of the switch.

index of nd = 2.2 and a large EO coefficient (dn/dE = 3.41 nm/V) [37].

*d*

**Figure 8.** (a) three-dimension and (b) two-dimensional schematic depiction of the proposed Electro-Plasmonic switch [34].

Stub filters, also could be employed in order to make plasmonic switches [32-34]. MIM plasmonic waveguide structure helps to shrink the size of a stub filter to nanoscale and the

Figures 8(a) and 8(b) show the three and two-dimensional schematic of the proposed electroplasmonic switch based on the MIM structure, respectively [34]. This switch is essentially an MIM stub filter, which comprises of a main waveguide and one or more stub(s) vertically connected to it. Several studies investigated the properties of the stub filters and many

The width of the waveguide and stubs, the length of the stubs and the displacement between them, and the refractive index of the core adjust the properties of the stub filter [19]. Any change in these factors will change the transfer function of the filter; therefore, one can control the transmittance valley or peak of the filter with a change in above parameters. Thus, through altering the refractive index of the core by applying an electric field to an EO material one could have an EO switch. In an MIM structure, the metal cladding boosts the electro optic effect in

Here, it is supposed that the entire of the switch filled with an EO material known as 4 dimethyl-amino-Nmethyl-4-stilbazolium tosylate (DAST) as the core, with a linear refractive

1 [22]. Although the properties of metals are quite different at optical wavelengths compared 2 to the microwave, designs that are qualitatively similar to their low frequency counterparts

4 In this method, a unit cell is formed by conceptually lling space with a network of 5 transmission-lines in such a way that the voltage and current give information on the 6 electric and magnetic elds [15]. A node represents the intersection point of the 7 transmission-lines. At each time step, voltage pulses come to the node from each of the 8 transmission-lines. These pulses are then scattered to produce a new set of pulses that 9 become incident on adjacent nodes at the next time step. The relationship between the 10 incident pulses and the scattered pulses is determined by the scattering matrix, which is set 11 to be consistent with Maxwell's equations. Additional elements, such as transmission-line 12 stubs, can be added to the node so that different material properties can be represented [15].

15 A stub is one of the key elements in microwave engineering and is employed in various 16 microwave devices to reduce their size [30]. Some research groups have proposed a 17 wavelength filter by using a stub structure in a photonic crystal waveguide [30, 31]. Such a 18 structure may be employed in a plasmon waveguide to perform as a wavelength selective

13 **6. A case study: electro-plasmonic switch based on MIM stub filter** 

*x*

Silver EO Material

*y*

*εm εd*

15

**Figure 9.** The real and imaginary part of the effective refractive index for MIM waveguide with the linear refractive index nd =2.2 and Wwg = 50nm [34].

#### *6.2.1. Modeling of the device using TLM*

The Transmission Line Method (TLM) helps to study the operation of the MIM stub filters more accurately and faster [35], in comparison to other methods. In this method, we convert the plasmonic MIM structure in figure 10a to an equivalent circuit diagram (see figure 10b). This equivalent circuit is formed by a parallel connection of the characteristic impedance of an infinite MIM waveguides (ZMIM) and the characteristic impedance of a finite and Lspp is the propagation length of the SPPs (equation (35)). The length of the stubs is the key point of the design of the stub filter and the electro-plasmonic switch [34]. Knowing that small increase in the real part of the refractive index leads to a red-shift in the transmittance spectrum of the filter, the transmission of the filter at wavelength λ=1550nm must be on the edge of a change, a falling edge of the magnitude for normally ON and a rising edge for normally OFF switch. Figure 11 shows the transmission of a typical 4-stub filter for different length of the stubs. The transmittance is defined as T=Pout/Pin and is calculated using Transmission Line Method [35]. In the computation of the spectrum, the width of the main waveguide and the width of the stubs has the same value Wwg= Wstub=50nm, the distances between stubs is d=100nm and the thickness of the silver layer (or the depth of the structure) is tAg=200nm.

MIM waveguide (ZS) terminated by ZL accounts for reflection of the SPPs from the stub [19]:

$$\mathbf{Z}\_S = \mathbf{Z}\_{\text{MM}} = \frac{\beta \{\mathbf{w} \} \mathbf{w}}{\alpha \mathbf{e}\_d \mathbf{e}\_m} \tag{49}$$

$$Z\_L = \sqrt{\frac{\mathcal{E}\_m}{\mathcal{E}\_d}} \times Z\_S \tag{50}$$

*β(*w*)* is the complex-valued propagation constant and describes the properties of the MIM waveguide. The circuit diagram in figure 10b may be modified to an equivalent form as shown in figure 10c by replacing ZMIM and ZS by an effective impedance (Zstub). The value of Zstub can be found from transmission-line theory and is given by [19]:

$$\mathbf{Z}\_{\rm stub} = \mathbf{Z}\_{\rm s} \frac{\mathbf{Z}\_{\rm L} - \mathrm{i}\mathbf{Z}\_{\rm s} \tan\left(\mathcal{J}\left(\boldsymbol{w}\right)\mathrm{L}\_{\rm stub}\right)}{\mathbf{Z}\_{\rm s} - \mathrm{i}\mathbf{Z}\_{\rm L} \tan\left(\mathcal{J}\left(\boldsymbol{w}\right)\mathrm{L}\_{\rm stub}\right)}\tag{51}$$

Now, we calculate the transmission spectra of an MIM stub filter with *N* identical stubs set apart by a distance *d*, the length of the stubs Lstub, the width of waveguide and stubs Wwg = Wstub = w [39]:

$$T = \left(P\_{\ast}^{N-1} \mathbf{Q}\_{\ast} - P\_{-}^{N-1} \mathbf{Q}\_{-}\right)^{-2} \exp(-\frac{L}{L\_{spp}}) \tag{52}$$

where

ఠఌఌ ܼ ൌ <sup>ට</sup>ఌ ൈ ܼௌ 3 (50) Applications of Nano-Scale Plasmonic Structures in Design of Stub Filters — A Step Towards Realization… http://dx.doi.org/10.5772/59877 113

$$P\_{\pm} = \frac{1}{2} \left[ 1 + \frac{Z\_{\text{MM}}}{2Z\_{\text{sdM}}} + \left( 1 - \frac{Z\_{\text{MM}}}{2Z\_{\text{sdM}}} \right) \exp\left( 2i\beta d \right) \pm R \right],$$

$$Q\_{\pm} = \frac{1}{2R} \left[ \left( 1 + \frac{Z\_{\text{MM}}}{2Z\_{\text{sdM}}} \right)^{2} - \left[ 1 + \left( \frac{Z\_{\text{MM}}}{2Z\_{\text{sdM}}} \right)^{2} \right] \exp\left( 2i\beta d \right) \right] \pm \frac{1}{2} \left( 1 + \frac{Z\_{\text{MM}}}{2Z\_{\text{sdM}}} \right)$$

$$R = \left\{ \left[ 1 + \frac{Z\_{\text{MM}}}{2Z\_{\text{sdM}}} + \left( 1 - \frac{Z\_{\text{MM}}}{2Z\_{\text{sdM}}} \right) \exp\left( 2i\beta d \right) \right]^{2} - 4 \exp\left( 2i\beta d \right) \right\}^{\frac{1}{2}},$$

18 Photonic Crystals

ܼௌ ൌ ܼெூெ ൌ ఉሺ௪ሻ௪

ఌ

1 MIM waveguide (ZS) terminated by ZL accounts for reflection of the SPPs from the stub [19]:

2 (49)

*6.2.1. Modeling of the device using TLM*

112 Photonic Crystals

The Transmission Line Method (TLM) helps to study the operation of the MIM stub filters more accurately and faster [35], in comparison to other methods. In this method, we convert the plasmonic MIM structure in figure 10a to an equivalent circuit diagram (see figure 10b). This equivalent circuit is formed by a parallel connection of the characteristic impedance of an infinite MIM waveguides (ZMIM) and the characteristic impedance of a finite and Lspp is the propagation length of the SPPs (equation (35)). The length of the stubs is the key point of the design of the stub filter and the electro-plasmonic switch [34]. Knowing that small increase in the real part of the refractive index leads to a red-shift in the transmittance spectrum of the filter, the transmission of the filter at wavelength λ=1550nm must be on the edge of a change, a falling edge of the magnitude for normally ON and a rising edge for normally OFF switch. Figure 11 shows the transmission of a typical 4-stub filter for different length of the stubs. The transmittance is defined as T=Pout/Pin and is calculated using Transmission Line Method [35]. In the computation of the spectrum, the width of the main waveguide and the width of the stubs has the same value Wwg= Wstub=50nm, the distances between stubs is d=100nm and the

MIM waveguide (ZS) terminated by ZL accounts for reflection of the SPPs from the stub [19]:

b

*m L S d Z Z* e

*β(*w*)* is the complex-valued propagation constant and describes the properties of the MIM waveguide. The circuit diagram in figure 10b may be modified to an equivalent form as shown in figure 10c by replacing ZMIM and ZS by an effective impedance (Zstub). The value of Zstub can

> tan tan *L S stub*

*Z iZ w L*

*Z iZ w L*

Now, we calculate the transmission spectra of an MIM stub filter with *N* identical stubs set apart by a distance *d*, the length of the stubs Lstub, the width of waveguide and stubs Wwg =

( ) <sup>2</sup> 1 1 exp( ) *N N*

*<sup>L</sup> T PQ PQ <sup>L</sup>* - - -

*S L stub*

b

b

( ( ) ) ( ( ) )

e

we e

( )

*d m w w*

= = (49)

= ´ (50)


*spp*

+ +- - =- - (52)

thickness of the silver layer (or the depth of the structure) is tAg=200nm.

*S MIM*

*Z Z*

be found from transmission-line theory and is given by [19]:

*stub S*

*Z Z*

Wstub = w [39]:

where

ܶ ൌ ሺܲା ேିଵܳା െ ܲି ேିଵܳିሻିଶሺെ ೞ 12 ሻ (52) 13 **Fig. 10.** The diagram of an MIM waveguide with a single stub coupled perpendicularly to 14 the waveguide axis. (b) The corresponding transmission-line representation. (c) The **Figure 10.** The diagram of an MIM waveguide with a single stub coupled perpendicularly to the waveguide axis. (b) The corresponding transmission-line representation. (c) The simplified circuit model. ZMIM and ZS correspond to the characteristic impedance of the MIM waveguide and the stub, respectively, ZL accounts for reflection of the SPPs from the stub end and Zstub is the effective stub impedance [19].

15 simplified circuit model. ZMIM and ZS correspond to the characteristic impedance of the MIM

**Figure 11.** The transmittance spectra of the 4-stub filter for different length of the stubs. The crossing points of the dashed-lines show the transmittance values for Lstub=300nm and Lstub=410nm when λ = 1550nm. By a red-shift in the spectrum, the magnitude of the transmittance rises at the point a while falls at the point b [34].

6 where

1

10

Running Title

2 **Fig. 11.** The transmittance spectra of the 4-stub filter for different length of the stubs. The 3 crossing points of the dashed-lines show the transmittance values for Lstub=300nm and 4 Lstub=410nm when λ = 1550nm. By a red-shift in the spectrum, the magnitude of the

> ቀͳ െ ಾಾ ଶೞೠ್

> > ቁ ሺʹ݅ߚ݀ሻቃ

20 Photonic Crystals

൨ ሺʹ݅ߚ݀ሻ൰ േ <sup>ଵ</sup>

ଶ

<sup>ଶ</sup> ቀͳ ಾಾ ଶೞೠ್

> భ మ

െ Ͷ ሺʹ݅ߚ݀ሻൠ

െ ͳ ቀ ಾಾ ଶೞೠ್ ቁ ଶ

5 transmittance rises at the point a while falls at the point b [34].

ܲേ ൌ <sup>ଵ</sup>

ଶோ ൬ቀͳ ಾಾ ଶೞೠ್ ቁ ଶ

ܴ ൌ ൜ቂͳ ಾಾ

ଶೞೠ್

ܳേ ൌ <sup>ଵ</sup>

<sup>ଶ</sup> ቂͳ ಾಾ ଶೞೠ್

8 ቁ,

 ቀͳ െ ಾಾ ଶೞೠ್

7 ቁ ሺʹ݅ߚ݀ሻ േ ܴቃ,

19

**Figure 12.** The transmittance spectra of the 4-stub filter for different refractive index with (a) Lstub= 410nm, and (b) Lstub= 300nm; the crossing of the dashed-lines indicate the transmittance of the filter with the correspond Lstub. Increment in the refractive index at 1550nm leads to (a) rise, (b) fall the magnitude of the transmittance [34]. 13 normally OFF switch. Figure 11 shows the transmission of a typical 4-stub filter for different 14 length of the stubs. The transmittance is defined as T=Pout/Pin and is calculated using 15 Transmission Line Method [35]. In the computation of the spectrum, the width of the main

12 edge of a change, a falling edge of the magnitude for normally ON and a rising edge for

16 waveguide and the width of the stubs has the same value Wwg= Wstub=50nm, the distances 17 between stubs is d=100nm and the thickness of the silver layer (or the depth of the structure) 18 is tAg=200nm. **Figure 13.** The transmittance spectra of the 4-stub filter with (a) Lstub= 410nm, and (b) Lstub= 300nm; in the calculation of the spectrums, 2D-TLM and 3D-FEM methods are used which show an acceptable conformity [34].

19 According to figure 11, by selecting the stub Lstub=300nm (point b) or Lstub=410nm (point a) 20 at the communication wavelength λ=1550nm, the normal state of the switch would be ON 21 or OFF, respectively. Slightly increment in the refractive index turns the switch OFF or ON, (a) (b) According to figure 11, by selecting the stub Lstub=300nm (point b) or Lstub=410nm (point a) at the communication wavelength λ=1550nm, the normal state of the switch would be ON or OFF, respectively. Slightly increment in the refractive index turns the switch OFF or ON, depending on the length of stubs (Lstub=300nm or Lstub=410nm, respectively). Figures 12a and 12b show the spectrum of the switch versus the refractive index for Lstub=300nm and Lstub=410nm, respectively. Figures 13a and 13b show the transmission spectrum for Lstub= 410nm and Lstub= 300nm and shows reasonable accuracy of three-dimensional FEM simulations [34]. These figures admit the red-shift effect on the ON/OFF status of the switch. As shown in figure 8, the silver cladding is connected to the voltage V and therefore the electric field surrounds the main waveguide and all the stubs. The relation between applied voltages and the refractive index of EO material is expressed using equation [37]:

(c) (d)

$$m = n\_0 + \frac{dn}{dE} \left(\frac{V}{w}\right) \tag{53}$$

20 Photonic Crystals

1 **Fig. 12.** The transmittance spectra of the 4-stub filter for different refractive index with (a) 2 Lstub= 410nm, and (b) Lstub= 300nm; the crossing of the dashed-lines indicate the 3 transmittance of the filter with the correspond Lstub. Increment in the refractive index at

5 **Fig. 13.** The transmittance spectra of the 4-stub filter with (a) Lstub= 410nm, and (b) Lstub= 6 300nm; in the calculation of the spectrums, 2D-TLM and 3D-FEM methods are used which

8 and Lspp is the propagation length of the SPPs (equation (35)). The length of the stubs is the 9 key point of the design of the stub filter and the electro-plasmonic switch [34]. Knowing that 10 small increase in the real part of the refractive index leads to a red-shift in the transmittance 11 spectrum of the filter, the transmission of the filter at wavelength λ=1550nm must be on the 12 edge of a change, a falling edge of the magnitude for normally ON and a rising edge for 13 normally OFF switch. Figure 11 shows the transmission of a typical 4-stub filter for different 14 length of the stubs. The transmittance is defined as T=Pout/Pin and is calculated using 15 Transmission Line Method [35]. In the computation of the spectrum, the width of the main

4 1550nm leads to (a) rise, (b) fall the magnitude of the transmittance [34].

7 show an acceptable conformity [34].

where *n*0 is the linear refractive index of the EO material that equals 2.2, V is the applied voltage, and w is the width of waveguides and stubs [34]. The applied electric field alters the linear refractive index of the switch and as a result causes the red-shift. Figures 14a-d demonstrate the distribution of the optical field in the structure of the electro-plasmonic switch. In figures 14a and 14c the applied voltage is 0V and corresponds to the normal-state of both switches, while figures 14b and 14d show the electro-optic switch when the voltage is applied to the structure. The applied electric field changes the refractive index of the EO material; therefore, the optical field reflected back to the input port. Figure 15 depicts the transmittance of the normally ON and normally OFF switch as a function of the applied voltage at the wavelength 1550nm. As can be seen, the threshold voltage for switching is 10V [34]. 16 waveguide and the width of the stubs has the same value Wwg= Wstub=50nm, the distances 17 between stubs is d=100nm and the thickness of the silver layer (or the depth of the structure) 18 is tAg=200nm. 19 According to figure 11, by selecting the stub Lstub=300nm (point b) or Lstub=410nm (point a) 20 at the communication wavelength λ=1550nm, the normal state of the switch would be ON 21 or OFF, respectively. Slightly increment in the refractive index turns the switch OFF or ON,

Running Title

2 **Fig. 11.** The transmittance spectra of the 4-stub filter for different length of the stubs. The 3 crossing points of the dashed-lines show the transmittance values for Lstub=300nm and 4 Lstub=410nm when λ = 1550nm. By a red-shift in the spectrum, the magnitude of the

> ቀͳ െ ಾಾ ଶೞೠ್

**Figure 12.** The transmittance spectra of the 4-stub filter for different refractive index with (a) Lstub= 410nm, and (b) Lstub= 300nm; the crossing of the dashed-lines indicate the transmittance of the filter with the correspond Lstub. Increment in

16 waveguide and the width of the stubs has the same value Wwg= Wstub=50nm, the distances 17 between stubs is d=100nm and the thickness of the silver layer (or the depth of the structure)

the spectrums, 2D-TLM and 3D-FEM methods are used which show an acceptable conformity [34].

**Figure 13.** The transmittance spectra of the 4-stub filter with (a) Lstub= 410nm, and (b) Lstub= 300nm; in the calculation of

19 According to figure 11, by selecting the stub Lstub=300nm (point b) or Lstub=410nm (point a) 20 at the communication wavelength λ=1550nm, the normal state of the switch would be ON 21 or OFF, respectively. Slightly increment in the refractive index turns the switch OFF or ON,

<sup>0</sup> ( ) *dn V*

*dE w*

= + (53)

According to figure 11, by selecting the stub Lstub=300nm (point b) or Lstub=410nm (point a) at the communication wavelength λ=1550nm, the normal state of the switch would be ON or OFF, respectively. Slightly increment in the refractive index turns the switch OFF or ON, depending on the length of stubs (Lstub=300nm or Lstub=410nm, respectively). Figures 12a and 12b show the spectrum of the switch versus the refractive index for Lstub=300nm and Lstub=410nm, respectively. Figures 13a and 13b show the transmission spectrum for Lstub= 410nm and Lstub= 300nm and shows reasonable accuracy of three-dimensional FEM simulations [34]. These figures admit the red-shift effect on the ON/OFF status of the switch. As shown in figure 8, the silver cladding is connected to the voltage V and therefore the electric field surrounds the main waveguide and all the stubs. The relation between applied voltages and

(a) (b)

(c) (d)

the refractive index of EO material is expressed using equation [37]:

*n n*

8 and Lspp is the propagation length of the SPPs (equation (35)). The length of the stubs is the 9 key point of the design of the stub filter and the electro-plasmonic switch [34]. Knowing that 10 small increase in the real part of the refractive index leads to a red-shift in the transmittance 11 spectrum of the filter, the transmission of the filter at wavelength λ=1550nm must be on the 12 edge of a change, a falling edge of the magnitude for normally ON and a rising edge for 13 normally OFF switch. Figure 11 shows the transmission of a typical 4-stub filter for different 14 length of the stubs. The transmittance is defined as T=Pout/Pin and is calculated using 15 Transmission Line Method [35]. In the computation of the spectrum, the width of the main

the refractive index at 1550nm leads to (a) rise, (b) fall the magnitude of the transmittance [34].

ቁ ሺʹ݅ߚ݀ሻቃ

20 Photonic Crystals

1 **Fig. 12.** The transmittance spectra of the 4-stub filter for different refractive index with (a) 2 Lstub= 410nm, and (b) Lstub= 300nm; the crossing of the dashed-lines indicate the 3 transmittance of the filter with the correspond Lstub. Increment in the refractive index at

5 **Fig. 13.** The transmittance spectra of the 4-stub filter with (a) Lstub= 410nm, and (b) Lstub= 6 300nm; in the calculation of the spectrums, 2D-TLM and 3D-FEM methods are used which

൨ ሺʹ݅ߚ݀ሻ൰ േ <sup>ଵ</sup>

ଶ

<sup>ଶ</sup> ቀͳ ಾಾ ଶೞೠ್

> భ మ

െ Ͷ ሺʹ݅ߚ݀ሻൠ

െ ͳ ቀ ಾಾ ଶೞೠ್ ቁ ଶ

5 transmittance rises at the point a while falls at the point b [34].

ܲേ ൌ <sup>ଵ</sup>

ଶோ ൬ቀͳ ಾಾ ଶೞೠ್ ቁ ଶ

ܴ ൌ ൜ቂͳ ಾಾ

ଶೞೠ್

**(a) (b)**

ܳേ ൌ <sup>ଵ</sup>

7 show an acceptable conformity [34].

18 is tAg=200nm.

<sup>ଶ</sup> ቂͳ ಾಾ ଶೞೠ್

8 ቁ,

9 ,

4 1550nm leads to (a) rise, (b) fall the magnitude of the transmittance [34].

 ቀͳ െ ಾಾ ଶೞೠ್

7 ቁ ሺʹ݅ߚ݀ሻ േ ܴቃ,

1

10

6 where

114 Photonic Crystals

19

**Figure 14.** The distribution of the optical field in the structure of the electro-optic switch for Lstub=410nm when (a) V = 0V (b) V = 10V and Lstub=300nm when (c) V = 0V (d) V = 10V [34].

**Figure 15.** The transmittance of the proposed electro-optic switch versus the applied voltage at the telecom wavelength 1550nm; the solid and dashed line corresponds to EO switch with the Lstub=410nm (Normally OFF) and the Lstub=300nm (Normally ON), respectively [34].

#### **Author details**

Hassan Kaatuzian\* and Ahmad Naseri Taheri

\*Address all correspondence to: hsnkato@aut.ac.ir

Photonics Research Lab., Amirkabir university of Technology, Iran

#### **References**


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**Author details**

116 Photonic Crystals

Hassan Kaatuzian\*

**References**

and Ahmad Naseri Taheri

Photonics Research Lab., Amirkabir university of Technology, Iran

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**Chapter 5**

## **Equal Frequency Surface**

G. Alagappan

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[34] Naseri Taheri A. and Kaatuzian H. Numerical Investigation of a Nano-Scale Electro-Plasmonic Switch Based on Metal-Insulator-Metal Stub Filter. Opt Quant Electron.

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118 Photonic Crystals

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Springer 2014.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59887

## **1. Introduction**

A photonic crystal (PC) forbids propagation of light in a spectral range called photonic band gap. Within this frequency gap, PC behaves like an insulator for light. The insulating properties of the PC have been widely used in the developments of waveguides, fibers and cavities. For frequencies outside the photonic band gaps though light can propagate, the propagation properties (i.e. conducting properties) are uniquely different. The distinct conducting prop‐ erties of the PC have led to discoveries of novel functionalities such as superprism effects, large angle polarization splitting, negative refraction, and superlensing.

The conducting properties of a PC can be best – analyzed using an equal frequency surface (EFS). The EFS is a surface in a three-dimensional (3D) PC, and a contour in a 2D PC. The gradient of the EFS plays a key role in determining the group velocity direction and hence, the propagation direction of light in the PC. Important developments on the EFS analysis for PCs can be found in [1-7].

EFS is typically obtained using a plane wave expansion methodology [8-9]. If the dielectric contrast of the PC is large, then a large number of plane waves is required to obtain EFSs with good accuracy. However, the distinct conducting properties, like superprism and beam splitting are normally well-pronounced in the PCs with a small dielectric modulation. [10-13]. For such PCs with small dielectric modulation, the requirement on the large number of plane waves to obtain the EFS can be relaxed.

In this chapter we will outline the theory of the EFS construction, and their applications in determining the conducting properties. For the sake of an easy understanding, throughout this chapter, we will use 2D PCs for illustrations and discussions. The chapter is organized as follows. The second section of the chapter focuses on the exact and approximate constructions of EFSs. Specifically, using one and two plane waves approximations [5], simple and handy analytical expressions for EFSs will be shown. It is worth mentioning that in the electronic

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

energy band theory, the one plane wave technique is used to approximate Fermi surfaces of a metal, and it is known in the name of Harrison's principles [14-15]. In the third section, we will outline a graphical technique that uses EFSs to determine light propagation directions in the PC. Fourth section of the chapter elaborates on the applications of the EFSs. Particularly, we show the usage of the EFSs to describe superprism, effective negative refractive index mediums, negative refractions, and superlens phenomena in PCs. Finally, fifth section of the chapter gives a summary of the whole chapter.

#### **2. EFS construction**

In order to construct an EFS for a specific frequency, firstly, a photonic band structure has to be calculated. The photonic band structure can be calculated using a plane wave expansion method. For the details of this method, see [8-9]. The photonic band structure should contain frequencies for a dense number wavevectors in the first Brillouin zone (BZ). For a 2D PC, the EFS can be obtained by projecting the 2D photonic band structure onto the wavevector plane (*kx*,*ky*), keeping a constant frequency.

In a homogenous dielectric medium of refractive index *n*, the dispersion relation is *ω*=*ck*/*n*, where *ω*, *c*, and *k* are the angular frequency, speed of light, and wavevector, respectively. In 2D, the dispersion relation is = *ω*<sup>2</sup> =*c*<sup>2</sup> (*kx* 2 +*ky* 2 )/*n*<sup>2</sup> . Thus, as shown in Figure 1, the EFSs of a homogeneous dielectric medium are circles. In a PC, the dispersion relation (i.e., the photonic band structure) is complicated, and therefore EFSs have to be calculated numerically.

**Figure 1.** photonic band structure, and EFSs of a homogenous dielectric medium

Let's take examples of EFSs in a 2D square lattice PC made of circular silicon rods (dielectric constant=12.1) in an air matrix (dielectric constant=1.0). In such a system, the two polarizations of light can be decoupled. The band structure of the *E* – polarization (electric field perpendic‐ ular to the 2D periodic plane) is shown in Figure 2, for a silicon rod radius of 0.2. The vertical axis in this figure represents the normalized frequency, *ω*=*a*/*λ*, with *a* and*λ* being the period and free space wavelength, respectively. As we can see from the 2D band structure, in the long wavelength limit (i.e., *a* << *λ*), the dispersion relationship is similar to a homogenous medium (Fig. 1).

**Figure 2.** Photonic band structure in 2D and EFSs (in 2D these are constant frequency contours) of a 2D PC [circular silicon rods with a radius of 0.2, air matrix]. The color bars indicate the normalized frequencies

#### **2.1. Approximation techniques**

energy band theory, the one plane wave technique is used to approximate Fermi surfaces of a metal, and it is known in the name of Harrison's principles [14-15]. In the third section, we will outline a graphical technique that uses EFSs to determine light propagation directions in the PC. Fourth section of the chapter elaborates on the applications of the EFSs. Particularly, we show the usage of the EFSs to describe superprism, effective negative refractive index mediums, negative refractions, and superlens phenomena in PCs. Finally, fifth section of the

In order to construct an EFS for a specific frequency, firstly, a photonic band structure has to be calculated. The photonic band structure can be calculated using a plane wave expansion method. For the details of this method, see [8-9]. The photonic band structure should contain frequencies for a dense number wavevectors in the first Brillouin zone (BZ). For a 2D PC, the EFS can be obtained by projecting the 2D photonic band structure onto the wavevector plane

In a homogenous dielectric medium of refractive index *n*, the dispersion relation is *ω*=*ck*/*n*, where *ω*, *c*, and *k* are the angular frequency, speed of light, and wavevector, respectively. In

homogeneous dielectric medium are circles. In a PC, the dispersion relation (i.e., the photonic

Let's take examples of EFSs in a 2D square lattice PC made of circular silicon rods (dielectric constant=12.1) in an air matrix (dielectric constant=1.0). In such a system, the two polarizations of light can be decoupled. The band structure of the *E* – polarization (electric field perpendic‐ ular to the 2D periodic plane) is shown in Figure 2, for a silicon rod radius of 0.2. The vertical axis in this figure represents the normalized frequency, *ω*=*a*/*λ*, with *a* and*λ* being the period and free space wavelength, respectively. As we can see from the 2D band structure, in the long

band structure) is complicated, and therefore EFSs have to be calculated numerically.

. Thus, as shown in Figure 1, the EFSs of a

=*c*<sup>2</sup> (*kx* 2 +*ky* 2 )/*n*<sup>2</sup>

**Figure 1.** photonic band structure, and EFSs of a homogenous dielectric medium

chapter gives a summary of the whole chapter.

**2. EFS construction**

120 Photonic Crystals

(*kx*,*ky*), keeping a constant frequency.

2D, the dispersion relation is = *ω*<sup>2</sup>

In a 3D PC made of either isotropic or anisotropic materials, it is known that the Maxwell equations corresponding to two independent polarizations of light are coupled [9,16-17]. For a 2D PC made of isotropic materials, we can always decouple the equations into two inde‐ pendent equations, corresponding to the two independent polarizations [9, 16]. On the other hand, for a 2D PC made of an anisotropic material, such a decoupling of polarizations is not possible in general. However, by restricting one of the principal axes of the anisotropic material to be perpendicular to the periodic plane of the 2D PC and the other two principal axes residing in the periodic plane, decoupling is possible [6,18-20]. Detail mathematical treatment of the polarization decoupling in a 2D PC made of anisotropic materials can be found in [6].

Assuming the polarizations can be decoupled, the photonic band structure of the *H*-polariza‐ tion (magnetic field perpendicular to the periodic plane) or the *E*-polarization can be found by solving the differential-equation in the form [9,6,16],

$$
\hat{D}[\mathbf{H}(\mathbf{r})] = (2\pi\alpha \wedge a)^2 [\mathbf{H}(\mathbf{r})],\tag{1}
$$

for a given wavevector in the first BZ. In Eqn. 1, **H**(**r**) is the position (**r**) dependent magnetic field vector, and *D* ^ is a differential operator that depends on the polarization, and the dielectric constant profile of the PC. Using the plane wave expansion method [9,6-7,16], we can transform Eqn. 1 to a matrix eigen-value problem. Consequently, the frequency containing term (2π*ω*/ *a*)2 =Ω in Eqn. 1, can be written as a matrix vector product,

$$
\Omega \mathbf{Q} = \mathbf{h} \cdot \hat{M} \mathbf{h}.\tag{2}
$$

In this equation, **h**=[*h*<sup>1</sup> *h*2...]', where *hn* is the Fourier expansion coefficient of **H**(**r**) associated with the reciprocal lattice vector **G***n*. The matrix element of *M* ^ [6] is defined as

$$M\_{mn} = \begin{cases} \left< k\_m \left| \tilde{\mathcal{J}}\_r(m-n) \right| k\_n \right>, & H-\text{polarization} \\ \left| k\_m \right| \left| k\_n \right| \mathcal{B}(m-n), & E-\text{polarization} \end{cases} \tag{3}$$

Here |*kn*>=**k** – **G***n*, with **k=**[*kx*, *ky*]'. In Eqn. 3, <...> represents the inner product of the vector, matrix and vector. For the *H* – polarization, *β*˜*<sup>r</sup>* (*n*) is the 2 by 2 matrix defining the inverse Fourier transform coefficient of the tensor dielectric function [associated with **G***n*], and similarly for the *E*-polarization, *β*(*n*) is the inverse Fourier transform coefficient of the scalar dielectric function.

Eqn. 2 defines an EFS for a given frequency, and has to be evaluated using the entire basis of the reciprocal space, which is infinite in number. In practice, the number of basis (i.e. the number of plane waves) is limited to a number for which the corresponding result achieves a required degree of accuracy. In general, larger the dielectric modulation, larger the number of required plane waves. For PCs with weak dielectric modulations, the number of plane waves can be significantly less. In the following sections, we will use one and two plane waves to approximate the EFS of the weakly modulated PCs, and as we shall see, such approximations lead to handy analytical expressions.

#### *2.1.1. One plane wave approximation*

Firstly, we will elaborate the one plane wave approximation [5] for the *H* – polarization. Consider a 2D PC made of two materials with permittivity tensors *ε*˜ *<sup>a</sup>* and *ε*˜ *<sup>b</sup>*, and assume the fill factor of the material with the permittivity tensor *ε*˜ *<sup>a</sup>* is *f*. The one plane wave technique assumes a very weak dielectric modulation such that *β*˜*<sup>r</sup>* (*m* - *n*), can be approximated with only one plane wave (i.e., one Fourier component) as *β*˜*<sup>r</sup>* (*m* - *n*)= *δmnε*˜ <sup>0</sup> -1 , where *ε*˜ <sup>0</sup> is the averaged dielectric tensor,

$$
\tilde{\varepsilon}\_0 = \tilde{\varepsilon}\_a f + (\mathbf{l} - f)\tilde{\varepsilon}\_b. \tag{4}
$$

Using this assumption, and *h*=[0.. 1... 0]' with 1 at the *m*-th position [Eqn. 2], we can show that,

for a given wavevector in the first BZ. In Eqn. 1, **H**(**r**) is the position (**r**) dependent magnetic

constant profile of the PC. Using the plane wave expansion method [9,6-7,16], we can transform Eqn. 1 to a matrix eigen-value problem. Consequently, the frequency containing term (2π*ω*/

In this equation, **h**=[*h*<sup>1</sup> *h*2...]', where *hn* is the Fourier expansion coefficient of **H**(**r**) associated

Here |*kn*>=**k** – **G***n*, with **k=**[*kx*, *ky*]'. In Eqn. 3, <...> represents the inner product of the vector,

transform coefficient of the tensor dielectric function [associated with **G***n*], and similarly for the *E*-polarization, *β*(*n*) is the inverse Fourier transform coefficient of the scalar dielectric

Eqn. 2 defines an EFS for a given frequency, and has to be evaluated using the entire basis of the reciprocal space, which is infinite in number. In practice, the number of basis (i.e. the number of plane waves) is limited to a number for which the corresponding result achieves a required degree of accuracy. In general, larger the dielectric modulation, larger the number of required plane waves. For PCs with weak dielectric modulations, the number of plane waves can be significantly less. In the following sections, we will use one and two plane waves to approximate the EFS of the weakly modulated PCs, and as we shall see, such approximations

Firstly, we will elaborate the one plane wave approximation [5] for the *H* – polarization. Consider a 2D PC made of two materials with permittivity tensors *ε*˜ *<sup>a</sup>* and *ε*˜ *<sup>b</sup>*, and assume the fill factor of the material with the permittivity tensor *ε*˜ *<sup>a</sup>* is *f*. The one plane wave technique

> e

(*m* - *n*)= *δmnε*˜ <sup>0</sup>


%% % = +- (1 ) . *a b f f* (4)

( ) , polarization. ( ), polarization

=Ω in Eqn. 1, can be written as a matrix vector product,

with the reciprocal lattice vector **G***n*. The matrix element of *M*

*mn*

matrix and vector. For the *H* – polarization, *β*˜*<sup>r</sup>*

lead to handy analytical expressions.

assumes a very weak dielectric modulation such that *β*˜*<sup>r</sup>*

0 ee

one plane wave (i.e., one Fourier component) as *β*˜*<sup>r</sup>*

*2.1.1. One plane wave approximation*

dielectric tensor,

*M*

b

*m n*

b

% *mr n*

ìï - - <sup>=</sup> <sup>í</sup> ï - - î

*k m nk H*

^ is a differential operator that depends on the polarization, and the dielectric

ˆ W= × **h h** *M* . (2)

*k k mn E* (3)

^ [6] is defined as

(*n*) is the 2 by 2 matrix defining the inverse Fourier

(*m* - *n*), can be approximated with only

, where *ε*˜ <sup>0</sup> is the averaged

field vector, and *D*

*a*)2

122 Photonic Crystals

function.

$$
\mathfrak{Q} = \left\langle k\_m \left| \tilde{\varepsilon}\_0^{-1} \right| k\_m \right\rangle. \tag{5}
$$

Eqn. 4 has to be true for every integer *m*. The tensor *ε*˜ 0 in general can be written as *ε*˜ <sup>0</sup> <sup>=</sup> *<sup>Q</sup> <sup>T</sup> <sup>ε</sup>*˜ *<sup>p</sup><sup>Q</sup>* [6], where *ε*˜ *<sup>p</sup>* is the dielectric tensor of the anisotropic material in its principal coordinate system, *Q* is a rotational operator (an orthogonal operator), and *QT* is the transpose of *Q*. With this *ε*˜ <sup>0</sup>, Eqn. 5 can be rewritten using the inner product properties [21] to,

$$
\Omega \Omega = \left\langle k\_m \left| \underline{Q}^T \tilde{\boldsymbol{\varepsilon}}\_p^{-1} \underline{Q} \right| k\_m \right\rangle = \left\langle \underline{Q} k\_m \left| \tilde{\boldsymbol{\varepsilon}}\_p^{-1} \right| \underline{Q} k\_m \right\rangle,\tag{6}
$$

where *Q*|*km*>=**k'** – **G'***m*=[*kx'* – *Gmx'*, *ky'* – *Gmy'*]' is the rotated vector of |*km*>=**k** – **G***m*=[*kx* – *Gmx*, *ky* – *Gmy*]'. If one of the material is assumed to be isotropic, and the other is assumed to be anisotropic which is often the case [6,19], then using Eqn. 4 we have,

$$
\tilde{\boldsymbol{\varepsilon}}\_{p}^{-1} = \begin{pmatrix}
1/\left\{\boldsymbol{\varepsilon}\_{a}f + \boldsymbol{\varepsilon}\_{b1}[\mathbf{l} - f]\right\} & \mathbf{0} \\
\mathbf{0} & \mathbf{1}/\left\{\boldsymbol{\varepsilon}\_{a}f + \boldsymbol{\varepsilon}\_{b2}[\mathbf{l} - f]\right\}
\end{pmatrix},\tag{7}
$$

where *ε*˜ *<sup>a</sup>* is taken as an identity matrix multiplied by a constant, *ε<sup>a</sup>* = *na* 2 (i.e., an isotropic material) and *ε*˜ *<sup>b</sup>* is assumed as the dielectric tensor of the anisotropic material in the principal coordinate system, with *εb*1 and *εb*2 being the principal dielectric constants. Substituting the expressions for *Q*|*km*> and *ε*˜ *<sup>p</sup>* -1 in Eqn. 6, we obtain the following equation for the EFS of the *H*-polarization in the rotated frame,

$$\frac{(k\_{\rm x^{\cdot}} - G\_{\rm nx^{\cdot}})^2}{\varepsilon\_a f + \varepsilon\_{b1} [1 - f]} + \frac{(k\_{\rm y^{\cdot}} - G\_{\rm my^{\cdot}})^2}{\varepsilon\_a f + \varepsilon\_{b2} [1 - f]} = \Omega. \tag{8}$$

Eqn. 8 describes an ellipse in the rotated frame (*kx'*, *ky'*) with the origin at **G'***m*=[*Gmx' Gmy'*]'. As the equation is true for every *m*, there will be repetitions of the same ellipse for each *m* [corresponding to each reciprocal lattice vector].

Similarly, for th*e E*-polarization, it can be shown that, the one plane wave approximation leads to |*km*|<sup>2</sup> =Ω*ε*0, where *β*(*m* - *n*)= *δmn* / *ε*0, has been used for Eqn. 3. Assuming |*km*>=**k** – **G***m*, we will arrive at the same equation as Eqn. 8, however the denominators at the left hand side of the Eqn. 8 are replaced with *ε*<sup>0</sup> = *ε<sup>a</sup> f* + *εb*3(1 - *f* ). Here, *εb*<sup>3</sup> is the principal dielectric constant of the anisotropic material experienced by the *E*-polarization. Note that for the *E* – polarization Eqn. 8 describes a circle.

To illustrate the EFS construction, let's first assume both materials are isotropic (i.e. *εb*<sup>1</sup> =*εb*<sup>2</sup> =*εb*<sup>3</sup> =*nb* 2 ) and consider a 2D PC with a square lattice, *na*=1.6, *f*=0.4, and a very small dielectric modulation (i.e. *na* ≈*nb*). Note that, with isotropic materials, and very small dielectric modulation, the distinction between the EFSs of *E* and *H* – polarizations vanishes.

For an instance of the EFS construction, consider *ω*=0.5. Using the radius given by Eqn. 8, let's draw circles at each reciprocal lattice point as shown in Figure 3(a). Each closed contour constitutes to an EFS of a particular band. The bands are indexed according to their positions in the extended zone diagram [14-15]. isotropic materials, and very small dielectric modulation, the distinction between the EFSs of *E* and *H*– polarizations vanishes.

each reciprocal lattice point as shown in **Figure 3(a)**. Each closed contour constitutes to an EFS of a particular band. The

= 0.5. Using the radius given by **Eqn. 8**, let's draw circles at

For an instance of the EFS construction, consider

bands are indexed according to their positions in the extended zone diagram [**14-15**].

blue, band 3 – red, band 4 – pink, band 5 – yellow. The thin lines indicate boundaries of the BZs (a) EFS construction for = 0.5 (b) EFS for = 0.5 (c) EFS for = 0.36 (d) EFS for = 0.67. Now let's demonstrate the validity of the EFS obtained from the one plane wave approximation, by comparing it with the numerically evaluated [**9**]. In **Figures 4(a) and 4(b),** we plot EFSs ( = 0.5) for *nb* = 2.0 and 2.6, respectively, and all other parameters are kept the same as in **Fig. 3**. The one plane wave construction is shown in the black dashed lines, **Figure 3.** EFS constructions of a square lattice PC with *f*=0.4, *na* = 1.6 and *na* ≈*nb*. Band indices: band 1 – black, band 2 – blue, band 3 – red, band 4 – pink, band 5 – yellow. The square boxes indicate boundaries of the BZs (a) EFS construc‐ tion for *ω*=0.5 (b) EFS for *ω*=0.5 (c) EFS for *ω*=0.36 (d) EFS for *ω*=0.67.

Figure 3 EFS constructions of a square lattice PC with *f* = 0.4, ݊ ൌ1.6 and݊ ൎ ݊ . Band indices: band 1 – black, band 2 –

whereas the EFS obtained from the numerical calculation is highlighted in color. As it is clear from **Fig. 4(a),** for the PC with weak dielectric modulation (i.e., *nb* = 2.0), the one plane wave construction agrees well with the numerically calculated EFS. However, when the dielectric modulation increases, the degree of disagreement increases [**Fig. 4(b)**]. Nevertheless, in PCs with large dielectric modulations, one plane wave approximation still can be used to gain a rough idea on the EFSs shapes, for frequencies far from the photonic band edge. It is important to note that one or two plane wave approximations (discussed in **Sec. 2.1.2**) fail when the frequency is close to the photonic band edge (see **Fig. 2**; **Sec. 4.2;** and [**2**]). Now let's demonstrate the validity of the EFS obtained from the one plane wave approxima‐ tion, by comparing it with the numerically evaluated. In Figures 4(a) and 4(b), we plot EFSs (*ω*=0.5) for *nb*=2.0 and 2.6, respectively, and all other parameters are kept the same as in Fig. 3. The one plane wave construction is shown in the black dashed lines, whereas the EFS obtained from the numerical calculation is highlighted in color. As it is clear from Fig. 4(a), for the PC with weak dielectric modulation (i.e., *nb*=2.0), the one plane wave construction agrees well with the numerically calculated EFS. However, when the dielectric modulation increases, the degree of disagreement increases (Fig. 4(b)). Nevertheless, in PCs with large dielectric modulations, one plane wave approximation still can be used to gain a rough idea on the EFSs shapes, for frequencies far from the photonic band edge. It is important to note that one or two plane wave approximations (discussed in Sec. 2**.**1**.**2) fail when the frequency is close to the photonic band edge (see Fig. 14; Sec. 4**.**2**;** and [2]).

To illustrate the EFS construction, let's first assume both materials are isotropic (i.e.

dielectric modulation (i.e. *na* ≈*nb*). Note that, with isotropic materials, and very small dielectric

For an instance of the EFS construction, consider *ω*=0.5. Using the radius given by Eqn. 8, let's draw circles at each reciprocal lattice point as shown in Figure 3(a). Each closed contour constitutes to an EFS of a particular band. The bands are indexed according to their positions

isotropic materials, and very small dielectric modulation, the distinction between the EFSs of *E* and *H*– polarizations

each reciprocal lattice point as shown in **Figure 3(a)**. Each closed contour constitutes to an EFS of a particular band. The

Figure 3 EFS constructions of a square lattice PC with *f* = 0.4, ݊ ൌ1.6 and݊ ൎ ݊ . Band indices: band 1 – black, band 2 – blue, band 3 – red, band 4 – pink, band 5 – yellow. The thin lines indicate boundaries of the BZs (a) EFS construction for

all other parameters are kept the same as in **Fig. 3**. The one plane wave construction is shown in the black dashed lines, whereas the EFS obtained from the numerical calculation is highlighted in color. As it is clear from **Fig. 4(a),** for the PC with weak dielectric modulation (i.e., *nb* = 2.0), the one plane wave construction agrees well with the numerically calculated EFS. However, when the dielectric modulation increases, the degree of disagreement increases [**Fig. 4(b)**]. Nevertheless, in PCs with large dielectric modulations, one plane wave approximation still can be used to gain a rough idea on the EFSs shapes, for frequencies far from the photonic band edge. It is important to note that one or two plane wave approximations (discussed in **Sec. 2.1.2**) fail when the frequency is close to the photonic band edge (see **Fig. 2**; **Sec. 4.2;** and [**2**]).

Now let's demonstrate the validity of the EFS obtained from the one plane wave approxima‐ tion, by comparing it with the numerically evaluated. In Figures 4(a) and 4(b), we plot EFSs (*ω*=0.5) for *nb*=2.0 and 2.6, respectively, and all other parameters are kept the same as in Fig. 3. The one plane wave construction is shown in the black dashed lines, whereas the EFS obtained from the numerical calculation is highlighted in color. As it is clear from Fig. 4(a), for the PC with weak dielectric modulation (i.e., *nb*=2.0), the one plane wave construction agrees well with the numerically calculated EFS. However, when the dielectric modulation increases,

(c) (d)

(a) (b)

= 0.5) for *nb* = 2.0 and 2.6, respectively, and

 = 0.67. Now let's demonstrate the validity of the EFS obtained from the one plane wave approximation, by comparing it

**Figure 3.** EFS constructions of a square lattice PC with *f*=0.4, *na* = 1.6 and *na* ≈*nb*. Band indices: band 1 – black, band 2 – blue, band 3 – red, band 4 – pink, band 5 – yellow. The square boxes indicate boundaries of the BZs (a) EFS construc‐

modulation, the distinction between the EFSs of *E* and *H* – polarizations vanishes.

For an instance of the EFS construction, consider

bands are indexed according to their positions in the extended zone diagram [**14-15**].

) and consider a 2D PC with a square lattice, *na*=1.6, *f*=0.4, and a very small

= 0.5. Using the radius given by **Eqn. 8**, let's draw circles at

*εb*<sup>1</sup> =*εb*<sup>2</sup> =*εb*<sup>3</sup> =*nb*

124 Photonic Crystals

2

in the extended zone diagram [14-15].

= 0.5 (b) EFS for

= 0.5 (c) EFS for

tion for *ω*=0.5 (b) EFS for *ω*=0.5 (c) EFS for *ω*=0.36 (d) EFS for *ω*=0.67.

with the numerically evaluated [**9**]. In **Figures 4(a) and 4(b),** we plot EFSs (

= 0.36 (d) EFS for

vanishes.

**Figure 4.** EFS constructions (dashed black line) and the numerically calculated EFS (color) of a square lattice PC with *<sup>f</sup>*=0.4 *na*=1.6, and *ω*=0.5. Band indices: band 3 – red, band 4 – pink, band 5 – yellow. (a) EFS for *nb*=2.0 (b) EFS for *nb*=2.6

Finally, we provide an EFS construction example for a 2D PC made of an anisotropic material. Consider a square lattice PC with an anisotropic material ( *εb*1=1.6, *εb*2=2.0), *na*=1.8, and *f*=0.2. The principal axis with the dielectric constant *εb*<sup>1</sup> is oriented 45° with respect to the *kx* – axis of the PC, as shown Figure 5(a). The *Q* operator in Eqn. 6 is a 45° anticlockwise rotational operator. Hence, based on Eqns. 6 – 8, we draw ellipse at each reciprocal lattice point in the rotated coordinate system as shown in Fig. 5(a). The numerically calculated EFS for the *H* – polarization with *ω*=0.5 is shown in Fig. 5(b), and as we can readily verify from the figure, both the construction and numerical evaluation share a good agreement.

**Figure 5.** EFS construction [*ω*=0.5] for the *H*-polarization (dashed black line), and the numerically calculated EFS (col‐ or) of a square lattice PC with an anisotropic material (*f*=0.2,*na*=1.8, *εb*1=1.6, *εb*2=2.0). Band indices: band 2 – blue, band 3 – red, band 4 – pink. (a) EFS construction (b) EFS with the band index assignment.

#### *2.1.2. Two plane wave approximation*

Although, one plane wave approximation seems to be good in approximating the EFS of a 2D PC with a weak dielectric modulation, a magnified version of the one plane wave EFS, will show that this approach is unable to approximate the edges of the EFS accurately. This problem can be addressed by using two plane waves approximation.

For the sake of simplicity, we will demonstrate the two plane waves approximation for 2D PCs made of isotropic materials. We will further assume the dielectric modulation is finite and weak, such that only two Fourier coefficients of *β*(*i*), *β*(0) and *β*(1), are significant. With these, we can approximate Eqn. 2 as,

$$
\Delta \Omega = h\_i M\_{\,i,l} h\_i + h\_i M\_{\,i,l+1} h\_{i+1} + h\_{i+1} M\_{\,i+1,l} h\_i + h\_{i+1} M\_{\,i+1,l+1} h\_{i+1},\tag{9}
$$

which has to be true for every *i*. We can choose *i*=1, and the eigenvalues of the resulting *M*, can be obtained by finding the determinant of the matrix,

$$
\begin{bmatrix}
\boldsymbol{M}\_{11} - \boldsymbol{\Omega} & \boldsymbol{M}\_{12} \\
\boldsymbol{M}\_{21} & \boldsymbol{M}\_{22} - \boldsymbol{\Omega}
\end{bmatrix}.
\tag{10}
$$

Using the fact *Mij*=*Mji*, and denoting *β*(*i*) as *β<sup>i</sup>* , we can show,

Hence, based on Eqns. 6 – 8, we draw ellipse at each reciprocal lattice point in the rotated coordinate system as shown in Fig. 5(a). The numerically calculated EFS for the *H* – polarization with *ω*=0.5 is shown in Fig. 5(b), and as we can readily verify from the figure, both the

**Figure 5.** EFS construction [*ω*=0.5] for the *H*-polarization (dashed black line), and the numerically calculated EFS (col‐ or) of a square lattice PC with an anisotropic material (*f*=0.2,*na*=1.8, *εb*1=1.6, *εb*2=2.0). Band indices: band 2 – blue,

Although, one plane wave approximation seems to be good in approximating the EFS of a 2D PC with a weak dielectric modulation, a magnified version of the one plane wave EFS, will show that this approach is unable to approximate the edges of the EFS accurately. This problem

band 3 – red, band 4 – pink. (a) EFS construction (b) EFS with the band index assignment.

can be addressed by using two plane waves approximation.

*2.1.2. Two plane wave approximation*

construction and numerical evaluation share a good agreement.

126 Photonic Crystals

$$
\Delta \Omega = \mathcal{J}\_0(\boldsymbol{k\_0}^2 + \boldsymbol{k\_1}^2) \pm \sqrt{\mathcal{J}\_0^2 \left(\boldsymbol{k\_0}^2 - \boldsymbol{k\_1}^2\right)^2 + \left(\mathcal{Q}\mathcal{J}\_0 \boldsymbol{k\_0}\boldsymbol{k\_1}\right)^2},\tag{11}
$$

where *ki kj* represents <*ki* |*kj* > and |*ki* ||*kj* | for the *H*-and *E*-polarizations, respectively. Note that for a weak dielectric modulation, *β*0 >> |*β*1| and *β*0 > 0.

Eqn. 11 can be evaluated using specific choices of wavevectors and reciprocal lattice vectors. For an instance, let's examine EFSs near the first band gap of the *E*-polarization, in the square lattice PC around the X(-*π*/*a*,0) point. With *g*=2π/*a*, **k**=[*kx ky*]', **G**0=[0 0]', and **G**1=[-*g* 0]', the terms in Eqn. 11 become,

$$\begin{aligned} \mathcal{B}\_0(k\_0^2 + k\_1^2) &= 2\mathcal{B}\_0[(k\_x + \mathbf{g}/2)^2 + k\_y^2 + \mathbf{g}^2/4] \\ \mathcal{B}\_0^2(k\_0^2 - k\_1^2)^2 &= \left[2\mathcal{B}\_0\mathbf{g}(k\_x + \mathbf{g}/2)\right]^2 \\ (\mathcal{B}\_0k\_0k\_1)^2 &= 4\mathcal{B}\_1^2[(k\_x + \mathbf{g}/2)^2 + k\_y^2 + \mathbf{g}^2/4]^2 - \left[2\mathcal{B}\_1(k\_x + \mathbf{g}/2)\right]^2. \end{aligned} \tag{12}$$

Let's move the origin of the reciprocal space from (0,0) to the X point, by writing *kX*=*kx*+*g*/2. In the new coordinate system, using Eqns. 11 and 12, we can show that,

$$
\Delta \Omega = \beta\_0 (\boldsymbol{k\_X}^2 + \boldsymbol{k\_y}^2 + \mathbf{g}^2 / 4) \pm \sqrt{(\beta\_0 \boldsymbol{k\_X} \mathbf{g})^2 + \beta\_1^2 \left(\boldsymbol{k\_X}^2 + \boldsymbol{k\_y}^2 + \mathbf{g}^2 / 4\right)^2 - \left(\beta\_1 \boldsymbol{k\_X} \mathbf{g}\right)^2}.\tag{13}
$$

Eqn. 13 does not provide any insight to the shape of the EFS. More useful information can be obtained if we can transform the equation to a simpler form. To do this, we first seek an approximation to the square root term in Eqn. 13, for a small *kX*. For a small *kX*, the square root term in Eqn. 13, can be written as *y* = *P* <sup>2</sup> + *Q* <sup>2</sup> , where *P* = |*β*0*kX g*| and *Q* = *β*1(*kX* <sup>2</sup> <sup>+</sup> *<sup>k</sup> <sup>y</sup>* <sup>2</sup> + *g* <sup>2</sup> / 4) (the term *β*1*kX g* in Eqn. 13 is neglected). We can approximate *y* using a binomial expansion as <sup>≈</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>P</sup>* <sup>2</sup> <sup>2</sup>*<sup>Q</sup>* . However, this requires *Q* >> *P*, and in fact this is a good assumption as we shall justify later in the discussion. Using the binomial expansion of *y*, for a small *kX* and *ky*, Eqn. 13 can be simplified to,

$$
\left[\Omega\_1 - \frac{\mathbf{g}^2}{4} \left(\mathcal{\beta}\_0 - |\mathcal{\beta}\_1|\right)\right] = k\_\chi \,^2 \left(\mathcal{\beta}\_0 - |\mathcal{\beta}\_1| - \frac{2\mathcal{\beta}\_0^{-2}}{|\mathcal{\beta}\_1|}\right) + k\_y \,^2 \left(\mathcal{\beta}\_0 - |\mathcal{\beta}\_1|\right), \tag{14}
$$

$$
\left[\Omega\_2 - \frac{\mathbf{g}^2}{4} \left(\mathcal{J}\_0 + \left|\mathcal{J}\_1\right|\right)\right] = k\_\chi^{-2} \left(\mathcal{J}\_0 + \left|\mathcal{J}\_1\right| + \frac{2\left|\mathcal{J}\_0\right|^2}{\left|\mathcal{J}\_1\right|}\right) + k\_\chi^{-2} \left(\mathcal{J}\_0 + \left|\mathcal{J}\_1\right|\right), \tag{15}
$$

where the two solutions of Eqn. 13 are denoted as Ω1 and Ω2. These two solutions correspond to frequencies in band 1 [Ω1] and band 2 [Ω2]. With *β*<sup>0</sup> > 0 and *β*0 > |*β*1|, we have *β*<sup>0</sup> <sup>+</sup> <sup>|</sup>*β*1| <sup>+</sup> <sup>2</sup>*β*<sup>0</sup> 2 |*β*1| > 0, in Eqn. 15. Therefore, EFSs for band 2 are elliptical (provided that *Q* >> *P*) with the lengths of the semi –major and –minor proportional to the reciprocals of *β*<sup>0</sup> <sup>+</sup> <sup>|</sup>*β*1| <sup>+</sup> <sup>2</sup>*β*<sup>0</sup> 2 <sup>|</sup>*β*1| and *β*<sup>0</sup> + |*β*1|, respectively. On the other hand, note that, the EFSs for band 1 (Eqn. 14, with *<sup>β</sup>*<sup>0</sup> - <sup>|</sup>*β*1| - <sup>2</sup>*β*<sup>0</sup> 2 <sup>|</sup>*β*1<sup>|</sup> < 0) are not elliptical.

Now let's examine the validity of the assumption *Q* > *P* in deriving Eqns. 14 and 15. The expressions *P* and *Q* are linear and quadratic functions of *kX*, respectively, and the minimum point of *Q* is *f* (*k <sup>y</sup>*) =|*β*1|(*k <sup>y</sup>* <sup>2</sup> <sup>+</sup> *<sup>g</sup>* <sup>2</sup> <sup>4</sup> ). Hence, a sufficient condition for *Q* > *P* is simply *f* (*k <sup>y</sup>*) >*P*. Inserting the expression for *P*, this condition becomes,

$$-\frac{\left|\left.\beta\_1\right|}{\beta\_0\mathcal{g}}(k\_y^{-2} + \frac{\mathcal{g}}{4}) < k\_X < \frac{\left|\left.\beta\_1\right|}{\beta\_0\mathcal{g}}(k\_y^{-2} + \frac{\mathcal{g}}{4})\right| $$

The conditions on *ky* can be obtained by solving the quadratic in-equality *Q* – *P* > 0. Solving this in-equality, we have,

$$k\_y > \frac{\mathcal{S}}{2 \parallel \beta\_1 \parallel} \sqrt{\beta\_0^2 - \beta\_1^2}.$$

Eqn. 13 does not provide any insight to the shape of the EFS. More useful information can be obtained if we can transform the equation to a simpler form. To do this, we first seek an approximation to the square root term in Eqn. 13, for a small *kX*. For a small *kX*, the square root

(the term *β*1*kX g* in Eqn. 13 is neglected). We can approximate *y* using a binomial expansion as

( ) ( ) <sup>2</sup> <sup>2</sup> 2 2 0

<sup>2</sup> , <sup>4</sup>

( ) ( ) <sup>2</sup> <sup>2</sup> 2 2 0

<sup>2</sup> , <sup>4</sup>

where the two solutions of Eqn. 13 are denoted as Ω1 and Ω2. These two solutions correspond to frequencies in band 1 [Ω1] and band 2 [Ω2]. With *β*<sup>0</sup> > 0 and *β*0 > |*β*1|, we have *β*<sup>0</sup> <sup>+</sup> <sup>|</sup>*β*1| <sup>+</sup> <sup>2</sup>*β*<sup>0</sup>

> 0, in Eqn. 15. Therefore, EFSs for band 2 are elliptical (provided that *Q* >> *P*) with the lengths

*β*<sup>0</sup> + |*β*1|, respectively. On the other hand, note that, the EFSs for band 1 (Eqn. 14, with

Now let's examine the validity of the assumption *Q* > *P* in deriving Eqns. 14 and 15. The expressions *P* and *Q* are linear and quadratic functions of *kX*, respectively, and the minimum

<sup>4</sup> )<*kX* <sup>&</sup>lt;

The conditions on *ky* can be obtained by solving the quadratic in-equality *Q* – *P* > 0. Solving

2| *<sup>β</sup>*<sup>1</sup> <sup>|</sup> *<sup>β</sup>*<sup>0</sup>


<sup>2</sup> −*β*<sup>1</sup> 2.

of the semi –major and –minor proportional to the reciprocals of *β*<sup>0</sup> <sup>+</sup> <sup>|</sup>*β*1| <sup>+</sup> <sup>2</sup>*β*<sup>0</sup>

1

1

*<sup>g</sup> k k* (15)

b

b

*<sup>g</sup> k k* (14)

b

b

1 01 0 1 0 1

2 01 0 1 0 1

 bb

ê ú W- + = + + + + ç ÷ ë û è ø *X y*

 bb

ê ú W- - = - - + - ç ÷ ë û è ø *X y*

é ù æ ö

é ù æ ö

<sup>2</sup>*<sup>Q</sup>* . However, this requires *Q* >> *P*, and in fact this is a good assumption as we shall justify later in the discussion. Using the binomial expansion of *y*, for a small *kX* and *ky*, Eqn.

, where *P* = |*β*0*kX g*| and *Q* = *β*1(*kX*

b b

b b

<sup>4</sup> ). Hence, a sufficient condition for *Q* > *P* is simply *f* (*k <sup>y</sup>*) >*P*.

<sup>2</sup> <sup>+</sup> *<sup>g</sup>* <sup>2</sup> 4 ) <sup>2</sup> <sup>+</sup> *<sup>k</sup> <sup>y</sup>*

<sup>2</sup> + *g* <sup>2</sup> / 4)

2 |*β*1|

2 <sup>|</sup>*β*1| and

term in Eqn. 13, can be written as *y* = *P* <sup>2</sup> + *Q* <sup>2</sup>

bb

bb

<sup>≈</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>P</sup>* <sup>2</sup>

128 Photonic Crystals

*<sup>β</sup>*<sup>0</sup> - <sup>|</sup>*β*1| - <sup>2</sup>*β*<sup>0</sup>

2

point of *Q* is *f* (*k <sup>y</sup>*) =|*β*1|(*k <sup>y</sup>*

this in-equality, we have,

<sup>|</sup>*β*1<sup>|</sup> < 0) are not elliptical.

<sup>2</sup> <sup>+</sup> *<sup>g</sup>* <sup>2</sup>

<sup>−</sup> <sup>|</sup> *<sup>β</sup>*<sup>1</sup> <sup>|</sup> *<sup>β</sup>*0*<sup>g</sup>* (*ky*

<sup>2</sup> <sup>+</sup> *<sup>g</sup>* <sup>2</sup>

*ky* <sup>&</sup>gt; *<sup>g</sup>*

Inserting the expression for *P*, this condition becomes,

13 can be simplified to,

**Figure 6.** EFS approximations using two plane waves approximation [square lattice PC with *na*=1.6, *nb*=1.8, and *f*=0.2]. The numerically calculated EFS (thin black lines) and the approximated EFS (thick blue lines) for frequencies in the (a) first band (b) second band.

To illustrate the EFS construction using the two plane waves approximation, consider a square lattice PC with *na*=1.6, *nb*=1.8, and *f*=0.2. The numerically calculated EFS and the approximated EFS, for frequencies in the first and the second band are shown in Figures 6(a) and 6(b), respectively. In both figures, we have evaluated Eqn. 11 for-π/*a* < *ky* <-π/*a* and-2π/*a* < *kx* < 0, and the results share a good agreement with those of numerical evaluation. As Eqn. 9 has to be true for every *i*, we can translate the EFS approximated for *i*=1 by a vector **G***<sup>i</sup>* . Furthermore the EFSs in the neighborhood at the other symmetrical X points [i.e. (*π*/*a*,0), (0,*π*/*a*) and (0,-*π*/*a*)] can be obtained using the EFS approximated in the neighborhood of X(-*π*/*a*,0) by point group – symmetry operations [9, 22].

**Figure 7.** The merit of the two plane waves approximation. EFSs for a square lattice PC with *na*=1.6, *f*=0.2. EFS (*ω*=0.31) obtained from the one plane wave approximation (dashed black line), two plane waves approximation (thick blue line), and the numerical calculation (thin red line), for (a) *nb*=1.8 (b) *nb*=2.0. Both *kx* and *ky* in (a) and (b) have units of 1/*a*.

In order to appreciate the ability and inability of the two plane waves approximation, we plot EFSs (band 2, *ω*=0.3) obtained from the one plane wave approximation, two plane waves approximation, and the full numerical calculation in Figures 7(a) and 7(b), for *nb*=1.8 and *nb*=2.0, respectively. The one plane wave – EFS is constructed using two circles originated from (0,0) and (-*g*, 0) points. Thus, it always predicts sharp edges for the EFS (see band 2 in Figs. 3(c), 7(a) and 7(b)). The sharp edges does not appear in the numerically calculated EFS, which is the accurate EFS. The failure of the one plane wave method is corrected in the two plane waves approximation. The two plane waves approximation perfectly matches the numerical coun‐ terpart when the dielectric modulation is small (Fig. 7(a)). As expected, when the dielectric modulation becomes large, the two plane waves approximation becomes poorer (Fig. 7(b)), however the approach still exhibits a better accuracy compared to the one plane wave approach.

#### **3. Propagation directions**

respectively. In both figures, we have evaluated Eqn. 11 for-π/*a* < *ky* <-π/*a* and-2π/*a* < *kx* < 0, and the results share a good agreement with those of numerical evaluation. As Eqn. 9 has to be

EFSs in the neighborhood at the other symmetrical X points [i.e. (*π*/*a*,0), (0,*π*/*a*) and (0,-*π*/*a*)] can be obtained using the EFS approximated in the neighborhood of X(-*π*/*a*,0) by point group

**Figure 7.** The merit of the two plane waves approximation. EFSs for a square lattice PC with *na*=1.6, *f*=0.2. EFS (*ω*=0.31) obtained from the one plane wave approximation (dashed black line), two plane waves approximation (thick blue line), and the numerical calculation (thin red line), for (a) *nb*=1.8 (b) *nb*=2.0. Both *kx* and *ky* in (a) and (b) have units of 1/*a*.

In order to appreciate the ability and inability of the two plane waves approximation, we plot EFSs (band 2, *ω*=0.3) obtained from the one plane wave approximation, two plane waves approximation, and the full numerical calculation in Figures 7(a) and 7(b), for *nb*=1.8 and *nb*=2.0, respectively. The one plane wave – EFS is constructed using two circles originated from (0,0) and (-*g*, 0) points. Thus, it always predicts sharp edges for the EFS (see band 2 in Figs. 3(c), 7(a)

. Furthermore the

true for every *i*, we can translate the EFS approximated for *i*=1 by a vector **G***<sup>i</sup>*

– symmetry operations [9, 22].

130 Photonic Crystals

When light crosses the boundary between two homogenous mediums (medium 1 and medium 2), light refracts. The light in medium 1 with an incident angle, *θ<sup>i</sup>* , can excite two symmetrical waves (with angles *θp*; see Figure 8) in medium 2. What happens to this refraction picture, when we replace the homogenous dielectric medium 2, with a PC?. When we replace the homogenous medium 2, with a PC, the simple refraction picture based on the Snell's law will disappear. A more general technique has to be used, in order to find light propagation directions. This section reviews the well – known method [1-4] of finding light propagation directions in PCs, based on their EFS. As a pre-requisite for the second part in this section, we will first give a brief review on anisotropic PCs [22].

**Figure 8.** Light propagation from a homogenous dielectric medium 1 to a homogenous dielectric medium 2

#### **3.1. Orientation parameter — Anisotropic PC**

A PC may be simply viewed as a lattice with a motif attached to each lattice point. If the reorientation of the motif causes the symmetry elements of the PC to change, then the corre‐ sponding PC is an anisotropic PC, as opposed to an isotropic PC, where the orientation of the motif is irrelevant to the symmetry of the PC. A 2D PC made of only isotropic dielectric materials, with a circular motif in a 2D lattice, is a clear example of a 2D isotropic PC. Aniso‐ tropic PCs, on the other hand, can be geometrically anisotropic or materially anisotropic. Figure 9(a) shows the geometry of a 2D hexagonal lattice PC with a square motif. The orien‐ tation of the square motif with respect to the underlying lattice plays a crucial role in deter‐ mining the optical properties of the 2D PC. In Fig. 9(b), instead of a square motif, we have a circular motif for which the corresponding orientation is irrelevant. If all materials are isotropic, then the geometries in Figs. 9(a) and 9(b) represent examples of geometrically anisotropic and isotropic PCs, respectively. In the presence of an anisotropic material, the optical properties of the PC will vary in accordance to the orientation of the anisotropic material (i.e., the orientation of the principal axes with respect to the lattice), and therefore the corre‐ sponding PC is defined as a materially anisotropic PC. A PC with a geometry in Fig. 9(b) constitutes to a materially anisotropic PC, if either the matrix or the circular cylinder is an anisotropic medium. On the other hand, in the presence of an anisotropic material, a PC with a geometry as shown in Fig. 9(a) has a mixed anisotropy (i.e., both geometrical and material's anisotropy exist).

As we will show in the next section, the orientation parameter of the anisotropic PC is crucial, as to determine the accurate light propagation direction.

**Figure 9.** Examples of geometries of the 2D hexagonal lattice PC. (a) square motif (b) circular motif

#### **3.2. Technique of determining light propagation direction**

The direction of light propagation in any medium is given by the direction of the group velocity, *vg*. Group velocity is defined as *vg* =∇**k***ω*, where ∇**k** is the gradient operator in the wavevector space. The light propagation direction at the frequency, *ω*, can be determined using the gradient of the corresponding EFS.

Consider the problem of light of propagation from medium 1 (*m*1) to medium 2 (*m*2), as in Fig. 8. Assume *m*1 to be a bulk isotropic dielectric material. For a general case, assume *m*<sup>2</sup> to be a 2D anisotropic PC, and for the sake of discussions, assume the anisotropic PC to be materially anisotropic. In finding the light propagation directions in *m*2, there are two important orien‐ tations. These are:


In Figure 10, we illustrate the definitions of *α* and *ϕ* via an example.

tropic PCs, on the other hand, can be geometrically anisotropic or materially anisotropic. Figure 9(a) shows the geometry of a 2D hexagonal lattice PC with a square motif. The orien‐ tation of the square motif with respect to the underlying lattice plays a crucial role in deter‐ mining the optical properties of the 2D PC. In Fig. 9(b), instead of a square motif, we have a circular motif for which the corresponding orientation is irrelevant. If all materials are isotropic, then the geometries in Figs. 9(a) and 9(b) represent examples of geometrically anisotropic and isotropic PCs, respectively. In the presence of an anisotropic material, the optical properties of the PC will vary in accordance to the orientation of the anisotropic material (i.e., the orientation of the principal axes with respect to the lattice), and therefore the corre‐ sponding PC is defined as a materially anisotropic PC. A PC with a geometry in Fig. 9(b) constitutes to a materially anisotropic PC, if either the matrix or the circular cylinder is an anisotropic medium. On the other hand, in the presence of an anisotropic material, a PC with a geometry as shown in Fig. 9(a) has a mixed anisotropy (i.e., both geometrical and material's

As we will show in the next section, the orientation parameter of the anisotropic PC is crucial,

**Figure 9.** Examples of geometries of the 2D hexagonal lattice PC. (a) square motif (b) circular motif

The direction of light propagation in any medium is given by the direction of the group velocity, *vg*. Group velocity is defined as *vg* =∇**k***ω*, where ∇**k** is the gradient operator in the wavevector space. The light propagation direction at the frequency, *ω*, can be determined using

Consider the problem of light of propagation from medium 1 (*m*1) to medium 2 (*m*2), as in Fig. 8. Assume *m*1 to be a bulk isotropic dielectric material. For a general case, assume *m*<sup>2</sup> to be a 2D anisotropic PC, and for the sake of discussions, assume the anisotropic PC to be materially anisotropic. In finding the light propagation directions in *m*2, there are two important orien‐

**3.2. Technique of determining light propagation direction**

the gradient of the corresponding EFS.

tations. These are:

as to determine the accurate light propagation direction.

anisotropy exist).

132 Photonic Crystals

**Figure 10.** An example of *m*2: A 2D hexagonal lattice PC with rectangular motifs [7]. The figures displays a microscopic view, with the details of the motif arrangement and orientation. The motifs are shown in the dark green color, while the unit cell constructions are in the blue color. The interface is normal to the dashed line (i.e., the normal line). The angle of the symmetrical axis of the motif (pink arrow) with respect to the normal line is defined as the angle of motif orientation. The angle between the symmetrical axis of the lattice (red arrows) and the normal line is defined as the lattice angle.

Figure 11(a) shows momentum space diagram that contains the EFS of the two mediums. The angles *α* and *ϕ* are shown in this diagram. EFS of *m*<sup>1</sup> is a circle, while EFS of *m*2 is arbitrarily assumed. When light crosses the boundary between the two mediums, the transversal momentum (i.e., momentum component that is parallel to the interface) of light is conserved. Hence a line of momentum conservation, being a vertical line parallel to the normal line (green dashed), determined by the incident angle, *θ<sup>i</sup>* , and passing through the EFS of *m*<sup>2</sup> produces two intersection points. The directions of the normal vector at the intersection points between the EFS of *m*2, and the line of momentum conservation give the propagation directions in *m*2. The normal vector with the sign of the vertical component same as the sign of the incident wavevector'svertical component is considered, as a forward wave and the other, is considered as a reverse wave [Fig. 11(a)] [23]. In Fig. 11(a), the sign of ∇*<sup>k</sup> ω* is assumed as negative. However, take note that, depending on the sign of ∇k*ω*, the gradient can be either inward or outward. The consequence of such changes in the sign of ∇k*ω* to the directions of forward and reverse waves is illustrated in Figure 11(b).

**Figure 11.** (a) In-plane propagation of light from *m*1 to *m*2. *m*1: isotropic bulk medium, *m*2: anisotropic PC (b) The sign of the group velocity, and the directions of the forward and reverse waves

## **4. Applications of the EFS analysis**

In this section we will use EFSs to describe few of the peculiar phenomena observed in PCs. Specifically, we will apply the concept of EFS to describe superprism, effective negative index mediums, negative refractions, and superlenses in PCs.

#### **4.1. Superprism effect**

two intersection points. The directions of the normal vector at the intersection points between the EFS of *m*2, and the line of momentum conservation give the propagation directions in *m*2. The normal vector with the sign of the vertical component same as the sign of the incident wavevector'svertical component is considered, as a forward wave and the other, is considered as a reverse wave [Fig. 11(a)] [23]. In Fig. 11(a), the sign of ∇*<sup>k</sup> ω* is assumed as negative. However, take note that, depending on the sign of ∇k*ω*, the gradient can be either inward or outward. The consequence of such changes in the sign of ∇k*ω* to the directions of forward and

**Figure 11.** (a) In-plane propagation of light from *m*1 to *m*2. *m*1: isotropic bulk medium, *m*2: anisotropic PC (b) The sign of

the group velocity, and the directions of the forward and reverse waves

reverse waves is illustrated in Figure 11(b).

134 Photonic Crystals

Superprism effect was first demonstrated in the PC by Kosaka *et al*. [1,24]. As the name suggests, a superprism is a special and a superior version of the ordinary prism. The super‐ prism has an extraordinary sensitivity to the incident wavelengths [24], and incident angles [1].

**Figure 12.** Angle – sensitive superprism effect

The angle sensitive superprism effect can be easily understood. In Figure 12, we show an example of the propagation angle versus incident angle plot at a specific frequency. The parameters for this plot can be found in [12]. The red curve corresponds to a bulk medium, while the blue curve is for a 2D hexagonal lattice PC. For the bulk medium, the propagation angle follows the Snell's Law, and the relationship between the incident angle and propagation angle is smooth (red curve). On the other hand, for the 2D PC, near the normal incidence, the propagation angles become highly sensitive to the incident angles (blue curve). Such a high sensitivity is due the sharp edge observed in the corresponding hexagonal like EFS of the 2D PC [see the blue EFS in the insert].

The wavelength dependent superprism effect is useful for the spatial filtering of multiple wavelengths. Therefore, it is very useful for implementation of a compact arrayed waveguide grating [24]. The operation principle of the wavelength dependent superprism effect is graphically shown in Figure 13. Fig. 13(a) shows an ordinary prism, while Fig. 13(b) shows a superprism.

**Figure 13.** Wavelength dependent superprism effect. (a) Ordinary prism; (b) Superprism [Note that the signs of ∇**<sup>k</sup>** *ω* for the red and blue EFSs, are positives and negatives, respectively.]

Let's consider two closely spaced wavelengths, *λ*1 and *λ*2. The EFSs for these two wavelengths are circles of different sizes in a bulk medium and in an ordinary prism. However, in a PC the EFSs can be very different. For superprism effects EFSs with sharp edges are preferred. By plotting the EFSs of the PC at various normalized frequencies (*a*/*λ*), EFSs with sharp edges can be identified. Then by adjusting *a*, we can design the superprism at the desired wavelengths. In Fig. 13(b), example of EFSs in a 2D hexagonal lattice PC are shown. These EFSs have sharp edges, and the EFSs have different sizes. By selecting the incident angle appropriately, we can design the corresponding EFSs gradient vectors of the two wavelengths, to be at two different curvatures. As shown in Fig. 13(b), this will cause a huge difference in the propagation angles of the two closely spaced wavelengths.

#### **4.2. Effective negative index mediums**

The behavior of an electron near the band edge of a semiconductor can be approximated as that of a free electron with an effective mass. The analogous concept in a PC was first shown by Notomi [2].

The EFS of a 2D PC with a symmetrical unit cell (i.e. the unit cell have a third or higher order rotational axes), becomes circular at the bottom edge of the first band (i.e., in the long wave‐ length limit) and for such a frequency range, the PC behaves like a bulk isotropic medium with an effective refractive index [25-26]. The effective refractive index (*neff*) can be assigned by fitting the EFS to the expression *ω*=*k*/|*neff*| [note that, *ω* and *k* are normalized]. The effective index for the first band is positive, since the first band has a positive sign of ∇**k***ω*. It is also reported that in such 2D PCs, when the modulation of the refractive index become stronger, the EFSs become circular for frequencies near the edges of the higher order bands. Thus for the corresponding range of frequencies, the 2D PC can be homogenized to an isotropic bulk medium with an effective refractive index. However, this effective index can be negative if the sign of ∇**k***ω* of the particular band is negative [2].

For an instance, a 2D photonic band structure of the hexagonal lattice PC with circular rods, and huge refractive index contrast is shown in Figure 14 (only the first two bands are shown) [2]. As we can see from this figure, the EFSs close to the bottom edge of the first band, and the top edge of the second band are circular. As frequency increases, the EFSs of the first band move outwards (increasing in size). Thus, the first band has a positive ∇**k***ω*, and the corre‐ sponding circular EFSs are defined with effective positive refractive indices. On the other hand, in the second band, as frequency increases the EFSs move inwards (decreasing in size). Hence, the second band has a negative ∇**k***ω*, and therefore their circular EFSs have effective negative refractive indices.

**Figure 14.** Example of circular EFSs of a PC with positive and negative effective refractive indices.

**Figure 13.** Wavelength dependent superprism effect. (a) Ordinary prism; (b) Superprism [Note that the signs of ∇**<sup>k</sup>** *ω*

Let's consider two closely spaced wavelengths, *λ*1 and *λ*2. The EFSs for these two wavelengths are circles of different sizes in a bulk medium and in an ordinary prism. However, in a PC the EFSs can be very different. For superprism effects EFSs with sharp edges are preferred. By plotting the EFSs of the PC at various normalized frequencies (*a*/*λ*), EFSs with sharp edges can be identified. Then by adjusting *a*, we can design the superprism at the desired wavelengths. In Fig. 13(b), example of EFSs in a 2D hexagonal lattice PC are shown. These EFSs have sharp edges, and the EFSs have different sizes. By selecting the incident angle appropriately, we can design the corresponding EFSs gradient vectors of the two wavelengths, to be at two different curvatures. As shown in Fig. 13(b), this will cause a huge difference in the propagation angles

The behavior of an electron near the band edge of a semiconductor can be approximated as that of a free electron with an effective mass. The analogous concept in a PC was first shown

The EFS of a 2D PC with a symmetrical unit cell (i.e. the unit cell have a third or higher order rotational axes), becomes circular at the bottom edge of the first band (i.e., in the long wave‐ length limit) and for such a frequency range, the PC behaves like a bulk isotropic medium with an effective refractive index [25-26]. The effective refractive index (*neff*) can be assigned by fitting the EFS to the expression *ω*=*k*/|*neff*| [note that, *ω* and *k* are normalized]. The effective

for the red and blue EFSs, are positives and negatives, respectively.]

of the two closely spaced wavelengths.

**4.2. Effective negative index mediums**

by Notomi [2].

136 Photonic Crystals

How to obtain effective anisotropic materials using PC?. Such investigations may lead to novel polarization splitting, and tunable devices.

It has been shown that, 2D materially anisotropic PCs, with symmetrically no rotational axes of order larger than two, exhibit elliptical EFSs in the long wavelength limit [25-26]. Near the higher order band edges, they do behave like a bulk anisotropic media, and if ∇**k***ω* <0, a set of effective negative principle refractive indices can be defined. The effective principal refractive indices can be defined by fitting to the expression, *k x* 2 *np*<sup>1</sup> <sup>2</sup> <sup>+</sup> *<sup>k</sup> <sup>y</sup>* 2 *np*<sup>2</sup> <sup>2</sup> = *ω* <sup>2</sup> , where *np*1 and *np*2 are the effective principal refractive indices (note that *ω*, *kx*, and *ky* are normalized). As an example, in Figure 15, we show the 2D photonic band structure and the EFSs (second band) for a 2D hexagonal lattice PC with rectangular motif [7]. From this figure, we can see that, near the top edge of the second band, the EFSs are elliptical, and the signs of ∇**<sup>k</sup>** *ω* < 0 (i.e., negative principal refractive indices).

**Figure 15.** (a) 2D photonic band structure of the anisotropic PC – rectangular motif in 2D hexagonal lattice. For the details of the structure, please see [7]. (b) The contour plot of the second band in (a);

#### **4.3. Negative refractions and superlenses**

In the usual (positive) refraction, when light crosses the interface of the two mediums (Fig. 8), light bends on the opposite sides of the normal line. However, if light bends on the same side of the normal line, then the corresponding bending is defined as a negative refraction. Negative refraction was first predicted by Veselago in 1968 [27]. Negative refractions have been observed in artificial mediums including PCs [28-30].

For an instance of negative refraction in the PC, take a look on Figs. 12(a) and 12(b). Both incident and propagation angles in the configuration of Fig. 12(a) are defined to be positive values. If the incident (or the propagation) angle goes the other side of the normal, then it is negative. Using this formalism, it easy to recognize that the red curve (corresponds to the bulk medium) shows propagation angles for the positive refractions, and the blue curve (corre‐ sponds to the PC) shows propagation angles for the negative refractions.

An important application of the negative refraction is a construction of a superlens [31]. Negative refraction alone does not satisfy the requirement to build a superlens. The corre‐ sponding structure also should exhibit an effective negative refractive index. Note that, a

indices can be defined by fitting to the expression,

refractive indices).

138 Photonic Crystals

*k x* 2 *np*<sup>1</sup> <sup>2</sup> <sup>+</sup> *<sup>k</sup> <sup>y</sup>* 2 *np*<sup>2</sup> <sup>2</sup> = *ω* <sup>2</sup>

effective principal refractive indices (note that *ω*, *kx*, and *ky* are normalized). As an example, in Figure 15, we show the 2D photonic band structure and the EFSs (second band) for a 2D hexagonal lattice PC with rectangular motif [7]. From this figure, we can see that, near the top edge of the second band, the EFSs are elliptical, and the signs of ∇**<sup>k</sup>** *ω* < 0 (i.e., negative principal

**Figure 15.** (a) 2D photonic band structure of the anisotropic PC – rectangular motif in 2D hexagonal lattice. For the

In the usual (positive) refraction, when light crosses the interface of the two mediums (Fig. 8), light bends on the opposite sides of the normal line. However, if light bends on the same side of the normal line, then the corresponding bending is defined as a negative refraction. Negative refraction was first predicted by Veselago in 1968 [27]. Negative refractions have

For an instance of negative refraction in the PC, take a look on Figs. 12(a) and 12(b). Both incident and propagation angles in the configuration of Fig. 12(a) are defined to be positive values. If the incident (or the propagation) angle goes the other side of the normal, then it is negative. Using this formalism, it easy to recognize that the red curve (corresponds to the bulk medium) shows propagation angles for the positive refractions, and the blue curve (corre‐

An important application of the negative refraction is a construction of a superlens [31]. Negative refraction alone does not satisfy the requirement to build a superlens. The corre‐ sponding structure also should exhibit an effective negative refractive index. Note that, a

details of the structure, please see [7]. (b) The contour plot of the second band in (a);

been observed in artificial mediums including PCs [28-30].

sponds to the PC) shows propagation angles for the negative refractions.

**4.3. Negative refractions and superlenses**

, where *np*1 and *np*2 are the

**Figure 16.** Rays from a point object (a) passing through a positive refractive index medium (b) passing through a nega‐ tive refractive index medium. In both (a) and (b), the reflected rays are not shown.

negative refraction does not guarantee a negative refractive index. A effective negative refractive index (*neff*) can be only defined when the EFS is circular (see Sec. 4**.**2). In Fig. 12(b), although we see negative refractions, the corresponding EFS (blue color) is not circular. Thus, in this case a negative refractive index cannot be defined. However, negative refractive indices can be defined for frequencies close to the top edge of the second band in Fig. 14. This frequency region has circular EFSs with ∇*<sup>k</sup> ω* < 0.

Superlens has flat surfaces, and they produce images with subwavelength resolutions. The details on the subwavelength resolution of the superlens can be found in [31]. Here, we would like to explain, how the flat surface can result in a focusing effect. The focusing effect of the superlens is graphically illustrated in Figure 16.

Consider a homogenous slab (as in Fig. 8), and a point object at some distance from the slab in an air ambience. Firstly, assume the slab has a positive refractive index (as for the bottom edge frequencies of the first band in Fig. 14). As shown in Fig. 16, the usual positive refraction in the slab, will cause the rays from the point object to diverge. If the slab has negative refractive index (as for the top edge frequencies of the second band in Fig. 14), the negative refractions will bring the rays from the point object to a focus (see the illustration in Fig. 16(b)).

#### **5. Summary**

EFS is the surface resulting from the projection of the photonic band structure onto the wavevector space, at a constant frequency. In PCs, EFSs can be accurately modeled using a plane wave expansion method. For 2D PCs with weak dielectric modulation, EFSs can be obtained using one and two plane wave approximation techniques [Sec. 2]. Though one plane wave approximation succeed to a large extend in predicting the EFS of a 2D PC with a weak dielectric modulation, the approximation is inaccurate for wavevectors near the boundaries of the BZ. The one plane wave approach often predicts a very sharp edge to the EFS. The deficiencies in the one plane wave approximation is corrected in the two plane waves ap‐ proach.

EFS is an essential tool in determining light propagation directions in the PC. The gradient of the EFS gives information about the group velocity direction and the sign of the propagation angle (Sec. 3). EFSs can be used to analyze various peculiar light propagations in the PC. Examples that cover superprism effects, negative index mediums, negative refractions, and superlens are discussed in Sec. 4.

#### **Author details**

#### G. Alagappan\*

Address all correspondence to: gandhi@ihpc.a-star.edu.sg

Department of Electronics and Photonics, Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A-STAR), Singapore

#### **References**


[4] Jiang W, Chen RT, Lu X. Theory of light refraction at the surface of a photonic crys‐ tal. Physical Review B 2005; 71(24) 245115.

**5. Summary**

140 Photonic Crystals

proach.

superlens are discussed in Sec. 4.

Address all correspondence to: gandhi@ihpc.a-star.edu.sg

Agency for Science, Technology and Research (A-STAR), Singapore

**Author details**

G. Alagappan\*

**References**

R10099.

2000; 62 (16) 10696-10705.

2005; 72(16) 165112.

EFS is the surface resulting from the projection of the photonic band structure onto the wavevector space, at a constant frequency. In PCs, EFSs can be accurately modeled using a plane wave expansion method. For 2D PCs with weak dielectric modulation, EFSs can be obtained using one and two plane wave approximation techniques [Sec. 2]. Though one plane wave approximation succeed to a large extend in predicting the EFS of a 2D PC with a weak dielectric modulation, the approximation is inaccurate for wavevectors near the boundaries of the BZ. The one plane wave approach often predicts a very sharp edge to the EFS. The deficiencies in the one plane wave approximation is corrected in the two plane waves ap‐

EFS is an essential tool in determining light propagation directions in the PC. The gradient of the EFS gives information about the group velocity direction and the sign of the propagation angle (Sec. 3). EFSs can be used to analyze various peculiar light propagations in the PC. Examples that cover superprism effects, negative index mediums, negative refractions, and

Department of Electronics and Photonics, Institute of High Performance Computing (IHPC),

[1] Kosaka H, Kawashima T, Tomita A, Notomi M, Tamamura T, Sato T, Kawakami S. Superprism phenomena in photonic crystals. Physical Review B 1998; 58(16) R10096-

[2] Notomi M. Theory of light propagation in strongly modulated photonic crystals: Re‐ fractionlike behavior in the vicinity of the photonic band gap. Physical Review B

[3] Foteinopoulou S, Soukoulis CM. Electromagnetic wave propagation in two-dimen‐ sional photonic crystals: A study of anomalous refractive effects. Physical Review B

