**2. Theory of photonic crystal**

Photonic crystals (PhC) are new class of optical material represented by natural or artificial structure with periodic modulation of the permittivity. Multiple interference of light on a periodic lattice leads to a photonic band gap and anomalous dispersion because light with a wavelength close to the period of modulation cannot propagate in certain directions. This

Optical Logic Devices Based on Photonic Crystal 65

The first Brillouin zone is defined as the region of the reciprocal space formed by the points which are closer to the origin than to any other vertex of the periodical lattice. Our structure has a diamond shaped Brillouin zone, which is illustrated in Figure 1d. The Irreducible Brillouin Zone (IBZ) is the triangular wedge in the bottom right corner and the rest of the Brillouin zone can be related to this wedge by rotational symmetry. The three special points , M and X correspond to (0, 0), (√2л/a, 0) and (√2л/a, √2л/a) respectively. Due to periodicity structure, the behavior of the entire crystal can be obtained by studying the unit

(a) (b)

Fig. 1. a) Two-dimensional photonic crystal with square lattice b) primitive vectors *a1* and *a2*

The dielectric function in wave vector is a function of spatial co-ordinates r and can be

() () *jkr*

where g(k) is the dielectric function in wave vector representation and k is the wave vector. In periodicity condition, the dielectric values of the function are repeated with lattice vector R in each direction and R= *m*a1+*n*a2 where *m* and *n* are integers. The dielectric constant in a

*r g k e dk* (6)

(c) (d)

in real space c) & d) First Brillouin Zone of square lattice

**2.2 Bloch-floquet theorem**

periodic structure is given as

represented by

2 *<sup>z</sup> <sup>b</sup> a* 

lattice in the IBZ.

2 2 ˆ

(5)

peculiar property leads to an opportunity for a number of applications. Depending on the geometry of the structure, PhC can be classified into one-dimensional (1D), two dimensional (2D) and three-dimensional (3D) structures. Two-dimensional photonic crystals impose periodicity of the permittivity in two directions, while in the third direction the medium is uniform. Because of the ease of fabrication and analyzing, 2D photonic crystals have attractive attention of large number of researchers and engineers.

The properties of photonic crystal can be engineered through the process of doping which is achieved by either adding or removing dielectric material in a certain area. The dielectric materials then act as a defect region that can be used to localize an electromagnetic wave. Upon incident radiation, the periodic scatterers, that is the periodic dielectric materials in the photonic crystal could reflect an incident radiation at the same frequency in all directions. Wherever in space the reflected radiation interferes constructively, sharp peaks would be observed. This portion of the radiation spectrum is then forbidden to propagate through the periodic structure, and this band of frequencies is called photonic bandgap. On the other hand, wherever in space an incident radiation destructively interferes with the periodic scatterers in a certain directions, this part of the radiation spectrum will propagate through the periodic structure with minimal attenuation and this band of frequencies is called pass band. Introducing point defect or line defect, strict periodicity in the PhC is broken and can form optical cavity with high Q factor and low mode volume or lossless optical waveguide.

#### **2.1 Two dimensional square lattice and Brillouin zone**

In two dimensional photonic crystals the permittivity is modulated in two directions, say in the x and z plane: ε(r) = ε(x,z). Periodicity in two dimensions can be realized in various geometries, the most common being the square and the triangular lattices. In our work, we consider square lattice of silicon rods embedded in air background. This square lattice having a starting period *a*'=√2 *a* along the x axis where '*a*' is lattice constant and oriented at 45° with respect to the x axis as shown in Figure 1a. Successive rods are shifted by δx along the x axis and by δz along the z axis, δx and δz values are given as

$$
\delta \mathfrak{d} \mathfrak{x} = \delta \mathfrak{y} = a \mathfrak{i} / \mathfrak{2} = a \mathfrak{i} / \sqrt{2} \tag{1}
$$

In this case, the direct lattice is formed by the primitive vectors *a1* and *a2* in the real space and given by the following equations

$$a\_1 = a \left(\frac{\hat{\mathbf{x}}}{\sqrt{2}} - \frac{\hat{z}}{\sqrt{2}}\right) \tag{2}$$

$$a\_2 = a \left(\frac{\hat{\mathbf{x}}}{\sqrt{2}} + \frac{\hat{z}}{\sqrt{2}}\right) \tag{3}$$

Where x and ˆ *z*ˆ are unit vectors along the x axis and z axis respectively.

Reciprocal lattice vectors b1 and b2 in the reciprocal space are written as

$$h\_1 = \frac{2\sqrt{2}\pi\hat{\mathbf{x}}}{a} \tag{4}$$

peculiar property leads to an opportunity for a number of applications. Depending on the geometry of the structure, PhC can be classified into one-dimensional (1D), two dimensional (2D) and three-dimensional (3D) structures. Two-dimensional photonic crystals impose periodicity of the permittivity in two directions, while in the third direction the medium is uniform. Because of the ease of fabrication and analyzing, 2D photonic crystals have

The properties of photonic crystal can be engineered through the process of doping which is achieved by either adding or removing dielectric material in a certain area. The dielectric materials then act as a defect region that can be used to localize an electromagnetic wave. Upon incident radiation, the periodic scatterers, that is the periodic dielectric materials in the photonic crystal could reflect an incident radiation at the same frequency in all directions. Wherever in space the reflected radiation interferes constructively, sharp peaks would be observed. This portion of the radiation spectrum is then forbidden to propagate through the periodic structure, and this band of frequencies is called photonic bandgap. On the other hand, wherever in space an incident radiation destructively interferes with the periodic scatterers in a certain directions, this part of the radiation spectrum will propagate through the periodic structure with minimal attenuation and this band of frequencies is called pass band. Introducing point defect or line defect, strict periodicity in the PhC is broken and can form

optical cavity with high Q factor and low mode volume or lossless optical waveguide.

In two dimensional photonic crystals the permittivity is modulated in two directions, say in the x and z plane: ε(r) = ε(x,z). Periodicity in two dimensions can be realized in various geometries, the most common being the square and the triangular lattices. In our work, we consider square lattice of silicon rods embedded in air background. This square lattice having a starting period *a*'=√2 *a* along the x axis where '*a*' is lattice constant and oriented at 45° with respect to the x axis as shown in Figure 1a. Successive rods are shifted by δx along

In this case, the direct lattice is formed by the primitive vectors *a1* and *a2* in the real space

ˆ ˆ 2 2 *x z*

ˆ ˆ <sup>2</sup> 2 2 *x z*

2 2 ˆ

*x ya a* '/ 2 / 2 (1)

(2)

(3)

(4)

attractive attention of large number of researchers and engineers.

**2.1 Two dimensional square lattice and Brillouin zone**

and given by the following equations

the x axis and by δz along the z axis, δx and δz values are given as

 

Where x and ˆ *z*ˆ are unit vectors along the x axis and z axis respectively. Reciprocal lattice vectors b1 and b2 in the reciprocal space are written as

1

*a a*

*a a*

1 *<sup>x</sup> <sup>b</sup> a* 

$$b\_2 = \frac{2\sqrt{2}\pi\hat{z}}{a} \tag{5}$$

The first Brillouin zone is defined as the region of the reciprocal space formed by the points which are closer to the origin than to any other vertex of the periodical lattice. Our structure has a diamond shaped Brillouin zone, which is illustrated in Figure 1d. The Irreducible Brillouin Zone (IBZ) is the triangular wedge in the bottom right corner and the rest of the Brillouin zone can be related to this wedge by rotational symmetry. The three special points , M and X correspond to (0, 0), (√2л/a, 0) and (√2л/a, √2л/a) respectively. Due to periodicity structure, the behavior of the entire crystal can be obtained by studying the unit lattice in the IBZ.

Fig. 1. a) Two-dimensional photonic crystal with square lattice b) primitive vectors *a1* and *a2* in real space c) & d) First Brillouin Zone of square lattice

#### **2.2 Bloch-floquet theorem**

The dielectric function in wave vector is a function of spatial co-ordinates r and can be represented by

$$
\omega(r) = \left[ \mathcal{g}(k)e^{\vec{j}kr} \, dk \right] \tag{6}
$$

where g(k) is the dielectric function in wave vector representation and k is the wave vector.

In periodicity condition, the dielectric values of the function are repeated with lattice vector R in each direction and R= *m*a1+*n*a2 where *m* and *n* are integers. The dielectric constant in a periodic structure is given as

Optical Logic Devices Based on Photonic Crystal 67

( ') ( ')exp( ( ). )

*G G HG ik G r*

 

( )exp( ( ). ) <sup>2</sup>

The eigen value equations for the Fourier expansion coefficients of electric field and

2

( ')( ) ( ') ( ') ( ) <sup>2</sup> *G G k G k G HG HG*

These are 'Master Equations' for 2D photonic crystals. Here G and G' are in-plane reciprocal lattice vectors, k is in-plane wave vector and ω is the eigen frequency of the TE polarization.

Rod type PhC consists of silicon dielectric rods with relative permittivity εa periodically embedded in air with a dielectric permittivity εb. For simplification, assume only one rod is present in the unit cell and the space dependence of the inverse of the permittivity χ in this

11 11 ( )( ) *r R*

where θ(r) is the Heaviside function and its value is 1 inside the rod and 0 outside the rod and χa =1/ εa , χb=1/εb. The expression for Fourier expansion coefficients of the dielectric

1 1 ( ) exp( ) *<sup>r</sup> jGr dr*

where Vo is the volume of the unit cell. In our structure, Vo= *a1* xˆ x *a2 z*ˆ + *a1* zˆ x *a2* xˆ .

1 111 ( ) ( )exp( ) ,0 *<sup>r</sup> r jGr dr <sup>G</sup> b V ab <sup>o</sup> Vo*

<sup>2</sup> 1 1 12 ( ) <sup>1</sup> ( ) ,0 *r J Gr <sup>a</sup> <sup>r</sup> <sup>G</sup> Gr b ab Vo*

set of the reciprocal lattice vectors should now be selected to provide correct Fourier

*<sup>2</sup>* is the cross-section area of the rod and *J*1(*Gr*) is the first order Bessel function. The

 

 

*r b ab R*

*V r <sup>o</sup> Vo*

 

where δG,0 =1 if G=0 and δG,0 =0 if G≠0. Using Bessel function Eq. 19 can be written as

 

( ')( ) ( ) ( ') ( ) <sup>2</sup> *G G k G k G EG EG*

*G c*

*G c*

The E(G) and H(G) can be projected onto the unit and orthogonal vectors.

Substituting Eq. 17 in Eq. 18, the following equation can be obtained

 

 

*c G*

*HG ik G r*

2

2

(17)

(18)

(19)

(20)

(15)

(16)

(14)

'

*GG*

elementary cell can be expressed as

function is represented by

where *πra*

magnetic field are

$$
\varepsilon(r) = \varepsilon(r+R) \tag{7}
$$

$$
\varepsilon(r+R) = \lceil g(k)e^{jkr}e^{j kR} \rceil \tag{8}
$$

The propagation of a wave in a periodic medium is governed by the Bloch-Floquet theorem which is the product of a plane wave with a periodic function and states that

$$E\_k(r) = e^{jkr} u\_k(r) \tag{9}$$

where uk(r) is a periodic envelope function on the lattice and uk(r) = uk(r + R).

#### **2.3 Photonic band gap - Plane Wave Expansion Method**

Photonic crystals have photonic band gap, which is the gap between the air-line and the dielectric line in the dispersion relation of PhC. Photonic crystals forbid the propagation of the range of frequency in the band gap, and allow the propagation of other frequencies with low loss. Photonic band diagram gives the information about the dispersion characteristics *w* (k) for the Eigen mode of the PhC.

The Plane Wave Expansion method (PWE), can be used to calculate the band structure using an eigen formulation of the Maxwell's equations, and thus solving for the eigen frequencies for each of the propagation directions of the wave vectors (Igor A. Sukhoivanov & Igor V. Guryev, 2009). The Helmholtz equations for TE and TM polarization can be derived from the fundamental Maxwell's equations,

$$\nabla \times \left[ \nabla \times \frac{1}{\varepsilon\_T(r)} (\varepsilon\_T(r)E) \right] = \frac{\alpha^2}{c^2} (\varepsilon\_T(r)E) \tag{10}$$

$$\nabla \times \left[\frac{1}{\varepsilon\_T(r)} \nabla \times H\right] = \frac{\alpha^2}{c^2} H \tag{11}$$

The dielectric function can be expanded to the Fourier series due to the periodicity

$$\frac{1}{\varpi(r)} = \sum\_{\mathbf{G}} \mathcal{X}(\mathbf{G}).\exp(j\mathbf{G}.r) \tag{12}$$

where G is a linear combination of reciprocal vector G= *l*b1+*n*b2 and χ(G) is Fourier expansion coefficient which depends on the reciprocal lattice vectors.

Substitution of Eq. 12 in Eq. 10 & 11 gives

$$\begin{aligned} \nabla \times \left[ \nabla \times \sum\_{\mathbf{G}} \sum\_{\mathbf{G}'} \mathcal{Z}(\mathbf{G} - \mathbf{G}') \mathcal{E}(\mathbf{G}') \exp(i(k+\mathbf{G}).r) \right] &= \\ \frac{a^2}{c^2} \sum\_{\mathbf{G}} \mathcal{E}(\mathbf{G}) \exp(i(k+\mathbf{G}).r) \end{aligned} \tag{13}$$

() ( ) *r rR* 

The propagation of a wave in a periodic medium is governed by the Bloch-Floquet theorem

() () *jkr Er e ur k k* (9)

Photonic crystals have photonic band gap, which is the gap between the air-line and the dielectric line in the dispersion relation of PhC. Photonic crystals forbid the propagation of the range of frequency in the band gap, and allow the propagation of other frequencies with low loss. Photonic band diagram gives the information about the dispersion characteristics

The Plane Wave Expansion method (PWE), can be used to calculate the band structure using an eigen formulation of the Maxwell's equations, and thus solving for the eigen frequencies for each of the propagation directions of the wave vectors (Igor A. Sukhoivanov & Igor V. Guryev, 2009). The Helmholtz equations for TE and TM polarization can be derived from

*r r c*

<sup>2</sup> <sup>1</sup> ( ) <sup>2</sup> *H H r r c*

<sup>1</sup> ( ).exp( . ) ( ) *G jG r*

where G is a linear combination of reciprocal vector G= *l*b1+*n*b2 and χ(G) is Fourier

( ') ( ')exp( ( ). )

*G G EG ik G r*

 

( )exp( ( ). ) <sup>2</sup>

2

*c G*

 

*r G* 

expansion coefficient which depends on the reciprocal lattice vectors.

'

*GG*

The dielectric function can be expanded to the Fourier series due to the periodicity

<sup>2</sup> <sup>1</sup> ( ()) ( ()) ( ) <sup>2</sup> *r r rE rE*

 

(10)

(12)

*EG ik G r*

(11)

(13)

(7)

*r R g k e e dk* (8)

which is the product of a plane wave with a periodic function and states that

where uk(r) is a periodic envelope function on the lattice and uk(r) = uk(r + R).

( ) () *jkr jkR*

**2.3 Photonic band gap - Plane Wave Expansion Method**

*w* (k) for the Eigen mode of the PhC.

the fundamental Maxwell's equations,

Substitution of Eq. 12 in Eq. 10 & 11 gives

$$\begin{aligned} \nabla \times \left[ \nabla \times \sum\_{\mathbf{G}} \sum\_{\mathbf{G}'} \mathcal{Z}(\mathbf{G} - \mathbf{G}') H(\mathbf{G}') \exp(i(\mathbf{k} + \mathbf{G}) \cdot \mathbf{r}) \right] &= \\ \frac{a^2}{c^2} \sum\_{\mathbf{G}} H(\mathbf{G}) \exp(i(\mathbf{k} + \mathbf{G}) \cdot \mathbf{r}) \end{aligned} \tag{14}$$

The eigen value equations for the Fourier expansion coefficients of electric field and magnetic field are

$$\sum\_{\mathbf{G}} \mathbb{Z}(\mathbf{G} - \mathbf{G}')(\mathbf{k} + \mathbf{G}) \times \left[ (\mathbf{k} + \mathbf{G}) \times E(\mathbf{G}') \right] = -\frac{\alpha^2}{c^2} E(\mathbf{G}) \tag{15}$$

$$\sum\_{\mathbf{G}} \mathbb{Z}(\mathbf{G} - \mathbf{G}')(k + \mathbf{G}) \times \left[ (k + \mathbf{G}') \times H(\mathbf{G}') \right] = -\frac{\alpha^2}{c^2} H(\mathbf{G}) \tag{16}$$

These are 'Master Equations' for 2D photonic crystals. Here G and G' are in-plane reciprocal lattice vectors, k is in-plane wave vector and ω is the eigen frequency of the TE polarization. The E(G) and H(G) can be projected onto the unit and orthogonal vectors.

Rod type PhC consists of silicon dielectric rods with relative permittivity εa periodically embedded in air with a dielectric permittivity εb. For simplification, assume only one rod is present in the unit cell and the space dependence of the inverse of the permittivity χ in this elementary cell can be expressed as

$$\frac{1}{\varepsilon\_T} = \frac{1}{\varepsilon\_\mathcal{b}} + \sum\_{\widetilde{\mathcal{R}}} (\frac{1}{\varepsilon\_\mathcal{a}} - \frac{1}{\varepsilon\_\mathcal{b}}) \theta(r - R) \tag{17}$$

where θ(r) is the Heaviside function and its value is 1 inside the rod and 0 outside the rod and χa =1/ εa , χb=1/εb. The expression for Fourier expansion coefficients of the dielectric function is represented by

$$\mathcal{X}(r) = \frac{1}{V\_{\mathcal{O}}} \int \frac{1}{\mathcal{E}\_{\mathcal{O}}} \exp(-jGr) dr \tag{18}$$

where Vo is the volume of the unit cell. In our structure, Vo= *a1* xˆ x *a2 z*ˆ + *a1* zˆ x *a2* xˆ . Substituting Eq. 17 in Eq. 18, the following equation can be obtained

$$\mathcal{L}\varphi(r) = \frac{1}{\varepsilon\_{\rm b}} \delta\_{\rm G,0} + \frac{1}{V\_{\rm o}} \left(\frac{1}{\varepsilon\_{\rm a}} - \frac{1}{\varepsilon\_{\rm b}}\right)\_{V\_{\rm o}} \theta(r) \exp(-jGr) dr \tag{19}$$

where δG,0 =1 if G=0 and δG,0 =0 if G≠0. Using Bessel function Eq. 19 can be written as

$$\chi(r) = \frac{1}{\varepsilon\_{\rm b}} \delta\_{\rm G,0} + \left(\frac{1}{\varepsilon\_{\rm a}} - \frac{1}{\varepsilon\_{\rm b}}\right) \frac{2\pi r\_{\rm a}^2 l\_1(Gr)}{V\_0 Gr} \tag{20}$$

where *πra <sup>2</sup>* is the cross-section area of the rod and *J*1(*Gr*) is the first order Bessel function. The set of the reciprocal lattice vectors should now be selected to provide correct Fourier

Optical Logic Devices Based on Photonic Crystal 69

close to each other the distortion of the original pulse will be minimal. If the group velocities are different the original pulse will widen. The group velocity and the group velocity dispersion are obtained from the dispersion diagram. The light propagation inside the PhC is governed by the Equifrequency Surface (EFS) which is the cross section of the band diagram at constant frequency. If the directions of the pulses components group velocities are perpendicular to the EFS, the widening of the original pulse is determined by the shape of the EFS. Each group velocity is locally perpendicular to local EFS. If the curvature of the EFS is large the original pulse will diverge or converge, depending on the sign of the curvature. So, depending on the EFS local curvature as well as on its evolution with the wavelength and the incident wave-vector, there are different types of effect are observed in PhC such as self-collimation, superlensing, negative refraction and superprism. These effects

Self collimation effect is a linear non-diffraction phenomenon, totally independent of light intensity (Kosaka et al., 1999). PhC are designed to have dispersion properties that allow the beam to propagate without spatial spreading. In the equifrequency contour, flat square contour with zero curvature can be used to latterly confine the light since all the pulse components propagate with the same group velocity. This effect is called self-collimation

Fig. 3. First band TM Equifrequency contour of photonic crystal dispersion surface in the

The Figure 3 shows the frequency contour for the square lattice PhC consisting of silicon rods embedded in air with rod radius 0.3*a*. In this contour map, the curves of the frequencies around 0.194(*a*/λ) can be identified as squares with round corners centered at the M point, where λ is the wavelength of incident radiation whose value is 1550 nm. So self-collimation phenomenon occurs around the normalized frequency 0.194. RSOFT BandSOLVE tool is used to calculate band diagram and equifrequency contour. When light is incident from a high refractive index (nh) medium onto a low refractive index (nl) medium, the incident

are used to control light propagation inside the PhC.

effect. It provides a mechanism to control the light as in a waveguide.

**2.5 Self-collimation effect** 

first Brillouin zone

expansion of the dielectric function and the Bloch functions. Square lattice of silicon rod in air is considered in our structure with εa = 11.56 and εb = 1.

Thus from Eqs. 15 & 16, for any given value for k leads to an infinite eigen value problem, these truncated by restricting G to a set of M vectors. The k-path within the first Brillouin zone are setting through , M and X correspond to (0,0), (√2л/a,0) and (√2л/a,√2л/a) respectively. The wave vector k describes the edge of the IBZ along the direction X, M and MX for reaching the extrema of ω(k) and this establishes the dispersion relation.

Fig. 2. Photonic band diagram of square lattice of silicon dielectric rod in the air

The Figure 2 shows the photonic band diagram of square lattice of silicon dielectric rod in the air. The radius of the silicon rod (r*a*) is 0.3*a*. In this rod type, only TM polarization exists. The first band gap lies in the normalized frequency region (ω*a*/2лc) 0.21 to 0.25.

#### **2.4 Dispersion properties of photonic crystals**

Dispersion of the Bloch modes is one of the most important properties of the photonic crystals and it determines the propagation of modes in the crystal. It depends on many parameters of the PhC such as lattice type, the refractive index contrast between the dielectric material and the host material and distribution of atoms in the primitive cell.

Light pulse in a photonic crystal can be represented as a superposition of the Bloch modes with different Bloch vectors and frequencies

$$
\mu(r,t) = \sum\_{m} \int f(k,m)\varphi\_{k,m}(r,t)dk\tag{21}
$$

where ψ k,m is *m*-th Bloch mode with the Bloch vector k and f(k,*m*) is the amplitude of the mode. The motion of the light pulse in photonic crystal is governed by the group velocity *<sup>v</sup>* = . (k) *<sup>g</sup> <sup>k</sup>* . Since the light pulse is constructed from a superposition of several pulses with different combinations of k and n, let us consider them independently. Each pulse component has the group velocity = . (k) k=k , *vg i <sup>i</sup> <sup>k</sup>* . If the group velocities Vg,i are

expansion of the dielectric function and the Bloch functions. Square lattice of silicon rod in

Thus from Eqs. 15 & 16, for any given value for k leads to an infinite eigen value problem, these truncated by restricting G to a set of M vectors. The k-path within the first Brillouin zone are setting through , M and X correspond to (0,0), (√2л/a,0) and (√2л/a,√2л/a) respectively. The wave vector k describes the edge of the IBZ along the direction X, M

and MX for reaching the extrema of ω(k) and this establishes the dispersion relation.

Fig. 2. Photonic band diagram of square lattice of silicon dielectric rod in the air

The first band gap lies in the normalized frequency region (ω*a*/2лc) 0.21 to 0.25.

**2.4 Dispersion properties of photonic crystals**

with different Bloch vectors and frequencies

pulse component has the group velocity = . (k) k=k , *vg i <sup>i</sup> <sup>k</sup>*

velocity *<sup>v</sup>* = . (k) *<sup>g</sup> <sup>k</sup>*

The Figure 2 shows the photonic band diagram of square lattice of silicon dielectric rod in the air. The radius of the silicon rod (r*a*) is 0.3*a*. In this rod type, only TM polarization exists.

Dispersion of the Bloch modes is one of the most important properties of the photonic crystals and it determines the propagation of modes in the crystal. It depends on many parameters of the PhC such as lattice type, the refractive index contrast between the dielectric material and the host material and distribution of atoms in the primitive cell.

Light pulse in a photonic crystal can be represented as a superposition of the Bloch modes

(,) (, ) (,) , *u r t f k m r t dk k m <sup>m</sup>*

where ψ k,m is *m*-th Bloch mode with the Bloch vector k and f(k,*m*) is the amplitude of the mode. The motion of the light pulse in photonic crystal is governed by the group

pulses with different combinations of k and n, let us consider them independently. Each

. Since the light pulse is constructed from a superposition of several

(21)

. If the group velocities Vg,i are

air is considered in our structure with εa = 11.56 and εb = 1.

close to each other the distortion of the original pulse will be minimal. If the group velocities are different the original pulse will widen. The group velocity and the group velocity dispersion are obtained from the dispersion diagram. The light propagation inside the PhC is governed by the Equifrequency Surface (EFS) which is the cross section of the band diagram at constant frequency. If the directions of the pulses components group velocities are perpendicular to the EFS, the widening of the original pulse is determined by the shape of the EFS. Each group velocity is locally perpendicular to local EFS. If the curvature of the EFS is large the original pulse will diverge or converge, depending on the sign of the curvature. So, depending on the EFS local curvature as well as on its evolution with the wavelength and the incident wave-vector, there are different types of effect are observed in PhC such as self-collimation, superlensing, negative refraction and superprism. These effects are used to control light propagation inside the PhC.

#### **2.5 Self-collimation effect**

Self collimation effect is a linear non-diffraction phenomenon, totally independent of light intensity (Kosaka et al., 1999). PhC are designed to have dispersion properties that allow the beam to propagate without spatial spreading. In the equifrequency contour, flat square contour with zero curvature can be used to latterly confine the light since all the pulse components propagate with the same group velocity. This effect is called self-collimation effect. It provides a mechanism to control the light as in a waveguide.

Fig. 3. First band TM Equifrequency contour of photonic crystal dispersion surface in the first Brillouin zone

The Figure 3 shows the frequency contour for the square lattice PhC consisting of silicon rods embedded in air with rod radius 0.3*a*. In this contour map, the curves of the frequencies around 0.194(*a*/λ) can be identified as squares with round corners centered at the M point, where λ is the wavelength of incident radiation whose value is 1550 nm. So self-collimation phenomenon occurs around the normalized frequency 0.194. RSOFT BandSOLVE tool is used to calculate band diagram and equifrequency contour. When light is incident from a high refractive index (nh) medium onto a low refractive index (nl) medium, the incident

Optical Logic Devices Based on Photonic Crystal 71

Iref is launched at the second input port. The photonic crystal structure with mirror and splitter performs a specific logic gate function by combining the reflected signal and the partially transmitted reference signal. The optical output is detected and converted into electrical signal by photo detector. This structure can be used for stand alone logic gates. In an integrated circuit the output value will be standardized using a PhC amplifier and given

to the input port of the next in sequence logic gate and so on.

Fig. 4. a Proposed structure of AND, NAND, NOR & XNOR logic gates

Fig. 4. b Proposed structure of XOR logic gate

Fig. 4. c Proposed structure of OR logic gate

wave is totally reflected back into the high refractive index medium at the interface, provided the incident angle is larger than the critical angle given by θc = sin-1(nl/nh) (Chul-Sik Kee et al., 2007) . Self-collimated beams can be totally reflected at the interface of a PhC and air because PhC and air correspond to high refractive and low refractive mediums respectively. When it undergoes total internal reflection, the field amplitude decays very rapidly into air and becomes negligible at a distance within one lattice constant. An air layer created by introducing a line defect by removing a few rods in a row is expected to give rise to total internal reflection. Reflection provides a mechanism for bending and splitting of self-collimated beams.
