**2. Optical nonlinearity in PCs**

Nonlinear optics field came to existence when (Franken et al., 1961) published their achievements on second harmonic generation. It was mainly due to acquiring the technology to produce high intensity coherent lights that could trigger the nonlinearity in materials. Such ability was facilitated by the introduction of laser by (Maiman, 1960). Since then many other optical phenomena such as parametric oscillation, four wave mixing, stimulated Raman and Brillouin scattering, phase conjugation etc. have been observed and studied (Boyd, 2003).

Nonlinear optical materials can be considered as the basic elements for all-optical processing systems. Although optical bistability in nonlinear materials has been extensively studied since 1980 (Saleh, 1991 and Kaatuzian, 2008), but since most materials exhibit pretty weak nonlinear characteristics, the large size and high operational power of such devices makes them unsuitable for all-optical integrated devices. Introduction of PCs, due to its unique characteristics (Yablonovitch, 1987 and John, 1987), helped to reduce both the operational power and size of nonlinear optical components and presented a new opportunity for their integration.

In a linear material the dipole moment density (or equally known as polarization) P(t) and electric field E(t) have a linear relationship:

$$P(t) = \varepsilon\_0 \chi \mid E(t). \tag{1}$$

The notation "χ" is usually referred to as susceptibility. In a nonlinear medium (1) can be generalized as follows (Boyd, 2003):

$$\begin{aligned} P(t) &= \varepsilon\_0 \left[ \chi^{(1)} E(t) + \chi^{(2)} E^2(t) + \chi^{(3)} E^3(t) + \dots \right] \\ &= P^1(t) + P^2(t) + P^3(t) + \dots \\ &= P^L + P^{NL} \end{aligned} \tag{2}$$

where, the notations χ(1), χ(2), χ(3) are called the first, second and third order susceptibility coefficients respectively. Also P1(t), P2(t), P3(t) are referred to as the first, second and third order polarizations. Therefore, P(t) can be considered to have a linear term PL and a nonlinear term PNL. In nonlinear optics, most of nonlinear observations are usually due to χ(2) and χ(3) which is mainly due to the fact that the higher order terms need much higher input intensities to show significant impact. In addition it is shown (Boyd, 2003) that χ(2) can be only observed in non-centrosymmetric crystals; therefore liquids, gas and many solid crystals do not display those phenomena that are originated by χ(2).

The ith component for the vector P can be expressed as (3) (Saleh, 1991):

$$P\_i = \varepsilon\_0 \sum\_{j=1}^3 \chi\_{ij}^{(1)} E\_j + \varepsilon\_0 \sum\_{j=1}^3 \sum\_{k=1}^3 \chi\_{ijk}^{(2)} E\_j E\_k + \varepsilon\_0 \sum\_{j=1}^3 \sum\_{k=1}^3 \sum\_{l=1}^3 \chi\_{ijkl}^{(3)} E\_j E\_k E\_l + \dots \tag{3}$$

where, i=1,2,3 (corresponding to x,y,z). For a linear material, only the first term of this equation is significant. One of the most important nonlinear optical effects which originates from the χ(3) coefficients is the Kerr effect (Weinberger, 2008). It was discovered in 1875 by John Kerr. Almost all materials show a Kerr effect. The Kerr effect is a change in the refractive index of a material in response to an external electric field. For materials that have a non-negligible Kerr effect, the third, χ(3) term is significant, with the even-order terms typically dropping out due to inversion symmetry of the Kerr medium. In the optical Kerr effect, an intense beam of light in a medium can itself provide the modulating electric field, without the need for an external field to be applied. In this case, the electric field is given by:

$$E = E\_{\text{co}} \text{Cost}(\text{out}).\tag{4}$$

where Eω is the amplitude of the wave. Combining this with the equation for the polarization will yield:

$$P = \varepsilon\_0 \left( \chi^{(1)} + \frac{3}{4} \chi^{(3)} \left| E\_{\rm co} \right|^2 \right) E\_{\rm co} \text{Cost(out)}.\tag{5}$$

As before:

124 Photonic Crystals – Innovative Systems, Lasers and Waveguides

In this chapter, first different types of optical nonlinearities are briefly explained in electromagnetics' terms. Thereafter, numerical and analytic methods for modeling optical nonlinearity in photonic crystals are reviewed. As a case study nonlinear optical Kerr effect is used to design a directional coupler switch and later as the second case study an structure

Nonlinear optics field came to existence when (Franken et al., 1961) published their achievements on second harmonic generation. It was mainly due to acquiring the technology to produce high intensity coherent lights that could trigger the nonlinearity in materials. Such ability was facilitated by the introduction of laser by (Maiman, 1960). Since then many other optical phenomena such as parametric oscillation, four wave mixing, stimulated Raman and Brillouin scattering, phase conjugation etc. have been observed and

Nonlinear optical materials can be considered as the basic elements for all-optical processing systems. Although optical bistability in nonlinear materials has been extensively studied since 1980 (Saleh, 1991 and Kaatuzian, 2008), but since most materials exhibit pretty weak nonlinear characteristics, the large size and high operational power of such devices makes them unsuitable for all-optical integrated devices. Introduction of PCs, due to its unique characteristics (Yablonovitch, 1987 and John, 1987), helped to reduce both the operational power and size of nonlinear optical components and presented a new opportunity for their

In a linear material the dipole moment density (or equally known as polarization) P(t) and

The notation "χ" is usually referred to as susceptibility. In a nonlinear medium (1) can be

(1) (2) (3) 2 3

( ) [ ( ) ( ) ( ) ...]

where, the notations χ(1), χ(2), χ(3) are called the first, second and third order susceptibility coefficients respectively. Also P1(t), P2(t), P3(t) are referred to as the first, second and third order polarizations. Therefore, P(t) can be considered to have a linear term PL and a nonlinear term PNL. In nonlinear optics, most of nonlinear observations are usually due to χ(2) and χ(3) which is mainly due to the fact that the higher order terms need much higher input intensities to show significant impact. In addition it is shown (Boyd, 2003) that χ(2) can be only observed in non-centrosymmetric crystals; therefore liquids, gas and many solid

*Pt Et E t E t*

123

*Pt Pt Pt*

0

, *L NL*

*P P*

crystals do not display those phenomena that are originated by χ(2).

The ith component for the vector P can be expressed as (3) (Saleh, 1991):

<sup>0</sup> *Pt Et* ( ) ( ). (1)

(2)

for all-optical AND gate operation is proposed.

electric field E(t) have a linear relationship:

generalized as follows (Boyd, 2003):

**2. Optical nonlinearity in PCs** 

studied (Boyd, 2003).

integration.

$$\mathcal{X} = \mathcal{X}\_{\text{L}} + \mathcal{X}\_{\text{NL}} = \mathcal{X}\_{\text{(1)}} + \frac{3}{4} \mathcal{X}\_{\text{(3)}} \left| E\_{\text{co}} \right|^2. \tag{6}$$

and since:

$$n = \sqrt{1 + \mathcal{X}} = \sqrt{1 + \mathcal{X}^{\perp} + \mathcal{X}^{\text{NL}}} = n\_0 \left( 1 + \frac{1}{2n\_0^2} \mathcal{X}^{\text{NL}} \right) \tag{7}$$

where n0=(1+χL)1/2 is the linear refractive index. Therefore:

$$n = n\_0 + \frac{3\chi^{(3)}}{8n\_0} \left| E\_{\rm eo} \right|^2 = n\_0 + n\_2 I\_{\prime} \tag{8}$$

where n2 is the second-order nonlinear refractive index, and I is the intensity of the wave. The refractive index change is thus proportional to the intensity of the light travelling through the medium.

#### **2.1 Modeling optical nonlinearity in PCs**

In order to numerically analyze linear PCs, various methods have been proposed in the literature. Among these methods, the most popular are: plane wave expansion (PWE), finite difference in time domain (FDTD), finite element method (FEM), multiple multimode method (MMP) and Wannier function method (WFM). In the nonlinear regime, in addition to the mentioned methods, complementary strategies have to be used to analyze the effect of the

*HIGH*

*ALLSPACE*

The mentioned change in the wavenumber can cause an optical phase difference between the linear and nonlinear states of a system. Many optical devices such as directional couplers or Mach-Zehnder interferometers are sensitive to the induced phase. Assuming that L is the waveguide length, *L k* shows the phase difference between the linear and nonlinear cases. According to (15) if the waveguide is designed to have a low group velocity, for a fixed amount of refractive index variation a larger *k* is obtained. It means that for a fixed a smaller device size L is needed; or equally for a fixed L a smaller *n* (which is proportional to operational power) is required. Since photonic crystals can be used to design waveguides with very low group velocities (Soljacic´ & Joannopoulos, 2004); therefore they provide the opportunity to reduce both the device size and the operational

The idea of a one dimensional PC all-optical switch was first proposed in (Scholz et al., 1998). Coupled cavity waveguides (CCW) with Kerr nonlinearity, were there after suggested for all-optical switching. The main drawback of CCW switches is that when the switch is in the OFF state, all the data signal is reflected back to the input port. Since the backscattered signal can affect other optical devices on an optical chip, it makes them unsuitable for all optical integrated circuit applications. A combination of directional couplers and nonlinear optical elements can be used to solve the mentioned problem (Yamamoto et al., 2006,

In a directional coupler based switch, according to the ON or OFF state of the switch, most of the data signal power is guided to either of the two output ports. Usually only a very small amount leaks to the input port. It gives the designer the ability of using the switch in

Here a PC directional coupler is designed first.The PC lattice used for this design is a two dimensional array of GaAs rods which is known to have Kerr type optical nonlinearity.

The band diagram of the PC, Which is obtained using Plane Wave Expansion (PWE) method, is shown in Fig. 1. The shaded region is the optical bandgap. No optical signal within the normalized frequency range of 0.28a/λ to 0.45a/λ; where a is the lattice constant and λ is free space wavelength, can propagate through the PC lattice. The value of a is chosen 635nm in our simulations. The radii of rods are equal to 0.2a and their refractive index is chosen equal to 3.4, which is equal to the refractive index of GaAs at the 1550nm

Introduction of defects into PC lattice is the first step in designing PC devices. These are usually classified in two different categories of point defects and line defects, which can create resonators and waveguides respectively. As an Instance, in the mention PC, removing a column of rods (Fig. 2a) creates a waveguide mode in the bandgap region between the

*v n dr r E r* 

0

*<sup>n</sup> <sup>k</sup>*

power; making such devices able to be integrated on a single chip.

**3. Case study 1: A PC all-optical switch** 

Cuesta-Soto et al., 2004, Rahmati & N. Granpayeh, 2009).

sequential optical circuits.

wavelength.

*g r*

2 3 0 0

*dr r E r*

2 3 0 0

() ()

.

(15)

() ()

nonlinear elements on the linear system. Transfer matrix method (TMM), perturbation theory, coupled mode theory (CMT) and FDTD can be used for this purpose. In order to model the Kerr effect using FDTD, several methods have been proposed. Here FDTD and perturbation theory are briefly reviewed. Also In section 4 CMT is used to analyze a PC limiter.

From the Maxwell equations, the wave equation for the nonlinear medium can be written as:

$$(\nabla^2 E - \mu\_0 \varepsilon\_0 \varepsilon\_r (\mathbf{x}\_\prime \mathcal{y}\_\prime \boldsymbol{z}\_\prime \mathbf{o}\_\prime \mathbf{e}\_\prime E) \frac{\partial^2 E}{\partial t^2} = 0. \tag{9}$$

It is shown in (Joseph & A Taflov, 1997) that (9) can be rewritten as:

$$
\Delta \nabla^2 E - \mu\_0 \varepsilon\_0 n\_0^2 \frac{\partial^2 E}{\partial t^2} = \mu\_0 \frac{\partial^2 P^{NL}}{\partial t^2}. \tag{10}
$$

Based on the type of nonlinearity different approaches have to be made to solve the mentioned equation. The FDTD model for Kerr-type materials assumes an instantaneous nonlinear response. The nonlinearity is modelled in the relation *D E* where:

$$\varepsilon = n^2 = (n\_0 + \frac{3\chi^{(3)}}{8n\_0} \left| E\_{\rm co} \right|^2)^2 = n\_0^{-2} + 2n\_0 \frac{3\chi^{(3)}}{8n\_0} \left| E \right|^2. \tag{11}$$

Therefore the relation ship between E and D can be iteratively determined using:

$$E = \frac{D}{m\_0^2 + \frac{\Im \mathcal{X}^{(3)}}{4} |E|^2}. \tag{12}$$

Perturbation theory is also a useful tool in engineering for analyzing systems with small nonlinearities. Using Maxwell equations, the following eigenvalue problem can be derived for linear time invariant PC systems (A waveguide is assumed in our case.). The Dirac notation *E*<sup>0</sup> specifies the Bloch eigenmode for the electric field.

$$\nabla \times \nabla \times \left| \vec{E}\_0 \right> = \left( \frac{\alpha\_0}{c} \right)^2 \varepsilon\_0(\vec{r}) \left| \vec{E}\_0 \right>. \tag{13}$$

It is shown in (Bravo-Abad et al., 2007) that a change in the dielectric constant can result in a variation in the original eigenvalue <sup>0</sup> as:

$$\Delta\alpha = -\frac{\alpha\_0}{2} \frac{\left< \vec{E}\_0 \, \middle| \, \Delta \varepsilon \left| \vec{E}\_0 \right>}{\left< \vec{E}\_0 \, \middle| \, \varepsilon\_0 \left| \vec{E}\_0 \right>} = -\frac{\alpha\_0}{2} \frac{\int d^3 r \Delta \varepsilon(\vec{r}) \left| \vec{E}\_0(\vec{r}) \right|^2}{\int d^3 r \varepsilon\_0(\vec{r}) \left| \vec{E}\_0(\vec{r}) \right|^2}. \tag{14}$$

But since for a waveguide *v k <sup>g</sup>* / then it can be shown that (Bravo-Abad et al., 2007) the following approximation is valid:

$$\Delta k = -\frac{\alpha\_0}{v\_\mathcal{S}} \frac{\Delta \mathcal{U}}{n\_r} \frac{d^3 r \varepsilon\_0(\vec{r}) \left| \vec{E}\_0(\vec{r}) \right|^2}{\int\_{ALSPACE} d^3 r \varepsilon\_0(\vec{r}) \left| \vec{E}\_0(\vec{r}) \right|^2}. \tag{15}$$

The mentioned change in the wavenumber can cause an optical phase difference between the linear and nonlinear states of a system. Many optical devices such as directional couplers or Mach-Zehnder interferometers are sensitive to the induced phase. Assuming that L is the waveguide length, *L k* shows the phase difference between the linear and nonlinear cases. According to (15) if the waveguide is designed to have a low group velocity, for a fixed amount of refractive index variation a larger *k* is obtained. It means that for a fixed a smaller device size L is needed; or equally for a fixed L a smaller *n* (which is proportional to operational power) is required. Since photonic crystals can be used to design waveguides with very low group velocities (Soljacic´ & Joannopoulos, 2004); therefore they provide the opportunity to reduce both the device size and the operational power; making such devices able to be integrated on a single chip.
