**1. Introduction**

122 Photonic Crystals – Innovative Systems, Lasers and Waveguides

Zielinski, E., Keppler, F., Hausser, S., Pilkuhn, M.H., Sauer, R. & Tsang, W.T., Optical Gain

25, no. 6, pp. 1407-1416, 1989

and Loss Processes in GaInAs/InP MQW Laser Structures, *J. Quant. Electron.*, vol.

Employing nonlinear elements in photonic crystals (PCs) opens up lots of new design opportunities. In comparison to ordinary linear PC structures, using optically nonlinear elements in PCs leads to the observation of many interesting phenomena that can be utilized to design all-optical devices. Optical bistability in PCs is among the mentioned observations, which due to its many applications has attracted the researchers' attention. Many optical devices such as limiters, switches, memories can be implemented when nonlinear elements are embedded in PCs. Kerr type nonlinearity is mostly used for this purpose. In this chapter we are going to discuss the benefits of using nonlinearity in photonic crystal devices. Since most of nonlinear optical devices are either based on directional couplers (DCs) or coupled cavity waveguides (CCWs), we have focused on these two groups to provide a better insight into their future prospects.

The first practical case of employing nonlinearity in PCs was for optical switches. Scholz and his colleagues were among the first ones to use optical Kerr nonlinearity in PCs (Scholz et al., 1998). They designed an all-optical switch using a one dimensional photonic crystal which was placed inside two cross waveguides. In their switch, a strong pump signal lateral to the PC layers made the crystal nonlinear and slightly shifted the position of the bandgap. The data signal wavelength was chosen on the bandgap edge such that it could not be placed in the shifted bandgap region (when the probe signal was present). Therefore; when no probe signal existed, the data signal could not pass through the PC layers, while when the probe was present, the shifted photonic bandgap would allow its transmission. Later the optical bistability in two dimensional nonlinear photonic crystal waveguides coupled to a micro-cavity was discussed in (Centeno & Felbacq, 2000) and it was suggested that the mentioned phenomena could be use to design all-optical switches. Mingaleev and Kivshar placed nonlinear elements in waveguides and bends to obtain optical limiters (Mingaleev & Kivshar, 2002). Meanwhile other structures were suggested by (Soljacic´ et al., 2002) and (Fan, 2002) for optical switching. Later in 2004 (Locatelli et al., 2004) and (Cuesta-Soto et al, 2004) suggested optical switches using directional couplers. Thereafter, some logic components were reported in literature such as: An optical AND gate by (Zhu et al., 2006), an all-optical PC on-chip memory implemented in (Shinya et. al., 2008), a PC half-adder structure designed in (Liu et al., 2008) and so forth.

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:

where Eω is the amplitude of the wave. Combining this with the equation for the

<sup>3</sup> ( ). <sup>4</sup>

(1) (3) 3 <sup>2</sup> . <sup>4</sup>

<sup>1</sup> 11 1 , <sup>2</sup>

<sup>3</sup> , <sup>8</sup> *n n E n nI n*

 *L NL NL n n*

> (3) <sup>2</sup> 0 0 2 0

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

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

 

(1) (3) 2

*P E E Cos t* 

0

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

(3)

*E E Cos t* ( ). (4)

(5)

*L NL <sup>E</sup>* (6)

(8)

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

0 2 0

3 3 3 333 (1) (2) (3)

1 1 1 1 11 ..., *<sup>i</sup> ij <sup>j</sup> ijk j k ijkl jkl*

*j j k jkl P E E E EEE* 

00 0

polarization will yield:

As before:

and since:

through the medium.

**2.1 Modeling optical nonlinearity in PCs** 

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 for all-optical AND gate operation is proposed.
