With PCs it thus becomes possible to manipulate the radiation dynamics of active materials embedded in them as well as the propagation characteristics of electromagnetics radiation. In fact, the seminal papers of Yablonovitch [1] and John [2] have been concerned with the possibility of inhibiting spontaneous emission [1] and the realization of Anderson-localization of light [2] in PCs, respectively. Since then, numerous novel effects regarding the radiation dynamics of active materials such as fractional localization near a photonic band edge [3] and all-optical transistor action [4] have been proposed. Similarly, the re-examination of classical nonlinaer effects such as nonlinear susceptibilities [5], and soliton propagation [6] in PCs have revealed a number of interesting results that await experimental confirmation.
In this presentation, we will give an introduction to photonic bandstructure computation with an emphasis on photonic bandstructure theory as a predicitve as well as interpretative tool for nonlinear optical effects in PCs. First, we will discuss the plane wave method (PWM) and its application to PCs whose constituent materials exhibit tunable electro-optical anisotropies [7]. The resulting tunability of the photonic bandstructure may greatly enhance the utility of these systems over and above conventional PCs as well as homogeneous electro-optical materials. Next, we describe a recently developed multigrid method (MGM) for photonic bandstructure computations [8]. This real space approach allows the efficient determination of photonic modestructures and avoids the PWM-inherent problem of truncating Fourier series. As a consequence, MGM is much better suited to the calculation of quantities that are relevant to nonlinear phenomena in PCs such as group velocities and effective nonlinearities. Finally, we will outline how the slowly varying envelope approximation may be generalized to the description of nonlinear phenomena in PCs. This may be achieved through combining elements from kp-perturbation theory of electronic semiconductors with a multi-scale analysis of the electromagnetic wave equation where the eigenmodes of the (linear) PC, the Bloch functions, represent the carrier waves.
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