Dependence of nonlinearity enhancement on power density in photonic crystals characterized by numerical Z -scan experiments based on the finite-difference time-domain technique

We investigate the enhancement of nonlinearity in one-dimensional (1D) photonic crystals (PCs) with Kerr nonlinearity by numerical Z-scan experiments based on the finite-difference time-domain technique. Focused Gaussian beams with well-defined waists and Rayleigh lengths necessary for Z-scan experiments are generated through a conjugated manipulation of the Gaussian beams propagating in free space. The Z-scan measurements used for bulk materials are naturally extended to 1D PCs after incorporating the frequency- and power-density-dependent reflections into their linear and nonlinear absorptions. The closed- and open-aperture Z-scan traces for the 1D PCs are obtained and a symmetric method is employed to modify the asymmetric closed-aperture traces. The nonlinearity enhancement factors at different frequencies in the first and second bands are derived numerically and analytically. A good agreement is found between the numerical and analytical results in the case of weak nonlinearity. Moreover, the dependences of the enhancement factor on the incident power density for different frequencies in the 1D PCs are extracted and they are found to be much different from those in bulk materials. It is revealed that the variation of the group velocity with increasing power density is responsible for the power-density dependence of the enhancement factor. It indicates that in practice one must deliberately choose the working frequency and power density of PC-based devices in order to achieve a maximum enhancement of nonlinearity.