Comparison of asymptotic and transfer matrix approaches for evaluating intensity-dependent dispersion in nonlinear photonic and phononic crystals

Abstract

This paper analyzes intensity dependent dispersion of acoustic and electromagnetic waves in nonlinear one-dimensional (1D) periodic media. Media considered include layered 1D crystals characterized by constitutive laws with cubic strain nonlinearities (phononic), and by cubic time-dependent Kerr nonlinearities (photonic). A recently presented perturbation approach is generalized such that it can be successfully applied to the analysis of wave propagation in weakly nonlinear elastic and electromagnetic systems. In both, the analysis predicts amplitude-dependent dispersion leading to tunable bandgaps. These predictions are compared with those obtained from a simplified analysis conducted by linearizing the governing equations about an assigned field intensity followed by application of the transfer matrix method, as is commonly done in nonlinear photonic crystal studies. Comparison of the two methods generally shows good dispersion agreement; however, in phononic systems the transfer matrix analysis does not accurately capture high-strain behavior that occurs for some configurations, such as those with thin nonlinear layers, and may even predict erroneous bandgaps. Numerical predictions resulting from a finite-element analysis of the nonlinear media confirm the improved accuracy of the perturbation approach, suggesting its continued use for predicting intensity-dependent dispersion in 2D and 3D nonlinear phononic and photonic crystals.