A simplified transfer function approach to beam propagation in anisotropic metamaterials

Abstract

The Berreman analysis for plane wave propagation through an anisotropic metamaterial is modified to derive the transfer function matrix for propagation in such a structure. The eigenvalues of the Berreman matrix, which determine the transfer function, depend on the anisotropy. Using the transfer function, propagation of TM and TE polarized beams in hyperbolic metamaterials is analyzed theoretically, and numerically based on the developed theory. For TM polarization, self-focusing of Gaussian beams in a hyperbolic metamaterial can be analytically explained using the q-parameter approach. Furthermore, the transmission coefficient for plane wave propagation through an anisotropic metamaterial slab is extended to a transmission transfer function, which is then used to analyze the propagation of beams through hyperbolic metamaterials and the ensuing spatial shifts. For TM polarized beams, negative refraction is verified. The main objective of this paper is to provide simple analytical tools for understanding self-focusing and negative refraction in anisotropic hyperbolic metamaterials as an alternative to, and check the validity of, rigorous numerical computations. Propagation of the longitudinally (z-) polarized optical field profile is derived. The transfer function approach should be useful for the analysis of arbitrary beam profiles through composite metamaterials.