Interfacial operator approach to computing band structures for photonic crystals of polar materials

Abstract

In this study, we propose an interfacial operator approach to compute surface phonon modes for one- and two-dimensional periodic arrays of polar materials in a finite-difference formulation. The key aspect of the approach is to introduce an interfacial variable along the interface between the polar material and the surrounding dielectric material, which represents the local strength of the surface phonon modes along the interface. In this approach, the apparently nonlinear eigenvalue problem can be reformulated as a quadratic eigensystem, and thus further reduced to a standard linear eigenvalue problem. Band structures can be computed directly without the need of examining transmission spectra as in the finite-difference time-domain method, or locating the mode frequency by testing an auxiliary function in other methods. Applying the method to four different types of photonic crystals of polar materials, we are able to uncover several interesting results by studying the effect of dimension, the size (filling ratio) effect, the effects of the transverse optical phonon frequency $({\ensuremath{\omega}}_{T})$, and longitudinal optical phonon frequency $({\ensuremath{\omega}}_{L})$ as well as the effect of shape or geometry of the polar material.