Optical Propagation in Three-Layer Lossy Media Using Transfer Matrix Method

Abstract

This paper presents a generalized transfer matrix method (TMM) for analyzing optical propagation in three-layer periodic structures composed of alternating transparent and complex refractive index media. Unlike conventional approaches, this work accounts for the decoupling of planes of constant amplitude and phase within lossy media, where the complex wave vector introduces non-orthogonal propagation characteristics. By separately applying Snell’s law to the amplitude and phase unit vectors, we derive explicit expressions for the effective phase and amplitude refractive indices. The transmission and dispersion relations for both TE and TM polarizations are established via boundary continuity conditions and Bloch’s theorem. Furthermore, we demonstrate that the derived transmission matrix for lossy media can be mathematically transformed into an equivalent lossless form through complex effective index and thickness renormalization. This provides a practical pathway for implementing lossy photonic crystals in commercial software such as COMSOL Multiphysics. Numerical validations confirm the accuracy and stability of the proposed method.