Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals

Abstract

A Legendre polynomial expansion of electromagnetic fields for analysis of layers with an inhomogeneous refractive index profile is reported. The solution of Maxwell's equations subject to boundary conditions is sought in a complete space spanned by Legendre polynomials. Also, the permittivity profile is interpolated by polynomials. Different cases including computation of reflection–transmission coefficients of inhomogeneous layers, band-structure extraction of one-dimensional photonic crystals whose unit-cell refractive index profiles are inhomogeneous, and inhomogeneous planar waveguide analysis are investigated. The presented approach can be used to obtain the transfer matrix of an arbitrary inhomogeneous monolayer holistically, and approximation of the refractive index or permittivity profile by dividing into homogeneous sublayers is not needed. Comparisons with other well-known methods such as the transfer-matrix method, WKB, and effective index method are made. The presented approach, based on a nonharmonic expansion, is efficient, shows fast convergence, is versatile, and can be easily and systematically employed to analyze different inhomogeneous structures.