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.

Surface electromagnetic waves supported by nano conducting layers with inhomogeneities in the conductivity profile

“…Here, the Helmholtz equation for both major polarizations is rather rigorously solved, where the Galerkin's method with Legendre polynomial basis functions is applied. This approach, being already employed in analysis of inhomogeneous optical structures Chamanzar et al 2006), is for the first time rephrased in a concise matrix format and includes TM polarized waves. The presented matrix formulation is general and can be similarly applied for analysis of inhomogeneous optical structures.…”

“…Furthermore, f 1 and f 2 can be interpolated by using polynomials of the hth degree (Chamanzar et al 2006):…”

“…Absorption of ξ in the mth Legendre polynomial basis function can be very easily accomplished by using the following equation (Chamanzar et al 2006):…”

Eigenvalue problems, spectral parameter power series, and modern applications

“…In spite of the fact that equations (69) and (70) describe the behavior of different components of an electromagnetic wave, corresponding to an electric and a magnetic field, respectively, there exists a simple transformation from (70) to (69) and vice versa (see, e.g., [77]). Namely, if v is a solution of (69), then U D v=n is a solution of the equation…”

“…where k 2 N 2 D k 2 n 2 C n 00 =n 2 .n 0 =n/ 2 . Thus, in both cases, the problem reduces to an equation of the form (69).…”