Three-dimensional diffraction analysis of gratings based on Legendre expansion of electromagnetic fields

摘要

Three-dimensional vectorial diffraction analysis of gratings is presented based on Legendre polynomial expansion of electromagnetic fields. In contrast to conventional rigorous coupled wave analysis (RCWA) in which the solution is obtained using state variables representation of the coupled wave amplitudes, here the solution of first-order coupled Maxwell's equations is expanded in terms of Legendre polynomials, where Maxwell's equations are analytically projected in the Hilbert space spanned by Legendre polynomials. This approach yields well-behaved algebraic equations for deriving diffraction efficiencies and electromagnetic field profiles without facing the problem of numerical instability. The proposed approach can be applied in the analysis of two cases: first, arbitrarily oriented planar gratings with slanted yet homogeneous fringes; second, nonslanted but longitudinally inhomogeneous gratings. The method is then applied to various test cases within the above-mentioned two categories, comparison to other methods already reported in the literature is made, and the presented approach is justified. Different aspects of the proposed method such as numerical stability and convergence rate are also investigated. Special attention is given to how the resonant frequency of frequency selective structures varies with introducing tilt angles and/or longitudinal inhomogenity in the permittivity profile.