Three-dimensional diffraction analysis of phase and amplitude gratings based on legendre expansion of electromagnetic fields

Abstract

Three-dimensional vectorial diffraction analysis of phase and amplitude gratings in conical mounting is presented based on Legendre expansion of electromagnetic fields. In the so-called conical mounting, different fields components are coupled and the solution is not separable in terms of independent TE and TM cases. In contrast to conventional RCWA in which the solution is obtained using state variables representation of the coupled wave amplitudes by expanding space harmonic amplitudes of the fields in terms of the eigenfunctions and eigenvectors of the coefficient matrix defined by rigorous coupled wave equations, here the solution of first order coupled Maxwell's equations is expanded in terms of Legendre polynomials. This approach yields well-behaved algebraic equations for deriving diffraction efficiencies and electromagnetic field profiles. It can nicely handle the cases in which conventional methods face the problem of numerical instability and inevitable round off errors; also, it yields accurate results to any desired level of accuracy. The method is applied to phase and amplitude gratings in conical mountings, comparison to other methods already reported in the literature is made, and the presented approach is justified and its usefulness in cases that other methods usually fail is demonstrated. This general method applies well even in such cases as thick gratings, non-Bragg incidence, and cases in which higher diffracted orders are needed to be retained, or evanescent orders corresponding to real eigenvalues have to be included. The efficacy of the proposed method relies on the fact that although Legendre polynomials span a complete space, they are not eigensolutions and hence each polynomial basis function bears a weighted projection of all eigenfunctions. Thus no modal information is completely missed in the ineluctable truncation process. In deriving the formulation, a rigorous approach is followed.