Planar Diffraction Analysis of Homogeneous and Longitudinally Inhomogeneous Gratings Based on Legendre Expansion of Electromagnetic Fields

Planar grating diffraction analysis based on Legendre expansion of electromagnetic fields is reported. In contrast to conventional RCWA in which the solution is obtained using state variables representation of the coupled wave amplitudes; here, the solution is expanded in terms of Legendre polynomials. This approach, without facing the problem of numerical instability and inevitable round off errors, yields well-behaved algebraic equations for deriving diffraction efficiencies, and can be employed for analysis of different types of gratings. Thanks to the recursive properties of Legendre polynomials, for longitudinally inhomogeneous gratings, wherein differential equations with non-constant coefficients are encountered, it can also be used to analyze the whole structure at one stroke. Although this is the only case for which the presented approach is efficient from both aspects of stability and computation load, the presented approach is applied to different test cases, and justified by comparison of the results to those obtained using previously reported methods. The method is general, and can handle many different cases like thick gratings, non-Bragg incidence, and cases in which higher diffracted orders or evanescent orders corresponding to real eigenvalues, have to be included in the solution of the Maxwell´s equations. In deriving the formulation, a rigorous approach is followed