NUMERICAL-SOLUTION OF OPTIMAL-DESIGN PROBLEMS FOR BINARY GRATINGS

In this paper we describe recent developments in the application of ma thematical and computational techniques to the problem of designing bi nary gratings on top of a multilayer stack in such a way that the prop agating modes have a specified intensity or phase pattern for a chosen range of wavelengths or incidence angles. The diffraction problems ar e transformed to strongly elliptic variational formulations of quasi p eriodic transmission problems for the Helmholtz equation in a bounded domain coupled with boundary integral representations in the exterior. We obtain analytic formulae for the gradients of cost functionals wit h respect to the parameters of the grating profile and the thickness o f the layers, so that the optimal design problems can be solved by min imization algorithms based on gradient descent. For the computation of diffraction efficiencies and gradients the variational problems are s olved by using a generalized finite element method with minimal pollut ion. We provide semi: numerical examples to demonstrate the convergenc e properties for evaluating diffraction efficiencies and gradients. Th e method is applied to optimal design problems for polarisation gratin gs and beam splitters. (C) 1998 Academic Press