DIFFRACTION IN PERIODIC STRUCTURES AND OPTIMAL-DESIGN OF BINARY GRATINGS - PART 1 - DIRECT PROBLEMS AND GRADIENT FORMULAS

The aim of the paper is to provide the mathematical foundation of effe ctive numerical algorithms for the optimal design of periodic binary g ratings. Special attention is paid to reliable methods for the computa tion of diffraction efficiencies and of the gradients of certain funct ionals with respect to the parameters of the non-smooth grating profil e. The methods are based on a generalized finite element discretizatio n of strongly elliptic variational formulations of quasi-periodic tran smission problems for the Helmholtz equation in a bounded domain coupl ed with boundary integral representations in the exterior. We prove un iqueness and existence results for quite general situations and analys e the convergence of the numerical solutions. Furthermore, explicit fo rmulas for the partial derivatives of the reflection and transmission coefficients with respect to the parameters of a binary grating profil e are derived. Finally, we briefly discuss the implementation of the g eneralized finite element method for solving direct and adjoint diffra ction problems and present some numerical results. (C) 1998 B. G. Teub ner Stuttgart-John Wiley & Sons, Ltd.