Journal of Computational Physics 155, 287–306 (1999) Article ID jcph.1999.6333, available online at http://www.idealibrary.com on Spectral Collocation Time-Domain Modeling of Diffractive Optical Elements J. S. Hesthaven, ∗ P. G. Dinesen,† and J. P. Lynov† ∗ Division of Applied Mathematics, Brown University, Box F, Providence, Rhode Island, 02912; †Department of Optics and Fluid Dynamics, RisøNational Laboratory, P.O. Box 49, DK-4000 Roskilde, Denmark E-mail: jansh@cfm.brown.edu, palle.dinesen@risoe.dk, jens-peter.lynov@risoe.dk Received December 9, 1998; revised June 9, 1999 A spectral collocation multi-domain scheme is developed for the accurate and efficient time-domain solution of Maxwell’s equations within multi-layered diffrac- tive optical elements. Special attention is being paid to the modeling of out-of-plane waveguide couplers. Emphasis is given to the proper construction of high-order schemes with the ability to handle very general problems of considerable geometric and material complexity. Central questions regarding efficient absorbing boundary conditions and time-stepping issues are also addressed. The efficacy of the overall scheme for the time-domain modeling of electrically large, and computationally chal- lenging, problems is illustrated by solving a number of plane as well as non-plane waveguide problems. c  1999 Academic Press Key Words: spectral collocation methods; multi-domain methods; computational electromagnetics; optical elements. 1. INTRODUCTION Diffractive optical elements (DOEs) comprising surface-relief gratings on thin film op- tical waveguides have become increasingly interesting for sensor applications as the fabri- cation technology for such devices has matured [22, 11]. A remaining challenge in the design of DOEs is to specify a surface-relief grating which will produce a desired free-space farfield pattern. A first step in this inverse design process is to solve the forward problem, i.e., to accurately determine the field pattern from a given relief profile. To that end analytic tools are not an option as they are limited to treating periodic structures of infinite extent. What is needed is a tool that allows for the analysis of devices of finite length with chirped, hence aperiodic, surface reliefs. An alternative to analytic methods is low-order numerical methods such as the finite difference time-domain method (FD-TD). While this approach is fairly straightforward it will in many cases lead to inaccurate results due to the inability to correctly reproduce the 287 0021-9991/99 $30.00 Copyright c  1999 by Academic Press All rights of reproduction in any form reserved.