Radio Science, Volume 31, Number 6, Pages 1863-1876,November-December 1996 Inverse scattering theory for optical waveguides and devices: Synthesis from rational and nonrational reflection coefficients Arthur K. Jordan Remote Sensing Division, Naval Research Laboratory, Washington, D.C. Lakshman S. Tamil Broadband Communications Laboratory, Erik Jonsson School of Engineering and Computer Science,University of Texas at Dallas, Richardson Abstract. Electromagnetic inverse scattering theory is applied to the synthesis of optical waveguidesand devices from specified reflection and transmission characteristics. The permittivity profiles in the inhomogeneous core regions of the devices are reconstructed from their transverse reflection coefficients. Two applications of this theory are demonstratedwith examplesusing specified reflection coefficients: design of a single-mode inhomogeneous optical waveguide and design of an optical logic gate. The previous inversescattering theory which used reflection coefficients that are rational functions of the transverse wavenumber has been supplemented with an inverse scattering theory that usesnonrational reflection coefficients.This inversescattering theory usesan iterative discretizedspace-time finite-difference method with the "leapfrog" algorithm. The initial values for the iterative method are obtained from the inversetheory that usesrational function reflection coefficients. The inversescattering calculations have been verified by finite-differencefrequencydomain direct scattering techniques. 1. Introduction With the advent of computer-controlledpho- tolithography, molecular beam epitaxy and re- lated technologies, it is possible to fabricate op- tical waveguides and integrated circuits directly from the computer programs for the detailed designs of these optical devices. These designs must meet general systemrequirements that will specify, for example, the reflectionand transmis- sion performance data of these devices. Electromagnetic scattering problems are usu- ally solved by assumingthat the geometry and the material characteristics of an object are known and the fields scattered by that object are then Copyright 1996 by the American Geophysical Union. Paper number 96RS02501. 0048-6604 / 96 / 96 RS- 02501 $11.00 calculated; this is the "direct" scattering the- ory. Electromagnetic inverse scattering theory reverses this procedure and seeks to reconstruct the physical characteristics of an object from in- formation containedin the scatteredfields. Syn- thesis is an application of inversescattering the- ory whereby the scattering data are specified and a device is designed that will produce these data. For example, the geometric and material characteristics of an optical integrated circuit need to be designedto meet system specifica- tions for operating wavelengths,pulse widths, modulation, etc. Using the general physical model for planar optical waveguides and devices, these data can be represented by the scatter- ing matrix S of the device that relates the inci- dent electromagnetic field Ein with the scattered field, Esca. t -- S. Ein. 1863