Boundary Integral Equation Methods for Simulation and Design of Photonic Devices

Abstract

This thesis presents novel boundary integral equation (BIE) and associated optimization methodologies for photonic devices. The simulation and optimization of such structures is a vast and rapidly growing engineering area, which impacts on design of optical devices such as waveguide splitters, tapers, grating couplers, and metamaterial structures, all of which are commonly used as elements in the field of integrated photonics. The design process has been significantly facilitated in recent years on the basis of a variety of methods in computational electromagnetic (EM) simulation and design. Unfortunately, however, the expense required by previous simulation tools has limited the extent and complexity of the structures that can be treated. The methods presented in this thesis represent the results of our efforts towards accomplishing the dual goals of 1) Accurate and efficient EM simulation for general, highly-complex three-dimensional problems, and 2) Development of effective optimization methods leading to an improved state of the art in EM design. One of the main proposed elements utilizes BIE in conjunction with a modified-search algorithm to obtain the modes of uniform waveguides with arbitrary cross sections. This method avoids spurious solutions by means of a certain normalization procedure for the fields within the waveguides. In order to handle problems including nonuniform waveguide structures, we introduce the windowed Green function (WGF) method, which used in conjunction with auxiliary integral representations for bound mode excitations, has enabled accurate simulation of a wide variety of waveguide problems on the basis of highly accurate and efficient BIE, in two and three spatial dimensions. The method provides the basic high-order singular-integration engine. Based on non-overlapping Chebyshev-discretized patches, the rectangular-polar method underlies the accuracy and efficiency of the proposed general-geometry three-dimensional BIE approach. Finally, we introduce a three-dimensional BIE framework for the efficient computation of sensitivities — i.e. gradients with respect to design parameters — via adjoint techniques. This methodology is then applied to the design of metalenses including up to a thousand parameters, where the overall optimization process takes in the order of three hours using five hundred computing cores. Forthcoming work along the lines of this effort seeks to extend and apply these methodologies to some of the most challenging and exciting design problems in electromagnetics in general, and photonics in particular.