Efficient uncertainty quantification of large two-dimensional optical systems with a parallelized stochastic Galerkin method.

Z Zubac,D Vande Ginste,J Fostier,D De Zutter

doi:10.1364/oe.23.030833

Abstract

It is well-known that geometrical variations due to manufacturing tolerances can degrade the performance of optical devices. In recent literature, polynomial chaos expansion (PCE) methods were proposed to model this statistical behavior. Nonetheless, traditional PCE solvers require a lot of memory and their computational complexity leads to prohibitively long simulation times, making these methods non-tractable for large optical systems. The uncertainty quantification (UQ) of various types of large, two-dimensional lens systems is presented in this paper, leveraging a novel parallelized Multilevel Fast Multipole Method (MLFMM) based Stochastic Galerkin Method (SGM). It is demonstrated that this technique can handle large optical structures in reasonable time, e.g., a stochastic lens system with more than 10 million unknowns was solved in less than an hour by using 3 compute nodes. The SGM, which is an intrusive PCE method, guarantees the accuracy of the method. By conjunction with MLFMM, usage of a preconditioner and by constructing and implementing a parallelized algorithm, a high efficiency is achieved. This is demonstrated with parallel scalability graphs. The novel approach is illustrated for different types of lens system and numerical results are validated against a collocation method, which relies on reusing a traditional deterministic solver. The last example concerns a Cassegrain system with five random variables, for which a speed-up of more than 12× compared to a collocation method is achieved.

Highlights

Variability analysis and uncertainty quantification (UQ) have become a major concern during the design step of optical systems and components as manufacturing tolerances and process variations can have a dramatic influence on the performance [1]
The full-wave character of the problem is described by means of a set of boundary integral equations (BIE) that are solved by means of the Method of Moments (MoM) [9]
Parallelization of Stochastic Galerkin Method (SGM) applied to elliptic partial differential equations is reported in [10], but as explained further it is still prohibitively slow to deal with the optical systems presented in this paper

Summary

Introduction

Variability analysis and uncertainty quantification (UQ) have become a major concern during the design step of optical systems and components as manufacturing tolerances and process variations can have a dramatic influence on the performance [1]. Even small variations of geometrical dimensions or material properties affect the electromagnetic behavior of the structures under design To model these variations, one may combine statistical analysis with traditional, deterministic, full-wave solvers. We present the parallelization of the full-wave intrusive SGM-MLFMM solver and we propose an effective preconditioning scheme to further accelerate the computations. Parallelization of SGM applied to elliptic partial differential equations is reported in [10], but as explained further it is still prohibitively slow to deal with the optical systems presented in this paper. Present paper is the first that proposes the parallelization of a full-wave SGM-MLFMM solver for Maxwell’s equations, capable of handling both dielectric and perfectly electric conducting (PEC) objects.

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Methods

Conclusion