On Methods for Bayesian Optimization of Least-Squares Problems and Optimization of Nanophotonic Devices

Abstract

The field of nanophotonics studies the interaction of light and matter at microscopic scales. It plays an important role in many modern technologies, such as efficient solar cells, fast and secure quantum communication devices, and many other applications. The development of these technologies in a primarily experimental environment is challenging. As such, nanophotonics is a computationally heavily involved discipline. It often involves numerical simulations, optimization algorithms, as well as advanced modeling techniques, that all complement each other.

This cumulative dissertation explores the application of a particular kind of machine learning surrogate model, Gaussian processes, to certain optimization and modeling problems that appear in nanophotonics. Gaussian processes---in the context of nanophotonics---are trained on results of numerical simulations and can afterwards be used to approximate these simulations. They can help reduce the computational impact because approximate solutions obtained using Gaussian processes are much cheaper than results from numerical simulations. In addition to an approximation of the training inputs, the predictions of a Gaussian process are also associated with a measure of the confidence in the approximation. We develop methods and approaches that rely on these properties, implement them as (typically) Python code, investigate their performance and properties, and finally also apply them to solve two particular problems that appear in nanophotonics.

First, we employ Gaussian processes to accelerate parameter reconstruction tasks in optical nanometrology that involve computationally expensive model functions. In addition to aiding in the reconstruction of the model parameters that best explain some experimental measurement, trained Gaussian processes are then further used to determine the confidence bounds of the found model parameters. We present the methodology and perform extensive benchmarks, measuring its performance with respect to other methods that have been used to perform parameter reconstructions. We additionally apply it to give insights into the impact of properly or improperly chosen numerical accuracy parameters during a parameter reconstruction effort.

Second, we employ Gaussian processes for the quantification and optimization of the robustness of nanophotonic devices under an assumed manufacturing uncertainty. Gaussian processes are well suited for this task due to their intrinsic ability to model uncertainties. We present approaches that use Monte Carlo methods to assess the robustness of design candidates using Gaussian processes. We apply these approaches to assess the manufacturability of a novel cavity design, as well as to optimize certain optical resonators in a way that incorporates manufacturing uncertainties, first on a comparatively small domain using a multi-objective function, and second on a much larger domain.

The methods we present here are developed and applied in the context of nanophotonics. Their applicability, however, is not limited to this field.