KEYWORDS: Mathematical optimization, Optical metrology, Monte Carlo methods, Education and training, Process modeling, Sand, Global Positioning System, Computer programming languages
Parameter reconstruction problems appear frequently in optical metrology. Here, one attempts to explain a set of K experimental measurements by fitting to them a parameterized forward model of the measurement process. We present a Bayesian target vector optimization scheme that can be used to perform this fit. It has been shown to be capable of outperforming established methods such as Levenberg-Marquardt, and can after a successful fit enable very efficient and accurate determination of the distribution of the reconstructed model parameters using Markov chain Monte Carlo sampling.
Parameter reconstruction is a common problem in optical nano metrology. It generally involves a set of measurements, to which one attempts to fit a numerical model of the measurement process. The model evaluation typically involves to solve Maxwell’s equations and is thus time consuming. This makes the reconstruction computationally demanding. Several methods exist for fitting the model to the measurements. On the one hand, Bayesian optimization methods for expensive black-box optimization enable an efficient reconstruction by training a machine learning model of the squared sum of deviations Χ2 . On the other hand, curve fitting algorithms, such as the Levenberg-Marquardt method, take the deviations between all model outputs and corresponding measurement values into account which enables a fast local convergence. In this paper we present a Bayesian Target Vector Optimization scheme which combines these two approaches. We compare the performance of the presented method against a standard Levenberg-Marquardt-like algorithm, a conventional Bayesian optimization scheme, and the L-BFGS-B and Nelder-Mead simplex algorithms. As a stand-in for problems from nano metrology, we employ a non-linear least-square problem from the NIST Standard Reference Database. We find that the presented method generally uses fewer calls of the model function than any of the competing schemes to achieve similar reconstruction performance.
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