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International Journal for Uncertainty Quantification

Impact factor: 0.967

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2015013050
pages 511-526

BAYESIAN APPROACH TO THE STATISTICAL INVERSE PROBLEM OF SCATTEROMETRY: COMPARISON OF THREE SURROGATE MODELS

Sebastian Heidenreich
Physikalisch-Technische Bundesanstalt, Abbestr 2-12, 10587 Berlin
Hermann Gross
Physikalisch-Technische Bundesanstalt, Abbestr 2-12, 10587 Berlin, Germany
Markus Bar
Physikalisch-Technische Bundesanstalt, Abbestr 2-12, 10587 Berlin, Germany

ABSTRACT

Scatterometry provides a fast indirect optical method for the determination of grating geometry parameters of photomasks and is used in mask metrology. To obtain a desired parameter, inverse methods like least squares or the maximum likelihood method are frequently used. A different method, the Bayesian approach, has many advantages against the others, but it is often not used for scatterometry due to the large computational costs. In this paper, we introduce different surrogate models to approximate computationally expensive calculations by fast function evaluations, which enable the Bayesian approach to scatterometry. We introduce the nearest neighbor interpolation, the response surface methodology and a method based on a polynomial chaos expansion. For every surrogate model, we discuss the approximation error and the convergence. Moreover, we apply Markov Chain Monte Carlo sampling to determine desired geometry parameters, and its uncertainties form simulated measurement values based on Bayesian inference. We show that the surrogate model involving polynomial chaos is the most effective.