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Proceedings Paper

An improved method to derive best-fit parameters and their uncertainties from depolarizing Mueller-matrices (Conference Presentation)
Author(s): Tobias Grunewald; Matthias Wurm; Sven Teichert; Bernd Bodermann; Johanna Reck; Uwe Richter

Paper Abstract

Spectroscopic ellipsometry is a versatile tool to measure dimensional or optical parameters of surface layers or surface structures. The determination of these parameters requires to solve an inverse problem. This is achieved by a fitting procedure, where a merit function is optimized. This merit function compares simulated with measured polarization quantities for each measurement configuration, i. e. for each wavelength and angle of incidence. For the simulation of unstructured samples Fresnel-equations can be used, while for structured surfaces the Maxwells equations must be solved using so-called Maxwell-solvers. For depolarizing samples Mueller ellipsometry must be used, because only the Mueller-stokes formalism is capable to describe depolarization. However, in common approaches to evaluate Mueller matrix measurements depolarization is treated in a nonoptimal way, which leads to inadequate measurement uncertainty estimations and in the worst case may lead to systematic measurement errors. Especially for surfaces with strong depolarization such as rough or textured surfaces this is a problem. To treat these issues, we developed an improved analysis method to attain reliable sample parameters and reliable and mathematically well-founded uncertainty estimations from Mueller ellipsometry measurements. While deterministic non-depolarising samples can be fully described within the well-known Jones formalism and can be simulated by Fresnel- or Maxwell-solvers, a complete simulation of depolarizing samples would require very high computational expenses which are too elaborate or even impossible in practice. Thus the question arises, how to treat depolarization in the optimization process. Commonly used merit functions treat it as a residual error, which is minimized and accept the fit solution with the smallest overall depolarization as the best approximation. However, depolarization is a sample property, which must not be minimized. Therefore, instead we use the depolarization to derive a weight-factor for each residual contribution to the merit-function. We will present the underlying mathematics. The evolved theory is applicated on measurement data obtained with a Sentech SE 850 system. The Mueller-matrices for each measurement configuration are computed from an overdetermined system of raw data achieved with analyzer-scans at discrete polarizer-steps (step-scan-mode). Thus the raw measurement data can additionally be exploited to get statistical information to the measured Mueller matrices and included in the merit function. Embedded into a Bayesian approach the best-fit values and their uncertainties are determined from the posterior distribution in a much more realistic and reliable way. Applied on both virtual and real measurement data, we demonstrate the advantages of this new method and compare the results with different standard analysis methods.

Paper Details

Date Published: 22 July 2019
Proc. SPIE 11057, Modeling Aspects in Optical Metrology VII, 110570S (22 July 2019); doi: 10.1117/12.2527866
Show Author Affiliations
Tobias Grunewald, Physikalisch-Technische Bundesanstalt (Germany)
Matthias Wurm, Physikalisch-Technische Bundesanstalt (Germany)
Sven Teichert, Physikalisch-Technische Bundesanstalt (Germany)
Bernd Bodermann, Physikalisch-Technische Bundesanstalt (Germany)
Johanna Reck, SENTECH Instruments GmbH (Germany)
Uwe Richter, SENTECH Instruments GmbH (Germany)

Published in SPIE Proceedings Vol. 11057:
Modeling Aspects in Optical Metrology VII
Bernd Bodermann; Karsten Frenner; Richard M. Silver, Editor(s)

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