Abstract

Multiple angles of incidence (MAI) have been employed in both ellipsometry and reflectometry as a means of obtaining several independent measurements where multiparameter estimation of optical constants is involved. However, there exists no systematic investigation into the determination of a set of incidence angles and polarizations that are optimal in terms of reducing both random and systematic errors and minimizing correlations among the measurements. Several researchers have found configurations that give good results for certain problems using physical arguments, but a general procedure that applies to all cases has not been developed. We demonstrate that the multi-dimensional Cramer-Rao (CR) bound of maximum likelihood estimation theory gives a quantitative measure of the random and systematic errors and correlations in one canonical expression; hence it is an ideal candidate for selecting an optimal experimental configuration in both MAI ellipsometry and reflectometry. Maximum likelihood (ML) estimation theory is used for the inversion problem because the estimation error is known to approach the CR bounds asymptotically. Numerical examples are given to demonstrate the procedure and illustrate how the effect of both random and systematic errors on the material estimates may be reduced.

© 1990 Optical Society of America