Task-based assessment of a proposed phase-shifting interferometer/ellipsometer

Abstract

In this dissertation, we offer a novel phase-shifting interferometer/ellipsometer. The uniqueness arises from the fact that this study is the consolidation of four distinct ideas drawn from the field of optics and the field of statistics. A conventional four-step phase-shifting interferometer is modified to allow for both TE and TM polarized measurements. Maximum-likelihood estimation theory is then used to extract the three parameters of interest, namely the real and imaginary components of the complex index of refraction and the surface profile. Finally, Cramer-Rao lower bounds serve as a quantitative means of assessing the particular system design at the task of estimating the three parameters in question. As we will show, the unknown parameters n, k and h are related to the measured irradiance in a complicated, nonlinear way. As such, no analytical expressions to estimate the unknown parameters from the measured data have been found. Rather, the unknown parameters are found numerically through a minimization program, developed and optimized specifically for this task. The results from our Monte Carlo simulations will show that conventional designs such as the Twyman-Green interferometer perform poorly at reconstructing n, k and h. The estimates on n and k exhibit bias, where the mean is not equal to the true value, and are non-efficient, where the standard deviation is greater than the Cramer-Rao lower bound. While the estimate of h is unbiased and efficient, the performance is an order of magnitude worse than the case where only h is to be estimated. By incorporating tilt in the design, the performance on all three parameters improves considerably. The estimates on n and k are shown to be unbiased and efficient, and the performance of the h estimator is equivalent to the h-only case. The dissertation culminates with the development of a Mach-Zehnder prototype. We demonstrate the feasibility of the proposed technique, and show how three system parameters, namely the incident amplitude and the relationship between the TE and TM polarized light in terms of amplitude and phase, affect the performance. We also show how quantization of the measured irradiance affects the performance.