Error function for interpretations of ellipsometric measurements

Abstract

As a rule, the results of physical measurements are interpreted by searching for a set of parameters, which minimize some error function (EF) (target function, figure of merit function). The latter is a measure of discrepancy (difference) between experimental data and model values. Interpretation of ellipsometric measurements involves a variety of EFs used for this purpose. In agreement with the known principle of maximum likelihood (MLH), one should choose such a set of parameters, which would provide a maximal probability to obtain the analyzed set of experimental data. In this case, EF is proportional to the logarithm of inverse probability of obtaining the given set of experimental data. Assuming that for the null measurement method the probability of obtaining the given values of ellipsometric angles is determined by the intensity of light at the detector input, we obtained a simple equation for the EF, meeting the requirement of MLH. This is a photometric EF: Φ=∣ E f∣ 2, E f= R p cos(Ψ E)− R s sin(Ψ E) exp( jΔ E). Here, E f is proportional to the electric field strength of the transmitted light wave at the output of the ellipsometer. The condition of complete extinction of transmitted light is equivalent to R p cos(Ψ E)= R s sin(Ψ E) exp( jΔ E). Thus, the so-called basic equation of ellipsometry: R p/ R s=tan(Ψ) exp( jΔ)—is a modified form of the equation describing the transmitted light extinction. It is also shown that ∣ E f∣ 2 is proportional to the square of distance between the points on the Poincare sphere corresponding to experimental data and model values. The coefficient of proportionality is equal to the coefficient of reflection of non-polarized light: R=(∣ R p∣ 2+∣ R s∣ 2)/2.