 # Ellipsometric function of a film–substrate system: detailed analysis and closed-form inversion

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## Abstract

The ellipsometric function ρ of a film–substrate system is analyzed through successive transformations from the plane of the two independent variables angle of incidence and film thickness ($ϕ–d$ plane) to the complex ρ plane. This analysis is achieved by introducing two intermediate planes: the unimodular plane ($Zi$ plane) and the translated ellipsometric plane ($ρ*$ plane). The analysis through the $Zi$ plane leads to classification of the film–substrate systems into two classes: clockwise and counterclockwise. The class of the film–substrate system governs the inversion from the $ρ*$ plane to the $Zi$-plane. It identifies the number of branch points of $ρ*-1$ from the $ρ*$ plane to the $Zi$ plane. The branch points of $ρ*-1$ and its preimage in the $ϕ–d$ plane are identified and studied. The domain of the double-valued function $ρ*-1$ is divided into two or four subdomains according to the class of the film–substrate system. In each of these subdomains, the single-valued branch of $ρ*-1$ is fixed, and we introduce a closed-form solution for the determination of the film thickness of the system. Mathematically, $ρ*-1$ exists in any domain that does not include the branch points. Hence the exceptive points are divided into two types: removable and essential. The closed-form inversion is obtained for the removable exceptive points. The conformality of both ρ and $ρ*,$ as well as their inverses, leads to identification of the two essential exceptive inversion points, which exist at $ϕ=0°$ and 90°. Accordingly, the closed-form solution is available throughout the ρ plane except at the two points ±1 (corresponding to $ϕ=0°$ and 90°). A study of the extrema of the magnitude and the phase of both ρ and $ρ*$ provides full information on the number of zeros and essential singularities for each of the three categories of film–substrate systems: negative, zero, and positive. Numerical examples are given to illustrate the introduced closed forms. Also, the table of transformation of regions between the $ϕ–d$ plane and the ρ plane induced by ρ and $ρ-1$ is given.

© 1999 Optical Society of America

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### Figures (18)

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### Equations (163)

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