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 ( plane) to the complex ρ plane. This analysis is achieved by introducing two intermediate planes: the unimodular plane ( plane) and the translated ellipsometric plane ( plane). The analysis through the 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 plane to the -plane. It identifies the number of branch points of from the plane to the plane. The branch points of and its preimage in the plane are identified and studied. The domain of the double-valued function 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 is fixed, and we introduce a closed-form solution for the determination of the film thickness of the system. Mathematically, 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 as well as their inverses, leads to identification of the two essential exceptive inversion points, which exist at and 90°. Accordingly, the closed-form solution is available throughout the ρ plane except at the two points ±1 (corresponding to and 90°). A study of the extrema of the magnitude and the phase of both ρ and 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 plane and the ρ plane induced by ρ and is given.
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