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 (ø–d plane) to the complex ρ plane. This analysis is achieved by introducing two intermediate planes: the unimodular plane (Z<sub>i</sub> plane) and the translated ellipsometric plane (ρ<sup>*</sup> plane). The analysis through the Z<sub>i</sub> 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<sup>*</sup> plane to the Z<sub>i</sub>-plane. It identifies the number of branch points of ρ<sup>*−1</sup> from the ρ<sup>*</sup> plane to the Z<sub>i</sub> plane. The branch points of ρ<sup>*−1</sup> and its preimage in the ø–d plane are identified and studied. The domain of the double-valued function ρ<sup>*−1</sup> 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<sup>*−1</sup> is fixed, and we introduce a closed-form solution for the determination of the film thickness of the system. Mathematically, ρ<sup>*−1</sup> 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<sup>*</sup>, as well as their inverses, leads to identification of the two essential exceptive inversion points, which exist at ø=0° and 90°. Accordingly, the closed-form solution is available throughout the ρ plane except at the two points ±1 (corresponding to ø=0° and 90°). A study of the extrema of the magnitude and the phase of both ρ and ρ<sup>*</sup> 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 ø–d plane and the ρ plane induced by ρ and ρ<sup>−1</sup> is given.
© 1999 Optical Society of America
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry
(120.4530) Instrumentation, measurement, and metrology : Optical constants
(120.5700) Instrumentation, measurement, and metrology : Reflection
(310.0310) Thin films : Thin films
A.-R. M. Zaghloul and M. S. A. Yousef, "Ellipsometric function of a film–substrate system:detailed analysis and closed-form inversion," J. Opt. Soc. Am. A 16, 2029-2044 (1999)