title: Unified analysis and mathematical representation of film-thickness behavior of film-substrate systems.
abstract: The ellipsometric function p of a film-substrate system is studied as the film thickness d is kept constant and the angle of incidence phi is changed. The generated constant-thickness contours (CTCs) are characterized by an introduced mathematical behavior indicator that represents a group of CTCs. The behavior of each group is developed and studied in the four planes phi-d, X, Z, and p, where X is the film-thickness exponential function and Z is a previously introduced intermediate plane. In the phi-d plane the film-thickness domain is identified and divided into a sequence of disconnected thickness subdomains (DTSs), depending on only N0 and N1, and their number depending on the range in which N0/N1 lies. The behavior of the CTCs in the successive planes X, Z, and p is then studied in each DTS, and the CTC's space is divided into disconnected subfamilies according to the behavior indicator. Equivalence classes that reduce the infinite number of subfamilies into a finite number are then introduced. The transformation from each plane to the next is studied with the origin of the Z plane mapped onto the point at infinity of the p plane, forming a singularity. A multiple-film-thickness inequality is derived to determine the unique solution of the film thickness. The type of reflection being internal or external at both ambient-film and film-substrate interfaces affects the analysis and is also considered. To conclude we introduce the design of polarization-preserving devices and a novel oscillating single-element ellipsometer to fully characterize zero film-substrate systems as examples of applying the knowledge developed here.
authors.affiliation: Ellipsometry Research and Applications Laboratories, School of Electrical and Computer Engineering, Georgia Institute of Technology, 210 Technology Circle, Savannah, Georgia 31407, USA. ar.zaghloul@gtsav.gatech.edu
authors.forename: A Rahman M
authors.initials: AR
authors.lastname: Zaghloul
authors.forename: Mohamed S A
authors.initials: MS
authors.lastname: Yousef