Unified analysis and mathematical representation of film-thickness behavior of film-substrate systems

The ellipsometric function rho of a film-substrate system is studied as the film thickness d is kept constant and the angle of incidence phiv 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 phiv-d,X,Z, and rho,where X is the film-thickness exponential function and Z is a previously introduced intermediate plane. In the phiv-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 rho is then studied in each DTS, and the CTCʹs 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 rho 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 filmsubstrate 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.