Principal angle, principal azimuth, and principal-angle ellipsometry of film-substrate systems

When the film thickness is considered as a parameter, a system composed of a transparent film on an absorbing substrate (in a transparent ambient) is characterized by a range of principal angle ϕ¯min⩽ϕ¯⩽ϕ¯max over which the associated principal azimuth ψ¯ varies between 0° and 90° (i.e., 0°⩽ψ¯⩽90°) and the reflection phase difference Δ assumes either one of the two values: +π/2 or −π/2. We determine the principal angle ϕ¯(d) and principal azimuth ψ¯(d) as functions of film thickness d for the vacuum-SiO2-Si system at several wavelengths as a concrete example. When the film thickness exceeds a certain minimum value, more than one principal angle becomes possible, as can be predicted by a simple graphical construction. We apply the results to principal-angle ellipsometry (PAE) of film-substrate systems; the relationship between ϕ¯ and ψ¯ during film growth is particularly interesting.