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 ${\overline{\varphi }}_{min}⩽\overline{\varphi }⩽{\overline{\varphi }}_{max}$ over which the associated principal azimuth $\overline{\psi }$ varies between 0° and 90° (i.e., $0°⩽\overline{\psi }⩽90°$) and the reflection phase difference Δ assumes either one of the two values: +π/2 or −π/2. We determine the principal angle $\overline{\varphi }(d)$ and principal azimuth $\overline{\psi }(d)$ 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 $\overline{\varphi }$ and $\overline{\psi }$ during film growth is particularly interesting.