Superconvergence relations and sum rules for reflection spectroscopy

Superconvergence relations for the complex reflectivity $\tilde{r}$(ω), its amplitude r(ω), and its phase θ(ω) are considered. It is shown that infinite families of sum rules can be systematically generated for various moments of Re ${\tilde{r}}^m$(ω) and Im ${\tilde{r}}^m$(ω); these rules are analogous to similar sum rules for the refractive index. They provide self-consistency checks on ellipsometric and interferometric measurements in which both phase and amplitude information are obtained experimentally. For the more common situation in which only the reflectivity amplitude is measured, it is customary to decouple amplitude and phase by considering the complex function ln $\tilde{r}$(ω) = ln r(ω) + iθ(ω) Simple superconvergence rules do not hold for this quantity because of the divergence of ln r(ω) at high frequencies where r(ω) approaches zero. However, this difficulty can be partially overcome by treating the logarithmic derivative of ln $\tilde{r}$ or by limiting consideration to finite energy intervals. The resulting rules are applicable to nonmetals and include a superconvergence relation for dr/dω and finite-energy reflectance-conservation and phase f-sum rules. The derivative rule is applicable to reflectance and modulated-reflectance studies, whereas the finite-energy rules provide direct self-consistency checks on infrared spectra. Examples of the application of these rules to the reststrahl spectra of an ionic solid are discussed.