Analysis of modulation spectra of stratified media

Abstract

A general formulation is obtained which relates the relative change of reflectance observed in a normal-incidence modulated-reflectance experiment to the dielectric properties of the system for laminar configurations consisting of a substrate, ambient, and possible surface dielectric overlayers. The two general types of modulation interactions, perturbation-induced changes of the substrate dielectric function or of the thickness of one of the dielectric layers, are discussed as limiting cases. The modulated-substrate expressions are evaluated numerically for the case of optically thick Ni and Ni–Al203 films on a Ge substrate, typical examples of Schottky-barrier and MOS electroreflectance configurations. It is shown in the latter case that the intrinsic sensitivity obtainable with shot-noise-limited measurement systems is neither enhanced nor impaired for modulation measurements performed in the vicinity of the reflectance minima characteristic of these field-effect configurations, although operation near these minima should be avoided for other reasons. The line-shape evolution observed in reflectance measurements of the exciton polariton is shown to follow naturally from the interference effect of the small dielectric discontinuity within the sample. The approximate integral expression describing the effect of inhomogeneous perturbations is also related to the more-general laminar solution. It is shown that, for small modulations, the integral formulation is equivalent to treating the substrate as being uniform with respect to propagation, but producing a net reflected component of the field, obtained by summing the values at the surface of the reflected components generated at the small dielectric discontinuities between the hypothetical laminations in the limit that the lamination thickness approaches zero.