Exact computation of the reflectance of a surface layer of arbitrary refractive-index profile and an approximate solution of the inverse problem

Abstract

The angular reflectance of a graded-index layer, of arbitrary refractive-index profile, on the surface of a uniform substrate is calculated exactly by direct numerical integration of the wave equation for both states of polarization of the incident light. A study of the results for a number of selected profiles shows that (1) oscillations in reflectance versus wave-number graphs have an almost constant period, and (2) the period, when plotted against angle of incidence, gives a curve that is quite sensitive to the shape of the refractive-index profile. This sensitivity is the basis of a simple graphical procedure by means of which the inverse problem (i.e., deducing the index profile from reflectance measurements) can be solved. The procedure is applied to published reflectance data to determine both the thickness of a graded-index surface layer and the refractive-index profile.