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Determining optical constants of thin film on substrate from transmission and reflection data

A computational method of determining spectral dependencies of optical constants, the refractive index n and the extinction coefficient k, for a thin film on a relatively thick and transparent substrate from measured reflectance and transmittance spectra is presented. While the reflectance R and transmittance T are single-valued functions of n and k, the optical constants are multivalued functions of R and T. Thus over a reasonable area of n-k space there is a number of (n, k) solutions for each measured reflectance-transmittance pair. To find solutions the Newton-Raphson numerical method is used. It is easy to find the (unphysical) solutions for low values of n and then we go step by step along the curve n (as a function of k) to the acceptable physical solution. In this way we avoid the possible jump to the solutions which are due to interferences. From time to time it happens that the starting coordinates are not good estimates of solutions and the numerical method failed. In such cases we use some variation of Monte Carlo method, when acceptable starting coordinates are found by chance in a small neighborhood of the original supposed starting point. The solutions are calculated for all measured frequencies. In order to simulate the role of the finite spectral resolution of the apparatus we smooth n and k by averaging their values at central point and its neighbors, with a reasonable step through the simulated spectral width. The method is applied to the semiconducting (beta) -iron disilicide film on silicon substrate.