Photometric methods for determining the optical constants and the thicknesses of thin absorbing films: selection of a combination of photometric quantities on the basis of error analysis

摘要

We select the best combinations of spectrophotometric quantities for the most accurate determination of the optical constants, n (refractive index), k (absorption coefficient), and the thicknesses of thin absorbing films. The basic comparative criteria used are the maximum absolute errors in the determination of n, k, and d that result from experimental errors in photometric measurements and in the optical constants of the substrates. We studied all possible combinations of photometric quantities T, T_(s)~(θ), T_(p)~(θ), R, R_(s)~(θ), R_(p)~(θ), R_(m), R_(ms)~(θ), and R_(mp)~(θ) at 0°<θ≤70°, where T denotes transmission; R, reflection; the subscripts s and p, s- and p-polarized light; m, reflection from a thin film coated upon an opaque substrate; and superscript θ, the angle of incidence of light. The absence of the subscripts s and p implies nonpolarized light; that of the subscript m, a nonabsorbing substrate; and that of superscript θ, normally incident light. The error analysis that is made admits the following conclusions: (1) The best double combinations are (TR), (TR_(m)), (TR_(p)~(70)), and (TR_(mp)~(70)); (2) the best triple combinations are (TRR_(m)), (TRR_(p)~(70)), (TRR_(mp)~(70)), (TR_(m)R_(p)~(70)), and (TR_(m)R_(mp)~(70)); (3) the methods indicated above, suitably combined, are quite sufficient to provide the maximum accuracy and reliability of n, k, and d for all practical situations; (4) TRR methods based on measurements with obliquely polarized light are more suitable for thin films with n<1, such as some metal films; (5) the regions of n, k, and d/λ with the highest and the lowest accuracies do not overlap in either the TR or the TRR methods. Hence more combinations, preferably all, should be applied for the most accurate determination of n, k (and d), and the errors should be evaluated as a criterion for the best combination.