Analytic determination of n, κ, and d of an absorbing film from polarimetric data in the thin-film limit

We obtain a solution of the three-phase model in the limit d∕λ⪡1, where the complex refractive index ñ=n+iκ and thickness d of an isotropic film on an isotropic substrate are given analytically at any single wavelength λ from polarimetric data Δρ∕ρ and ΔR∕R, where ρ is the complex reflectance ratio and R is either the p- or s-polarized reflectance. We describe several procedures for extending the range of validity of the solution. Analysis of correlations shows that the uncertainty δ(ΔR∕R) of ΔR∕R is significantly more important than the δ(Δρ∕ρ) of Δρ∕ρ, which allows us to obtain an expression for the uncertainties δn, δκ, and δd of n, κ, and d, and to identify conditions that optimize the determination of the layer parameters. We find that the relative uncertainties δn∕n and δd∕d are not equal, as would be expected if they were determined by the optical thickness nd measured by ellipsometry, but that ΔR∕R breaks the connection. We verify our results by measurements of H2O reversibly physisorbed on oxidized GaAs, finding, for example, that for our conditions δκ is determined more accurately than δn, and δn more accurately than δd. These data and model calculations show that fluctuations in parameters, particularly d, are asymmetric, leading in principle to inaccurate average values. However, we show that the importance of the ΔR∕R data together with the remaining high correlation between n and d allows us to define a characteristic curve that can be used to correct the results for this nonlinearity. Finally, we extend our analysis to determine the orthogonal linear combinations of n, κ, and d that the data actually determine, which explains why the data fit the characteristic curve so well. Our results will be useful in various contexts for the analysis of films less than 1nm thick, for example, in applications involving preparation of next-generation electronic and optoelectronic devices with complicated multilayer structures, real-time control of deposition, and the identification of physisorbed and chemisorbed layers on the monolayer scale.