Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air

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Abstract
The ratio ρt = Tp/Ts of the complex amplitude transmission coefficients for the p and s polarizations of a transparent unbacked or embedded thin film is examined as a function of the film thickness-to-wavelength ratio d/λ and the angle of incidence ϕ for a given film refractive index N. The maximum value of the differential transmission phase shift (or retardance), Δt = argρt, is determined, for given N and ϕ, by a simple geometrical construction that involves the iso–ϕ circle locus of ρt in the complex plane. The upper bound on this maximum equals arctan{[N − (1/N)]/2} and is attained in the limit of grazing incidence. An analytical noniterative method is developed for determining N and d of the film from ρt measured by transmission ellipsometry (TELL) at ϕ = 45°. An explicit expression for Δt of an ultrathin film, d/λ ≪ 1, is derived in product form that shows the dependence of Δt on N, ϕ, and d/λ separately. The angular dependence is given by an obliquity factor, fo(ϕ) = 21/2 sinϕ tanϕ, which is verified experimentally by TELL measurements on a stable planar soap film in air at λ = 633 nm. The singularity of fo at ϕ = 90° is resolved; Δt is shown to have a maximum just short of grazing incidence and drops to 0 at ϕ = 90°. Because N and d/λ are inseparable for an ultrathin film, N is determined by a Brewster angle measurement and d/λ is subsequently obtained from Δt. Finally, the ellipsometric function in reflection ρr is related to that in transmission ρt.