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

Abstract

The ratio Pt = Tp/T, 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/X and the angle of incidence 0 for a given film refractive index N. The maximum value of the differential transmission phase shift (or retardance), At = argPt, is determined, for given N and X, by a simple geometrical construction that involves the iso-k circle locus of Pt in the complex plane. The upper bound on this maximum equals arctan[N -(1N)]I21 and is attained in the limit of grazing incidence. An analytical noniterative method is developed for determining N and d of the film from Pt measured by transmission ellipsometry (TELL) at (p = 45°. An explicit expression for At of an ultrathin film, d/X << 1, is derived in product form that shows the dependence of At on N, , and d/X separately. The angular dependence is given by an obliquity factor, fo(0) = 21/2 sino tan, which is verified experimentally by TELL measurements on a stable planar soap film in air at X = 633 nm. The singularity of f at 4, = 90° is resolved; At is shown to have a maximum just short of grazing incidence and drops to 0 at 4 = 900. Because N and d/ are inseparable for an ultrathin film, N is determined by a Brewster angle measurement and d/X is subsequently obtained from At.Finally, the ellipsometric function in reflection Pr is related to that in transmission Pt.

Transmission polarization devices using an unsupported film/pellicle: closed-form design formula

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Tilted bilayer membranes as simple transmission quarter-wave retardation plates

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Inversion of transmission ellipsometric data for transparent films

“…Transmission ellipsometry measures the real and (1) imaginary parts of T. Section 2 shows how the unknowns E and Az can be determined analytically from Eq. (1) and the experimental data.…”