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

The ratio rho(t) = T(p)/T(s) 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/lambda and the angle of incidence Phi for a given film refractive index N. The maximum value of the differential transmission phase shift (or retardance), Delta(t) = argrho(t), is determined, for given N and Phi, by a simple geometrical construction that involves the iso-Phi circle locus of rho(t) 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 rho(t) measured by transmission ellipsometry (TELL) at Phi = 45 degrees . An explicit expression for d Delta(t) of an ultrathin film, d/lambda << 1, is derived in product form that shows the dependence of Delta(t) on N, Phi, and d/lambda separately. The angular dependence is given by an obliquity factor, f(0)(Phi) = 2((1/2)) sinPhi tanPhi, which is verified experimentally by TELL measurements on a stable planar soap film in air at lambda = 633 nm. The singularity of f(0) at Phi = 90 degrees is resolved; Delta(t) is shown to have a aximum just short of grazing incidence and drops to 0 at Phi = 90 degrees . Because N and d/lambda are inseparable for an ultrathin film, N is determined by a Brewster angle measurement and d/lambdais subsequently obtained from Delta(t) Finally, the ellipsometric function in reflection rho(r) is related to that in transmission P(t).