Reflection and transmission ellipsometry of a uniform layer

We examine the general properties of the ellipsometric quantities ρ = r_p/r_s and τ = t_p/t_s for a nonabsorbing uniform film. At fixed angle of incidence and for variable thickness, τ moves periodically along circles (as is known) and ρ along quartics in the complex plane. For the special case of a film between like media we derive envelopes (and corresponding bounds on τ and ρ) as functions of the dielectric constants of the layer and its bounding medium. In the case of unlike media on either side of the film, bounds are found for Re τ and Im τ for ∊² > ∊₁∊₂ (where ∊₁, ∊, and ∊₂ are, respectively, the dielectric constants of the medium from which light is incident, of the film, and of the substrate), we find an approximate upper bound for the ellipsometric quantity $\overline{\rho }$ (defined as Im ρ when Re ρ = 0).