Complex reflection coefficients for the parallel and perpendicular polarizations of a film–substrate system

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Abstract
The complex reflection coefficients Rν(ϕ,ζ) of a film–substrate system for the parallel (ν = p) and perpendicular (ν = s) polarizations are examined in detail as functions of the angle of incidence ϕ(0 ≤ ϕ ≤ 90°) and the reduced normalized film thickness ζ(0 ≤ ζ < 1). For definiteness, the reflection of light of wavelength λ = 0.6328 μm by the air–SiO2–Si system is assumed. Families of circles that represent the constant-angle-of-incidence contours, their envelopes, and the associated constant-thickness contours of Rp and Rs are all presented in the complex plane. Furthermore, the amplitude-reflectance and phase-shift functions, |Rν|(ϕ,ζ) and argRν(ϕ,ζ), are plotted vs ζ with ϕ constant and vs ϕ with ζ constant. It is shown that Rp or Rs can assume the same complex value at two different angles of incidence (i.e., the film–substrate system can have identical reflection characteristics for a given polarization at two angles) for certain ranges of film thickness. The distinct case of internal reflection is represented by a separate example.