The locus of all points in the complex plane of the dielectric function ∊[∊r + j∊i, = |∊| exp(jθ)], that represent all possible interfaces characterized by the same pseudo-Brewster angle ϕpB of minimum p reflectance, is derived in the polar form: |∊| = l cos(ζ/3), where l = 2(tan2ϕpB)k, ζ = arccos(− cosθ cos2ϕpB/k3), and k = (1 − ⅔ sin2ϕpB)1/2. Families of iso-ϕpB contours for (I) 0° ≤ ϕpB 45° and (II) 45° ≤ ϕpB ≤ 75° are presented. In range I, an iso-ϕpB contour resembles a cardioid. In range II, the contour gradually transforms toward a circle centered on the origin as ϕpB increases. However, the deviation from a circle is still substantial. Only near grazing incidence ϕpB > 80°) is the iso-ϕpB contour accurately approximated as a circle. We find that |∊| < 1 for ϕpB < 37.23°, and |∊| > 1 for ϕpB > 45°. The optical constants n,k (where n+jk = ϕ1/2 is the complex refractive index) are determined from the normal incidence reflectance R0 and ϕpB graphically and analytically. Nomograms that consist of iso-R0 and iso-ϕpB families of contours in the nk plane are presented. Equations that permit the reader to produce his own version of the same nomogram are also given. Valid multiple solutions (n,k) for a given measurement set (Ro, ϕpB) are possible in the domain of fractional optical constants. An analytical solution of the (Ro,ϕpB) → (n,k) inversion problem is developed that involves an exact (noniterative) solution of a quartic equation in |∊|. Finally, a graphic representation is developed for the determination of complex ∊ from two pseudo-Brewster angles measured in two different media of incidence.
© 1989 Optical Society of AmericaFull Article | PDF Article
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