Contours of constant pseudo-Brewster angle in the complex є plane and an analytical method for the determination of optical constants

The locus of all points in the complex plane of the dielectric function є[єr + jєi = |є| exp(jθ)], that represent all possible interfaces characterized by the same pseudo-Brewster angle θpB of minimum p reflectance, is derived in the polar form: |є| = l cos(ζ/3), where l = 2(tan2ΦpB)k, ζ = arccos(- cosθ cos2ΦpB/k3), and k = (1 - 2/3 sin2ΦpB)½. 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 \u3e 80°) is the iso-ΦpB contour accurately approximated as a circle. We find that |є| \u3c 1 for ΦpB \u3c 37.23°, and |є| \u3e 1 for ΦpB \u3e 45°. The optical constants n,k (where n + jk = є½ 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 (R0,φpB) are possible in the domain of fractional optical constants. An analytical solution of the (R0,ΦpB) → (n,k) inversion problem is developed that involves an exact (noniterative) solution of a quartic equation in |є|. Finally, a graphic representation is developed for the determination of complex є from two pseudo-Brewster angles measured in two different media of incidence