Contours of constant pseudo-Brewster angle in the complex ? plane and an analytical method for the determination of optical constants.
The locus of all points in the complex plane of the dielectric function ?[?(r) + j?(i) = |?| exp(jtheta)], that represent all possible interfaces characterized by the same pseudo-Brewster angle theta(p)B of minimum p reflectance, is derived in the polar form: |?| = l cos(zeta/3), where l = 2(tan(2)Phi(p)B)k, zeta = arccos(- costheta cos(2)Phi(p)B/k(3)), and k = (1 - 2/3 sin(2)Phi(p)B)(1/2). Families of iso-Phi(p)B contours for (I) 0 degrees </= Phi(p)B </= 45 degrees and (II) 45 degrees </= Phi(p)B </= 75 degrees are presented. In range I, an iso-Phi(p)B contour resembles a cardioid. In range II, the contour gradually transforms toward a circle centered on the origin as Phi(p)B increases. However, the deviation from a circle is still substantial. Only near grazing incidence (Phi(p)B > 80 degrees ) is the iso-Phi(p)B contour accurately approximated as a circle. We find that |?| < 1 for Phi(p)B < 37.23 degrees , and |?| > 1 for Phi(p)B > 45 degrees . The optical constants n,k (where n + jk = ?((1/2)) is the complex refractive index) are determined from the normal incidence reflectance R(0) and Phi(p)B graphically and analytically. Nomograms that consist of iso-R(0) and iso-Phi(p)B 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 (R(0),phi(p)B) are possible in the domain of fractional optical constants. An analytical solution of the (R(0),Phi(p)B) ? (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.
Azzam
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Ugbo
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