## Abstract

The functions, *w* = *f*(*z*), that describe the transformation of Fresnel’s reflection coefficients of parallel and perpendicularly polarized light between normal and oblique incidence, as well as their inverses, *z* = *g*(*w*), are studied in detail as conformal mappings between the complex *z* and *w* planes for angles of incidence of 15, 30, 45, 60, and 75°. New nomograms are obtained for the determination of optical properties of absorbing isotropic and anisotropic media from measurements of reflectances of *s*- or *p*-polarized light at normal and oblique incidence.

© 1980 Optical Society of America

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### Equations (12)

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(1)
$$w=\frac{\left(1-z\right)-{\left(1-az+{z}^{2}\right)}^{1/2}}{\left(1-z\right)+{\left(1-az+{z}^{2}\right)}^{1/2}},$$
(2)
$$a=4\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{2}\varphi +2.$$
(3)
$$z=\frac{\left(a+2p\right)\pm {\left[\left({a}^{2}-4\right)+4\left(a+2\right)p\right]}^{1/2}}{2\left(1-p\right)},$$
(4)
$$p={\left[\left(1-w\right)/\left(1+w\right)\right]}^{2},$$
(5)
$$w=\frac{{\left(1+z\right)}^{2}-\left(1-z\right){\left(1+az+{z}^{2}\right)}^{1/2}}{{\left(1+z\right)}^{2}+\left(1-z\right){\left(1+az+{z}^{2}\right)}^{1/2}},$$
(6)
$${w}_{p}={w}_{s}\left({w}_{s}-\text{cos}2\varphi \right)/\left(1-{w}_{s}\phantom{\rule{0.2em}{0ex}}\text{cos}2\varphi \right),$$
(7)
$${w}_{s}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\phantom{\rule{0.2em}{0ex}}\text{cos}2\varphi \left(1-{w}_{p}\right)\pm {\left[{w}_{p}+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{2}2\varphi {\left(1-{w}_{p}\right)}^{2}\right]}^{1/2}.$$
(8)
$${w}_{p}=\frac{\u220a\text{cos}\varphi -{\left(\u220a-{\text{sin}}^{2}\varphi \right)}^{1/2}}{\u220a\text{cos}\varphi +{\left(\u220a-{\text{sin}}^{2}\varphi \right)}^{1/2}},$$
(9)
$$\u220a=\left[1\pm {\left(1-{Q}_{p}\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}2\varphi \right)}^{1/2}\right]/2{Q}_{p}\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{2}\varphi ,$$
(10)
$${Q}_{p}={\left[\left(1-{w}_{p}\right)/\left(1+{w}_{p}\right)\right]}^{2}.$$
(11)
$${z}_{p}=\left({\u220a}^{1/2}-1\right)/\left({\u220a}^{1/2}+1\right).$$
(12)
$$N=\pm n\left(1-z\right)/\left(1+z\right),$$