## Abstract

We have found a significant relation, *r*_{p} = *r*_{s}(*r*_{s} − cos2ϕ)/(1 − *r*_{s} cos2ϕ), between Fresnel’s interface complex-amplitude reflection coefficients *r*_{p} and *r*_{s} for the parallel (*p*) and perpendicular (*s*) polarizations at the same angle of incidence ϕ. This relation is universal in that it applies to reflection at *all* interfaces between homogeneous isotropic media *collectively* and, of course, throughout the electromagnetic spectrum. We investigate the properties of this function, *r*_{p} = *f*(*r*_{s}), and its inverse, *r*_{s} = *g*(*r*_{p}) conformal mappings between the complex planes of *r*_{s} and *r*_{p}. A related function, ρ = (*r*_{s} − cos2ϕ)/(1 − *r*_{s} cos2ϕ), which is a bilinear transformation, is also studied, where ρ = *r*_{p}/r_{s} is the (ellipsometric) ratio of reflection coefficients. Several previously described reflection characteristics come out readily as specific results of this work. Simple explicit analytical and graphical solutions are provided to determine reflection phase shifts and the dielectric function from measured *p* and *s* reflectances at the same angle of incidence. We also show that when *r*_{p} is real, negative, and in the range −1 ≤ *r*_{p} ≤ −tan^{2}(*ϕ* − 45°), *r*_{s} is complex and its locus in the complex plane is an arc of a circle with center on the real axis at sec2ϕ and radius of | tan2*ϕ* |. Under these conditions, we also find the interesting result that |*r*_{s}| = |*r*_{p}|^{1/2}.

© 1979 Optical Society of America

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### Equations (30)

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(1)
$${r}_{p}=\frac{\u220acos\varphi -{(\u220a-{sin}^{2}\varphi )}^{1/2}}{\u220acos\varphi +{(\u220a-{sin}^{2}\varphi )}^{1/2}},$$
(2)
$${r}_{s}=\frac{cos\varphi -{(\u220a-{sin}^{2}\varphi )}^{1/2}}{cos\varphi +{(\u220a-{sin}^{2}\varphi )}^{1/2}},$$
(3)
$$x={(\u220a-{sin}^{2}\varphi )}^{1/2},$$
(4)
$${r}_{p}=\frac{cos\varphi ({x}^{2}+{sin}^{2}\varphi )-x}{cos\varphi ({x}^{2}+{sin}^{2}\varphi )+x},$$
(5)
$${r}_{s}=(cos\varphi -x)/(cos\varphi +x).$$
(6)
$${r}_{p}={r}_{s}({r}_{s}-cos2\varphi )/(1-{r}_{s}cos2\varphi ).$$
(7)
$${r}_{s}=cos2\varphi ,$$
(8)
$$\frac{\partial {r}_{p}}{\partial {r}_{s}}=(-{r}_{s}^{2}cos2\varphi +2{r}_{s}-cos2\varphi )/{(1-{r}_{s}cos2\varphi )}^{2},$$
(9)
$${r}_{s}=sec2\varphi -tan2\varphi =-tan(\varphi -45\xb0).$$
(10)
$${r}_{p}=-{(sec2\varphi -tan2\varphi )}^{2}=-{tan}^{2}(\varphi -45\xb0)=-{r}_{s}^{2}.$$
(11)
$${r}_{s}=\frac{1}{2}cos2\varphi (1-{r}_{p})+{\left({r}_{p}+\frac{1}{4}{cos}^{2}2\varphi {(1-{r}_{p})}^{2}\right)}^{1/2}.$$
(12)
$$\rho ={r}_{p}/{r}_{s}$$
(13)
$$\rho =({r}_{s}-cos2\varphi )/(1-{r}_{s}cos2\varphi ).$$
(14)
$${r}_{s}=(\rho +cos2\varphi )/(1+\rho cos2\varphi ),$$
(15)
$${r}_{s}=[\rho -cos2(90\xb0-\varphi )/[1-\rho cos2(90\xb0-\varphi )].$$
(16)
$$\begin{array}{cc}R={|{r}_{l}|}^{2},& l=p,s.\end{array}$$
(17)
$${R}_{p}={R}_{s}\frac{{R}_{s}+{cos}^{2}2\varphi -2{R}_{s}^{1/2}cos2\varphi cos{\delta}_{s}}{1+{R}_{s}+{cos}^{2}2\varphi -2{R}_{s}^{1/2}cos2\varphi cos{\delta}_{s}},$$
(18)
$$cos{\delta}_{s}=\frac{({R}_{s}^{2}-{r}_{p})+{R}_{s}(1-{R}_{p}){cos}^{2}2\varphi}{2{R}_{s}^{1/2}({R}_{s}-{R}_{p})cos2\varphi}.$$
(19)
$$\u220a={sin}^{2}\varphi +{cos}^{2}\varphi {[(1-{r}_{s})/(1+{r}_{s})]}^{2},$$
(20)
$$tan\mathrm{\Delta}=\frac{{R}_{s}^{1/2}sin{\delta}_{s}{sin}^{2}2\varphi}{{R}_{s}^{1/2}cos{\delta}_{s}(1+{cos}^{2}2\varphi )-{R}_{s}cos2\varphi},$$
(21)
$$|\rho cos2\varphi |=|{r}_{s}-cos2\varphi |/|{r}_{s}-sec2\varphi |.$$
(22)
$$-1\le {r}_{p}\le -{tan}^{2}(\varphi -45\xb0),$$
(23)
$$Q={r}_{p}+(1/4){cos}^{2}2\varphi {(1-{r}_{p})}^{2},$$
(25)
$${r}_{p}=-{(sec2\varphi \mp tan2\varphi )}^{2}=-{tan}^{2}(\varphi \mp 45\xb0).$$
(27)
$$\begin{array}{cc}x=(1/2)cos2\varphi (1-{r}_{p}),& y={(-Q)}^{1/2}.\end{array}$$
(28)
$${(x-sec2\varphi )}^{2}+{y}^{2}={tan}^{2}2\varphi ,$$
(29)
$$|{r}_{s}|={|{r}_{p}|}^{1/2}.$$
(30)
$${R}_{p}={{R}_{s}}^{2},$$