## Abstract

The complex Fresnel reflection coefficients ${r}_{p}$ and ${r}_{s}$ of $p$- and $s$-polarized light and their ratio $\rho ={r}_{p}/{r}_{s}$ at the pseudo-Brewster angle (PBA) ${\varphi}_{pB}$ of a dielectric–conductor interface are evaluated for all possible values of the complex relative dielectric function $\epsilon =|\epsilon |\mathrm{exp}(-j\theta )={\epsilon}_{r}-j{\epsilon}_{i}$, ${\epsilon}_{i}>0$ that share the same ${\varphi}_{pB}$. Complex-plane trajectories of ${r}_{p}$, ${r}_{s}$, and $\rho $ at the PBA are presented at discrete values of ${\varphi}_{pB}$ from 5° to 85° in equal steps of 5° as $\theta $ is increased from 0° to 180°. It is shown that for ${\varphi}_{pB}>70\xb0$ (high-reflectance metals in the IR) ${r}_{p}$ at the PBA is essentially pure negative imaginary and the reflection phase shift ${\delta}_{p}=\mathrm{arg}({r}_{p})\approx -90\xb0$. In the domain of fractional optical constants (vacuum UV or light incidence from a high-refractive-index immersion medium) $0<{\varphi}_{pB}<45\xb0$ and ${r}_{p}$ is pure real negative (${\delta}_{p}=\pi $) when $\theta ={\mathrm{tan}}^{-1}(\sqrt{\mathrm{cos}(2{\varphi}_{pB})})$, and the corresponding locus of $\epsilon $ in the complex plane is obtained. In the limit of ${\epsilon}_{i}=0$, ${\epsilon}_{r}<0$ (interface between a dielectric and plasmonic medium) the total reflection phase shifts ${\delta}_{p}$, ${\delta}_{s}$, $\mathrm{\Delta}={\delta}_{p}-{\delta}_{s}=\mathrm{arg}(\rho )$ are also determined as functions of ${\varphi}_{pB}$.

© 2013 Optical Society of America

Full Article |

PDF Article

**OSA Recommended Articles**
### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

### Equations (17)

Equations on this page are rendered with MathJax. Learn more.

(1)
$${r}_{p}=\frac{\epsilon \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\varphi -{(\epsilon -{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\varphi )}^{1/2}}{\epsilon \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\varphi +{(\epsilon -{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\varphi )}^{1/2}},$$
(2)
$${r}_{s}=\frac{\mathrm{cos}\text{\hspace{0.17em}}\varphi -{(\epsilon -{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\varphi )}^{1/2}}{\mathrm{cos}\text{\hspace{0.17em}}\varphi +{(\epsilon -{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\varphi )}^{1/2}}.$$
(3)
$${\epsilon}_{r}=|\epsilon |\mathrm{cos}\text{\hspace{0.17em}}\theta ,\phantom{\rule[-0.0ex]{2em}{0.0ex}}{\epsilon}_{i}=|\epsilon |\mathrm{sin}\text{\hspace{0.17em}}\theta ,$$
(4)
$$|\epsilon |=\ell \text{\hspace{0.17em}}\mathrm{cos}(\varsigma /3),\ell =2u{(1-\frac{2}{3}u)}^{1/2}/(1-u),\varsigma ={\mathrm{cos}}^{-1}[-(1-u)\mathrm{cos}\text{\hspace{0.17em}}\theta /{(1-\frac{2}{3}u)}^{3/2}],u={\mathrm{sin}}^{2}\text{\hspace{0.17em}}{\varphi}_{pB},0\le \theta \le 180\xb0.$$
(5)
$$\theta ({\delta}_{p}=\pi )={\mathrm{tan}}^{-1}\left(\sqrt{\mathrm{cos}(2{\varphi}_{pB})}\right).$$
(6)
$$\epsilon ={\epsilon}_{r}=-\frac{1}{2}\text{\hspace{0.17em}}{\mathrm{tan}}^{2}\text{\hspace{0.17em}}{\varphi}_{pB}[1+{(9-8\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\text{\hspace{0.17em}}{\varphi}_{pB})}^{1/2}]$$
(7)
$$\rho ={r}_{p}/{r}_{s}=\frac{\mathrm{sin}\text{\hspace{0.17em}}\varphi \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\varphi -{(\epsilon -{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\varphi )}^{1/2}}{\mathrm{sin}\text{\hspace{0.17em}}\varphi \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\varphi +{(\epsilon -{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\varphi )}^{1/2}}.$$
(A1)
$${y}^{2}=a+{({a}^{2}-bx)}^{1/2}-{x}^{2},$$
(A2)
$$a={u}^{2}(1.5-u)/{(1-u)}^{2},\phantom{\rule{0ex}{0ex}}b={u}^{3}/{(1-u)}^{2},\phantom{\rule{0ex}{0ex}}u={\mathrm{sin}}^{2}\text{\hspace{0.17em}}{\varphi}_{pB}.$$
(A3)
$${y}^{2}=2ux-{x}^{2}.$$
(A4)
$${({a}^{2}-bx)}^{1/2}=2ux-a.$$
(A5)
$$4{u}^{2}{x}^{2}=(4au-b)x.$$
(A6)
$$x=(4au-b)/(4{u}^{2}).$$
(A8)
$$y=u\sqrt{1-2u}/(1-u).$$
(A9)
$$\mathrm{tan}\text{\hspace{0.17em}}\theta =y/x=\sqrt{1-2u}.$$
(A10)
$$\theta ({\delta}_{p}=\pi )={\mathrm{tan}}^{-1}\left(\sqrt{\mathrm{cos}(2{\varphi}_{pB})}\right).$$