## 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

## 1. INTRODUCTION

A salient feature of the reflection of collimated monochromatic $p$ (TM)-polarized light at a planar interface between a transparent medium of incidence (dielectric) and an absorbing medium of refraction (conductor) is the appearance of a reflectance minimum at the pseudo-Brewster angle (PBA) ${\varphi}_{pB}$. If the medium of refraction is also transparent, the minimum reflectance is zero and ${\varphi}_{pB}$ reverts back to the usual Brewster angle ${\varphi}_{B}={\mathrm{tan}}^{-1}\text{\hspace{0.17em}}n={\mathrm{tan}}^{-1}\sqrt{{\epsilon}_{r}}$. The PBA ${\varphi}_{pB}$ is determined by the complex relative dielectric function $\epsilon ={\epsilon}_{1}/{\epsilon}_{0}={\epsilon}_{r}-j{\epsilon}_{i}$, ${\epsilon}_{i}>0$, where ${\epsilon}_{0}$ and ${\epsilon}_{1}$ are the real and complex permittivities of the dielectric and conductor, respectively, by solving a cubic equation in $u={\mathrm{sin}}^{2}\text{\hspace{0.17em}}{\varphi}_{pB}$ [1–5]. Measurement of ${\varphi}_{pB}$ and of reflectance at that angle or at normal incidence enables the determination of complex $\epsilon $ [1,6–9]. It is also possible to determine $\epsilon $ of an optically thick absorbing film from two PBAs measured in transparent ambient and substrate media that sandwich the thick film [10]. Reflection at the PBA has also had other interesting applications [11,12].

For light reflection at any angle of incidence $\varphi $ the complex-amplitude Fresnel reflection coefficients (see, e.g., [13]) of the $p$ and $s$ polarizations are given by

In this paper, loci of all possible values of complex ${r}_{p}=|{r}_{p}|\mathrm{exp}(j{\delta}_{p})$, ${r}_{s}=|{r}_{s}|\mathrm{exp}(j{\delta}_{s})$, and $\rho ={r}_{p}/{r}_{s}=|\rho |\mathrm{exp}(j\mathrm{\Delta})$ at the PBA are determined at discrete values of ${\varphi}_{pB}$ from 5° to 85° in equal steps of 5° and as $\theta =-\mathrm{arg}(\epsilon )$ covers the full range $0\xb0\le \theta \le 180\xb0$. These results are presented in Sections 2, 3, and 4, respectively, and lead to interesting conclusions. In particular, questions related to phase shifts that accompany the reflection of $p$- and $s$-polarized light at the PBA (e.g., [12]) are settled. Section 5 summarizes the essential conclusions of this paper.

## 2. COMPLEX REFLECTION COEFFICIENT OF THE $p$ POLARIZATION AT THE PBA

Figure 1 shows the loci of complex ${r}_{p}$ as $\theta $ increases from 0° to 180° at constant values of ${\varphi}_{pB}$ from 5° to 85° in equal steps of 5°. All constant-${\varphi}_{pB}$ contours begin at the origin $O$ ($\theta =0$) as a common point, that represents zero reflection at an ideal Brewster angle, and end on the 90° arc of the unit circle in the third quadrant (shown as a dotted line) that represents total reflection $|{r}_{p}|=1$ at $\theta =180\xb0$ $({\epsilon}_{i}=0,{\epsilon}_{r}<0)$. A quick conclusion from Fig. 1 is that for ${\varphi}_{pB}>70\xb0$ (high-reflectance metals) ${r}_{p}$ at the PBA is essentially pure negative imaginary, and ${\delta}_{p}\approx -90\xb0$.

In Fig. 1 the constant-${\varphi}_{pB}$ contours of ${r}_{p}$ for $0<{\varphi}_{pB}<45\xb0$ spill over into a limited range of the second quadrant of the complex plane and each contour intersects the negative real axis. In Appendix A it is shown that $\theta $ at the point of intersection, where ${\delta}_{p}=\mathrm{arg}({r}_{p})=\pi $, is given by the remarkably simple formula

The locus of complex $\epsilon $ such that ${\delta}_{p}=\mathrm{arg}({r}_{p})=\pi $ at the PBA [as determined by Eqs. (3)–(5)] falls in the domain of fractional optical constants and is shown in Fig. 3. The end points (0, 0) and (1, 0) of this trajectory correspond to ${\varphi}_{pB}=0$ and 45°, respectively. At $\epsilon =(0.6,0.3)$, a point that falls exactly on the curve very near to its peak, ${\varphi}_{pB}=37.761\xb0$.

For small PBAs, ${\varphi}_{pB}\le 5\xb0$, the upper limit on $|\epsilon |$ is calculated from $\ell $ of Eq. (4), $|\epsilon |=\ell \le 0.0153$, and represents the domain of so-called epsilon-near-zero (ENZ) materials [16].

Negative real values of $\epsilon $ at $\theta =180\xb0$ [14] are given by

In Fig. 5 ${\delta}_{p}$ is plotted as a function of $\theta $ for ${\varphi}_{pB}$ from 10° to 40° in equal steps of 10°. Vertical transitions from $+180\xb0$ to $-180\xb0$ are located at $\theta $ values that agree with Eq. (5).

Another family of ${\delta}_{p}$-versus-$\theta $ curves for ${\varphi}_{pB}$ from 45° to 85° in equal steps of 5° is shown in Fig. 6. For ${\varphi}_{pB}>45\xb0$ the ${\delta}_{p}$-versus-$\theta $ curve first exhibits a minimum then reaches saturation as $\theta \to 180\xb0$. The saturated value of ${\delta}_{p}$ is a function of ${\varphi}_{pB}$ and is shown in Fig. 4.

## 3. COMPLEX REFLECTION COEFFICIENT OF THE $s$ POLARIZATION AT THE PBA

Figure 7 shows the loci of complex ${r}_{s}$ as $\theta $ increases from 0° to 180° at discrete values of ${\varphi}_{pB}$ from 5° to 85° in equal steps of 5°. All curves start on the real axis at $\theta =0$, ${r}_{s}=\mathrm{cos}(2{\varphi}_{pB})$, which is the $s$ amplitude reflectance at the Brewster angle of a dielectric–dielectric interface [17], and terminate on the upper half of the unit circle (dotted line) that represents total reflection ${r}_{s}=\mathrm{exp}(j{\delta}_{s})$ at $\theta =180\xb0$ $({\epsilon}_{i}=0,{\epsilon}_{r}<0)$. The associated total reflection phase shift ${\delta}_{s}$ along the dotted semicircle is a function of ${\varphi}_{pB}$ as shown in Fig. 4.

Although we are locked on the PBA, all possible values of complex ${r}_{s}$ (within the upper half of the unit circle) are generated at that angle. This is not the case of complex ${r}_{p}$ at the PBA (Fig. 1) which is squeezed mostly in the third quadrant of the unit circle. Recall that the unconstrained domain of ${r}_{p}$ for light reflection at all dielectric–conductor interfaces is on and inside the full unit circle [17].

## 4. RATIO OF COMPLEX REFLECTION COEFFICIENTS OF THE $p$ AND $s$ POLARIZATIONS AT THE PBA

The ratio of complex $p$ and $s$ reflection coefficients, also known as the ellipsometric function $\rho =\mathrm{tan}\text{\hspace{0.17em}}\psi \text{\hspace{0.17em}}\mathrm{exp}(j\mathrm{\Delta})$ [13], is obtained from Eqs. (1) and (2) as

Figure 8 shows loci of complex $\rho $ as $\theta $ increases from 0° to 180° at constant values of ${\varphi}_{pB}$ from 5° to 85° in equal steps of 5°. All contours begin at the origin $O$ (as a common point that represents the ideal Brewster-angle condition of ${r}_{p}=0$ at $\theta =0$), then fan out and terminate on the 90° arc of the unit circle in the second quadrant of the complex plane (dotted line), so that $\rho =\mathrm{exp}(j\mathrm{\Delta})$ at $\theta =180\xb0$ $({\epsilon}_{i}=0,{\epsilon}_{r}<0)$. The differential reflection phase shift $\mathrm{\Delta}={\delta}_{p}-{\delta}_{s}+360\xb0$ at $\theta =180\xb0$ decreases monotonically from 180° to 90° as ${\varphi}_{pB}$ increases from 0° to 90° as shown in Fig. 4.

## 5. SUMMARY

The Fresnel complex reflection coefficients ${r}_{p}$, ${r}_{s}$ and their ratio $\rho ={r}_{p}/{r}_{s}$ are evaluated at the PBA ${\varphi}_{pB}$ of a dielectric–conductor interface for all possible values of the complex relative dielectric function $\epsilon =|\epsilon |\mathrm{exp}(-j\theta )={\epsilon}_{r}-j{\epsilon}_{i}$, ${\epsilon}_{i}>0$. Complex-plane loci of ${r}_{p}$, ${r}_{s}$, and $\rho $ at the PBA are obtained at discrete values of ${\varphi}_{pB}$ from 5° to 85° in equal steps of 5° and as $\theta $ increases from 0° to 180°; these are presented in Figs. 1, 7, and 8, respectively. The reflection phase shift ${\delta}_{p}$ of the $p$ polarization at the PBA is plotted as function of $\theta $ in Figs. 5 and 6 for two different sets of ${\varphi}_{pB}$. For ${\varphi}_{pB}>70\xb0$ (e.g., high-reflectance metals in the IR), ${r}_{p}$ at the PBA is essentially pure negative imaginary and ${\delta}_{p}=\mathrm{arg}({r}_{p})\approx -90\xb0$. In the domain of fractional optical constants (vacuum UV or light incidence from a high-refractive-index immersion medium) $0\xb0<{\varphi}_{pB}<45\xb0$, and ${r}_{p}$ is pure real negative (${\delta}_{p}=\pi $) at $\theta ={\mathrm{tan}}^{-1}(\sqrt{\mathrm{cos}(2{\varphi}_{pB})})$. The associated locus of complex $\epsilon $ is shown in Fig. 3. Finally, the total reflection phase shifts ${\delta}_{p}$, ${\delta}_{s}$, $\mathrm{\Delta}=\mathrm{arg}(\rho )$ at an ideal dielectric–plasmonic medium interface $({\epsilon}_{i}=0,{\epsilon}_{r}<0)$, are shown as functions of ${\varphi}_{pB}$ in Fig. 4.

## APPENDIX A

By setting ${\epsilon}_{r}=x$ and ${\epsilon}_{i}=y$, the Cartesian equation of a constant-${\varphi}_{pB}$ contour (a cardioid [8]) takes the form [10]

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