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 ${\varphi }_B={\tan }^{-1}n={\tan }^{-1}\sqrt{{\epsilon }_r}$. 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 $u={\sin }^2{\varphi }_{pB}$ [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

 

Fig. 1. Complex-plane trajectories of $r_p$ at discrete values of the PBA ${\varphi }_{pB}$ from 5° to 85° in equal steps of 5° as $\theta =-\arg (\epsilon )$ covers the full range $0°\le \theta \le 180°$.

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In this paper, loci of all possible values of complex $r_p=|r_p|\exp (j{\delta }_p)$, $r_s=|r_s|\exp (j{\delta }_s)$, and $\rho =r_p/r_s=|\rho |\exp (j\mathrm{\Delta })$ at the PBA are determined at discrete values of ${\varphi }_{pB}$ from 5° to 85° in equal steps of 5° and as $\theta =-\arg (\epsilon )$ covers the full range $0°\le \theta \le 180°$. 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 $\theta =180°$ $({\epsilon }_i=0,{\epsilon }_r<0)$. A quick conclusion from Fig. 1 is that for ${\varphi }_{pB}>70°$ (high-reflectance metals) $r_p$ at the PBA is essentially pure negative imaginary, and ${\delta }_p\approx -90°$.

In Fig. 1 the constant-${\varphi }_{pB}$ contours of $r_p$ for $0<{\varphi }_{pB}<45°$ 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 ${\delta }_p=\arg (r_p)=\pi$, is given by the remarkably simple formula

 

Fig. 2. Graph of the function of Eq. (5). Both ${\varphi }_{pB}$ and $\theta$ are in degrees.

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The locus of complex $\epsilon$ such that ${\delta }_p=\arg (r_p)=\pi$ 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, ${\varphi }_{pB}=37.761°$.

 

Fig. 3. Locus of all possible values of complex $\epsilon$ such that ${\delta }_p=\arg (r_p)=\pi$ at the PBA.

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For small PBAs, ${\varphi }_{pB}\le 5°$, 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 $\theta =180°$ [14] are given by

 

Fig. 4. Total reflection phase shifts ${\delta }_p$, ${\delta }_s$, and $\mathrm{\Delta }={\delta }_p-{\delta }_s+360°$ at the interface between a dielectric and plasmonic medium in the limit as $\theta \to 180°$ $({\epsilon }_i=0,{\epsilon }_r<0)$ are plotted as a functions of ${\varphi }_{pB}$. All angles are in degrees.

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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 $+180°$ to $-180°$ are located at $\theta$ values that agree with Eq. (5).

 

Fig. 5. Family of ${\delta }_p$ versus $\theta$ curves for ${\varphi }_{pB}$ from 10° to 40° in equal steps of 10°. Both $\theta$ and ${\delta }_p$ are in degrees.

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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 ${\varphi }_{pB}>45°$ the ${\delta }_p$-versus-$\theta$ curve first exhibits a minimum then reaches saturation as $\theta \to 180°$. The saturated value of ${\delta }_p$ is a function of ${\varphi }_{pB}$ and is shown in Fig. 4.

 

Fig. 6. Family of ${\delta }_p$ versus $\theta$ curves for ${\varphi }_{pB}$ from 45° to 85° in equal steps of 5°. Both $\theta$ and ${\delta }_p$ are in degrees.

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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 $\theta =0$, $r_s=\cos (2{\varphi }_{pB})$, 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 $r_s=\exp (j{\delta }_s)$ at $\theta =180°$ $({\epsilon }_i=0,{\epsilon }_r<0)$. The associated total reflection phase shift ${\delta }_s$ along the dotted semicircle is a function of ${\varphi }_{pB}$ as shown in Fig. 4.

 

Fig. 7. Complex-plane contours of $r_s$ at discrete values of the PBA ${\varphi }_{pB}$ from 5° to 85° in equal steps of 5° as $\theta =-\arg (\epsilon )$ covers the full range from 0° to 180°.

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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 $\rho =\tan \psi \exp (j\mathrm{\Delta })$ [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 $\rho =\exp (j\mathrm{\Delta })$ at $\theta =180°$ $({\epsilon }_i=0,{\epsilon }_r<0)$. The differential reflection phase shift $\mathrm{\Delta }={\delta }_p-{\delta }_s+360°$ at $\theta =180°$ decreases monotonically from 180° to 90° as ${\varphi }_{pB}$ increases from 0° to 90° as shown in Fig. 4.

 

Fig. 8. Complex-plane trajectories of the ratio $\rho =r_p/r_s$ at discrete values of the PBA ${\varphi }_{pB}$ from 5° to 85° in equal steps of 5° as $\theta =-\arg (\epsilon )$ covers the full range $0°\le \theta \le 180°$.

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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 $\epsilon =|\epsilon |\exp (-j\theta )={\epsilon }_r-j{\epsilon }_i$, ${\epsilon }_i>0$. 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 ${\varphi }_{pB}>70°$ (e.g., high-reflectance metals in the IR), $r_p$ at the PBA is essentially pure negative imaginary and ${\delta }_p=\arg (r_p)\approx -90°$. In the domain of fractional optical constants (vacuum UV or light incidence from a high-refractive-index immersion medium) $0°<{\varphi }_{pB}<45°$, and $r_p$ is pure real negative (${\delta }_p=\pi$) at $\theta ={\tan }^{-1}(\sqrt{\cos (2{\varphi }_{pB})})$. The associated locus of complex $\epsilon$ is shown in Fig. 3. Finally, the total reflection phase shifts ${\delta }_p$, ${\delta }_s$, $\mathrm{\Delta }=\arg (\rho )$ at an ideal dielectric–plasmonic medium interface $({\epsilon }_i=0,{\epsilon }_r<0)$, are shown as functions of ${\varphi }_{pB}$ in Fig. 4.