The complex Fresnel reflection coefficients and of - and -polarized light and their ratio at the pseudo-Brewster angle (PBA) of a dielectric–conductor interface are evaluated for all possible values of the complex relative dielectric function , that share the same . Complex-plane trajectories of , , and at the PBA are presented at discrete values of from 5° to 85° in equal steps of 5° as is increased from 0° to 180°. It is shown that for (high-reflectance metals in the IR) at the PBA is essentially pure negative imaginary and the reflection phase shift . In the domain of fractional optical constants (vacuum UV or light incidence from a high-refractive-index immersion medium) and is pure real negative () when , and the corresponding locus of in the complex plane is obtained. In the limit of , (interface between a dielectric and plasmonic medium) the total reflection phase shifts , , are also determined as functions of .
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A salient feature of the reflection of collimated monochromatic (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) . If the medium of refraction is also transparent, the minimum reflectance is zero and reverts back to the usual Brewster angle . The PBA is determined by the complex relative dielectric function , , where and are the real and complex permittivities of the dielectric and conductor, respectively, by solving a cubic equation in [1–5]. Measurement of and of reflectance at that angle or at normal incidence enables the determination of complex [1,6–9]. It is also possible to determine of an optically thick absorbing film from two PBAs measured in transparent ambient and substrate media that sandwich the thick film . Reflection at the PBA has also had other interesting applications [11,12].
For light reflection at any angle of incidence the complex-amplitude Fresnel reflection coefficients (see, e.g., ) of the and polarizations are given by8,14]: 15] and also as is evident in Fig. 1 of Section 2.
In this paper, loci of all possible values of complex , , and at the PBA are determined at discrete values of from 5° to 85° in equal steps of 5° and as covers the full range . 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 - and -polarized light at the PBA (e.g., ) are settled. Section 5 summarizes the essential conclusions of this paper.
2. COMPLEX REFLECTION COEFFICIENT OF THE POLARIZATION AT THE PBA
Figure 1 shows the loci of complex as increases from 0° to 180° at constant values of from 5° to 85° in equal steps of 5°. All constant- contours begin at the origin () 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 at . A quick conclusion from Fig. 1 is that for (high-reflectance metals) at the PBA is essentially pure negative imaginary, and .
In Fig. 1 the constant- contours of for 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 at the point of intersection, where , is given by the remarkably simple formula5) is shown in Fig. 2.
The locus of complex such that 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 and 45°, respectively. At , a point that falls exactly on the curve very near to its peak, .
Negative real values of at  are given by1) is obtained from Eqs. (1) and (6) and is plotted as a function of in Fig. 4. In Fig. 4 increases monotonically from to as increases from 0° to 90°. The initial rise of with respect to is linear for and then transitions to saturation at , in accord with Fig. 1.
Another family of -versus- curves for from 45° to 85° in equal steps of 5° is shown in Fig. 6. For the -versus- curve first exhibits a minimum then reaches saturation as . The saturated value of is a function of and is shown in Fig. 4.
3. COMPLEX REFLECTION COEFFICIENT OF THE POLARIZATION AT THE PBA
Figure 7 shows the loci of complex as increases from 0° to 180° at discrete values of from 5° to 85° in equal steps of 5°. All curves start on the real axis at , , which is the amplitude reflectance at the Brewster angle of a dielectric–dielectric interface , and terminate on the upper half of the unit circle (dotted line) that represents total reflection at . The associated total reflection phase shift along the dotted semicircle is a function of as shown in Fig. 4.
Although we are locked on the PBA, all possible values of complex (within the upper half of the unit circle) are generated at that angle. This is not the case of complex at the PBA (Fig. 1) which is squeezed mostly in the third quadrant of the unit circle. Recall that the unconstrained domain of for light reflection at all dielectric–conductor interfaces is on and inside the full unit circle .
4. RATIO OF COMPLEX REFLECTION COEFFICIENTS OF THE AND POLARIZATIONS AT THE PBA
Figure 8 shows loci of complex as increases from 0° to 180° at constant values of from 5° to 85° in equal steps of 5°. All contours begin at the origin (as a common point that represents the ideal Brewster-angle condition of at ), 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 at . The differential reflection phase shift at decreases monotonically from 180° to 90° as increases from 0° to 90° as shown in Fig. 4.
The Fresnel complex reflection coefficients , and their ratio are evaluated at the PBA of a dielectric–conductor interface for all possible values of the complex relative dielectric function , . Complex-plane loci of , , and at the PBA are obtained at discrete values of from 5° to 85° in equal steps of 5° and as increases from 0° to 180°; these are presented in Figs. 1, 7, and 8, respectively. The reflection phase shift of the polarization at the PBA is plotted as function of in Figs. 5 and 6 for two different sets of . For (e.g., high-reflectance metals in the IR), at the PBA is essentially pure negative imaginary and . In the domain of fractional optical constants (vacuum UV or light incidence from a high-refractive-index immersion medium) , and is pure real negative () at . The associated locus of complex is shown in Fig. 3. Finally, the total reflection phase shifts , , at an ideal dielectric–plasmonic medium interface , are shown as functions of in Fig. 4.
APPENDIX A18] A1) and (A3) are satisfied simultaneously if their right-hand sides are equal; this gives A4) we obtain A5) is obviously satisfied when , and from Eq. (A3) one gets and . The more significant solution of Eq. (A5) is A2) in Eq. (A6) leads to the simple result A3) as A7) and (A8) by A9) gives 5).
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