## Abstract

For reflection at interfaces between transparent optically isotropic media, the difference between the Brewster angle ${\varphi}_{\mathrm{B}}$ of zero reflectance for incident $p$-polarized light and the angle ${\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}}$ of minimum reflectance for incident unpolarized or circularly polarized light is considered as function of the relative refractive $n$ in external and internal reflection. We determine the following. (i) ${\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}}<{\varphi}_{\mathrm{B}}$ for all values of $n$. (ii) In external reflection $(n>1)$, the maximum difference ${({\varphi}_{\mathrm{B}}-{\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}})}_{\mathrm{max}}=75\xb0$ at $n=2+\sqrt{3}$. (iii) In internal reflection and $0<n\le 2-\sqrt{3}$, ${({\varphi}_{\mathrm{B}}-{\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}})}_{\mathrm{max}}=15\xb0$ at $n=2-\sqrt{3}$; for $2-\sqrt{3}<n<1$, ${\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}}=0$, and ${({\varphi}_{\mathrm{B}}-{\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}})}_{\mathrm{max}}=45\xb0$ as $n\to 1$. (iv) For $2-\sqrt{3}\le n\le 2+\sqrt{3}$, the intensity reflectance ${R}_{0}$ at normal incidence is in the range $0\le {R}_{0}\le 1/3$, ${\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}}=0$, and ${\varphi}_{\mathrm{B}}-{\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}}={\varphi}_{\mathrm{B}}$. (v) For internal reflection and $0<n<2-\sqrt{3}$, ${\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}}$ exhibits an unexpected maximum (= 12.30°) at $n=0.24265$. Finally, (vi) for $1/3\le {R}_{0}<1$, ${R}_{u\text{\hspace{0.17em}}\mathrm{min}}$ at ${\varphi}_{u\text{\hspace{0.17em}}\mathrm{min}}$ is limited to the range $1/3\le {R}_{u\text{\hspace{0.17em}}\mathrm{min}}<1/2$.

© 2015 Optical Society of America

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