PX284 - S2b - brewster's angle

considering a boundary with $n_{2} > n_{1}$
by snell's law:

\sin t = \frac{n_{1}}{n_{2}} \sin i ⟹ t \leq i

for normal incidence, ie. $i = t = 0 :$

r_{s} = \frac{n_{1} - n_{2}}{n_{1} + n_{2}} = r_{p} < 0

for oblique incidence $(i = 90 °) :$

\begin{matrix} r_{s} = - \frac{n_{2} \cos t}{n_{2} \cos t} = - 1 \\ r_{p} = \frac{n_{1} \cos t}{n_{1} \cos t} = + 1 \end{matrix}

the brewster angle is the value of $i$ at which $r_{p} = 0$

\begin{matrix} n_{1} \cos t_{B} = n_{2} \cos i_{B} (fresnel) \\ n_{1} \sin i_{B} = n_{2} \sin t_{B} (snell) \\ n_{1}^{2} \cos^{2} t_{B} = n_{2}^{2} \cos^{2} i_{B} \\ n_{1}^{2} \sin^{2} i_{B} = n_{2}^{2} \sin^{2} t_{B} \\ ⟹ n_{1}^{2} n_{2}^{2} = n_{1}^{4} \sin^{2} i_{B} + n_{2}^{4} \cos^{2} i_{B} \\ n_{1}^{2} n_{2}^{2} (1 + \tan^{2} i_{B}) = n_{1}^{4} \tan^{2} i_{B} + n_{2}^{4} \\ n_{1}^{2} \tan^{2} i_{B} (n_{2}^{2} - n_{1}^{2}) = (- n_{1}^{2} + n_{2}^{2}) n_{2}^{2} \\ ∴ \tan i_{B} = \frac{n_{2}}{n_{1}} \end{matrix}

at $i_{B}$ , $r_{p} = 0$
therefore, any reflected light must be s-polarized
for air-to-glass: $n_{1} / n_{2} \approx 1 / 1.5 ⟹ i_{B} \approx 56.3 °$
for air-to-water: $n_{1} / n_{2} \approx 1 / 1.33 ⟹ i_{B} \approx 53.1 °$
the applications:
- polarized sunglasses: road glare, mostly s-polarized, filter to allow only p-polarization to pass through
- polarized filters on cameras