Light reflecting off a flat surface comes back partly polarized — its vibrations biased into one plane. At one special angle of incidence the bias is total: the reflected beam is completely polarized, and a filter turned across it can erase the glare outright. That angle is Brewster's angle, and it obeys $\tan\theta_B = \dfrac{n_2}{n_1}$ — read aloud tan theta-B equals n-two over n-one. $\theta_B$ is the polarizing angle in degrees, measured (as always) from the normal. $n_1$ is the index of the medium the light arrives through — usually air, 1.00 — and $n_2$ is the index of the reflecting surface. Same subscript map as Snell's law: 1 where the light comes from, 2 where it is going.

The trap is one keystroke wide. Every other formula in this chapter uses a sine; this one uses a tangent, and it is the only one that does. Reach for arcsin here and you will compute the critical angle for a boundary that has none. Air to crown glass gives $\theta_B = \arctan(1.50) \approx 56.3°$ — comfortably past 45°, which is worth remembering as a sanity rail, because $n_2 > n_1$ forces a tangent greater than 1.

One elegance to carry away: at Brewster's angle the reflected and refracted rays leave at exactly 90° to each other. That is the physical reason the effect exists — the reflected ray would have to vibrate along its own direction of travel to carry the other polarization, and light will not do that. Polarized sunglasses are cut to block the horizontal bias thrown up by roads and water, which is why they clear a windshield's glare and do nothing for a streetlight.