Last updated: 2026-06-16

Brewster's angle

Brewster's angle is the unique angle of incidence at which light reflected from a flat interface between two transparent media is completely plane-polarized perpendicular to the plane of incidence. Named after the Scottish physicist David Brewster, who discovered the relationship in 1815, it arises from the electromagnetic nature of light and the way oscillating electric dipoles in a medium radiate — or fail to radiate — in particular directions.

Why reflected light becomes polarized

Light is a transverse electromagnetic wave whose electric field oscillates in a plane perpendicular to the direction of travel. That oscillation can be decomposed into two components: one lying in the plane of incidence (the p-polarization) and one perpendicular to it (the s-polarization).

When light strikes a surface, the transmitted ray drives electric dipoles in the second medium to oscillate parallel to the transmitted beam's electric field. A dipole does not radiate along its own oscillation axis. At the special angle where the reflected ray would have to align with the direction of the refracted ray — that is, where the two rays are 90° apart — the p-polarized component cannot appear in the reflected beam at all. Only the s-component reflects, giving completely polarized light.

Formula

The condition that the reflected and refracted rays are perpendicular, combined with Snell's law $n_1 \sin\theta_B = n_2 \sin\theta_t$ and the geometric constraint $\theta_B + \theta_t = 90°$, yields

and therefore

The transmitted angle at Brewster incidence follows immediately:

Worked example

A ray of light in air ($n_1 = 1.00$) strikes a flat glass surface ($n_2 = 1.50$).

The transmitted angle is $\theta_t = 90° - 56.3° = 33.7°$. At any other angle of incidence the reflected light is only partially polarized; at exactly 56.3° the reflected beam is 100% s-polarized.

Applications

Polarizing filters. Camera and sunglass lenses exploit Brewster reflection. Glare from water, wet roads, or glass surfaces consists largely of s-polarized light reflected near Brewster's angle. A filter that transmits only p-polarization cuts this glare while passing light from other directions.

Brewster windows in lasers. Laser resonator tubes are often sealed with windows tilted at Brewster's angle to the beam. The p-polarized intracavity beam passes through with zero reflection loss, improving efficiency, while the transmitted beam emerges linearly polarized. This is standard in helium-neon and carbon-dioxide lasers.

Ellipsometry. Surface scientists use polarized light near Brewster's angle to measure thin-film thickness and optical constants with sub-nanometre precision, because the reflected intensity is extremely sensitive to surface conditions when the p-component is nearly extinguished.

Limits of the model

The formula $\theta_B = \arctan(n_2/n_1)$ applies to a perfectly flat, abrupt interface between two isotropic, non-absorbing media. Rough or curved surfaces spread the angular dependence, so polarization is partial rather than complete. Metals and other absorbing materials have complex refractive indices, and the analogue of Brewster's angle (the principal angle) only minimizes, rather than eliminates, the reflected p-component. The formula also assumes monochromatic light; because refractive index varies with wavelength, Brewster's angle is slightly different for each color.