Brewster angle

At an interface between two transparent media of refractive indices $n_1$ (incident side) and $n_2$, the Brewster angle is

At ${\theta }_B$, the reflectivity for p-polarized light (electric field in the plane of incidence) is exactly zero. The reflectivity for s-polarized light (electric field perpendicular to the plane of incidence) is nonzero and substantial. Light reflected from a surface near the Brewster angle is therefore strongly s-polarized, even from unpolarized incident light. This is the principle behind polarizing sunglasses and dielectric pile-of-plates polarizers.

Geometric origin: at ${\theta }_B$, the reflected and refracted beams are exactly perpendicular (${\theta }_B+{\theta }_r=90\mathrm{°}$). The p-polarized dipoles driven in the second medium oscillate perpendicular to the refracted beam, which at ${\theta }_B$ is along the reflected direction; a dipole does not radiate along its own axis, so the p-polarized reflected beam vanishes.

Typical Brewster angles (from air, $n_1=1$):

Applications in laser systems:

• Brewster windows in gas lasers (HeNe, argon-ion, dye lasers): glass plates tilted at Brewster angle inside the cavity transmit p-polarized light with zero loss while reflecting s-polarization out of the cavity. The intracavity field is forced to oscillate purely p-polarized.
• Edge facets of solid-state lasers sometimes use Brewster-cut geometry to eliminate one polarization mode and enforce single-polarization output.
• Polarizing prisms such as the Glan-Taylor are cut so that the transmitted polarization meets the internal air gap near the Brewster angle while the other polarization undergoes total internal reflection.

Brewster transmission applies only to dielectric interfaces. Metals do not have a true Brewster angle; instead they have a pseudo-Brewster angle where p-reflection reaches a minimum (but nonzero) value. See Fresnel equations for the full reflectivity expressions.

The Snell's law calculator plots the s and p reflectance against angle, with Brewster's angle marked, and the Fresnel equations calculator adds the phase on reflection and the pseudo-Brewster angle of absorbing surfaces.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017). M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).