Brewster's angle is the angle of incidence at which light reflecting off a boundary between two transparent media becomes completely polarized. Optics students, photographers, and anyone working with polarizing filters or laser cavities run into this relationship constantly — this article explains where the formula comes from and what it's used for.
How the formula works
Brewster's angle satisfies θB = arctan(n2/n1), where n1 and n2 are the refractive indices of the two media light travels between. The relationship falls directly out of Snell's law (n1·sinθ1 = n2·sinθ2) combined with the geometric condition that, at θB, the reflected ray and refracted ray are exactly perpendicular to each other (they sum to 90°).
Because the reflected ray's electric field would have to oscillate along its own direction of travel to radiate a p-polarized component under this geometry — which classical dipole radiation forbids — the p-polarized component of the reflection vanishes entirely at θB. Only the s-polarized component (oscillating perpendicular to the plane of incidence) survives, so the reflected beam is fully polarized.
Inputs and what they mean
n1 and n2 are refractive indices — dimensionless numbers with typical values of 1.0 for air/vacuum, about 1.33 for water, and about 1.5 for common glass. The order matters: n1 is the medium the light starts in, n2 is the medium it's entering. Brewster's angle is always measured from the normal (the line perpendicular to the surface), not from the surface itself.
The Solve Index tab expects a measured Brewster's angle in degrees, strictly between 0° and 90°, along with the known index of the first medium.
Limits and edge cases
The formula assumes both media are transparent, non-absorbing dielectrics — it does not directly apply to metals or other absorbing/conducting materials, where the reflectance minimum (the pseudo-Brewster angle) is typically not a true zero and involves a more complex Fresnel calculation with complex refractive indices.
Brewster's angle also only fully polarizes the specularly reflected beam — light scattered diffusely off a rough surface, or transmitted through multiple internal reflections, is not fully polarized even near θB. For metals, high-precision optics, or multilayer coatings, consult full Fresnel-equation modeling rather than this single-interface formula.