About Brewster's Angle
When unpolarized light strikes a boundary between two transparent media — air and glass, air and water, glass and plastic — most of the time both the reflected and transmitted beams come out partially polarized. But at one specific angle of incidence, something special happens: the component of light polarized parallel to the plane of incidence (p-polarized) is transmitted with zero reflection. Only the perpendicular component (s-polarized) reflects, so the reflected beam is completely, purely polarized. That special angle is called Brewster's angle, or the polarizing angle, named for Scottish physicist Sir David Brewster, who published the relationship in 1815.
The formula
Brewster's angle θ_B is found from the refractive indices of the two media:
θ_B = arctan(n₂ / n₁)
where n₁ is the refractive index of the medium the light starts in (the incident side) and n₂ is the refractive index of the medium beyond the boundary (the transmission side). For ordinary light going from air (n₁ ≈ 1.00) into window glass (n₂ ≈ 1.50), θ_B works out to about 56.3°.
Why it works
Brewster's angle has a neat geometric signature: at that angle of incidence, the reflected ray and the refracted (transmitted) ray are exactly perpendicular to each other — they sum to 90°. Combining that fact with Snell's law, n₁ sin(θ_B) = n₂ sin(θ_t), and the identity θ_t = 90° − θ_B, gives n₁ sin(θ_B) = n₂ cos(θ_B), which rearranges directly to tan(θ_B) = n₂ / n₁. Physically, the p-polarized reflection vanishes because the oscillating dipoles in the second medium that would radiate the reflected wave are aligned along the direction the reflected ray would have to travel — a dipole cannot radiate along its own axis, so no p-polarized light is reflected in that direction.
Knowing the limits
The formula above assumes both media are transparent, non-absorbing dielectrics (like glass, water, or air) and that the incident light is a simple plane wave hitting a flat, smooth interface. It does not directly apply to metals or other conductive/absorbing materials, where the refractive index is complex and the "Brewster angle" (if it exists at all) behaves differently. It also does not account for multiple layers or coatings, which need the full Fresnel equations layer by layer. Refractive index itself varies slightly with the wavelength of light (dispersion), so for precise work use the index value that matches your light source.