Brewster Angle Calculator
Find Brewster's angle, θB = arctan(n₂/n₁), where reflected light is fully polarized — with the refraction angle, s-polarized reflectance, the critical angle and values for water, glass and diamond.
How it works
tan θB = n₂ ÷ n₁ · θB + θt = 90° (reflected and refracted rays are perpendicular) · Rs = ((n₁cos θB − n₂cos θt) ÷ (n₁cos θB + n₂cos θt))² · Rp = 0 at θB · critical angle θc = arcsin(n₂ ÷ n₁) when n₁ > n₂
At Brewster's angle (David Brewster, 1815) the reflected and refracted rays are at right angles, and the Fresnel reflection coefficient for p-polarized light (field in the plane of incidence) falls to zero. Whatever is reflected is therefore purely s-polarized, which is why polarized sunglasses cut glare from water and roads and why laser windows are tilted at Brewster's angle. Refractive indices change slightly with wavelength, so θB does too.
Worked example
Light travelling in air (n₁ = 1.000) hits glass with n₂ = 1.5. tan θB = 1.5 ÷ 1 gives θB = 56.31°, and the refracted ray travels at 33.69° inside the glass. At that angle p-polarized light is not reflected at all, while 14.79% of s-polarized light is. Going from the glass back into air, Brewster’s angle is 33.69° and the critical angle is 41.81°.
Sources: HyperPhysics — Brewster's angle · Wikipedia — Brewster's angle
Frequently asked questions
What is Brewster's angle?
It is the angle of incidence at which light with polarization in the plane of incidence is fully transmitted, so the reflected light is completely polarized perpendicular to that plane. It satisfies tan θB = n₂/n₁.
What is the Brewster angle of water?
From air, tan θB = 1.333, so θB ≈ 53.1°. Glare off a lake is most strongly polarized when the sun is about 37° above the horizon.
What is the Brewster angle of glass?
For typical glass with n ≈ 1.5 it is about 56.3° from air, and 33.7° when light inside the glass hits the glass–air surface.
How is the Brewster angle related to the critical angle?
Both depend only on n₁ and n₂, but the critical angle (sin θc = n₂/n₁) exists only going into a less dense medium, and Brewster’s angle is always smaller than it.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.