Hong Life CalculatorsUnderstanding the Brewster's angle calculator — concept and principle
At one particular angle of incidence, light reflected from a transparent surface is completely polarised. This calculator finds that Brewster's angle from the two refractive indices, together with the refraction angle, the Brewster angle from the other side and the critical angle for total internal reflection.
How it is calculated
- Brewster's law: tan θB = n₂ ÷ n₁.
- At θB the reflected and refracted rays are 90° apart, so the refraction angle = 90° − θB.
- From the denser side: θB' = arctan(n₁ ÷ n₂).
- Critical angle = arcsin(nlow ÷ nhigh).
The principle
Reflected light comes from electrons in the second medium oscillating under the refracted wave. Oscillating electrons cannot radiate along their own direction of motion. When the reflected and refracted rays are perpendicular, the component of light polarised in the plane of incidence (p-polarised) would have to be radiated along that direction, so it is not reflected at all. Only s-polarised light remains in the reflection. Combining this 90° condition with Snell's law gives tan θB = n₂ ÷ n₁.
Worked example
Default inputs: n₁ = 1.000 (air), n₂ = 1.50 (typical glass).
- θB = arctan(1.50 ÷ 1.000) ≈ 56.31°, or 0.9828 rad.
- Refraction angle = 90 − 56.31 = 33.69°.
- Brewster angle from inside the glass = arctan(1 ÷ 1.5) ≈ 33.69°.
- Critical angle = arcsin(1 ÷ 1.5) ≈ 41.81°.
How to read the result
Polarised sunglasses and camera polarising filters exploit this effect: glare from water or roads is strongly polarised near Brewster's angle and can be blocked. Lasers use Brewster windows to pass one polarisation without reflection loss. Because the refractive index varies slightly with wavelength, θB does too.
Common mistakes
- Swapping n₁ and n₂.
- Confusing Brewster's angle with the critical angle; the critical angle is always larger.
- Measuring angles from the surface instead of from the normal.
- Using a refractive index for the wrong wavelength.
Frequently asked questions
What is Brewster's angle for glass?
About 56° for glass with n ≈ 1.5, coming from air.
Which polarisation is reflected at Brewster's angle?
Only s-polarised light (perpendicular to the plane of incidence); p-polarised light is fully transmitted.
Why do polarised sunglasses reduce glare?
Glare from horizontal surfaces is mostly horizontally polarised, and the lenses block that direction.
Sources: Hecht, E., Optics, 5th ed. Pearson (Brewster's law) · Born, M. & Wolf, E., Principles of Optics. Cambridge University Press (Fresnel equations)