Brewster Angle Calculator
Calculate Brewster angle from two refractive indices.
At this incidence reflected light is fully polarized — used in optics, photography, and laser design.
Brewster’s Angle
When light strikes a transparent boundary at a particular angle, the reflected beam becomes completely polarized perpendicular to the plane of incidence (s-polarization). This angle, named after Sir David Brewster, is the only angle at which the parallel-polarized component (p-polarization) reflects with zero intensity.
Formula
θ_B = arctan(n₂ / n₁)
Where:
- n₁ = refractive index of the medium light is coming from
- n₂ = refractive index of the medium light is entering
The complementary refraction angle is exactly 90° − θ_B, so the reflected and refracted rays are mutually perpendicular.
Worked Example — Air to Water
- n₁ = 1.000 (air)
- n₂ = 1.333 (water)
- θ_B = arctan(1.333 / 1.000) = 53.1°
Light from above hitting a calm pond at 53° from vertical reflects with all its glare polarized horizontally — exactly what circular and linear polarizers are designed to suppress.
Worked Example — Air to Glass
- n₁ = 1.000, n₂ = 1.50
- θ_B = arctan(1.5) = 56.3°
This is why photographers’ polarizing filters are most effective on storefront windows photographed at roughly 30–60° from normal — not directly head-on.
Common Brewster Angles (from air, n₁ = 1)
| Material | n₂ | θ_B |
|---|---|---|
| Water | 1.333 | 53.1° |
| Crown glass | 1.52 | 56.7° |
| Flint glass | 1.62 | 58.3° |
| Diamond | 2.42 | 67.5° |
| Plexiglas | 1.49 | 56.1° |
| Quartz | 1.54 | 56.9° |
Applications
| Use | How Brewster’s Angle Helps |
|---|---|
| Polarizing filter | Reduces glare from water, glass, and dielectric surfaces |
| Brewster windows in lasers | Allow p-polarized cavity light to pass with no loss |
| Sunglasses | Polarized lenses block horizontal Brewster-reflected glare |
| Ellipsometry | Measures thin-film thickness via Brewster shift |
| Optical coatings | Anti-reflection design starts from Brewster geometry |
Brewster Angle Is Asymmetric
Going from glass back into air uses n₂ < n₁, giving θ_B = arctan(1 / 1.52) = 33.3° — the internal Brewster angle. The external (air to glass) angle was 56.7°. These two angles are complementary: 33.3° + 56.7° = 90°.
Why p-Polarization Goes to Zero
At Brewster’s angle, the dipoles induced in the second medium oscillate along the direction of the would-be reflected p-ray. A dipole cannot radiate along its own axis, so the p-reflection vanishes exactly. This is the deep physical origin of the formula and is why Brewster’s law is exact for ideal dielectrics.
Caveats
The formula assumes both media are non-absorbing and non-magnetic. For metals and absorbing materials, the “Brewster angle” generalizes into the principal angle with non-zero p-reflection at minimum intensity. For magnetic materials, an analogous Brewster condition exists but with a different formula involving magnetic permeability.