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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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