Brewster's Angle
Reference for Brewster's angle θ_B = arctan(n2/n1) at which reflected light is fully polarized.
Covers glass, water, polarized sunglasses, and camera filters.
The Formula
θ_B = arctan(n₂ / n₁)
Brewster's angle is the specific angle of incidence at which reflected light is completely polarized. At this angle, the reflected and refracted rays are exactly 90° apart.
Variables
| Symbol | Meaning |
|---|---|
| θ_B | Brewster's angle (degrees) |
| n₁ | Refractive index of the first medium (where light comes from) |
| n₂ | Refractive index of the second medium (where light enters) |
Example 1
Find Brewster's angle for light going from air to glass (n=1.52)
θ_B = arctan(1.52 / 1.00)
θ_B = arctan(1.52) ≈ 56.7°
Example 2
Find Brewster's angle for light going from air to water (n=1.33)
θ_B = arctan(1.33 / 1.00)
θ_B = arctan(1.33) ≈ 53.1°
When to Use It
Use Brewster's angle when:
- Designing polarizing filters and anti-glare coatings
- Building laser cavities with minimal reflection loss
- Understanding why polarized sunglasses reduce glare
- Optimizing window angles in optical instruments
Limitations
- At Brewster's angle, the reflected beam is completely polarized, but the transmitted beam is only partially polarized — multiple stacked glass plates are needed for a well-polarized transmitted beam
- The formula assumes a perfectly flat, smooth interface — rough or curved surfaces scatter light and reduce polarization effectiveness
- Brewster's angle is wavelength-independent for non-dispersive materials, but for strongly dispersive media the refractive index varies with wavelength and so does the angle
Key Notes
- Formula: tan(θ_B) = n₂ / n₁: θ_B is Brewster's angle, n₁ is the refractive index of the incident medium, and n₂ is that of the transmitted medium. For glass (n≈1.5) in air (n=1): θ_B ≈ 56.3°.
- Reflected light is completely polarized: At Brewster's angle, the reflected beam contains only s-polarization (electric field perpendicular to the plane of incidence). The transmitted beam is partially polarized.
- The reflected and refracted rays are perpendicular: At Brewster's angle, the reflected and refracted rays make a 90° angle — this is actually the geometric reason why p-polarization is absent from the reflected beam.
- Used in laser optics: Laser cavities often use Brewster windows — glass plates oriented at Brewster's angle — to transmit one polarization without reflection losses, resulting in a linearly polarized output beam.
- Photography application: Polarizing filters reduce glare from water and glass by blocking the s-polarized reflected light. Rotating the filter to block light at Brewster's angle maximizes glare reduction.