Playground
SVG diagram showing incident, reflected (s-polarized), and refracted rays at Brewster's angle, with the reflected and refracted rays perpendicular.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Brewster's angleoutput Angle of incidence at which reflected light is fully polarized | rad | 1 | 0 – 1.5708 | |
| Refractive index (medium 1) Refractive index of the incident medium | dimensionless | 1 | 1 – 2.5 | |
| Refractive index (medium 2) Refractive index of the second medium | dimensionless | 1 | 1 – 2.5 |
Deep dive
Derivation
At Brewster's angle, the reflected and refracted rays are perpendicular: θ_B + θ_r = 90°. From Snell's law: n1·sin(θ_B) = n2·sin(θ_r) = n2·sin(90° - θ_B) = n2·cos(θ_B). Dividing both sides by cos(θ_B): n1·tan(θ_B) = n2, giving tan(θ_B) = n2/n1. Physically, the reflected ray cannot contain the p-polarization component because the oscillating dipoles in the surface would have to radiate along their own axis, which is impossible.
Experimental verification
Brewster measured polarization of reflected light from glass surfaces (1815). Modern: verified using polarimeters and laser sources at dielectric interfaces. Brewster windows in gas lasers exploit this — they transmit p-polarized light with zero reflection loss.
Common misconceptions
- All reflected light is polarized at Brewster's angle — only the reflected light is fully s-polarized; the transmitted beam is partially polarized.
- Brewster's angle works for metals — metals have complex refractive indices; reflected light is never fully polarized.
- Polarized sunglasses use Brewster's angle — they use dichroic filters, but they work because glare from horizontal surfaces is s-polarized near Brewster's angle.
Real-world applications
- Laser cavities: Brewster windows eliminate reflection losses for p-polarized light.
- Polarized photography: photographers use polarizing filters to cut glare at near-Brewster angles.
- Ellipsometry: measuring Brewster's angle determines the refractive index of thin films.
- Fiber optic connectors: angled polishing at ~8° reduces back-reflection.
Worked examples
Brewster's angle for water surface
Given:
- n_1:
- 1
- n_2:
- 1.33
Find: theta_B
Solution
θ_B = arctan(n2/n1) = arctan(1.33/1.0) = arctan(1.33) = 53.06° = 0.926 rad
Brewster's angle for glass
Given:
- n_1:
- 1
- n_2:
- 1.52
Find: theta_B
Solution
θ_B = arctan(1.52) = 56.66° = 0.989 rad
Scenarios
What if…
- scenario:
- What if you approach from the denser medium?
- answer:
- Brewster's angle from glass to air: θ_B = arctan(1.0/1.52) = 33.3°. The two Brewster angles (from each side) are complementary: they sum to 90°.
- scenario:
- What if the surface is a metal?
- answer:
- Metals have complex refractive indices. The reflected light is elliptically polarized at all angles — there is no true Brewster angle for conductors.
- scenario:
- What if n1 = n2?
- answer:
- tan(θ_B) = 1, so θ_B = 45°. But with no actual interface (same medium), there's no reflection at all.
Limiting cases
- condition:
- n_1 = n_2
- result:
- theta_B = 45°
- explanation:
- Same medium: arctan(1) = 45°, but no interface exists to reflect.
- condition:
- n_2 → ∞
- result:
- theta_B → 90°
- explanation:
- Extremely dense medium requires near-grazing incidence.
- condition:
- n_2 → n_1
- result:
- theta_B → 45°
- explanation:
- As media become similar, Brewster angle approaches 45 degrees.
Context
David Brewster · 1815
Brewster discovered empirically that reflected light is fully polarized when the reflected and refracted rays are perpendicular.
Hook
Why do polarized sunglasses cut the glare off a lake but not off a metal car hood?
Find the incidence angle at which reflected light from water (n=1.33) is perfectly polarized. Use theta_B = arctan(n2/n1).
Dimensions:
- lhs:
- tan(θ_B) → [1] (dimensionless)
- rhs:
- n₂/n₁ → [1]/[1] = [1]
- check:
- Both sides are dimensionless. ✓
Validity: Valid for dielectric (non-conducting) surfaces. Does not apply to metallic surfaces where free electrons prevent full polarization of reflected light.