Optics
Brewster's Angle
At a special incidence angle, reflected light is 100% polarized perpendicular to the plane of incidence
Brewster's angle θ_B is the angle of incidence at which light reflected from a surface is completely polarized perpendicular to the plane of incidence. The condition: θ_B + θ_t = 90° (reflected and transmitted rays are perpendicular), giving tan θ_B = n₂/n₁. Examples: glass (n=1.5) has θ_B ≈ 56° for incidence from air; water (n=1.33) θ_B ≈ 53°. Discovered by David Brewster (1815). At Brewster's angle, the reflected p-polarized component (parallel to plane of incidence) vanishes — only s-polarization (perpendicular) is reflected. Applications: polarized sunglasses (block glare reflected from horizontal surfaces near θ_B ≈ 53°), Brewster windows in laser cavities (eliminate cavity loss for one polarization), and polarimetry in remote sensing. Fresnel equations explain it via the boundary conditions for E and H fields.
- Formulatan θ_B = n₂/n₁
- Typical anglesGlass 56°, water 53°
- ReflectionOnly s-polarized survives
- DiscoveredBrewster 1815
- ApplicationsSunglasses, laser windows
- GeometryReflected + transmitted at 90°
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Why Brewster's angle matters
- Polarized sunglasses. Horizontal surfaces (water, asphalt, car hoods) at typical viewing angles reflect light near Brewster's angle, producing strongly horizontally polarized glare. Vertical-axis polarizer lenses cut that glare by 90% while preserving the rest of the scene.
- Laser cavity windows. Glass plates tilted at θ_B inside gas laser tubes (HeNe, argon-ion, CO₂) transmit one polarization with essentially zero loss while introducing ~15% loss to the orthogonal polarization. After thousands of cavity round trips, the laser is naturally polarized — no separate polarizer needed.
- Remote sensing and polarimetry. The polarization state of sunlight reflected from the ocean, atmosphere, or planetary surfaces carries information about surface roughness, refractive index, and aerosol content. Brewster-angle behavior calibrates the inversion.
- Photography polarizing filters. Circular polarizers on camera lenses suppress reflections from glass storefronts, water, foliage gloss — particularly effective at oblique angles approaching θ_B. They deepen blue skies (Rayleigh-scattered light is partially polarized).
- Refractometry. Measuring θ_B and inverting tan θ_B = n₂/n₁ gives the refractive index. The classical Brewster-angle method works for liquids, surfaces, and (with phase-sensitive detection) thin films of nm thickness.
- Optical coatings and ellipsometry. Thin-film monitoring exploits the strong angular dependence of r_p near Brewster's angle. Ellipsometers measure complex r_s/r_p ratios across the spectrum to determine film thickness, refractive index, and absorption.
- Microscopy and biology. Brewster-angle microscopy images Langmuir-Blodgett monolayers floating on water by exploiting the contrast between bare water (zero p-reflection) and a covered surface (small but nonzero p-reflection from the film).
Common misconceptions
- "Brewster's angle is always 56°." Only for crown glass at 589 nm. It depends on the index ratio: water 53°, diamond 67°, silicon ~75°. It also varies slightly with wavelength because n(λ) varies (chromatic Brewster shift).
- "Polarized sunglasses remove all glare." Only the horizontally polarized portion. Reflections at angles far from θ_B are weakly polarized — sunglasses help less. Curved or wavy surfaces have local Brewster angles in many directions.
- "Brewster's angle magnifies or sharpens the image." No — it only changes polarization composition. The geometric image is unaffected; only the reflected light's vector character changes.
- "Reflection at Brewster's angle is total." Reverse: at θ_B, the p-polarized reflection is zero. The s-polarization still reflects normally. Total reflection is total internal reflection at angles beyond the critical angle.
- "There's only one Brewster's angle for any pair of media." Light from medium 1 to medium 2 has θ_B₁→₂ = arctan(n₂/n₁); from medium 2 back to 1 has θ_B₂→₁ = arctan(n₁/n₂). They sum to 90° (consequence of Snell's law).
- "Brewster's angle requires non-magnetic media." The simple tan θ_B = n₂/n₁ assumes μ_r ≈ 1. For magnetic media at optical frequencies this is essentially always true; for radio-frequency reflection off magnetic materials, generalized Brewster conditions involve permeability ratios.
Frequently asked questions
Why does p-polarization fully transmit at Brewster's angle?
p-polarized light has its electric field oscillating in the plane of incidence. At the boundary, the molecules of the second medium are driven by the transmitted field and re-radiate as oscillating dipoles. A dipole emits zero radiation along its own oscillation axis. At Brewster's angle, the reflected ray direction would be exactly along the dipole axis of the medium-2 dipoles driven by the p-polarized field — so the reflection amplitude is zero. s-polarization, oscillating perpendicular to the plane, has no such cancellation and reflects normally.
What's s and p polarization?
Define a plane of incidence: the plane containing the incoming ray and the surface normal. p-polarization (parallel) has E-field oscillating in that plane. s-polarization (senkrecht — German for perpendicular) has E-field oscillating perpendicular to that plane (parallel to the surface). Any linear polarization decomposes into s and p components, and the Fresnel equations give different reflection coefficients r_s and r_p for each. r_p hits zero at Brewster's angle while r_s continues smoothly.
How do polarized sunglasses use this?
Sunlight reflecting off horizontal surfaces (water, road, car hoods) at angles near Brewster's angle (≈53° from normal for water-to-air, the typical viewing geometry on a sunny day) becomes strongly horizontally polarized — that's the glare. Polarized sunglasses incorporate a vertical-axis polarizer, blocking the horizontal component. Glare is suppressed by 90% or more while the rest of the scene loses only half its light. It's why anglers can see fish below the surface and drivers see less windshield reflection.
What's a Brewster window in a laser?
Tilt a piece of glass at Brewster's angle inside a laser cavity. p-polarized light traverses with essentially zero reflection loss per pass; s-polarized light loses ~15% per pass. After thousands of round trips, only p-polarization survives — the laser self-polarizes. Brewster windows are universal in HeNe, argon-ion, dye, and CO₂ gas lasers: they seal the gas tube while imposing single-polarization output without an explicit polarizer that would heat up. Multipass amplifier chains use the same trick.
Why does the reflection coefficient go to zero (Fresnel)?
The Fresnel equation for p-polarization is r_p = (n₂ cos θ₁ − n₁ cos θ₂) / (n₂ cos θ₁ + n₁ cos θ₂). Combined with Snell's law n₁ sin θ₁ = n₂ sin θ₂, the numerator vanishes exactly when θ₁ + θ₂ = 90°, i.e., the reflected and refracted rays are perpendicular. Solving gives tan θ_B = n₂/n₁. The s-polarization Fresnel coefficient r_s = (n₁ cos θ₁ − n₂ cos θ₂)/(n₁ cos θ₁ + n₂ cos θ₂) has no analogous zero in the real-angle range.
How is Brewster's angle measured?
Shine unpolarized light onto a sample at variable incidence angle, view reflection through a polarizer, and rotate the polarizer between vertical and horizontal at each angle. Plot the ratio of detected intensities; the angle at which the p-component drops to zero is θ_B. Inverting tan θ_B = n₂/n₁ gives the refractive index — Brewster-angle refractometry is a classical way to measure n of liquid surfaces and thin films. Modern ellipsometers extend this to phase information for nm-scale film thickness.