What is Brewster's angle?

Brewster's angle θ_B = arctan(n₂/n₁) is the angle of incidence at which light reflected from a surface is completely s-polarized. The p-polarized component has zero reflectance at this angle. For air-to-glass (n=1.5), θ_B ≈ 56.3°.

What are practical uses of Brewster's angle?

Brewster windows in laser cavities minimize reflection losses for p-polarized light. Polarizing sunglasses and camera filters exploit polarization-by-reflection at Brewster geometry. Solar panels are optimized for low reflection near Brewster's angle. Anti-glare coatings consider the angle dependence.

Why does p-polarization disappear at Brewster's angle?

At Brewster's angle, reflected and refracted rays are exactly perpendicular. The transmitted dipoles oscillate parallel to the surface (perpendicular to the refracted ray). For p-polarized light, the reflected ray direction would be exactly along the dipole axis — but classical dipoles don't radiate along their own axis. So no p-polarized reflection occurs.

How is Brewster's angle related to the refractive index?

θ_B = arctan(n₂/n₁). For air going into a denser medium: more dense = larger Brewster's angle. Glass (n=1.5): 56°. Diamond (n=2.4): 67°. Conversely, very low contrast (n₂ slightly greater than n₁) gives Brewster's angle near 45°.

What's the difference between Brewster's angle and critical angle?

Brewster's angle (any direction) is where reflection of p-polarized light is zero. Critical angle (only going from high to low n) is where transmitted light goes to 90° — beyond critical angle, all light reflects (TIR). These are entirely different phenomena, though both involve angles at interfaces.

Does Brewster's angle work for all colors of light?

Approximately. Since refractive index varies slightly with wavelength (dispersion), Brewster's angle varies slightly too. For BK7 glass: 56.5° at 700 nm, 56.7° at 400 nm — a tiny difference for most purposes. Broadband Brewster windows work effectively across the visible spectrum.

How do polarizing sunglasses use Brewster's effect?

When sunlight reflects off horizontal surfaces (water, roads, snow) at angles near Brewster's angle (~53°), the reflected glare is strongly horizontally polarized. Polarized sunglasses have their transmission axis vertical, blocking the horizontal glare while letting through more of the unpolarized direct light. The effect is most pronounced at solar elevations near 37° (90° − 53°).

Brewster's Angle Calculator

Determine Brewster's angle (polarizing angle) for an optical interface. At this angle, the reflected beam is completely s-polarized and the reflected and refracted beams are perpendicular.

Brewster's angle is the angle of incidence at which reflected light is perfectly polarized — the p-polarization component reflects zero, leaving only s-polarized reflection. This isn't a trick; it's a consequence of Fresnel's equations applied to the boundary condition. At Brewster's angle the reflected and refracted rays make exactly 90°, and the oscillating electric dipoles in the second medium can't radiate in the direction of the reflected ray for p-polarization, so none reflects.

The formula is simple: θ_B = arctan(n₂/n₁). For light going from air (n=1) into glass (n=1.5), Brewster's angle is arctan(1.5) ≈ 56.3°. This is the angle of polarizing sunglasses and camera circular polarizers — at this angle of reflection from horizontal surfaces (water, roads, glass), the reflected glare is entirely horizontally polarized, and vertically-aligned polarizing filters block it completely.

Brewster's angle has practical uses everywhere reflection needs to be controlled. Laser cavity windows are mounted at Brewster's angle so the p-polarized laser mode passes through unreflected (saving the cost of AR coatings on high-power optics). Photographers use polarizers to cut glare from water surfaces and reduce sky reflections in car windshields. And the underlying polarization-by-reflection mechanism is what makes glare more visible on horizontal surfaces (lakes, pavement) than vertical ones (walls) at typical viewing angles.

Inputs

Refractive Index n₁ (incident medium)

Refractive Index n₂ (transmitted medium)

Results

Brewster's Angle

56.31°

Refraction Angle

33.69°

s-pol Reflectance

14.79%

Brewster's Angle Results

Parameter	Value
Brewster's Angle θB	56.3099°
Refraction Angle at θB	33.6901°
θB + θ_refracted	90.0000° (should be 90°)
p-Polarization Reflectance	0.0000% (by definition)
s-Polarization Reflectance	14.7929%
n₁ (incident)	1
n₂ (transmitted)	1.5
Formula	θB = arctan(n₂/n₁)

Last updated: May 29, 2026

Formula

**Brewster's angle:** θ_B = arctan(n₂ / n₁) Where n₁ is the refractive index of the medium light comes from, and n₂ is the medium it goes into. **Geometric property:** At θ_B, the reflected and refracted rays are exactly perpendicular: θ_B + θ_refracted = 90°. **Polarization at Brewster's angle:** - **s-polarized** (perpendicular to plane of incidence): reflects with substantial intensity. - **p-polarized** (parallel to plane of incidence): reflects with zero intensity at θ_B. For unpolarized incident light, the reflected beam is 100% s-polarized. **Worked example: air → glass (n₁ = 1.0, n₂ = 1.517)** θ_B = arctan(1.517 / 1.0) = arctan(1.517) = **56.6°** At this angle, reflection of p-polarized light is exactly zero; reflection of s-polarized light is about 15%. **Worked example: air → water (n=1.333)** θ_B = arctan(1.333) = **53.1°** The sun at about 37° above horizon reflects off lake surfaces at an angle of incidence of 53° — Brewster's angle. Reflected glare is highly polarized horizontally; vertically-polarized sunglasses block it. **Common Brewster's angles:** | Interface (light going from / to) | θ_B | |---|---| | Air / water | 53.1° | | Air / acrylic (n=1.49) | 56.1° | | Air / glass BK7 (n=1.517) | 56.6° | | Air / fused silica (n=1.458) | 55.6° | | Air / flint glass SF11 (n=1.78) | 60.7° | | Air / sapphire (n=1.77) | 60.5° | | Air / diamond (n=2.42) | 67.5° | | Water / glass BK7 (n=1.0 + n=1.517 relative) | 48.7° | **Fresnel reflectance at Brewster's angle (analytical):** R_p = 0 (exactly zero by definition) R_s = ((n₁² + n₂²) − 2 n₁ n₂) / (n₁² + n₂² + 2 n₁ n₂) = (n₂ − n₁)² × (n₁² + n₂²) / (n₁² + n₂²)² (approximately) For typical glass: R_s at Brewster's angle ≈ 15%. So a Brewster window passes p-pol unattenuated and reflects ~15% of s-pol — a very crude polarizer when used alone, but a much better one when combined with multiple Brewster surfaces in a stack. **Why p-polarization disappears at θ_B:** The transmitted ray's electric dipoles oscillate perpendicular to the refracted ray direction. P-polarization causes dipoles to oscillate in the plane of incidence. At Brewster's angle, the reflected ray would be exactly along the dipole oscillation axis — but dipoles don't radiate along their own axis (a classical EM result). So no p-polarized reflection occurs.

How to use this calculator

Enter n₁ (incident medium refractive index) and n₂ (transmitted medium).
The calculator returns Brewster's angle in degrees, measured from the surface normal.
For air → most materials, n₁ = 1.0 and Brewster's angle is around 55–60° for typical glasses.
For polarizing camera filters, set them so the transmission axis is perpendicular to surface reflection direction (vertical for horizontal water surfaces).
For laser Brewster windows, mount the window with its plane perpendicular to the laser beam, tilted to make Brewster's angle.
Remember Brewster's angle eliminates ONLY p-polarization reflection; s-polarization still reflects significantly.

Worked examples

Polarizing sunglasses on the water

**Scenario:** Fishing on a lake at noon. The sun reflects off the water surface and creates blinding glare. Why do polarized sunglasses help? **Calculation:** Sunlight at the lake hits water at various angles, but the most painful glare comes from rays near Brewster's angle (53° for water). At this angle, reflected light is 100% s-polarized — horizontally polarized for a horizontal water surface. Sunglasses with vertical transmission axes block this horizontal light completely. **Result:** Polarized sunglasses block ~95% of glare from horizontal surfaces (water, ice, pavement, car hoods). They do nothing for glare off vertical surfaces (walls, buildings) where reflection geometry doesn't favor strong polarization. For fishing, polarized glasses also reveal underwater detail by removing surface glare.

Brewster window in a HeNe laser

**Scenario:** A helium-neon laser uses external mirrors with the gain tube terminated by Brewster windows (fused silica, n=1.458). What angle? Why? **Calculation:** θ_B = arctan(1.458) = 55.6°. The windows tilt at this angle to the laser axis. P-polarized light passes through with essentially zero reflection loss; s-polarized light suffers significant loss on each pass. **Result:** The laser self-selects p-polarization because it has the lowest cavity loss. The output beam is therefore linearly polarized in the plane of incidence of the Brewster windows. This polarization control is "free" — no separate polarizer needed. Brewster windows are essential for low-loss laser cavities, especially at wavelengths where AR coatings are difficult.

Photography polarizer for car windshield

**Scenario:** Photographing through a windshield in bright sunlight; the windshield reflects sky and dashboard. Can a polarizing filter help? **Calculation:** Windshield is glass, n ≈ 1.5. Brewster's angle ≈ 56°. But the camera is typically looking nearly straight through (angle near 0–30°), so Brewster effects are weak. The reflected sky light at small angles is only partially polarized. Polarizing filter helps a moderate amount but doesn't eliminate the reflections completely. **Result:** Polarizing filters are very effective at Brewster's angle (water at 53°, ground at moderate angles) but less effective at the small angles typical of windshields when shooting from the driver's seat. For maximum effect, shoot from the side of the car looking through the windshield at ~55° angle of incidence. From the front, the polarizer's effect is partial.

When to use this calculator

**Use Brewster's angle calculations for:**

- **Photography**: choosing polarizing filter usage, predicting how strong glare reduction will be. - **Laser cavity design**: positioning windows and intracavity optics for zero p-polarization loss. - **Polarization analysis**: determining whether reflected light from a surface is polarized. - **Coating design**: AR coatings have minimal performance impact near Brewster's angle (because R_p is already low). - **Solar panel anti-reflection**: at Brewster's angle (which the sun crosses twice a day), reflection is minimized. - **Underwater photography**: polarizing filters at the water surface cut glare; effects depend on angle. - **Glasses and visor design**: polarized eyewear, helicopter visors, racing goggles.

**Practical photography guidance:**

- **Sky polarization**: maximum at 90° from sun direction (Rayleigh scattering). Use polarizer to deepen blue sky (but not when shooting wide-angle — uneven sky tone). - **Water and ground glare**: polarizer most effective at Brewster's angle (53° water, 56° glass) — typically when sun is 37° above horizon. - **Leaves and wet surfaces**: polarizer reduces specular highlights, saturates colors. - **Cannot be used on color sensors with strong sensor angle dependence**: extremely rare problem on modern sensors but historically affected some early DSLRs.

**Brewster windows in laser systems:**

- **Why used**: zero loss for p-polarization without AR coatings. - **Disadvantage**: introduce astigmatism in convergent beams; only fully transparent at exact θ_B. - **Bandwidth**: works at all wavelengths (no chromatic AR sensitivity), but n varies slightly with λ. - **Tilt sensitivity**: small angular deviation (< 1°) negligible; large deviation introduces s-polarization loss.

**Combining multiple Brewster surfaces:**

A stack of N Brewster plates passes p-polarization with essentially no loss while reflecting/absorbing increasing fractions of s-polarization. For N = 10 plates, p-pol transmission is ~99% while s-pol transmission is reduced to ~5%. This is the basis of "pile of plates" polarizers, used in some specialized applications.

**Brewster's angle at non-traditional interfaces:**

- **Air → metal**: complex refractive indices; "pseudo-Brewster's angle" exists where reflection is minimized but not zero. - **Anisotropic materials (crystals)**: two Brewster's angles, one for each polarization eigenstate. - **Total internal reflection (n₁ > n₂)**: Brewster's angle still exists at θ < θ_critical; useful in waveguides.

Common mistakes to avoid

Thinking Brewster's angle blocks all reflection. Only p-polarization reflection is zero; s-polarization still reflects ~15% or more.
Using polarizing sunglasses for vertical-surface glare. They're optimized for horizontal surfaces and Brewster geometry.
Confusing Brewster's angle with critical angle. Critical angle is for total internal reflection (n₁ > n₂). Brewster's angle is about polarization (any n combo).
Computing θ_B with wrong order of indices. arctan(n₂/n₁) — outgoing index over incoming index. Reversed gives wrong angle.
Forgetting wavelength dependence. n varies with wavelength, so θ_B is slightly different for different colors of light.
Trying to use a Brewster window for unpolarized light. The window passes p-polarization with no loss but reflects s-polarization — separating, not transmitting, both.
Applying Brewster geometry to metals. Pseudo-Brewster's angle on metals minimizes p-pol reflection but doesn't eliminate it; the complex refractive index changes the behavior.

Frequently Asked Questions

Sources & further reading

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