Brewster’s Angle Calculator — Glare’s End

At one special angle of incidence, a boundary stops reflecting one polarization entirely — glare dies, and the reflected ray emerges perfectly polarized. Enter the two refractive indices, and this calculator returns that angle: Brewster’s angle, the physics inside every pair of polarized sunglasses.

The Angle Where Glare Gives Up

Reflection at a boundary treats the two polarizations unequally, and there is an incidence angle — found by David Brewster in 1815 — where the inequality becomes total: the p-polarized component's reflection falls to zero, because at that geometry the reflected ray would have to be radiated by dipoles pointing along their own axis, which dipoles cannot do. The condition reduces to the cleanest formula in polarization optics, tan θ_B = n₂ ÷ n₁, with the equivalent geometric signature the widget prints: at Brewster incidence the reflected and refracted rays stand exactly 90° apart. Every polarizing filter photograph, every glare-free shop window and every laser cavity window tilted "at Brewster's angle" (to pass one polarization losslessly) is this arctangent, hired.

Worked Example: Air to Crown Glass

The defaults price the textbook boundary — air (1.00) to glass (1.50): tan θ_B = 1.5, so θ_B = 56.31° from the normal, with the transmitted ray running at 33.69° (the two summing to 90°, as the geometry guarantees). Stand a shop window or a lake surface so the sun strikes near 56° and the reflected glare is almost entirely one polarization — which is the quantitative secret of polarized sunglasses: they block that plane, and the glare from roads and water at driving and fishing geometries (near Brewster for those indices) collapses while the scene's unpolarized remainder merely dims. Water's own angle, air to 1.33, is 53.1° — the fisherman's number, three degrees off the glass default, computed by the same line.

Why the Reflected Ray Is Perfectly Polarized

The dipole argument earns its paragraph: reflection is re-radiation from electrons in the second medium, driven to oscillate along the transmitted ray's electric field. For the p-component, that oscillation direction, at Brewster geometry, points exactly along where the reflected ray would travel — and an oscillating charge radiates nothing along its own axis (the radiation pattern is a doughnut around the oscillation, with a hole through the middle). Zero radiation along the would-be ray means zero p-reflection: not a small coefficient but an exact extinction written into electrodynamics, holding for any pair of transparent media. The s-component, oscillating perpendicular to the plane, faces no such ban and reflects as usual (about 15% of its intensity at the glass default) — so the reflected beam is purely s, a polarizer made of nothing but an angle.

Brewster in the Camera Bag

Photographers meet this angle twice: first in the circular polarizer, whose whole effect peaks when the subject's reflective surfaces are lit near their Brewster geometry (which is why the filter transforms a scene at mid-morning and merely dims it at noon), and second in the folk rule that polarizers work best at 90° to the sun — an approximation of keeping the camera-to-surface-to-sun angle near the polarization angle for the scene's dominant surfaces. The calculator gives the exact figure per material, which the professional uses in the studio, where product photography on glass and acrylic is a negotiation with reflections that this arctangent settles: tilt the product (or the lights) by the computed angle, thread the polarizer, and the glare leaves on physics' instruction rather than by trial.

The Sky Itself Is Polarized

Sunlight scattered by air molecules arrives partially polarized, strongest in the band of sky 90° from the sun — a fact bees navigate by, reading the polarization pattern as a compass when the sun itself is hidden, and which Viking navigators may have exploited with "sunstone" crystals to locate the sun through cloud. The phenomenon is the same family physics as this page's boundary polarization (anisotropy selecting a preferred electric-field direction), and landscape photographers exploit the sky's version with the same polarizer: darken the blue at right angles to the sun, saturate the scene, delete a window's reflection at the computed Brewster tilt. One arctangent prices the glass case exactly, and the sky case teaches the deeper lesson — wherever light changes direction decisively, polarization is the signature it leaves.

Glare on the Water: The Angler's Computation

The fisherman's sunglasses work because the geometry conspires: the sun low, the water flat, the sight-line shallow — incidence angles grazing the 53.1° that this page computes for the air-water pair across a whole morning's fishing. Cut the reflected sheet away with a horizontal-blocking polarizer and the surface turns transparent; fish, weed beds and the bottom's structure appear as if the water had been drained of its mirror. The calculator's contribution to the sport is precision about when the trick peaks — as the sun climbs, the incidence at a standing angler's downward glance steepens away from Brewster's angle, the reflection's polarization dilutes, and by noon the glasses are merely dark. Physics schedules the miracle; the angler who has run the number knows the hours, and buys the glasses for the geometry rather than the brand.

Related Tools

More from the Physics Calculators hub: the Index of Refraction Calculator — the indices this angle is built from, and the Snell bending they also govern — and the Velocity Calculator for the light doing all of this at c.

Frequently Asked Questions

How do polarized sunglasses use this, exactly?

Glare from horizontal surfaces — water, roads, snowfields — arrives at your eyes after reflecting at angles near Brewster for those materials, so it is predominantly polarized with its electric field horizontal (the s-direction for a horizontal surface). Sunglass lenses are polarizers oriented to block the horizontal component: the glare's main carrier is extinguished while roughly half of the ordinary, unpolarized light passes. The calculator's angle explains the lenses' uneven performance — strongest exactly at Brewster geometry (mid-morning and mid-afternoon sun on water), weakest at steep noon angles where the reflection is far from the polarization angle and the glare is barely polarized to begin with.

What is a Brewster window in a laser?

Gas-laser tubes are sealed with windows tilted at Brewster's angle for the lasing polarization, so the beam passes through both windows, round trip after round trip, with zero reflection loss for that polarization — while the other polarization is reflected away and dies. The tube thus selects its polarization by pure geometry, no coatings required, and the technique generalises: any optical element that must pass one polarization losslessly (in cavities, interferometers and high-power beams where coatings would burn) is tilted to the angle this page computes. Enter the window glass's index pair and the arctangent gives the machinist's tilt directly.

Can Brewster's angle exceed 45 degrees?

Whenever light travels into the denser medium (n₂ > n₁) — which is most everyday cases, air into anything — tan θ_B > 1 and the angle exceeds 45°, running from 53° (water) through 56° (glass) toward 67.5° for diamond-index material. Going the other way (glass into air) the angle mirrors down to 33.7° — note the complement: the internal Brewster angle and the external one sum to 90°, the same pairing as the reflected-transmitted geometry. The calculator is direction-aware through the order of the two indices; swap them and it prices the return journey, a check worth performing because boundaries, unlike sunglasses wearers, face both ways.

Does the angle depend on wavelength?

Slightly, through dispersion: the indices themselves vary with colour (the refraction page's caveat), so Brewster's angle for blue light sits a fraction of a degree off the red value — the reflected beam at a single "Brewster" setting for green is perfectly polarized only for the green. Precision polarimetry works monochromatically for this reason; sunglasses do not care. The calculator consumes whatever indices you supply, so entering the values for your wavelength prices your colour exactly — the formula is achromatic, the materials are not, and the distinction is the entire content of optical engineering's fine print.

What happens to the transmitted light at Brewster's angle?

It passes enriched in the p-component (which lost nothing to reflection) but still mixed — a single pass polarizes only the reflected beam perfectly; the transmitted beam needs help (a stack of plates at Brewster incidence, each skimming the s-component into reflection, gradually purifies the transmitted beam too — the "pile of plates" polarizer of the pre-film era). The transmitted angle the widget prints, 90° minus the incidence, is exact at Brewster geometry by the same dipole condition, and it is the working figure for laying out those plate stacks and Brewster windows in real instruments.

Is there a Brewster angle for metals?

Not a clean one: metals absorb, their indices are complex numbers, and the p-reflection dips near a "pseudo-Brewster" angle without reaching zero — the reflected light becomes strongly but imperfectly polarized, and ellipsometry (the science of measuring thin films by exactly this dip) turns the imperfection into an instrument. For the transparent dielectrics of this page — glass, water, plastic, ice — the extinction is exact and the calculator's angle is the whole story. The metal caveat matters mostly as a reminder that formulas have jurisdictions, and this one's is printed, implicitly, in its two innocent index fields. Within that jurisdiction, though, the result is one of optics' few absolutes: a reflection that is not approximately but exactly single-polarization, gift-wrapped by geometry.