How this instrument works
Étienne-Louis Malus noticed in 1808 that sunlight bouncing off a palace window came back polarised. Seven years later David Brewster pinned down which tilt does it best: aim a beam so that it meets a surface at arctan(n₂/n₁) from the normal, and every trace of light vibrating in the plane of incidence vanishes from what reflects. Only s-polarised light survives, vibrating parallel to that surface.
Geometry explains why. At this particular tilt, reflected and refracted rays leave at right angles to one another. Electrons driven by a refracted wave then oscillate along a line pointing straight back down the reflected path — and an oscillating dipole radiates nothing along its own axis. So no p-polarised light can be emitted in that direction. It is a null enforced by radiation pattern rather than by any special material property.
Two assumptions hold this up: both media must be transparent dielectrics with real, non-magnetic indices, and their boundary must be optically smooth. Aim at aluminium or silicon and index turns complex, so p-reflectance dips to a shallow minimum instead of zero — an effect known as pseudo-Brewster. Dispersion matters too, since n₂ drifts across visible wavelengths; for crown glass blue polarises roughly an eighth of a degree away from red.
- Put whatever your beam travels through first into Refractive index, first medium — 1.000 for air, 1.333 if you are already underwater.
- Enter what is being struck under Refractive index, second medium: 1.50 for crown glass, 1.46 for fused silica, 2.42 for diamond.
- Read Brewster angle (degrees), measured from the surface normal — never from a surface itself.
- Swap both entries to describe a reverse crossing; those two answers always add to 90°.
Worked example — air onto a crown glass pane
A photographer wants glare off a shop window gone. Air comes first, so Refractive index, first medium = 1; crown glass next, so Refractive index, second medium = 1.5. That gives θ_B = arctan(1.5 ⁄ 1) = 56.309932474°, which Brewster angle (degrees) reports as 56.309932 at six decimals.
Stand so your camera axis meets that pane 56.3° off its normal and reflected glare is entirely horizontal in polarisation. Rotate a linear polariser until its transmission axis runs vertical and reflection all but disappears, while goods behind glass stay bright. Windscreen glare obeys an identical rule, which is why driving sunglasses are cut with a vertical axis.
Questions
Is Brewster's angle measured from the surface or from the normal?
From the normal. Every Fresnel result, this one included, references a perpendicular dropped onto that interface. So 56.3° for glass sits only 33.7° above a pane itself, which is why glare punishes you at shallow, grazing views of wet tarmac. Quoting 33.7° as Brewster's angle is probably the most common slip made with this quantity.
Does light passing through become polarised as well?
Only partly. At θ_B all p-polarised light is transmitted, but so is roughly 85% of s-polarised light, leaving a transmitted beam mildly biased rather than pure. Stacking many plates at Brewster's angle — a pile-of-plates polariser — strips a little more s-light at each surface and eventually yields usable polarised output. Laser windows exploit exactly this, cut at θ_B so that one polarisation crosses with no loss at all.
How does this differ from a critical angle?
Different formula, different physics. Critical angle is arcsin(n₂/n₁) and exists only when light heads from dense toward rare medium, marking onset of total internal reflection. Brewster's angle is arctan(n₂/n₁), exists for any index pair, and marks disappearance of one polarisation rather than disappearance of transmission. Water shows 48.6° critical seen from below, 53.1° Brewster seen from above.
What happens at metal or other absorbing surfaces?
Reflectance for p-polarisation dips but never reaches zero, because a complex refractive index throws reflected and refracted oscillations out of phase. Practitioners call that minimum a pseudo-Brewster angle; for polished aluminium it lands near 80°. Entering a metal's real index part alone here marks roughly where that dip sits, not a genuine null.
Why do polarising sunglasses cut glare so effectively?
Glare off roads, water and car bonnets reflects near Brewster's angle from a horizontal plane, so surviving light vibrates horizontally. A filter whose transmission axis stands vertical rejects it while passing most of everything else. Anglers rely on an identical trick: killing horizontally polarised surface reflection at 53° reveals fish below.
Does wavelength change my answer?
Slightly, through dispersion. Borosilicate crown glass has n₂ ≈ 1.5224 at 486 nm and 1.5143 at 656 nm, shifting Brewster's angle from 56.70° down to 56.56° — about an eighth of a degree across visible light. Negligible for photography, worth tracking for a laser window or an ellipsometer, where you should enter n₂ at your working wavelength.