Polarization

In an electromagnetic wave E and B are both perpendicular to the direction of travel, and by convention the polarization is named after E. That leaves a whole plane of possible directions for a beam going one way, and which one it uses is a genuine physical property. A laser or a radio transmitter emits with E along a fixed line: linear polarization. A filament lamp or the Sun does not. Its light comes from enormous numbers of atoms radiating independently, each wave train lasting of order 10⁻⁸ s before the direction resets at random, so over any measurable time every orientation appears equally often. That is unpolarized light — not light with no direction, but light whose direction is reshuffled far faster than any instrument can track. A longitudinal wave has no such freedom: sound displaces air along its own direction of travel, so there is no second axis to distinguish and no rotation of any filter could change what gets through.

An ideal polarizing sheet has one transmission axis. Resolve the incoming field into a component along that axis and a component across it; the sheet absorbs the second and passes the first. Since the field is a vector, the surviving amplitude is E₀ cos θ, where θ is the angle between the incident polarization and the axis. Intensity carries the square of the field, so I = I₀ cos²θ. That squaring is where most of the arithmetic errors live. At θ = 45° the field keeps cos 45° = 0.707 of its size, but the intensity keeps only 0.500. At θ = 30° the field keeps 0.866 and the intensity 0.750; at θ = 60° the field keeps 0.500 and the intensity just 0.250. Notice also what leaves the sheet: light polarized along the sheet's own axis, whatever direction it arrived from. Nothing about the incident direction survives except the single factor cos θ in the amplitude.

Malus's law needs a definite θ, and unpolarized light has none. Take instead the average over every orientation. Because cos²θ swings between 0 and 1 symmetrically about ½, its mean over a full turn is exactly ½, so the first sheet always transmits I₁ = ½ I₀ no matter how it is oriented — turning a single polarizer in the beam of a filament lamp dims nothing, while skylight, already partly polarized by scattering, does dim as the sheet turns. What the sheet does change is the state: the light emerging is linearly polarized along its axis. From that point on the beam has a definite direction, and every later angle must be measured from that axis rather than from the lab vertical. Put 200 W m⁻² of unpolarized light into a vertical sheet and 100 W m⁻² leaves, polarized vertically; send it through a second sheet 35° away and 100 cos²35° = 67.1 W m⁻² survives, polarized at 35°. Rotating the first sheet alone changes nothing, because θ is a relative angle.

Two sheets at 90° to each other pass nothing: the first prepares the light along its axis, and cos²90° = 0. Now insert a third sheet between them, its axis at θ to the first. Sheet 1 gives ½I₀. Sheet 2 is θ away, so it transmits ½I₀ cos²θ and re-prepares the light at θ. Sheet 3 is now 90° − θ from that, giving ½I₀ cos²θ cos²(90° − θ) = ½I₀ cos²θ sin²θ, which the identity sin θ cos θ = ½ sin 2θ tidies to I₃ = (I₀/8) sin²(2θ). At θ = 45° the output peaks at I₀/8: a beam of 400 W m⁻² delivers 50 W m⁻² through a pair of sheets that on their own delivered nothing. Adding an absorber increased the transmission. That is only paradoxical if you think the sheets remove rays; it is inevitable if each one projects the field onto a new axis.

Shine unpolarized light onto glass or water and the reflected beam comes back partially polarized parallel to the surface, and at one special incidence completely so. That is Brewster's angle, tan θ_B = n₂/n₁. Air onto water, n₂/n₁ = 1.33, gives θ_B = 53.1°; air onto crown glass at 1.52 gives 56.7°. The mechanism is worth having. At θ_B the reflected and refracted rays are exactly 90° apart — for water, 53.1° + 36.9° = 90° — and the reflected light is radiated by charges in the surface driven along the refracted beam's own field direction. A dipole radiates nothing along its own axis, so the component that would have been polarized in the plane of incidence has no way to leave in the reflected direction, and only the component parallel to the surface survives. Glare from a road, a lake or a car roof is therefore horizontally polarized, which is why polarizing sunglasses are cut with a vertical transmission axis.

Calcite, quartz and stretched plastic are birefringent: the refractive index depends on the polarization direction. A uniaxial crystal has an ordinary index nₒ and an extraordinary index nₑ, so the two components of a field resolved onto the fast and slow axes travel at different speeds and arrive with a phase difference Δφ = 2π (nₑ − nₒ) d / λ₀. Nothing is absorbed — the components are simply re-timed. Choose d so that Δφ = π and you have a half-wave plate, which flips the sign of the slow component and so reflects the polarization direction in the fast axis: light at α to the fast axis leaves at −α, rotated by 2α. Choose Δφ = π/2 and the two components leave a quarter cycle apart, turning linear polarization into circular. Quartz at 589 nm has nₒ = 1.5443 and nₑ = 1.5534, so Δn = 0.0091 and a quarter-wave plate needs only d = 16.2 µm. Because λ₀ sits in the denominator, a wave plate is exact at one wavelength alone.