PLANE WAVES AND POLARIZATION

Last topic pulled a wave equation out of Maxwell's four laws and read the propagation speed, c = 1/√(μ₀ε₀), straight off the coefficients. That told us how fast a disturbance moves through empty space. It did not tell us what one looks like. This topic does.

Start with the simplest shape: a plane wave. Every point on any surface perpendicular to a single direction k̂ sees the same field at the same moment; the surfaces of constant phase are infinite flat planes marching at speed c. No real source produces one, but far from any source every wave looks locally plane, and every EM wave in vacuum is a sum of plane-wave pieces.

For that shape, Maxwell's vacuum equations lock three rules into place, no exceptions:

Those three rules are the skeleton of every wave that ever travelled through empty space: sunlight, the Wi-Fi in your walls, the photons that carried this page to your screen. The only remaining freedom is the direction E is allowed to point inside the plane perpendicular to k. That freedom is called polarization.

The scene renders a plane wave along +x̂. Magenta is E, cyan is B, amber is k. At every point along the ray the three arrows satisfy E · k̂ = 0 and B · k̂ = 0 — the two fields live inside the plane perpendicular to propagation, which is the meaning of transverse. Cross-check the phase: when E peaks, B peaks; when E crosses zero, so does B. That is the vacuum signature. (Inside matter those two fields do get shuffled out of phase — that is §09.1's problem.)

The polarisation is the trajectory the tip of E traces in that transverse plane over one period. Slide ellipticity from 0 to 1 and phase δ from 0 to π, and watch the shape morph:

There is no fourth kind. Linear and circular are the two degenerate edges of the ellipse family — one family, two real parameters, the whole space of what light in vacuum can be. That is the point worth sitting with. A single plane wave, with a fixed wavelength and a fixed direction of travel, carries an extra bit of structure that the scalar wave equation you met in §04 did not have. Scalar waves have amplitude and phase. Vector waves have amplitude, phase, and a polarisation. The polariser in front of your sunglasses is a device that cares about the difference.

Underneath the morph is a piece of 2D bookkeeping. Fix k̂ = x̂. Then E(x, t) lives inside the (ŷ, ẑ) plane and splits into two orthogonal real oscillators:

Three real numbers describe the polarisation: the amplitudes E_y and E_z, and the phase delay δ between them. Everything follows from those three:

The kernel exposes a classifier polarizationState({Ex_amplitude, Ey_amplitude, phaseDelta}) that returns exactly those three labels by reducing δ to (−π, π] and bucketing against the special cases. It drives both scenes above and below. A companion helper, isTransverse(k, E), checks that a candidate E is perpendicular to k to within a tolerance — the tests use it to verify that every E chosen in the transverse plane for k = x̂ actually satisfies Gauss's constraint.

Light from the sun, from a bulb, from a fire is unpolarised — a chaotic superposition of modes with all polarisations averaged incoherently. A linear polariser transmits the component of E along its polarisation axis and blocks the perpendicular component. The beam emerges linearly polarised, at half the original intensity (the average of cos²θ over random θ is ½).

Put a second polariser behind the first, with its axis at angle θ to the first. The component of the now-linearly-polarised E along the second axis is E₀·cos(θ). The intensity — proportional to E² — that emerges is E₀²·cos²(θ). This is Malus's law, written down in 1809 by the French army engineer Étienne-Louis Malus while, the legend goes, watching sunset reflect off the Luxembourg Palace through Iceland spar:

When the analyser lines up with the incoming polarisation (θ = α), everything gets through. When it is crossed (θ = α ± π/2), nothing does. Between those, cos² squeezes the beam continuously — a clean experimental handle on a property a person cannot see directly.

Polaroid sunglasses use this. The filter axis is vertical, so the horizontally-polarised glare reflected off roads and water (preferentially so, per §09.3's Fresnel equations) is attenuated. The vertically-polarised part of skylight still gets through, so the world stays visible. Two crossed polarisers in sequence let nothing through; insert a third polariser between them at 45° and suddenly a sixteenth of the original intensity emerges — the quantum-mechanical version of this puzzle is the standard first-lecture trick, but the classical cos²-law explanation has already delivered the answer.

Any 2D polarisation state can be specified by a Jones vector — a two-component complex amplitude (E_y, E_z·e^) — or, equivalently, by the real triple (|E_y|, |E_z|, δ). Normalise by the larger amplitude and two parameters remain: the ratio r = |E_z|/|E_y| and the phase delay δ. Every plane-polarised state in vacuum sits at one point of that 2D map.

Three features of the map are worth holding in your head:

The jonesVector(ellipticity, orientation) helper goes the other way: given χ ∈ [0, 1] and a major-axis angle θ, it builds the (|Ex|, |Ey|, δ) triple. χ = 0 gives linear along θ; χ = 1 gives circular; in between lives the ellipse continuum. The tests verify the round-trip: feed jonesVector into polarizationState, get the label you asked for.

One last sanity check, because the transversality rule is the spine of everything above. Take a candidate plane-wave along a general direction:

Take the divergence. The gradient of the cosine brings down a factor of k, so ∇·E = −k·E₀·sin(k·r − ωt). In vacuum ∇·E = 0 identically, at every r and t. The sine oscillates; the only way to keep the product zero everywhere is to force k·E₀ = 0. E₀ ⊥ k — not an approximation, just Gauss's law applied to a plane wave. The same argument on Gauss-for-magnetism gives B₀ ⊥ k, and Faraday's law locks the magnitude and phase together. What remains is exactly two real numbers in the transverse plane — the polarisation.

Polarisation is structural. It becomes operational the moment the wave hits a boundary. In §09.3 (Fresnel's equations) the components parallel and perpendicular to the plane of incidence reflect and refract with completely different coefficients — which is why glare off water is polarised, why polarising filters work, and how Brewster's angle exists. In §09.9 (polarisation phenomena) Malus meets birefringence, optical rotation, and the fact that sugar water can rotate a beam's polarisation by a measurable angle per centimetre.

Circular polarisation comes back as the natural basis for photon spin, antenna handedness, and the way radar distinguishes round raindrops from jagged ice crystals — circular light survives the scattering, linear light doesn't. For now, §08.2's payoff is the reveal: a plane wave is two fields and a direction, locked perpendicular, locked in phase, with one remaining degree of design. The rest of classical optics is what you do with it.