One Hidden Direction: How Polarisation Explains Sunglasses, Screens and the Sky — Epoche B2
Introduction Many people believe that polarising sunglasses are simply dark grey filters: they cut brightness, and nothing more. This essay argues that the belief is mistaken. Light is a transverse wave, which means its oscillation is sideways to its motion, as on a shaken rope, and not along the direction of travel, as in a sound wave in air. What oscillates is the electric field $\vec{E}$ — the quantity that would push on a charge placed in the beam — and it points at right angles to the direction of travel. That leaves two independent sideways directions to choose between, and the orientation of the field among them, its polarisation, is a real physical property of the beam. Ordinary light from the Sun or a lamp is unpolarised : it is emitted by enormous numbers of atoms acting independently, so the field direction changes randomly millions of times faster than any detector can follow, with no direction favoured on average. Unpolarised light is therefore not light without polarisation but light whose polarisation is scrambled — and it can be unscrambled. By examining three apparently unrelated observations, we can conclude, by induction, that this single hidden direction explains them all. Example 1: Glare and Brewster's angle Fix some vocabulary first. The plane of incidence is the plane containing the incoming ray and the perpendicular to the surface — for light bouncing off a lake into your eyes, a vertical plane. Any polarisation can be split into two components relative to it: p , lying in that plane, and s , perpendicular to it (from the German senkrecht ). For a horizontal surface, s-polarisation is horizontal. This bookkeeping matters because a dielectric — a transparent insulator such as water or glass, which has no free electrons but whose bound charges shift slightly under an applied field — reflects the two components by different amounts. David Brewster established in 1815 that at one particular angle of incidence the p-component is not reflected at all, so the reflected beam is purely s-polarised. The condition is $$\tan\theta_B = \frac{n_2}{n_1},$$ where the refractive index $n$ of a medium is the factor by which light is slowed in it, $n = c/v$. The reason is worth seeing, because it is the same reason that will explain the sky. Light entering the water sets its molecules oscillating; each oscillating charge is a miniature aerial, and the reflected beam is nothing but the combined radiation of those aerials sent back into the air. Now the key fact about an aerial: an oscillating charge radiates nothing along its own line of oscillation, because the radiated field must be transverse, and in that one direction there is no sideways component of the acceleration to supply it. For p-polarised light the molecular dipoles oscillate in the plane of incidence, perpendicular to the refracted ray. If it happens that the reflected direction is perpendicular to the refracted direction, then the reflected direction lies exactly along the dipole axis — and no p-polarised light can be sent that way at all. That geometric condition, $\theta_B + \theta_t = 90^\circ$ with $\theta_t$ the angle of the refracted ray, gives the formula in two lines. Snell's law of refraction says $n_1 \sin\theta_B = n_2 \sin\theta_t$; substituting $\theta_t = 90^\circ - \theta_B$ and using $\sin(90^\circ - \theta) = \cos\theta$ gives $n_1 \sin\theta_B = n_2 \cos\theta_B$, and dividing by $\cos\theta_B$ yields $\tan\theta_B = n_2/n_1$. For light in air ($n_1 = 1.00$) striking water ($n_2 = 1.33$), $\theta_B = \arctan(1.33) = 53.1^\circ$ from the vertical — a shallow, late-afternoon sort of angle, which is why glare off a lake is worst when the Sun is low. Glare from roads and water is therefore mostly horizontally polarised: not perfectly, since only at exactly $\theta_B$ does the p-component vanish, but strongly across a wide band of angles either side. Polarising sunglasses contain a filter whose transmission axis is vertical. Such a filter passes only the field component along its axis; if the incoming field has amplitude $E_0$ at angle $\theta$ to that axis, the surviving amplitude is the projection $E_0\cos\theta$, and since a wave's intensity goes as the square of its amplitude, the transmitted intensity is $I = I_0\cos^2\theta$. This is Malus's law, published by Étienne-Louis Malus in 1809. Put in the numbers: horizontal glare meets a vertical axis at $\theta = 90^\circ$, and $\cos^2 90^\circ = 0$, so it is removed almost completely. Ordinary scenery arrives unpolarised, and averaging $\cos^2\theta$ over all angles gives $\tfrac{1}{2}$, so it is merely halved. A plain grey filter cutting brightness in half would dim glare and scenery alike; the polariser destroys one and dims the other. The glasses select a direction. Example 2: Liquid crystal displays A liquid crystal is a fluid made of rod-shaped molecules which flow like a liquid yet, like a crystal, tend to line up with one another. Two consequences follow. First, the aligned material is birefringent : light polarised along the rods travels at a different speed from light polarised across them, because the molecules are easier to polarise along their length. Second, the alignment can be steered — by rubbing the containing glass in one direction, so the molecules touching it lie along the grooves, and by applying an electric field, since a rod-shaped molecule in a field feels a twisting force that ends only when it lines up. The twisted nematic cell, introduced by Schadt and Helfrich in 1971, exploits both. Its liquid crystal layer sits between two polarising filters whose axes are crossed at $90^\circ$ — an arrangement that by Malus's law transmits nothing, since $\cos^2 90^\circ = 0$. The two glass surfaces are rubbed at right angles to each other, so the molecules execute a quarter-turn helix from one face to the other. With no voltage applied, light that has passed the first polariser enters this helix and its polarisation is carried round with