Polarisation

Put on a pair of good sunglasses on a bright day by the sea and something quietly magical happens: the harsh white glare bouncing off the water melts away, and you can suddenly see into the waves. Tilt your head right over to one side, though, and the glare floods straight back. Nothing about the light's colour or brightness explains that — it is a clue about a hidden property of light that we haven't met yet: its polarisation.

You already know that light is a transverse wave — one of the whole family that makes up the electromagnetic spectrum. This page shows what "transverse" really buys us. Because the wobble in a transverse wave is across the direction of travel, we can ask a brand-new question that makes no sense for sound: which way across? Pinning the wobble down to a single plane is what polarisation means — and it turns out to be one of the sharpest pieces of evidence that light is transverse in the first place.

Only transverse waves can be polarised

Picture shaking a rope through a narrow slot in a fence. If you wobble your hand up and down and the slot is vertical, the wave sails through. Turn the slot horizontal and the same up-and-down wave is stopped dead — the fence only lets through the wobble that lines up with its slot. A transverse wave has a direction of wobble, so a slot can accept or reject it.

Ordinary light from a bulb or the Sun is unpolarised: it is a jumble of countless little waves wobbling in every plane at once — up–down, left–right, and every diagonal in between — all mixed together. When we force that jumble down to wobbling in just one plane, we call the result plane-polarised (or simply polarised) light.

Now the key point. A longitudinal wave, like sound, wobbles back and forth along its direction of travel. There is only one such direction — there is no "which way across" to choose — so a longitudinal wave cannot be polarised at all. This is exactly why polarisation matters so much in physics: the plain fact that light can be polarised proves it must be transverse. Sound can never do it.

The polarising filter (Polaroid)

The optical version of the slotted fence is a polarising filter, often called a Polaroid. It has a transmission axis — a special direction — and it transmits only the part of the wave's oscillation that lies along that axis, soaking up the rest. Shine unpolarised light at it and what comes out the other side is plane-polarised, wobbling neatly in the direction of the transmission axis.

Because unpolarised light is a fair mix of every direction, an ideal filter passes exactly half of it — on average, half the wobbling lines up with the axis and half doesn't. So one filter both polarises the light and halves its intensity:

I_{\text{after one filter}} = \tfrac{1}{2}\,I_0 \quad\text{(unpolarised light in).}

The real fun starts with a second filter, called the analyser. The light reaching it is already polarised, so how much gets through now depends entirely on the angle between the analyser's transmission axis and the polarised light — and that relationship has a beautiful, exact law.

Malus's law: how much gets through the analyser

Send plane-polarised light of intensity I_0 into an analyser whose transmission axis makes an angle \theta with the light's plane of polarisation. Only the component of the electric-field oscillation that lies along the axis gets through. The field amplitude scales as \cos\theta, and since intensity is proportional to amplitude squared, the transmitted intensity scales as \cos^2\theta:

For plane-polarised light of intensity I_0 passing through an analyser at angle \theta to its plane of polarisation, the transmitted intensity is I = I_0\cos^2\theta.
When the axes are aligned (\theta = 0), \cos^2 0 = 1, so all the light passes: I = I_0.
When the filters are crossed (\theta = 90^\circ), \cos^2 90^\circ = 0, so no light passes: I = 0.

Worked example 1 — a single angle. Plane-polarised light of intensity I_0 = 8\ \text{W/m}^2 meets an analyser at \theta = 30^\circ. Since \cos 30^\circ = \tfrac{\sqrt3}{2}, we have \cos^2 30^\circ = \tfrac{3}{4}, so

I = I_0\cos^2\theta = 8 \times \tfrac{3}{4} = 6\ \text{W/m}^2.

Worked example 2 — start from an unpolarised beam. Unpolarised light of intensity I_0 = 12\ \text{W/m}^2 passes through a polariser and then an analyser set at 60^\circ to the first. The first filter halves the intensity; the analyser then applies Malus's law:

I = \tfrac{1}{2}I_0 \cos^2 60^\circ = \tfrac{1}{2}\times 12 \times \left(\tfrac{1}{2}\right)^2 = 6 \times \tfrac{1}{4} = 1.5\ \text{W/m}^2.

Worked example 3 — the crossed-filter case. Two filters are set at right angles (\theta = 90^\circ). Whatever polarised light of intensity I_0 reaches the analyser,

I = I_0\cos^2 90^\circ = I_0 \times 0 = 0.

The field, being wholly perpendicular to the transmission axis, has no component along it — so the second filter goes completely black. Slowly rotate one filter and you watch the view smoothly darken from bright at 0^\circ to pitch dark at 90^\circ, brighten again to 180^\circ, and so on — a soft \cos^2 pulsing.

See it: turn the analyser

Below, vertical plane-polarised light arrives at an analyser whose transmission axis you can rotate with the slider. On the left, watch the incoming vertical oscillation and the component that survives along the tilted axis (it shrinks as \cos\theta). In the middle, the patch shows how bright the view actually looks. On the right, the bar tracks the transmitted intensity I = I_0\cos^2\theta. Drag to 90^\circ and everything goes dark — crossed filters.

Only TRANSVERSE waves can be polarised. A longitudinal wave such as sound oscillates back and forth along its travel, so there is no "sideways plane" to restrict — it can never be polarised. If you ever see a wave being polarised, it must be transverse. This is the classic exam question, and it is the historical evidence that light is a transverse wave.
Crossed filters (θ = 90°) block light completely, not just a bit. \cos^2 90^\circ = 0, so the transmitted intensity is exactly zero. Students often guess "half" — but half is what a single filter passes from unpolarised light, which is a different situation.
An ideal polariser passes HALF of unpolarised light — but Malus's law is for POLARISED light. Don't apply I_0\cos^2\theta to the first filter facing an unpolarised beam (there's no single \theta to use). Use \tfrac{1}{2}I_0 for that first step, then Malus's law for every filter after it.

Polarisation by reflection — and why sunglasses beat glare

Light doesn't only get polarised by filters. When light reflects off a shiny non-metal surface — water, wet roads, glass, a car bonnet — it comes away partially polarised, with its oscillation biased into the horizontal plane (parallel to the surface). That horizontally-polarised flash is the blinding glare that hides the fish under the water and the lane markings on a wet road.

The cure is elegant. Polarising sunglasses have their transmission axis set vertical. The glare is horizontal, so it meets the lenses effectively crossed — \theta \approx 90^\circ — and is almost entirely blocked, while the useful, more randomly-polarised light from the rest of the scene still gets through. That's why tilting your head sideways ruins the effect: you rotate the lens axis towards the glare's horizontal plane, \theta drops, and the glare pours back in.

Sunglasses — cut reflected glare off water, snow and roads.
Photography — a rotatable polarising filter on a lens deepens blue skies and kills reflections in shop windows and ponds.
LCD screens — every liquid-crystal display is built from two crossed polarisers with the light steered between them (see below).
Stress analysis — clear plastic models viewed between crossed polarisers show rainbow fringes that map where the material is under strain.
3D cinema — the two eyes' images are projected with different polarisations, and the glasses let each eye see only its own.

Hold a phone screen up to polarising sunglasses and rotate it — at one angle the screen goes black. That's because a liquid-crystal display is a polarisation trick from top to bottom. Behind the pixels sits a backlight and a first polariser; in front sits a second polariser crossed at 90° to it. On their own, crossed filters would let nothing through and every pixel would be dark.

The magic is the liquid crystal sandwiched between them. In its resting state its molecules form a gentle twist that rotates the plane of polarisation by exactly 90^\circ as light passes — so the light arrives lined up with the second filter and the pixel glows bright. Apply a small voltage and the crystal untwists; the light is no longer rotated, meets the crossed filter head-on, and the pixel turns dark. Millions of these tiny \cos^2\theta gates, switching by voltage, paint the whole picture you are reading right now.