Polarisation of Light: Malus's Law and the Vanishing Beam

Light oscillates sideways, not along its path

Light is a transverse electromagnetic wave: its electric field oscillates perpendicular to the direction the light is travelling, not along it. Because there are infinitely many directions perpendicular to a given line of travel, that electric field can point anywhere in the plane transverse to the beam. Ordinary sunlight or an incandescent bulb emits a chaotic, rapidly changing mix of every possible orientation at once — unpolarised light. Light whose electric field oscillates along one fixed direction (or traces out a fixed, repeating pattern such as a circle) is called polarised.

A polarising filter is built from long-chain molecules aligned in one direction that absorb the component of the electric field parallel to the chains and let the perpendicular component pass through largely unaffected. Send unpolarised light through one ideal polariser and, on average, exactly half its intensity survives, all of it now oscillating along the polariser's single transmission axis — every direction has been reduced to one.

Malus's law: the second polariser does the interesting part

Once light is polarised, a second polariser (traditionally called the analyser) whose transmission axis sits at an angle θ to the first no longer simply blocks or passes it wholesale. Only the component of the electric field that projects onto the analyser's axis gets through, and since intensity depends on the square of the field amplitude, the transmitted intensity follows a cosine-squared law discovered by Étienne-Louis Malus in 1808:

Malus's law:

  I(theta) = I_0 * cos^2(theta)

  theta = 0      ->  I = I_0        (axes aligned, everything passes)
  theta = 45 deg ->  I = I_0 / 2
  theta = 90 deg ->  I = 0           (crossed polarisers, no light at all)

Two polarisers with their axes at 90° to each other — "crossed polarisers" — block essentially all light, which is easy to demonstrate and often shown as a party trick, but it is also the operating principle behind LCD screens: each pixel sits between two crossed polarisers, and a liquid-crystal layer in between twists the light's polarisation by a controllable amount, letting the display dial the transmitted brightness continuously from zero to maximum with no moving parts.

Where polarisation shows up in nature

Reflection and scattering both preferentially polarise light. Light reflecting off a non-metallic surface such as water or glass becomes partially polarised parallel to the surface, and at one special angle — Brewster's angle, where the reflected and refracted rays would be perpendicular to each other — the reflected light becomes completely polarised. Sky light is polarised too, because sunlight scattering off air molecules (Rayleigh scattering) produces a partial polarisation pattern across the sky that is strongest at 90° from the sun; several insect and bird species can see this pattern directly and use it to navigate even under overcast skies, something the human eye cannot do unaided.

Circular and elliptical polarisation

Linear polarisation, where the field oscillates back and forth along one fixed line, is only the simplest case. Combine two linearly polarised waves of equal amplitude, oriented perpendicular to each other, with a 90° phase difference between them, and the resulting electric field vector traces out a circle as the wave propagates — circular polarisation. Any other phase difference or amplitude ratio traces out an ellipse — elliptical polarisation, the general case, with linear and circular polarisation as its two special limits. Materials called wave plates, made from birefringent crystals that slow down light differently depending on its polarisation direction, are the standard tool for converting linear polarisation into circular and back, and are essential components in laser optics, 3D-cinema glasses and many quantum optics experiments.

Frequently asked questions

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