Polarising filters and Malus's law

A polarising filter (Polaroid) lets through only the component of $\vec{E}$ along its characteristic axis, the transmission axis, absorbing the rest. Unpolarised light incident on a single filter emerges linearly polarised along the filter’s axis, with halved intensity (the other half is absorbed).

Malus's law

If a wave already linearly polarised with intensity $I_0$ hits a second filter whose axis is rotated by $\theta$ with respect to the wave’s polarisation direction, the transmitted intensity is $\ev{I = I_0\,\cos^2\theta}$

Why $\cos^2$? After the first filter, the electric field has amplitude $E_0$ along the filter’s axis. Through the second filter only the component parallel to its axis passes, i.e. $E_0\cos\theta$. The intensity, proportional to the square of the electric field, becomes $I_0\cos^2\theta$.

Example — Two crossed Polaroids

Unpolarised light of intensity $I_\text{nat}$ passes through two filters; the second is rotated by $\theta = 60^\circ$ with respect to the first. The first lets through $I_1 = I_\text{nat}/2$. By Malus’s law: $I_2 = I_1\cos^2 60^\circ = \frac{I_\text{nat}}{2}\cdot\frac{1}{4} = \frac{I_\text{nat}}{8}$ One eighth of the initial light survives.

The trick of the third Polaroid

Two Polaroids crossed at $90^\circ$ block all the light ($\cos^2 90^\circ = 0$). Inserting between the two a third Polaroid rotated by $45^\circ$, surprisingly the light gets through again. The calculation: after the first, $I/2$; after the intermediate one, $\tfrac{I}{2}\cos^2 45^\circ = I/4$; after the second, $\tfrac{I}{4}\cos^2 45^\circ = I/8$. The intermediate Polaroid re-projects the polarisation onto its axis, offering the second filter a new target. It is one of the few elementary experiments in which “adding an obstacle helps”, and it will reappear, in a quantum guise, in the theory of measurement on photons.

Example — Reflections on water and polarised sunglasses

Light reflected off the surface of a lake is partially polarised in the horizontal direction (Brewster’s angle gives the maximum polarisation). Polarised sunglasses have a vertical transmission axis: they block the horizontal component, drastically reducing glare without darkening the landscape too much (Bloomfield 2016).

The polarisation of EM waves is the natural bridge to the quantum polarising filter: the same $\cos^2\theta$ reappears in a completely different context, the theory of measurement on single photons.

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Topics: Electromagnetic waves Concepts: Polarisation and Malus’s law · Intensity of a wave

Related exercises: Problem — Crossed polarisers · Problem — Crossed polarisers with one in between · Problem — WiFi intensity at 10 metres