Malus's law

Remarks

The ideal polarizer keeps the component of the electric field along its axis and removes the orthogonal component. The kept amplitude is $E_0 \cos\theta$, and the intensity is proportional to the square of the amplitude. At $\theta = 0$ all light passes; at $\theta = 90^\circ$ none passes, which is the case of crossed polarizers. With three polarizers at $0^\circ$, $45^\circ$ and $90^\circ$, light polarized at $0^\circ$ leaves with intensity $I_0 \cdot \tfrac12 \cdot \tfrac12 = I_0/4$, although the first and last are crossed. The hypothesis of linear polarization matters: for unpolarized light, the average of $\cos^2\theta$ over all angles gives $I_0/2$ for every orientation.

Going deeper

For a student

Malus's law is the simplest case of Mueller calculus and of Jones calculus: it is the projection of a Jones vector onto the axis of the polarizer. It is a quick way to measure the angle of linear polarization and is the basis of polarimetry. The same cosine-squared rule appears in quantum mechanics as the probability that a single photon passes the polarizer.