Malus's law

A polarizer passes the component of the electric field along its transmission axis. For linearly polarized light at an angle $\theta$ to that axis, the transmitted field is $E\cos\theta$, and the transmitted intensity is

which Étienne-Louis Malus found in 1809 by looking at light reflected from a window through a calcite crystal. The law gives 75% at 30°, 50% at 45°, 25% at 60° and nothing at 90°, the crossed position. Unpolarized light, averaged over all angles, is transmitted at half its intensity by an ideal polarizer, and a second polarizer then follows Malus's law with the first one's axis as the reference.

With real polarizers

A real polarizer leaks a small fraction of the blocked polarization, described by its extinction ratio ER, the ratio of transmission along the pass axis to transmission along the blocked axis. The transmitted fraction becomes $\cos^2\theta + \sin^2\theta/\mathrm{ER}$, so a polarizer with an extinction ratio of 10⁴ (40 dB) transmits 10⁻⁴ of the light when crossed, not zero. The angle also has to be right: a crossed pair misaligned by 1° transmits $\sin^2 1°$ = 3.0 × 10⁻⁴, a 35 dB extinction, so an angular error of one degree already limits a 40 dB measurement. Measuring an extinction ratio therefore means finding the true minimum by rotating in small steps.

Uses

Two polarizers with one rotating form a continuously variable attenuator, and a half-wave plate followed by a fixed polarizing beamsplitter does the same without moving the output polarization; the fraction then varies as $\cos^2 2\phi$ with the plate angle $\phi$. A rotating analyzer in front of a detector measures the polarization state of a beam, the basis of polarimetry. The law also explains the familiar demonstration that inserting a third polarizer at 45° between two crossed ones lets light through: each stage passes $\cos^2 45°$ = 1/2, so 1/8 of the original unpolarized intensity emerges.

Measurement

Malus's law is checked by rotating an analyzer in front of a detector and fitting the transmitted power against angle to $A\cos^2(\theta - \theta_0) + B$; the offset $B$ relative to $A$ gives the combined extinction of the source's polarization and the analyzer, and $\theta_0$ locates the axis.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 8.