Last updated: 2026-06-16

Malus's law

Malus's law, discovered by Étienne-Louis Malus in 1809, describes how the intensity of linearly polarized light changes when it passes through a linear polarizer. If the light has intensity $I_0$ and the angle between its polarization direction and the polarizer's transmission axis is $\theta$, the transmitted intensity is

$I = I_0 \cos^2\theta.$

The transmittance — the fraction of intensity that passes through — is therefore $T = \cos^2\theta$, independent of $I_0$.

Physical mechanism

A linear polarizer transmits only the component of the electric field $E$ that lies along its transmission axis. If the incoming field has amplitude $E_0$ and makes angle $\theta$ with the transmission axis, the transmitted amplitude is $E_0 \cos\theta$. Because intensity is proportional to the square of the amplitude, the transmitted intensity is

$I = I_0 \cos^2\theta.$

The $\cos^2$ dependence is the direct consequence of projecting the field onto one axis and then squaring.

Formula reference

Worked example

A beam of linearly polarized light has intensity $I_0 = 100\ \text{W m}^{-2}$ and its polarization axis makes an angle $\theta = 30°$ with the transmission axis of a linear polarizer.

Three-quarters of the incident intensity passes through the polarizer.

Unpolarized light and the first polarizer

Malus's law applies to light that is already linearly polarized. When unpolarized light passes through a polarizer for the first time, the transmitted intensity is half the incident intensity regardless of the orientation of the polarizer:

$I = \frac{I_0}{2}.$

This factor of one-half arises because the electric field directions in unpolarized light are uniformly distributed, and the average of $\cos^2\theta$ over all angles is one-half. The transmitted beam is then linearly polarized and Malus's law governs any further polarizers downstream.

Crossed polarizers

When two polarizers are oriented at $\theta = 90°$ to each other, the transmittance drops to $\cos^2(90°) = 0$ and virtually no light passes. Such a pair is called crossed polarizers. A third polarizer inserted between them at an intermediate angle (for example 45°) restores some transmission, because it partially rotates the polarization direction before the second filter.

Applications

Malus's law underpins several practical technologies.

LCD screens place a liquid-crystal layer between two crossed polarizers. Applying a voltage to the liquid crystal rotates the polarization direction, allowing varying amounts of light through each pixel and producing the full range of brightnesses needed for an image.

Photography filters use circular polarizing filters to reduce reflections from water, glass, and foliage, because reflected light is partially polarized. Rotating the filter changes $\theta$ and controls how much of the reflected glare is blocked.

Stress analysis (photoelasticity) passes polarized light through a transparent model under mechanical load. Stresses in the material rotate the polarization locally, creating colored fringes between crossed polarizers that reveal the stress distribution inside the part.

Limits of the model

Malus's law assumes an ideal linear polarizer with perfect extinction at 90° and zero absorption at 0°. Real polarizers have a finite extinction ratio (the ratio of maximum to minimum transmission) and some absorption even in the pass direction. The law also assumes coherent, collimated, monochromatic light; in practice, broadband or diffuse light behaves the same way on average, but the angle $\theta$ must be defined carefully for extended or non-collimated sources.