Malus's law

Malus's law states that when linearly polarised light of intensity I₀ passes through an ideal polariser whose transmission axis makes angle θ with the incident polarisation direction, the transmitted intensity is I = I₀ cos²θ. At θ = 0 the full intensity is transmitted; at θ = 90° (crossed polarisers) the intensity falls to zero; at θ = 45° the intensity is I₀/2. The law is easily derived: the incident field E₀ decomposes into a component E₀ cos θ along the transmission axis (which passes through) and E₀ sin θ perpendicular to it (which is absorbed); the transmitted intensity is proportional to the squared transmitted amplitude, giving cos²θ.

Étienne-Louis Malus discovered the law in 1809 while accidentally looking at reflected sunlight through a calcite crystal from his Paris window: he noticed that as he rotated the crystal, the reflected light alternately appeared and disappeared. The sunlight had been linearly polarised by the glancing reflection off a building across the street, and the calcite was acting as a linear-polarisation analyser. The cos² modulation sealed the geometric regularity. Malus's law is the quantitative basis of every polarisation measurement: an ellipsometer scans θ and fits I(θ) to extract the polarisation state of an unknown beam. It also underlies the operation of quantum-mechanical polarisation measurements — the photon version gives the Born-rule probability of a single-photon transmission, which is the basis of every Bell-inequality test in quantum optics.