Result

Malus's law

I = I_{0} cos^{2} θ

Statement

The intensity transmitted through a polarizer, given the angle θ between the incoming light's polarization axis and the polarizer's axis.

Symbols and units

$I$	transmitted intensity	W/m²
$I_{0}$	incident intensity	W/m²
$θ$	angle between polarization axes	degrees

When it applies

Ideal, lossless linear polarizers.

Don't use when

Assumes ideal, lossless linear polarizers — a real polarizer absorbs some of the transmitted component too, reducing intensity below this ideal prediction.

Derivation

Only the component of the electric field aligned with the polarizer's axis passes through; intensity depends on the square of that component's amplitude:

I = I_{0} cos^{2} θ

Worked examples

Intensity at 45°

I = I_{0} cos^{2} (4 5^{\circ}) = \frac{1}{2} I_{0}

Rotating a polarizer a quarter of the way to fully crossed already cuts the intensity in half — the falloff is far from proportional to the angle.

Watch out for

Two polarizers at 90° (crossed) transmit exactly zero light, since cos²(90°) = 0. At 45°, only half the intensity passes — a nonlinear falloff, not a proportional one, so intuition based on the rotation angle alone is misleading.