Malus's Law Calculator

It gives the intensity of polarized light after passing through a polarizer, equal to the incident intensity times the square of the cosine of the angle.

At a 45 degree angle exactly half the polarized light is transmitted, because the cosine squared of 45 degrees is one half.

The field that survives scales with the cosine, and intensity scales with the square of the field, so the transmitted intensity follows cosine squared.

This calculator assumes the incoming light is already polarized. Unpolarized light first loses half its intensity passing the first polarizer, so through a second polarizer the transmitted intensity is I = 0.5·I_unpol·cos²θ, where θ is the angle between the two polarizer axes. To model that case, enter half of your unpolarized intensity as the incident value.

Malus's law describes an ideal filter. A real polarizer has two transmittance factors: k1, the fraction passed when the light is aligned with the axis (typically 0.8 to 0.95 for sheet polarizers, limited by absorption and surface reflection), and k2, the small fraction that leaks through when crossed (often 0.0001 or less). For already-polarized light the transmission is closer to I = I₀·[(k1 - k2)·cos²θ + k2], so even at θ = 0° only k1 gets through and a crossed pair never reaches perfect darkness. For most classroom problems the ideal I = I₀·cos²θ is a good approximation.

Two polarizers crossed at 90° block all light because cos²90° = 0. Inserting a third polarizer at 45° between them re-projects the polarization at each stage: light passing the middle filter is dimmed by cos²45° = 0.5, then by cos²45° = 0.5 again at the last filter, so I = I₀ × 0.5 × 0.5 = 0.25·I₀ - a quarter of the light gets through a pair that was blocking everything. It follows directly from applying Malus's law at each successive filter.