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Malus's Law Calculator

Use Malus's law, I = I₀ cos²θ, to find how much polarized or unpolarized light passes through a polarizer and analyser at any angle, with a table of intensity from 0° to 90°.

Incident intensity (I₀)

Angle between the polarization and the polarizer axis (θ)

Incident light

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How it works

I = I₀ cos²θ · unpolarized light through a first polarizer: I₁ = I₀ ÷ 2, then through an analyser at θ: I = (I₀ ÷ 2) cos²θ

Malus's law (Étienne-Louis Malus, 1809) says that an ideal polarizer passes only the component of the electric field along its transmission axis, so the field drops by cos θ and the intensity, which goes as the field squared, by cos²θ. Crossed polarizers (θ = 90°) block everything. Unpolarized light, such as sunlight or a bulb, loses half its intensity at the first polarizer whatever its orientation. Real polarizers also absorb a few percent even along their axis.

Worked example

Polarized light of 100 W/m² meets a polarizer turned 30° from its polarization direction. cos²30° = 0.7500, so I = 100 × 0.7500 = 75.00 W/m² passes through (75.00%) and 25.00 W/m² is blocked. If the same 100 W/m² were unpolarized, the first polarizer would pass 50 W/m² and an analyser at 30° would pass 37.50 W/m².

Sources: HyperPhysics — Polarization and Malus's law · Wikipedia — Polarizer (Malus's law)

Frequently asked questions

What is Malus's law?

It gives the intensity of polarized light after a polarizer: I = I₀ cos²θ, where θ is the angle between the light’s polarization direction and the polarizer’s transmission axis.

What fraction of light passes two polarizers at 45°?

cos²45° = 0.5. Starting from unpolarized light, the first polarizer passes 50% and the second passes half of that, so 25% of the original intensity gets through.

Why does a third polarizer between crossed polarizers let light through?

With polarizers at 0° and 90° nothing passes. Inserting one at 45° between them gives (I₀/2) × cos²45° × cos²45° = I₀/8, so 12.5% of unpolarized light now gets through.

Does Malus's law work for unpolarized light?

Not directly. Unpolarized light has no single polarization angle, so an ideal polarizer always passes half of it. Apply cos²θ only from the second polarizer on.

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This calculator is for educational purposes. Double-check important results before acting on them.