Malus Law Calculator

With this Malus law calculator, you can find the intensity of light that passes through the polarizer, taking into account the direction of its initial polarization.

Transmitted intensity750 W/m²I₀cos²30° = 75% of the incident

Transmitted fraction0.75note the SQUARE — intensity goes as cos², while the amplitude goes as cos

Amplitude fraction0.866025

At 45°50%exactly half, since cos²45° = ½

At 90°0%crossed polarisers block everything — and inserting a THIRD polariser between them at 45° lets light through again, which is genuinely counter-intuitive

Crossed pair with a 45° polariser inserted25% of the incidentcos²45° twice — a quarter gets through where none did before, because each polariser reprojects the polarisation rather than merely filtering it

Same total rotation over 2 polarisers75% transmitted30° per stage — spreading the rotation over more stages passes MORE light, because cos² of a small angle is close to 1

The formula

I = I₀cos²θ (Malus); A = −log₁₀(T)

Polarisers reproject, they do not just filter

Malus's law gives the transmitted intensity as I₀cos²θ. The square matters: at 45° exactly half the intensity passes, not 71% — that figure is the amplitude, and intensity goes as amplitude squared.

The three-polariser result is the one worth sitting with. Two crossed polarisers pass nothing. Insert a third between them at 45° and a quarter of the light gets through. Nothing was added — so the middle polariser cannot be merely blocking. What it does is reproject the polarisation onto a new axis, and light polarised at 45° to the final filter can then pass it. This is a genuinely quantum-mechanical behaviour showing up in a classroom demonstration.

The same reprojection is why spreading a rotation over more polarisers transmits more light: many small cos² factors near 1 multiply to more than one large one. With enough stages you can rotate polarisation through 90° with almost no loss.

Absorbance is a base-ten logarithmic scale, so absorbance 1 transmits 10%, absorbance 2 transmits 1%, and the absorbances of stacked filters add rather than multiply — which is exactly why the scale is defined that way.