Malus's Law Polarized Light Calculator

When polarized light passes through a second polarizer (analyzer) at angle θ to the polarization axis, the transmitted intensity follows Malus’s Law:

I = I₀ cos²(θ)

Where:

• I = Transmitted intensity (W/m² or any relative unit)
• I₀ = Incident (initial) intensity of the polarized light
• θ = Angle between the polarization direction and the analyzer axis

Key values:

Natural light and polarization:

Unpolarized natural light can be thought of as containing all polarization angles equally. When it passes through a linear polarizer, half the intensity is transmitted: I_after_first_polarizer = I₀/2

Then Malus’s law applies for any subsequent polarizers.

Applications:

• Sunglasses: Polarized lenses block horizontally polarized light from road glare and water surfaces
• LCD screens: Two crossed polarizers with liquid crystal molecules that rotate polarization
• Photography: Polarizing filters reduce glare and reflections
• Stress analysis: transparent materials under stress rotate polarization, which shows up as coloured fringes in polarized light (photoelasticity)
• 3D cinema: Left and right eye images use opposite circular polarization

The three-polarizer trick

Here is the result that makes people stop and stare. Cross two polarizers at 90° and nothing gets through, which the formula gives you directly since cos²(90°) = 0. Now slide a third polarizer between them at 45°. Light comes back.

Work it through. The first filter passes half the unpolarized light. The middle one is 45° from it, so cos²(45°) = 0.5 and half survives. The last one is another 45° on, so half again. You end up with 12.5% of the original where you had exactly zero, purely by adding another filter.

The reason it is not a paradox: a polarizer does not filter light so much as re-orient what passes. Whatever emerges from the middle filter is polarized at 45°, and 45° is only 45° away from the final axis instead of the full 90°. Insert the middle filter at 0° or 90° instead and the screen goes black again, which the formula predicts and which is worth checking yourself with three cheap polarizing filters.

A practical note on sunglasses

Glare off a road or a lake is mostly horizontally polarized, which is why polarized lenses are cut to pass only the vertical component. It also explains an annoyance: tilt your head sideways while wearing them and the glare comes back, because you have rotated θ away from the angle the lens was designed for. The same effect makes some car dashboards and phone screens go dark through polarized lenses, since those displays emit polarized light of their own.