Malus’s Law Calculator (light through one or more polarizers, I = I₀ cos²θ)

Understanding the Malus's law calculator — concept and principle

This calculator follows light through one or more polarizers using Malus's law, starting from unpolarized or polarized light. It gives the intensity after the first and last polarizer, the share transmitted and the final polarization direction. It is used in optics labs, photography and for understanding polarized sunglasses and LCD screens.

The first sheet halves unpolarized light; each 45° turn after that keeps cos² 45° = half again.

How it is calculated

I = I₀ cos² θ

θ is the angle between the light's polarization and the polarizer's transmission axis.
Unpolarized light loses exactly half at the first polarizer: I₁ = I₀ ÷ 2.
Each further polarizer multiplies by cos² of the angle between its axis and the previous one.
Axes are listed in order, in degrees, separated by commas; up to 20 polarizers.

The principle

Light is a transverse wave whose electric field points in some direction across the beam. An ideal polarizer passes only the field component along its axis, which is the field times cos θ; intensity goes with the field squared, giving cos² θ. Unpolarized light contains all directions equally, and the average of cos² θ over all angles is ½. After each polarizer the light is polarized along that polarizer's axis, so the next angle is measured from there.

Worked example

Default inputs: intensity 100, unpolarized, axes 0°, 45°, 90°.

After the first polarizer (0°): 100 ÷ 2 = 50.
Second polarizer at 45°: 50 × cos² 45° = 50 × 0.5 = 25.
Third polarizer at 90°, 45° from the second: 25 × 0.5 = 12.5.
Share transmitted = 12.5%; there are 3 polarizers; final polarization direction 90°.

Each point is the calculator re-run with only “Intensity before the first polarizer” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Intensity before the first polarizer	Intensity after the last polarizer
25	3.125
50	6.25
75	9.375
100	12.5
125	15.63
150	18.75
200	25

How to read the result

Two crossed polarizers (0° and 90°) block all light, yet inserting a third at 45° between them lets 12.5% through, a famous demonstration. Splitting a rotation into many small steps passes more light: in the limit of many polarizers, almost all of the polarized light can be rotated by 90°. Real polarizers also absorb some of the passing component.

Common mistakes

Using cos θ instead of cos² θ.
Measuring every angle from the first polarizer instead of from the previous one.
Forgetting the factor ½ for unpolarized light.

Keep in mind: The calculation assumes ideal polarizers. Real filters transmit less than 100% along their axis and leak a little when crossed.

Frequently asked questions

Why does unpolarized light lose half?

The average of cos² θ over all directions is ½.

How do polarized sunglasses reduce glare?

Glare reflected from water or roads is partly polarized horizontally, and the lenses block that direction.

Can a third polarizer increase transmission?

Yes, between two crossed polarizers, by rotating the polarization in two steps.

What is the final polarization direction?

The axis of the last polarizer.

Sources: OpenStax, University Physics Volume 3 (2016), section 1.7 "Polarization"

Checked 9 Nov 2026. OpenStax, University Physics Volume 3 (2016), §1.7 “Polarization”: Malus’s law I = I₀ cos²θ, θ = angle between the polarization direction and the filter axis; a polarizing filter halves the intensity of unpolarized light.

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