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Optics

Malus Law Calculator

Calculate polarized light intensity using I = I₀cos²θ.

Interactive calculator

Malus Law Calculator

Calculate polarized light intensity using Malus's law: I = I₀ cos²θ. Find transmitted intensity, initial intensity, angle, or transmission coefficient.

Try an example

Your result will appear here.

Choose a calculation mode, fill in the known values, and click Calculate.

Quick Guide

  • Choose: transmitted intensity, initial intensity, polarizer angle, or transmission coefficient.
  • At 45°, half the intensity passes.
  • At 90°, complete extinction (ideal polarizer).

Table of Contents

Key Takeaways

  • Malus's law: I = I₀ cos²θ, where θ is the angle between the polarization direction and the analyzer axis.
  • At θ = 0°, 100% of light is transmitted; at θ = 90°, zero light passes (ideal polarizer).
  • At θ = 45°, exactly 50% of the polarized intensity is transmitted.
  • The law has 180° periodicity: cos²θ = cos²(θ + 180°).
  • Unpolarized light through a first polarizer loses 50% intensity before Malus's law applies.

What Is Malus's Law?

Discovered by Étienne-Louis Malus in 1808, Malus’s law describes the intensity of linearly polarized light after passing through an analyzer (second polarizer). The transmitted intensity depends on the square of the cosine of the angle between the polarization direction and the analyzer’s transmission axis.

Formula & Derivation

I=I0cos⁡2θI = I_0 \cos^2\theta

The electric field component transmitted by the analyzer is E = E₀ cos θ. Since intensity is proportional to E², we get I = I₀ cos²θ.

θ=arccos⁡II0\theta = \arccos\sqrt{\frac{I}{I_0}}

Solving for the angle when you know both intensities.

Applications

  • Polarized sunglasses — block glare by filtering horizontally polarized reflected light.
  • LCD displays — liquid crystals rotate polarization to control pixel brightness.
  • Photography — polarizing filters reduce reflections and enhance contrast.
  • Laser systems — variable optical attenuators use rotating polarizers.
  • Stress analysis — photoelasticity uses crossed polarizers to visualize stress in transparent materials.

How to Use

  1. Select what you want to solve for.
  2. Enter known values (intensity in W/m² and angle in degrees).
  3. Click Calculate for results with transmission percentage.

Examples

Sunglasses at 60°

I = 1000 × cos²(60°) = 1000 × 0.25 = 250 W/m² (75% blocked)

Half-intensity angle

θ = arccos(√0.5) = 45° — the “half-power angle”

FAQ

What is Malus's law?›

Malus's law describes how the intensity of linearly polarized light changes when it passes through a polarizing filter (analyzer). The transmitted intensity equals the initial intensity multiplied by cos²θ, where θ is the angle between the light's polarization direction and the filter's transmission axis.

Why does I = I₀/2 for unpolarized light?›

Unpolarized light has electric field oscillations in all directions. A polarizer transmits only the component along its axis. Averaging cos²θ over all angles gives 1/2, so the first polarizer transmits exactly half the unpolarized intensity.

Can I use Malus's law with non-ideal polarizers?›

Real polarizers have some leakage at 90° (they don't achieve perfect extinction). For non-ideal polarizers, the formula becomes I = I₀[T_max·cos²θ + T_min·sin²θ], where T_max and T_min are the principal transmittances.

How is Malus's law used in LCD displays?›

LCD displays use two crossed polarizers with liquid crystals between them. By applying voltage, the crystals rotate the polarization direction, controlling how much light passes through the second polarizer per Malus's law — this controls pixel brightness.

Sources

  1. Hecht, E. – Optics (5th ed.)
  2. HyperPhysics – Polarization
  3. Wikipedia – Malus's Law
Manish Kumar

Author & technical reviewer

Manish Kumar

PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.

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