Malus's Law — Polarized Light Calculator
Calculate transmitted polarized light intensity through a filter using Malus's Law: I = I0*cos^2(theta).
Solve for angle, initial, or output intensity.
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:
| Angle θ | cos²(θ) | Transmission |
|---|---|---|
| 0° | 1.000 | 100% (full transmission) |
| 30° | 0.750 | 75% |
| 45° | 0.500 | 50% |
| 60° | 0.250 | 25% |
| 90° | 0.000 | 0% (complete extinction) |
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 — visible in polarized light (photoelasticity)
- 3D cinema: Left and right eye images use opposite circular polarization
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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