Malus's Law
Calculate transmitted light intensity through a polarizer using Malus's Law I = I₀ cos²θ. Free online optics and polarization calculator with interactive charts.
About This Calculator
Malus's Law (also called Malus's Law of Polarization) describes the intensity of linearly polarized light after it passes through an ideal polarizer. Named after Étienne-Louis Malus who discovered it in 1808, the law states that the transmitted intensity I equals the initial intensity I₀ multiplied by the square of the cosine of the angle θ between the light's polarization direction and the polarizer's transmission axis: I = I₀ cos²θ.
This calculator is designed for physics students, optics researchers, photographers, and anyone working with polarized light. Enter the incident intensity and the angle to instantly compute the transmitted intensity, cos(θ), cos²(θ), and the transmission percentage. The interactive chart shows how transmitted intensity varies across the full 0°–90° range, helping you visualize the cosine-squared relationship.
Formula
I = I₀ × cos²(θ) where I₀ = incident intensity, θ = angle between polarization direction and polarizer axis (in degrees), I = transmitted intensity.
The angle θ ranges from 0° (polarization aligned with polarizer axis — maximum transmission) to 90° (polarization perpendicular to polarizer axis — minimum transmission).
Regional Notes
Malus's Law is a universal physics principle and applies identically worldwide. The calculator accepts any consistent intensity units (W/m², mW/cm², lux, or arbitrary units). Key applications include:
- Global: Polarized sunglasses, camera filters, LCD displays, optical instruments
- Education: Standard topic in high school and university optics curricula across all countries
- Research: Used in spectroscopy, ellipsometry, and materials characterization worldwide
Frequently Asked Questions
What is Malus's Law?
Malus's Law states that the intensity of linearly polarized light transmitted through an ideal polarizer is given by I = I₀ cos²θ, where I₀ is the incident intensity and θ is the angle between the light's polarization direction and the polarizer's transmission axis.
How do I use the Malus's Law calculator?
Enter the initial intensity of polarized light and the angle between the polarization direction and the polarizer's axis. Click Calculate to see the transmitted intensity, transmission percentage, cosine values, and an interactive chart showing how intensity varies with angle.
What are the units for intensity in Malus's Law?
Intensity (irradiance) is typically measured in watts per square meter (W/m²). However, the calculator works with any consistent unit since the formula uses ratios — you can use arbitrary units and the transmission percentage will be the same.
What happens when the angle is 0° or 90°?
At 0°, cos(0) = 1, so cos²(0) = 1 and I = I₀ (full transmission). At 90°, cos(90) = 0, so cos²(90) = 0 and I = 0 (no light passes through). At 45°, cos²(45) = 0.5, so exactly half the light is transmitted.
Is Malus's Law only for ideal polarizers?
The formula I = I₀ cos²θ assumes an ideal polarizer with 100% transmission when aligned and 0% when crossed. Real polarizers have some absorption and leakage, but Malus's Law accurately describes the angular dependence of transmission for typical sheet polarizers used in sunglasses, photography filters, and laboratory optics.
Can this calculator be used for unpolarized light?
No, Malus's Law applies only to linearly polarized light. For unpolarized light passing through a single polarizer, the transmitted intensity is always I₀/2 (50%), regardless of the polarizer's orientation. To get polarized light from unpolarized light, you need a polarizer first, then Malus's Law applies to subsequent polarizers.
Where is Malus's Law used in real life?
Malus's Law is essential for designing polarized sunglasses (which block glare from horizontal surfaces), camera polarizing filters (adjustable by rotation), liquid crystal displays (LCDs), optical isolators in laser systems, and scientific instruments that analyze light polarization in materials research and astronomy.
How do polarized sunglasses use Malus's Law?
Light reflecting off horizontal surfaces like water, snow, or roads becomes partially horizontally polarized. Polarized sunglasses have a vertical transmission axis, so at 90° to the horizontal polarization, cos²(90°) = 0 and glare is nearly eliminated. Rotating the glasses would let more glare through, exactly as Malus's Law predicts.