Malus's Law Calculator
Enter the initial light intensity and the angle between the polarizer axis and the polarization plane to compute the transmitted intensity using Malus's law (I = I₀ cos²θ). Switch solve modes to work backward from a known transmitted intensity to find the required angle or source intensity. The chart shows how output intensity varies as the angle sweeps from 0° to 90°.
Formula
Worked example
Incident intensity I₀ = 100 W/m², polarizer angle θ = 45°. cos²(45°) = 0.5. Transmitted: I = 100 × 0.5 = 50 W/m². Reverse: to find the angle when I = 25 W/m² and I₀ = 100 W/m²: θ = arccos(√0.25) = arccos(0.5) = 60°.
What is Malus's law?
Malus's law describes how the intensity of linearly polarized light changes after it passes through a polarizing filter. Discovered by Etienne-Louis Malus 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 polarization axis of the incoming light and the transmission axis of the polarizer: I = I₀ cos²(θ). When θ = 0° the two axes are aligned and all light passes through. When θ = 90° the axes are perpendicular (crossed polarizers) and theoretically no light is transmitted. At 45° exactly half the intensity is passed.
How to use this calculator
Select the quantity you want to solve for using the 'Solve for' menu. In forward mode (transmitted intensity), enter the initial irradiance in W/m² and the angle in degrees. Toggle 'Unpolarized source' if the incoming beam has not yet been polarized - the first ideal polarizer will halve the intensity before Malus's law is applied. To find the angle needed to reach a target intensity, switch to 'Angle' mode and enter both I₀ and the desired output I. To find the source intensity implied by a measurement, choose 'Initial intensity' mode. The chart (in forward mode) shows how transmitted intensity varies across the full 0° to 90° sweep for the current I₀.
Unpolarized vs. polarized light
Natural light sources such as the sun, incandescent bulbs, and LEDs emit unpolarized light, meaning the electric field oscillates in all transverse directions equally. When unpolarized light passes through an ideal linear polarizer it emerges with the polarization axis of the filter, and the transmitted intensity is I₀/2, regardless of the filter's orientation. A second polarizer (the analyzer) placed at angle θ to the first then obeys Malus's law: I = (I₀/2) cos²(θ). Many practical applications, including polarized sunglasses and LCD displays, use this two-polarizer arrangement.
Real-world applications
Polarizing filters control glare in photography and sunglasses by blocking horizontally polarized reflections from water and roads. LCD screens sandwich a liquid-crystal layer between two crossed polarizers - applying voltage rotates the crystal's polarization to let light through. Brewster angle optics exploit the fact that reflected light is partially polarized at a specific angle, enabling low-loss transmission in lasers. Polarimetry measures the rotation of polarization by optically active substances such as sugars, used in food quality testing. Stress analysis in engineering (photoelasticity) uses crossed polarizers to visualize stress concentrations in transparent materials.
Malus's law: key angle reference values
| Angle θ (°) | cos²(θ) | Transmission (%) |
|---|---|---|
| 0 | 1.0000 | 100.00 |
| 15 | 0.9330 | 93.30 |
| 30 | 0.7500 | 75.00 |
| 45 | 0.5000 | 50.00 |
| 60 | 0.2500 | 25.00 |
| 75 | 0.0670 | 6.70 |
| 90 | 0.0000 | 0.00 |
Transmission factor cos²(θ) and percentage for common polarizer angles.
Frequently asked questions
What is Malus's law formula?
Malus's law states that I = I₀ cos²(θ), where I is the transmitted intensity, I₀ is the initial intensity of the polarized light, and θ is the angle between the polarization direction of the incident light and the transmission axis of the polarizer. At θ = 0° all light is transmitted; at θ = 90° no light is transmitted.
What happens at 45 degrees?
At θ = 45°, cos²(45°) = 0.5, so exactly half of the incident polarized intensity is transmitted. This is a common result used to split a beam into two equal-intensity paths in optical setups.
Does the law apply to unpolarized light?
Malus's law applies to already-polarized light. If the incident light is unpolarized, the first polarizer transmits I₀/2 regardless of its orientation, and then any subsequent polarizer (analyzer) follows Malus's law at the angle between the two polarizer axes.
Why do two crossed polarizers block all light?
At θ = 90° (crossed polarizers), cos²(90°) = 0, so the theoretical transmitted intensity is zero. In practice, a tiny amount leaks through due to imperfections in the polarizer material and edge diffraction, but a quality linear polarizer pair at 90° achieves extinction ratios of 1000:1 or better.
How do I stack multiple polarizers?
Apply Malus's law successively. If polarizer 1 is at angle θ₁ to the source polarization and polarizer 2 is at angle θ₂ relative to polarizer 1, the final intensity is I = I₀ cos²(θ₁) cos²(θ₂). Adding a polarizer at an intermediate angle between two crossed polarizers will actually let some light through - a counterintuitive result that follows directly from the formula.