Malus's Law Calculator — Polarised Light Intensity
When polarised light passes through a second polariser (the analyser), its transmitted intensity follows Malus's law: I = I₀ cos²(θ). Enter the initial intensity and the angle between the polariser axes to find how much light gets through.
W/m²
°
Intensity after passing through the analyser polariser
- 1
Convert angle to radians
θ = 45° × π ÷ 180 = 0.785398 - 2
Cosine of θ
cos(45°) = 0.707107 - 3
cos²(θ)
0.707107² = 0.5 - 4
Transmitted intensity I = I₀ × cos²(θ)
100 × 0.5 = 50
How does this calculator work?
Malus's law: I = I₀ cos²(θ), where I₀ is the intensity of polarised light striking the analyser and θ is the angle between the polarisation direction and the analyser axis. At 0° all light passes; at 90° none does; at 45° exactly half is transmitted. The unit of intensity is preserved (W/m² in, W/m² out).
Formula
How this is calculated
Malus's law (Étienne-Louis Malus, 1808) describes how a linear polariser reduces the intensity of already-polarised light. Linearly polarised light with intensity I₀ hitting a second polariser whose transmission axis is at angle θ to the first emerges with intensity I = I₀ cos²(θ). At θ = 0° the polarisers are aligned and all light passes; at θ = 90° they are crossed and no light passes; at θ = 45° exactly half the intensity is transmitted (cos² 45° = 0.5).
The physical reason is that only the component of the electric-field amplitude parallel to the analyser axis passes through, reducing the amplitude by cos(θ). Because intensity is proportional to amplitude squared, the transmitted intensity drops by cos²(θ).
The law applies to ideal linear polarisers and perfectly polarised monochromatic light. Real polarisers have extinction ratios less than infinite, and if the incoming light is only partially polarised, the effective I₀ is the polarised component only.
Frequently asked questions
At θ = 45°, cos²(45°) = 0.5, so exactly 50% of the incident intensity is transmitted. This is why 45° is used as a convenient calibration point when aligning polarisers.
At θ = 90° the polarisers are "crossed" — cos²(90°) = 0 — so in theory no light passes. Real polarisers have a small leakage (finite extinction ratio), so a tiny amount may still be visible.
No — Malus's law applies to linearly polarised light. For circular or elliptical polarisation the transmitted intensity depends on the polarisation state and requires Stokes parameter analysis or Jones calculus.
Also known as
TG we-Calculate Editorial Team. (2026). Malus's Law Calculator — Polarised Light Intensity [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/malus-law-calculator
TG we-Calculate Editorial Team. "Malus's Law Calculator — Polarised Light Intensity." TG we-Calculate. 2026. https://we-calculate.com/calculator/malus-law-calculator.
TG we-Calculate Editorial Team, "Malus's Law Calculator — Polarised Light Intensity," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/malus-law-calculator
@misc{wecalculate_malus_law_calculator, title = {Malus's Law Calculator — Polarised Light Intensity}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/malus-law-calculator}}, year = {2026}, note = {TG we-Calculate} }
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