Formula and Method for Malus's Law
Malus's Law, discovered in 1808 by French engineer Étienne-Louis Malus, describes how the intensity of polarized light changes as it passes through a polarizing filter (an analyzer). When light that is already linearly polarized reaches an analyzer whose transmission axis is rotated by an angle θ relative to the light's polarization direction, the transmitted intensity is I = I₁cos²θ. This calculator also handles the common two-polarizer setup, where unpolarized light first passes through a polarizer — cutting its intensity to I₁ = I₀/2 — and then through a second polarizer, the analyzer, held at angle θ.
How the calculation works
Enter the incident intensity I₀, choose whether that light is already polarized or unpolarized, and set the angle θ between the analyzer's transmission axis and the light's polarization direction (in degrees or radians). If the source is unpolarized, the calculator first halves the intensity to get I₁ = I₀/2, the intensity leaving the first polarizer; if the source is already polarized, I₁ simply equals I₀. It then applies Malus's Law, I = I₁ × cos²θ, to find the intensity leaving the analyzer, and also reports the overall fraction of the original I₀ that survives (I/I₀) and the attenuation in decibels, −10log₁₀(I/I₀).
Common mistakes
- Skipping the halving step: Malus's Law itself (I = I₁cos²θ) only applies to light that is already polarized when it reaches the analyzer — for unpolarized light passing through a first polarizer, apply the I₀/2 factor before applying cos²θ.
- Mixing angle units: cos²θ is very sensitive to whether θ is in degrees or radians; entering 30 as radians (≈1719°) instead of degrees gives a drastically wrong answer.
- Confusing axis angle with angle of incidence: θ is the angle between the light's polarization direction and the analyzer's transmission axis, not the angle at which the light beam strikes the filter.
Real-world applications
- Polarizing sunglasses and camera filters use angled polarizers to cut glare from light reflected off water, glass, and roads.
- LCD screens sandwich liquid crystals between two polarizers; rotating the light's polarization pixel by pixel controls how much light Malus's Law lets through.
- Photography and optical instruments use polarizing filters to dim light precisely without shifting its color, unlike absorptive neutral-density filters.
- Photoelasticity and stress analysis use crossed polarizers to reveal stress patterns in transparent plastics and glass.