50
0.5
50
%
0.5
50
W/m²
0.785398
rad
50
0.5
50
%
0.5
50
W/m²
0.785398
rad
The Malus's Law Calculator solves the fundamental polarization equation I = I₀cos²θ for any of its three variables: transmitted intensity, incident intensity, or the angle between the polarizer and analyzer axes. It also computes the transmission percentage, extinction ratio, and absorbed intensity for complete analysis of polarizer performance.
Discovered by Etienne-Louis Malus in 1809 while observing sunlight reflected from a window through a calcite crystal, this law quantifies how the intensity of linearly polarized light depends on the relative orientation of a polarizing element. It is the most frequently used equation in polarimetry and underpins technologies from LCD screens to optical communications to glare-reducing sunglasses.
When linearly polarized light of intensity I₀ encounters a linear polarizer (analyzer), only the component of the electric field parallel to the analyzer's transmission axis passes through. Since intensity is proportional to the square of the electric field amplitude:
$$I = I_0\cos^2\theta$$
where θ is the angle between the incoming polarization direction and the analyzer's transmission axis.
Special cases:
To find the angle from measured intensities:
$$\theta = \arccos\sqrt{\frac{I}{I_0}}$$
To find the original intensity:
$$I_0 = \frac{I}{\cos^2\theta}$$
The extinction ratio I/I₀ = cos²θ characterizes the polarizer system's ability to block light. In practice, real polarizers have a finite extinction ratio even at 90° due to imperfect alignment, scattering, and material absorption. High-quality Glan-Thompson prisms achieve extinction ratios of 10⁻⁶ or better.
The absorbed intensity I₀ − I represents the energy removed from the beam. In a real polarizer, this energy is either absorbed (dichroic polarizers) or deflected into an extraordinary beam (birefringent polarizers).
The transmission percentage directly tells you what fraction of incident light passes through. At 0° you get 100% (parallel alignment), and at 90° you get 0% (crossed). The cos² dependence means transmission drops slowly near 0° but rapidly near 90° — at 60° you have already lost 75% of the light. The absorbed intensity shows how much energy the analyzer removes; for high-power laser applications, this absorbed energy can cause heating damage.
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Polarized light at 100 W/m² passing through an analyzer tilted 45° transmits exactly 50 W/m². This 50% transmission at 45° is a key reference point for calibrating polarimeters.
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If 500 W/m² of polarized light is reduced to 125 W/m² by an analyzer, the angle between them is arccos(√0.25) = 60°. The analyzer absorbs 375 W/m², or 75% of the incident intensity.
Malus's law states that the intensity of linearly polarized light transmitted through a linear analyzer is I = I₀cos²θ, where I₀ is the incident polarized intensity and θ is the angle between the polarization direction and the analyzer's transmission axis. Maximum transmission occurs at θ = 0° and complete extinction at θ = 90°.
Not directly. For unpolarized light, the first polarizer transmits half the intensity (I = I₀/2) because averaging cos²θ over all random polarization directions gives 1/2. Malus's law then applies to subsequent polarizers acting on the now-polarized beam.
The extinction ratio is the ratio of transmitted to incident intensity: I/I₀ = cos²θ. For crossed polarizers it should be zero, but real polarizers have a small leakage. High-quality polarizers achieve extinction ratios of 10⁻⁵ to 10⁻⁶, meaning they block 99.999% of cross-polarized light.
An LCD uses two crossed polarizers with a liquid crystal layer between them. Without an applied voltage, the liquid crystal rotates the polarization by 90°, allowing light through (bright pixel). Applying a voltage untwists the crystal, blocking light (dark pixel). Intermediate voltages produce gray levels following Malus's law.
No. Malus's law applies only to linearly polarized light. Circularly polarized light passed through a linear polarizer always transmits 50%, regardless of the polarizer orientation, because the electric field vector rotates uniformly.
For ideal thin polarizers, Malus's law is exact. Real polarizers deviate slightly due to absorption, surface reflections (typically 4% per surface), wavelength dependence, and acceptance angle. High-quality sheet polarizers follow the law to within 1% across most of the visible spectrum.
Roboculator Team
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