Malus Law Calculator

Created by Wojciech Sas, PhD
Reviewed by Bogna Szyk
Last updated: Nov 05, 2022

Malus law calculator is a helpful tool that explains the functional principle of polarizers. Depending on the relative position between the direction of light polarization and the axis of a polarizer, the different amount of light can be transmitted.

By knowing the initial light intensity and polarization orientation, you can easily find the output intensity, which comes out of the polarizer, based on Malus law.

🙋 Learn how to choose the ideal polarized sunglasses with Omni's sunglasses size calculator.

Light polarization

We can describe light as mutually oscillating electric and magnetic fields forming an electromagnetic wave. These oscillations are perpendicular to each other and simultaneously perpendicular to the direction of wave propagation.

The light is said to be linearly polarized if the orientation of oscillations is constant in time. It is common to say that the direction of light polarization is the same as that of electric field oscillation.

We can obtain polarized light due to reflection on the boundary of two media with different refractive indexes if the angle of incidence equals Brewster's angle or by passing light through some dichroic crystal. Check our Brewster's angle calculator to find the exact angle value.

How does a polarizer work?

Polarizers are usually made out of oblong-shaped molecules, all aligned in the same direction. It turns out that if the polarization of the incident beam is the same as the alignment orientation, then the light is most likely to be absorbed.

If the polarization is perpendicular to the long axis of molecules, then it is transmitted almost entirely, and that direction is the axis of the polarizer. If the angle of polarization is something in between, it passes through only partially, and its initial irradiance decreases. We can determine the exact value thanks to the Malus law.

The change of light polarization after passing trough the polarizer.

Malus law formula

Irradiance (intensity) of light is the amount of power received on the unit of surface, so we usually measure it in W/m². Considering the wavelengths of photons, we can also convert them into luxes using our lumen calculator, obtaining the units of illuminance. We can calculate the light intensity, which passes through the ideal polarizer, as follows:

intensity = initial_intensity × cos²θ,

where:

  • initial_intensity is the irradiance of the incident light,
  • θ is the angle between the light polarization and the axis of the polarizer, and
  • intensity is the output irradiance.

Irradiance changes under rotation - an example

Let's say that you want to check how the intensity of polarized light changes while you rotate your polarizer:

  1. Choose a few different values of the axis of polarizer orientation with respect to the polarization of incident rays, e.g. θ₁=20°, θ₂=45°, θ₃=70°;
  2. Determine cosθ of these angles, which are 0.939, 0.707, and 0.342, respectively;
  3. Find squares of these values: 0.883, 0.5, 0.117;
  4. Multiply them by the initial intensity, say I₀=5 W/m²: I₁=4.415 W/m², I₂=2.5 W/m², I₃=0.585 W/m²;
  5. Or simply enter data into this Malus law calculator and save your time!

You can always express obtained results as the percentages of the initial intensity.

Polarizers in everyday life

Have you ever wondered why it is possible to remove light reflections on the water surface only by rotating the polarizing filter in your camera? Now you know the secret and, what's more, you can easily predict the effect using the Malus law calculator. Polarized sunglasses are also based on this phenomenon so that they are able to block a vast majority of polarized light during sunny days.

Wojciech Sas, PhD
Initial Intensity
W/
Angle of incidence
deg
Intensity
W/
ft²
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