Malus's Law Calculator
The Malus's 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, different amounts of light can be transmitted. By knowing the initial light intensity and polarization orientation, you can easily find the output intensity that comes from the polarizer, based on Malus's law.
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 the 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.
How does a polarizer work?
Polarizers are usually made 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 the 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 Malus's law.
Malus's law formula
Irradiance (intensity) of light is the amount of power received on a unit of surface, so we usually measure it in W/m² (metric) or W/ft² (US customary). We can calculate the light intensity that passes through an ideal polarizer as follows:
intensity = initial intensity × cos²θ
Where:
- initial intensity (I₀) — the irradiance of the incident light;
- θ — the angle between the light polarization and the axis of the polarizer; and
- intensity (I) — the output irradiance.
Irradiance changes under rotation — an example
Let's say you want to check how the intensity of polarized light changes while you rotate your polarizer:
- Choose a few different values of the polarizer axis orientation with respect to the polarization of the incident rays, e.g., θ₁ = 20°, θ₂ = 45°, θ₃ = 70°.
- Determine cos θ of these angles, which are 0.939, 0.707, and 0.342, respectively.
- Find the squares of these values: 0.883, 0.5, and 0.117.
- 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.
Or simply enter the data into this Malus's law calculator and save your time! You can always express the obtained results as percentages of the initial intensity.
| Angle θ | cos θ | cos²θ (transmittance) | Output for I₀ = 5 W/m² |
|---|---|---|---|
| 0° | 1.000 | 1.000 | 5.000 W/m² |
| 20° | 0.939 | 0.883 | 4.415 W/m² |
| 30° | 0.866 | 0.750 | 3.750 W/m² |
| 45° | 0.707 | 0.500 | 2.500 W/m² |
| 60° | 0.500 | 0.250 | 1.250 W/m² |
| 70° | 0.342 | 0.117 | 0.585 W/m² |
| 90° | 0.000 | 0.000 | 0.000 W/m² |
Polarizers in everyday life
Polarizers are everywhere around us. Polarized sunglasses cut the glare reflected from water, snow, and roads, because reflected light is strongly polarized. Photographers use polarizing filters to deepen blue skies and reduce reflections from glass and water. LCD screens rely on two crossed polarizers to switch pixels on and off, and 3D cinema glasses use polarization to send a slightly different image to each eye.
Unit systems supported
This calculator works with both the metric system (irradiance in W/m²) and the US customary system (irradiance in W/ft²). Because Malus's law depends only on the ratio cos²θ, the transmittance and percentage results are identical in any unit system, while intensities are shown in the units you select (1 W/m² = 0.0929 W/ft²).