Polarization of Light

Light is an electromagnetic wave with electric and magnetic fields oscillating perpendicular to its direction of travel. In everyday unpolarized light, the electric field vectors point in random orientations. A polarizer is a material that allows only one orientation of electric field to pass. Malus's law describes how the intensity of a polarized beam changes when it encounters a polarizer that is not aligned with its polarization direction. The calculator on this page implements Malus's law for up to three consecutive polarizers, letting you explore the cumulative effect of each stage.

Malus's Law Formula

The essence of Malus's law can be captured with a single expression:

$I=\mathrm{I\_0}{\cos }^2\theta$

Here, $I$ is the transmitted intensity after the polarizer, $\mathrm{I\_0}$ is the incident intensity, and $\theta$ is the angle between the light's initial polarization direction and the axis of the polarizer. The cosine term is squared because the transmitted amplitude is proportional to the projection of the electric field onto the polarizer axis, and intensity is proportional to amplitude squared. When the polarizer aligns with the incoming polarization ($\theta =\; 0$), the cosine term equals one and the full intensity passes through. When the polarizer is perpendicular ($\theta =\; 90°$), the cosine term vanishes and no light emerges.

Sequential Polarizers

Malus's law applies separately to each polarizer. If you have multiple polarizers, the output of one becomes the input to the next. Our calculator allows you to specify up to three polarizers by entering angles for θ₁, θ₂, and θ₃. The second angle is measured relative to the first polarizer's axis, and the third is measured relative to the second. The transmitted intensity after the first polarizer is

$\mathrm{I\_1}=\mathrm{I\_0}{\cos }^2\mathrm{\theta \_1}$

After the second polarizer, the intensity becomes

$\mathrm{I\_2}=\mathrm{I\_1}{\cos }^2\mathrm{\theta \_2}$

and similarly for the third. By allowing optional second and third angles, the tool demonstrates how adding intermediate polarizers can restore some light even when the first and last polarizers are crossed at ninety degrees—a counterintuitive result that has fascinated students for centuries.

Historical Notes

The law bears the name of Étienne-Louis Malus, a French engineer and physicist who studied the polarization of light by reflection in the early nineteenth century. While observing sunlight reflected off a window, Malus noticed that rotating a crystal of calcite caused the intensity of the transmitted beam to vary. He deduced the cosine-squared relationship and published it in 1809. Malus's work laid the foundation for much of modern optics, from the design of sunglasses to the operation of liquid crystal displays.

Physical Interpretation

Polarization can be visualized using vectors. Suppose an electric field oscillates horizontally. A polarizer oriented at 30° to the horizontal only transmits the component of the field along its axis. The projection formula from vector mathematics tells us that the component along the axis is the magnitude of the original vector times the cosine of the angle between them. Because intensity scales with the square of the field, the transmitted intensity is proportional to the square of that projection. Malus's law therefore emerges naturally from the combination of vector projections and the energy content of electromagnetic waves.

The law assumes ideal polarizers that perfectly transmit one polarization and completely absorb the orthogonal component. Real polarizing films have finite extinction ratios and may absorb slightly less than ideal. Nonetheless, the cosine-squared dependence remains an excellent approximation for most practical purposes.

Applications

Polarizers and Malus's law appear across science and technology:

• Sunglasses and glare reduction: Polarized sunglasses reduce reflections from horizontal surfaces such as water or roads. The polarizer is oriented to block horizontally polarized light, which is preferentially produced by reflection.
• Photography: Photographers use polarizing filters to control reflections and enhance contrast in skies. Rotating the filter changes the amount of polarized light reaching the camera sensor according to Malus's law.
• Liquid crystal displays (LCDs): LCD screens rely on polarizers combined with electrically controlled liquid crystals. By rotating the polarization of light in each pixel, the screen modulates brightness using the same cosine-squared principle.
• Quantum optics experiments: Entangled photons and polarization measurements often employ polarizers. Malus's law helps predict correlation rates and test fundamental principles of quantum mechanics.

Sample Calculations

The table below illustrates the effect of a single polarizer on light of unit intensity for various angles:

Angle θ	Transmitted Intensity I/I0
0°	1.0
30°	0.75
60°	0.25
90°	0.0

Notice how the intensity drops sharply as the angle increases. The squared cosine ensures that even moderate misalignment can significantly reduce light transmission. For multiple polarizers, the effect compounds, yet clever orientation can achieve unexpected outcomes. For instance, three polarizers each rotated by 45° relative to the previous one transmit 25% of the original light even though the first and last are crossed.

Advanced Considerations

While Malus's law treats light as a classical wave, it also arises from quantum principles. Photons carry spin angular momentum, and a polarizer acts like a measurement device selecting one spin state. The probability that a photon passes through a polarizer is the square of the cosine of the angle between its polarization state and the polarizer axis, mirroring the intensity formula. This quantum view explains why individual photons obey the same law as classical light beams.

Real optical systems may require additional factors. Birefringent materials can split and rotate polarization states, while dichroic polarizers may introduce wavelength dependence. In fiber optics, polarization mode dispersion can degrade signals. Nevertheless, Malus's law remains the starting point for analyzing how polarizers influence beam intensity.

Using the Calculator

To explore these effects, enter the initial intensity and the angle of the first polarizer. Optionally, add angles for a second and third polarizer. The script computes the transmitted intensity after each stage using the cosine-squared rule. All calculations occur client-side in your browser, ensuring quick results and privacy. The output displays each intermediate intensity so you can see how the light dims step by step.

Historical Experiments

One classic classroom demonstration involves arranging three polarizing filters. The first and third are crossed at 90°, blocking all light. Inserting a third filter at 45° between them allows some light to pass. Malus's law predicts this surprising result: the first filter transmits half the light, the middle filter transmits half of that again, and the final filter transmits half once more, yielding one eighth of the initial intensity. Despite the final polarizer being perpendicular to the first, the intermediate filter rotates the polarization so that some light survives.

Limitations

Our calculator assumes perfectly polarized input light. Unpolarized light incident on the first polarizer will have half its intensity transmitted, independent of angle, because only one polarization component is allowed through. After that initial reduction, subsequent polarizers follow Malus's law relative to the axis defined by the first. Additionally, the formula neglects depolarization effects and reflections at the surfaces of the polarizers, which can be relevant in high-precision instruments.

Conclusion

Malus's law elegantly links geometry with optics, showing how the simple act of rotating a polarizer changes the brightness of transmitted light. By providing a hands-on calculator with rich explanations, this page aims to deepen intuition about polarization, whether you are a student, photographer, or curious experimenter. Adjust the angles, stack polarizers, and watch how light obeys the cosine-squared rule—a fundamental cornerstone of wave physics.