Malus's Law Calculator

Introduction: Malus's Law and Polarized Light

This Malus's law calculator models how a polarized light beam loses intensity as it passes through one, two, or three polarizers. 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 applies the law stage by stage so you can examine the cumulative effect of relative polarizer angles.

Malus's Law Formula for Polarizer Transmission

For the polarized-light transmission calculated here, Malus's law is expressed as:

Formula: I = I_0 cos^2 θ

Here, is the transmitted intensity after a polarizer, is the initial intensity entered in the calculator, and is the angle between the incoming polarization direction and the polarizer axis. The cosine term is squared because the transmitted amplitude is proportional to the projection of the electric field onto the polarizer axis, while intensity is proportional to amplitude squared. When the polarizer aligns with the incoming polarization (), the cosine term equals one and the full entered intensity passes through. When the polarizer is perpendicular (), the cosine term vanishes and no light emerges.

Sequential Polarizers in This Malus's Law Calculator

For sequential polarizers, this Malus's law calculator uses the output of each stage as the input to the next. Enter θ₁ as the first polarizer's angle relative to the initial polarization direction; θ₂ is measured relative to the first polarizer's axis, and θ₃ relative to the second. The transmitted intensity after the first polarizer is

Formula: I_1 = I_0 cos^2 θ_1

For the second polarizer, the calculator multiplies that first-stage result by another cosine-squared factor:

Formula: I_2 = I_1 cos^2 θ_2

The same multiplication is applied if you enter a third relative angle. This stage-by-stage setup shows why an intermediate polarizer can transmit some light between two crossed axes: each polarizer establishes the polarization direction used by the following stage.

Historical Notes on Malus's Polarization Law

Malus's law is named for 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. His work helped establish polarization as a central part of optics and remains relevant to polarizing filters and liquid crystal displays.

Physical Interpretation of Malus's Law

Malus's law in this calculator follows directly from a vector projection. Suppose an electric field oscillates horizontally. A polarizer oriented at 30° to the horizontal transmits only the component of the field along its axis. The component along that axis is the original field magnitude times the cosine of the angle between them. Because intensity scales with the square of the field, the transmitted intensity follows the square of that projection. The calculator therefore converts each entered angle into a cosine-squared transmission factor.

The model uses ideal polarizers: each one perfectly transmits its selected polarization and completely blocks the orthogonal component. Real polarizing films have finite extinction ratios and can absorb light beyond the idealized projection loss. Even so, cosine-squared transmission is a useful approximation for many introductory and practical comparisons.

Applications of Polarizer Intensity Calculations

Malus's law and the polarizer-intensity relationships calculated on this page 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.

Worked example: Malus's Law Transmission at Common Angles

This Malus's law table shows the single-polarizer transmission fraction for light with unit initial intensity:

The Malus's law transmission fraction falls as the relative angle increases from 0° toward 90°. With multiple polarizers, the calculator compounds those factors. For example, if the first angle is 0° and each of the next two polarizers is 45° relative to the preceding one, the final result is one quarter of the entered initial intensity; the two 45° stages each contribute a factor of one half.

Advanced Considerations for Polarized-Light Transmission

Malus's law describes both the classical wave picture used by this calculator and a related quantum probability result. For a photon with a defined polarization, the probability of passage through an ideal polarizer is the square of the cosine of the angle between its polarization state and the polarizer axis. A beam's intensity ratio is the corresponding aggregate behavior of many photons.

Real optical systems can require effects beyond the calculator's ideal-polarizer model. Birefringent materials can split and rotate polarization states, dichroic polarizers may have wavelength dependence, and fiber-optic systems can experience polarization mode dispersion. Malus's law remains a starting point for estimating how relative polarizer orientation changes beam intensity.

How to use: Malus's Law Polarizer Inputs

To calculate Malus's law transmission, enter a non-negative initial intensity and the first polarizer angle. You may then add relative angles for a second and third polarizer. The script converts degrees to radians and calculates the cosine-squared intensity at every entered stage. The result lists the intensity after each polarizer, allowing you to track where the beam is attenuated most strongly.

Historical Experiments with Crossed Polarizers

A classic polarization demonstration uses two polarizing filters with axes crossed by 90°, which block light that has already been polarized by the first filter. Adding a third filter between them at intermediate relative angles can allow some transmission. In this calculator, set the first angle according to the initial polarization direction, then enter the relative rotations for the later filters; the displayed stage values reveal how each cosine-squared factor changes the remaining intensity. The demonstration illustrates that an intermediate polarizer changes the polarization direction before the final filter, rather than simply adding another blocker.

Limitations of This Malus's Law Calculator

This Malus's law calculator begins with the intensity and polarization reference you supply, so its first cosine-squared factor models a polarized input beam. For unpolarized light, an ideal first polarizer transmits half the incident intensity regardless of its absolute orientation; that initial one-half reduction is not added automatically here. The model also omits imperfect extinction, depolarization, wavelength effects, and reflections at polarizer surfaces, all of which can matter in precision optical measurements.

Capturing Malus's Law Calculation Results

After the calculator displays the transmitted intensity at each polarizer, use the copy button to place those displayed results on your clipboard. Recording output for several relative-angle combinations can help students compare an ideal Malus's law prediction with laboratory measurements.

Photographers and hobbyists can also retain copied intensity ratios in notes when comparing polarizing-filter orientations, while remembering that actual camera exposure can include additional optical and scene-dependent effects.

Conclusion: Using Malus's Law to Compare Polarizer Angles

This Malus's law calculator connects relative polarizer orientation to transmitted intensity through the cosine-squared rule. Entering a first angle and optional later relative angles makes the successive losses visible rather than treating a polarizer stack as a single black box. Use the stage results to check alignments, explore crossed-polarizer arrangements, and build intuition for the role of polarization in optical systems.