Polarization of Light and Malus's Law

Light is a transverse wave — its electric field oscillates perpendicular to the direction of travel. But which perpendicular direction? In ordinary (unpolarized) light from the sun or a light bulb, the electric field vibrates in all transverse directions randomly and rapidly. There's no preferred oscillation direction; every orientation is equally likely from one wave cycle to the next.

A polarizer is a filter that transmits only the component of the electric field along one specific direction (the transmission axis). After passing through a polarizer, the light is linearly polarized — its electric field oscillates in a single, well-defined plane.

What happens step by step

Step 1 — Unpolarized light hits the first polarizer: The unpolarized beam has intensity $I_{0}$ and every possible polarization direction. The polarizer selects only the component along its transmission axis. On average, exactly half the intensity gets through:

$I_{\text{after first}} = \dfrac{I_{0}}{2}$

The factor of 1/2 comes from averaging $\cos^{2}\theta$ over all angles: $\langle\cos^{2}\theta\rangle = 1/2$.

Step 2 — Polarized light hits the second polarizer (analyzer): The light is now linearly polarized with intensity $I_{0}/2$ and oscillation direction aligned with the first polarizer's axis. The second polarizer is rotated by angle $\theta$ relative to the first.

Only the component of the electric field along the analyzer's axis passes through. If $E$ is the amplitude, the transmitted component is $E\cos\theta$, and since intensity is proportional to $E^{2}$:

$I = \dfrac{I_{0}}{2}\cos^{2}\theta$

This is Malus's law, named after Étienne-Louis Malus who discovered it in 1809 while looking at sunlight reflected from the windows of the Palais du Luxembourg in Paris through a calcite crystal.

Numerical results

• $\theta = 0°$: $I = \frac{I_{0}}{2}\cos^{2}0° = \frac{I_{0}}{2}(1) = \frac{I_{0}}{2} = 50\%$. Axes aligned — maximum transmission after the first filter.
• $\theta = 30°$: $I = \frac{I_{0}}{2}\cos^{2}30° = \frac{I_{0}}{2}(0.75) = \frac{3I_{0}}{8} = 37.5\%$.
• $\theta = 45°$: $I = \frac{I_{0}}{2}\cos^{2}45° = \frac{I_{0}}{2}(0.5) = \frac{I_{0}}{4} = 25\%$.
• $\theta = 60°$: $I = \frac{I_{0}}{2}\cos^{2}60° = \frac{I_{0}}{2}(0.25) = \frac{I_{0}}{8} = 12.5\%$.
• $\theta = 90°$: $I = \frac{I_{0}}{2}\cos^{2}90° = 0$. Axes perpendicular — no light gets through (crossed polarizers).

The crossed-polarizer surprise

At $\theta = 90°$, two crossed polarizers block all light. Here's the surprise: insert a third polarizer between them at 45° to both. Now, the first polarizer transmits $I_{0}/2$, the 45° filter passes $I_{0}/4$, and the last filter passes $I_{0}/8$. You went from zero transmission to 12.5% — inserting an obstacle increased the light!

This is not paradoxical once you understand that each polarizer changes the polarization direction of the transmitted light. The middle polarizer rotates the polarization by 45°, giving the light a component along the third polarizer's axis.

Polarization in nature and technology

• Polaroid sunglasses: Block horizontally polarized glare from reflective surfaces (roads, water, snow). Road-reflected sunlight is partially horizontally polarized (Brewster's law), so vertical polarizers cut the glare while transmitting most vertically-polarized light.
• LCD screens: Liquid crystals rotate the polarization of light. Two crossed polarizers sandwich a layer of liquid crystal that, depending on voltage, either rotates light 90° (pixel on — light passes) or doesn't (pixel off — light blocked). Every pixel on your screen is a tiny Malus's law experiment.
• 3D cinema: Projects two images with perpendicular polarizations (one for each eye). Polarized glasses let each eye see only its intended image, creating the illusion of depth.
• Stress analysis: Stressed transparent materials (plastic, glass) become birefringent — they polarize different colors differently. Viewed between crossed polarizers, colorful fringe patterns reveal internal stress distributions. Engineers use this to check bridge components and aircraft windows.
• Brewster's angle: When light reflects from a surface at a specific angle $\theta_{B} = \arctan(n_{2}/n_{1})$, the reflected beam is perfectly polarized. Malus discovered this by accident — the Luxembourg windows reflected polarized light.

Common mistakes

• Forgetting the $1/2$ from the first polarizer. Malus's law is often stated as $I = I_{0}\cos^{2}\theta$, which is only valid when $I_{0}$ is the intensity of already-polarized light entering the analyzer. If the source is unpolarized, the first polarizer cuts the intensity in half before $\cos^{2}\theta$ applies.
• Thinking crossed polarizers always block everything. They do for a single pair. But adding more polarizers between them at intermediate angles can let light through.
• Confusing polarization with color filtering. Colored filters absorb certain wavelengths. Polarizers absorb certain orientations of the electric field, regardless of wavelength.

Try it in the visualization

Rotate the analyzer angle from 0° to 90° and watch the transmitted intensity drop to zero. Turn on the third polarizer at 45° between the crossed pair and see light reappear. Toggle the E-field wave animation to visualize the oscillation direction changing at each polarizer. The intensity gauge and $\cos^{2}\theta$ curve update in real time.