AP Physics 2: Polarization of Light

Polarization is the definitive proof that light is a transverse wave, a concept central to modern optics and countless technologies. Understanding how to control the oscillation direction of light's electric field is not just an academic exercise; it explains why your sunglasses cut glare, how 3D movies work, and the basic function of every LCD screen.

The Nature of Polarized and Unpolarized Light

Ordinary light from the sun or a lamp is unpolarized. This means its electromagnetic waves oscillate in all possible directions perpendicular to its direction of travel. Imagine a rope being shaken up and down, then side-to-side, then at every angle in between—all simultaneously. For light, it's the electric field vector that performs these oscillations.

A polarizing filter (or polarizer) acts as a gatekeeper for these oscillations. It has a specific molecular or structural alignment called its transmission axis. Only the component of the electric field oscillating parallel to this axis is transmitted. The perpendicular component is absorbed. Therefore, when unpolarized light passes through an ideal polarizer, it emerges as linearly polarized light—light oscillating in only one plane—and its intensity is reduced by half. You can visualize this by thinking of the polarizer as a picket fence: only the vibrations parallel to the slits can get through.

Malus's Law: Quantifying Transmission Through Polarizers

What happens when you pass linearly polarized light through a second polarizer, often called an analyzer? The intensity of the transmitted light depends on the angle between the polarization direction of the incoming light and the transmission axis of the analyzer. This relationship is described by Malus's Law:

$$I=I_0{\cos }^2\theta$$

Here, $I$ is the transmitted intensity, $I_0$ is the intensity of the polarized light incident on the analyzer, and $\theta$ is the angle between the polarization direction of the incident light and the analyzer's transmission axis.

Let's walk through a key example. Suppose you have two polarizers in sequence. Unpolarized light of intensity $S_0$ strikes the first polarizer. The light emerging from it is polarized and has an intensity of $I_1=\frac{1}{2}{S}_0$ (the half-intensity rule). If the second polarizer's axis is at a 30° angle to the first, the intensity after the second polarizer is calculated using Malus's Law: $I_2=I_1{\cos }^2(30°)=(\frac{1}{2}{S}_0)\times (\frac{\sqrt{3}}{2}{)}^2=(\frac{1}{2}{S}_0)\times \frac{3}{4}=\frac{3}{8}{S}_0$. If the two polarizers are crossed (θ = 90°), then ${\cos }^2(90°)=0$ and no light is transmitted. Inserting a third polarizer at a 45° angle between two crossed polarizers allows some light to pass, demonstrating how Malus's Law applies step-by-step through each filter.

Polarization by Reflection and Brewster's Angle

Light reflecting off a non-metallic surface, like water or glass, becomes partially polarized. The component oscillating parallel to the surface (the s-polarization) reflects more strongly than the component oscillating perpendicular to it (the p-polarization). At one specific angle of incidence, the reflected light becomes completely polarized. This special angle is called Brewster's angle (${\theta }_B$).

At Brewster's angle, the reflected ray and the refracted ray are perpendicular to each other. From this geometry and Snell's Law, we derive the condition for Brewster's angle:

$$\tan {\theta }_B=n$$

Where $n$ is the index of refraction of the reflecting medium (assuming the incident medium is air, $n_{air}\approx 1$). For example, if light reflects off water ($n\approx 1.33$), then ${\theta }_B=\arctan (1.33)\approx 53°$. Polarizing sunglasses exploit this principle. Their lenses are vertically polarized to block the horizontally polarized glare that dominates when sunlight reflects off horizontal surfaces like roads or water.

Polarization by Scattering

Why is the sky blue and polarized? Sunlight scattering off air molecules in the atmosphere, a process called Rayleigh scattering, also polarizes light. When unpolarized sunlight strikes a molecule, it causes the electrons in the molecule to oscillate. These accelerating electrons then re-radiate light, but most strongly in a direction perpendicular to their oscillation. If you look at a point in the sky 90° away from the sun, the scattered light you see has traveled a path perpendicular to the original sunlight's direction. This geometry leads to the light being preferentially polarized perpendicular to the plane containing you, the sun, and the scattering molecule. This is why photographers use polarizing filters to darken blue skies; by rotating the filter, they can block this scattered, polarized skylight.

Common Pitfalls

1. Misapplying the Half-Intensity Rule: A common error is applying the $I=\frac{1}{2}{I}_0$ rule to any light entering a polarizer. This rule only applies to unpolarized incident light. If the light is already polarized, you must use Malus's Law ($I=I_0{\cos }^2\theta$) directly, starting with the full incident intensity $I_0$.

1. Confusing the Angle in Malus's Law: The angle $\theta$ in $I=I_0{\cos }^2\theta$ is not necessarily the angle from the vertical. It is specifically the angle between the polarization direction of the incident light and the transmission axis of the analyzer. Always identify these two directions first before calculating θ.

1. Forgetting the Transverse Wave Implication: Students sometimes memorize polarization phenomena without connecting them to the core principle: polarization is only possible for transverse waves. If light were a longitudinal wave (like sound in air), it could not be polarized. This fact is a key piece of evidence in the historical development of wave optics.

1. Misidentifying Brewster's Angle Conditions: Remember, at Brewster's angle, the reflected light is completely polarized, but the transmitted (refracted) light is only partially polarized. Also, the condition for maximum polarization of the reflected ray is that it is perpendicular to the refracted ray.

Summary

• Polarization is a filter for light's oscillation direction, proving light is a transverse wave. A polarizer transmits only the component of light's electric field parallel to its transmission axis.
• Malus's Law, $I=I_0{\cos }^2\theta$, quantitatively describes how the intensity of polarized light changes after passing through a second polarizer (analyzer), where $\theta$ is the angle between the polarization direction and the analyzer's axis.
• Polarization by reflection occurs at non-metallic surfaces, with Brewster's angle ($\tan {\theta }_B=n$) being the specific angle where the reflected light is 100% polarized. This principle is used in polarizing sunglasses to eliminate glare.
• Polarization by scattering, such as Rayleigh scattering from atmospheric molecules, explains why skylight is both blue and polarized, especially at angles 90° from the sun.