Suppose a polarized light wave of intensity $I_0$ approaches a polarizing filter. What is the intensity of the light that passes through the filter? FIGURE 31.29 shows that an oscillating electric field can be decomposed into components parallel and perpendicular to the polarizer axis. If we call the polarizer axis the y-axis, then the incident electric field is

where $\theta$ is the angle between the incident plane of polarization and the polarizer axis.

FIGURE 31.29 An incident electric field can be decomposed into components parallel and perpendicular to a polarizer axis.

If the polarizer is ideal, meaning that light polarized parallel to the axis is 100% transmitted and light perpendicular to the axis is 100% blocked, then the electric field of the light transmitted by the filter is

Because the intensity depends on the square of the electric field amplitude, you can see that the transmitted intensity is related to the incident intensity by

This result, which was discovered experimentally in 1809, is called Malus’s law.

FIGURE 31.30a shows that Malus’s law can be demonstrated with two polarizing filters. The first, called the polarizer, is used to produce polarized light of intensity $I_0$. The second, called the analyzer, is rotated by angle $\theta$ relative to the polarizer. As the photographs of FIGURE 31.30b show, the transmission of the analyzer is (ideally) 100% when $\theta =0\text{°}$ and steadily decreases to zero when $\theta =90\text{°}$. Two polarizing filters with perpendicular axes, called crossed polarizers, block all the light.

FIGURE 31.30 The intensity of the transmitted light depends on the angle between the polarizing filters.

Suppose the light incident on a polarizing filter is unpolarized, as is the light incident from the left on the polarizer in Figure 31.30a. The electric field of unpolarized light varies randomly through all possible values of $\theta$. Because the average value of ${\text{cos}}^2\theta$ is ${\displaystyle \frac{1}{2}}$, the intensity transmitted by a polarizing filter is

In other words, a polarizing filter passes 50% of unpolarized light and blocks 50%.

In polarizing sunglasses, the polymer grid is aligned horizontally (when the glasses are in the normal orientation) so that the glasses transmit vertically polarized light. Most natural light is unpolarized, so the glasses reduce the light intensity by 50%. But glare—the reflection of the sun and the skylight from roads and other horizontal surfaces—has a strong horizontal polarization. This light is almost completely blocked by the Polaroid, so the sunglasses “cut glare” without affecting the main scene you wish to see.

You can test whether your sunglasses are polarized by holding them in front of you and rotating them as you look at the glare reflecting from a horizontal surface. Polarizing sunglasses substantially reduce the glare when the glasses are “normal” but not when the glasses are $90\text{°}$ from normal. (You can also test them against a pair of sunglasses known to be polarizing by seeing if all light is blocked when the lenses of the two pairs are crossed.)