Polarization and Malus's Law
Have you ever put on polarized sunglasses at a lake and watched the blinding glare on the water suddenly vanish? Tilt your head sideways and the glare comes back. How can a piece of tinted plastic tell glare from other light, and why does tilting your head matter? By the end of this page you will be able to answer that.
What polarization means
Light is a transverse wave: the electric field vibrates perpendicular to the direction the light travels. But "perpendicular" leaves many choices. Around a beam heading toward you, the field could vibrate up and down, left and right, or at any angle in between.
(The dot in the middle is the beam coming straight out of the page at you.)
A polarizing filter (polarizer) contains long aligned molecules that act like the slats of a fence. It has a transmission axis. Field components along that axis pass through; components perpendicular to it are absorbed.
Why only transverse waves? Only a wave that vibrates sideways can have different "sideways directions" to choose from. Sound in air is longitudinal: the air vibrates back and forth along the direction of travel, so there is no sideways direction to filter. Sound cannot be polarized.
Explain in your own words what it means for light to be "polarized." Then state whether sound waves in air can be polarized, and give the reason for your answer.
Explain in your own words what it means for light to be "polarized." Then state whether sound waves in air can be polarized, and give the reason for your answer.
Unpolarized light and Malus's law
Unpolarized light through an ideal polarizer. On average, the vibrations are spread evenly over all directions, so about half the energy is along the axis. The polarizer absorbs the rest:
$$I = \frac{I_0}{2}$$The light that emerges is now polarized along the transmission axis.
Polarized light through a polarizer. Suppose the light is polarized at angle $\theta$ to the transmission axis. Split the electric field $E_0$ into two components: $E_0\cos\theta$ along the axis (passes) and $E_0\sin\theta$ across it (absorbed).
Intensity is proportional to the square of the field amplitude. Squaring $E_0\cos\theta$ gives:
Explore the graph below: it shows the transmitted fraction $I/I_0$ for each angle $\theta$ from $0^\circ$ to $180^\circ$.
Unpolarized light of intensity $I_0 = 80\ \text{W/m}^2$ strikes an ideal polarizer. What is the intensity of the light that emerges? Is the emerging light polarized?
Unpolarized light of intensity $I_0 = 80\ \text{W/m}^2$ strikes an ideal polarizer. What is the intensity of the light that emerges? Is the emerging light polarized?
Polarized light of intensity $50\ \text{W/m}^2$ reaches an ideal polarizer. Find the transmitted intensity when the angle between the light's polarization direction and…
Polarized light of intensity $50\ \text{W/m}^2$ reaches an ideal polarizer. Find the transmitted intensity when the angle between the light's polarization direction and the polarizer's transmission axis is (a) $0^\circ$ and (b) $90^\circ$.
Polarized light of intensity $I_0 = 120\ \text{W/m}^2$ passes through an ideal polarizer whose axis makes an angle of $30^\circ$ with the light's polarization direction.…
Polarized light of intensity $I_0 = 120\ \text{W/m}^2$ passes through an ideal polarizer whose axis makes an angle of $30^\circ$ with the light's polarization direction. Then it passes through a second ideal polarizer whose axis is at $60^\circ$ to the first polarizer's axis. Find the intensity after each polarizer.
Using Malus's law with data
In a lab you can check Malus's law with two filters, a lamp, and a light sensor. The first filter polarizes the lamp light. You rotate the second filter and record the intensity at each angle.
Tip for checking a law with a table: divide each reading by the $0^\circ$ reading and compare with $\cos^2\theta$. At $45^\circ$ the ratio should be $0.50$, and at $90^\circ$ it should be $0$.
Emma holds two polarizing filters from a lab kit in front of a lamp. Unpolarized light hits the first filter. She rotates the second filter and records the transmitted…
Emma holds two polarizing filters from a lab kit in front of a lamp. Unpolarized light hits the first filter. She rotates the second filter and records the transmitted intensity:
| Angle between axes | Intensity after second filter (W/m²) |
|---|---|
| $0^\circ$ | 40 |
| $45^\circ$ | 20 |
| $90^\circ$ | 0 |
(a) What was the intensity of the lamp light before the first filter? (b) Show that the data follow Malus's law. (c) Predict the intensity at $60^\circ$.
Polarization in everyday life
Polarized sunglasses. Light reflecting off a flat surface like a lake or a road becomes partly polarized horizontally. Polarized sunglasses have their transmission axes vertical, so they absorb much of that horizontal glare while passing about half of ordinary light. Tilt your head $90^\circ$ and the axes become horizontal: now the glare passes freely.
3-D movie glasses. A passive 3-D system projects two images with perpendicular polarizations. Each lens has an axis matching one image and blocking the other ($\theta = 90^\circ$ gives $I = 0$). Each eye sees a slightly different view, and your brain builds depth.
LCD screens. An LCD sits between two crossed polarizing filters (axes at $90^\circ$). Each pixel holds a liquid crystal that can twist the polarization of the light by $90^\circ$, so light passes the second filter. An electric voltage untwists the crystal, so that pixel gets darker. The light leaving the screen is already polarized, so polarized sunglasses can darken a screen completely if the glasses' axes are perpendicular to the screen's.
Sunlight reflected from a lake is partially polarized horizontally. Explain how polarized sunglasses reduce glare from the water. State which direction the lenses'…
Sunlight reflected from a lake is partially polarized horizontally. Explain how polarized sunglasses reduce glare from the water. State which direction the lenses' transmission axes should point (vertical or horizontal) and why. Then explain what happens to the glare if Jayden tilts his head $90^\circ$.
Two ideal polarizers have axes at $90^\circ$ to each other, so no light gets through. Unpolarized light of intensity $I_0$ enters the first polarizer. A third ideal…
Two ideal polarizers have axes at $90^\circ$ to each other, so no light gets through. Unpolarized light of intensity $I_0$ enters the first polarizer. A third ideal polarizer is then inserted between them, with its axis at $45^\circ$ to the first.
(a) Find the final intensity as a fraction of $I_0$. (b) Explain why adding a filter can increase the light that gets through. (c) Find the angle for the middle polarizer that gives the largest final intensity, and compute that intensity as a fraction of $I_0$.
At a theater, Sofia watches a 3-D movie with passive polarized glasses. The projector sends two images with perpendicular polarizations, and each lens of the glasses…
At a theater, Sofia watches a 3-D movie with passive polarized glasses. The projector sends two images with perpendicular polarizations, and each lens of the glasses transmits only one of them.
(a) Explain how the glasses let each eye see a different image. (b) Sofia tilts her head by $30^\circ$. Using Malus's law, find the fraction of the intended image's intensity that reaches the correct eye and the fraction of the other image that leaks into it. (c) Explain why she sees ghosting. (d) Explain why an LCD screen viewed through polarized sunglasses can look dark when the glasses are rotated to certain angles.
Back to the start: polarized sunglasses have vertical axes, so they block the horizontally polarized glare off the lake. Tilting your head turns the axes and lets the glare back in.