Polarisation

Polaroids and Malus’s law

Narrated video. Transcript:

Shake one end of a long string up and down. A transverse wave runs along it, and every point of the string moves only up and down, in one plane. Such a wave is called linearly polarised, or plane polarised.

Pass the string through a vertical slit. Up-and-down waves go straight through. Shake it sideways, and the slit stops the wave. The slit lets through only vibrations along its own direction.

Light is a transverse wave too: its electric field vibrates at right angles to its direction of travel. In light from a bulb or the Sun, the direction of the field changes randomly and very fast. Such light is unpolarised.

A polaroid is a thin film with long-chain molecules all lined up in one direction. The electric field along the molecules is absorbed; the component perpendicular to them passes. That direction is the pass axis of the polaroid.

Send unpolarised light through a polaroid P₁. Only the component along the pass axis gets through, so the intensity is halved. Turn P₁ round: the light coming out stays just as bright.

Now put a second polaroid, P₂, in the beam, and turn it. When its pass axis is parallel to P₁’s, all the light from P₁ passes. When the two axes are at right angles, crossed, no light gets through.

In between, let the pass axes make an angle θ. The field E₀ from P₁ has a component E₀ cos θ along the axis of P₂. Intensity goes as the square of the field, so I = I₀ cos²θ, where I₀ is the intensity after P₁. This is Malus’s law.

Example: unpolarised light of intensity 32 W/m² falls on P₁, and P₂ is at 60° to it. After P₁, the intensity is 16 W/m². After P₂, it is 16 × cos² 60° = 16 × ¼ = 4 W/m².

Polaroids are used to control the intensity of light: in sunglasses, in window panes, in photographic cameras, and in 3D movie cameras and glasses.

Polarisation by scattering

Narrated video. Transcript:

Look at the clear blue sky through a polaroid, and turn the polaroid. The sky becomes brighter and darker. So light from the sky is partly polarised, although sunlight itself is unpolarised.

Sunlight falls on the molecules of the air. Its electric field makes the electrons in each molecule oscillate. These oscillating charges send out light in all directions: the light is scattered.

Look at a molecule from a direction at 90° to the sunlight. Draw the incoming field as two components: one in the plane of the picture, shown by double arrows, and one perpendicular to it, shown by dots.

Charges oscillating along the double arrows move along our line of sight, towards and away from us. An oscillating charge sends no light along its own direction of oscillation. So this part does not reach us.

Only the charges oscillating along the dots send light towards us. So the light scattered at 90° to the sunlight is polarised, with its electric field perpendicular to the plane of the picture.

The effect depends on the direction you look. Light scattered at 90° to the sunlight is the most strongly polarised. Light from the sky close to the Sun is hardly polarised at all.

The same scattering gives the sky its colour. Air molecules scatter short wavelengths much more strongly than long ones, so the blue part of sunlight is scattered across the sky.

Try it: look at a patch of sky at 90° from the Sun through a polaroid, and turn it through a full circle. The sky becomes much darker at two positions, 180° apart, and brightest halfway between them.

Polarisation by reflection: Brewster’s law

Narrated video. Transcript:

Light reflected from water or glass is also partly polarised. That is why polaroid sunglasses can cut the glare from a wet road or a lake.

Let unpolarised light fall on a glass surface. Show its electric field as two parts: dots, perpendicular to the plane of incidence, and double arrows, in the plane of incidence.

Part of the light is reflected, and part refracted. At most angles, the reflected light has both parts, but more dots than arrows: it is partly polarised.

At one special angle, the reflected and refracted rays are exactly at right angles. Then the reflected light has only dots: it is completely plane polarised, with its electric field perpendicular to the plane of incidence.

Why? The electrons in the glass that oscillate along the double arrows would have to send light along the reflected ray, but that is their own direction of oscillation. They cannot radiate that way, so the reflected ray gets only dots.

This angle of incidence is called Brewster’s angle, i_B. Because the two rays are perpendicular, i_B + r = 90°, so r = 90° − i_B.

By Snell’s law, n = sin i_B / sin r = sin i_B / sin(90° − i_B) = sin i_B / cos i_B. So tan i_B = n. This is Brewster’s law.

Example: for glass with n = 1.5, i_B = tan⁻¹ 1.5 ≈ 56.3°. For water, n = 1.33, so i_B = tan⁻¹ 1.33 ≈ 53°.

Glare from a road or a lake is reflected light with its field mostly horizontal. Sunglasses with a vertical pass axis block it, while still letting through much of the rest of the light.

Try it: polaroids and reflection

Interactive simulation. Things to try:

• Drag the orange knob on a polaroid disc to turn its pass axis. The circles above the beam show the electric field in each part of the beam.
• Turn P₁ alone: the light after it stays at I₀/2. Turn P₂: the graph traces (I₀/2) cos²θ.
• Cross P₁ and P₂, then put P₃ in the middle and turn it: light comes back!
• Switch to “Reflection” and find the angle where the reflected ray carries only dots. Check that tan i = n there.

A polaroid between crossed polaroids

Unpolarised light of intensity I passes through two crossed polaroids P₁ and P₂. A third polaroid P₃ is then placed between them, with its pass axis at 45° to that of P₁. What intensity now comes out of P₂?

1. 0
2. I/8
3. I/4
4. I/2

Glare from a lake

Sunlight is reflected from the surface of a lake (n = 1.33). At what angle of incidence is the reflected light completely plane polarised?

1. About 37°
2. About 49°
3. About 53°
4. 90°

Summary

• Unpolarised light through one polaroid: I = I₀/2
• Malus’s law: I = I₀ cos²θ
• Brewster’s law: tan i_B = n, i_B + r = 90°

• In plane polarised light, the electric field vibrates in one direction only. Light from ordinary sources is unpolarised.
• A polaroid passes only the field component along its pass axis. Unpolarised light comes out polarised, at half the intensity.
• Through a second polaroid at angle θ, the intensity is I₀ cos²θ (Malus’s law); crossed polaroids let no light through.
• Light scattered at 90° by air molecules is polarised, which is why the blue sky looks brighter or darker through a turning polaroid.
• Reflected light is completely polarised at Brewster’s angle, when the reflected and refracted rays are perpendicular.