Polarised light and Malus' law

Sunglasses and a rope

Hold a long rope that passes through a narrow vertical gap in a fence. Shake the rope up and down, and the waves go through the gap. Shake it sideways, and the waves stop. A transverse wave can have its vibration in different directions, and a gap in one direction lets only that direction through. Light does the same, with a Polaroid in place of the gap.

Light is a transverse wave. Its electric field vibrates at right angles to the direction of travel. The light of a lamp or the Sun is unpolarised: it is a mixture of vibrations in every direction at right angles to the beam. Plane polarised light (also called linearly polarised light) vibrates in one direction only. Sound in air is a longitudinal wave, so it has no direction at right angles to its motion, and it cannot be polarised.

The Polaroid

A Polaroid is a plastic sheet with long molecules lined up in one direction. They absorb the part of the electric field along them. The light that leaves has its vibration along one direction, the pass direction (or transmission axis) of the sheet.

For unpolarised light, every direction is present equally. Half of the intensity of the unpolarised light goes through, whatever the direction of the pass axis. So

The light that comes out is plane polarised.

Malus' law

Now let plane polarised light of amplitude $E_0$ and intensity $I_p$ fall on a Polaroid, called the analyser, whose pass direction makes an angle $\theta$ with the direction of the vibration. Only the part of the field along the pass direction, $E_0\cos\theta$, goes through. The intensity goes as the square of the amplitude:

This is Malus' law. At $\theta = 0$ the whole beam passes. At $\theta = 90^\circ$ the Polaroids are crossed, and no light passes.

With the middle sheet at $45^\circ$, what is the intensity after the third sheet? It is the greatest, $I_0/8$. Two crossed Polaroids give darkness, but a third one between them lets light through again. The middle sheet turns the vibration, so the last sheet has a part of it along its pass direction.

A worked example. Unpolarised light of intensity $96\ \text{W/m}^2$ falls on three Polaroids in a row. The first has its pass direction vertical, the second is turned by $30^\circ$ from the first, and the third is horizontal. Find the intensity after each sheet.

1. After the first sheet. $I_1 = \dfrac{I_0}{2} = \dfrac{96\ \text{W/m}^2}{2} = 48\ \text{W/m}^2$.
2. After the second sheet. $I_2 = I_1\cos^2 30^\circ = 48\ \text{W/m}^2 \times 0.75 = 36.0\ \text{W/m}^2$.
3. After the third sheet. The third sheet is $90^\circ - 30^\circ = 60^\circ$ from the second. So $I_3 = I_2\cos^2 60^\circ = 36.0\ \text{W/m}^2 \times 0.25 = 9.0\ \text{W/m}^2$.
4. Check. With only the first and the third sheets, nothing would get through. The middle sheet lets $9.4\ \%$ of the first beam out.

Partly polarised light

Light can be a mixture: an unpolarised part with intensity $I_u$ and a plane polarised part with intensity $I_p$. Turn an analyser round in this light. The unpolarised part always passes with $I_u/2$. The polarised part follows Malus' law. So

The degree of polarisation is $\dfrac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}$. It is $1$ for plane polarised light and $0$ for unpolarised light.

Other books. The pass direction is also called the transmission axis. A-Level and IB write Malus' law as $I = I_0\cos^2\theta$, with $I_0$ the intensity of the polarised light that falls on the analyser. A Polaroid is one kind of polariser. The light can also be polarised by reflection (next lesson), by scattering and by double refraction in a crystal such as calcite.

Practise with new numbers

Try it in Python

The program turns an analyser through half a circle in a beam that is partly polarised, and finds the greatest and least intensities and the degree of polarisation.

Common mistakes

• Using $\cos\theta$ in place of $\cos^2\theta$. The amplitude goes as $\cos\theta$. The intensity goes as its square.
• Forgetting the halving. Unpolarised light loses half of its intensity in the first Polaroid. Malus' law is for light that is already polarised.
• Applying the halving twice. The light after the first sheet is polarised. The second and third sheets use $\cos^2\theta$ only.
• Measuring the angle from the wrong direction. Use the angle between the vibration of the light that arrives and the pass direction of the sheet it meets.

Remember these

• Light is a transverse wave and can be polarised. Sound in air is longitudinal and cannot.
• Unpolarised light through a Polaroid: $I = I_0/2$.
• Malus' law: $I = I_p\cos^2\theta$.
• Partly polarised: $I_{\max} = I_p + I_u/2$ and $I_{\min} = I_u/2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Can the sound of a bell be polarised? Say why.

2. Unpolarised light goes through two Polaroids that are crossed. What is the intensity of the light that comes out?

3. Polarised light of intensity $I_p$ falls on an analyser at $60^\circ$. What fraction of $I_p$ comes out?

4. Unpolarised light of intensity $I_0$ falls on one Polaroid. What is the intensity after it, and what does it do if the Polaroid is turned round the beam?

References

• OpenStax, University Physics Volume 3, Chapter 1 (The Nature of Light), polarization and Malus' law.
• NCERT Physics Class 12, Chapter 10 (Wave Optics): polarisation, Polaroids and Malus' law.