What is polarisation?
Light is an electromagnetic wave. Its electric field E shakes at right angles to the direction of travel, so light is a transverse wave.
In unpolarised light (sun, bulb, candle) the direction of shaking keeps changing at random. In plane polarised light the field shakes in only one plane. Changing unpolarised light into polarised light is called polarisation.
Only transverse waves can be polarised. Sound in air is longitudinal, so it cannot be polarised. This is the strongest proof that light is transverse.
Polaroids and Malus's law
A polaroid is a thin plastic sheet with long molecules lined up in one direction. It passes the part of E along its pass axis and absorbs the rest.
- Unpolarised light of intensity I₀ through one polaroid → I₀/2, whatever way the polaroid faces.
- Polarised light of intensity I₁ through a second polaroid (the analyser) at angle θ → I = I₁ cos²θ. This is Malus's law.
Why squared? The amplitude that passes is E cosθ, and intensity depends on amplitude squared.
Crossed polaroids
At θ = 90°, cos 90° = 0, so no light passes. Turning the analyser once round gives two bright and two dark positions.
Polarisation by reflection and scattering
When light reflects from glass or water, the reflected light is partly polarised. At one special angle, the Brewster angle θB, it is fully polarised and the reflected and refracted rays are at 90° to each other. Brewster's law: tan θB = n (refractive index). For glass, n = 1.5 gives θB ≈ 56°; for water, n = 1.33 gives about 53°.
Sunlight scattered by air molecules is also partly polarised, strongest at 90° from the Sun. Bees use this to find direction.
Uses of polarisation
- Polarised sunglasses and camera filters cut glare.
- LCD screens in phones, calculators and watches use two polarisers with liquid crystal between them.
- 3D cinema glasses send a different polarised picture to each eye.
- Engineers view plastic models between crossed polaroids to see stress as colour bands.
- Chemists measure sugar concentration by how much a solution turns the plane of polarisation.
Try it at home
Look at a phone or laptop screen through polarised sunglasses and slowly rotate the glasses. Note the angle at which the screen looks darkest. That is 90° between the screen's polariser and the glasses.
Key formulas and definitions
- Unpolarised I₀ through one polaroid: I = I₀ / 2
- Malus's law: I = I₁ cos²θ (θ = angle between pass axes)
- Crossed polaroids (θ = 90°): I = 0
- Brewster's law: tan θ_B = n; at θ_B, reflected ⟂ refracted ray (θ_B + r = 90°)
Worked examples
1. Unpolarised light of intensity 80 W m⁻² falls on a polaroid. What comes out?
One polaroid passes half: I = 80 / 2 = 40 W m⁻².
2. Polarised light of 40 W m⁻² meets an analyser at 60°. Find the intensity.
cos 60° = 0.5, cos² 60° = 0.25. I = 40 × 0.25 = 10 W m⁻².
3. Unpolarised light I₀ passes two polaroids at 45°. What fraction comes out?
After first: I₀/2. After second: (I₀/2) × cos²45° = (I₀/2) × ½ = I₀/4. So one quarter.
4. Find the Brewster angle for water (n = 1.33).
tan θ_B = 1.33 → θ_B = tan⁻¹(1.33) ≈ 53°.
5. Three polaroids: first and last are crossed, the middle is at 45° to both. Unpolarised I₀ enters. Find the output.
After 1st: I₀/2. After 2nd (45°): I₀/2 × ½ = I₀/4. After 3rd (45° to 2nd): I₀/4 × ½ = I₀/8. Light gets through even though 1st and 3rd are crossed!
6. At what angle should the analyser be set so that the intensity is 75% of the polarised light falling on it?
cos²θ = 0.75 → cosθ = 0.866 → θ = 30°.
Common mistakes
- Using cosθ instead of cos²θ in Malus's law.
- Forgetting that the first polaroid halves unpolarised light (Malus's law does not apply to unpolarised light).
- Thinking sound can be polarised – only transverse waves can.
- Thinking crossed polaroids at 90° always stop light even when a third polaroid is placed between them.