← Wave Systems

8 Polarisation

Polarisation restricts a wave's oscillations to a single plane. It is the standard piece of evidence that light is a transverse wave — a longitudinal wave like sound has no direction perpendicular to travel to restrict, so it cannot be polarised at all.

What a polariser does

  • Unpolarised light oscillates in every direction perpendicular to travel. A polarising filter only transmits the component of oscillation aligned with its transmission axis.
  • One polariser: transmits roughly half the intensity (the component along its axis), and the transmitted light is now polarised.
  • A SECOND polariser (an "analyser") then only transmits the component of THAT polarised light along its own axis.

Malus's law

Malus's law I=I_0\cos^2\theta I₀ = intensity after the first polariser. θ = angle between the two polarisers' transmission axes.

The two limiting cases

  • Axes PARALLEL (θ = 0°): cos²0° = 1, so all the already-polarised light passes through — maximum transmitted intensity.
  • Axes CROSSED at 90°: cos²90° = 0, so NO light passes through at all — this is why "crossed polarisers block all light" and is the classic exam demonstration.
  • At any angle in between, intensity varies smoothly following cos²θ, not linearly with θ.
Worked example Intensity through two polarisers Unpolarised light of intensity 120 W m⁻² passes through a polariser, then through a second polariser whose axis is at 40° to the first. Find the intensity that emerges. Through the first polariser, unpolarised light loses half its intensity: I₁ = 120/2 = 60 W m⁻². Apply Malus's law for the second polariser: I = I₁ cos²θ = 60 × cos²(40°). cos(40°) = 0.766, so cos²(40°) = 0.587. I = 60 × 0.587 = 35.2 W m⁻². ✓ I ≈ 35 W m⁻². ✕

Malus's law only applies to the SECOND polariser onward

The halving of intensity through the FIRST polariser (unpolarised light → polarised light) is a separate, simpler rule — it always removes half the intensity regardless of the filter's orientation, because unpolarised light has no preferred axis to measure an angle against. Malus's law, I = I₀cos²θ, only describes what happens from the SECOND polariser onward, where the incoming light is already polarised and θ is a genuine, meaningful angle. Applying cos²θ to the first polariser (or halving intensity at every polariser regardless of order) is the standard mixed-up answer. Polarising sunglasses use this to cut reflected glare (light reflecting off water or roads is partially polarised horizontally, so a vertical transmission axis blocks most of it); LCD screens use crossed polarisers with a layer between them that rotates the polarisation, controlled electronically pixel by pixel.

Practise just this