Difficulty: Intermediate | Prerequisites: EM wave basics (displacement current, plane wave equations, intensity), trigonometric identities.
Polarisation describes the direction in which the electric field oscillates in an electromagnetic wave. It matters because most natural light is unpolarised, and understanding what happens when it passes through polarising filters or special materials is tested heavily. This builds directly on the plane-wave description from the displacement current / EM waves material. If you are comfortable with the E-field of a harmonic plane wave and know what intensity means, you are ready for this.
Polarisation is the orientation of the electric-field oscillation in an EM wave. Passing unpolarised light through a polariser halves its intensity; passing already-polarised light through a second polariser reduces it by cos²(θ) (Malus's law). Linear and circular polarisation differ by the phase relationship between the x- and y-components, and birefringent materials can convert between them.
Polarisation
The direction in which the electric field vector oscillates in a transverse electromagnetic wave. Only transverse waves can be polarised.
In simple terms: it tells you which way the E-field is waving as the light moves forward.
Unpolarised light
Light whose electric field oscillates in all directions perpendicular to propagation, with no preferred orientation. Sunlight and most artificial light sources produce unpolarised light.
Malus's law
For polarised light of intensity I₀ passing through a polariser whose transmission axis is at angle θ relative to the polarisation direction: I_final = I₀ cos²(θ₁ − θ₂). Named after Etienne-Louis Malus.
Think of it as: the closer the polariser's axis is to the light's polarisation direction, the more light gets through. At 90° nothing passes.
Linear polarisation
The case where the E-field oscillates along a single fixed direction. The phase difference between the x- and y-components is δ = φ_x − φ_y = 0 (or π).
Circular polarisation
The case where the E-field vector traces out a circle as the wave propagates. Requires equal-amplitude x- and y-components with a phase difference δ = ±π/2.
In simple terms: instead of waving back and forth in a line, the E-field corkscrews around the propagation axis.
Birefringent material
A material with two different refractive indices (and therefore two different wave speeds) for light polarised along different axes. It introduces a phase difference between the fast and slow components: Δφ = ωd(1/v_fast − 1/v_slow).
Think of it as: the material slows down one polarisation component more than the other, which can turn linearly polarised light into circularly polarised light (or vice versa) depending on the thickness.
The general E-field of a polarised wave: E = E₀ sin(kz − ωt + φ)
The direction of E defines the polarisation. For a wave travelling in the z-direction, E can oscillate in any direction in the x-y plane.
A polariser transmits only the component of E along its transmission axis.
Unpolarised light through a single polariser: I_final = ½ I₀. The factor of ½ comes from averaging cos² over all angles.
Already-polarised light through a polariser: apply Malus's law, I_final = I₀ cos²(θ₁ − θ₂), where the angles are those of the incoming polarisation and the polariser axis.
Both x- and y-components of E oscillate in phase (δ = φ_x − φ_y = 0).
The tip of the E-field vector traces a straight line.
The direction of that line is set by the relative amplitudes of E_x and E_y.
Requires equal amplitudes in the x- and y-components and a phase difference δ = ±π/2.
+π/2 gives right-circular polarisation; −π/2 gives left-circular (convention varies by textbook, so check yours).
The tip of the E-field vector traces a circle.
A birefringent crystal has a "fast axis" and a "slow axis" with different refractive indices.
Each component of E along these axes travels at a different speed: phase accumulated = ωd/v.
The phase difference introduced: Δφ = φ_f − φ_s = ωd(1/v_fast − 1/v_slow).
A quarter-wave plate is a birefringent slab cut so that Δφ = π/2, converting linearly polarised light (at 45° to the axes) into circularly polarised light.
Polarised wave: E = E₀ sin(kz − ωt + φ)
Unpolarised light through one polariser: I_final = ½ I₀
Malus's law: I_final = I₀ cos²(θ₁ − θ₂)
Linear polarisation condition: δ = φ_x − φ_y = 0
Circular polarisation condition: δ = φ_x − φ_y = ±π/2
Phase in birefringent material: φ = ωd / v_slow/fast
Phase difference from birefringence: Δφ = φ_f − φ_s = ωd(1/v_fast − 1/v_slow)
Polarising sunglasses work by blocking horizontally polarised light, which is the dominant polarisation of glare reflected from flat surfaces like roads and water. LCD screens rely on polarisers and birefringent liquid-crystal layers to control which light passes through each pixel. 3D cinema glasses use circular polarisation (one handedness per eye) so that tilting your head does not ruin the effect.
Students often apply Malus's law to unpolarised light. It only applies when the incident light is already polarised. For unpolarised light hitting its first polariser, the rule is I_final = ½ I₀.
A common error is forgetting to square the cosine. Malus's law gives cos², not cos.
Students sometimes think circular polarisation requires different amplitudes in x and y. It requires equal amplitudes; it is the phase difference (±π/2) that matters.
Confusing the sign convention for right- vs left-circular polarisation. This varies between textbooks, so always check which convention your course uses.
⚠️ Multi-polariser problems are common: remember to apply the ½ rule for the first (unpolarised → polarised) stage, then Malus's law for each subsequent polariser.
⚠️ Know the phase-difference conditions cold: δ = 0 for linear, δ = ±π/2 for circular.
⚠️ Birefringent-material problems typically give you ω, d, v_fast, and v_slow and ask for the resulting polarisation state. Compute Δφ and compare to 0, π/2, π.
⚠️ Be careful with angle conventions. The angle in Malus's law is between the polarisation direction of the incoming light and the transmission axis of the analyser.
True or false: unpolarised light through a single polariser has intensity I₀ cos²θ.
Fill in the blank: circular polarisation requires equal amplitudes and a phase difference of ______.
True or false: Malus's law involves cos² (not cos) of the angle between polarisation direction and transmission axis.
Fill in the blank: a birefringent material introduces a phase difference because the two polarisation components travel at ______ speeds.
True or false: if δ = π, the light is circularly polarised.
Q: Unpolarised light of intensity 120 W/m² passes through two polarisers. The first is vertical; the second is at 30° from vertical. What is the final intensity?
A: After the first polariser: I₁ = ½ × 120 = 60 W/m². After the second (Malus's law): I₂ = 60 × cos²(30°) = 60 × 0.75 = 45 W/m².
Q: What is the phase difference δ for linearly polarised light?
A: δ = φ_x − φ_y = 0 (or an integer multiple of π for the general case, but the standard answer for this course is 0).
Q: Light passes through a birefringent slab of thickness d. The angular frequency is ω, v_fast = 2.0 × 10⁸ m/s, v_slow = 1.5 × 10⁸ m/s. Write an expression for the phase difference Δφ.
A: Δφ = ωd(1/v_fast − 1/v_slow) = ωd(1/(2.0 × 10⁸) − 1/(1.5 × 10⁸)). Note Δφ is negative here, meaning the slow component accumulates more phase.
Q: Two polarisers are crossed (90° apart). No light gets through. A third polariser is inserted between them at 45° to both. What fraction of the original unpolarised intensity emerges?
A: After polariser 1: ½ I₀. After polariser 2 (at 45°): ½ I₀ × cos²(45°) = ½ I₀ × ½ = ¼ I₀. After polariser 3 (another 45°): ¼ I₀ × cos²(45°) = ¼ I₀ × ½ = ⅛ I₀. So ⅛ of the original intensity passes through.
Polarisation builds directly on the plane-wave description from the displacement current and EM waves material. The energy relations (intensity halving, Malus's law) rely on the same intensity formula I = ½ cε₀E₀². Birefringence connects to optics and crystallography, and circular polarisation becomes important again in quantum mechanics when discussing photon spin states.
Polarisation, polarization, Malus's law, Law of Malus, unpolarised light, linear polarisation, circular polarisation, right-circular, left-circular, birefringence, birefringent material, quarter-wave plate, half-wave plate, fast axis, slow axis, phase difference, crossed polarisers, polarising filter, PHYS 212, University Physics Electricity and Magnetism