Polarization & Jones Vectors (IPhO Optics)

Polarization is phase geometry: it is about the relative amplitude and relative phase of two perpendicular field components. Jones vectors turn that into a compact complex-vector algebra, and most IPhO problems then reduce to “multiply a few matrices and square a magnitude”.

Practice feedback

• A quarter-wave plate gives circular output only when the incident field has equal-magnitude components along its fast and slow axes and the plate supplies a π/2 phase difference.
• Orthogonal polarisations have zero analyser-free interference cross term. A common analyser projects both fields onto one axis, allowing their projected components to interfere when coherence is retained.

1. Definitions (Must Know)

• Jones vector: a complex 2-vector of field amplitudes, E = (Eₓ; E_y), defined up to an overall (global) phase.
• Linear polarization at angle α: E ∝ (cos α; sin α).
• Circular polarization: equal amplitudes with a quarter-cycle phase difference:
  E ∝ 1/(square root of 2) (1; ± i) .
• Elliptical polarization: the general case with unequal amplitudes and a nonzero relative phase.
• Intensity (for a given Jones vector): I ∝ |Eₓ|² + |E_y|².
• Rotation matrix: R(α) = (cos α, - sin α; sin α, cos α).
• Ideal polarizer (transmits x): Pₓ = (1, 0; 0, 0), and at angle α: P(α) = R(-α)Pₓ R(α).
• Wave plate retardance δ (fast axis x): W(δ) = (1, 0; 0, e^iδ), and at angle α: W(α,δ) = R(-α)W(δ)R(α).

2. Key Ideas (What Earns Marks)

• Global phase does not matter; only the relative phase between components changes the polarization ellipse.
• If you only need intensity after a polarizer, you can often treat it as a projection problem (Malus’ law) instead of full matrix multiplication.
• Quarter-wave plates change linear into circular only when the input has equal components along the fast and slow axes.
• Half-wave plates rotate linear polarization: the output angle is mirrored about the fast axis, giving a rotation by twice the misalignment angle.
• Two beams do not produce interference fringes in total intensity if their polarizations are orthogonal, but an analyzer can project them onto a common polarization and restore fringes.

3. Detailed Explanations

3.1 Jones vectors and the meaning of complex components

Write the transverse electric field as

If Eₓ and E_y have a phase difference, the tip of the real field vector traces an ellipse. Multiplying both components by the same phase factor e^iφ does not change the physical polarization.

3.2 Polarizers and wave plates as matrices

If an element is linear and deterministic, it acts as

If you chain elements, multiply matrices in the order the light experiences them.

3.3 Malus’ law from Jones vectors

If the input is linear at angle α and the analyzer transmits angle β, the transmitted field is proportional to the projection:

So the transmitted intensity is

3.4 Fast recognition patterns

• QWP at 45 degrees: linear input at 45 degrees to the fast axis gives circular output.
• HWP: linear input at angle α becomes linear at angle 2θ-α if the fast axis is at angle θ.

4. Common Mistakes

• Treating Jones vectors as valid for unpolarized light without switching to Stokes/Mueller methods.
• Forgetting that intensity depends on squared magnitude, not on the complex amplitude itself.
• Mixing up the fast axis and slow axis (sign of the relative phase shift).
• Dropping the overall normalization and then comparing absolute intensities incorrectly.
• Assuming orthogonal polarizations always mean “no interference” even after an analyzer is introduced.

5. Exam Tips

6. Worked Examples

7. Mind Stretchers

8. Practice