Polarization; quarter and half wave plates

Polarization of light is the process of restricting the vibrations of the electric vector of light to a single direction (or a regular pattern) perpendicular to the direction of propagation. It proves that light is a transverse wave.

Importance in optical instruments

• Polaroid sunglasses and camera filters cut glare from roads and water.
• LCD screens work by controlling polarized light with liquid crystals.
• Polarimeters/saccharimeters measure optical rotation to find sugar concentration.
• Photoelasticity shows stress patterns in transparent models.
• Polarizing microscopes identify minerals and crystals; 3D cinema uses polarized glasses; polarized lasers and optical isolators are used in optical communication.

Wave plates

A plate cut parallel to the optic axis from a uniaxial crystal produces a path difference (μ_e − μ_o)t between the e and o rays (i.e. a phase difference δ = 2π(μ_e − μ_o)t/λ).

1. Quarter-wave plate (path difference λ/4, δ = π/2; lowest order):

(μ_e − μ_o) t = λ/4   →   t = λ / [4 (μ_e − μ_o)]

For any order: (μ_e − μ_o)t = (2m+1)λ/4. It changes plane polarized light into circularly polarized light (vibration at 45° to the axis) or elliptically polarized light.

2. Half-wave plate (path difference λ/2, δ = π):

(μ_e − μ_o) t = λ/2   →   t = λ / [2 (μ_e − μ_o)]

For any order: (μ_e − μ_o)t = (2m+1)λ/2. It rotates the plane of vibration of plane polarized light by 2θ, where θ is the angle between the incident vibration and the optic axis; for circularly polarized light it reverses the sense of rotation.

In both cases the modulus |μ_e − μ_o| is used (positive crystal, e.g. quartz; negative crystal, e.g. calcite). Example: quartz (μ_e − μ_o = 0.009) at 5890 Å gives t ≈ 16.4 μm for a quarter-wave plate.