Double refraction, linear and circular light

Double refraction is the splitting of a ray of unpolarized light into two plane-polarized rays on entering an anisotropic crystal such as calcite or quartz. The ordinary (o) ray obeys Snell's law and has the same speed in all directions. The extraordinary (e) ray does not obey Snell's law and its speed depends on direction. Their vibrations are perpendicular. Along the optic axis there is no double refraction.

Linearly polarized light

1. Pass natural light through a calcite crystal (or Nicol prism). The two emergent rays are plane polarized perpendicular to each other.
2. In a Nicol prism the o-ray is removed by total internal reflection at the Canada balsam layer, so only the e-ray, which is plane polarized, emerges.
3. Alternative: use a tourmaline plate, which absorbs the o-ray.

Circularly polarized light

 unpolarized -> [Nicol]-> plane pol. -> [QWP, axis at 45°] -> circular
                         (A)               a = b = A/√2, δ = 90°

1. Plane polarized light of amplitude A falls on a quarter-wave plate (a crystal plate cut parallel to the optic axis), with its vibration at θ = 45° to the optic axis.
2. It splits into e and o components: a = A cos 45° and b = A sin 45°, so a = b = A/√2.
3. The plate thickness is chosen so that the path difference is λ/4, i.e. phase difference δ = π/2: t = λ / [4(μ_o − μ_e)].
4. With x = a cos ωt and y = a sin ωt (quadrature), the resultant is x² + y² = a², a circle.

Hence the emergent light is circularly polarized (right or left handed depending on which ray is faster). If θ ≠ 45° with the same plate, the light is elliptically polarized.