Production of plane, circular, elliptical light

Polarization is the property of transverse waves by which the vibrations are restricted to a particular direction or pattern. In plane (linear) polarized light the electric vector vibrates in one plane; in circular and elliptical light its tip traces a circle or ellipse.

Principle of double refraction

When unpolarized light enters a doubly refracting (anisotropic) crystal like calcite or quartz, it splits into two rays:

• Ordinary (o) ray: obeys Snell's law, has speed v_o in all directions, vibrates perpendicular to the principal section.
• Extraordinary (e) ray: does not obey Snell's law, speed varies with direction, vibrates in the principal section.

Both are plane polarized with mutually perpendicular vibrations. Along the optic axis both travel with the same speed. (Huygens: o gives spherical, e gives ellipsoidal wavefront.)

Production of polarized light

 plane-polarized   optic axis (OA)
 light, amplitude A  |
 -------->[ crystal plate, thickness t ]--> E and O rays
              angle θ between vibration and OA

1. Plane polarized light of amplitude A falls normally on a plate cut parallel to the optic axis, vibrating at angle θ to it.
2. It splits into e-component (along axis) a = A cos θ and o-component (perpendicular) b = A sin θ; they travel with different speeds.
3. On emerging, path difference = (μ_o − μ_e) t, so phase difference δ = (2π/λ)(μ_o − μ_e) t.
4. Take x = a cos ωt, y = b cos(ωt + δ). Eliminating t:

x²/a² + y²/b² − (2xy/ab) cos δ = sin²δ

5. Special cases:

Thus a quarter-wave plate with θ = 45° gives circular light, with the thickness t = λ/[4(μ_o − μ_e)].