A half-wave plate (or any arbitrary-wavelength plate) works on the principle of Birefringence, which is that the crystal in question has a different refractive index for a different polarization of light.
Most materials that you deal with in basic physics and engineering classes are isotropic, meaning that the refractive index is the same no matter which direction the light travels and no matter what its polarization is.
Most wave plates are made with uniaxial crystals — the refractive index for one polarization direction is different from for other polarization directions. They can also be made with biaxial crystals, but those are usually more expensive. If memory serves me correctly, the most common material for wave plates in inexpensive optics is Calcite since it has decent uniaxial birefringence (that is, the axis with a different refractive index is more than just a tiny bit different from the other axes).
The wave plate consists of a crystal that is cut along its optic axes, so that the optic axis (the axis with a different refractive index) is in the plane of the crystal, so that there will be no separate refraction.
One of the meanings of refractive index is relative phase velocity of light — speed of light is equal to vacuum speed of light divided by refractive index — v = c/n.
Any light entering the wave plate at normal incidence will have two polarization components — the component along the optic axis, and the component orthogonal to the optic axis. The component along the axis with higher refractive index will travel slower than the component with a lower refractive index. The axis with lower refractive index is called the fast axis, and the one with higher refractive index is called the slow axis.
If the input light is polarized in a direction that is not exactly along one of the two crystal axes will have one of the polarization components be slower than the other, which changes the relative phase between them through the crystal.
It starts with linear polarization, then the phase retardation changes it to elliptical, then (if it is linearly polarized with equal fast and slow axis components) to circular, then back to elliptical, and finally to linear but rotated to its supplementary angle. This is described with nothing but the maximum components of the two orthogonal polarization directions and the phase difference between the fast and slow polarizations.
The phase difference is accumulated over length.
The phase difference is equal to the distance times the difference between the fast and slow wavenumbers:
phase difference phi = d*(k_slow – k_fast).
k = n * 2*pi/wavelength
(or k_fast uses the lower n, and k_fast uses the higher n).
At a phase difference of 0 it is linear, and at a phase difference of pi/2 it is linear but at a complementary angle to the 0 phase difference. If the magnitudes of the two orthogonal polarizations are the same, then a phase difference of pi/4 then it will be circular polarization.
Half-wave plates will rotate arbitrarily rotate polarizations according to how you rotate the plate — that is, they add a phase difference of pi/2. Quarter wave plates change 45 degree linearly polarized light into circularly polarized light.
This article comes from quora edit released