Editing Waveplate (section)

== Principles of operation ==
[[File:Polarization change in uniaxial crystal.gif|300px|thumb|right|A wave in a uniaxial crystal will separate in two components, one parallel and one perpendicular to the optic axis, that will accumulate phase at different rates. This can be used to manipulate the polarization state of the wave.]]
[[Image:Optical-waveplate-1inch.jpg|thumb|A waveplate mounted in a rotary mount]]

A waveplate works by shifting the [[phase (waves)|phase]] between two perpendicular polarization components of the light wave. A typical waveplate is simply a [[Birefringence|birefringent]] crystal with a carefully chosen orientation and thickness. The crystal is cut into a plate, with the orientation of the cut chosen so that the [[Optic axis of a crystal|optic axis]] of the crystal is parallel to the surfaces of the plate. This results in two axes in the plane of the cut: the ''ordinary axis'', with index of refraction ''n''<sub>o</sub>, and the ''extraordinary axis'', with index of refraction ''n''<sub>e</sub>. The ordinary axis is perpendicular to the optic axis. The extraordinary axis is parallel to the optic axis. For a light wave normally incident upon the plate, the polarization component along the ordinary axis travels through the crystal with a speed ''v''<sub>o</sub> = ''c''/''n''<sub>o</sub>, while the polarization component along the extraordinary axis travels with a speed ''v''<sub>e</sub> = ''c''/''n''<sub>e</sub>. This leads to a phase difference between the two components as they exit the crystal. When ''n''<sub>e</sub>&nbsp;< ''n''<sub>o</sub>, as in [[calcite]], the extraordinary axis is called the ''fast axis'' and the ordinary axis is called the ''slow axis''. For ''n''<sub>e</sub>&nbsp;> ''n''<sub>o</sub> the situation is reversed.

Depending on the thickness of the crystal, light with polarization components along both axes will emerge in a different polarization state. The waveplate is characterized by the amount of relative phase, Γ, that it imparts on the two components, which is related to the birefringence Δ''n'' and the thickness ''L'' of the crystal by the formula

:<math>\Gamma = \frac{2 \pi\, \Delta n\, L}{\lambda_0},</math>

where λ<sub>0</sub> is the vacuum wavelength of the light.

Waveplates in general, as well as [[polarizer]]s, can be described using the [[Jones calculus|Jones matrix]] formalism, which uses a vector to represent the polarization state of light and a matrix to represent the linear transformation of a waveplate or polarizer.

Although the birefringence Δ''n'' may vary slightly due to [[dispersion (optics)|dispersion]], this is negligible compared to the variation in phase difference according to the wavelength of the light due to the fixed path difference (λ<sub>0</sub> in the denominator in the above equation). Waveplates are thus manufactured to work for a particular range of wavelengths. The phase variation can be minimized by stacking two waveplates that differ by a tiny amount in thickness back-to-back, with the slow axis of one along the fast axis of the other. With this configuration, the relative phase imparted can be, for the case of a quarter-wave plate, one-fourth a wavelength rather than three-fourths or one-fourth plus an integer. This is called a ''zero-order waveplate''.

For a single waveplate changing the wavelength of the light introduces a linear error in the phase. Tilt of the waveplate enters via a factor of 1/cos θ (where θ is the angle of tilt) into the path length and thus only quadratically into the phase. For the extraordinary polarization the tilt also changes the refractive index to the ordinary via a factor of cos θ, so combined with the path length, the phase shift for the extraordinary light due to tilt is zero.

A polarization-independent phase shift of zero order needs a plate with thickness of one wavelength.
For calcite the refractive index changes in the first decimal place, so that a true zero order plate is ten times as thick as one wavelength.
For [[quartz]] and [[magnesium fluoride]] the refractive index changes in the second decimal place and true zero order plates are common for wavelengths above 1&nbsp;μm.