# Waveplate A half-wave plate. Linearly polarized light entering a waveplate can be resolved into two waves, parallel (shown as green) and perpendicular (blue) to the optical axis of the waveplate. In the plate, the parallel wave propagates slightly slower than the perpendicular one. At the far side of the plate, the parallel wave is exactly half of a wavelength delayed relative to the perpendicular wave, and the resulting combination (red) is a mirror-image of the entry polarization state (relative to the optical axis).

A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. Two common types of waveplates are the half-wave plate, which shifts the polarization direction of linearly polarized light, and the quarter-wave plate, which converts linearly polarized light into circularly polarized light and vice versa. A quarter wave plate can be used to produce elliptical polarization as well.

Waveplates are constructed out of a birefringent material (such as quartz or mica), for which the index of refraction is different for different orientations of light passing through it. The behavior of a waveplate (that is, whether it is a half-wave plate, a quarter-wave plate, etc.) depends on the thickness of the crystal, the wavelength of light, and the variation of the index of refraction. By appropriate choice of the relationship between these parameters, it is possible to introduce a controlled phase shift between the two polarization components of a light wave, thereby altering its polarization.

## Principles of operation

A waveplate works by shifting the phase between two perpendicular polarization components of the light wave. A typical waveplate is simply a 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 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 no, and the extraordinary axis, with index of refraction ne. For a light wave normally incident upon the plate, polarization component along the ordinary axis travels through the crystal with a speed vo = c/no, while the polarization component along the extraordinary axis travels with a speed ve = c/ne. This leads to a phase difference between the two components as they exit the crystal. When ne < no, as in calcite, the extraordinary axis is called the fast axis and the ordinary axis is called the slow axis. For ne > no 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 Δn and the thickness L of the crystal by the formula

$\Gamma ={\frac {2\pi \,\Delta n\,L}{\lambda _{0}}},$ where λ0 is the vacuum wavelength of the light.

Waveplates in general as well as polarizers can be described using the 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 Δn may vary slightly due to dispersion, this is negligible compared to the variation in phase difference according to the wavelength of the light due to the fixed path difference (λ0 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 µm.

## Half-wave plate

For a half-wave plate, the relationship between L, Δn, and λ0 is chosen so that the phase shift between polarization components is Γ = π. Now suppose a linearly polarized wave with polarization vector ${\mathbf {\hat {p}} }$ is incident on the crystal. Let θ denote the angle between ${\mathbf {\hat {p}} }$ and ${\mathbf {\hat {f}} }$ , where ${\mathbf {\hat {f}} }$ is the vector along the waveplate's fast axis. Let z denote the propagation axis of the wave. The electric field of the incident wave is

${\mathbf {E} }\,{\mathrm {e} }^{i(kz-\omega t)}=E\,{\mathbf {\hat {p}} }\,{\mathrm {e} }^{i(kz-\omega t)}=E(\cos \theta \,{\mathbf {\hat {f}} }+\sin \theta \,{\mathbf {\hat {s}} }){\mathrm {e} }^{i(kz-\omega t)},$ where ${\mathbf {\hat {s}} }$ lies along the waveplate's slow axis. The effect of the half-wave plate is to introduce a phase shift term eiΓ = eiπ = −1 between the f and s components of the wave, so that upon exiting the crystal the wave is now given by

$E(\cos \theta \,{\mathbf {\hat {f}} }-\sin \theta \,{\mathbf {\hat {s}} }){\mathrm {e} }^{i(kz-\omega t)}=E[\cos(-\theta ){\mathbf {\hat {f}} }+\sin(-\theta ){\mathbf {\hat {s}} }]{\mathrm {e} }^{i(kz-\omega t)}.$ If ${\mathbf {\hat {p}} }'$ denotes the polarization vector of the wave exiting the waveplate, then this expression shows that the angle between ${\mathbf {\hat {p}} }'$ and ${\mathbf {\hat {f}} }$ is −θ. Evidently, the effect of the half-wave plate is to mirror the wave's polarization vector through the plane formed by the vectors ${\mathbf {\hat {f}} }$ and $\mathbf {\hat {z}}$ . For linearly polarized light, this is equivalent to saying that the effect of the half-wave plate is to rotate the polarization vector through an angle 2θ; however, for elliptically polarized light the half-wave plate also has the effect of inverting the light's handedness.

## Quarter-wave plate

For a quarter-wave plate, the relationship between L, Δn, and λ0 is chosen so that the phase shift between polarization components is Γ = π/2. Now suppose a linearly polarized wave is incident on the crystal. This wave can be written as

$(E_{f}\mathbf {\hat {f}} +E_{s}\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)},$ where the f and s axes are the quarter-wave plate's fast and slow axes, respectively, the wave propagates along the z axis, and Ef and Es are real. The effect of the quarter-wave plate is to introduce a phase shift term eiΓ =eiπ/2 = i between the f and s components of the wave, so that upon exiting the crystal the wave is now given by

$(E_{f}{\mathbf {\hat {f}} }+iE_{s}{\mathbf {\hat {s}} }){\mathrm {e} }^{i(kz-\omega t)}.$ The wave is now elliptically polarized.

If the axis of polarization of the incident wave is chosen so that it makes a 45° with the fast and slow axes of the waveplate, then Ef = Es ≡ E, and the resulting wave upon exiting the waveplate is

$E({\mathbf {\hat {f}} }+i{\mathbf {\hat {s}} }){\mathrm {e} }^{i(kz-\omega t)},$ and the wave is circularly polarized.

If the axis of polarization of the incident wave is chosen so that it makes a 0° with the fast or slow axes of the waveplate, then the polarization will not change, so remains linear. If the angle is in between 0° and 45° the resulting wave has an elliptical polarization.

A circulating polarization looks strange, but can be easier imagined as the sum of two linear polarizations with a phase difference of 90°. The output depends on the polarization of the input. Suppose polarization axes x and y parallel with the fast and slow axis of the wave plate:

The polarization of the incoming photon (or beam) can be resolved as two polarizations on the x and y axis. If the input polarization is parallel to the fast or slow axis, then there is no polarization of the other axis, so the output polarization is the same as the input (only the phase more or less delayed). If the input polarization is 45° to the fast and slow axis, the polarization on those axes are equal. But the phase of the output of the slow axis will be delayed 90° with the output of the fast axis. If not the amplitude but both sinus values are displayed, then x and y combined will describe a circle. With other angles than 0° or 45° the values in fast and slow axis will differ and their resultant output will describe an ellipse.
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