Waveplate

A wave passing through a 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 ${\displaystyle \mathbf {\hat {p}} }$ is incident on the crystal. Let θ denote the angle between ${\displaystyle \mathbf {\hat {p}} }$ and ${\displaystyle \mathbf {\hat {f}} }$, where ${\displaystyle \mathbf {\hat {f}} }$ 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

${\displaystyle \mathbf {E} \,\mathrm {e} ^{i(kz-\omega t)}=E\,\mathbf {\hat {p}} \,\mathrm {e} ^{i(kz-\omega t)}=E(\cos \theta \,\mathbf {\hat {f}} +\sin \theta \,\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)},}$

where ${\displaystyle \mathbf {\hat {s}} }$ lies along the waveplate's slow axis. The effect of the half-wave plate is to introduce a phase shift term e^iΓ = e^iπ = −1 between the f and s components of the wave, so that upon exiting the crystal the wave is now given by

${\displaystyle E(\cos \theta \,\mathbf {\hat {f}} -\sin \theta \,\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)}=E[\cos(-\theta )\mathbf {\hat {f}} +\sin(-\theta )\mathbf {\hat {s}} ]\mathrm {e} ^{i(kz-\omega t)}.}$

If ${\displaystyle \mathbf {\hat {p}} '}$ denotes the polarization vector of the wave exiting the waveplate, then this expression shows that the angle between ${\displaystyle \mathbf {\hat {p}} '}$ and ${\displaystyle \mathbf {\hat {f}} }$ 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 ${\displaystyle \mathbf {\hat {f}} }$ and ${\displaystyle \mathbf {\hat {z}} }$. 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.[1]

Two waves differing by a quarter-phase shift for one axis.

Creating circular polarization using a quarter-wave plate and a polarizing filter

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

${\displaystyle (E_{f}\mathbf {\hat {f}} +E_{s}\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)},}$

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 E_f and E_s are real. The effect of the quarter-wave plate is to introduce a phase shift term e^iΓ =e^iπ/2 = i between the f and s components of the wave, so that upon exiting the crystal the wave is now given by

${\displaystyle (E_{f}\mathbf {\hat {f}} +iE_{s}\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)}.}$

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 E_f = E_s ≡ E, and the resulting wave upon exiting the waveplate is

${\displaystyle E(\mathbf {\hat {f}} +i\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)},}$

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 can be visualized 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 waveplate:

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 sine 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.

A full-wave plate introduces a phase difference of exactly one wavelength between the two polarization directions, for one wavelength of light. In optical mineralogy, it is common to use a full-wave plate designed for green light (wavelength = 540 nm). Linearly polarized white light which passes through the plate becomes elliptically polarized, except for 540 nm light which will remain linear. If a linear polarizer oriented perpendicular to the original polarization is added, this green wavelength is fully extinguished but elements of the other colors remain. This means that under these conditions the plate will appear an intense shade of red-violet, sometimes known as "sensitive tint".[3] This gives rise to this plate's alternative names, the sensitive-tint plate or (less commonly) red-tint plate. These plates are widely used in mineralogy to aid in identification of minerals in thin sections of rocks.[2]

A multiple-order waveplate is made from a single birefringent crystal that produces an integer multiple of the rated retardance (for example, a multiple-order half-wave plate may have an absolute retardance of 37λ/2). By contrast, a zero-order waveplate produces exactly the specified retardance. This can be accomplished by combining two multiple-order wave plates such that the difference in their retardances yields the net (true) retardance of the waveplate. Zero-order waveplates are less sensitive to temperature and wavelength shifts, but are more expensive than multiple-order ones.[4]