What is a Waveplate?
A waveplate or retarder is an optical device that alters the polarization state of a light wave traveling through it. Waveplates, transmit light and modify its polarization state without attenuating, deviating, or displacing the beam. They do this by retarding or delaying one component of polarization to its orthogonal component. In unpolarized light, waveplates are equivalent to windows – they are both flat optical components through which light passes. Waveplates are made from birefringent materials, most commonly crystal quartz. Birefringent materials have slightly different indices of refraction for light polarized in different orientations. As such, they separate incident unpolarized light into its parallel and orthogonal components. The figure below shows a birefringent calcite crystal separating unpolarized light.
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.
Linearly polarized light entering a half-wave plate can be resolved into two waves, parallel and perpendicular to the optic 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 is a mirror-image of the entry polarization state relative to the optic axis as shown in the figure below.
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. A common use of waveplates—particularly the sensitive-tint full-wave and quarter-wave plates—is in optical mineralogy. The addition of plates between the polarizers of a petrographic microscope makes the optical identification of minerals in thin sections of rocks easier, in particular by allowing the deduction of the shape and orientation of the optical indicatrices within the visible crystal sections. Optical indicatrices are imaginary ellipsoidal surfaces whose axes represent the refractive indices of a crystal for light following different directions to the crystal axes. This alignment can allow discrimination between minerals that otherwise appear very similar in plane-polarized and cross-polarized light.
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 the index of refraction no, and the extraordinary axis, with the index of refraction ne. 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 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
where λ0 is the vacuum wavelength of the light.
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.
Half-wave plate
The most commonly used waveplates are the half-waveplate (Γ = π) and the quarter-waveplate (Γ = π/2). Half-waveplates can be used to rotate the plane of linearly polarized light as shown in the figure above.
Suppose a linearly polarized wave is normally incident on a waveplate, and its plane of polarization is at an angle θ to the fast axis. To see what happens, resolve the incident field into components polarized along the fast and slow axes, as shown in the figure above. After passing through the plate, pick a point in the wave where the fast component passes through a maximum. Since the slow component is retarded by one half-wave, it will also be a maximum, but 180° out of phase, or points along the negative slow axis. If we follow the wave further, we see that the slow component remains exactly 180° out of phase with the original slow component, relative to the fast component. This describes a linearly polarized wave but making an angle θ on the opposite side of the fast axis. The original polarization axis has been rotated through an angle 2θ. The same result will be found if the incident wave makes an angle θ to the slow axis.
A half-waveplate is very helpful in rotating the plane of polarization from a polarized laser to any other desired plane especially if the laser is too large to rotate. Most large ion lasers are vertically polarized, for example, so to obtain horizontal polarization, simply place a half-waveplate in the beam with its fast or slow axis 45° to the vertical. If the half-waveplate being used does not have marked axes or if the markings are hidden by the mount, place a linear polarizer in the beam first and orient it for extinction (horizontally polarized), then interpose the half-waveplate normally to the beam and rotate it around the beam axis, so that the beam remains extinct - one of the axes has now been found. Then, rotate the half-waveplate exactly 45° around the beam axis in either direction from this position, and the polarization of the beam will have been rotated by 90°. Check this by rotating the polarizer 90° to see that extinction occurs again. If you need some other angle, instead of 90° polarization rotation, simply rotate the waveplate by half the angle you desire
Quarter-wave plate
Quarter-waveplates are used to turn linearly polarized light into circularly polarized light and vice versa as shown in the figure above. To do this, the waveplate must be oriented so that equal amounts of fast and slow waves are excited. This is achieved by orienting an incident linearly polarized wave at 45° to the fast or slow axis, as shown in the figure above.
On the other side of the plate, examine the wave at a point where the fast-polarized component is at maximum. At this point, the slow-polarized component will be passing through zero, since it has been retarded by a quarter-wave or 90° in phase. Moving wavelength farther, we will note that the two are the same magnitude, but the fast component is decreasing and the slow component is increasing. Moving another wave, we find the slow component is at maximum and the fast component is zero. If the tip of the total electric vector is traced, we find it traces out a helix, with a period of just one wavelength. This describes circularly polarized light. The electric field vector is a mathematical description of the magnitude and direction of the electric field. The right-hand circularly polarized light is shown in the figure; the helix wraps in the opposite sense for the left hand. Left-hand polarized light is produced by rotating either the waveplate or the plane of polarization of the incident light 90° in the figure above.
Setting up a waveplate to produce circularly polarized light proceeds the same as described for rotating 90° with a half-waveplate: first, cross a polarizer in the beam to find the plane of polarization. Next, insert the quarter-waveplate between the source and the polarizer and rotate the waveplate around the beam axis to find the orientation that retains the extinction. Then rotate the waveplate 45° from this position. Half of the incident light should now be passing through the polarizer the other half being absorbed or deflected, depending on which kind of polarizer is being used. The quality of the circularly polarized light can be checked by rotating the polarizer - the intensity of light passing through the polarizer should remain unchanged. If it varies somewhat, it means that the light is actually elliptically polarized, and the waveplate isn’t exactly a quarter-waveplate at the operating wavelength of your interest. This may be corrected by tilting the waveplate about its fast or slow axes slightly while rotating the polarizer to check for constancy.
The result is elliptically polarized light - where the amount of ellipticity is a function of the retardation of the incident plane wave, and the tilt of the axis is a function of the tilt of the incident plane wave. The exception is half-wave retardation, in which case the ellipse degenerates into a plane wave making an angle of 2θ with the fast axis. Note that the quarter-waveplate does not produce circularly polarized light here, because equal amounts of fast and slow wave components were not used; the incident tilt angle must be exactly 45° to the fast or slow axis to make these components equal.
Use of waveplates in mineralogy and optical petrology
The full-wave and quarter-wave plates are widely used in the field of optical mineralogy. The addition of plates between the polarizers of a petrographic microscope makes easier the optical identification of minerals in thin sections of rocks, in particular by allowing the deduction of the shape and orientation of the optical indicatrices within the visible crystal sections. The plate is inserted between the perpendicular polarizers at an angle of 45 degrees. This allows two different procedures to be carried out to investigate the mineral under the crosshairs of the microscope. Firstly, in ordinary cross-polarized light, the plate can be used to distinguish the orientation of the optical indicatrix relative to crystal elongation – that is, whether the mineral is "length slow" or "length fast" – based on whether the visible interference colors increase or decrease by one order when the plate is added. Secondly, a slightly more complex procedure allows for a tint plate to be used in conjunction with interference figure techniques to allow measurement of the optic angle of the mineral. The optic angle can both be diagnostic of mineral type, as well as in some cases reveal information about the variation of chemical composition within a single mineral type.
Waveplate Material and Practice
Many naturally occurring crystals exhibit birefringence, and could, in principle, be used for waveplates. Calcite and crystalline quartz are typical materials used. While they are durable and of high optical quality, the refractive index difference, nslow - nfast is so large that a true half-waveplate would be too thin to polish, therefore impractical to create.
It is also possible to induce small amounts of birefringence into a normally isotropic material through stress. For example, most polymers exhibit birefringence from the stress applied in the manufacture. Polymer waveplate material is available in half or quarter-wave retardation. This material can be sandwiched between two high-quality windows to make precision zero-order waveplates.
