Waveplate

Last updated Electric field parallel to optic axis  Electric field perpendicular to axis  The combined field 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).

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. Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties. Polarization is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves in solids. Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to visible light, which is the visible spectrum that is visible to the human eye and is responsible for the sense of sight. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), or 4.00 × 10−7 to 7.00 × 10−7 m, between the infrared and the ultraviolet. This wavelength means a frequency range of roughly 430–750 terahertz (THz).

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Waveplates are constructed out of a birefringent material (such as quartz or mica, or even plastic), for which the index of refraction is different for light linearly polarized along one or the other of two certain perpendicular crystal axes. 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. Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent. The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress. Quartz is a mineral composed of silicon and oxygen atoms in a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical formula of SiO2. Quartz is the second most abundant mineral in Earth's continental crust, behind feldspar. The mica group of sheet silicate (phyllosilicate) minerals includes several closely related materials having nearly perfect basal cleavage. All are monoclinic, with a tendency towards pseudohexagonal crystals, and are similar in chemical composition. The nearly perfect cleavage, which is the most prominent characteristic of mica, is explained by the hexagonal sheet-like arrangement of its atoms.

A common use of waveplates—particularly the sensitive-tint (full-wave) and quarter-wave plates—is in optical mineralogy. 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 deduction of the shape and orientation of the optical indicatrices within the visible crystal sections. This alignment can allow discrimination between minerals which otherwise appear very similar in plane polarized and cross polarized light. Optical mineralogy is the study of minerals and rocks by measuring their optical properties. Most commonly, rock and mineral samples are prepared as thin sections or grain mounts for study in the laboratory with a petrographic microscope. Optical mineralogy is used to identify the mineralogical composition of geological materials in order to help reveal their origin and evolution. A petrographic microscope is a type of optical microscope used in petrology and optical mineralogy to identify rocks and minerals in thin sections. The microscope is used in optical mineralogy and petrography, a branch of petrology which focuses on detailed descriptions of rocks. The method is called "polarized light microscopy" (PLM). Depending on the grade of observation required, petrological microscopes are derived from conventional brightfield microscopes of similar basic capabilities by: A mineral is, broadly speaking, a solid chemical compound that occurs naturally in pure form. Minerals are most commonly associated with rocks due to the presence of minerals within rocks. These rocks may consist of one type of mineral, or may be an aggregate of two or more different types of minerals, spacially segregated into distinct phases. Compounds that occur only in living beings are usually excluded, but some minerals are often biogenic and/or are organic compounds in the sense of chemistry. Moreover, living beings often synthesize inorganic minerals that also occur in rocks.

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. 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. In physics and mathematics, the phase of a periodic function of some real variable is the relative value of that variable within the span of each full period.

An optic axis of a crystal is a direction in which a ray of transmitted light suffers no birefringence. An optical axis is a direction rather than a single line: all rays that are parallel to that direction exhibit the same lack of birefringence. Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate (CaCO3). The Mohs scale of mineral hardness, based on scratch hardness comparison, defines value 3 as "calcite".

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.

In optics, polarized light can be described using the Jones calculus, discovered by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light. Note that Jones calculus is only applicable to light that is already fully polarized. Light which is randomly polarized, partially polarized, or incoherent must be treated using Mueller calculus.

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. In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.

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.

Plate types

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 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

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

Full-wave, or sensitive-tint plate

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".  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. 

Multiple-order vs. zero-order waveplates

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. 

Use of waveplates in mineralogy and optical petrology

The sensitive-tint (full-wave) and quarter-wave plates are widely used in the field of optical mineralogy. 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 deduction of the shape and orientation of the optical indicatrices within the visible crystal sections.

In practical terms, 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. More simply, 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. 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 (often notated as "2V") can both be diagnostic of mineral type, as well as in some cases revealing information about the variation of chemical composition within a single mineral type.

Related Research Articles The Fresnel equations describe the reflection and transmission of light when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface. Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (values of atomic electric fields, typically 108 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds. In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light travels through the material. It is defined as Total Internal Reflection (TIR) is the phenomenon that makes the water-to-air surface in a fish-tank look like a perfectly silvered mirror when viewed from below the water level (Fig. 1). Technically, TIR is the total reflection of a wave incident at a sufficiently oblique angle on the interface between two media, of which the second ("external") medium is transparent to such waves but has a higher wave velocity than the first ("internal") medium. TIR occurs not only with electromagnetic waves such as light waves and microwaves, but also with other types of waves, including sound and water waves. In the case of a narrow train of waves, such as a laser beam, we tend to speak of the total internal reflection of a "ray" (Fig. 2). Optical rotation or optical activity is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Optical activity occurs only in chiral materials, those lacking microscopic mirror symmetry. Unlike other sources of birefringence which alter a beam's state of polarization, optical activity can be observed in fluids. This can include gases or solutions of chiral molecules such as sugars, molecules with helical secondary structure such as some proteins, and also chiral liquid crystals. It can also be observed in chiral solids such as certain crystals with a rotation between adjacent crystal planes or metamaterials. Rotation of light's plane of polarization may also occur through the Faraday effect which involves a static magnetic field, however this is a distinct phenomenon that is not usually classified under "optical activity." In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electric field of the wave has a constant magnitude but its direction rotates at a constant rate in a plane perpendicular to the direction of the wave.

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A depolarizer or depolariser is an optical device used to scramble the polarization of light. An ideal depolarizer would output randomly polarized light whatever its input, but all practical depolarizers produce pseudo-random output polarization. A polarization rotator is an optical device that rotates the polarization axis of a linearly polarized light beam by an angle of choice. Such devices can be based on the Faraday effect, on birefringence, or on total internal reflection. Rotators of linearly polarized light have found widespread applications in modern optics since laser beams tend to be linearly polarized and it is often necessary to rotate the original polarization to its orthogonal alternative.

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