# Birefringence

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Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. [1] These optically anisotropic materials are said to be birefringent (or birefractive). 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.

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.

In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light propagates through the material. It is defined as

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. In some types of transverse waves, the wave displacement is limited to a single direction, so these also do not exhibit polarization; for example, in surface waves in liquids, the wave displacement of the particles is always in a vertical plane.

## Contents

Birefringence is responsible for the phenomenon of double refraction whereby a ray of light, when incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. This effect was first described by the Danish scientist Rasmus Bartholin in 1669, who observed it [2] in calcite, a crystal having one of the strongest birefringences. However it was not until the 19th century that Augustin-Jean Fresnel described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarizations (perpendicular to the direction of the wave vector).

In optics a ray is an idealized model of light, obtained by choosing a line that is perpendicular to the wavefronts of the actual light, and that points in the direction of energy flow. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very complex optical systems to be analyzed mathematically or simulated by computer. Ray tracing uses approximate solutions to Maxwell's equations that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light's wavelength. Ray theory does not describe phenomena such as interference and diffraction, which require wave theory.

Denmark, officially the Kingdom of Denmark, is a Nordic country and the southernmost of the Scandinavian nations. Denmark lies southwest of Sweden and south of Norway, and is bordered to the south by Germany. The Kingdom of Denmark also comprises two autonomous constituent countries in the North Atlantic Ocean: the Faroe Islands and Greenland. Denmark proper consists of a peninsula, Jutland, and an archipelago of 443 named islands, with the largest being Zealand, Funen and the North Jutlandic Island. The islands are characterised by flat, arable land and sandy coasts, low elevation and a temperate climate. Denmark has a total area of 42,924 km2 (16,573 sq mi), land area of 42,394 km2 (16,368 sq mi), and the total area including Greenland and the Faroe Islands is 2,210,579 km2 (853,509 sq mi), and a population of 5.8 million.

Rasmus Bartholin was a Danish scientist, physician and grammarian. As part of his studies, he travelled in Europe for ten years. Professor at the University of Copenhagen, first in Geometry, later in Medicine. He was a younger brother of Thomas Bartholin. He wrote, in Latin, the first grammar of the Danish language, the 1657 De studio lingvæ danicæ.

## Explanation

A mathematical description of wave propagation in a birefringent medium is presented below. Following is a qualitative explanation of the phenomenon.

### Uniaxial materials

The simplest type of birefringence is described as uniaxial, meaning that there is a single direction governing the optical anisotropy whereas all directions perpendicular to it (or at a given angle to it) are optically equivalent. Thus rotating the material around this axis does not change its optical behavior. This special direction is known as the optic axis of the material. Light whose polarization is perpendicular to the optic axis is governed by a refractive index no (for "ordinary"). Light whose polarization is in the direction of the optic axis sees an optical index ne (for "extraordinary"). For any ray direction there is a linear polarization direction perpendicular to the optic axis, and this is called an ordinary ray. However, for ray directions not parallel to the optic axis, the polarization direction perpendicular to the ordinary ray's polarization will be partly in the direction of the optic axis, and this is called an extraordinary ray. I.e., when unpolarized light enters an uniaxial birefringent material it is split into two beams travelling different directions; the ordinary ray doesn't change direction while the extraordinary ray bends (is refracted) as it travels through the material. The ordinary ray will always experience a refractive index of no, whereas the refractive index of the extraordinary ray will be in between no and ne, depending on the ray direction as described by the index ellipsoid. The magnitude of the difference is quantified by the birefringence:

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.

In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal .

${\displaystyle \Delta n=n_{\mathrm {e} }-n_{\mathrm {o} }\,.}$

The propagation (as well as reflection coefficient) of the ordinary ray is simply described by no as if there were no birefringence involved. However the extraordinary ray, as its name suggests, propagates unlike any wave in a homogenous optical material. Its refraction (and reflection) at a surface can be understood using the effective refractive index (a value in between no and ne). However it is in fact an inhomogeneous wave whose power flow (given by the Poynting vector) is not exactly in the direction of the wave vector. This causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of calcite as photographed above. Rotating the calcite crystal will cause one of the two images, that of the extraordinary ray, to rotate slightly around that of the ordinary ray, which remains fixed.

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.

In physics, the Poynting vector represents the directional energy flux of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m2). It is named after its discoverer John Henry Poynting who first derived it in 1884. Oliver Heaviside also discovered it independently.

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

When the light propagates either along or orthogonal to the optic axis, such a lateral shift does not occur. In the first case, both polarizations see the same effective refractive index, so there is no extraordinary ray. In the second case the extraordinary ray propagates at a different phase velocity (corresponding to ne) but is not an inhomogeneous wave. A crystal with its optic axis in this orientation, parallel to the optical surface, may be used to create a waveplate, in which there is no distortion of the image but an intentional modification of the state of polarization of the incident wave. For instance, a quarter-wave plate is commonly used to create circular polarization from a linearly polarized source.

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.

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 with time at a steady rate in a plane perpendicular to the direction of the wave.

### Biaxial materials

The case of so-called biaxial crystals is substantially more complex. [3] These are characterized by three refractive indices corresponding to three principal axes of the crystal. For most ray directions, both polarizations would be classified as extraordinary rays but with different effective refractive indices. Being extraordinary waves, however, the direction of power flow is not identical to the direction of the wave vector in either case.

The two refractive indices can be determined using the index ellipsoids for given directions of the polarization. Note that for biaxial crystals the index ellipsoid will not be an ellipsoid of revolution ("spheroid") but is described by three unequal principle refractive indices nα, nβ and nγ. Thus there is no axis around which a rotation leaves the optical properties invariant (as there is with uniaxial crystals whose index ellipsoid is a spheroid).

Although there is no axis of symmetry, there are two optical axes or binormals which are defined as directions along which light may propagate without birefringence, i.e., directions along which the wavelength is independent of polarization. [3] For this reason, birefringent materials with three distinct refractive indices are called biaxial. Additionally, there are two distinct axes known as optical ray axes or biradials along which the group velocity of the light is independent of polarization.

### Double refraction

When an arbitrary beam of light strikes the surface of a birefringent material, the polarizations corresponding to the ordinary and extraordinary rays generally take somewhat different paths. Unpolarized light consists of equal amounts of energy in any two orthogonal polarizations, and even polarized light (except in special cases) will have some energy in each of these polarizations. According to Snell's law of refraction, the angle of refraction will be governed by the effective refractive index which is different between these two polarizations. This is clearly seen, for instance, in the Wollaston prism which is designed to separate incoming light into two linear polarizations using a birefringent material such as calcite.

The different angles of refraction for the two polarization components are shown in the figure at the top of the page, with the optic axis along the surface (and perpendicular to the plane of incidence), so that the angle of refraction is different for the p polarization (the "ordinary ray" in this case, having its polarization perpendicular to the optic axis) and the s polarization (the "extraordinary ray" with a polarization component along the optic axis). In addition, a distinct form of double refraction occurs in cases where the optic axis is not along the refracting surface (nor exactly normal to it); in this case the electric polarization of the birefringent material is not exactly in the direction of the wave's electric field for the extraordinary ray. The direction of power flow (given by the Poynting vector) for this inhomogenous wave is at a finite angle from the direction of the wave vector resulting in an additional separation between these beams. So even in the case of normal incidence, where the angle of refraction is zero (according to Snell's law, regardless of effective index of refraction), the energy of the extraordinary ray may be propagated at an angle. This is commonly observed using a piece of calcite cut appropriately with respect to its optic axis, placed above a paper with writing, as in the above two photographs.

## Terminology

Much of the work involving polarization preceded the understanding of light as a transverse electromagnetic wave, and this has affected some terminology in use. Isotropic materials have symmetry in all directions and the refractive index is the same for any polarization direction. An anisotropic material is called "birefringent" because it will generally refract a single incoming ray in two directions, which we now understand correspond to the two different polarizations. This is true of either a uniaxial or biaxial material.

In a uniaxial material, one ray behaves according to the normal law of refraction (corresponding to the ordinary refractive index), so an incoming ray at normal incidence remains normal to the refracting surface. However, as explained above, the other polarization can be deviated from normal incidence, which cannot be described using the law of refraction. This thus became known as the extraordinary ray. The terms "ordinary" and "extraordinary" are still applied to the polarization components perpendicular to and not perpendicular to the optic axis respectively, even in cases where no double refraction is involved.

A material is termed uniaxial when it has a single direction of symmetry in its optical behavior, which we term the optic axis. It also happens to be the axis of symmetry of the index ellipsoid (a spheroid in this case). The index ellipsoid could still be described according to the refractive indices, nα, nβ and nγ, along three coordinate axes, however in this case two are equal. So if nα = nβ corresponding to the x and y axes, then the extraordinary index is nγ corresponding to the z axis, which is also called the optic axis in this case.

However materials in which all three refractive indices are different are termed biaxial and the origin of this term is more complicated and frequently misunderstood. In a uniaxial crystal, different polarization components of a beam will travel at different phase velocities, except for rays in the direction of what we call the optic axis. Thus the optic axis has the particular property that rays in that direction do not exhibit birefringence, with all polarizations in such a beam experiencing the same index of refraction. It is very different when the three principal refractive indices are all different; then an incoming ray in any of those principle directions will still encounter two different refractive indices. But it turns out that there are two special directions (at an angle to all of the 3 axes) where the refractive indices for different polarizations are again equal. For this reason, these crystals were designated as biaxial, with the two "axes" in this case referring to ray directions in which propagation does not experience birefringence.

### Fast and slow rays

In a birefringent material, a wave consists of two polarization components which generally are governed by different effective refractive indices. The so-called slow ray is the component for which the material has the higher effective refractive index (slower phase velocity), while the fast ray is the one with a lower effective refractive index. When a beam is incident on such a material from air (or any material with a lower refractive index), the slow ray is thus refracted more towards the normal than the fast ray. In the figure at the top of the page, it can be seen that refracted ray with s polarization in the direction of the optic axis (thus the extraordinary ray) is the slow ray in this case.

Using a thin slab of that material at normal incidence, one would implement a waveplate. In this case there is essentially no spatial separation between the polarizations, however the phase of the wave in the parallel polarization (the slow ray) will be retarded with respect to the perpendicular polarization. These directions are thus known as the slow axis and fast axis of the waveplate.

### Positive or negative

Uniaxial birefringence is classified as positive when the extraordinary index of refraction ne is greater than the ordinary index no. Negative birefringence means that Δn = neno is less than zero. [4] In other words, the polarization of the fast (or slow) wave is perpendicular to the optic axis when the birefringence of the crystal is positive (or negative, respectively). In the case of biaxial crystals, all three of the principal axes have different refractive indices, so this designation does not apply. But for any defined ray direction one can just as well designate the fast and slow ray polarizations.

## Sources of optical birefringence

While birefringence is usually obtained using an anisotropic crystal, it can result from an optically isotropic material in a few ways:

• Stress birefringence results when isotropic materials are stressed or deformed (i.e., stretched or bent) causing a loss of physical isotropy and consequently a loss of isotropy in the material's permittivity tensor.
• Circular birefringence in liquids where there is an enantiomeric excess in a solution containing a molecule which has stereo isomers.
• Form birefringence, whereby structure elements such as rods, having one refractive index, are suspended in a medium with a different refractive index. When the lattice spacing is much smaller than a wavelength, such a structure is described as a metamaterial.
• By the Kerr effect, whereby an applied electric field induces birefringence at optical frequencies through the effect of nonlinear optics;
• By the Faraday effect, where a magnetic field causes some materials to become circularly birefringent (having slightly different indices of refraction for left- and right-handed circular polarizations), making the material optically active until the field is removed;
• By the self or forced alignment into thin films of amphiphilic molecules such as lipids, some surfactants or liquid crystals

## Common birefringent materials

The best characterized birefringent materials are crystals. Due to their specific crystal structures their refractive indices are well defined. Depending on the symmetry of a crystal structure (as determined by one of the 32 possible crystallographic point groups), crystals in that group may be forced to be isotropic (not birefringent), to have uniaxial symmetry, or neither in which case it is a biaxial crystal. The crystal structures permitting uniaxial and biaxial birefringence are noted in the two tables, below, listing the two or three principal refractive indices (at wavelength 590 nm) of some better known crystals. [5]

Many plastics are birefringent because their molecules are "frozen" in a stretched conformation when the plastic is molded or extruded. [6] For example, ordinary cellophane is birefringent. Polarizers are routinely used to detect stress in plastics such as polystyrene and polycarbonate.

Cotton fiber is birefringent because of high levels of cellulosic material in the fiber's secondary cell wall.

Polarized light microscopy is commonly used in biological tissue, as many biological materials are birefringent. Collagen, found in cartilage, tendon, bone, corneas, and several other areas in the body, is birefringent and commonly studied with polarized light microscopy. [7] Some proteins are also birefringent, exhibiting form birefringence. [8]

Inevitable manufacturing imperfections in optical fiber leads to birefringence, which is one cause of pulse broadening in fiber-optic communications. Such imperfections can be geometrical (lack of circular symmetry), due to stress applied to the optical fiber and/or due to bending of the fiber. Birefringence is intentionally introduced (for instance, by making the cross-section elliptical) in order to produce polarization-maintaining optical fibers.

In addition to anisotropy in the electric polarizability (electric susceptibility), anisotropy in the magnetic polarizability (magnetic permeability) can also cause birefringence. However, at optical frequencies, values of magnetic permeability for natural materials are not measurably different from µ0, so this is not a source of optical birefringence in practice.

## Measurement

Birefringence and other polarization-based optical effects (such as optical rotation and linear or circular dichroism) can be measured by measuring the changes in the polarization of light passing through the material. These measurements are known as polarimetry. Polarized light microscopes, which contain two polarizers that are at 90° to each other on either side of the sample, are used to visualize birefringence. The addition of quarter-wave plates permit examination of circularly polarized light. Birefringence measurements have been made with phase-modulated systems for examining the transient flow behavior of fluids. [9] [10]

Birefringence of lipid bilayers can be measured using dual polarization interferometry. This provides a measure of the degree of order within these fluid layers and how this order is disrupted when the layer interacts with other biomolecules.

## Applications

Birefringence is used in many optical devices. Liquid-crystal displays, the most common sort of flat panel display, cause their pixels to become lighter or darker through rotation of the polarization (circular birefringence) of linearly polarized light as viewed through a sheet polarizer at the screen's surface. Similarly, light modulators modulate the intensity of light through electrically induced birefringence of polarized light followed by a polarizer. The Lyot filter is a specialized narrowband spectral filter employing the wavelength dependence of birefringence. Wave plates are thin birefringent sheets widely used in certain optical equipment for modifying the polarization state of light passing through it.

Birefringence also plays an important role in second-harmonic generation and other nonlinear optical components, as the crystals used for this purpose are almost always birefringent. By adjusting the angle of incidence, the effective refractive index of the extraordinary ray can be tuned in order to achieve phase matching, which is required for efficient operation of these devices.

### Medicine

Birefringence is utilized in medical diagnostics. One powerful accessory used with optical microscopes is a pair of crossed polarizing filters. Light from the source is polarized in the x direction after passing through the first polarizer, but above the specimen is a polarizer (a so-called analyzer) oriented in the y direction. Therefore, no light from the source will be accepted by the analyzer, and the field will appear dark. However areas of the sample possessing birefringence will generally couple some of the x-polarized light into the y polarization; these areas will then appear bright against the dark background. Modifications to this basic principle can differentiate between positive and negative birefringence.

For instance, needle aspiration of fluid from a gouty joint will reveal negatively birefringent monosodium urate crystals. Calcium pyrophosphate crystals, in contrast, show weak positive birefringence. [11] Urate crystals appear yellow, and calcium pyrophosphate crystals appear blue when their long axes are aligned parallel to that of a red compensator filter, [12] or a crystal of known birefringence is added to the sample for comparison.

Birefringence can be observed in amyloid plaques such as are found in the brains of Alzheimer's patients when stained with a dye such as Congo Red. Modified proteins such as immunoglobulin light chains abnormally accumulate between cells, forming fibrils. Multiple folds of these fibers line up and take on a beta-pleated sheet conformation. Congo red dye intercalates between the folds and, when observed under polarized light, causes birefringence.

In ophthalmology, binocular retinal birefringence screening of the Henle fibers (photoreceptor axons that go radially outward from the fovea) provides a reliable detection of strabismus and possibly also of anisometropic amblyopia. [13] Furthermore, scanning laser polarimetry utilises the birefringence of the optic nerve fibre layer to indirectly quantify its thickness, which is of use in the assessment and monitoring of glaucoma.

Birefringence characteristics in sperm heads allow the selection of spermatozoa for intracytoplasmic sperm injection. [14] Likewise, zona imaging uses birefringence on oocytes to select the ones with highest chances of successful pregnancy. [15] Birefringence of particles biopsied from pulmonary nodules indicates silicosis.

Dermatologists use dermatascopes to view skin lesions. Dermatascopes use polarized light, allowing the user to view crystalline structures corresponding to dermal collagen in the skin. These structures may appear as shiny white lines or rosette shapes and are only visible under polarized dermoscopy.

### Stress-induced birefringence

Isotropic solids do not exhibit birefringence. However, when they are under mechanical stress, birefringence results. The stress can be applied externally or is "frozen in" after a birefringent plastic ware is cooled after it is manufactured using injection molding. When such a sample is placed between two crossed polarizers, colour patterns can be observed, because polarization of a light ray is rotated after passing through a birefringent material and the amount of rotation is dependent on wavelength. The experimental method called photoelasticity used for analyzing stress distribution in solids is based on the same principle. There has been recent research on using stress induced birefringence in a glass plate to generate an Optical vortex and full Poincare beams (optical beams that have every possible polarization states across its cross-section). [16]

### Other cases of birefringence

Birefringence is observed in anisotropic elastic materials. In these materials, the two polarizations split according to their effective refractive indices, which are also sensitive to stress.

The study of birefringence in shear waves traveling through the solid Earth (the Earth's liquid core does not support shear waves) is widely used in seismology.

Birefringence is widely used in mineralogy to identify rocks, minerals, and gemstones.

## Theory

In an isotropic medium (including free space) the so-called electric displacement (D) is just proportional to the electric field (E) according to D = ɛE where the material's permittivity ε is just a scalar (and equal to n2 ε0 where n is the index of refraction). However, in an anisotropic material exhibiting birefringence, the relationship between D and E must now be described using a tensor equation:

${\displaystyle \mathbf {D} ={\boldsymbol {\varepsilon }}\mathbf {E} }$

(1)

where ε is now a 3 × 3 permittivity tensor. We assume linearity and no magnetic permeability in the medium: μ = μ0 . The electric field of a plane wave of angular frequency ω can be written in the general form:

${\displaystyle \mathbf {E} =\mathbf {E} _{0}e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}}$

(2)

where r is the position vector, t is time, and E0 is a vector describing the electric field at r = 0, t = 0. Then we shall find the possible wave vectors k. By combining Maxwell's equations for ∇ × E and ∇ × H, we can eliminate H = 1/μ0B to obtain:

${\displaystyle -\nabla \times \nabla \times \mathbf {E} =\mu _{0}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {D} }$

(3a)

With no free charges, Maxwell's equation for the divergence of D vanishes:

${\displaystyle \nabla \cdot \mathbf {D} =0}$

(3b)

We can apply the vector identity ∇ × (∇ × A) = ∇(∇ ⋅ A) − ∇2A to the left hand side of eq. 3a , and use the spatial dependence in which each differentiation in x (for instance) results in multiplication by ikx to find:

${\displaystyle -\nabla \times \nabla \times \mathbf {E} =(\mathbf {k} \cdot \mathbf {E} )\mathbf {k} -(\mathbf {k} \cdot \mathbf {k} )\mathbf {E} }$

(3c)

The right hand side of eq. 3a can be expressed in terms of E through application of the permittivity tensor ε and noting that differentiation in time results in multiplication by , eq. 3a then becomes:

${\displaystyle (-\mathbf {k} \cdot \mathbf {k} )\mathbf {E} +(\mathbf {k} \cdot \mathbf {E} )\mathbf {k} =-\mu _{0}\omega ^{2}({\boldsymbol {\varepsilon }}\mathbf {E} )}$

(4a)

Applying the differentiation rule to eq. 3b we find:

${\displaystyle \mathbf {k} \cdot \mathbf {D} =0}$

(4b)

Eq. 4b indicates that D is orthogonal to the direction of the wavevector k, even though that is no longer generally true for E as would be the case in an isotropic medium. Eq. 4b will not be needed for the further steps in the following derivation.

Finding the allowed values of k for a given ω is easiest done by using Cartesian coordinates with the x, y and z axes chosen in the directions of the symmetry axes of the crystal (or simply choosing z in the direction of the optic axis of a uniaxial crystal), resulting in a diagonal matrix for the permittivity tensor ε:

${\displaystyle \mathbf {\varepsilon } =\varepsilon _{0}{\begin{bmatrix}n_{x}^{2}&0&0\\0&n_{y}^{2}&0\\0&0&n_{z}^{2}\end{bmatrix}}}$

(4c)

where the diagonal values are squares of the refractive indices for polarizations along the three principal axes x, y and z. With ε in this form, and substituting in the speed of light c using c2 = 1/μ0ε0, eq. 4a becomes

${\displaystyle \left(-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}\right)E_{x}+k_{x}^{2}E_{x}+k_{x}k_{y}E_{y}+k_{x}k_{z}E_{z}=-{\frac {\omega ^{2}n_{x}^{2}}{c^{2}}}E_{x}}$

(5a)

where Ex, Ey, Ez are the components of E (at any given position in space and time) and kx, ky, kz are the components of k. Rearranging, we can write (and similarly for the y and z components of eq. 4a )

${\displaystyle \left(-k_{y}^{2}-k_{z}^{2}+{\frac {\omega ^{2}n_{x}^{2}}{c^{2}}}\right)E_{x}+k_{x}k_{y}E_{y}+k_{x}k_{z}E_{z}=0}$

(5b)

${\displaystyle k_{x}k_{y}E_{x}+\left(-k_{x}^{2}-k_{z}^{2}+{\frac {\omega ^{2}n_{y}^{2}}{c^{2}}}\right)E_{y}+k_{y}k_{z}E_{z}=0}$

(5c)

${\displaystyle k_{x}k_{z}E_{x}+k_{y}k_{z}E_{y}+\left(-k_{x}^{2}-k_{y}^{2}+{\frac {\omega ^{2}n_{z}^{2}}{c^{2}}}\right)E_{z}=0}$

(5d)

This is a set of linear equations in Ex, Ey, Ez, so it can have a nontrivial solution (that is, one other than E = 0) as long as the following determinant is zero:

${\displaystyle {\begin{vmatrix}\left(-k_{y}^{2}-k_{z}^{2}+{\frac {\omega ^{2}n_{x}^{2}}{c^{2}}}\right)&k_{x}k_{y}&k_{x}k_{z}\\k_{x}k_{y}&\left(-k_{x}^{2}-k_{z}^{2}+{\frac {\omega ^{2}n_{y}^{2}}{c^{2}}}\right)&k_{y}k_{z}\\k_{x}k_{z}&k_{y}k_{z}&\left(-k_{x}^{2}-k_{y}^{2}+{\frac {\omega ^{2}n_{z}^{2}}{c^{2}}}\right)\end{vmatrix}}=0}$

(6)

Evaluating the determinant of eq. 6 , and rearranging the terms, we obtain

${\displaystyle {\frac {\omega ^{4}}{c^{4}}}-{\frac {\omega ^{2}}{c^{2}}}\left({\frac {k_{x}^{2}+k_{y}^{2}}{n_{z}^{2}}}+{\frac {k_{x}^{2}+k_{z}^{2}}{n_{y}^{2}}}+{\frac {k_{y}^{2}+k_{z}^{2}}{n_{x}^{2}}}\right)+\left({\frac {k_{x}^{2}}{n_{y}^{2}n_{z}^{2}}}+{\frac {k_{y}^{2}}{n_{x}^{2}n_{z}^{2}}}+{\frac {k_{z}^{2}}{n_{x}^{2}n_{y}^{2}}}\right)\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right)=0}$

(7)

In the case of a uniaxial material, choosing the optic axis to be in the z direction so that nx = ny = no and nz = ne, this expression can be factored into

${\displaystyle \left({\frac {k_{x}^{2}}{n_{\mathrm {o} }^{2}}}+{\frac {k_{y}^{2}}{n_{\mathrm {o} }^{2}}}+{\frac {k_{z}^{2}}{n_{\mathrm {o} }^{2}}}-{\frac {\omega ^{2}}{c^{2}}}\right)\left({\frac {k_{x}^{2}}{n_{\mathrm {e} }^{2}}}+{\frac {k_{y}^{2}}{n_{\mathrm {e} }^{2}}}+{\frac {k_{z}^{2}}{n_{\mathrm {o} }^{2}}}-{\frac {\omega ^{2}}{c^{2}}}\right)=0}$

(8)

Setting either of the factors in eq. 8 to zero will define an ellipsoidal surface [note 1] in the space of wavevectors k that are allowed for a given ω. The first factor being zero defines a sphere; this is the solution for so-called ordinary rays, in which the effective refractive index is exactly no regardless of the direction of k. The second defines a spheroid symmetric about the z axis. This solution corresponds to the so-called extraordinary rays in which the effective refractive index is in between no and ne, depending on the direction of k. Therefore, for any arbitrary direction of propagation (other than in the direction of the optic axis), two distinct wavevectors k are allowed corresponding to the polarizations of the ordinary and extraordinary rays.

For a biaxial material a similar but more complicated condition on the two waves can be described; [17] the allowed k vectors in a specified direction now lie on one of two ellipsoids. By inspection one can see that eq. 6 is generally satisfied for two positive values of ω. Or, for a specified optical frequency ω and direction normal to the wavefronts k/|k|, it is satisfied for two wavenumbers (or propagation constants) |k| (and thus effective refractive indices) corresponding to the propagation of two linear polarizations in that direction.

When those two propagation constants are equal then the effective refractive index is independent of polarization, and there is consequently no birefringence encountered by a wave traveling in that particular direction. For a uniaxial crystal, this is the optic axis, the z direction according to the above construction. But when all three refractive indices (or permittivities), nx, ny and nz are distinct, it can be shown that there are exactly two such directions (where the two ellipsoids intersect); these directions are not at all obvious and do not lie along any of the three principal axes (x, y, z according to the above convention). Historically that accounts for the use of the term "biaxial" for such crystals, as the existence of exactly two such special directions (considered "axes") was discovered well before polarization and birefringence were understood physically. However these two special directions are generally not of particular interest; biaxial crystals are rather specified by their three refractive indices corresponding to the three axes of symmetry.

A general state of polarization launched into the medium can always be decomposed into two waves, one in each of those two polarizations, which will then propagate with different wavenumbers |k|. Applying the different phase of propagation to those two waves over a specified propagation distance will result in a generally different net polarization state at that point; this is the principle of the waveplate for instance. However, when you have a wave launched into a birefringent material at non-normal incidence, the problem is yet more complicated since the two polarization components will now not only have distinct wavenumbers but the k vectors will not even be in exactly the same direction (see figure at the top of the page). In this case the two k vectors are rather solutions of eq. 6 constrained by the boundary condition which requires that the components of the two transmitted waves' k vectors, and the k vector of the incident wave, as projected onto the surface of the interface, must all be identical.

## Notes

1. Although related, note that this is not the same as the index ellipsoid.

## Related Research Articles

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.

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."

An optical isolator, or optical diode, is an optical component which allows the transmission of light in only one direction. It is typically used to prevent unwanted feedback into an optical oscillator, such as a laser cavity.

Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone–Dale relation. Crystals are often naturally anisotropic, and in some media it is possible to induce anisotropy by applying an external electric field.

A Wollaston prism is an optical device, invented by William Hyde Wollaston, that manipulates polarized light. It separates light into two separate linearly polarized outgoing beams with orthogonal polarization. The two beams will be polarized according to the optical axis of the two right angle prisms.

A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well-defined polarization, that is polarized light. The common types of polarizers are linear polarizers and circular polarizers. Polarizers are used in many optical techniques and instruments, and polarizing filters find applications in photography and LCD technology. Polarizers can also be made for other types of electromagnetic waves besides light, such as radio waves, microwaves, and X-rays.

Vauxite is a phosphate mineral with the chemical formula Fe2+Al2(PO4)2(OH)2·6(H2O). It belongs to the laueite – paravauxite group, paravauxite subgroup, although Mindat puts it as a member of the vantasselite Al4(PO4)3(OH)3·9H2O group. There is no similarity in structure between vauxite and paravauxite Fe2+Al2(PO4)2(OH)2·8H2O or metavauxite Fe3+Al2(PO4)2(OH)2·8H2O, even though they are closely similar chemically, and all minerals occur together as secondary minerals. Vauxite was named in 1922 for George Vaux Junior (1863–1927), an American attorney and mineral collector.

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 Fresnel rhomb is an optical prism that introduces a 90° phase difference between two perpendicular components of polarization, by means of two total internal reflections. If the incident beam is linearly polarized at 45° to the plane of incidence and reflection, the emerging beam is circularly polarized, and vice versa. If the incident beam is linearly polarized at some other inclination, the emerging beam is elliptically polarized with one principal axis in the plane of reflection, and vice versa. Thus the rhomb functions as if it were a wideband quarter-wave plate: whereas a birefringent quarter-wave plate introduces a 90° phase difference for a single wavelength of light, the phase difference introduced by the rhomb depends only on its refractive index, which typically varies only slightly over the visible range of wavelengths. The material of which the rhomb is made is specifically not birefringent.

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.

The Sénarmont prism is a type of polariser. It is made from two prisms of a birefringent material such as calcite, usually cemented together. The Sénarmont prism is named after Henri Hureau de Sénarmont. It is similar to the Rochon and Wollaston prisms.

Uniaxial crystals are transmissive optical elements in which the refractive index of one crystal axis is different from the other two crystal axes (i.e. ninj = nk). This unique axis is called the extraordinary axis and is also referred to as the optic axis. Light travels with a higher phase velocity through an axis that has the smallest refractive index and this axis is called the fast axis. Similarly, an axis which has the highest refractive index is called a slow axis since the phase velocity of light is the lowest along this axis. The optic axis can be the fast or the slow axis for the crystal depending upon the material. Negative uniaxial crystals (e.g. calcite CaCO3, ruby Al2O3) have ne < no so for these crystals, the extraordinary axis (optic axis) is the fast axis whereas for positive uniaxial crystals (e.g. quartz SiO2, sellaite (magnesium fluoride) MgF2, rutile TiO2), ne > n o and thus the extraordinary axis (optic axis) is the slow axis.These crystals show birefringent property.

A conoscopic interference pattern or interference figure is a pattern of birefringent colours crossed by dark bands, which can be produced using a geological petrographic microscope for the purposes of mineral identification and investigation of mineral optical and chemical properties. The figures are produced by optical interference when diverging light rays travel through an optically non-isotropic substance - that is, one in which the substance's refractive index varies in different directions within it. The figure can be thought of as a "map" of how the birefringence of a mineral would vary with viewing angle away from perpendicular to the slide, where the central colour is the birefringence seen looking straight down, and the colours further from the centre equivalent to viewing the mineral at ever increasing angles from perpendicular. The dark bands correspond to positions where optical extinction would be seen. In other words, the interference figure presents all possible birefringence colours for the mineral at once.

Sarabauite (sar-a-bau'-ite) is a red monoclinic sulfide mineral with the chemical formula: CaSb10O10S6.

Carminite (PbFe3+2(AsO4)2(OH)2) is an anhydrous arsenate mineral containing hydroxyl. It is a rare secondary mineral that is structurally related to palermoite (Li2SrAl4(PO4)4(OH)4). Sewardite (CaFe3+2(AsO4)2(OH)2) is an analogue of carminite, with calcium in sewardite in place of the lead in carminite. Mawbyite is a dimorph (same formula, different structure) of carminite; mawbyite is monoclinic and carminite is orthorhombic. It has a molar mass of 639.87 g. It was discovered in 1850 and named for the characteristic carmine colour.

The term plane of polarization refers to the direction of polarization of linearly-polarized light or other electromagnetic radiation. Unfortunately the term is used with two contradictory meanings. As originally defined by Étienne-Louis Malus in 1811, the plane of polarization happened to coincide with the plane containing the direction of propagation and the magnetic vector; but this was not known at the time. In modern literature, the term plane of polarization, if it is used at all, more often refers to the plane containing the direction of propagation and the electric vector, because the electric field has the greater propensity to interact with matter. That propensity, together with Malus's definition and Fresnel's speculations on the luminiferous aether, led early investigators to define the "plane of vibration" as perpendicular to the plane of polarization and containing the direction of propagation.

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