Ordinary Polarization
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Birefringence is the
optical 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, ultravio ...
property of a material having a
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
that depends on the polarization and propagation direction of
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
. These optically
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
materials are said to be birefringent (or birefractive). The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material.
Crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
s with non-cubic
crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
s are often birefringent, as are
plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
s under
mechanical stress In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...
. 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 Danish scientist
Rasmus Bartholin Rasmus Bartholin (; Latinized: ''Erasmus Bartholinus''; 13 August 1625 – 4 November 1698) was a Danish physician and grammarian. Biography Bartholin was born in Roskilde. He was the son of Caspar Bartholin the Elder (1585–1629) and Anna ...
in 1669, who observed it in
calcite Calcite is a Carbonate minerals, carbonate mineral and the most stable Polymorphism (materials science), polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on ...
, a crystal having one of the strongest birefringences. In the 19th century
Augustin-Jean Fresnel Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular th ...
described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarization (perpendicular to the direction of the wave vector).A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" et seq., ''Annales de Chimie et de Physique'', Ser. 2, vol. 17, pp. 102–11 (May 1821), 167–96 (June 1821), 312–15 ("Postscript", July 1821); reprinted (with added section nos.) in Fresnel, 1866–70, vol. 1, pp. 609–48; translated as "On the calculation of the tints that polarization develops in crystalline plates, & postscript", (Creative Commons), 2021; §14.A. Fresnel, "Extrait d'un Mémoire sur la double réfraction", ''Annales de Chimie et de Physique'', Ser. 2, vol. 28, pp. 263–79 (March 1825); reprinted as "Extrait du second Mémoire sur la double réfraction" in Fresnel, 1866–70, vol. 2, pp. 465–78; translated as "Extract of a econdmemoir on double refraction", , 2021 (open access).


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 behaviour. This special direction is known as the
optic axis An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight. The optical axis is an imaginary line that defines the path along which light propagat ...
of the material. Light propagating parallel to the optic axis (whose polarization is always ''perpendicular'' to the optic axis) is governed by a refractive index (for "ordinary") regardless of its specific polarization. For rays with any other propagation direction, there is one linear polarization that would be perpendicular to the optic axis, and a ray with that polarization is called an ''ordinary ray'' and is governed by the same refractive index value . For a ray propagating in the same direction but with a polarization perpendicular to that of the ordinary ray, the polarization direction will be partly in the direction of the optic axis, and this ''extraordinary ray'' will be governed by a different, ''direction-dependent'' refractive index. Because the index of refraction depends on the polarization when unpolarized light enters a uniaxial birefringent material, it is split into two beams travelling in different directions, one having the polarization of the ordinary ray and the other the polarization of the extraordinary ray. The ordinary ray will always experience a refractive index of , whereas the refractive index of the extraordinary ray will be in between and , depending on the ray direction as described by the
index ellipsoid In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polariz ...
. The magnitude of the difference is quantified by the birefringence: :\Delta n=n_\mathrm-n_\mathrm\,. The propagation (as well as
reflection coefficient In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected w ...
) of the ordinary ray is simply described by as if there were no birefringence involved. The extraordinary ray, as its name suggests, propagates unlike any wave in an isotropic optical material. Its refraction (and reflection) at a surface can be understood using the effective refractive index (a value in between and ). Its power flow (given by the Poynting vector) is not exactly in the direction of the
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
. This causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of
calcite Calcite is a Carbonate minerals, carbonate mineral and the most stable Polymorphism (materials science), polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on ...
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. 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 are perpendicular to the optic axis and 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 ) but still has the power flow in the direction of the
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
. A crystal with its optic axis in this orientation, parallel to the optical surface, may be used to create a
waveplate A waveplate or retarder is an optics, optical device that alters the Polarization (waves), polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization directio ...
, 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 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 ligh ...
is commonly used to create
circular polarization In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
from a linearly polarized source.


Biaxial materials

The case of so-called biaxial crystals is substantially more complex.Landau, L. D., and Lifshitz, E. M., ''Electrodynamics of Continuous Media'', Vol. 8 of the ''Course of Theoretical Physics'' 1960 (Pergamon Press), §79 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, 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 ellipsoid In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polariz ...
s for given directions of the polarization. Note that for biaxial crystals the index ellipsoid will ''not'' be an ellipsoid of revolution ("
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
") but is described by three unequal principle refractive indices , and . 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. 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 at non-normal incidence, the polarization component normal to the optic axis (ordinary ray) and the other linear polarization (extraordinary ray) will be refracted toward somewhat different paths. Natural light, so-called
unpolarized light Unpolarized light is light with a random, time-varying polarization. Natural light, like most other common sources of visible light, produced independently by a large number of atoms or molecules whose emissions are uncorrelated. This term is so ...
, consists of equal amounts of energy in any two orthogonal polarizations. Even linearly polarized light has some energy in both polarizations, unless aligned along one of the two axes of birefringence. According to
Snell's law Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
of refraction, the two angles of refraction are governed by the effective
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of each of these two polarizations. This is clearly seen, for instance, in the
Wollaston prism 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 a ...
which separates incoming light into two linear polarizations using prisms composed of a birefringent material such as
calcite Calcite is a Carbonate minerals, carbonate mineral and the most stable Polymorphism (materials science), polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on ...
. The different angles of refraction for the two polarization components are shown in the figure at the top of this page, with the optic axis along the surface (and perpendicular to the
plane of incidence In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation. (In wave opt ...
), so that the angle of refraction is different for the polarization (the "ordinary ray" in this case, having its electric vector perpendicular to the optic axis) and the polarization (the "extraordinary ray" in this case, whose electric field polarization includes a component in the direction of the optic axis). In addition, a distinct form of double refraction occurs, even with normal incidence, in cases where the optic axis is not along the refracting surface (nor exactly normal to it); in this case, the di
electric polarization In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
of the birefringent material is not exactly in the direction of the wave's
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
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 In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
resulting in an additional separation between these beams. So even in the case of normal incidence, where one would compute the angle of refraction as zero (according to Snell's law, regardless of the effective index of refraction), the energy of the extraordinary ray is propagated at an angle. If exiting the crystal through a face parallel to the incoming face, the direction of both rays will be restored, but leaving a ''shift'' between the two beams. This is commonly observed using a piece of calcite cut along its natural cleavage, placed above a paper with writing, as in the above photographs. On the contrary,
waveplate A waveplate or retarder is an optics, optical device that alters the Polarization (waves), polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization directio ...
s specifically have their optic axis ''along'' the surface of the plate, so that with (approximately) normal incidence there will be no shift in the image from light of either polarization, simply a relative
phase shift In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...
between the two light waves.


Terminology

Much of the work involving polarization preceded the understanding of light as a transverse
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
, 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. As explained above, the other polarization can deviate 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, , and , along three coordinate axes; in this case two are equal. So if corresponding to the and axes, then the extraordinary index is corresponding to the axis, which is also called the ''optic axis'' in this case. 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 principal 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 example figure at top of this page, it can be seen that refracted ray with ''s'' polarization (with its electric vibration along the direction of the optic axis, thus called the extraordinary ray) is the slow ray in given scenario. Using a thin slab of that material at normal incidence, one would implement a
waveplate A waveplate or retarder is an optics, optical device that alters the Polarization (waves), polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization directio ...
. In this case, there is essentially no spatial separation between the polarizations, 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 is greater than the ordinary index . Negative birefringence means that is less than zero. 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 the best known source of birefringence is the entrance of light into an anisotropic crystal, it can result in otherwise optically isotropic materials in a few ways: * '' Stress birefringence'' results when a normally isotropic solid is stressed and deformed (i.e., stretched or bent) causing a loss of physical isotropy and consequently a loss of isotropy in the material's permittivity tensor; * 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 A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...
; * By the Pockels or Kerr effect, whereby an applied electric field induces birefringence due to
nonlinear optics 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 typic ...
; * By the self or forced alignment into thin films of
amphiphilic An amphiphile (from the Greek αμφις amphis, both, and φιλíα philia, love, friendship), or amphipath, is a chemical compound possessing both hydrophilic (''water-loving'', polar) and lipophilic (''fat-loving'') properties. Such a compo ...
molecules such as
lipids Lipids are a broad group of naturally-occurring molecules which includes fats, waxes, sterols, fat-soluble vitamins (such as vitamins A, D, E and K), monoglycerides, diglycerides, phospholipids, and others. The functions of lipids include ...
, some
surfactants Surfactants are chemical compounds that decrease the surface tension between two liquids, between a gas and a liquid, or interfacial tension between a liquid and a solid. Surfactants may act as detergents, wetting agents, emulsifiers, foaming ...
or
liquid crystals Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. Th ...
; * '' Circular birefringence'' takes place generally not in materials which are anisotropic but rather ones which are
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...
. This can include liquids where there is an
enantiomeric excess In stereochemistry, enantiomeric excess (ee) is a measurement of purity used for chiral substances. It reflects the degree to which a sample contains one enantiomer in greater amounts than the other. A racemic mixture has an ee of 0%, while a si ...
of a chiral molecule, that is, one that has stereo isomers; * By the
Faraday effect The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect (MOFE), is a physical magneto-optical phenomenon. The Faraday effect causes a polarization rotation which is proportional to the projection of the ...
, where a longitudinal magnetic field causes some materials to become '' circularly birefringent'' (having slightly different indices of refraction for left- and right-handed
circular polarization In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
s), similar to optical activity while the field is applied.


Common birefringent materials

The best characterized birefringent materials are
crystals A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macrosc ...
. Due to their specific
crystal structures A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macrosc ...
their refractive indices are well defined. Depending on the symmetry of a crystal structure (as determined by one of the 32 possible
crystallographic point group In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal u ...
s), 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. In addition to induced birefringence while under stress, many
plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
s obtain permanent birefringence during manufacture due to stresses which are "frozen in" due to mechanical forces present when the plastic is molded or extruded. For example, ordinary
cellophane Cellophane is a thin, transparent sheet made of regenerated cellulose. Its low permeability to air, oils, greases, bacteria, and liquid water makes it useful for food packaging. Cellophane is highly permeable to water vapour, but may be coated ...
is birefringent. Polarizers are routinely used to detect stress, either applied or frozen-in, in plastics such as
polystyrene Polystyrene (PS) is a synthetic polymer made from monomers of the aromatic hydrocarbon styrene. Polystyrene can be solid or foamed. General-purpose polystyrene is clear, hard, and brittle. It is an inexpensive resin per unit weight. It is a ...
and
polycarbonate Polycarbonates (PC) are a group of thermoplastic polymers containing carbonate groups in their chemical structures. Polycarbonates used in engineering are strong, tough materials, and some grades are optically transparent. They are easily work ...
.
Cotton Cotton is a soft, fluffy staple fiber that grows in a boll, or protective case, around the seeds of the cotton plants of the genus ''Gossypium'' in the mallow family Malvaceae. The fiber is almost pure cellulose, and can contain minor perce ...
fiber is birefringent because of high levels of cellulosic material in the fibre's secondary cell wall which is directionally aligned with the cotton fibers. Polarized light microscopy is commonly used in biological tissue, as many biological materials are linearly or circularly birefringent. Collagen, found in cartilage, tendon, bone, corneas, and several other areas in the body, is birefringent and commonly studied with polarized light microscopy. Some proteins are also birefringent, exhibiting form birefringence. Inevitable manufacturing imperfections in
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
leads to birefringence, which is one cause of
pulse broadening In optics, and by analogy other branches of physics dealing with wave propagation, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency; sometimes the term chromatic dispersion is used for specificity to o ...
in
fiber-optic communication Fiber-optic communication is a method of transmitting information from one place to another by sending pulses of infrared light through an optical fiber. The light is a form of carrier wave that is modulated to carry information. Fiber is pref ...
s. Such imperfections can be geometrical (lack of circular symmetry), or due to unequal lateral stress applied to the optical fibre. Birefringence is ''intentionally'' introduced (for instance, by making the cross-section elliptical) in order to produce polarization-maintaining optical fibers. Birefringence can be induced (or corrected!) in optical fibers through bending them which causes anisotropy in form and stress given the axis around which it is bent and radius of curvature. In addition to anisotropy in the electric polarizability that we have been discussing, anisotropy in the
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
could be a source of birefringence. At optical frequencies, there is no measurable magnetic polarizability (''μ''= ''μ''0) of natural materials, so this is not an actual source of birefringence at optical wavelengths.


Measurement

Birefringence and other polarization-based optical effects (such as
optical rotation Optical rotation, also known as polarization rotation or circular birefringence, 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. Circul ...
and linear or circular dichroism) can be observed by measuring any change in the polarization of light passing through the material. These measurements are known as
polarimetry Polarimetry is the measurement and interpretation of the polarization of transverse waves, most notably electromagnetic waves, such as radio or light waves. Typically polarimetry is done on electromagnetic waves that have traveled through or ...
. 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, since light that has ''not'' been affected by birefringence remains in a polarization that is totally rejected by the second polarizer ("analyzer"). The addition of quarter-wave plates permits examination using circularly polarized light. Determination of the change in polarization state using such an apparatus is the basis of
ellipsometry Ellipsometry is an optical technique for investigating the dielectric properties (complex refractive index or dielectric function) of thin films. Ellipsometry measures the change of polarization upon reflection or transmission and compares it t ...
, by which the optical properties of specular surfaces can be gauged through reflection. Birefringence measurements have been made with phase-modulated systems for examining the transient flow behaviour of fluids. Birefringence of
lipid bilayers The lipid bilayer (or phospholipid bilayer) is a thin polar membrane made of two layers of lipid molecules. These membranes are flat sheets that form a continuous barrier around all cell (biology), cells. The cell membranes of almost all organis ...
can be measured using
dual-polarization interferometry Dual-polarization interferometry (DPI) is an analytical technique that probes molecular layers adsorbed to the surface of a waveguide using the evanescent wave of a laser beam. It is used to measure the conformational change in proteins, or othe ...
. 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. For the 3D measurement of birefringence, a technique based on holographic tomograph

can be used.


Applications

Birefringence is used in many optical devices.
Liquid-crystal display A liquid-crystal display (LCD) is a flat-panel display A flat-panel display (FPD) is an electronic display used to display visual content such as text or images. It is present in consumer, medical, transportation, and industrial equipmen ...
s, the most common sort of
flat-panel display A flat-panel display (FPD) is an electronic display used to display visual content such as text or images. It is present in consumer, medical, transportation, and industrial equipment. Flat-panel displays are thin, lightweight, provide better l ...
, 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 A Lyot filter, named for its inventor Bernard Lyot, is a type of optical filter that uses birefringence to produce a narrow passband of transmitted wavelengths. Lyot filters are often used in astronomy, particularly for solar astronomy. A Lyot ...
is a specialized narrowband spectral filter employing the wavelength dependence of birefringence.
Waveplate A waveplate or retarder is an optics, optical device that alters the Polarization (waves), polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization directio ...
s 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 Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy o ...
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 Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
, which is required for the 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 direction after passing through the first polarizer, but above the specimen is a polarizer (a so-called ''analyzer'') oriented in the direction. Therefore, no light from the source will be accepted by the analyzer, and the field will appear dark. Areas of the sample possessing birefringence will generally couple some of the -polarized light into the 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
gout Gout ( ) is a form of inflammatory arthritis characterized by recurrent attacks of a red, tender, hot and swollen joint, caused by deposition of monosodium urate monohydrate crystals. Pain typically comes on rapidly, reaching maximal intensit ...
y joint will reveal negatively birefringent monosodium
urate Uric acid is a heterocyclic compound of carbon, nitrogen, oxygen, and hydrogen with the formula C5H4N4O3. It forms ions and salts known as urates and acid urates, such as ammonium acid urate. Uric acid is a product of the metabolic breakdown o ...
crystals.
Calcium pyrophosphate Calcium pyrophosphate (Ca2P2O7) is a chemical compound, an insoluble calcium salt containing the pyrophosphate anion. There are a number of forms reported: an anhydrous form, a dihydrate, Ca2P2O7·2H2O and a tetrahydrate, Ca2P2O7·4H2O. Deposition ...
crystals, in contrast, show weak positive birefringence. Urate crystals appear yellow, and calcium pyrophosphate crystals appear blue when their long axes are aligned parallel to that of a red compensator filter, or a crystal of known birefringence is added to the sample for comparison. The birefringence of tissue inside a living human thigh was measured using polarization-sensitive optical coherence tomography at 1310 nm and a single mode fiber in a needle. Skeletal muscle birefringence was Δn = 1.79 × 10−3± 0.18×10−3, adipose Δn = 0.07 × 10−3 ± 0.50 × 10−3, superficial aponeurosis Δn = 5.08 × 10−3 ± 0.73 × 10−3 and interstitial tissue Δn = 0.65 ×10−3 ±0.39 ×10−3. These measurements may be important for the development of a less invasive method to diagnose
Duchenne muscular dystrophy Duchenne muscular dystrophy (DMD) is a severe type of muscular dystrophy that primarily affects boys. Muscle weakness usually begins around the age of four, and worsens quickly. Muscle loss typically occurs first in the thighs and pelvis fol ...
. Birefringence can be observed in
amyloid Amyloids are aggregates of proteins characterised by a Fibril, fibrillar morphology of 7–13 Nanometer, nm in diameter, a beta sheet (β-sheet) Secondary structure of proteins, secondary structure (known as cross-β) and ability to be Staining, ...
plaques such as are found in the brains of
Alzheimer's Alzheimer's disease (AD) is a neurodegenerative disease that usually starts slowly and progressively worsens. It is the cause of 60–70% of cases of dementia. The most common early symptom is difficulty in remembering recent events. As t ...
patients when stained with a dye such as Congo Red. Modified proteins such as
immunoglobulin An antibody (Ab), also known as an immunoglobulin (Ig), is a large, Y-shaped protein used by the immune system to identify and neutralize foreign objects such as pathogenic bacteria and viruses. The antibody recognizes a unique molecule of the ...
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 Congo red is an organic compound, the sodium salt of 3,3′-( ,1′-biphenyl4,4′-diyl)bis(4-aminonaphthalene-1-sulfonic acid). It is an azo dye. Congo red is water-soluble, yielding a red colloidal solution; its solubility is greater in organic ...
dye intercalates between the folds and, when observed under polarized light, causes birefringence. In
ophthalmology Ophthalmology ( ) is a surgical subspecialty within medicine that deals with the diagnosis and treatment of eye disorders. An ophthalmologist is a physician who undergoes subspecialty training in medical and surgical eye care. Following a medic ...
, binocular retinal birefringence screening of the ''Henle fibers'' (photoreceptor axons that go radially outward from the fovea) provides a reliable detection of
strabismus Strabismus is a vision disorder in which the eyes do not properly align with each other when looking at an object. The eye that is focused on an object can alternate. The condition may be present occasionally or constantly. If present during a ...
and possibly also of anisometropic amblyopia. In healthy subjects, the maximum retardation induced by the Henle fiber layer is approximately 22 degrees at 840 nm. Furthermore, scanning laser polarimetry uses the birefringence of the
optic nerve In neuroanatomy, the optic nerve, also known as the second cranial nerve, cranial nerve II, or simply CN II, is a paired cranial nerve that transmits visual information from the retina to the brain. In humans, the optic nerve is derived fro ...
fibre layer to indirectly quantify its thickness, which is of use in the assessment and monitoring of
glaucoma Glaucoma is a group of eye diseases that result in damage to the optic nerve (or retina) and cause vision loss. The most common type is open-angle (wide angle, chronic simple) glaucoma, in which the drainage angle for fluid within the eye rem ...
. Polarization-sensitive optical coherence tomography measurements obtained from healthy human subjects have demonstrated a change in birefringence of the retinal nerve fiber layer as a function of location around the optic nerve head. The same technology was recently applied in the living human retina to quantify the polarization properties of vessel walls near the optic nerve. Birefringence characteristics in
sperm head The acrosome is an organelle that develops over the anterior (front) half of the head in the spermatozoa (sperm cells) of many animals including humans. It is a cap-like structure derived from the Golgi apparatus. In placental mammals the acrosome ...
s allow the selection of spermatozoa for
intracytoplasmic sperm injection Intracytoplasmic sperm injection (ICSI ) is an in vitro fertilization (IVF) procedure in which a single sperm cell is injected directly into the cytoplasm of an egg. This technique is used in order to prepare the gametes for the obtention of emb ...
. Likewise, '' zona imaging'' uses birefringence on
oocyte An oocyte (, ), oöcyte, or ovocyte is a female gametocyte or germ cell involved in reproduction. In other words, it is an immature ovum, or egg cell. An oocyte is produced in a female fetus in the ovary during female gametogenesis. The female ...
s to select the ones with highest chances of successful pregnancy. Birefringence of particles biopsied from pulmonary nodules indicates silicosis. Dermatologists use dermatoscopes to view skin lesions. Dermoscopes 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 Dermatoscopy also known as dermoscopy or epiluminescence microscopy, is the examination of skin lesions with a dermatoscope. It is a tool similar to a camera to allow for inspection of skin lesions unobstructed by skin surface reflections. The de ...
.


Stress-induced birefringence

Isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
solids do not exhibit birefringence. When they are under
mechanical stress In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...
, 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 Injection moulding (U.S. spelling: injection molding) is a manufacturing process for producing parts by injecting molten material into a mould, or mold. Injection moulding can be performed with a host of materials mainly including metals (for ...
. 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 Photoelasticity describes changes in the optical properties of a material under mechanical deformation. It is a property of all dielectric media and is often used to experimentally determine the stress distribution in a material, where it gives ...
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 An optical vortex (also known as a photonic quantum vortex, screw dislocation or phase singularity) is a zero of an optical field; a point of zero Intensity (physics), intensity. The term is also used to describe a beam of light that has such a ze ...
and full Poincare beams (optical beams that have every possible polarization state across a cross-section).


Other cases of birefringence

Birefringence is observed in anisotropic
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
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 wave __NOTOC__ In seismology and other areas involving elastic waves, S waves, secondary waves, or shear waves (sometimes called elastic S waves) are a type of elastic wave and are one of the two main types of elastic body waves, so named because th ...
s traveling through the solid Earth (the Earth's liquid core does not support shear waves) is widely used in
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
. 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 In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
() is just proportional to the electric field () according to where the material's
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
is just a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
(and equal to where is the
index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
). In an anisotropic material exhibiting birefringence, the relationship between and must now be described using a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
equation: where is now a 3 × 3 permittivity tensor. We assume linearity and no
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
in the medium: . The electric field of a plane wave of angular frequency can be written in the general form: where is the position vector, is time, and is a vector describing the electric field at , . Then we shall find the possible
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
s . By combining
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
for and , we can eliminate to obtain: With no free charges, Maxwell's equation for the divergence of vanishes: We can apply the vector identity to the left hand side of , and use the spatial dependence in which each differentiation in (for instance) results in multiplication by to find: The right hand side of can be expressed in terms of through application of the permittivity tensor and noting that differentiation in time results in multiplication by , then becomes: Applying the differentiation rule to we find: indicates that is orthogonal to the direction of the wavevector , even though that is no longer generally true for as would be the case in an isotropic medium. will not be needed for the further steps in the following derivation. Finding the allowed values of for a given is easiest done by using Cartesian coordinates with the , and axes chosen in the directions of the symmetry axes of the crystal (or simply choosing in the direction of the optic axis of a uniaxial crystal), resulting in a diagonal matrix for the permittivity tensor : where the diagonal values are squares of the refractive indices for polarizations along the three principal axes , and . With in this form, and substituting in the speed of light using , the component of the vector equation becomes where , , are the components of (at any given position in space and time) and , , are the components of . Rearranging, we can write (and similarly for the and components of ) This is a set of linear equations in , , , so it can have a nontrivial solution (that is, one other than ) as long as the following
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
is zero: Evaluating the determinant of , and rearranging the terms according to the powers of \frac, the constant terms cancel. After eliminating the common factor \frac from the remaining terms, we obtain In the case of a uniaxial material, choosing the optic axis to be in the direction so that and , this expression can be factored into Setting either of the factors in to zero will define an
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
al surfaceAlthough related, note that this is not the same as the
index ellipsoid In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polariz ...
.
in the space of wavevectors that are allowed for a given . 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 regardless of the direction of . The second defines a
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
symmetric about the axis. This solution corresponds to the so-called extraordinary rays in which the effective refractive index is in between and , depending on the direction of . Therefore, for any arbitrary direction of propagation (other than in the direction of the optic axis), two distinct wavevectors 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;Born & Wolf, 2002, §15.3.3 the locus of allowed vectors (the ''wavevector surface'') is a 4th-degree two-sheeted surface, so that in a given direction there are generally two permitted vectors (and their opposites).M.V. Berry and M.R. Jeffrey, "Conical diffraction: Hamilton's diabolical point at the heart of crystal optics", in E. Wolf (ed.), ''Progress in Optics'', vol.50, Amsterdam: Elsevier, 2007, , , at . By inspection one can see that is generally satisfied for two positive values of . Or, for a specified optical frequency and direction normal to the wavefronts , it is satisfied for two
wavenumbers In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
(or propagation constants) (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), , and are distinct, it can be shown that there are exactly two such directions, where the two sheets of the wave-vector surface touch; these directions are not at all obvious and do not lie along any of the three principal axes (, , 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. 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 . 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 A waveplate or retarder is an optics, optical device that alters the Polarization (waves), polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization directio ...
for instance. With a waveplate, there is no spatial displacement between the two rays as their vectors are still in the same direction. That is true when each of the two polarizations is either normal to the optic axis (the ordinary ray) or parallel to it (the extraordinary ray). In the more general case, there ''is'' a difference not only in the magnitude but the direction of the two rays. For instance, the photograph through a calcite crystal (top of page) shows a shifted image in the two polarizations; this is due to the optic axis being neither parallel nor normal to the crystal surface. And even when the optic axis ''is'' parallel to the surface, this will occur for waves launched at non-normal incidence (as depicted in the explanatory figure). In these cases the two vectors can be found by solving constrained by the boundary condition which requires that the components of the two transmitted waves' vectors, and the vector of the incident wave, as projected onto the surface of the interface, must all be identical. For a uniaxial crystal it will be found that there is ''not'' a spatial shift for the ordinary ray (hence its name) which will refract as if the material were non-birefringent with an index the same as the two axes which are not the optic axis. For a biaxial crystal neither ray is deemed "ordinary" nor would generally be refracted according to a refractive index equal to one of the principal axes.


See also

*
Cotton–Mouton effect In physical optics, the Cotton–Mouton effect is the birefringence in a liquid in the presence of a constant transverse magnetic field. It is a similar but stronger effect than the Voigt effect (in which the medium is a gas instead of a liquid ...
*
Crystal optics Crystal optics is the branch of optics that describes the behaviour of light in '' anisotropic media'', that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. The index of refrac ...
* Dichroism *
Iceland spar Iceland spar, formerly called Iceland crystal ( is, silfurberg , ) and also called optical calcite, is a transparent variety of calcite, or crystallized calcium carbonate, originally brought from Iceland, and used in demonstrating the polarizati ...
*
Index ellipsoid In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polariz ...
* John Kerr *
Optical rotation Optical rotation, also known as polarization rotation or circular birefringence, 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. Circul ...
*
Periodic poling Periodic poling is a formation of layers with alternate orientation in a birefringent material. The domains are regularly spaced, with period in a multiple of the desired wavelength of operation. The structure is designed to achieve quasi-phase-mat ...
*
Pleochroism Pleochroism (from Greek πλέων, ''pléōn'', "more" and χρῶμα, ''khrôma'', "color") is an optical phenomenon in which a substance has different colors when observed at different angles, especially with polarized light. Backgrou ...


Notes


References


Bibliography

* M. Born and E. Wolf, 2002, ''
Principles of Optics ''Principles of Optics'', colloquially known as ''Born and Wolf'', is an optics textbook written by Max Born and Emil Wolf that was initially published in 1959 by Pergamon Press. After going through six editions with Pergamon Press, the book wa ...
'', 7th Ed., Cambridge University Press, 1999 (reprinted with corrections, 2002). * A. Fresnel, 1827, "Mémoire sur la double réfraction", ''Mémoires de l'Académie Royale des Sciences de l'Institut de France'', vol. (for 1824, printed 1827)
pp.45–176
reprinted as "Second mémoire..." in Fresnel, 1866–70, vol. 2
pp.479–596
translated by A.W. Hobson a
"Memoir on double refraction"
in R.Taylor (ed.), ''Scientific Memoirs'', vol. (London: Taylor & Francis, 1852), pp.238–333. (Cited page numbers are from the translation.) * A. Fresnel (ed. H. de Sénarmont, E. Verdet, and L. Fresnel), 1866–70, ''Oeuvres complètes d'Augustin Fresnel'' (3 volumes), Paris: Imprimerie Impériale
vol. 1 (1866)vol. 2 (1868)vol. 3 (1870)


External links


Stress Analysis Apparatus (based on Birefringence theory)



Video of stress birefringence in Polymethylmethacrylate (PMMA or Plexiglas).

Artist Austine Wood Comarow employs birefringence to create kinetic figurative images.
*
The Birefringence of Thin Ice (Tom Wagner, photographer)
{{Authority control Polarization (waves) Optical mineralogy