2020:. 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. While retinal vessel walls become thicker and less birefringent in patients who suffer from hypertension, hinting at a decrease in vessel wall condition, the vessel walls of diabetic patients do not experience a change in thickness, but do see an increase in birefringence, presumably due to fibrosis or inflammation.
1907:
564:
1898:
765:
5427:
4833:
43:
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4947:
882:
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31:
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576:, 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.
2104:
5422:{\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}
901:), 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.
1787:
4828:{\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}
5813:{\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}
1798:) is horizontally polarized (5) and passes through the liquid-crystal modulator (3) sandwiched in between transparent layers (2, 4) containing electrodes. Horizontally polarized light is blocked by the vertically oriented polarizer (1), except where its polarization has been rotated by the liquid crystal (3), appearing bright to the viewer.
6272:, Ser. 2, vol. 17, pp. 102–111 (May 1821), 167–96 (June 1821), 312–15 ("Postscript", July 1821); reprinted (with added section nos.) in Fresnel, 1866–70, vol. 1, pp. 609–648; translated as "On the calculation of the tints that polarization develops in crystalline plates, & postscript",
2983:
933:
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
755:
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
675:
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.
517:
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
481:
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
1953:
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 ± 0.18×10, adipose Δn = 0.07 × 10 ± 0.50 × 10, superficial aponeurosis
1752:
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
696:
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
538:
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
360:
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
3617:
708:. 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.
6112:
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
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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
579:
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
567:
Comparison of positive and negative birefringence : In positive birefringence (figure 1), the ordinary ray (p-polarisation in this case w.r.t. magenta-coloured plane of incidence), perpendicular to optic axis A is the fast ray (F) while the extraordinary ray (s-polarisation in this case and
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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
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180:, 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
218:
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
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157:
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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.
2048:
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
2087:. 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
591:
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,
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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
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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
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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
792:
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
6295:, Ser. 2, vol. 28, pp. 263–279 (March 1825); reprinted as "Extrait du second Mémoire sur la double réfraction" in Fresnel, 1866–70, vol. 2, pp. 465–478; translated as "Extract of a memoir on double refraction",
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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.
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197:(for "ordinary") regardless of its specific polarization. For rays with any other propagation direction, there is one linear polarization that is perpendicular to the optic axis, and a ray with that polarization is called an
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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
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584:. 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.
486:, 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
676:
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
210:. 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 (parallel to) the optic axis, and this
2283:
1815:, 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,
308:
789:
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;
1853:. The crystals used for these purposes 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
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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
957:. 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.
6048:
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
3046:
1924:
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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.
667:
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,
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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).
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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,
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505:
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
908:
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
148:
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).
50:
polarization (which means perpendicular to plane of incidence – and so in this example becomes "parallel polarisation" to optic axis, thus is called extraordinary ray) sees a greater
6964:
Afsharan, Hadi; Hackmann, Michael J.; Wang, Qiang; Navaeipour, Farzaneh; Jayasree, Stephy Vijaya Kumar; Zawadzki, Robert J.; Silva, Dilusha; Joo, Chulmin; Cense, Barry (2021-07-01).
2091:
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
1889:
polarization; these areas will then appear bright against the dark background. Modifications to this basic principle can differentiate between positive and negative birefringence.
1748:. 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
132:
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
1954:Δn = 5.08 × 10 ± 0.73 × 10 and interstitial tissue Δn = 0.65 × 10 ±0.39 × 10. These measurements may be important for the development of a less invasive method to diagnose
6833:
Reed M. Jost; Joost Felius; Eileen E. Birch (August 2014). "High sensitivity of binocular retinal birefringence screening for anisometropic amblyopia without strabismus".
6045:
directions are generally not of particular interest; biaxial crystals are rather specified by their three refractive indices corresponding to the three axes of symmetry.
1915:
Gout and pseudogout crystals viewed under a microscope with a red compensator, which slows red light in one orientation (labeled "polarized light axis"). Urate crystals (
2978:{\displaystyle (-\mathbf {k} \cdot \mathbf {k} )\mathbf {E} +(\mathbf {k} \cdot \mathbf {E} )\mathbf {k} =-\mu _{0}\omega ^{2}({\boldsymbol {\varepsilon }}\mathbf {E} )}
1923:
appear yellow when their long axis is parallel to the slow transmission axis of the red compensator and appear blue when perpendicular. The opposite colors are seen in
6248:"An account of sundry experiments made and communicated by that learn'd mathematician, Dr. Erasmus Bartholin, upon a chrystal-like body, sent to him out of Island,"
160:
Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images.
2340:
688:
In a birefringent material, a wave consists of two polarization components which generally are governed by different effective refractive indices. The so-called
551:
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
930:
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.
6653:
Gilbert, Larry; Weber, M.F.; Strharsky, R.J.; Stover, C.A.; Nevitt, T.J.; Ouderkirk, A.J. (2001). "Giant birefringent optics in multilayer polymer filters".
1768:. 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.
7074:"Non-Invasive Retinal Blood Vessel Wall Measurements with Polarization-Sensitive Optical Coherence Tomography for Diabetes Assessment: A Quantitative Study"
6359:
Born & Wolf, 2002, pp. 807–808. (In 19th-century terminology, the ordinary ray is said to be polarized in the plane of the optic axis; but this "
3612:{\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}}
2083:, birefringence results. The stress can be applied externally or is "frozen in" after a birefringent plastic ware is cooled after it is manufactured using
6108:
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
3274:{\displaystyle \mathbf {\varepsilon } =\varepsilon _{0}{\begin{bmatrix}n_{x}^{2}&0&0\\0&n_{y}^{2}&0\\0&0&n_{z}^{2}\end{bmatrix}}}
701:
polarization (with its electric vibration along the direction of the optic axis, thus called the extraordinary ray) is the slow ray in given scenario.
5858:. 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
949:. Such imperfections can be geometrical (lack of circular symmetry), or due to unequal lateral stress applied to the optical fibre. Birefringence is
6383:
6084:
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
2813:{\displaystyle -\nabla \times \nabla \times \mathbf {E} =(\mathbf {k} \cdot \mathbf {E} )\mathbf {k} -(\mathbf {k} \cdot \mathbf {k} )\mathbf {E} }
445:. Thus there is no axis around which a rotation leaves the optical properties invariant (as there is with uniaxial crystals whose index ellipsoid
378:, 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
7397:
7281:
M.V. Berry and M.R. Jeffrey, "Conical diffraction: Hamilton's diabolical point at the heart of crystal optics", in E. Wolf (ed.),
2146:
2123:
materials. In these materials, the two polarizations split according to their effective refractive indices, which are also sensitive to stress.
6168:
954:
1760:
Birefringence measurements have been made with phase-modulated systems for examining the transient flow behaviour of fluids. Birefringence of
6670:
6545:
6919:"Thickness and Birefringence of Healthy Retinal Nerve Fiber Layer Tissue Measured with Polarization-Sensitive Optical Coherence Tomography"
2253:
1973:
light chains abnormally accumulate between cells, forming fibrils. Multiple folds of these fibers line up and take on a beta-pleated sheet
7025:"Hypertension-associated changes in retinal blood vessel walls measured in vivo with polarization-sensitive optical coherence tomography"
2008:. In healthy subjects, the maximum retardation induced by the Henle fiber layer is approximately 22 degrees at 840 nm. Furthermore,
671:
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
1831:
are thin birefringent sheets widely used in certain optical equipment for modifying the polarization state of light passing through it.
2041:
to select the ones with highest chances of successful pregnancy. Birefringence of particles biopsied from pulmonary nodules indicates
1744:) can be observed by measuring any change in the polarization of light passing through the material. These measurements are known as
897:
their refractive indices are well defined. Depending on the symmetry of a crystal structure (as determined by one of the 32 possible
6324:
353:. This causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of
6860:
Cense, Barry; Wang, Qiang; Lee, Sangyeol; Zhao, Liang; Elsner, Ann E.; Hitzenberger, Christoph K.; Miller, Donald T. (2013-11-01).
256:
7133:"Birefringence characteristics in sperm heads allow for the selection of reacted spermatozoa for intracytoplasmic sperm injection"
6458:
Wolman, M.; Kasten, F. H. (1986). "Polarized light microscopy in the study of the molecular structure of collagen and reticulin".
6058:. Applying the different phase of propagation to those two waves over a specified propagation distance will result in a generally
5907:. Therefore, for any arbitrary direction of propagation (other than in the direction of the optic axis), two distinct wavevectors
756:
designation does not apply. But for any defined ray direction one can just as well designate the fast and slow ray polarizations.
2028:
1765:
7174:"Automatic user-independent zona pellucida imaging at the oocyte stage allows for the prediction of preimplantation development"
1993:
961:
4320:{\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}
4113:{\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}
3906:{\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}
776:
While the best known source of birefringence is the entrance of light into an anisotropic crystal, it can result in otherwise
539:
exiting the crystal through a face parallel to the incoming face, the direction of both rays will be restored, but leaving a
6214:
1865:
Birefringence is utilized in medical diagnostics. One powerful accessory used with optical microscopes is a pair of crossed
1906:
563:
7447:
5883:
axis. This solution corresponds to the so-called extraordinary rays in which the effective refractive index is in between
6966:"Polarization properties of retinal blood vessel walls measured with polarization sensitive optical coherence tomography"
6561:
Frattini, P. L.; Fuller, G. G. (1984). "A note on phase-modulated flow birefringence: a promising rheo-optical method".
5990:(and thus effective refractive indices) corresponding to the propagation of two linear polarizations in that direction.
5916:
For a biaxial material a similar but more complicated condition on the two waves can be described; the locus of allowed
1982:
942:
898:
101:. The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material.
680:, with the two "axes" in this case referring to ray directions in which propagation does not experience birefringence.
7452:
6268:
A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" et seq.,
3136:
in the direction of the optic axis of a uniaxial crystal), resulting in a diagonal matrix for the permittivity tensor
3015:
1955:
498:
which separates incoming light into two linear polarizations using prisms composed of a birefringent material such as
2602:{\displaystyle -\nabla \times \nabla \times \mathbf {E} =\mu _{0}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {D} }
692:
is the component for which the material has the higher effective refractive index (slower phase velocity), while the
764:
7023:
Afsharan, Hadi; Anilkumar, Vidyalakshmi; Silva, Dilusha; Dwivedi, Girish; Joo, Chulmin; Cense, Barry (2024-01-01).
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2009:
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polarization (the "ordinary ray" in this case, having its electric vector perpendicular to the optic axis) and the
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46:
In this example, optic axis along the surface is shown perpendicular to plane of incidence. Incoming light in the
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where the diagonal values are squares of the refractive indices for polarizations along the three principal axes
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Sandwiched in between crossed polarizers, clear polystyrene cutlery exhibits wavelength-dependent birefringence
1835:
772:
with coloured fringes due to stress birefringence of partially polarised skylight through a circular polariser
374:. A crystal with its optic axis in this orientation, parallel to the optical surface, may be used to create a
2095:
and full
Poincare beams (optical beams that have every possible polarization state across a cross-section).
1816:
964:
could be a source of birefringence. At optical frequencies, there is no measurable magnetic polarizability (
519:
181:
2130:
traveling through the solid Earth (the Earth's liquid core does not support shear waves) is widely used in
5998:
direction according to the above construction. But when all three refractive indices (or permittivities),
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1966:
1808:
1791:
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1839:
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To manufacture polarizers with high transmittance, birefringent crystals are used in devices such as the
960:
In addition to anisotropy in the electric polarizability that we have been discussing, anisotropy in the
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direction after passing through the first polarizer, but above the specimen is a polarizer (a so-called
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383:
145:
7359:(London: Taylor & Francis, 1852), pp. 238–333. (Cited page numbers are from the translation.)
6862:"Henle fiber layer phase retardation measured with polarization-sensitive optical coherence tomography"
6096:
constrained by the boundary condition which requires that the components of the two transmitted waves'
568:
parallel to optic axis A) is the slow ray (S). In negative birefringence (figure 2), it is the reverse.
531:
6390:
First published in StoneChat, the
Journal of the UK Facet Cutter's Guild. January–March. edition 2005.
5926:) is a 4th-degree two-sheeted surface, so that in a given direction there are generally two permitted
2016:
fiber layer to indirectly quantify its thickness, which is of use in the assessment and monitoring of
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polarization (which becomes ordinary ray because "perpendicular polarisation" to optic axis) and so
42:
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2226:
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970:
844:
6917:
Cense, Barry; Chen, Teresa C.; Park, B. Hyle; Pierce, Mark C.; Boer, Johannes F. de (2004-08-01).
1827:
is a specialized narrowband spectral filter employing the wavelength dependence of birefringence.
410:
for given directions of the polarization. Note that for biaxial crystals the index ellipsoid will
398:
refractive indices corresponding to three principal axes of the crystal. For most ray directions,
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1974:
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The case of so-called biaxial crystals is substantially more complex. These are characterized by
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74:
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6426:"The use of birefringence for predicting the stiffness of injection molded polycarbonate discs"
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for instance. With a waveplate, there is no spatial displacement between the two rays as their
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2000:(photoreceptor axons that go radially outward from the fovea) provides a reliable detection of
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Experimenta crystalli islandici disdiaclastici quibus mira & infolita refractio detegitur
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6695:
6658:
6609:
6578:
6518:
6467:
6440:
6153:
2202:
1854:
1850:
1737:
1627:
881:
869:
860:
835:
809:
777:
769:
716:
Uniaxial birefringence is classified as positive when the extraordinary index of refraction
572:
Much of the work involving polarization preceded the understanding of light as a transverse
491:
184:
of the material. Light propagating parallel to the optic axis (whose polarization is always
133:
78:
51:
7340:
2405:{\displaystyle \mathbf {E} =\mathbf {E} _{0}e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}}
1842:
and other variants. Layered birefringent polymer sheets can also be used for this purpose.
6425:
6387:
6189:
6158:
6143:
2088:
1306:
1148:
848:
828:
527:
495:
407:
346:
247:
6686:
Frances Lixey, Mary (1983-06-01). "Inexpensive
Compensator for a Polarizing Microscope".
1757:, by which the optical properties of specular surfaces can be gauged through reflection.
7379:
7230:
7108:
7073:
7040:
6629:
6574:
6514:
2145:
7413:
7000:
6894:
6810:
6747:
6722:
6128:
5913:
are allowed corresponding to the polarizations of the ordinary and extraordinary rays.
2137:
Birefringence is widely used in mineralogy to identify rocks, minerals, and gemstones.
2092:
1970:
1820:
1761:
996:
916:
are routinely used to detect stress, either applied or frozen-in, in plastics such as
855:
839:
takes place generally not in materials which are anisotropic but rather ones which are
801:
523:
487:
63:
30:
7407:
Artist
Austine Wood Comarow employs birefringence to create kinetic figurative images.
7298:
6614:
6597:
3130:
axes chosen in the directions of the symmetry axes of the crystal (or simply choosing
953:
introduced (for instance, by making the cross-section elliptical) in order to produce
7441:
7190:
7173:
7149:
7132:
6522:
1989:
1031:
938:
921:
164:
A mathematical description of wave propagation in a birefringent medium is presented
7328:, 7th Ed., Cambridge University Press, 1999 (reprinted with corrections, 2002).
7049:
7024:
6487:
2103:
6138:
2193:
2033:
1754:
1668:
1635:
1608:
794:
137:
129:
6782:
McBride, Jeffrey M.; Hackmann, Michael J.; Nimphius, Sophia; Cense, Barry (2022).
1985:
between the folds and, when observed under polarized light, causes birefringence.
7375:
7371:
6846:
2229:). In an anisotropic material exhibiting birefringence, the relationship between
219:
extraordinary ray. The ordinary ray will always experience a refractive index of
7421:
6163:
5977:
4378:
2464:
2013:
1824:
1745:
917:
824:
805:
552:
535:
371:
350:
90:
17:
580:
cannot be described using the law of refraction. This thus became known as the
494:
of each of these two polarizations. This is clearly seen, for instance, in the
144:) crystals which have one of the strongest birefringences. In the 19th century
7401:
6772:
Author: Alan N. Baer; Chief Editor: Herbert S. Diamond. Updated: Nov 22, 2010.
6699:
2131:
2127:
2050:
2024:
2001:
1969:
patients when stained with a dye such as Congo Red. Modified proteins such as
1624:
1579:
1541:
1500:
1470:
1381:
1359:
1328:
1261:
1182:
909:
816:
704:
Using a thin slab of that material at normal incidence, one would implement a
7248:
7099:
7058:
6991:
6942:
6885:
6707:
6596:
Doyle, P. S.; Shaqfeh, E. S. G.; McKinley, G. H.; Spiegelberg, S. H. (1998).
1776:
38:
crystal laid upon a graph paper with blue lines showing the double refraction
7402:
6133:
6063:
5843:
2076:
2070:
2042:
2005:
1978:
1866:
1828:
1795:
1706:
1558:
913:
864:(having slightly different indices of refraction for left- and right-handed
705:
544:
473:
along which the group velocity of the light is independent of polarization.
379:
375:
7256:
7199:
7158:
7117:
7009:
6950:
6903:
6819:
522:
of the birefringent material is not exactly in the direction of the wave's
228:, whereas the refractive index of the extraordinary ray will be in between
7131:
Gianaroli L.; Magli M. C.; Ferraretti A. P.; et al. (December 2008).
7072:
Afsharan, Hadi; Silva, Dilusha; Joo, Chulmin; Cense, Barry (August 2023).
6835:
Journal of
American Association for Pediatric Ophthalmology and Strabismus
6756:
6479:
979:) of natural materials, so this is not an actual source of birefringence.
490:
of refraction, the two angles of refraction are governed by the effective
7239:
7090:
6934:
6877:
6662:
6367:
to the vibration; cf. Fresnel, 1827, tr. Hobson, p. 318.)
6319:. Vol. 1. Palo Alto: Blackwell Scientific Publications. p. 28.
5874:
5445:
In the case of a uniaxial material, choosing the optic axis to be in the
2307:
is now a 3 × 3 permittivity tensor. We assume linearity and no
2017:
1334:
1290:
1277:
1236:
1207:
1157:
1106:
1044:
890:
663:
Materials in which all three refractive indices are different are termed
415:
6983:
6801:
6738:
6471:
1962:
1786:
1684:
1595:
1520:
1097:
905:
499:
354:
141:
110:
102:
35:
6444:
3093:
will not be needed for the further steps in the following derivation.
858:, where a longitudinal magnetic field causes some materials to become
7343:; reprinted as "Second mémoire..." in Fresnel, 1866–70, vol. 2,
7213:
Beckley, Amber M.; Brown, Thomas G.; Alonso, Miguel A. (2010-05-10).
6783:
6582:
6296:
6273:
2854:
and noting that differentiation in time results in multiplication by
2278:{\displaystyle \mathbf {D} ={\boldsymbol {\varepsilon }}\mathbf {E} }
2242:
2108:
2038:
1350:
1252:
1198:
927:
840:
820:
526:
for the extraordinary ray. The direction of power flow (given by the
250:. The magnitude of the difference is quantified by the birefringence
6301:
6278:
1736:
Birefringence and other polarization-based optical effects (such as
6598:"Relaxation of dilute polymer solutions following extensional flow"
6062:
net polarization state at that point; this is the principle of the
2111:
observed in different polarizations using a rotating polarizer (or
7333:
Mémoires de l'Académie Royale des
Sciences de l'Institut de France
6406:
2709:, and use the spatial dependence in which each differentiation in
2144:
2102:
2060:
1857:, which is required for the efficient operation of these devices.
1785:
1651:
1444:
1060:
880:
763:
562:
418:") but is described by three unequal principle refractive indices
155:
86:
41:
29:
4898:, the constant terms cancel. After eliminating the common factor
808:, whereby an applied electric field induces birefringence due to
619:, along three coordinate axes; in this case two are equal. So if
7366:
H. de Sénarmont, E. Verdet, and L. Fresnel), 1866–70,
4370:, so it can have a nontrivial solution (that is, one other than
1938:
1920:
1516:
1223:
2625:
With no free charges, Maxwell's equation for the divergence of
934:
proteins are also birefringent, exhibiting form birefringence.
361:
ray propagates at a different phase velocity (corresponding to
5932:
vectors (and their opposites). By inspection one can see that
1122:
904:
In addition to induced birefringence while under stress, many
7392:
6501:
Sano, Y (1988). "Optical anistropy of bovine serum albumin".
6291:
A. Fresnel, "Extrait d'un Mémoire sur la double réfraction",
303:{\displaystyle \Delta n=n_{\mathrm {e} }-n_{\mathrm {o} }\,.}
7432:
2163:
In an isotropic medium (including free space) the so-called
982:
168:. Following is a qualitative explanation of the phenomenon.
6784:"In vivo PS-OCT needle probe scan of human skeletal muscle"
2326:. The electric field of a plane wave of angular frequency
1925:
calcium pyrophosphate dihydrate crystal deposition disease
7406:
7393:
Stress
Analysis Apparatus (based on Birefringence theory)
6251:
2153:
vectors for a fixed frequency for a biaxial crystal (see
2079:
solids do not exhibit birefringence. When they are under
1931:
image): blue when parallel and yellow when perpendicular.
7433:
7172:
Ebner T.; Balaban B.; Moser M.; et al. (May 2009).
6188:
Although related, note that this is not the same as the
509:), so that the angle of refraction is different for the
4857:, and rearranging the terms according to the powers of
1794:. Light reflected by the surface (6) (or coming from a
406:
The two refractive indices can be determined using the
370:) but still has the power flow in the direction of the
7331:
A. Fresnel, 1827, "Mémoire sur la double réfraction",
6723:"Acute gout and the accident and emergency department"
4398:
3177:
7277:
7275:
5501:
4950:
4904:
4863:
4392:
4144:
3937:
3730:
3383:
3321:
in this form, and substituting in the speed of light
3153:
3018:
2878:
2735:
2642:
2531:
2343:
2256:
465:. Additionally, there are two distinct axes known as
259:
246:, depending on the ray direction as described by the
188:
to the optic axis) is governed by a refractive index
6381:
Birefringence for facetors I: what is birefringence?
3701:. Rearranging, we can write (and similarly for the
3081:, even though that is no longer generally true for
1869:filters. Light from the source is polarized in the
815:By the self or forced alignment into thin films of
201:and is governed by the same refractive index value
176:The simplest type of birefringence is described as
6540:. New York: Oxford University Press. p. 202.
5938:is generally satisfied for two positive values of
5812:
5421:
4931:
4890:
4827:
4319:
4112:
3905:
3611:
3273:
3040:
2977:
2812:
2662:
2601:
2404:
2277:
889:The best characterized birefringent materials are
302:
3075:is orthogonal to the direction of the wavevector
785:
452:Although there is no axis of symmetry, there are
6923:Investigative Ophthalmology & Visual Science
6338:
6336:
3041:{\displaystyle \mathbf {k} \cdot \mathbf {D} =0}
2065:Color pattern of a plastic box with "frozen in"
1937:For instance, needle aspiration of fluid from a
1823:of polarized light followed by a polarizer. The
7370:(3 volumes), Paris: Imprimerie Impériale;
6400:
6398:
6396:
2848:through application of the permittivity tensor
1807:Birefringence is used in many optical devices.
534:is at a finite angle from the direction of the
3668:(at any given position in space and time) and
2173:) is just proportional to the electric field (
1845:Birefringence also plays an important role in
1775:, a technique based on holographic tomography
941:leads to birefringence, which is one cause of
3087:as would be the case in an isotropic medium.
2449:is a vector describing the electric field at
843:. This can include liquids where there is an
317:) of the ordinary ray is simply described by
8:
7285:, vol. 50, Amsterdam: Elsevier, 2007,
4932:{\displaystyle {\frac {\omega ^{2}}{c^{2}}}}
4891:{\displaystyle {\frac {\omega ^{2}}{c^{2}}}}
2715:(for instance) results in multiplication by
847:of a chiral molecule, that is, one that has
6243:(Copenhagen, Denmark: Daniel Paulli, 1669).
6213:Abramowitz, Mortimer; Davidson, Michael W.
2663:{\displaystyle \nabla \cdot \mathbf {D} =0}
1965:plaques such as are found in the brains of
6113:index equal to one of the principal axes.
1941:joint will reveal negatively birefringent
1399:
986:
937:Inevitable manufacturing imperfections in
7238:
7189:
7148:
7107:
7089:
7048:
6999:
6893:
6809:
6746:
6694:(6). Oxford University Press (OUP): 382.
6613:
6300:
6277:
5791:
5781:
5775:
5764:
5758:
5757:
5747:
5742:
5736:
5725:
5719:
5718:
5708:
5703:
5697:
5686:
5680:
5679:
5669:
5664:
5658:
5640:
5630:
5624:
5613:
5607:
5606:
5596:
5591:
5585:
5574:
5568:
5567:
5557:
5552:
5546:
5535:
5529:
5528:
5518:
5513:
5507:
5500:
5402:
5397:
5384:
5379:
5366:
5361:
5338:
5333:
5323:
5318:
5307:
5302:
5296:
5284:
5279:
5269:
5264:
5253:
5248:
5242:
5230:
5225:
5215:
5210:
5199:
5194:
5188:
5167:
5162:
5151:
5146:
5133:
5128:
5121:
5110:
5105:
5094:
5089:
5076:
5071:
5064:
5053:
5048:
5037:
5032:
5019:
5014:
5007:
4994:
4984:
4978:
4967:
4957:
4951:
4949:
4921:
4911:
4905:
4903:
4880:
4870:
4864:
4862:
4798:
4787:
4782:
4772:
4765:
4756:
4751:
4738:
4733:
4713:
4703:
4691:
4681:
4667:
4657:
4638:
4627:
4622:
4612:
4605:
4596:
4591:
4578:
4573:
4553:
4543:
4529:
4519:
4507:
4497:
4478:
4467:
4462:
4452:
4445:
4436:
4431:
4418:
4413:
4393:
4391:
4305:
4288:
4277:
4272:
4262:
4255:
4246:
4241:
4228:
4223:
4202:
4192:
4182:
4169:
4159:
4149:
4143:
4098:
4088:
4078:
4065:
4048:
4037:
4032:
4022:
4015:
4006:
4001:
3988:
3983:
3962:
3952:
3942:
3936:
3891:
3881:
3871:
3858:
3848:
3838:
3825:
3808:
3797:
3792:
3782:
3775:
3766:
3761:
3748:
3743:
3729:
3603:
3591:
3580:
3575:
3565:
3558:
3546:
3536:
3526:
3513:
3503:
3493:
3480:
3470:
3465:
3452:
3437:
3432:
3419:
3414:
3401:
3396:
3382:
3257:
3252:
3223:
3218:
3189:
3184:
3172:
3166:
3154:
3152:
3027:
3019:
3017:
2967:
2962:
2953:
2943:
2928:
2920:
2912:
2901:
2893:
2885:
2877:
2805:
2797:
2789:
2778:
2770:
2762:
2751:
2734:
2649:
2641:
2594:
2585:
2571:
2565:
2559:
2547:
2530:
2383:
2375:
2368:
2358:
2353:
2344:
2342:
2270:
2265:
2257:
2255:
2119:Birefringence is observed in anisotropic
518:exactly normal to it); in this case, the
349:) is not exactly in the direction of the
296:
289:
288:
274:
273:
258:
6770:The Approach to the Painful Joint Workup
6602:Journal of Non-Newtonian Fluid Mechanics
6317:Optical Mineralogy: Theory and Technique
5944:. Or, for a specified optical frequency
1819:modulate the intensity of light through
6205:
6181:
5950:and direction normal to the wavefronts
5490:, this expression can be factored into
2963:
2266:
2027:allow the selection of spermatozoa for
955:polarization-maintaining optical fibers
62:polarization ray is undergoing greater
6169:Huygens principle of double refraction
643:axes, then the extraordinary index is
6246:Erasmus Bartholin (January 16, 1671)
4343:This is a set of linear equations in
3001:Applying the differentiation rule to
120:Birefringence is responsible for the
7:
7368:Oeuvres complètes d'Augustin Fresnel
7347:; translated by A.W. Hobson as
6342:Landau, L. D., and Lifshitz, E. M.,
6215:"Olympus Microscopy Resource Center"
5846:surface in the space of wavevectors
5492:
4941:
4939:from the remaining terms, we obtain
4383:
4135:
3928:
3721:
3374:
3144:
3009:
2869:
2726:
2633:
2522:
2334:
2332:can be written in the general form:
2247:
734:. Negative birefringence means that
66:on entering and exiting the crystal.
6344:Electrodynamics of Continuous Media
6257: : 2041–2048 (pdf pp 282–289).
725:is greater than the ordinary index
547:specifically have their optic axis
5759:
5720:
5681:
5608:
5569:
5530:
2745:
2739:
2643:
2578:
2568:
2541:
2535:
2463:. Then we shall find the possible
1821:electrically induced birefringence
988:Uniaxial crystals, at 590 nm
386:from a linearly polarized source.
290:
275:
260:
25:
6433:Polymer Engineering & Science
5836:Setting either of the factors in
3366:component of the vector equation
2686:We can apply the vector identity
2023:Birefringence characteristics in
1961:Birefringence can be observed in
1401:Biaxial crystals, at 590 nm
214:will be governed by a different,
7191:10.1016/j.fertnstert.2009.03.106
7150:10.1016/j.fertnstert.2008.10.024
7029:Optics and Lasers in Engineering
6721:Hardy RH, Nation B (June 1984).
6293:Annales de Chimie et de Physique
6270:Annales de Chimie et de Physique
6090:vectors can be found by solving
6075:In the more general case, there
5901:, depending on the direction of
3028:
3020:
2968:
2929:
2921:
2913:
2902:
2894:
2886:
2806:
2798:
2790:
2779:
2771:
2763:
2752:
2650:
2595:
2548:
2384:
2376:
2354:
2345:
2271:
2258:
2029:intracytoplasmic sperm injection
1905:
1896:
1766:dual-polarization interferometry
760:Sources of optical birefringence
414:be an ellipsoid of revolution ("
345:). Its power flow (given by the
27:Refractive property of materials
7050:10.1016/j.optlaseng.2023.107838
5867:regardless of the direction of
1994:retinal birefringence screening
1773:3D measurement of birefringence
656:axis, which is also called the
109:are often birefringent, as are
7269:Born & Wolf, 2002, §15.3.3
6282:(Creative Commons), 2021; §14.
4851:Evaluating the determinant of
3096:Finding the allowed values of
2972:
2959:
2925:
2909:
2898:
2879:
2802:
2786:
2775:
2759:
2397:
2372:
2241:must now be described using a
2126:The study of birefringence in
2012:uses the birefringence of the
1:
7349:"Memoir on double refraction"
7299:10.1016/S0079-6638(07)50002-8
6655:Optical Interference Coatings
6615:10.1016/S0377-0257(97)00113-4
6348:Course of Theoretical Physics
5852:that are allowed for a given
2842:can be expressed in terms of
899:crystallographic point groups
877:Common birefringent materials
555:between the two light waves.
85:and propagation direction of
7351:, in R. Taylor (ed.),
6847:10.1016/j.jaapos.2014.07.017
6536:Hobbs, Peter Victor (2010).
6523:10.1016/0021-9797(88)90178-6
3715:
3368:
3089:
3065:
3003:
2863:
2838:
2705:
2099:Other cases of birefringence
2057:Stress-induced birefringence
313:The propagation (as well as
136:in 1669, who observed it in
7322:M. Born and E. Wolf, 2002,
6092:
5980:(or propagation constants)
5934:
5838:
4853:
4377:) as long as the following
2155:
2069:placed between two crossed
1956:Duchenne muscular dystrophy
1851:nonlinear optical processes
1790:Reflective twisted-nematic
872:while the field is applied.
382:is commonly used to create
165:
93:materials are described as
7474:
7339:(for 1824, printed 1827),
6630:"Birefringent_ Polarizers"
6386:December 14, 2013, at the
6350:1960 (Pergamon Press), §79
6315:Ehlers, Ernest G. (1987).
6305:, 2021 (open access).
6217:. Olympus Life Science Inc
5976:, it is satisfied for two
2010:scanning laser polarimetry
1885:-polarized light into the
1847:second-harmonic generation
1811:, the most common sort of
947:fiber-optic communications
6971:Biomedical Optics Express
6866:Biomedical Optics Express
6789:Biomedical Optics Express
6411:The Physics Hypertextbook
3108:is easiest done by using
2703:to the left hand side of
1943:monosodium urate crystals
780:materials in a few ways:
7426:University of Nottingham
6503:J. Colloid Interface Sci
2434:is the position vector,
893:. Due to their specific
7380:vol. 3 (1870)
7376:vol. 2 (1868)
7372:vol. 1 (1866)
6700:10.1093/labmed/14.6.382
5873:. The second defines a
5842:to zero will define an
2836:The right hand side of
2149:Surface of the allowed
2006:anisometropic amblyopia
1809:Liquid-crystal displays
861:circularly birefringent
520:dielectric polarization
77:of a material having a
5814:
5423:
4933:
4892:
4829:
4321:
4114:
3907:
3695:are the components of
3662:are the components of
3613:
3275:
3042:
2979:
2814:
2664:
2603:
2406:
2279:
2160:
2116:
2073:
2037:uses birefringence on
1799:
1792:liquid-crystal display
1764:can be measured using
886:
866:circular polarizations
836:Circular birefringence
773:
569:
315:reflection coefficient
304:
161:
67:
39:
7412:Merrifield, Michael.
7215:"Full Poincaré beams"
6424:Neves, N. M. (1998).
6405:Elert, Glenn (2021).
6361:plane of polarization
5815:
5424:
4934:
4893:
4830:
4322:
4115:
3908:
3614:
3276:
3110:Cartesian coordinates
3043:
2980:
2815:
2665:
2604:
2407:
2309:magnetic permeability
2280:
2192:where the material's
2165:electric displacement
2148:
2106:
2064:
2004:and possibly also of
1947:Calcium pyrophosphate
1789:
962:magnetic permeability
884:
767:
652:corresponding to the
635:corresponding to the
587:A material is termed
566:
384:circular polarization
305:
159:
146:Augustin-Jean Fresnel
45:
33:
7448:Polarization (waves)
7325:Principles of Optics
7240:10.1364/OE.18.010777
7091:10.3390/biom13081230
6935:10.1167/iovs.03-1160
6878:10.1364/BOE.4.002296
6663:10.1364/OIC.2001.FA2
6657:. OSA. p. FA2.
6124:Cotton–Mouton effect
5877:symmetric about the
5499:
4948:
4902:
4861:
4390:
4142:
3935:
3728:
3381:
3151:
3016:
2876:
2733:
2640:
2529:
2341:
2254:
786:Stress birefringence
768:View from under the
712:Positive or negative
574:electromagnetic wave
257:
81:that depends on the
7231:2010OExpr..1810777B
7225:(10): 10777–10785.
7041:2024OptLE.17207838A
6822:– via Optica.
6688:Laboratory Medicine
6575:1984JRheo..28...61F
6563:Journal of Rheology
6515:1988JCIS..124..403S
6237:Erasmus Bartholin,
5769:
5752:
5730:
5713:
5691:
5674:
5618:
5601:
5579:
5562:
5540:
5523:
5407:
5389:
5371:
5343:
5328:
5312:
5289:
5274:
5258:
5235:
5220:
5204:
5172:
5156:
5138:
5115:
5099:
5081:
5058:
5042:
5024:
4792:
4761:
4743:
4632:
4601:
4583:
4472:
4441:
4423:
4282:
4251:
4233:
4042:
4011:
3993:
3802:
3771:
3753:
3585:
3475:
3442:
3424:
3406:
3262:
3228:
3194:
2490:, we can eliminate
2474:Maxwell's equations
2227:index of refraction
1836:Glan–Thompson prism
1402:
989:
845:enantiomeric excess
778:optically isotropic
216:direction-dependent
7453:Optical mineralogy
7353:Scientific Memoirs
7283:Progress in Optics
6984:10.1364/BOE.426079
6802:10.1364/BOE.446169
6739:10.1136/emj.1.2.89
6472:10.1007/bf00508652
5924:wavevector surface
5810:
5753:
5738:
5714:
5699:
5675:
5660:
5602:
5587:
5563:
5548:
5524:
5509:
5451:direction so that
5419:
5393:
5375:
5357:
5329:
5314:
5298:
5275:
5260:
5244:
5221:
5206:
5190:
5158:
5142:
5124:
5101:
5085:
5067:
5044:
5028:
5010:
4929:
4888:
4825:
4813:
4778:
4747:
4729:
4618:
4587:
4569:
4458:
4427:
4409:
4317:
4268:
4237:
4219:
4110:
4028:
3997:
3979:
3903:
3788:
3757:
3739:
3609:
3571:
3461:
3428:
3410:
3392:
3271:
3265:
3248:
3214:
3180:
3038:
2975:
2810:
2660:
2599:
2402:
2275:
2161:
2117:
2074:
1877:) oriented in the
1813:flat-panel display
1800:
1742:circular dichroism
1400:
1331:(complex silicate)
1174:magnesium fluoride
987:
895:crystal structures
887:
819:molecules such as
774:
684:Fast and slow rays
570:
507:plane of incidence
380:quarter-wave plate
300:
172:Uniaxial materials
162:
107:crystal structures
89:. These optically
68:
54:than light in the
40:
7345:pp. 479–596
6872:(11): 2296–2306.
6672:978-1-55752-682-3
6547:978-0-19-958771-1
6445:10.1002/pen.10347
6439:(10): 1770–1777.
6102:vectors, and the
5834:
5833:
5797:
5770:
5731:
5692:
5646:
5619:
5580:
5541:
5443:
5442:
5345:
5291:
5237:
5173:
5116:
5059:
5000:
4973:
4927:
4886:
4849:
4848:
4804:
4644:
4484:
4341:
4340:
4294:
4134:
4133:
4054:
3927:
3926:
3814:
3633:
3632:
3597:
3295:
3294:
3062:
3061:
2999:
2998:
2834:
2833:
2684:
2683:
2623:
2622:
2592:
2426:
2425:
2299:
2298:
2085:injection molding
2081:mechanical stress
2067:mechanical stress
1840:Glan–Taylor prism
1728:
1727:
1721:
1720:
1396:
1395:
1375:zircon, low ZrSiO
912:is birefringent.
582:extraordinary ray
532:inhomogenous wave
484:unpolarized light
477:Double refraction
390:Biaxial materials
212:extraordinary ray
126:double refraction
115:mechanical stress
16:(Redirected from
7465:
7429:
7365:
7358:
7341:pp. 45–176
7338:
7310:
7308:
7306:
7292:
7290:
7279:
7270:
7267:
7261:
7260:
7242:
7210:
7204:
7203:
7193:
7169:
7163:
7162:
7152:
7128:
7122:
7121:
7111:
7093:
7069:
7063:
7062:
7052:
7020:
7014:
7013:
7003:
6978:(7): 4340–4362.
6961:
6955:
6954:
6929:(8): 2606–2612.
6914:
6908:
6907:
6897:
6857:
6851:
6850:
6830:
6824:
6823:
6813:
6796:(3): 1386–1397.
6779:
6773:
6767:
6761:
6760:
6750:
6718:
6712:
6711:
6683:
6677:
6676:
6650:
6644:
6643:
6641:
6640:
6626:
6620:
6619:
6617:
6593:
6587:
6586:
6583:10.1122/1.549768
6558:
6552:
6551:
6533:
6527:
6526:
6498:
6492:
6491:
6455:
6449:
6448:
6430:
6421:
6415:
6414:
6402:
6391:
6374:
6368:
6357:
6351:
6346:, Vol. 8 of the
6340:
6331:
6330:
6312:
6306:
6304:
6289:
6283:
6281:
6266:
6260:
6232:
6226:
6225:
6223:
6222:
6210:
6193:
6186:
6154:Optical rotation
6107:
6101:
6089:
6071:
6057:
6055:
6043:
6037:
6031:
6024:
6015:
6006:
5989:
5987:
5975:
5974:
5972:
5971:
5969:
5962:
5959:
5949:
5943:
5931:
5921:
5912:
5906:
5900:
5891:
5882:
5872:
5866:
5857:
5851:
5828:
5819:
5817:
5816:
5811:
5803:
5799:
5798:
5796:
5795:
5786:
5785:
5776:
5771:
5768:
5763:
5762:
5751:
5746:
5737:
5732:
5729:
5724:
5723:
5712:
5707:
5698:
5693:
5690:
5685:
5684:
5673:
5668:
5659:
5652:
5648:
5647:
5645:
5644:
5635:
5634:
5625:
5620:
5617:
5612:
5611:
5600:
5595:
5586:
5581:
5578:
5573:
5572:
5561:
5556:
5547:
5542:
5539:
5534:
5533:
5522:
5517:
5508:
5493:
5489:
5473:
5450:
5437:
5428:
5426:
5425:
5420:
5412:
5408:
5406:
5401:
5388:
5383:
5370:
5365:
5351:
5347:
5346:
5344:
5342:
5337:
5327:
5322:
5311:
5306:
5297:
5292:
5290:
5288:
5283:
5273:
5268:
5257:
5252:
5243:
5238:
5236:
5234:
5229:
5219:
5214:
5203:
5198:
5189:
5179:
5175:
5174:
5171:
5166:
5157:
5155:
5150:
5137:
5132:
5122:
5117:
5114:
5109:
5100:
5098:
5093:
5080:
5075:
5065:
5060:
5057:
5052:
5043:
5041:
5036:
5023:
5018:
5008:
5001:
4999:
4998:
4989:
4988:
4979:
4974:
4972:
4971:
4962:
4961:
4952:
4942:
4938:
4936:
4935:
4930:
4928:
4926:
4925:
4916:
4915:
4906:
4897:
4895:
4894:
4889:
4887:
4885:
4884:
4875:
4874:
4865:
4843:
4834:
4832:
4831:
4826:
4818:
4817:
4810:
4806:
4805:
4803:
4802:
4793:
4791:
4786:
4777:
4776:
4766:
4760:
4755:
4742:
4737:
4718:
4717:
4708:
4707:
4696:
4695:
4686:
4685:
4672:
4671:
4662:
4661:
4650:
4646:
4645:
4643:
4642:
4633:
4631:
4626:
4617:
4616:
4606:
4600:
4595:
4582:
4577:
4558:
4557:
4548:
4547:
4534:
4533:
4524:
4523:
4512:
4511:
4502:
4501:
4490:
4486:
4485:
4483:
4482:
4473:
4471:
4466:
4457:
4456:
4446:
4440:
4435:
4422:
4417:
4384:
4376:
4369:
4360:
4351:
4335:
4326:
4324:
4323:
4318:
4310:
4309:
4300:
4296:
4295:
4293:
4292:
4283:
4281:
4276:
4267:
4266:
4256:
4250:
4245:
4232:
4227:
4207:
4206:
4197:
4196:
4187:
4186:
4174:
4173:
4164:
4163:
4154:
4153:
4136:
4128:
4119:
4117:
4116:
4111:
4103:
4102:
4093:
4092:
4083:
4082:
4070:
4069:
4060:
4056:
4055:
4053:
4052:
4043:
4041:
4036:
4027:
4026:
4016:
4010:
4005:
3992:
3987:
3967:
3966:
3957:
3956:
3947:
3946:
3929:
3921:
3912:
3910:
3909:
3904:
3896:
3895:
3886:
3885:
3876:
3875:
3863:
3862:
3853:
3852:
3843:
3842:
3830:
3829:
3820:
3816:
3815:
3813:
3812:
3803:
3801:
3796:
3787:
3786:
3776:
3770:
3765:
3752:
3747:
3722:
3712:
3706:
3700:
3694:
3685:
3676:
3667:
3661:
3652:
3643:
3627:
3618:
3616:
3615:
3610:
3608:
3607:
3598:
3596:
3595:
3586:
3584:
3579:
3570:
3569:
3559:
3551:
3550:
3541:
3540:
3531:
3530:
3518:
3517:
3508:
3507:
3498:
3497:
3485:
3484:
3474:
3469:
3457:
3456:
3447:
3443:
3441:
3436:
3423:
3418:
3405:
3400:
3375:
3365:
3359:
3358:
3356:
3355:
3341:
3338:
3326:
3320:
3314:
3308:
3302:
3289:
3280:
3278:
3277:
3272:
3270:
3269:
3261:
3256:
3227:
3222:
3193:
3188:
3171:
3170:
3158:
3145:
3141:
3135:
3129:
3123:
3117:
3107:
3101:
3086:
3080:
3074:
3056:
3047:
3045:
3044:
3039:
3031:
3023:
3010:
2993:
2984:
2982:
2981:
2976:
2971:
2966:
2958:
2957:
2948:
2947:
2932:
2924:
2916:
2905:
2897:
2889:
2870:
2860:
2853:
2847:
2828:
2819:
2817:
2816:
2811:
2809:
2801:
2793:
2782:
2774:
2766:
2755:
2727:
2723:
2714:
2702:
2678:
2669:
2667:
2666:
2661:
2653:
2634:
2630:
2617:
2608:
2606:
2605:
2600:
2598:
2593:
2591:
2590:
2589:
2576:
2575:
2566:
2564:
2563:
2551:
2523:
2519:
2515:
2513:
2512:
2504:
2501:
2489:
2482:
2471:
2462:
2455:
2448:
2439:
2433:
2420:
2411:
2409:
2408:
2403:
2401:
2400:
2387:
2379:
2363:
2362:
2357:
2348:
2335:
2331:
2325:
2306:
2293:
2284:
2282:
2281:
2276:
2274:
2269:
2261:
2248:
2240:
2234:
2224:
2218:
2200:
2191:
2178:
2172:
1909:
1900:
1888:
1884:
1880:
1872:
1817:light modulators
1738:optical rotation
1439:
1429:
1419:
1403:
1026:
1018:
1008:
990:
983:
978:
943:pulse broadening
870:optical activity
810:nonlinear optics
770:Sky Pool, London
754:
733:
724:
655:
651:
642:
638:
634:
618:
609:
600:
516:
512:
492:refractive index
467:optical ray axes
456:optical axes or
444:
435:
426:
408:index ellipsoids
369:
344:
335:
325:
309:
307:
306:
301:
295:
294:
293:
280:
279:
278:
245:
236:
227:
209:
196:
134:Rasmus Bartholin
79:refractive index
75:optical property
61:
57:
52:refractive index
49:
21:
18:Uniaxial crystal
7473:
7472:
7468:
7467:
7466:
7464:
7463:
7462:
7438:
7437:
7414:"Birefringence"
7411:
7389:
7363:
7362:A. Fresnel (ed.
7356:
7336:
7319:
7314:
7313:
7304:
7302:
7288:
7286:
7280:
7273:
7268:
7264:
7212:
7211:
7207:
7171:
7170:
7166:
7130:
7129:
7125:
7071:
7070:
7066:
7022:
7021:
7017:
6963:
6962:
6958:
6916:
6915:
6911:
6859:
6858:
6854:
6832:
6831:
6827:
6781:
6780:
6776:
6768:
6764:
6720:
6719:
6715:
6685:
6684:
6680:
6673:
6652:
6651:
6647:
6638:
6636:
6628:
6627:
6623:
6608:(1–3): 79–110.
6595:
6594:
6590:
6560:
6559:
6555:
6548:
6535:
6534:
6530:
6500:
6499:
6495:
6457:
6456:
6452:
6428:
6423:
6422:
6418:
6404:
6403:
6394:
6388:Wayback Machine
6375:
6371:
6363:" is the plane
6358:
6354:
6341:
6334:
6327:
6314:
6313:
6309:
6290:
6286:
6267:
6263:
6233:
6229:
6220:
6218:
6212:
6211:
6207:
6202:
6197:
6196:
6190:index ellipsoid
6187:
6183:
6178:
6173:
6159:Periodic poling
6144:Index ellipsoid
6119:
6103:
6097:
6085:
6067:
6051:
6049:
6039:
6033:
6027:
6022:
6017:
6013:
6008:
6004:
5999:
5983:
5981:
5965:
5963:
5960:
5955:
5954:
5952:
5951:
5945:
5939:
5927:
5917:
5908:
5902:
5899:
5893:
5890:
5884:
5878:
5868:
5865:
5859:
5853:
5847:
5826:
5787:
5777:
5657:
5653:
5636:
5626:
5506:
5502:
5497:
5496:
5488:
5480:
5475:
5472:
5464:
5457:
5452:
5446:
5435:
5356:
5352:
5313:
5259:
5205:
5187:
5183:
5123:
5066:
5009:
5006:
5002:
4990:
4980:
4963:
4953:
4946:
4945:
4917:
4907:
4900:
4899:
4876:
4866:
4859:
4858:
4841:
4812:
4811:
4794:
4768:
4767:
4725:
4721:
4719:
4709:
4699:
4697:
4687:
4677:
4674:
4673:
4663:
4653:
4651:
4634:
4608:
4607:
4565:
4561:
4559:
4549:
4539:
4536:
4535:
4525:
4515:
4513:
4503:
4493:
4491:
4474:
4448:
4447:
4405:
4401:
4394:
4388:
4387:
4371:
4367:
4362:
4358:
4353:
4349:
4344:
4333:
4301:
4284:
4258:
4257:
4215:
4211:
4198:
4188:
4178:
4165:
4155:
4145:
4140:
4139:
4126:
4094:
4084:
4074:
4061:
4044:
4018:
4017:
3975:
3971:
3958:
3948:
3938:
3933:
3932:
3919:
3887:
3877:
3867:
3854:
3844:
3834:
3821:
3804:
3778:
3777:
3735:
3731:
3726:
3725:
3708:
3702:
3696:
3692:
3687:
3683:
3678:
3674:
3669:
3663:
3659:
3654:
3650:
3645:
3641:
3636:
3625:
3599:
3587:
3561:
3560:
3542:
3532:
3522:
3509:
3499:
3489:
3476:
3448:
3388:
3384:
3379:
3378:
3361:
3354:
3348:
3342:
3339:
3336:
3335:
3333:
3328:
3322:
3316:
3310:
3304:
3298:
3287:
3264:
3263:
3246:
3241:
3235:
3234:
3229:
3212:
3206:
3205:
3200:
3195:
3173:
3162:
3149:
3148:
3137:
3131:
3125:
3119:
3113:
3103:
3097:
3082:
3076:
3070:
3069:indicates that
3054:
3014:
3013:
2991:
2949:
2939:
2874:
2873:
2855:
2849:
2843:
2826:
2731:
2730:
2721:
2716:
2710:
2687:
2676:
2638:
2637:
2626:
2615:
2581:
2577:
2567:
2555:
2527:
2526:
2511:
2505:
2502:
2499:
2498:
2496:
2491:
2484:
2477:
2472:. By combining
2467:
2457:
2450:
2447:
2441:
2435:
2429:
2418:
2364:
2352:
2339:
2338:
2327:
2323:
2312:
2311:in the medium:
2302:
2291:
2252:
2251:
2236:
2230:
2220:
2216:
2206:
2196:
2180:
2179:) according to
2174:
2168:
2143:
2101:
2089:photoelasticity
2059:
1935:
1934:
1933:
1932:
1912:
1911:
1910:
1902:
1901:
1886:
1882:
1878:
1870:
1863:
1805:
1803:Optical devices
1784:
1734:
1729:
1702:
1698:
1694:
1690:
1665:
1661:
1657:
1631:
1605:
1601:
1576:
1572:
1568:
1564:
1538:
1534:
1530:
1526:
1496:
1492:
1466:
1462:
1458:
1454:
1450:
1438:
1432:
1428:
1422:
1418:
1412:
1409:Crystal system
1378:
1356:
1307:silicon carbide
1287:
1283:
1258:
1233:
1229:
1204:
1179:
1154:
1149:lithium niobate
1128:
1103:
1078:
1074:
1070:
1066:
1041:
1037:
1021:
1017:
1011:
1007:
1001:
976:
965:
879:
829:liquid crystals
762:
753:
746:
735:
732:
726:
723:
717:
714:
686:
653:
650:
644:
640:
636:
633:
626:
620:
617:
611:
608:
602:
599:
593:
561:
528:Poynting vector
514:
510:
496:Wollaston prism
479:
443:
437:
434:
428:
425:
419:
392:
368:
362:
347:Poynting vector
343:
337:
334:
328:
324:
318:
284:
269:
255:
254:
248:index ellipsoid
244:
238:
235:
229:
226:
220:
208:
202:
195:
189:
174:
154:
105:with non-cubic
59:
55:
47:
28:
23:
22:
15:
12:
11:
5:
7471:
7469:
7461:
7460:
7455:
7450:
7440:
7439:
7436:
7435:
7430:
7409:
7404:
7399:
7395:
7388:
7387:External links
7385:
7384:
7383:
7360:
7329:
7318:
7315:
7312:
7311:
7271:
7262:
7219:Optics Express
7205:
7184:(3): 913–920.
7178:Fertil. Steril
7164:
7143:(3): 807–813.
7137:Fertil. Steril
7123:
7064:
7015:
6956:
6909:
6852:
6825:
6774:
6762:
6727:Arch Emerg Med
6713:
6678:
6671:
6645:
6621:
6588:
6553:
6546:
6528:
6509:(2): 403–407.
6493:
6460:Histochemistry
6450:
6416:
6392:
6369:
6352:
6332:
6325:
6307:
6284:
6261:
6259:
6258:
6244:
6227:
6204:
6203:
6201:
6198:
6195:
6194:
6180:
6179:
6177:
6174:
6172:
6171:
6166:
6161:
6156:
6151:
6146:
6141:
6136:
6131:
6129:Crystal optics
6126:
6120:
6118:
6115:
6020:
6011:
6002:
5897:
5888:
5863:
5832:
5831:
5822:
5820:
5809:
5806:
5802:
5794:
5790:
5784:
5780:
5774:
5767:
5761:
5756:
5750:
5745:
5741:
5735:
5728:
5722:
5717:
5711:
5706:
5702:
5696:
5689:
5683:
5678:
5672:
5667:
5663:
5656:
5651:
5643:
5639:
5633:
5629:
5623:
5616:
5610:
5605:
5599:
5594:
5590:
5584:
5577:
5571:
5566:
5560:
5555:
5551:
5545:
5538:
5532:
5527:
5521:
5516:
5512:
5505:
5486:
5478:
5470:
5462:
5455:
5441:
5440:
5431:
5429:
5418:
5415:
5411:
5405:
5400:
5396:
5392:
5387:
5382:
5378:
5374:
5369:
5364:
5360:
5355:
5350:
5341:
5336:
5332:
5326:
5321:
5317:
5310:
5305:
5301:
5295:
5287:
5282:
5278:
5272:
5267:
5263:
5256:
5251:
5247:
5241:
5233:
5228:
5224:
5218:
5213:
5209:
5202:
5197:
5193:
5186:
5182:
5178:
5170:
5165:
5161:
5154:
5149:
5145:
5141:
5136:
5131:
5127:
5120:
5113:
5108:
5104:
5097:
5092:
5088:
5084:
5079:
5074:
5070:
5063:
5056:
5051:
5047:
5040:
5035:
5031:
5027:
5022:
5017:
5013:
5005:
4997:
4993:
4987:
4983:
4977:
4970:
4966:
4960:
4956:
4924:
4920:
4914:
4910:
4883:
4879:
4873:
4869:
4847:
4846:
4837:
4835:
4824:
4821:
4816:
4809:
4801:
4797:
4790:
4785:
4781:
4775:
4771:
4764:
4759:
4754:
4750:
4746:
4741:
4736:
4732:
4728:
4724:
4720:
4716:
4712:
4706:
4702:
4698:
4694:
4690:
4684:
4680:
4676:
4675:
4670:
4666:
4660:
4656:
4652:
4649:
4641:
4637:
4630:
4625:
4621:
4615:
4611:
4604:
4599:
4594:
4590:
4586:
4581:
4576:
4572:
4568:
4564:
4560:
4556:
4552:
4546:
4542:
4538:
4537:
4532:
4528:
4522:
4518:
4514:
4510:
4506:
4500:
4496:
4492:
4489:
4481:
4477:
4470:
4465:
4461:
4455:
4451:
4444:
4439:
4434:
4430:
4426:
4421:
4416:
4412:
4408:
4404:
4400:
4399:
4397:
4365:
4356:
4347:
4339:
4338:
4329:
4327:
4316:
4313:
4308:
4304:
4299:
4291:
4287:
4280:
4275:
4271:
4265:
4261:
4254:
4249:
4244:
4240:
4236:
4231:
4226:
4222:
4218:
4214:
4210:
4205:
4201:
4195:
4191:
4185:
4181:
4177:
4172:
4168:
4162:
4158:
4152:
4148:
4132:
4131:
4122:
4120:
4109:
4106:
4101:
4097:
4091:
4087:
4081:
4077:
4073:
4068:
4064:
4059:
4051:
4047:
4040:
4035:
4031:
4025:
4021:
4014:
4009:
4004:
4000:
3996:
3991:
3986:
3982:
3978:
3974:
3970:
3965:
3961:
3955:
3951:
3945:
3941:
3925:
3924:
3915:
3913:
3902:
3899:
3894:
3890:
3884:
3880:
3874:
3870:
3866:
3861:
3857:
3851:
3847:
3841:
3837:
3833:
3828:
3824:
3819:
3811:
3807:
3800:
3795:
3791:
3785:
3781:
3774:
3769:
3764:
3760:
3756:
3751:
3746:
3742:
3738:
3734:
3713:components of
3690:
3681:
3672:
3657:
3648:
3639:
3631:
3630:
3621:
3619:
3606:
3602:
3594:
3590:
3583:
3578:
3574:
3568:
3564:
3557:
3554:
3549:
3545:
3539:
3535:
3529:
3525:
3521:
3516:
3512:
3506:
3502:
3496:
3492:
3488:
3483:
3479:
3473:
3468:
3464:
3460:
3455:
3451:
3446:
3440:
3435:
3431:
3427:
3422:
3417:
3413:
3409:
3404:
3399:
3395:
3391:
3387:
3352:
3346:
3293:
3292:
3283:
3281:
3268:
3260:
3255:
3251:
3247:
3245:
3242:
3240:
3237:
3236:
3233:
3230:
3226:
3221:
3217:
3213:
3211:
3208:
3207:
3204:
3201:
3199:
3196:
3192:
3187:
3183:
3179:
3178:
3176:
3169:
3165:
3161:
3157:
3060:
3059:
3050:
3048:
3037:
3034:
3030:
3026:
3022:
2997:
2996:
2987:
2985:
2974:
2970:
2965:
2961:
2956:
2952:
2946:
2942:
2938:
2935:
2931:
2927:
2923:
2919:
2915:
2911:
2908:
2904:
2900:
2896:
2892:
2888:
2884:
2881:
2867:then becomes:
2832:
2831:
2822:
2820:
2808:
2804:
2800:
2796:
2792:
2788:
2785:
2781:
2777:
2773:
2769:
2765:
2761:
2758:
2754:
2750:
2747:
2744:
2741:
2738:
2719:
2682:
2681:
2672:
2670:
2659:
2656:
2652:
2648:
2645:
2621:
2620:
2611:
2609:
2597:
2588:
2584:
2580:
2574:
2570:
2562:
2558:
2554:
2550:
2546:
2543:
2540:
2537:
2534:
2509:
2445:
2424:
2423:
2414:
2412:
2399:
2396:
2393:
2390:
2386:
2382:
2378:
2374:
2371:
2367:
2361:
2356:
2351:
2347:
2321:
2297:
2296:
2287:
2285:
2273:
2268:
2264:
2260:
2214:
2205:(and equal to
2142:
2139:
2100:
2097:
2093:optical vortex
2058:
2055:
1971:immunoglobulin
1914:
1913:
1904:
1903:
1895:
1894:
1893:
1892:
1891:
1862:
1859:
1855:phase matching
1804:
1801:
1783:
1780:
1762:lipid bilayers
1740:and linear or
1733:
1730:
1726:
1725:
1719:
1718:
1715:
1712:
1709:
1704:
1700:
1696:
1692:
1688:
1681:
1680:
1677:
1674:
1671:
1666:
1663:
1659:
1655:
1648:
1647:
1644:
1641:
1638:
1633:
1629:
1621:
1620:
1617:
1614:
1611:
1606:
1603:
1599:
1592:
1591:
1588:
1585:
1582:
1577:
1574:
1570:
1566:
1562:
1554:
1553:
1550:
1547:
1544:
1539:
1536:
1532:
1528:
1524:
1513:
1512:
1509:
1506:
1503:
1498:
1494:
1490:
1483:
1482:
1479:
1476:
1473:
1468:
1464:
1460:
1456:
1452:
1448:
1441:
1440:
1436:
1430:
1426:
1420:
1416:
1410:
1407:
1397:
1394:
1393:
1390:
1387:
1384:
1379:
1376:
1372:
1371:
1368:
1365:
1362:
1357:
1354:
1347:
1346:
1343:
1340:
1337:
1332:
1325:
1324:
1321:
1318:
1315:
1310:
1303:
1302:
1299:
1296:
1293:
1288:
1285:
1281:
1274:
1273:
1270:
1267:
1264:
1259:
1256:
1249:
1248:
1245:
1242:
1239:
1234:
1231:
1227:
1220:
1219:
1216:
1213:
1210:
1205:
1202:
1195:
1194:
1191:
1188:
1185:
1180:
1177:
1170:
1169:
1166:
1163:
1160:
1155:
1152:
1145:
1144:
1141:
1138:
1135:
1130:
1126:
1119:
1118:
1115:
1112:
1109:
1104:
1101:
1094:
1093:
1090:
1087:
1084:
1079:
1076:
1072:
1068:
1064:
1057:
1056:
1053:
1050:
1047:
1042:
1039:
1035:
1028:
1027:
1019:
1015:
1009:
1005:
999:
997:Crystal system
994:
981:
974:
878:
875:
874:
873:
868:), similar to
856:Faraday effect
852:
849:stereo isomers
832:
813:
798:
790:
761:
758:
751:
744:
730:
721:
713:
710:
685:
682:
660:in this case.
648:
631:
624:
615:
606:
597:
560:
557:
524:electric field
478:
475:
441:
432:
423:
391:
388:
366:
341:
332:
322:
311:
310:
299:
292:
287:
283:
277:
272:
268:
265:
262:
242:
233:
224:
206:
193:
173:
170:
153:
150:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7470:
7459:
7456:
7454:
7451:
7449:
7446:
7445:
7443:
7434:
7431:
7427:
7423:
7419:
7418:Sixty Symbols
7415:
7410:
7408:
7405:
7403:
7400:
7398:
7396:
7394:
7391:
7390:
7386:
7381:
7377:
7373:
7369:
7361:
7355:, vol.
7354:
7350:
7346:
7342:
7335:, vol.
7334:
7330:
7327:
7326:
7321:
7320:
7316:
7300:
7296:
7284:
7278:
7276:
7272:
7266:
7263:
7258:
7254:
7250:
7246:
7241:
7236:
7232:
7228:
7224:
7220:
7216:
7209:
7206:
7201:
7197:
7192:
7187:
7183:
7179:
7175:
7168:
7165:
7160:
7156:
7151:
7146:
7142:
7138:
7134:
7127:
7124:
7119:
7115:
7110:
7105:
7101:
7097:
7092:
7087:
7083:
7079:
7075:
7068:
7065:
7060:
7056:
7051:
7046:
7042:
7038:
7034:
7030:
7026:
7019:
7016:
7011:
7007:
7002:
6997:
6993:
6989:
6985:
6981:
6977:
6973:
6972:
6967:
6960:
6957:
6952:
6948:
6944:
6940:
6936:
6932:
6928:
6924:
6920:
6913:
6910:
6905:
6901:
6896:
6891:
6887:
6883:
6879:
6875:
6871:
6867:
6863:
6856:
6853:
6848:
6844:
6840:
6836:
6829:
6826:
6821:
6817:
6812:
6807:
6803:
6799:
6795:
6791:
6790:
6785:
6778:
6775:
6771:
6766:
6763:
6758:
6754:
6749:
6744:
6740:
6736:
6732:
6728:
6724:
6717:
6714:
6709:
6705:
6701:
6697:
6693:
6689:
6682:
6679:
6674:
6668:
6664:
6660:
6656:
6649:
6646:
6635:
6631:
6625:
6622:
6616:
6611:
6607:
6603:
6599:
6592:
6589:
6584:
6580:
6576:
6572:
6568:
6564:
6557:
6554:
6549:
6543:
6539:
6532:
6529:
6524:
6520:
6516:
6512:
6508:
6504:
6497:
6494:
6489:
6485:
6481:
6477:
6473:
6469:
6465:
6461:
6454:
6451:
6446:
6442:
6438:
6434:
6427:
6420:
6417:
6412:
6408:
6401:
6399:
6397:
6393:
6389:
6385:
6382:
6378:
6373:
6370:
6366:
6365:perpendicular
6362:
6356:
6353:
6349:
6345:
6339:
6337:
6333:
6328:
6326:0-86542-323-7
6322:
6318:
6311:
6308:
6303:
6298:
6294:
6288:
6285:
6280:
6275:
6271:
6265:
6262:
6256:
6252:
6249:
6245:
6242:
6241:
6236:
6235:
6231:
6228:
6216:
6209:
6206:
6199:
6191:
6185:
6182:
6175:
6170:
6167:
6165:
6162:
6160:
6157:
6155:
6152:
6150:
6147:
6145:
6142:
6140:
6137:
6135:
6132:
6130:
6127:
6125:
6122:
6121:
6116:
6114:
6111:
6106:
6100:
6095:
6094:
6088:
6083:
6078:
6073:
6070:
6065:
6061:
6054:
6046:
6042:
6036:
6030:
6023:
6014:
6005:
5997:
5991:
5986:
5979:
5968:
5958:
5948:
5942:
5937:
5936:
5930:
5925:
5922:vectors (the
5920:
5914:
5911:
5905:
5896:
5887:
5881:
5876:
5871:
5862:
5856:
5850:
5845:
5841:
5840:
5830:
5823:
5821:
5807:
5804:
5800:
5792:
5788:
5782:
5778:
5772:
5765:
5754:
5748:
5743:
5739:
5733:
5726:
5715:
5709:
5704:
5700:
5694:
5687:
5676:
5670:
5665:
5661:
5654:
5649:
5641:
5637:
5631:
5627:
5621:
5614:
5603:
5597:
5592:
5588:
5582:
5575:
5564:
5558:
5553:
5549:
5543:
5536:
5525:
5519:
5514:
5510:
5503:
5495:
5494:
5491:
5485:
5481:
5469:
5465:
5458:
5449:
5439:
5432:
5430:
5416:
5413:
5409:
5403:
5398:
5394:
5390:
5385:
5380:
5376:
5372:
5367:
5362:
5358:
5353:
5348:
5339:
5334:
5330:
5324:
5319:
5315:
5308:
5303:
5299:
5293:
5285:
5280:
5276:
5270:
5265:
5261:
5254:
5249:
5245:
5239:
5231:
5226:
5222:
5216:
5211:
5207:
5200:
5195:
5191:
5184:
5180:
5176:
5168:
5163:
5159:
5152:
5147:
5143:
5139:
5134:
5129:
5125:
5118:
5111:
5106:
5102:
5095:
5090:
5086:
5082:
5077:
5072:
5068:
5061:
5054:
5049:
5045:
5038:
5033:
5029:
5025:
5020:
5015:
5011:
5003:
4995:
4991:
4985:
4981:
4975:
4968:
4964:
4958:
4954:
4944:
4943:
4940:
4922:
4918:
4912:
4908:
4881:
4877:
4871:
4867:
4856:
4855:
4845:
4838:
4836:
4822:
4819:
4814:
4807:
4799:
4795:
4788:
4783:
4779:
4773:
4769:
4762:
4757:
4752:
4748:
4744:
4739:
4734:
4730:
4726:
4722:
4714:
4710:
4704:
4700:
4692:
4688:
4682:
4678:
4668:
4664:
4658:
4654:
4647:
4639:
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4628:
4623:
4619:
4613:
4609:
4602:
4597:
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4588:
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4574:
4570:
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4550:
4544:
4540:
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4526:
4520:
4516:
4508:
4504:
4498:
4494:
4487:
4479:
4475:
4468:
4463:
4459:
4453:
4449:
4442:
4437:
4432:
4428:
4424:
4419:
4414:
4410:
4406:
4402:
4395:
4386:
4385:
4382:
4380:
4374:
4368:
4359:
4350:
4337:
4330:
4328:
4314:
4311:
4306:
4302:
4297:
4289:
4285:
4278:
4273:
4269:
4263:
4259:
4252:
4247:
4242:
4238:
4234:
4229:
4224:
4220:
4216:
4212:
4208:
4203:
4199:
4193:
4189:
4183:
4179:
4175:
4170:
4166:
4160:
4156:
4150:
4146:
4138:
4137:
4130:
4123:
4121:
4107:
4104:
4099:
4095:
4089:
4085:
4079:
4075:
4071:
4066:
4062:
4057:
4049:
4045:
4038:
4033:
4029:
4023:
4019:
4012:
4007:
4002:
3998:
3994:
3989:
3984:
3980:
3976:
3972:
3968:
3963:
3959:
3953:
3949:
3943:
3939:
3931:
3930:
3923:
3916:
3914:
3900:
3897:
3892:
3888:
3882:
3878:
3872:
3868:
3864:
3859:
3855:
3849:
3845:
3839:
3835:
3831:
3826:
3822:
3817:
3809:
3805:
3798:
3793:
3789:
3783:
3779:
3772:
3767:
3762:
3758:
3754:
3749:
3744:
3740:
3736:
3732:
3724:
3723:
3720:
3718:
3717:
3711:
3705:
3699:
3693:
3684:
3675:
3666:
3660:
3651:
3642:
3629:
3622:
3620:
3604:
3600:
3592:
3588:
3581:
3576:
3572:
3566:
3562:
3555:
3552:
3547:
3543:
3537:
3533:
3527:
3523:
3519:
3514:
3510:
3504:
3500:
3494:
3490:
3486:
3481:
3477:
3471:
3466:
3462:
3458:
3453:
3449:
3444:
3438:
3433:
3429:
3425:
3420:
3415:
3411:
3407:
3402:
3397:
3393:
3389:
3385:
3377:
3376:
3373:
3371:
3370:
3364:
3351:
3345:
3331:
3325:
3319:
3313:
3307:
3301:
3291:
3284:
3282:
3266:
3258:
3253:
3249:
3243:
3238:
3231:
3224:
3219:
3215:
3209:
3202:
3197:
3190:
3185:
3181:
3174:
3167:
3163:
3159:
3155:
3147:
3146:
3143:
3140:
3134:
3128:
3122:
3116:
3111:
3106:
3100:
3094:
3092:
3091:
3085:
3079:
3073:
3068:
3067:
3058:
3051:
3049:
3035:
3032:
3024:
3012:
3011:
3008:
3006:
3005:
2995:
2988:
2986:
2954:
2950:
2944:
2940:
2936:
2933:
2917:
2906:
2890:
2882:
2872:
2871:
2868:
2866:
2865:
2859:
2852:
2846:
2841:
2840:
2830:
2823:
2821:
2794:
2783:
2767:
2756:
2748:
2742:
2736:
2729:
2728:
2725:
2722:
2713:
2708:
2707:
2701:
2700:
2696:
2692:
2680:
2673:
2671:
2657:
2654:
2646:
2636:
2635:
2632:
2629:
2619:
2612:
2610:
2586:
2582:
2572:
2560:
2556:
2552:
2544:
2538:
2532:
2525:
2524:
2521:
2518:
2508:
2494:
2488:
2481:
2475:
2470:
2466:
2460:
2453:
2444:
2440:is time, and
2438:
2432:
2422:
2415:
2413:
2394:
2391:
2388:
2380:
2369:
2365:
2359:
2349:
2337:
2336:
2333:
2330:
2324:
2320:
2315:
2310:
2305:
2295:
2288:
2286:
2262:
2250:
2249:
2246:
2244:
2239:
2233:
2228:
2223:
2217:
2213:
2209:
2204:
2199:
2195:
2190:
2187:
2183:
2177:
2171:
2166:
2158:
2157:
2152:
2147:
2140:
2138:
2135:
2133:
2129:
2124:
2122:
2114:
2110:
2107:Birefringent
2105:
2098:
2096:
2094:
2090:
2086:
2082:
2078:
2072:
2068:
2063:
2056:
2054:
2052:
2046:
2044:
2040:
2036:
2035:
2030:
2026:
2021:
2019:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1990:ophthalmology
1986:
1984:
1980:
1976:
1972:
1968:
1964:
1959:
1957:
1951:
1948:
1944:
1940:
1930:
1927:(pseudogout,
1926:
1922:
1918:
1908:
1899:
1890:
1876:
1868:
1860:
1858:
1856:
1852:
1848:
1843:
1841:
1837:
1832:
1830:
1826:
1822:
1818:
1814:
1810:
1802:
1797:
1793:
1788:
1781:
1779:
1778:can be used.
1777:
1774:
1769:
1767:
1763:
1758:
1756:
1751:
1747:
1743:
1739:
1731:
1724:
1716:
1713:
1710:
1708:
1705:
1686:
1683:
1682:
1678:
1675:
1672:
1670:
1667:
1653:
1650:
1649:
1645:
1642:
1639:
1637:
1634:
1632:
1626:
1623:
1622:
1618:
1615:
1612:
1610:
1607:
1597:
1594:
1593:
1589:
1586:
1583:
1581:
1578:
1560:
1556:
1555:
1551:
1548:
1545:
1543:
1540:
1522:
1518:
1515:
1514:
1510:
1507:
1504:
1502:
1499:
1488:
1485:
1484:
1480:
1477:
1474:
1472:
1469:
1446:
1443:
1442:
1435:
1431:
1425:
1421:
1415:
1411:
1408:
1405:
1404:
1398:
1391:
1388:
1385:
1383:
1380:
1374:
1373:
1369:
1366:
1363:
1361:
1358:
1352:
1349:
1348:
1344:
1341:
1338:
1336:
1333:
1330:
1327:
1326:
1322:
1319:
1316:
1314:
1311:
1308:
1305:
1304:
1300:
1297:
1294:
1292:
1289:
1279:
1276:
1275:
1271:
1268:
1265:
1263:
1260:
1254:
1251:
1250:
1246:
1243:
1240:
1238:
1235:
1225:
1222:
1221:
1217:
1214:
1211:
1209:
1206:
1200:
1197:
1196:
1192:
1189:
1186:
1184:
1181:
1175:
1172:
1171:
1167:
1164:
1161:
1159:
1156:
1150:
1147:
1146:
1142:
1139:
1136:
1134:
1131:
1124:
1121:
1120:
1116:
1113:
1110:
1108:
1105:
1099:
1096:
1095:
1091:
1088:
1085:
1083:
1080:
1062:
1059:
1058:
1054:
1051:
1048:
1046:
1043:
1033:
1032:barium borate
1030:
1029:
1025:
1020:
1014:
1010:
1004:
1000:
998:
995:
992:
991:
985:
984:
980:
977:
973:
968:
963:
958:
956:
952:
951:intentionally
948:
944:
940:
939:optical fiber
935:
931:
929:
925:
923:
922:polycarbonate
919:
915:
911:
907:
902:
900:
896:
892:
883:
876:
871:
867:
863:
862:
857:
853:
850:
846:
842:
838:
837:
833:
830:
826:
822:
818:
814:
811:
807:
803:
799:
796:
791:
788:
787:
783:
782:
781:
779:
771:
766:
759:
757:
750:
743:
739:
729:
720:
711:
709:
707:
702:
700:
695:
691:
683:
681:
679:
674:
670:
666:
661:
659:
647:
630:
623:
614:
605:
596:
590:
585:
583:
577:
575:
565:
558:
556:
554:
550:
546:
542:
537:
533:
529:
525:
521:
508:
503:
501:
497:
493:
489:
485:
476:
474:
472:
468:
464:
459:
455:
450:
449:a spheroid).
448:
440:
431:
422:
417:
413:
409:
404:
401:
397:
389:
387:
385:
381:
377:
373:
365:
358:
356:
352:
348:
340:
331:
321:
316:
297:
285:
281:
270:
266:
263:
253:
252:
251:
249:
241:
232:
223:
217:
213:
205:
200:
192:
187:
186:perpendicular
183:
179:
171:
169:
167:
158:
151:
149:
147:
143:
139:
135:
131:
127:
123:
118:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
72:
71:Birefringence
65:
53:
44:
37:
32:
19:
7417:
7367:
7352:
7332:
7323:
7317:Bibliography
7282:
7265:
7222:
7218:
7208:
7181:
7177:
7167:
7140:
7136:
7126:
7081:
7078:Biomolecules
7077:
7067:
7032:
7028:
7018:
6975:
6969:
6959:
6926:
6922:
6912:
6869:
6865:
6855:
6841:(4): e5–e6.
6838:
6834:
6828:
6793:
6787:
6777:
6765:
6733:(2): 89–95.
6730:
6726:
6716:
6691:
6687:
6681:
6654:
6648:
6637:. Retrieved
6633:
6624:
6605:
6601:
6591:
6569:(1): 61–70.
6566:
6562:
6556:
6537:
6531:
6506:
6502:
6496:
6466:(1): 41–49.
6463:
6459:
6453:
6436:
6432:
6419:
6410:
6407:"Refraction"
6376:
6372:
6364:
6355:
6347:
6343:
6316:
6310:
6292:
6287:
6269:
6264:
6254:
6250:
6239:
6230:
6219:. Retrieved
6208:
6184:
6139:Iceland spar
6109:
6104:
6098:
6091:
6086:
6081:
6076:
6074:
6068:
6059:
6052:
6047:
6040:
6034:
6028:
6018:
6009:
6000:
5995:
5992:
5984:
5966:
5956:
5946:
5940:
5933:
5928:
5923:
5918:
5915:
5909:
5903:
5894:
5885:
5879:
5869:
5860:
5854:
5848:
5837:
5835:
5824:
5483:
5476:
5467:
5460:
5453:
5447:
5444:
5433:
4852:
4850:
4839:
4372:
4363:
4354:
4345:
4342:
4331:
4124:
3917:
3714:
3709:
3703:
3697:
3688:
3679:
3670:
3664:
3655:
3646:
3637:
3634:
3623:
3367:
3362:
3349:
3343:
3329:
3323:
3317:
3311:
3305:
3299:
3296:
3285:
3138:
3132:
3126:
3120:
3114:
3104:
3102:for a given
3098:
3095:
3088:
3083:
3077:
3071:
3064:
3063:
3052:
3002:
3000:
2989:
2862:
2857:
2850:
2844:
2837:
2835:
2824:
2717:
2711:
2704:
2698:
2694:
2690:
2685:
2674:
2627:
2624:
2613:
2516:
2506:
2492:
2486:
2479:
2468:
2465:wave vectors
2458:
2451:
2442:
2436:
2430:
2427:
2416:
2328:
2318:
2313:
2303:
2300:
2289:
2237:
2231:
2221:
2211:
2207:
2197:
2194:permittivity
2188:
2185:
2181:
2175:
2169:
2162:
2154:
2150:
2136:
2125:
2118:
2112:
2075:
2047:
2034:zona imaging
2032:
2031:. Likewise,
2022:
1998:Henle fibers
1997:
1992:, binocular
1987:
1983:intercalates
1975:conformation
1960:
1952:
1936:
1928:
1916:
1874:
1864:
1844:
1833:
1806:
1782:Applications
1772:
1770:
1759:
1755:ellipsometry
1749:
1735:
1722:
1669:Orthorhombic
1636:Orthorhombic
1609:Orthorhombic
1433:
1423:
1413:
1353:, high ZrSiO
1023:
1012:
1002:
971:
966:
959:
950:
936:
932:
926:
903:
888:
859:
834:
795:metamaterial
784:
775:
748:
741:
737:
727:
718:
715:
703:
698:
693:
689:
687:
677:
672:
668:
664:
662:
657:
645:
628:
621:
612:
603:
594:
588:
586:
581:
578:
571:
548:
540:
504:
480:
470:
466:
462:
457:
453:
451:
446:
438:
429:
420:
411:
405:
399:
395:
393:
363:
359:
338:
329:
319:
312:
239:
230:
221:
215:
211:
203:
199:ordinary ray
198:
190:
185:
177:
175:
163:
138:Iceland spar
125:
119:
99:birefractive
98:
95:birefringent
94:
83:polarization
70:
69:
7422:Brady Haran
7084:(8): 1230.
6538:Ice physics
6164:Pleochroism
6080:optic axis
5978:wavenumbers
5844:ellipsoidal
4379:determinant
2520:to obtain:
2128:shear waves
2025:sperm heads
2014:optic nerve
1967:Alzheimer's
1825:Lyot filter
1746:polarimetry
1732:Measurement
918:polystyrene
825:surfactants
817:amphiphilic
806:Kerr effect
559:Terminology
553:phase shift
536:wave vector
530:) for this
488:Snell's law
372:wave vector
351:wave vector
152:Explanation
91:anisotropic
7442:Categories
7035:: 107838.
6639:2024-03-15
6221:2021-07-21
6200:References
2693:) = ∇(∇ ⋅
2631:vanishes:
2245:equation:
2201:is just a
2132:seismology
2071:polarizers
2051:dermoscopy
2002:strabismus
1919:image) in
1867:polarizing
1849:and other
1829:Waveplates
1625:perovskite
1580:Monoclinic
1542:Monoclinic
1501:Monoclinic
1487:epsom salt
1471:Monoclinic
1382:Tetragonal
1360:Tetragonal
1329:tourmaline
1262:Tetragonal
1183:Tetragonal
914:Polarizers
910:cellophane
658:optic axis
545:waveplates
182:optic axis
128:whereby a
122:phenomenon
64:refraction
7458:Asymmetry
7249:1094-4087
7100:2218-273X
7059:0143-8166
6992:2156-7085
6943:1552-5783
6886:2156-7085
6708:0007-5027
6377:Brad Amos
6149:John Kerr
6134:Dichroism
6064:waveplate
6060:different
5779:ω
5773:−
5628:ω
5622:−
4982:ω
4976:−
4955:ω
4909:ω
4868:ω
4770:ω
4745:−
4727:−
4610:ω
4585:−
4567:−
4450:ω
4425:−
4407:−
4381:is zero:
4260:ω
4235:−
4217:−
4020:ω
3995:−
3977:−
3780:ω
3755:−
3737:−
3563:ω
3556:−
3426:−
3408:−
3390:−
3164:ε
3156:ε
3112:with the
3025:⋅
3007:we find:
2964:ε
2951:ω
2941:μ
2937:−
2918:⋅
2891:⋅
2883:−
2795:⋅
2784:−
2768:⋅
2749:×
2746:∇
2743:×
2740:∇
2737:−
2724:to find:
2689:∇ × (∇ ×
2647:⋅
2644:∇
2579:∂
2569:∂
2557:μ
2545:×
2542:∇
2539:×
2536:∇
2533:−
2392:ω
2389:−
2381:⋅
2267:ε
2077:Isotropic
2043:silicosis
1979:Congo red
1796:backlight
1707:Triclinic
1559:muscovite
1406:Material
1313:Hexagonal
1133:Hexagonal
1082:Hexagonal
993:Material
706:waveplate
697:ray with
471:biradials
458:binormals
376:waveplate
282:−
261:Δ
7424:for the
7257:20588931
7200:19439291
7159:19064263
7118:37627295
7109:10452597
7010:34457418
6951:15277483
6904:24298395
6820:35414965
6488:25214054
6384:Archived
6117:See also
5875:spheroid
3372:becomes
2113:analyzer
2018:glaucoma
1875:analyzer
1861:Medicine
1771:For the
1523:K(Mg,Fe)
1335:Trigonal
1291:Trigonal
1278:sapphire
1237:Trigonal
1208:Trigonal
1158:Trigonal
1143:+0.0014
1107:Trigonal
1055:−0.1242
1045:Trigonal
906:plastics
891:crystals
694:fast ray
690:slow ray
589:uniaxial
416:spheroid
178:uniaxial
111:plastics
103:Crystals
7364:
7305:
7289:
7227:Bibcode
7037:Bibcode
7001:8367251
6895:3829392
6811:8973164
6757:6536274
6748:1285204
6571:Bibcode
6511:Bibcode
6480:3733471
6302:5442206
6299::
6279:4058004
6276::
5973:
5953:
3357:
3334:
3315:. With
2514:
2497:
2225:is the
2121:elastic
2039:oocytes
1996:of the
1963:amyloid
1685:ulexite
1598:(Mg,Fe)
1596:olivine
1573:)(F,OH)
1535:)(F,OH)
1521:biotite
1392:+0.047
1370:+0.055
1345:−0.031
1323:+0.046
1301:−0.008
1272:+0.287
1247:−0.008
1218:+0.009
1193:+0.006
1168:−0.085
1117:−0.172
1098:calcite
1092:−0.045
854:By the
823:, some
802:Pockels
800:By the
678:biaxial
665:biaxial
500:calcite
463:biaxial
355:calcite
142:calcite
73:is the
36:calcite
7255:
7247:
7198:
7157:
7116:
7106:
7098:
7057:
7008:
6998:
6990:
6949:
6941:
6902:
6892:
6884:
6818:
6808:
6755:
6745:
6706:
6669:
6544:
6486:
6478:
6323:
6297:Zenodo
6274:Zenodo
6056:|
6050:|
5988:|
5982:|
5970:|
5964:|
3716:eq. 4a
3635:where
3369:eq. 4a
3360:, the
3327:using
3090:Eq. 4b
3066:Eq. 4b
3004:eq. 3b
2864:eq. 3a
2839:eq. 3a
2706:eq. 3a
2428:where
2301:where
2243:tensor
2219:where
2203:scalar
2141:Theory
2109:rutile
1723:
1717:1.520
1679:1.627
1662:(F,OH)
1646:2.380
1619:1.680
1590:1.601
1557:mica,
1552:1.640
1511:1.461
1481:1.472
1351:zircon
1253:rutile
1199:quartz
1140:1.3104
1137:1.3090
1052:1.5534
1049:1.6776
928:Cotton
841:chiral
821:lipids
669:except
113:under
7307:20–21
7301:, at
7291:13–50
6484:S2CID
6429:(PDF)
6234:See:
6176:Notes
6093:eq. 6
5935:eq. 6
5839:eq. 8
4854:eq. 6
2697:) − ∇
2156:eq. 7
1939:gouty
1929:right
1714:1.510
1711:1.490
1687:NaCaB
1676:1.620
1673:1.618
1652:topaz
1643:2.340
1640:2.300
1628:CaTiO
1616:1.660
1613:1.640
1587:1.596
1584:1.563
1565:(AlSi
1549:1.640
1546:1.595
1527:(AlSi
1508:1.455
1505:1.433
1478:1.469
1475:1.447
1459:)(OH)
1445:borax
1389:1.967
1386:1.920
1367:2.015
1364:1.960
1342:1.638
1339:1.669
1320:2.693
1317:2.647
1298:1.760
1295:1.768
1269:2.903
1266:2.616
1244:1.762
1241:1.770
1215:1.553
1212:1.544
1190:1.385
1187:1.380
1165:2.187
1162:2.272
1151:LiNbO
1114:1.486
1111:1.658
1089:1.557
1086:1.602
1061:beryl
549:along
541:shift
396:three
166:below
87:light
7253:PMID
7245:ISSN
7196:PMID
7155:PMID
7114:PMID
7096:ISSN
7055:ISSN
7006:PMID
6988:ISSN
6947:PMID
6939:ISSN
6900:PMID
6882:ISSN
6816:PMID
6753:PMID
6704:ISSN
6667:ISBN
6634:Home
6542:ISBN
6476:PMID
6321:ISBN
6016:and
5892:and
5474:and
3707:and
3309:and
3124:and
2485:∇ ×
2483:and
2478:∇ ×
2476:for
2235:and
1981:dye
1921:gout
1917:left
1695:(OH)
1517:mica
1489:MgSO
1224:ruby
1100:CaCO
1071:(SiO
920:and
639:and
610:and
436:and
400:both
336:and
237:and
117:.
7337:VII
7303:pp.
7295:doi
7287:pp.
7235:doi
7186:doi
7145:doi
7104:PMC
7086:doi
7045:doi
7033:172
6996:PMC
6980:doi
6931:doi
6890:PMC
6874:doi
6843:doi
6806:PMC
6798:doi
6743:PMC
6735:doi
6696:doi
6659:doi
6610:doi
6579:doi
6519:doi
6507:124
6468:doi
6441:doi
6110:not
4375:= 0
2461:= 0
2454:= 0
2134:.
1988:In
1750:not
1699:·5H
1658:SiO
1602:SiO
1561:KAl
1493:·7H
1463:·8H
1309:SiC
1255:TiO
1201:SiO
1176:MgF
1123:ice
1034:BaB
945:in
827:or
804:or
673:not
469:or
454:two
412:not
130:ray
124:of
97:or
7444::
7420:.
7416:.
7378:,
7374:,
7293:,
7274:^
7251:.
7243:.
7233:.
7223:18
7221:.
7217:.
7194:.
7182:94
7180:.
7176:.
7153:.
7141:93
7139:.
7135:.
7112:.
7102:.
7094:.
7082:13
7080:.
7076:.
7053:.
7043:.
7031:.
7027:.
7004:.
6994:.
6986:.
6976:12
6974:.
6968:.
6945:.
6937:.
6927:45
6925:.
6921:.
6898:.
6888:.
6880:.
6868:.
6864:.
6839:18
6837:.
6814:.
6804:.
6794:13
6792:.
6786:.
6751:.
6741:.
6729:.
6725:.
6702:.
6692:14
6690:.
6665:.
6632:.
6606:76
6604:.
6600:.
6577:.
6567:28
6565:.
6517:.
6505:.
6482:.
6474:.
6464:85
6462:.
6437:38
6435:.
6431:.
6409:.
6395:^
6379:.
6335:^
6253:,
6082:is
6077:is
6038:,
6032:,
6007:,
5482:=
5466:=
5459:=
4361:,
4352:,
4334:5d
4127:5c
3920:5b
3719:)
3686:,
3677:,
3653:,
3644:,
3626:5a
3332:=
3303:,
3288:4c
3142::
3118:,
3055:4b
2992:4a
2861:,
2858:iω
2827:3c
2718:ik
2677:3b
2616:3a
2495:=
2456:,
2316:=
2184:=
2159:).
2053:.
2045:.
1977:.
1958:.
1945:.
1838:,
1654:Al
1571:10
1533:10
1519:,
1451:(B
1447:Na
1280:Al
1226:Al
1067:Al
1063:Be
969:=
924:.
747:−
740:=
627:=
601:,
502:.
447:is
427:,
34:A
7428:.
7382:.
7357:V
7309:.
7297::
7259:.
7237::
7229::
7202:.
7188::
7161:.
7147::
7120:.
7088::
7061:.
7047::
7039::
7012:.
6982::
6953:.
6933::
6906:.
6876::
6870:4
6849:.
6845::
6800::
6759:.
6737::
6731:1
6710:.
6698::
6675:.
6661::
6642:.
6618:.
6612::
6585:.
6581::
6573::
6550:.
6525:.
6521::
6513::
6490:.
6470::
6447:.
6443::
6413:.
6329:.
6255:5
6224:.
6192:.
6105:k
6099:k
6087:k
6069:k
6053:k
6041:z
6035:y
6029:x
6026:(
6021:z
6019:n
6012:y
6010:n
6003:x
6001:n
5996:z
5994:±
5985:k
5967:k
5961:/
5957:k
5947:ω
5941:ω
5929:k
5919:k
5910:k
5904:k
5898:e
5895:n
5889:o
5886:n
5880:z
5870:k
5864:o
5861:n
5855:ω
5849:k
5829:)
5827:8
5825:(
5808:0
5805:=
5801:)
5793:2
5789:c
5783:2
5766:2
5760:o
5755:n
5749:2
5744:z
5740:k
5734:+
5727:2
5721:e
5716:n
5710:2
5705:y
5701:k
5695:+
5688:2
5682:e
5677:n
5671:2
5666:x
5662:k
5655:(
5650:)
5642:2
5638:c
5632:2
5615:2
5609:o
5604:n
5598:2
5593:z
5589:k
5583:+
5576:2
5570:o
5565:n
5559:2
5554:y
5550:k
5544:+
5537:2
5531:o
5526:n
5520:2
5515:x
5511:k
5504:(
5487:e
5484:n
5479:z
5477:n
5471:o
5468:n
5463:y
5461:n
5456:x
5454:n
5448:z
5438:)
5436:7
5434:(
5417:0
5414:=
5410:)
5404:2
5399:z
5395:k
5391:+
5386:2
5381:y
5377:k
5373:+
5368:2
5363:x
5359:k
5354:(
5349:)
5340:2
5335:y
5331:n
5325:2
5320:x
5316:n
5309:2
5304:z
5300:k
5294:+
5286:2
5281:z
5277:n
5271:2
5266:x
5262:n
5255:2
5250:y
5246:k
5240:+
5232:2
5227:z
5223:n
5217:2
5212:y
5208:n
5201:2
5196:x
5192:k
5185:(
5181:+
5177:)
5169:2
5164:x
5160:n
5153:2
5148:z
5144:k
5140:+
5135:2
5130:y
5126:k
5119:+
5112:2
5107:y
5103:n
5096:2
5091:z
5087:k
5083:+
5078:2
5073:x
5069:k
5062:+
5055:2
5050:z
5046:n
5039:2
5034:y
5030:k
5026:+
5021:2
5016:x
5012:k
5004:(
4996:2
4992:c
4986:2
4969:4
4965:c
4959:4
4923:2
4919:c
4913:2
4882:2
4878:c
4872:2
4844:)
4842:6
4840:(
4823:0
4820:=
4815:|
4808:)
4800:2
4796:c
4789:2
4784:z
4780:n
4774:2
4763:+
4758:2
4753:y
4749:k
4740:2
4735:x
4731:k
4723:(
4715:z
4711:k
4705:y
4701:k
4693:z
4689:k
4683:x
4679:k
4669:z
4665:k
4659:y
4655:k
4648:)
4640:2
4636:c
4629:2
4624:y
4620:n
4614:2
4603:+
4598:2
4593:z
4589:k
4580:2
4575:x
4571:k
4563:(
4555:y
4551:k
4545:x
4541:k
4531:z
4527:k
4521:x
4517:k
4509:y
4505:k
4499:x
4495:k
4488:)
4480:2
4476:c
4469:2
4464:x
4460:n
4454:2
4443:+
4438:2
4433:z
4429:k
4420:2
4415:y
4411:k
4403:(
4396:|
4373:E
4366:z
4364:E
4357:y
4355:E
4348:x
4346:E
4336:)
4332:(
4315:0
4312:=
4307:z
4303:E
4298:)
4290:2
4286:c
4279:2
4274:z
4270:n
4264:2
4253:+
4248:2
4243:y
4239:k
4230:2
4225:x
4221:k
4213:(
4209:+
4204:y
4200:E
4194:z
4190:k
4184:y
4180:k
4176:+
4171:x
4167:E
4161:z
4157:k
4151:x
4147:k
4129:)
4125:(
4108:0
4105:=
4100:z
4096:E
4090:z
4086:k
4080:y
4076:k
4072:+
4067:y
4063:E
4058:)
4050:2
4046:c
4039:2
4034:y
4030:n
4024:2
4013:+
4008:2
4003:z
3999:k
3990:2
3985:x
3981:k
3973:(
3969:+
3964:x
3960:E
3954:y
3950:k
3944:x
3940:k
3922:)
3918:(
3901:0
3898:=
3893:z
3889:E
3883:z
3879:k
3873:x
3869:k
3865:+
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3856:E
3850:y
3846:k
3840:x
3836:k
3832:+
3827:x
3823:E
3818:)
3810:2
3806:c
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3710:z
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3658:z
3656:E
3649:y
3647:E
3640:x
3638:E
3628:)
3624:(
3605:x
3601:E
3593:2
3589:c
3582:2
3577:x
3573:n
3567:2
3553:=
3548:z
3544:E
3538:z
3534:k
3528:x
3524:k
3520:+
3515:y
3511:E
3505:y
3501:k
3495:x
3491:k
3487:+
3482:x
3478:E
3472:2
3467:x
3463:k
3459:+
3454:x
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3439:2
3434:z
3430:k
3421:2
3416:y
3412:k
3403:2
3398:x
3394:k
3386:(
3363:x
3353:0
3350:ε
3347:0
3344:μ
3340:/
3337:1
3330:c
3324:c
3318:ε
3312:z
3306:y
3300:x
3290:)
3286:(
3267:]
3259:2
3254:z
3250:n
3244:0
3239:0
3232:0
3225:2
3220:y
3216:n
3210:0
3203:0
3198:0
3191:2
3186:x
3182:n
3175:[
3168:0
3160:=
3139:ε
3133:z
3127:z
3121:y
3115:x
3105:ω
3099:k
3084:E
3078:k
3072:D
3057:)
3053:(
3036:0
3033:=
3029:D
3021:k
2994:)
2990:(
2973:)
2969:E
2960:(
2955:2
2945:0
2934:=
2930:k
2926:)
2922:E
2914:k
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2907:+
2903:E
2899:)
2895:k
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2880:(
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2851:ε
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2825:(
2807:E
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2753:E
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2712:x
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2675:(
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2655:=
2651:D
2628:D
2618:)
2614:(
2596:D
2587:2
2583:t
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2549:E
2517:B
2510:0
2507:μ
2503:/
2500:1
2493:H
2487:H
2480:E
2469:k
2459:t
2452:r
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2329:ω
2322:0
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2304:ε
2294:)
2292:1
2290:(
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2263:=
2259:D
2238:E
2232:D
2222:n
2215:0
2212:ε
2208:n
2198:ε
2189:E
2186:ɛ
2182:D
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2167:(
2151:k
2115:)
1887:y
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1879:y
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1703:O
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1697:6
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1630:3
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1600:2
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1434:n
1427:β
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1024:n
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1016:e
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967:μ
851:;
831:;
812:;
797:;
752:o
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