Birefringence, also called double refraction, is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are described as birefringent or birefractive. The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress.
Birefringence is responsible for the phenomenon of double refraction whereby a ray of light, when incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. This effect was first described by Danish scientist Rasmus Bartholin in 1669, who observed itSee: Erasmus Bartholin, Experimenta crystalli islandici disdiaclastici quibus mira & infolita refractio detegitur [Experiments on birefringent Icelandic crystal through which is detected a remarkable and unique refraction] (Copenhagen, Denmark: Daniel Paulli, 1669). Erasmus Bartholin (January 16, 1671) "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," Philosophical Transactions of the Royal Society of London, 5 : 2041–2048 (pdf pp 282–289). in Iceland spar (calcite) crystals, which have one of the strongest birefringences. In the 19th century, Augustin-Jean Fresnel described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarization (perpendicular to the direction of the wave vector).A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" et seq., Annales de Chimie et de Physique, Ser. 2, vol. 17, pp. 102–111 (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", (Creative Commons), 2021; §14.A. Fresnel, "Extrait d'un Mémoire sur la double réfraction", Annales de Chimie et de Physique, Ser. 2, vol. 28, pp. 263–279 (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 [second] memoir on double refraction", , 2021 (open access).
Explanation
A mathematical description of wave propagation in a birefringent medium is presented below. Following is a qualitative explanation of the phenomenon.
Uniaxial materials The simplest type of birefringence is described as uniaxial, meaning that there is a single direction governing the optical anisotropy whereby all directions perpendicular to it (or at a given angle to it) are optically equivalent. Thus rotating the material around this axis does not change its optical behaviour. This special direction is known as the optic axis of the material. Light propagating parallel to the optic axis (whose polarization is always perpendicular to the optic axis) is governed by a refractive index (for "ordinary") regardless of its specific polarization. For rays with any other propagation direction, there is one linear polarization that is perpendicular to the optic axis, and a ray with that polarization is called an ordinary ray and is governed by the same refractive index value . For a ray propagating in the same direction but with a polarization perpendicular to that of the ordinary ray, the polarization direction will be partly in the direction of (parallel to) the optic axis, and this extraordinary ray will be governed by a different, direction-dependent refractive index. Because the index of refraction depends on the polarization when unpolarized light enters a uniaxial birefringent material, it is split into two beams travelling in different directions, one having the polarization of the ordinary ray and the other the polarization of the extraordinary ray. The ordinary ray will always experience a refractive index of , whereas the refractive index of the extraordinary ray will be in between and , depending on the ray direction as described by the index ellipsoid. The magnitude of the difference is quantified by the birefringence
The propagation (as well as reflection coefficient) of the ordinary ray is simply described by as if there were no birefringence involved. The extraordinary ray, as its name suggests, propagates unlike any wave in an isotropic optical material. Its refraction (and reflection) at a surface can be understood using the effective refractive index (a value in between and ). Its power flow (given by the Poynting vector) is not exactly in the direction of the wave vector. This causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of calcite as photographed above. Rotating the calcite crystal will cause one of the two images, that of the extraordinary ray, to rotate slightly around that of the ordinary ray, which remains fixed.
When the light propagates either along or orthogonal to the optic axis, such a lateral shift (or double refracted image) does not occur. In the first case, both polarizations are perpendicular to the optic axis and see the same effective refractive index, so there is no extraordinary ray. In the second case the extraordinary ray propagates at a different phase velocity (corresponding to ) but still has the power flow in the direction of the wave vector. A crystal with its optic axis parallel to the optical surface, may be used to create a waveplate, in which there is no distortion of the image but an intentional modification of the state of polarization of the incident wave. For instance, a quarter-wave plate is commonly used to create circular polarization from a linearly polarized source.
Biaxial materials The case of so-called biaxial crystals is substantially more complex.Landau, L. D., and Lifshitz, E. M., Electrodynamics of Continuous Media, Vol. 8 of the Course of Theoretical Physics 1960 (Pergamon Press), §79 These are characterized by three refractive indices corresponding to three principal axes of the crystal. For most ray directions, both polarizations would be classified as extraordinary rays but with different effective refractive indices. Being extraordinary waves, the direction of power flow is not identical to the direction of the wave vector in either case.
The two refractive indices can be determined using the index ellipsoids for given directions of the polarization. Note that for biaxial crystals the index ellipsoid will not be an ellipsoid of revolution ("spheroid") but is described by three unequal principle refractive indices , and . Thus there is no axis around which a rotation leaves the optical properties invariant (as there is with uniaxial crystals whose index ellipsoid is a spheroid).
Although there is no axis of symmetry, there are two optical axes or binormals which are defined as directions along which light may propagate without birefringence, i.e., directions along which the wavelength is independent of polarization. For this reason, birefringent materials with three distinct refractive indices are called biaxial. Additionally, there are two distinct axes known as optical ray axes or biradials along which the group velocity of the light is independent of polarization.
Double refraction
When an arbitrary beam of light strikes the surface of a birefringent material at non-normal incidence, the polarization component normal to the optic axis (ordinary ray) and the other linear polarization (extraordinary ray) will be refracted toward somewhat different paths. Natural light, so-called unpolarized light, consists of equal amounts of energy in any two orthogonal polarizations. Even linearly polarized light has some energy in both polarizations, unless aligned along one of the two axes of birefringence. According to Snell's law of refraction, the two angles of refraction are governed by the effective refractive index of each of these two polarizations. This is clearly seen, for instance, in the Wollaston prism which separates incoming light into two linear polarizations using prisms composed of a birefringent material such as calcite.
The different angles of refraction for the two polarization components are shown in the figure at the top of this page, with the optic axis along the surface (and perpendicular to the plane of incidence), so that the angle of refraction is different for the polarization (the "ordinary ray" in this case, having its electric vector perpendicular to the optic axis) and the polarization (the "extraordinary ray" in this case, whose electric field polarization includes a component in the direction of the optic axis). A distinct form of double refraction occurs, even with normal incidence, in cases where the optic axis is not along the refracting surface (nor exactly normal to it); in this case, the dielectric polarization of the birefringent material is not exactly in the direction of the wave's electric field for the extraordinary ray. The direction of power flow (given by the Poynting vector) for this inhomogenous wave is at a finite angle from the direction of the wave vector resulting in an additional separation between these beams. So even in the case of normal incidence, where one would compute the angle of refraction as zero (according to Snell's law, regardless of the effective index of refraction), the energy of the extraordinary ray is propagated at an angle. If exiting the crystal through a face parallel to the incoming face, the direction of both rays will be restored, but leaving a shift between the two beams. This is commonly observed using a piece of calcite cut along its natural cleavage, placed above a paper with writing, as in the above photograph. Hence, waveplates specifically have their optic axis along the surface of the plate, so that with (approximately) normal incidence there will be no shift in the image from light of either polarization, simply a relative phase shift between the two light waves.
Terminology
Much of the work involving polarization preceded the understanding of light as a transverse electromagnetic wave, and this has affected some terminology in use. Isotropic materials have symmetry in all directions and the refractive index is the same for any polarization direction. An anisotropic material is called "birefringent" because it will generally refract a single incoming ray in two directions, which we now understand correspond to the two different polarizations. This is true of either a uniaxial or biaxial material.
In a uniaxial material, one ray behaves according to the normal law of refraction (corresponding to the ordinary refractive index), so an incoming ray at normal incidence remains normal to the refracting surface. As explained above, the other polarization can deviate from normal incidence, which cannot be described using the law of refraction. This thus became known as the extraordinary ray. The terms "ordinary" and "extraordinary" are still applied to the polarization components perpendicular to and not perpendicular to the optic axis respectively, even in cases where no double refraction is involved.
A material is termed uniaxial when it has a single direction of symmetry in its optical behavior, which we term the optic axis. It also happens to be the axis of symmetry of the index ellipsoid (a spheroid in this case). The index ellipsoid could still be described according to the refractive indices, , and , along three coordinate axes; in this case two are equal. So if corresponding to the and axes, then the extraordinary index is corresponding to the axis, which is also called the optic axis in this case.