Uniaxial Anisotropy
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In
crystal optics Crystal optics is the branch of optics that describes the behaviour of light in '' anisotropic media'', that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. The index of refrac ...
, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polarizations of light, as functions of the orientation of the
wavefront In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
, in a doubly-refractive
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
(provided that the crystal does not exhibit optical rotation). When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called a ''central section'' or ''diametral section'') is an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
whose major and minor semiaxes have lengths equal to the two refractive indices for that orientation of the wavefront, and have the directions of the respective polarizations as expressed by the electric displacement vector . The principal semiaxes of the index ellipsoid are called the ''principal refractive indices''. It follows from the sectioning procedure that each principal semiaxis of the ellipsoid is generally ''not'' the refractive index for propagation in the direction of that semiaxis, but rather the refractive index for wavefronts tangential to that direction, with the vector parallel to that direction, propagating perpendicular to that direction. Thus the direction of propagation (normal to the wavefront) to which each principal refractive index applies is in the plane perpendicular to the associated principal semiaxis.


Terminology

The index ellipsoid is not to be confused with the ''index surface'', whose radius vector (from the origin) in any direction is indeed the refractive index for propagation in that direction; for a birefringent medium, the index surface is the two-sheeted surface whose two radius vectors in any direction have lengths equal to the major and minor semiaxes of the diametral section of the index ellipsoid by a plane ''normal'' to that direction. If we let n_a, n_b, n_c denote the principal semiaxes of the index ellipsoid, and choose a Cartesian coordinate system in which these semiaxes are respectively in the x, y, and z directions, the equation of the index ellipsoid is If the index ellipsoid is ''triaxial'' (meaning that its principal semiaxes are all unequal), there are two cutting planes for which the diametral section reduces to a circle. For wavefronts parallel to these planes, all polarizations are permitted and have the same refractive index, hence the same wave speed. The directions ''normal'' to these two planes—that is, the directions of a single wave speed for all polarizations—are called the ''binormal axes'' or ''optic axes'', and the medium is therefore said to be ''biaxial''.Or, in older literature, ''biaxal''. Thus, paradoxically, if the index ellipsoid of a medium is ''tri''axial, the medium itself is called ''bi''axial. If two of the principal semiaxes of the index ellipsoid are equal (in which case their common length is called the ''ordinary'' index, and the third length the ''extraordinary'' index), the ellipsoid reduces to a
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
(ellipsoid of revolution), and the two optic axes merge, so that the medium is said to be ''uniaxial''.Or, in older literature, ''uniaxal''. As the index ellipsoid reduces to a spheroid, the two-sheeted index ''surface'' constructed therefrom reduces to a sphere and a spheroid touching at opposite ends of their common axis, which is parallel to that of the index ellipsoid; but the principal axes of the spheroidal index ellipsoid and the spheroidal sheet of the index surface are interchanged. In the well-known case of
calcite Calcite is a Carbonate minerals, carbonate mineral and the most stable Polymorphism (materials science), polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on ...
, for example, the index ellipsoid is an oblate spheroid, so that one sheet of the index surface is a sphere touching that oblate spheroid at the equator, while the other sheet of the index surface is a ''
prolate A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
'' spheroid touching the sphere at the poles, with an equatorial radius (extraordinary index) equal to the polar radius of the oblate spheroidal index ellipsoid.Yariv & Yeh (1984, pp. 86–7) give an example of the contrary kind, in which the index surface is prolate (Figure 4.4), and the associated index surface (which they call the "normal surface") comprises a sphere and an oblate spheroid touching at the poles. In both examples the proportions of the extraordinary wavefront expanding from a point source in the crystal are inverse to those of the index surface, because the refractive index is inversely proportional to the normal velocity of the wavefront. If all three principal semi-axes of the index ellipsoid are equal, it reduces to a sphere: all diametral sections of the index ellipsoid are circular, whence all polarizations are permitted for all directions of propagation, with the same refractive index for all directions, and the index surface merges with the (spherical) index ellipsoid; in short, the medium is ''optically
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
''. Cubic crystals exhibit this property as well as amorphous transparent media such as glass and water.


History

A surface analogous to the index ellipsoid can be defined for the wave speed (normal to the wavefront) instead of the refractive index. Let denote the length of the radius vector from the origin to a general point on the index ellipsoid. Then dividing equation () by gives where \cos\xi, \cos\eta, and \cos\zeta are the direction cosines of the radius vector. But is also the refractive index for a wavefront parallel to a diametral section of which the radius vector is major or minor semiaxis. If that wavefront has speed v, we have n=c_0/v, where c_0 is the speed of light in a vacuum.Or sometimes it is convenient to use air instead of a vacuum as the reference medium; ''cf.'' Zernike & Midwinter, 1973, p. 2. For the principal semiaxes of the index ellipsoid, for which takes the values n_a, n_b, n_c, let v take the values respectively, so that n_a=c_0/a,  n_b=c_0/b, and n_c=c_0/c. Making these substitutions in () and canceling the common factor c_0^2, we obtain This equation was derived by
Augustin-Jean Fresnel Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular th ...
in January 1822. If v is the length of the radius vector, the equation describes a surface with the property that the major and minor semiaxes of any diametral section have lengths equal to the wave-normal speeds of wavefronts parallel to that section, and the directions of what Fresnel called the "vibrations" (which we now recognize as oscillations of ). Whereas the surface described by () is in index space (in which the coordinates are dimensionless numbers), the surface described by () is in velocity space (in which the coordinates have the units of velocity). Whereas the former surface is of the 2nd degree, the latter is of the 4th degree, as may be verified by redefining x,y,z as the components of velocity and putting \cos\xi=x/v, etc; thus the latter surface () is generally not an ellipsoid, but another sort of ''ovaloid''. And as the index ellipsoid generates the index surface, so the surface (), by the same process, generates what we call the ''normal-velocity surface''.That is, the surface whose radius vector in any direction is the wave-normal velocity in that direction. Jenkins & White (1976, pp. 555–6) call this the ''normal-velocity surface''. Born & Wolf (2002, p. 803) call it the ''normal surface''. But Yariv & Yeh (1984) use the term ''normal surface'' for the index surface (p. 87) or the corresponding surface for the wave vector (p. 73). Hence the surface () might reasonably be called the "normal-velocity ovaloid". Fresnel, however, called it the ''surface of elasticity'', because he derived it by supposing that light waves were transverse elastic waves, that the medium had three perpendicular directions in which a displacement of a molecule produced a restoring force in exactly the opposite direction, and that the restoring force due to a vector sum of displacements was the vector sum of the restoring forces due to the separate displacements. Fresnel soon realized that the ellipsoid constructed on the same principal semi-axes as the surface of elasticity has the same relation to the ray velocities that the surface of elasticity has to the wave-normal velocities. Fresnel's ellipsoid is now called the ''ray ellipsoid''. Thus, in modern terms, the ray ellipsoid generates the ray velocities as the index ellipsoid generates the refractive indices. The major and minor semiaxes of the diametral section of the ray ellipsoid are in the permitted directions of the electric field vector . The term ''index surface'' was coined by
James MacCullagh James MacCullagh (1809 – 24 October 1847) was an Irish mathematician. Early Life MacCullagh was born in Landahaussy, near Plumbridge, County Tyrone, Ireland, but the family moved to Curly Hill, Strabane when James was about 10. He was the e ...
in 1837. In a previous paper, read in 1833, MacCullagh had called this surface the "surface of refraction" and shown that it is generated by the major and minor semiaxes of a diametral section of an ellipsoid which has principal semiaxes inversely proportional to those of Fresnel's ellipsoid, and which MacCullagh later called the "ellipsoid of indices". In 1891,
Lazarus Fletcher Sir Lazarus Fletcher (3 March 1854 – 6 January 1921) was a British geologist. He was elected a Fellow of the Royal Society in 1889 and won the Wollaston Medal of the Geological Society in 1912. Fletcher was knighted in 1916. Fletcher was Keepe ...
called this ellipsoid the ''optical indicatrix''.


Electromagnetic interpretation

Deriving the index ellipsoid and its generating property from electromagnetic theory is non-trivial. ''Given'' the index ellipsoid, however, we can easily relate its parameters to the electromagnetic properties of the medium. The
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in a vacuum is c_0=1\big/\sqrt\,, where \mu_0 and \epsilon_0 are respectively the magnetic permeability and the electric
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the vacuum. For a transparent material medium, we can still reasonably assume that the magnetic permeability is \mu_0 (especially at optical frequencies), but \epsilon_0 must be replaced by \epsilon_r\epsilon_0\;\!, where \epsilon_r is the ''
relative permittivity The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insulat ...
'' (also called the ''dielectric constant''), so that the wave speed becomes v=1\big/\sqrt\,. Dividing c_0 by v, we obtain the refractive index: n=\sqrt\,. This derivation treats \epsilon_r as a scalar, which is valid in an isotropic medium. In an ''
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
'' medium, the result holds only for those combinations of propagation direction and polarization which avoid the anisotropy—that is, for those cases in which the electric displacement vector is parallel to the electric field vector , as in an isotropic medium. In view of the symmetry of the index ellipsoid, these must be the cases in which is in the direction of one of the axes. So, denoting the relative permittivities in the x, y, and z directions by \epsilon_x\;\!,\epsilon_y,\epsilon_z (the so-called ''principal dielectric constants''), and recalling that n_a, n_b, n_c denote the refractive indices for these directions of , we must have n_a=\sqrt ~;~~ n_b=\sqrt ~;~~ n_c=\sqrt ~, indicating that the semiaxes of the index ellipsoid are the square roots of the principal dielectric constants. Substituting these expressions into (), we obtain the equation of the index ellipsoid in the alternative formBorn & Wolf, 2002, p. 799; Jenkins & White, 1976, p. 560; Landau & Lifshitz, 1960, p. 320. \frac+\frac+\frac=1\,, which explains why it is also called the ''dielectric'' ellipsoid.


See also

* Birefringence *
Complex refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
*
Crystal optics Crystal optics is the branch of optics that describes the behaviour of light in '' anisotropic media'', that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. The index of refrac ...
*
D-DIA The D-DIA or deformation-DIA is an apparatus used for high pressure and high temperature deformation experiments. The advantage of this apparatus is the ability to apply pressures up to approximately 15 GPa while independently creating uniaxial ...
*
Mathematical descriptions of opacity When an electromagnetic wave travels through a medium in which it gets attenuated (this is called an "opaque" or " attenuating" medium), it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to ...
.


Notes


References


Bibliography

* M. Born and E. Wolf, 2002, ''Principles of Optics'', 7th Ed., Cambridge University Press, 1999 (reprinted with corrections, 2002), . * A. Fresnel (ed.  H. de Senarmont, E. Verdet, and L. Fresnel), 1868, ''Oeuvres complètes d'Augustin Fresnel'', Paris: Imprimerie Impériale (3 vols., 1866–70)
vol. 2 (1868)
* F.A. Jenkins and H.E. White, 1976, ''Fundamentals of Optics'', 4th Ed., New York: McGraw-Hill, . * L.D. Landau and E.M. Lifshitz (tr.  J.B. Sykes & J.S. Bell), 1960, ''Electrodynamics of Continuous Media'' (vol. 8 of ''Course of Theoretical Physics''), London: Pergamon Press. * A. Yariv and P. Yeh, 1984, ''Optical Waves in Crystals: Propagation and control of laser radiation'', New York: Wiley, . * F. Zernike and J.E. Midwinter, 1973,
Applied Nonlinear Optics
', New York: Wiley (reprinted Mineola, NY: Dover, 2006). {{DEFAULTSORT:Index Ellipsoid Optics Physical optics Polarization (waves) Surfaces Optical mineralogy