Index ellipsoid

Incrystal optics,theindex ellipsoid(also known as theoptical indicatrixor sometimes as thedielectric ellipsoid) is a geometric construction which concisely represents therefractive indicesand associatedpolarizationsof light, as functions of the orientation of thewavefront,in adoubly-refractivecrystal(provided that the crystal does not exhibitoptical rotation). When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called acentral sectionordiametral section) is anellipsewhose 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 theelectric displacementvectorD.The principal semiaxes of the index ellipsoid are called theprincipal refractive indices.

It follows from the sectioning procedure that each principal semiaxis of the ellipsoid is generallynotthe refractive index for propagation in the direction of that semiaxis, but rather the refractive index for propagationperpendicularto that semiaxis, with theDvector parallel to that semiaxis (and parallel to the wavefront). 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 theindex 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 planenormalto 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

${\frac {x^{2}}{n_{a}^{2}}}+{\frac {y^{2}}{n_{b}^{2}}}+{\frac {z^{2}}{n_{c}^{2}}}=1\,.$(1)

If the index ellipsoid istriaxial(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 directionsnormalto these two planes—that is, the directions of a single wave speed for all polarizations—are called thebinormal axesoroptic axes,and the medium is therefore said to bebiaxial.Thus, paradoxically, if the index ellipsoid of a medium istriaxial, the medium itself is calledbiaxial.

If two of the principal semiaxes of the index ellipsoid are equal (in which case their common length is called theordinaryindex, and the third length theextraordinaryindex), the ellipsoid reduces to aspheroid(ellipsoid of revolution), and the two optic axes merge, so that the medium is said to beuniaxial.As the index ellipsoid reduces to a spheroid, the two-sheeted indexsurfaceconstructed 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 ofcalcite,for example, the index ellipsoid is anoblate 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 aprolatespheroid touching the sphere at the poles, with an equatorial radius (extraordinary index) equal to the polar radius of the oblate spheroidal index ellipsoid.

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 isopticallyisotropic.Cubic crystals exhibit this propertyas 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. Letndenote the length of the radius vector from the origin to a general point on the index ellipsoid. Then dividing equation (1) byn²gives

${\frac {\cos ^{2}\xi }{n_{a}^{2}}}+{\frac {\cos ^{2}\eta }{n_{b}^{2}}}+{\frac {\cos ^{2}\zeta }{n_{c}^{2}}}={\frac {1}{n^{2}}}\,,$(2)

where$\cos \xi$,$\cos \eta$,and$\cos \zeta$are thedirection cosinesof the radius vector. Butnis 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.For the principal semiaxes of the index ellipsoid, for whichntakes the values$n_{a},n_{b},n_{c},$let$v$take the valuesa,b,c,respectively, so that$n_{a}=c_{0}/a,$$n_{b}=c_{0}/b,$and$n_{c}=c_{0}/c$.Making these substitutions in (2) and canceling the common factor$c_{0}^{2}$,we obtain

$v^{2}=a^{2}\cos ^{2}\xi +b^{2}\cos ^{2}\eta +c^{2}\cos ^{2}\zeta \,.$(3)

This equation was derived byAugustin-Jean Fresnelin 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 ofD).

Whereas the surface described by (1) is in index space (in which the coordinates are dimensionless numbers), the surface described by (3) 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 (3) is generally not an ellipsoid, but another sort ofovaloid.And as the index ellipsoid generates the index surface, so the surface (3), by the same process, generates what we call thenormal-velocity surface.Hence the surface (3) might reasonably be called the "normal-velocity ovaloid". Fresnel, however, called it thesurface 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 theray 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 theelectric fieldvectorE.

The termindex surfacewas coined byJames MacCullaghin 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 Fletchercalled this ellipsoid theoptical indicatrix.

Electromagnetic interpretation

Deriving the index ellipsoid and its generating property from electromagnetic theory is non-trivial.Giventhe index ellipsoid, however, we can easily relate its parameters to the electromagnetic properties of the medium.

Thespeed of lightin a vacuum is $$c_{0}=1{\big /}{\sqrt {\mu _{0}\epsilon _{0}}}\,,$$ where$\mu _{0}$and$\epsilon _{0}$are respectively the magneticpermeabilityand the electricpermittivityof 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 therelative permittivity(also called thedielectric constant), so that the wave speed becomes $$v=1{\big /}{\sqrt {\mu _{0}\epsilon _{r}\epsilon _{0}}}\,.$$ Dividing$c_{0}$by$v$,we obtain the refractive index: $$n={\sqrt {\epsilon _{r}}}\,.$$ This derivation treats$\epsilon _{r}$as a scalar, which is valid in an isotropic medium. In ananisotropicmedium, 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 vectorDis parallel to the electric field vectorE,as in an isotropic medium. In view of the symmetry of the index ellipsoid, these must be the cases in whichDis 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-calledprincipal dielectric constants), and recalling that$n_{a},n_{b},n_{c}$denote the refractive indices for these directions ofD,we must have $$n_{a}={\sqrt {\epsilon _{x}}}~;~~n_{b}={\sqrt {\epsilon _{y}}}~;~~n_{c}={\sqrt {\epsilon _{z}}}~,$$ indicating that the semiaxes of the index ellipsoid are the square roots of the principal dielectric constants.Substituting these expressions into (1), we obtain the equation of the index ellipsoid in the alternative form $${\frac {x^{2}}{\epsilon _{x}}}+{\frac {y^{2}}{\epsilon _{y}}}+{\frac {z^{2}}{\epsilon _{z}}}=1\,,$$ which explains why it is also called thedielectricellipsoid.

See also

• Birefringence
• Complex refractive index
• Crystal optics
• D-DIA
• Mathematical descriptions of opacity

Bibliography

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• L.D. Landau and E.M. Lifshitz (tr. J.B. Sykes & J.S. Bell), 1960,Electrodynamics of Continuous Media(vol. 8 ofCourse of Theoretical Physics), London: Pergamon Press.
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