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{{short description|Relation between the refractive indices and orientations of light in a birefringent crystal}}

In [[crystal optics]], the '''index&nbsp;ellipsoid''' (also known as the '''optical&nbsp;indicatrix'''<ref>Born & Wolf, 2002, p.&nbsp;799; Yariv & Yeh, 1984, p.&nbsp;77.</ref> or sometimes as the '''dielectric&nbsp;ellipsoid'''<ref>Jenkins & White, 1976, pp.&nbsp;560–561.</ref>) is a geometric construction which concisely represents the [[refractive index|refractive&nbsp;indices]] and associated [[polarization (waves)|polarizations]] of light, as functions of the orientation of the [[wavefront]], in a [[birefringence|doubly-refractive]] [[crystal]] (provided that the crystal does not exhibit [[optical&nbsp;rotation]]).  When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called a ''central&nbsp;section'' or ''diametral&nbsp;section'') is an [[ellipse]] 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 field|electric displacement]] vector {{math|'''D'''}}.<ref>Born & Wolf, 2002, pp.&nbsp;799–800; Landau & Lifshitz, 1960, p.&nbsp;320; Yariv & Yeh, 1984, pp.&nbsp;77–78.</ref> The principal semiaxes of the index ellipsoid are called the ''principal refractive indices''.<ref>Zernike & Midwinter, 1973, p.&nbsp;11.</ref>

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 propagation ''perpendicular'' to that semiaxis, with the {{math|'''D'''}} vector 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 the ''index&nbsp;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 <math>n_\text{a}, n_\text{b}, n_\text{c}</math> denote the principal semiaxes of the index ellipsoid, and choose a Cartesian coordinate system in which these semiaxes are respectively in the <math>x</math>, <math>y</math>, and <math>z</math> directions, the equation of the index ellipsoid is
{{NumBlk|:|<math>
  \frac{x^2}{n_\text{a}^2}+\frac{y^2}{n_\text{b}^2}+\frac{z^2}{n_\text{c}^2}=1\,.
</math>|{{EquationRef|1}}}}

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&nbsp;axes''<ref>Landau & Lifshitz, 1960, p.&nbsp;326.</ref> or ''optic axes'',<ref>Born & Wolf, 2002, p.&nbsp;801; Jenkins & White, 1976, pp.&nbsp;562; Yariv & Yeh, 1984, p.&nbsp;73.</ref> and the medium is therefore said to be ''biaxial''.<ref group=Note>Or, in older literature, ''biaxal''.</ref> 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]] (ellipsoid of revolution), and the two optic axes merge, so that the medium is said to be ''uniaxial''.<ref group=Note>Or, in older literature, ''uniaxal''.</ref> 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;<ref>''Cf.'' Yariv & Yeh, 1984, pp.&nbsp;82,&nbsp;84.</ref> 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]], for example, the index ellipsoid is an [[oblate&nbsp;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 spheroid|prolate]]'' spheroid touching the sphere at the poles, with an equatorial radius (extraordinary index) equal to the polar radius of the oblate spheroidal index ellipsoid.<ref group=Note>Yariv & Yeh (1984, pp.&nbsp;86–87) give an example of the contrary kind, in which the index surface is prolate (Figure&nbsp;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.</ref>

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 [[isotropy|isotropic]]''. Cubic crystals exhibit this property<ref>Landau & Lifshitz, 1960, p.&nbsp;321; Yariv & Yeh, 1984, pp.&nbsp;82–83; Zernike & Midwinter, 1973, p.&nbsp;12.</ref> as well as amorphous transparent media such as glass and water.<ref>Born & Wolf, 2002, p.&nbsp;805.</ref>

== 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 {{math|''n''}} denote the length of the radius vector from the origin to a general point on the index ellipsoid. Then dividing equation&nbsp;({{EquationNote|1}}) by {{math|''n''<sup>2</sup>}} gives
{{NumBlk|:|<math>
  \frac{\cos^2\xi}{n_\text{a}^2}+\frac{\cos^2\eta}{n_\text{b}^2}+\frac{\cos^2\zeta}{n_\text{c}^2}
  = \frac{1}{n^2}\,,
</math>|{{EquationRef|2}}}}
where <math>\cos\xi</math>, <math>\cos\eta</math>, and <math>\cos\zeta</math> are the [[direction&nbsp;cosine]]s of the radius vector. But {{math|''n''}} 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 <math>v</math>, we have <math>n=c_0/v</math>, where <math>c_0</math> is the speed of light in vacuum.<ref group=Note>Or sometimes it is convenient to use air instead of vacuum as the reference medium; ''cf.'' Zernike & Midwinter, 1973, p.&nbsp;2.</ref> For the principal semiaxes of the index ellipsoid, for which {{math|''n''}} takes the values <math>n_\text{a}, n_\text{b,} n_\text{c},</math> let <math>v</math> take the values {{math|''a,b,c'',}} respectively, so that <math>n_\text{a}=c_0/a,</math>&nbsp; <math>n_\text{b}=c_0/b,</math> and <math>n_\text{c}=c_0/c</math>. Making these substitutions in ({{EquationNote|2}}) and canceling the common factor <math>c_0^2</math>, we obtain
{{NumBlk|:|<math>
  v^2=a^2\cos^2\xi+b^2\cos^2\eta+c^2\cos^2\zeta\,.
</math>|{{EquationRef|3}}}}
This equation was derived by [[Augustin-Jean Fresnel]] in January 1822.<ref name=fresnel-1822ax/> If <math>v</math> 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 {{math|'''D'''}}).

Whereas the surface described by ({{EquationNote|1}}) is in index space (in which the coordinates are dimensionless numbers), the surface described by ({{EquationNote|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 <math>x,y,z</math> as the components of velocity and putting <math>\cos\xi=x/v</math>, etc.; thus the latter surface&nbsp;({{EquationNote|3}}) is generally not an ellipsoid, but another sort of ''ovaloid''. And as the index ellipsoid generates the index surface, so the surface&nbsp;({{EquationNote|3}}), by the same process, generates what we call the ''normal-velocity surface''.<ref group=Note>That is, the surface whose radius vector in any direction is the wave-normal velocity in that direction. Jenkins & White (1976, pp.&nbsp;555–556) call this the ''normal-velocity surface''. Born & Wolf (2002, p.&nbsp;803) call it the ''normal&nbsp;surface''. But Yariv & Yeh (1984) use the term ''normal&nbsp;surface'' for the index surface (p.&nbsp;87) or the corresponding surface for the wave vector {{math|'''k'''}} (p.&nbsp;73).</ref> Hence the surface&nbsp;({{EquationNote|3}}) 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.<ref name=fresnel-1822ax/>

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.<ref name=fresnel-1822-5/><ref>Fresnel, 1868, pp.&nbsp;395–396 (written no later than 31&nbsp;March 1822; see p.&nbsp;442).</ref> Fresnel's ellipsoid is now called the ''ray&nbsp;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&nbsp;field]] vector {{math|'''E'''}}.<ref>Born & Wolf, 2002, p.&nbsp;802.</ref>

The term ''index&nbsp;surface'' was coined by [[James&nbsp;MacCullagh]] in 1837.<ref name=macCullagh-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,<ref name=macCullagh-1833/> and which MacCullagh later called the "ellipsoid of indices".<ref>''Proceedings of the Royal Irish Academy'', no.&nbsp;49 (13&nbsp;January 1845), [https://books.google.com/books?id=SEn77pYVKdAC&pg=PA49 pp.&nbsp;49–51].</ref> In 1891, [[Lazarus&nbsp;Fletcher]] called this ellipsoid the ''optical&nbsp;indicatrix''.<ref name=fletcher-1891/>

== Electromagnetic interpretation ==

Deriving the index ellipsoid and its generating property from electromagnetic theory is non-trivial.<ref>See e.g. Born & Wolf, 2002, pp.&nbsp;790–801; Jenkins & White, 1976, pp.&nbsp;559–562; Landau & Lifshitz, 1960, pp.&nbsp;313–320; Yariv & Yeh, 1984, pp.&nbsp;69–79; Zernike & Midwinter, 1973, pp.&nbsp;6–12. Of these, only Yariv & Yeh use [[SI&nbsp;units]]; the others use the lesser-known [[Gaussian&nbsp;units]], which change the forms of some equations.</ref> ''Given'' the index ellipsoid, however, we can easily relate its parameters to the electromagnetic properties of the medium.

The [[speed of light]] in vacuum is
<math display=block>c_0=1\big/\sqrt{\mu_0\epsilon_0}\,,</math>
where <math>\mu_0</math> and <math>\epsilon_0</math> are respectively the magnetic [[permeability (electromagnetism)|permeability]] and the electric [[permittivity]] of the vacuum. For a transparent material medium, we can still reasonably assume that the magnetic permeability is <math>\mu_0</math> (especially at optical frequencies),<ref>Landau & Lifshitz, 1960, pp.&nbsp;251–253 (§60). The authors use [[Gaussian units]], in which the magnetic permeability of vacuum is 1.</ref> but <math>\epsilon_0</math> must be replaced by
<math>\epsilon_\text{r}\epsilon_0\;\!,</math> where <math>\epsilon_\text{r}</math>
is the ''[[relative&nbsp;permittivity]]'' (also called the ''dielectric&nbsp;constant''), so that the wave speed becomes
<math display=block>v=1\big/\sqrt{\mu_0\epsilon_\text{r}\epsilon_0}\,.</math>
Dividing <math>c_0</math> by <math>v</math>, we obtain the refractive index:
<math display=block>n=\sqrt{\epsilon_\text{r}}\,.</math>
This derivation treats <math>\epsilon_\text{r}</math> as a scalar, which is valid in an isotropic medium. In an ''[[anisotropy|anisotropic]]'' 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 {{math|'''D'''}} is parallel to the electric field vector {{math|'''E'''}}, as in an isotropic medium. In view of the symmetry of the index ellipsoid, these must be the cases in which {{math|'''D'''}} is in the direction of one of the axes. So, denoting the relative permittivities in the <math>x</math>, <math>y</math>, and <math>z</math> directions by <math>\epsilon_x\;\!,\epsilon_y,\epsilon_z</math> (the so-called ''principal dielectric constants''), and recalling that <math>n_\text{a}, n_\text{b}, n_\text{c}</math> denote the refractive indices for these directions of {{math|'''D'''}}, we must have
<math display=block>
  n_\text{a}=\sqrt{\epsilon_x} ~;~~
  n_\text{b}=\sqrt{\epsilon_y} ~;~~
  n_\text{c}=\sqrt{\epsilon_z} ~,
</math>
indicating that the semiaxes of the index ellipsoid are the square roots of the principal dielectric constants.<ref>Born & Wolf, 2002, p.&nbsp;799; Jenkins & White, 1976, p.&nbsp;560.</ref> Substituting these expressions into ({{EquationNote|1}}), we obtain the equation of the index ellipsoid in the alternative form<ref>Born & Wolf, 2002, p.&nbsp;799; Jenkins & White, 1976, p.&nbsp;560; Landau & Lifshitz, 1960, p.&nbsp;320.</ref>
<math display=block>
  \frac{x^2}{\epsilon_x}+\frac{y^2}{\epsilon_y}+\frac{z^2}{\epsilon_z}=1\,,
</math>
which explains why it is also called the ''dielectric'' ellipsoid.

== See also ==
* [[Birefringence]]
* [[Complex refractive index]]
* [[Crystal optics]]
* [[D-DIA]]
* [[Mathematical descriptions of opacity]]

== Notes ==

{{reflist|group=Note}}

== References ==

{{reflist|refs=

<ref name=fletcher-1891>L. Fletcher, "The Optical Indicatrix and the Transmission of Light in Crystals" (read 16&nbsp;June 1891), ''Mineralogical Magazine and Journal of the Mineralogical Society'', vol.&nbsp;9, [https://books.google.com/books?id=mlMPAAAAIAAJ&pg=PA278 pp.&nbsp;278–388] (Dec.&nbsp;1891); [https://archive.org/details/opticalindicatri00fletrich reprinted] London: Oxford University Press Warehouse, 1892; reviewed by "R.T.G." in ''Nature'', vol.&nbsp;46, no.&nbsp;1199 (20&nbsp;Oct.&nbsp;1892), [https://archive.org/details/paper-doi-10_1038_046581a0/mode/1up pp.&nbsp;581–582].</ref>

<ref name=fresnel-1822ax>A. Fresnel, "Extrait du Supplément au Mémoire sur la double réfraction" (read 13&nbsp;Jan.&nbsp;1822&nbsp;?), printed in Fresnel, 1868, pp.&nbsp;335–342; translated as "''Extrait'' of the Supplement to the Memoir on double refraction", {{Zenodo|5886692}}, 2022.</ref>

<ref name=fresnel-1822-5>A. Fresnel, "Extrait d'un Mémoire sur la double réfraction", ''Annales de Chimie et de Physique'', Ser.&nbsp;2, vol.&nbsp;28, pp.&nbsp;263–279 (March&nbsp;1825); reprinted as "Extrait du second Mémoire sur la double réfraction" in Fresnel, 1868, pp.&nbsp;465–478; translated as "''Extrait'' of a [second] memoir on double refraction", {{Zenodo|5442206}}, 2022. (An earlier version of this paper appeared in ''Bulletin des Sciences par la Société Philomatique de Paris'', vol.&nbsp;9, pp.&nbsp;63–71, May&nbsp;1822.)</ref>

<ref name=macCullagh-1833>J. MacCullagh, "Geometrical propositions applied to the wave theory of light" (read 24&nbsp;June 1833), ''Transactions of the Royal Irish Academy'', vol.&nbsp;17 (nominally for 1831), pp.&nbsp;241–263, {{JSTOR|30078792}}, at&nbsp;p.&nbsp;260.</ref>

<ref name=macCullagh-1837>J. MacCullagh, "On the laws of crystalline reflexion and refraction" (read 9&nbsp;Jan.&nbsp;1837), ''Transactions of the Royal Irish Academy'', vol.&nbsp;18 (1839), pp.&nbsp;31–74, {{JSTOR|30078974}}, at&nbsp;p.&nbsp;38.</ref>

}}

== Bibliography ==

* M. Born and E. Wolf, 2002, ''Principles of Optics'', 7th&nbsp;Ed., Cambridge University Press, 1999 (reprinted with corrections, 2002), {{ISBN|978-0-521-64222-4}}.
* A. Fresnel (ed.&nbsp;&nbsp;H.&nbsp;de&nbsp;Senarmont, E.&nbsp;Verdet, and L.&nbsp;Fresnel), 1868, ''Oeuvres complètes d'Augustin Fresnel'', Paris: Imprimerie Impériale (3&nbsp;vols., 1866–70), [https://books.google.com/books?id=g6tzUG7JmoQC vol.&nbsp;2&nbsp;(1868)].
* F.A. Jenkins and H.E. White, 1976, ''Fundamentals of Optics'', 4th&nbsp;Ed., New&nbsp;York: McGraw-Hill, {{ISBN|0-07-032330-5}}.
* L.D. Landau and E.M.&nbsp;Lifshitz (tr.&nbsp;&nbsp;J.B.&nbsp;Sykes & J.S.&nbsp;Bell), 1960, ''Electrodynamics of Continuous Media'' (vol.&nbsp;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, {{ISBN|0-471-09142-1}}.
* F. Zernike and J.E. Midwinter, 1973, ''[https://books.google.com/books?id=Y3Wsyo7TgdkC Applied Nonlinear Optics]'', New York: Wiley (reprinted Mineola, NY: Dover, 2006).

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[[Category:Optics]]
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