2867:
2879:
1394:, 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.
111:, 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
423:
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
1270:
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
1423:
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
441:
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
2419:
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
636:
376:(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
1397:
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
2726:
See e.g. Born & Wolf, 2002, pp. 790–801; Jenkins & White, 1976, pp. 559–62; Landau & Lifshitz, 1960, pp. 313–20; Yariv & Yeh, 1984, pp. 69–79; Zernike & Midwinter, 1973, pp. 6–12. Of these, only Yariv & Yeh use
2339:
2228:
354:
1402:. 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
98:
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.
1228:
2420:
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.
1783:
483:
1517:
2037:
2239:
2105:
67:
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
1292:) 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
1675:
1906:
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
1871:
2594:, Ser. 2, vol. 28, pp. 263–79 (March 1825); reprinted as "Extrait du second Mémoire sur la double réfraction" in Fresnel, 1868, pp. 465–78; translated as "
890:
2094:
169:
1019:
968:
1898:
1702:
1633:
1571:
1362:
1067:
239:
730:
704:
2567:
A. Fresnel, "Extrait du Supplément au Mémoire sur la double réfraction" (read 13 Jan. 1822 ?), printed in
Fresnel, 1868, pp. 335–42; translated as "
1105:
801:
678:
1606:
1544:
1810:
1322:
828:
2442:
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
1982:
1962:
1942:
1830:
1717:
1268:
910:
760:
742:
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
229:
209:
189:
438:
spheroid touching the sphere at the poles, with an equatorial radius (extraordinary index) equal to the polar radius of the oblate spheroidal index ellipsoid.
1115:
1454:
2230:
indicating that the semiaxes of the index ellipsoid are the square roots of the principal dielectric constants. Substituting these expressions into (
432:, 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
631:{\displaystyle {\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}}}\,,}
171:
denote the principal semiaxes of the index ellipsoid, and choose a
Cartesian coordinate system in which these semiaxes are respectively in the
457:
A surface analogous to the index ellipsoid can be defined for the wave speed (normal to the wavefront) instead of the refractive index. Let
1987:
1835:
463:
denote the length of the radius vector from the origin to a general point on the index ellipsoid. Then dividing equation (
2788:
55:). When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called a
2374:
2546:
Landau & Lifshitz, 1960, p. 321; Yariv & Yeh, 1984, pp. 82–3; Zernike & Midwinter, 1973, p. 12.
2498:
Born & Wolf, 2002, pp. 799–800; Landau & Lifshitz, 1960, p. 320; Yariv & Yeh, 1984, pp. 77–8.
88:
the refractive index for propagation in the direction of that semiaxis, but rather the refractive index for propagation
2840:
2814:
2766:
Born & Wolf, 2002, p. 799; Jenkins & White, 1976, p. 560; Landau & Lifshitz, 1960, p. 320.
1638:
1581:
of the vacuum. For a transparent material medium, we can still reasonably assume that the magnetic permeability is
1444:
the index ellipsoid, however, we can easily relate its parameters to the electromagnetic properties of the medium.
403:
If two of the principal semiaxes of the index ellipsoid are equal (in which case their common length is called the
2866:
2334:{\displaystyle {\frac {x^{2}}{\epsilon _{x}}}+{\frac {y^{2}}{\epsilon _{y}}}+{\frac {z^{2}}{\epsilon _{z}}}=1\,,}
2223:{\displaystyle n_{a}={\sqrt {\epsilon _{x}}}~;~~n_{b}={\sqrt {\epsilon _{y}}}~;~~n_{c}={\sqrt {\epsilon _{z}}}~,}
1574:
2525:
Born & Wolf, 2002, p. 801; Jenkins & White, 1976, pp. 562; Yariv & Yeh, 1984, p. 73.
1918:, as in an isotropic medium. In view of the symmetry of the index ellipsoid, these must be the cases in which
2909:
2697:
L. Fletcher, "The
Optical Indicatrix and the Transmission of Light in Crystals" (read 16 June 1891),
2919:
68:
2660:
J. MacCullagh, "Geometrical propositions applied to the wave theory of light" (read 24 June 1833),
839:
380:
to these two planes—that is, the directions of a single wave speed for all polarizations—are called the
2857:
2046:
449:. Cubic crystals exhibit this property as well as amorphous transparent media such as glass and water.
121:
2883:
1440:
Deriving the index ellipsoid and its generating property from electromagnetic theory is non-trivial.
349:{\displaystyle {\frac {x^{2}}{n_{a}^{2}}}+{\frac {y^{2}}{n_{b}^{2}}}+{\frac {z^{2}}{n_{c}^{2}}}=1\,.}
1286:) is in index space (in which the coordinates are dimensionless numbers), the surface described by (
973:
922:
84:
It follows from the sectioning procedure that each principal semiaxis of the ellipsoid is generally
2904:
2639:
J. MacCullagh, "On the laws of crystalline reflexion and refraction" (read 9 Jan. 1837),
2359:
1876:
1680:
1611:
1549:
1327:
1024:
2794:
A. Fresnel (ed. H. de Senarmont, E. Verdet, and L. Fresnel), 1868,
2714:
709:
2820:
L.D. Landau and E.M. Lifshitz (tr. J.B. Sykes & J.S. Bell), 1960,
830:
is the speed of light in a vacuum. For the principal semiaxes of the index ellipsoid, for which
683:
1078:
765:
657:
2702:
1584:
1522:
2685:
1706:
1247:
1390:) might reasonably be called the "normal-velocity ovaloid". Fresnel, however, called it the
1788:
1732:
1476:
1295:
806:
37:
2621:
Fresnel, 1868, pp. 395–6 (written no later than 31 March 1822; see p. 442).
415:(ellipsoid of revolution), and the two optic axes merge, so that the medium is said to be
8:
2706:
1924:
is in the direction of one of the axes. So, denoting the relative permittivities in the
2665:
2644:
1967:
1947:
1927:
1815:
1253:
895:
745:
214:
194:
174:
2429:
Or sometimes it is convenient to use air instead of a vacuum as the reference medium;
2914:
2836:
2810:
2784:
2783:, 7th Ed., Cambridge University Press, 1999 (reprinted with corrections, 2002),
2454:
for the index surface (p. 87) or the corresponding surface for the wave vector
1223:{\displaystyle v^{2}=a^{2}\cos ^{2}\xi +b^{2}\cos ^{2}\eta +c^{2}\cos ^{2}\zeta \,.}
1425:
1420:
733:
434:
52:
33:
2899:
2848:
2799:
429:
2871:
2745:
2732:
2364:
1448:
1403:
1374:. And as the index ellipsoid generates the index surface, so the surface (
17:
2893:
2354:
45:
2757:
Born & Wolf, 2002, p. 799; Jenkins & White, 1976, p. 560.
1578:
1778:{\displaystyle v=1{\big /}{\sqrt {\mu _{0}\epsilon _{r}\epsilon _{0}}}\,.}
2709:
London: Oxford
University Press Warehouse, 1892; reviewed by "R.T.G." in
2236:), we obtain the equation of the index ellipsoid in the alternative form
2669:
2648:
1902:
2480:
Born & Wolf, 2002, p. 799; Yariv & Yeh, 1984, p. 77.
419:. As the index ellipsoid reduces to a spheroid, the two-sheeted index
2833:
Optical Waves in
Crystals: Propagation and control of laser radiation
442:
merges with the (spherical) index ellipsoid; in short, the medium is
41:
2604:
2577:
2744:
Landau & Lifshitz, 1960, pp. 251–3 (§60). The authors use
2728:
1424:
which MacCullagh later called the "ellipsoid of indices". In 1891,
445:
412:
1512:{\displaystyle c_{0}=1{\big /}{\sqrt {\mu _{0}\epsilon _{0}}}\,,}
425:
64:
48:
2699:
Mineralogical
Magazine and Journal of the Mineralogical Society
2599:
2572:
77:. The principal semiaxes of the index ellipsoid are called the
2590:
A. Fresnel, "Extrait d'un Mémoire sur la double réfraction",
2369:
2032:{\displaystyle \epsilon _{x}\;\!,\epsilon _{y},\epsilon _{z}}
392:. Thus, paradoxically, if the index ellipsoid of a medium is
32:) is a geometric construction which concisely represents the
2610:
Bulletin des
Sciences par la Société Philomatique de Paris
1900:
as a scalar, which is valid in an isotropic medium. In an
2713:, vol. 46, no. 1199 (20 Oct. 1892),
2852:, New York: Wiley (reprinted Mineola, NY: Dover, 2006).
2798:, Paris: Imprimerie Impériale (3 vols., 1866–70),
2571:
of the
Supplement to the Memoir on double refraction",
2748:, in which the magnetic permeability of a vacuum is 1.
2664:, vol. 17 (nominally for 1831), pp. 241–63,
2608:, 2022. (An earlier version of this paper appeared in
2096:
denote the refractive indices for these directions of
2855:
2242:
2108:
2049:
1990:
1970:
1950:
1930:
1879:
1838:
1818:
1791:
1720:
1683:
1641:
1614:
1587:
1552:
1525:
1457:
1370:) is generally not an ellipsoid, but another sort of
1330:
1298:
1256:
1118:
1081:
1027:
976:
925:
898:
842:
809:
768:
748:
712:
686:
660:
486:
242:
217:
197:
177:
124:
1380:), by the same process, generates what we call the
231:directions, the equation of the index ellipsoid is
107:The index ellipsoid is not to be confused with the
2446:. Born & Wolf (2002, p. 803) call it the
2333:
2222:
2088:
2031:
1976:
1956:
1936:
1892:
1865:
1824:
1804:
1777:
1696:
1669:
1627:
1600:
1565:
1538:
1511:
1356:
1316:
1262:
1222:
1099:
1061:
1013:
962:
904:
884:
822:
795:
754:
724:
698:
672:
630:
348:
223:
203:
183:
163:
2563:
2561:
2002:
1663:
1435:
40:of light, as functions of the orientation of the
2891:
2654:
2633:
1670:{\displaystyle \epsilon _{r}\epsilon _{0}\;\!,}
2612:, vol. 9, pp. 63–71, May 1822.)
2584:
2558:
2691:
2809:, 4th Ed., New York: McGraw-Hill,
2537:Yariv & Yeh, 1984, pp. 82, 84.
1866:{\displaystyle n={\sqrt {\epsilon _{r}}}\,.}
51:(provided that the crystal does not exhibit
2735:, which change the forms of some equations.
2489:Jenkins & White, 1976, pp. 560–61.
2507:Zernike & Midwinter, 1973, p. 11.
2450:. But Yariv & Yeh (1984) use the term
2001:
1662:
1324:as the components of velocity and putting
2603:
2576:
2516:Landau & Lifshitz, 1960, p. 326.
2433:Zernike & Midwinter, 1973, p. 2.
2341:which explains why it is also called the
2327:
1912:is parallel to the electric field vector
1859:
1771:
1608:(especially at optical frequencies), but
1505:
1216:
624:
428:, for example, the index ellipsoid is an
388:, and the medium is therefore said to be
342:
2662:Transactions of the Royal Irish Academy
2643:, vol. 18 (1839), pp. 31–74,
2641:Transactions of the Royal Irish Academy
2892:
2684:, no. 49 (13 January 1845),
2682:Proceedings of the Royal Irish Academy
1364:, etc.; thus the latter surface (
2846:F. Zernike and J.E. Midwinter, 1973,
2796:Oeuvres complètes d'Augustin Fresnel
2598:of a memoir on double refraction",
2375:Mathematical descriptions of opacity
1109:
477:
233:
2822:Electrodynamics of Continuous Media
2805:F.A. Jenkins and H.E. White, 1976,
2630:Born & Wolf, 2002, p. 802.
2555:Born & Wolf, 2002, p. 805.
411:index), the ellipsoid reduces to a
396:axial, the medium itself is called
13:
2731:; the others use the lesser-known
1832:, we obtain the refractive index:
1714:), so that the wave speed becomes
1280:Whereas the surface described by (
1075:) and canceling the common factor
885:{\displaystyle n_{a},n_{b},n_{c},}
14:
2931:
2089:{\displaystyle n_{a},n_{b},n_{c}}
1271:now recognize as oscillations of
1069:. Making these substitutions in (
164:{\displaystyle n_{a},n_{b},n_{c}}
2877:
2865:
2592:Annales de Chimie et de Physique
407:index, and the third length the
2773:
2760:
2751:
2738:
2720:
2675:
2624:
2615:
2549:
2540:
2436:
2423:
2413:
2528:
2519:
2510:
2501:
2492:
2483:
2474:
2400:
2387:
2041:principal dielectric constants
1573:are respectively the magnetic
1436:Electromagnetic interpretation
1014:{\displaystyle n_{b}=c_{0}/b,}
963:{\displaystyle n_{a}=c_{0}/a,}
102:
1:
2826:Course of Theoretical Physics
2467:
1893:{\displaystyle \epsilon _{r}}
1697:{\displaystyle \epsilon _{r}}
1628:{\displaystyle \epsilon _{0}}
1566:{\displaystyle \epsilon _{0}}
1357:{\displaystyle \cos \xi =x/v}
1246:This equation was derived by
1062:{\displaystyle n_{c}=c_{0}/c}
79:principal refractive indices
7:
2831:A. Yariv and P. Yeh, 1984,
2779:M. Born and E. Wolf, 2002,
2348:
2232:
1386:
1376:
1366:
1288:
1282:
1236:
1071:
725:{\displaystyle \cos \zeta }
644:
465:
362:
92:to that semiaxis, with the
10:
2936:
2828:), London: Pergamon Press.
1707:relative permittivity
1428:called this ellipsoid the
1384:. Hence the surface (
736:of the radius vector. But
699:{\displaystyle \cos \eta }
452:
372:If the index ellipsoid is
2406:Or, in older literature,
2393:Or, in older literature,
1100:{\displaystyle c_{0}^{2}}
796:{\displaystyle n=c_{0}/v}
673:{\displaystyle \cos \xi }
30:dielectric ellipsoid
2849:Applied Nonlinear Optics
2380:
2360:Complex refractive index
1712:dielectric constant
1601:{\displaystyle \mu _{0}}
1539:{\displaystyle \mu _{0}}
2800:vol. 2 (1868)
2444:normal-velocity surface
1873:This derivation treats
1430:optical indicatrix
1382:normal-velocity surface
34:refractive indices
26:optical indicatrix
2807:Fundamentals of Optics
2672:, at p. 260.
2335:
2224:
2090:
2043:), and recalling that
2033:
1978:
1958:
1938:
1894:
1867:
1826:
1806:
1779:
1698:
1671:
1629:
1602:
1567:
1540:
1513:
1358:
1318:
1264:
1224:
1101:
1063:
1015:
964:
919:respectively, so that
906:
886:
824:
797:
756:
734:direction cosines
726:
700:
674:
632:
350:
225:
205:
185:
165:
61:diametral section
2651:, at p. 38.
2336:
2225:
2091:
2034:
1979:
1959:
1939:
1895:
1868:
1827:
1807:
1805:{\displaystyle c_{0}}
1780:
1699:
1672:
1630:
1603:
1568:
1541:
1514:
1426:Lazarus Fletcher
1421:James MacCullagh
1392:surface of elasticity
1359:
1319:
1317:{\displaystyle x,y,z}
1265:
1248:Augustin-Jean Fresnel
1225:
1102:
1064:
1016:
965:
907:
887:
825:
823:{\displaystyle c_{0}}
798:
757:
727:
701:
675:
633:
351:
226:
206:
186:
166:
69:electric displacement
53:optical rotation
2910:Polarization (waves)
2781:Principles of Optics
2240:
2106:
2047:
1988:
1968:
1948:
1928:
1877:
1836:
1816:
1789:
1718:
1681:
1639:
1635:must be replaced by
1612:
1585:
1550:
1523:
1455:
1328:
1296:
1254:
1250:in January 1822. If
1116:
1079:
1025:
974:
923:
896:
840:
807:
766:
746:
710:
684:
658:
484:
430:oblate spheroid
240:
215:
195:
175:
122:
57:central section
28:or sometimes as the
22:index ellipsoid
2835:, New York: Wiley,
2733:Gaussian units
2452:normal surface
2448:normal surface
1404:electric field
1096:
601:
561:
521:
333:
301:
269:
115:to that direction.
24:(also known as the
2920:Optical mineralogy
2884:History of science
2705:(Dec. 1891);
2331:
2220:
2086:
2029:
1974:
1954:
1934:
1890:
1863:
1822:
1802:
1775:
1694:
1667:
1625:
1598:
1563:
1536:
1509:
1417:index surface
1400:ray ellipsoid
1354:
1314:
1260:
1220:
1097:
1082:
1059:
1011:
960:
902:
882:
820:
793:
752:
722:
696:
670:
628:
587:
547:
507:
382:binormal axes
346:
319:
287:
255:
221:
201:
181:
161:
109:index surface
2789:978-0-521-64222-4
2319:
2292:
2265:
2216:
2212:
2186:
2183:
2177:
2173:
2147:
2144:
2138:
2134:
1977:{\displaystyle z}
1957:{\displaystyle y}
1937:{\displaystyle x}
1857:
1825:{\displaystyle v}
1769:
1710:(also called the
1577:and the electric
1503:
1263:{\displaystyle v}
1244:
1243:
905:{\displaystyle v}
836:takes the values
755:{\displaystyle v}
652:
651:
622:
602:
562:
522:
370:
369:
334:
302:
270:
224:{\displaystyle z}
204:{\displaystyle y}
184:{\displaystyle x}
46:doubly-refractive
2927:
2882:
2881:
2880:
2870:
2869:
2861:
2824:(vol. 8 of
2767:
2764:
2758:
2755:
2749:
2742:
2736:
2724:
2718:
2703:pp. 278–388
2695:
2689:
2679:
2673:
2658:
2652:
2637:
2631:
2628:
2622:
2619:
2613:
2607:
2588:
2582:
2580:
2565:
2556:
2553:
2547:
2544:
2538:
2532:
2526:
2523:
2517:
2514:
2508:
2505:
2499:
2496:
2490:
2487:
2481:
2478:
2461:
2459:
2440:
2434:
2427:
2421:
2417:
2411:
2404:
2398:
2391:
2340:
2338:
2337:
2332:
2320:
2318:
2317:
2308:
2307:
2298:
2293:
2291:
2290:
2281:
2280:
2271:
2266:
2264:
2263:
2254:
2253:
2244:
2229:
2227:
2226:
2221:
2214:
2213:
2211:
2210:
2201:
2196:
2195:
2184:
2181:
2175:
2174:
2172:
2171:
2162:
2157:
2156:
2145:
2142:
2136:
2135:
2133:
2132:
2123:
2118:
2117:
2101:
2095:
2093:
2092:
2087:
2085:
2084:
2072:
2071:
2059:
2058:
2038:
2036:
2035:
2030:
2028:
2027:
2015:
2014:
2000:
1999:
1983:
1981:
1980:
1975:
1963:
1961:
1960:
1955:
1943:
1941:
1940:
1935:
1923:
1917:
1911:
1899:
1897:
1896:
1891:
1889:
1888:
1872:
1870:
1869:
1864:
1858:
1856:
1855:
1846:
1831:
1829:
1828:
1823:
1811:
1809:
1808:
1803:
1801:
1800:
1784:
1782:
1781:
1776:
1770:
1768:
1767:
1758:
1757:
1748:
1747:
1738:
1736:
1735:
1703:
1701:
1700:
1695:
1693:
1692:
1676:
1674:
1673:
1668:
1661:
1660:
1651:
1650:
1634:
1632:
1631:
1626:
1624:
1623:
1607:
1605:
1604:
1599:
1597:
1596:
1572:
1570:
1569:
1564:
1562:
1561:
1545:
1543:
1542:
1537:
1535:
1534:
1518:
1516:
1515:
1510:
1504:
1502:
1501:
1492:
1491:
1482:
1480:
1479:
1467:
1466:
1411:
1363:
1361:
1360:
1355:
1350:
1323:
1321:
1320:
1315:
1276:
1269:
1267:
1266:
1261:
1238:
1229:
1227:
1226:
1221:
1209:
1208:
1199:
1198:
1180:
1179:
1170:
1169:
1151:
1150:
1141:
1140:
1128:
1127:
1110:
1106:
1104:
1103:
1098:
1095:
1090:
1068:
1066:
1065:
1060:
1055:
1050:
1049:
1037:
1036:
1020:
1018:
1017:
1012:
1004:
999:
998:
986:
985:
969:
967:
966:
961:
953:
948:
947:
935:
934:
918:
912:take the values
911:
909:
908:
903:
891:
889:
888:
883:
878:
877:
865:
864:
852:
851:
835:
829:
827:
826:
821:
819:
818:
802:
800:
799:
794:
789:
784:
783:
761:
759:
758:
753:
741:
731:
729:
728:
723:
705:
703:
702:
697:
679:
677:
676:
671:
646:
637:
635:
634:
629:
623:
621:
620:
608:
603:
600:
595:
586:
579:
578:
568:
563:
560:
555:
546:
539:
538:
528:
523:
520:
515:
506:
499:
498:
488:
478:
474:
462:
364:
355:
353:
352:
347:
335:
332:
327:
318:
317:
308:
303:
300:
295:
286:
285:
276:
271:
268:
263:
254:
253:
244:
234:
230:
228:
227:
222:
210:
208:
207:
202:
190:
188:
187:
182:
170:
168:
167:
162:
160:
159:
147:
146:
134:
133:
97:
76:
2935:
2934:
2930:
2929:
2928:
2926:
2925:
2924:
2905:Physical optics
2890:
2889:
2888:
2878:
2876:
2864:
2856:
2776:
2771:
2770:
2765:
2761:
2756:
2752:
2743:
2739:
2725:
2721:
2701:, vol. 9,
2696:
2692:
2680:
2676:
2659:
2655:
2638:
2634:
2629:
2625:
2620:
2616:
2589:
2585:
2566:
2559:
2554:
2550:
2545:
2541:
2533:
2529:
2524:
2520:
2515:
2511:
2506:
2502:
2497:
2493:
2488:
2484:
2479:
2475:
2470:
2465:
2464:
2455:
2441:
2437:
2428:
2424:
2418:
2414:
2405:
2401:
2392:
2388:
2383:
2351:
2313:
2309:
2303:
2299:
2297:
2286:
2282:
2276:
2272:
2270:
2259:
2255:
2249:
2245:
2243:
2241:
2238:
2237:
2206:
2202:
2200:
2191:
2187:
2167:
2163:
2161:
2152:
2148:
2128:
2124:
2122:
2113:
2109:
2107:
2104:
2103:
2102:, we must have
2097:
2080:
2076:
2067:
2063:
2054:
2050:
2048:
2045:
2044:
2039:(the so-called
2023:
2019:
2010:
2006:
1995:
1991:
1989:
1986:
1985:
1969:
1966:
1965:
1949:
1946:
1945:
1929:
1926:
1925:
1919:
1913:
1907:
1884:
1880:
1878:
1875:
1874:
1851:
1847:
1845:
1837:
1834:
1833:
1817:
1814:
1813:
1796:
1792:
1790:
1787:
1786:
1763:
1759:
1753:
1749:
1743:
1739:
1737:
1731:
1730:
1719:
1716:
1715:
1688:
1684:
1682:
1679:
1678:
1656:
1652:
1646:
1642:
1640:
1637:
1636:
1619:
1615:
1613:
1610:
1609:
1592:
1588:
1586:
1583:
1582:
1557:
1553:
1551:
1548:
1547:
1530:
1526:
1524:
1521:
1520:
1497:
1493:
1487:
1483:
1481:
1475:
1474:
1462:
1458:
1456:
1453:
1452:
1451:in a vacuum is
1438:
1407:
1346:
1329:
1326:
1325:
1297:
1294:
1293:
1272:
1255:
1252:
1251:
1204:
1200:
1194:
1190:
1175:
1171:
1165:
1161:
1146:
1142:
1136:
1132:
1123:
1119:
1117:
1114:
1113:
1091:
1086:
1080:
1077:
1076:
1051:
1045:
1041:
1032:
1028:
1026:
1023:
1022:
1000:
994:
990:
981:
977:
975:
972:
971:
949:
943:
939:
930:
926:
924:
921:
920:
913:
897:
894:
893:
873:
869:
860:
856:
847:
843:
841:
838:
837:
831:
814:
810:
808:
805:
804:
785:
779:
775:
767:
764:
763:
747:
744:
743:
737:
711:
708:
707:
685:
682:
681:
659:
656:
655:
616:
612:
607:
596:
591:
574:
570:
569:
567:
556:
551:
534:
530:
529:
527:
516:
511:
494:
490:
489:
487:
485:
482:
481:
470:
458:
455:
328:
323:
313:
309:
307:
296:
291:
281:
277:
275:
264:
259:
249:
245:
243:
241:
238:
237:
216:
213:
212:
196:
193:
192:
176:
173:
172:
155:
151:
142:
138:
129:
125:
123:
120:
119:
105:
93:
72:
36:and associated
12:
11:
5:
2933:
2923:
2922:
2917:
2912:
2907:
2902:
2887:
2886:
2874:
2854:
2853:
2844:
2829:
2818:
2803:
2792:
2775:
2772:
2769:
2768:
2759:
2750:
2746:Gaussian units
2737:
2719:
2715:pp. 581–2
2690:
2686:pp. 49–51
2674:
2653:
2632:
2623:
2614:
2583:
2557:
2548:
2539:
2527:
2518:
2509:
2500:
2491:
2482:
2472:
2471:
2469:
2466:
2463:
2462:
2435:
2422:
2412:
2399:
2385:
2384:
2382:
2379:
2378:
2377:
2372:
2367:
2365:Crystal optics
2362:
2357:
2350:
2347:
2330:
2326:
2323:
2316:
2312:
2306:
2302:
2296:
2289:
2285:
2279:
2275:
2269:
2262:
2258:
2252:
2248:
2219:
2209:
2205:
2199:
2194:
2190:
2180:
2170:
2166:
2160:
2155:
2151:
2141:
2131:
2127:
2121:
2116:
2112:
2083:
2079:
2075:
2070:
2066:
2062:
2057:
2053:
2026:
2022:
2018:
2013:
2009:
2005:
1998:
1994:
1984:directions by
1973:
1953:
1933:
1887:
1883:
1862:
1854:
1850:
1844:
1841:
1821:
1799:
1795:
1774:
1766:
1762:
1756:
1752:
1746:
1742:
1734:
1729:
1726:
1723:
1691:
1687:
1666:
1659:
1655:
1649:
1645:
1622:
1618:
1595:
1591:
1560:
1556:
1533:
1529:
1508:
1500:
1496:
1490:
1486:
1478:
1473:
1470:
1465:
1461:
1449:speed of light
1437:
1434:
1419:was coined by
1353:
1349:
1345:
1342:
1339:
1336:
1333:
1313:
1310:
1307:
1304:
1301:
1259:
1242:
1241:
1232:
1230:
1219:
1215:
1212:
1207:
1203:
1197:
1193:
1189:
1186:
1183:
1178:
1174:
1168:
1164:
1160:
1157:
1154:
1149:
1145:
1139:
1135:
1131:
1126:
1122:
1094:
1089:
1085:
1058:
1054:
1048:
1044:
1040:
1035:
1031:
1010:
1007:
1003:
997:
993:
989:
984:
980:
959:
956:
952:
946:
942:
938:
933:
929:
901:
881:
876:
872:
868:
863:
859:
855:
850:
846:
817:
813:
792:
788:
782:
778:
774:
771:
751:
721:
718:
715:
695:
692:
689:
669:
666:
663:
650:
649:
640:
638:
627:
619:
615:
611:
606:
599:
594:
590:
585:
582:
577:
573:
566:
559:
554:
550:
545:
542:
537:
533:
526:
519:
514:
510:
505:
502:
497:
493:
454:
451:
368:
367:
358:
356:
345:
341:
338:
331:
326:
322:
316:
312:
306:
299:
294:
290:
284:
280:
274:
267:
262:
258:
252:
248:
220:
200:
180:
158:
154:
150:
145:
141:
137:
132:
128:
104:
101:
18:crystal optics
9:
6:
4:
3:
2:
2932:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2901:
2898:
2897:
2895:
2885:
2875:
2873:
2868:
2863:
2862:
2859:
2851:
2850:
2845:
2842:
2841:0-471-09142-1
2838:
2834:
2830:
2827:
2823:
2819:
2816:
2815:0-07-032330-5
2812:
2808:
2804:
2801:
2797:
2793:
2790:
2786:
2782:
2778:
2777:
2763:
2754:
2747:
2741:
2734:
2730:
2729:SI units
2723:
2716:
2712:
2708:
2704:
2700:
2694:
2687:
2683:
2678:
2671:
2667:
2663:
2657:
2650:
2646:
2642:
2636:
2627:
2618:
2611:
2606:
2601:
2597:
2593:
2587:
2579:
2574:
2570:
2564:
2562:
2552:
2543:
2536:
2531:
2522:
2513:
2504:
2495:
2486:
2477:
2473:
2460:(p. 73).
2458:
2453:
2449:
2445:
2439:
2432:
2426:
2416:
2409:
2403:
2396:
2390:
2386:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2355:Birefringence
2353:
2352:
2346:
2344:
2328:
2324:
2321:
2314:
2310:
2304:
2300:
2294:
2287:
2283:
2277:
2273:
2267:
2260:
2256:
2250:
2246:
2235:
2234:
2217:
2207:
2203:
2197:
2192:
2188:
2178:
2168:
2164:
2158:
2153:
2149:
2139:
2129:
2125:
2119:
2114:
2110:
2100:
2081:
2077:
2073:
2068:
2064:
2060:
2055:
2051:
2042:
2024:
2020:
2016:
2011:
2007:
2003:
1996:
1992:
1971:
1951:
1931:
1922:
1916:
1910:
1905:
1904:
1885:
1881:
1860:
1852:
1848:
1842:
1839:
1819:
1797:
1793:
1772:
1764:
1760:
1754:
1750:
1744:
1740:
1727:
1724:
1721:
1713:
1709:
1708:
1689:
1685:
1664:
1657:
1653:
1647:
1643:
1620:
1616:
1593:
1589:
1580:
1576:
1558:
1554:
1531:
1527:
1506:
1498:
1494:
1488:
1484:
1471:
1468:
1463:
1459:
1450:
1445:
1443:
1433:
1431:
1427:
1422:
1418:
1413:
1410:
1405:
1401:
1395:
1393:
1389:
1388:
1383:
1379:
1378:
1373:
1369:
1368:
1351:
1347:
1343:
1340:
1337:
1334:
1331:
1311:
1308:
1305:
1302:
1299:
1291:
1290:
1285:
1284:
1278:
1275:
1257:
1249:
1240:
1233:
1231:
1217:
1213:
1210:
1205:
1201:
1195:
1191:
1187:
1184:
1181:
1176:
1172:
1166:
1162:
1158:
1155:
1152:
1147:
1143:
1137:
1133:
1129:
1124:
1120:
1112:
1111:
1108:
1092:
1087:
1083:
1074:
1073:
1056:
1052:
1046:
1042:
1038:
1033:
1029:
1008:
1005:
1001:
995:
991:
987:
982:
978:
957:
954:
950:
944:
940:
936:
931:
927:
916:
899:
879:
874:
870:
866:
861:
857:
853:
848:
844:
834:
815:
811:
790:
786:
780:
776:
772:
769:
749:
740:
735:
719:
716:
713:
693:
690:
687:
667:
664:
661:
648:
641:
639:
625:
617:
613:
609:
604:
597:
592:
588:
583:
580:
575:
571:
564:
557:
552:
548:
543:
540:
535:
531:
524:
517:
512:
508:
503:
500:
495:
491:
480:
479:
476:
473:
468:
467:
461:
450:
448:
447:
439:
437:
436:
431:
427:
422:
418:
414:
410:
409:extraordinary
406:
401:
399:
395:
391:
387:
383:
379:
375:
366:
359:
357:
343:
339:
336:
329:
324:
320:
314:
310:
304:
297:
292:
288:
282:
278:
272:
265:
260:
256:
250:
246:
236:
235:
232:
218:
198:
178:
156:
152:
148:
143:
139:
135:
130:
126:
116:
114:
110:
100:
96:
91:
90:perpendicular
87:
82:
80:
75:
70:
66:
62:
58:
54:
50:
47:
43:
39:
38:polarizations
35:
31:
27:
23:
19:
2847:
2832:
2825:
2821:
2806:
2795:
2780:
2774:Bibliography
2762:
2753:
2740:
2722:
2710:
2698:
2693:
2681:
2677:
2661:
2656:
2640:
2635:
2626:
2617:
2609:
2595:
2591:
2586:
2568:
2551:
2542:
2534:
2530:
2521:
2512:
2503:
2494:
2485:
2476:
2456:
2451:
2447:
2443:
2438:
2430:
2425:
2415:
2407:
2402:
2394:
2389:
2342:
2231:
2098:
2040:
1920:
1914:
1908:
1901:
1711:
1705:
1579:permittivity
1575:permeability
1446:
1441:
1439:
1429:
1416:
1414:
1408:
1399:
1396:
1391:
1385:
1381:
1375:
1371:
1365:
1287:
1281:
1279:
1273:
1245:
1234:
1107:, we obtain
1070:
914:
832:
738:
653:
642:
471:
464:
459:
456:
443:
440:
433:
420:
416:
408:
404:
402:
397:
393:
389:
385:
381:
377:
373:
371:
360:
117:
112:
108:
106:
94:
89:
85:
83:
78:
73:
60:
56:
29:
25:
21:
15:
2345:ellipsoid.
1903:anisotropic
103:Terminology
2894:Categories
2468:References
2343:dielectric
762:, we have
444:optically
386:optic axes
118:If we let
2707:reprinted
2311:ϵ
2284:ϵ
2257:ϵ
2204:ϵ
2165:ϵ
2126:ϵ
2021:ϵ
2008:ϵ
1993:ϵ
1882:ϵ
1849:ϵ
1785:Dividing
1761:ϵ
1751:ϵ
1741:μ
1686:ϵ
1654:ϵ
1644:ϵ
1617:ϵ
1590:μ
1555:ϵ
1528:μ
1495:ϵ
1485:μ
1415:The term
1338:ξ
1335:
1214:ζ
1211:
1185:η
1182:
1156:ξ
1153:
720:ζ
717:
694:η
691:
668:ξ
665:
584:ζ
581:
544:η
541:
504:ξ
501:
446:isotropic
42:wavefront
2915:Surfaces
2670:30078792
2649:30078974
2349:See also
803:, where
732:are the
417:uniaxial
413:spheroid
405:ordinary
374:triaxial
63:) is an
2872:Physics
2858:Portals
2605:5442206
2602::
2596:Extrait
2581:, 2022.
2578:5886692
2575::
2569:Extrait
2408:uniaxal
1704:is the
1406:vector
1372:ovaloid
970:
453:History
435:prolate
426:calcite
421:surface
400:axial.
390:biaxial
71:vector
65:ellipse
49:crystal
44:, in a
2900:Optics
2839:
2813:
2787:
2711:Nature
2668:
2647:
2600:Zenodo
2573:Zenodo
2395:biaxal
2215:
2185:
2182:
2176:
2146:
2143:
2137:
1964:, and
1677:where
1519:where
706:, and
654:where
475:gives
378:normal
211:, and
113:normal
20:, the
2666:JSTOR
2645:JSTOR
2381:Notes
2370:D-DIA
1442:Given
915:a,b,c
469:) by
2837:ISBN
2811:ISBN
2785:ISBN
1546:and
1447:The
1021:and
892:let
2535:Cf.
2431:cf.
1812:by
1332:cos
1277:).
1202:cos
1173:cos
1144:cos
714:cos
688:cos
662:cos
572:cos
532:cos
492:cos
394:tri
384:or
86:not
59:or
16:In
2896::
2560:^
1944:,
1432:.
1412:.
680:,
398:bi
191:,
81:.
2860::
2843:.
2817:.
2802:.
2791:.
2717:.
2688:.
2457:k
2410:.
2397:.
2329:,
2325:1
2322:=
2315:z
2305:2
2301:z
2295:+
2288:y
2278:2
2274:y
2268:+
2261:x
2251:2
2247:x
2233:1
2218:,
2208:z
2198:=
2193:c
2189:n
2179:;
2169:y
2159:=
2154:b
2150:n
2140:;
2130:x
2120:=
2115:a
2111:n
2099:D
2082:c
2078:n
2074:,
2069:b
2065:n
2061:,
2056:a
2052:n
2025:z
2017:,
2012:y
2004:,
1997:x
1972:z
1952:y
1932:x
1921:D
1915:E
1909:D
1886:r
1861:.
1853:r
1843:=
1840:n
1820:v
1798:0
1794:c
1773:.
1765:0
1755:r
1745:0
1733:/
1728:1
1725:=
1722:v
1690:r
1665:,
1658:0
1648:r
1621:0
1594:0
1559:0
1532:0
1507:,
1499:0
1489:0
1477:/
1472:1
1469:=
1464:0
1460:c
1409:E
1387:3
1377:3
1367:3
1352:v
1348:/
1344:x
1341:=
1312:z
1309:,
1306:y
1303:,
1300:x
1289:3
1283:1
1274:D
1258:v
1239:)
1237:3
1235:(
1218:.
1206:2
1196:2
1192:c
1188:+
1177:2
1167:2
1163:b
1159:+
1148:2
1138:2
1134:a
1130:=
1125:2
1121:v
1093:2
1088:0
1084:c
1072:2
1057:c
1053:/
1047:0
1043:c
1039:=
1034:c
1030:n
1009:,
1006:b
1002:/
996:0
992:c
988:=
983:b
979:n
958:,
955:a
951:/
945:0
941:c
937:=
932:a
928:n
917:,
900:v
880:,
875:c
871:n
867:,
862:b
858:n
854:,
849:a
845:n
833:n
816:0
812:c
791:v
787:/
781:0
777:c
773:=
770:n
750:v
739:n
647:)
645:2
643:(
626:,
618:2
614:n
610:1
605:=
598:2
593:c
589:n
576:2
565:+
558:2
553:b
549:n
536:2
525:+
518:2
513:a
509:n
496:2
472:n
466:1
460:n
365:)
363:1
361:(
344:.
340:1
337:=
330:2
325:c
321:n
315:2
311:z
305:+
298:2
293:b
289:n
283:2
279:y
273:+
266:2
261:a
257:n
251:2
247:x
219:z
199:y
179:x
157:c
153:n
149:,
144:b
140:n
136:,
131:a
127:n
95:D
74:D
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