25:
912:
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1823:
cause a variation of a medium's permittivity tensor when an external electric field is applied, proportional (to lowest order) to the strength of the field. This causes a rotation of the principal axes of the medium and alters the behaviour of light travelling through it; the effect can be used to
426:
907:{\displaystyle {\begin{pmatrix}P_{x}\\P_{y}\\P_{z}\end{pmatrix}}=\varepsilon _{0}{\begin{pmatrix}\chi _{xx}&\chi _{xy}&\chi _{xz}\\\chi _{yx}&\chi _{yy}&\chi _{yz}\\\chi _{zx}&\chi _{zy}&\chi _{zz}\end{pmatrix}}{\begin{pmatrix}E_{x}\\E_{y}\\E_{z}\end{pmatrix}}}
1440:{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} =\varepsilon _{0}\mathbf {E} +\varepsilon _{0}{\boldsymbol {\chi }}\mathbf {E} =\varepsilon _{0}(I+{\boldsymbol {\chi }})\mathbf {E} =\varepsilon _{0}{\boldsymbol {\varepsilon }}\mathbf {E} .}
1018:
326:
604:
555:. In a physical picture, this can be thought of as the dipoles induced in the medium by the electric field having certain preferred directions, related to the physical structure of the crystal. This can be written as:
1711:
1592:
204:
1803:. Light polarised at some angle to the axes will experience a different phase velocity for different polarization components, and cannot be described by a single index of refraction. This is often depicted as an
301:
480:
1270:
1206:
1142:
537:
1850:
A dielectric tensor that is not
Hermitian gives rise to complex eigenvalues, which corresponds to a material with gain or absorption at a particular frequency.
421:{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\chi \varepsilon _{0}\mathbf {E} =\varepsilon _{0}(1+\chi )\mathbf {E} =\varepsilon \mathbf {E} }
922:
561:
1843:
are complex-valued vectors, corresponding to elliptically polarized light, and time-reversal symmetry can be broken. This can be used to design
233:
present in the medium). Physically, the polarization field can be regarded as the response of the medium to the electric field of the light.
1603:
1484:
161:
1892:
Amnon Yariv, Pochi Yeh. (2006). Photonics optical electronics in modern communications (6th ed.). Oxford
University Press. pp. 30-31.
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Thus these waves will see two different refractive indices and travel at different speeds. This phenomenon is known as
1919:
68:
46:
133:, which means that light behaves the same way no matter which direction it is travelling in the medium. In terms of
39:
1840:
222:
1066:
the tensor by choosing the appropriate set of coordinate axes, zeroing all components of the tensor except χ
107:. The index of refraction depends on both composition and crystal structure and can be calculated using the
506:
142:
108:
1470:
such the electric field of the wave is parallel to the x-axis. The wave experiences a susceptibility χ
1929:
1924:
33:
1466:
of the medium must also be a tensor. Consider a light wave propagating along the z principal axis
1283:
tensor are real, corresponding to a case in which the refractive index is real in all directions.
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134:
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8:
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1013:{\displaystyle P_{i}=\varepsilon _{0}\sum _{j\in \{x,y,z\}}\chi _{ij}E_{j}\quad .}
1904:
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1804:
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599:{\displaystyle \mathbf {P} =\varepsilon _{0}{\boldsymbol {\chi }}\mathbf {E} .}
149:
112:
103:) in which light behaves differently depending on which direction the light is
1780:) for light polarised in the x or y directions, and an "extraordinary" index (
115:) it is possible to induce anisotropy by applying an external electric field.
1913:
1859:
1718:
1279:
of the medium. Note that these axes will be orthogonal if all entries in the
1787:) for polarisation in the z direction. A uniaxial crystal is "positive" if n
1773:. A uniaxial crystal exhibits two refractive indices, an "ordinary" index (
226:
214:
94:
138:
111:. Crystals are often naturally anisotropic, and in some media (such as
1706:{\displaystyle n_{yy}=(1+\chi _{yy})^{1/2}=(\varepsilon _{yy})^{1/2}.}
1587:{\displaystyle n_{xx}=(1+\chi _{xx})^{1/2}=(\varepsilon _{xx})^{1/2}.}
199:{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }
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242:
129:
547:
In an anisotropic medium, such as a crystal, the polarisation field
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100:
551:
is not necessarily aligned with the electric field of the light
1727:
614:
246:
85:
1831:, some materials can have a dielectric tensor that is complex-
296:{\displaystyle \mathbf {P} =\chi \varepsilon _{0}\mathbf {E} }
124:
89:
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The directions x, y and z are in this case known as the
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635:
475:{\displaystyle \varepsilon =\varepsilon _{0}(1+\chi )}
1606:
1487:
1303:
1265:{\displaystyle P_{z}=\varepsilon _{0}\chi _{zz}E_{z}}
1215:
1201:{\displaystyle P_{y}=\varepsilon _{0}\chi _{yy}E_{y}}
1151:
1137:{\displaystyle P_{x}=\varepsilon _{0}\chi _{xx}E_{x}}
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253:is proportional and parallel to the electric field
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489:of the medium. The value 1+χ is called the
1722:and occurs in some common crystals such as
1038:In nonmagnetic and transparent materials, χ
621:. In terms of components in 3 dimensions:
236:
1597:For a wave polarised in the y direction:
69:Learn how and when to remove this message
141:, this gives a relationship between the
32:This article includes a list of general
1425:
1399:
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1912:
493:of the medium, and is related to the
1835:; this is called a gyro-magnetic or
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312:of the medium. The relation between
18:
1078:. This gives the set of relations:
916:or using the summation convention:
532:{\displaystyle n={\sqrt {1+\chi }}}
13:
123:Typical transparent media such as
118:
38:it lacks sufficient corresponding
14:
1941:
1905:A virtual polarization microscope
1898:
1031:is not necessarily colinear with
1810:
1478:. The refractive index is thus:
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1407:
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1331:
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1305:
613:is not a number as before but a
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249:medium, this polarization field
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88:that describes the behaviour of
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1294:are also related by a tensor:
619:electric susceptibility tensor
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399:
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1:
1880:
500:, for non-magnetic media, by
16:Sub-branch of Optical Physics
1456:relative permittivity tensor
7:
1853:
143:electric displacement field
10:
1946:
1824:produce light modulators.
1745:, the crystal is known as
99:, that is, media (such as
1062:, it is thus possible to
1058:. In accordance with the
1920:Condensed matter physics
1751:Optic axis of a crystal
309:electric susceptibility
237:Electric susceptibility
231:electric dipole moments
109:Gladstone–Dale relation
53:more precise citations.
1819:phenomena such as the
1769:the crystal is called
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1839:. In this case, the
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1837:magneto-optic effect
1821:electro-optic effect
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1474:and a permittivity ε
1462:. Consequently, the
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1795:and "negative" if n
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135:Maxwell's equations
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1827:In response to a
1817:nonlinear optical
1460:dielectric tensor
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543:Anisotropic media
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229:corresponding to
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227:vector field
223:polarization
218:
215:permittivity
208:
152:
145:
128:
122:
93:
81:
80:
65:
59:January 2013
56:
37:
1064:diagonalise
1050:, i.e. the
105:propagating
95:anisotropic
51:introducing
1914:Categories
1881:References
139:dielectric
34:references
1833:Hermitian
1671:ε
1634:χ
1552:ε
1515:χ
1468:polarised
1426:ε
1416:ε
1400:χ
1381:ε
1368:χ
1358:ε
1340:ε
1314:ε
1241:χ
1231:ε
1177:χ
1167:ε
1113:χ
1103:ε
1056:symmetric
985:χ
958:∈
951:∑
941:ε
831:χ
816:χ
801:χ
784:χ
769:χ
754:χ
737:χ
722:χ
707:χ
689:ε
585:χ
575:ε
525:χ
467:χ
449:ε
442:ε
411:ε
397:χ
379:ε
361:ε
357:χ
340:ε
320:is thus:
280:ε
276:χ
243:isotropic
175:ε
130:isotropic
1854:See also
1815:Certain
1749:. (See
1747:uniaxial
148:and the
101:crystals
1771:biaxial
1753:.) If χ
1724:calcite
485:is the
213:is the
209:where ε
125:glasses
47:improve
1799:< n
1791:> n
1728:quartz
1023:Since
615:tensor
431:where
247:linear
241:In an
86:optics
36:, but
1875:Prism
1761:and χ
1450:Here
1074:and χ
609:Here
225:(the
137:in a
97:media
90:light
1733:If χ
1726:and
1290:and
316:and
245:and
127:are
1765:≠ χ
1757:≠ χ
1741:≠ χ
1737:= χ
1458:or
1070:, χ
1044:= χ
92:in
1916::
1807:.
1767:zz
1763:yy
1759:yy
1755:xx
1743:zz
1739:yy
1735:xx
1730:.
1476:xx
1472:xx
1076:zz
1072:yy
1068:xx
1047:ji
1041:ij
1035:.
257::
155::
1801:o
1797:e
1793:o
1789:e
1785:e
1782:n
1778:o
1775:n
1701:.
1696:2
1692:/
1688:1
1684:)
1678:y
1675:y
1667:(
1664:=
1659:2
1655:/
1651:1
1647:)
1641:y
1638:y
1630:+
1627:1
1624:(
1621:=
1616:y
1613:y
1609:n
1582:.
1577:2
1573:/
1569:1
1565:)
1559:x
1556:x
1548:(
1545:=
1540:2
1536:/
1532:1
1528:)
1522:x
1519:x
1511:+
1508:1
1505:(
1502:=
1497:x
1494:x
1490:n
1452:ε
1435:.
1431:E
1420:0
1412:=
1408:E
1404:)
1396:+
1393:I
1390:(
1385:0
1377:=
1373:E
1362:0
1354:+
1350:E
1344:0
1336:=
1332:P
1328:+
1324:E
1318:0
1310:=
1306:D
1292:E
1288:D
1281:χ
1258:z
1254:E
1248:z
1245:z
1235:0
1227:=
1222:z
1218:P
1194:y
1190:E
1184:y
1181:y
1171:0
1163:=
1158:y
1154:P
1130:x
1126:E
1120:x
1117:x
1107:0
1099:=
1094:x
1090:P
1052:χ
1033:E
1029:P
1025:χ
1008:.
1002:j
998:E
992:j
989:i
979:}
976:z
973:,
970:y
967:,
964:x
961:{
955:j
945:0
937:=
932:i
928:P
900:)
892:z
888:E
878:y
874:E
864:x
860:E
853:(
846:)
838:z
835:z
823:y
820:z
808:x
805:z
791:z
788:y
776:y
773:y
761:x
758:y
744:z
741:x
729:y
726:x
714:x
711:x
700:(
693:0
685:=
680:)
672:z
668:P
658:y
654:P
644:x
640:P
633:(
611:χ
594:.
590:E
579:0
571:=
567:P
553:E
549:P
522:+
519:1
514:=
511:n
498:n
470:)
464:+
461:1
458:(
453:0
445:=
415:E
408:=
404:E
400:)
394:+
391:1
388:(
383:0
375:=
371:E
365:0
354:+
350:E
344:0
336:=
332:D
318:E
314:D
290:E
284:0
273:=
269:P
255:E
251:P
219:P
211:0
193:P
189:+
185:E
179:0
171:=
167:D
153:E
146:D
72:)
66:(
61:)
57:(
43:.
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