1916:
1872:
1807:
1162:
67:. Temporal dispersion represents memory effects in systems, commonly seen in optics and electronics. Spatial dispersion on the other hand represents spreading effects and is usually significant only at microscopic length scales. Spatial dispersion contributes relatively small perturbations to optics, giving weak effects such as
1793:
Nearby crystal surfaces and boundaries, it is no longer valid to describe system response in terms of wavevectors. For a full description it is necessary to return to a full nonlocal response function (without translational symmetry), however the end effect can sometimes be described by "additional
1741:
47:
of any signals (such as light or sound) being considered. Since these small spatial structures cannot be resolved by the waves, only indirect effects (e.g. wavevector dependence) remain detectable. An example of spatial dispersion is that of visible light propagating through a crystal such as
79:
The origin of spatial dispersion can be modelled as a nonlocal response, where response to a force field appears at many locations, and can appear even in locations where the force is zero. This usually arises due to a spreading of effects by the hidden microscopic degrees of freedom.
365:
1513:
1845:
in solutions of chiral molecules. In isotropic materials without optical activity, the permittivity tensor can be broken down to transverse and longitudinal components, referring to the response to electric fields either perpendicular or parallel to the wavevector.
56:. In such a case, although the light cannot resolve the individual atoms, they nevertheless can as an aggregate affect how the light propagates. Another common mechanism is that the (e.g.) light is coupled to an excitation of the material, such as a
2022:. The values of the permeability and permittivity are different in this alternative representation, however this leads to no observable differences in real quantities such as electric field, magnetic flux density, magnetic moments, and current.
968:
574:
1568:
1902:
In plasma physics, a wave can be collisionlessly damped by particles in the plasma whose velocity matches the wave's phase velocity. This is typically represented as a spatially dispersive loss in the plasma's permittivity.
1840:
Although symmetry demands that the permittivity is isotropic for zero wavevector, this restriction does not apply for nonzero wavevector. The non-isotropic permittivity for nonzero wavevector leads to effects such as
1348:
833:
744:
448:
909:
1285:
2001:
1788:
1209:
956:
196:
609:
1560:
184:
151:
116:
2071:, especially in solids, spatial dispersion can be significant for wavelengths comparable to the lattice spacing, which typically occurs at very high frequencies (
1790:
can lead to strange phenomena, such as the existence of multiple modes at the same frequency and wavevector direction, but with different wavevector magnitudes.
1157:{\displaystyle {\tilde {\sigma }}(k,\omega )=\int _{-\infty }^{-\infty }dx''\int _{-\infty }^{-\infty }dt''\,e^{-ikx''+i\omega t''}\sigma _{\rm {sym}}(x'',t'').}
653:
633:
1736:{\displaystyle \omega ^{2}\mu _{0}\epsilon ({\vec {k}},\omega ){\vec {E}}-({\vec {k}}\cdot {\vec {k}}){\vec {E}}+({\vec {k}}\cdot {\vec {E}}){\vec {k}}=0.}
2098:
can be seen as a spatial dispersion in the restoring forces, from the "hidden" non-mechanical degree of freedom that is the electromagnetic field.
2095:
40:. Normally, such a dependence is assumed to be absent for simplicity, however spatial dispersion exists to varying degrees in all materials.
2161:
1508:{\displaystyle P_{i}({\vec {k}},\omega )=\sum _{j}(\epsilon _{ij}({\vec {k}},\omega )-\epsilon _{0}\delta _{ij})E_{j}({\vec {k}},\omega ),}
750:
661:
464:
2189:
Agranovich, Vladimir M.; Gartstein, Yu.N. (2006). "REVIEWS OF TOPICAL PROBLEMS: Spatial dispersion and negative refraction of light".
43:
The underlying physical reason for the wavevector dependence is often that the material has some spatial structure smaller than the
2048:
844:
1957:
2232:
Portigal, D. L.; Burstein, E. (1968). "Acoustical
Activity and Other First-Order Spatial Dispersion Effects in Crystals".
186:, but this breaks down if the system has memory (temporal dispersion) or spreading (spatial dispersion). The most general
2056:
52:, where the refractive index depends on the direction of travel (the orientation of the wavevector) with respect to the
2179:
Agranovich & Ginzburg . Crystal Optics with
Spatial Dispersion, and Excitons . 978-3-662-02408-9, 978-3-662-02406-5
2281:
2109:. For example, there is acoustical activity — the rotation of the polarization plane of transverse sound waves — in
1218:
1749:
1533:
inside such crystals. These occur when the following relationship is satisfied for a nonzero electric field vector
1170:
917:
458:
373:
360:{\displaystyle J(x,t)=\int _{-\infty }^{-\infty }dx'\int _{-\infty }^{-\infty }dt'\,\sigma (x,x',t,t')E(x',t'),}
33:
1334:
Inside crystals there may be a combination of spatial dispersion, temporal dispersion, and anisotropy. The
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2153:
1335:
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2198:
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2008:
1951:
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can instead be alternatively represented by a spatially dispersive contribution to permittivity
1536:
160:
1837:
In materials that have no relevant crystalline structure, spatial dispersion can be important.
838:
which yields a remarkably simple relationship between the two plane waves' complex amplitudes:
2257:
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53:
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1308:
1304:
121:
86:
68:
21:
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2110:
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1315:. Spatial dispersion also plays an important role in the understanding of electromagnetic
187:
2245:
2202:
63:
Spatial dispersion can be compared to temporal dispersion, the latter often just called
2145:
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1862:
1288:
638:
618:
1915:
1871:
1806:
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2106:
2004:
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2210:
461:) and invariant in space (space translation symmetry), then we can simplify because
2044:
1320:
1316:
29:
154:
1530:
2014:
What this means is that at nonzero frequency, any contribution to permeability
2137:
612:
44:
37:
2261:
2011:
induced by the electric field, though with a highly dispersive relationship.
2253:
2072:
2068:
1954:. Moreover, since the electric and magnetic fields are directly related by
71:. Spatial dispersion and temporal dispersion may occur in the same system.
2055:
are used, and debate over the reality of "negative permeability" seen in
1850:
57:
49:
28:
is usually described as a phenomenon where material parameters such as
17:
153:, which is varying in space (x) and time (t). Simplified laws such as
1519:
1307:, spatial dispersion plays a role in a few material effects such as
828:{\displaystyle E(x,t)=\operatorname {Re} (E_{0}e^{ikx-i\omega t})}
739:{\displaystyle J(x,t)=\operatorname {Re} (J_{0}e^{ikx-i\omega t})}
569:{\displaystyle \sigma (x,x',t,t')=\sigma _{\rm {sym}}(x-x',t-t')}
1518:
i.e., the permittivity is a wavevector- and frequency-dependent
2043:. There is some discussion over whether this is appropriate in
1910:
1866:
1801:
157:
would say that these are directly proportional to each other,
2025:
As a result, it is most common at optical frequencies to set
2007:
induced by a magnetic field can be represented instead as a
1211:
has spatial dispersion if it is dependent on the wavevector
904:{\displaystyle J_{0}={\tilde {\sigma }}(k,\omega )E_{0}.}
1946:
At nonzero frequencies, it is possible to represent all
1907:
Permittivity–permeability ambiguity at nonzero frequency
2090:
which relates stress and strain. For polar vibrations (
1927:
1883:
1818:
1996:{\displaystyle \nabla \times E=-\partial B/\partial t}
2096:
distinction between longitudinal and transverse modes
1960:
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376:
199:
163:
124:
89:
2105:
effects from spatial dispersion find an analogue in
1849:
For frequencies nearby an absorption line (e.g., an
1995:
1853:), spatial dispersion can play an important role.
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110:
118:that is driven in response to an electric field
2086:of sound is due to a spatial dispersion in the
1280:{\displaystyle \sigma _{\rm {sym}}(x-x',t-t')}
2078:In solids, the difference in propagation for
1783:{\displaystyle \epsilon ({\vec {k}},\omega )}
1204:{\displaystyle {\tilde {\sigma }}(k,\omega )}
951:{\displaystyle {\tilde {\sigma }}(k,\omega )}
8:
2175:
2173:
2039:and only consider a dispersive permittivity
1319:. Most commonly, the spatial dispersion in
443:{\displaystyle \sigma (x,x',t,t')dx'\,dt'}
1982:
1959:
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88:
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1299:Spatial dispersion in electromagnetism
1215:. This occurs if the spatial function
962:of the space-time response function:
7:
457:If the system is invariant in time (
83:As an example, consider the current
2150:Electrodynamics of Continuous Media
604:{\displaystyle \sigma _{\rm {sym}}}
1987:
1976:
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14:
2113:, analogous to optical activity.
1914:
1870:
1805:
2211:10.1070/PU2006v049n10ABEH006067
2063:Spatial dispersion in acoustics
2049:effective medium approximations
2152:. Vol. 8 (2nd ed.).
1794:boundary conditions" (ABC's).
1777:
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1529:, one can find the plane wave
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93:
1:
2057:negative index metamaterials
576:for some convolution kernel
2084:longitudinal acoustic modes
2298:
1860:
1555:{\displaystyle {\vec {E}}}
1342:vector can be written as:
1167:The conductivity function
179:{\displaystyle J=\sigma E}
2080:transverse acoustic modes
459:time translation symmetry
75:Origin: nonlocal response
2254:10.1103/PhysRev.170.673
611:. We can also consider
454:conductivity function.
1997:
1784:
1746:Spatial dispersion in
1737:
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1509:
1281:
1205:
1158:
952:
905:
829:
740:
649:
629:
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570:
444:
361:
180:
147:
146:{\displaystyle E(x,t)}
112:
111:{\displaystyle J(x,t)}
2154:Butterworth-Heinemann
1998:
1785:
1738:
1557:
1510:
1336:constitutive relation
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1206:
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906:
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741:
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181:
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2103:electromagnetic wave
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122:
87:
2246:1968PhRv..170..673P
2203:2006PhyU...49.1029A
2031:vacuum permeability
1527:Maxwell's equations
1054:
1022:
914:where the function
273:
241:
36:have dependence on
2282:Physical phenomena
1993:
1926:. You can help by
1882:. You can help by
1817:. You can help by
1798:In isotropic media
1780:
1733:
1552:
1505:
1398:
1313:doppler broadening
1287:is not pointlike (
1277:
1201:
1154:
1034:
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825:
736:
645:
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601:
566:
440:
357:
253:
221:
176:
143:
108:
26:spatial dispersion
2163:978-0-7506-2634-7
2088:elasticity tensor
1944:
1943:
1900:
1899:
1835:
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1768:
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1709:
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1661:
1646:
1628:
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1427:
1389:
1374:
1183:
981:
960:Fourier transform
930:
870:
648:{\displaystyle J}
628:{\displaystyle E}
54:crystal structure
2289:
2266:
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2229:
2223:
2222:
2186:
2180:
2177:
2168:
2167:
2134:
2111:chiral materials
2002:
2000:
1999:
1994:
1986:
1950:as time-varying
1939:
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1911:
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1843:optical activity
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1397:
1376:
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1361:
1360:
1326:is of interest.
1309:optical activity
1305:electromagnetism
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69:optical activity
22:continuous media
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2234:Physical Review
2231:
2230:
2226:
2191:Physics-Uspekhi
2188:
2187:
2183:
2178:
2171:
2164:
2146:L.P. Pitaevskii
2136:
2135:
2124:
2119:
2092:optical phonons
2065:
2038:
1956:
1955:
1940:
1934:
1931:
1924:needs expansion
1909:
1896:
1890:
1887:
1880:needs expansion
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1815:needs expansion
1800:
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188:linear response
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84:
77:
12:
11:
5:
2295:
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2274:
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2267:
2240:(3): 673–678.
2224:
2181:
2169:
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2121:
2120:
2118:
2115:
2107:acoustic waves
2064:
2061:
2036:
1992:
1989:
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1978:
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1972:
1969:
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1963:
1948:magnetizations
1942:
1941:
1921:
1919:
1908:
1905:
1898:
1897:
1877:
1875:
1863:Landau damping
1861:Main article:
1858:
1857:Landau damping
1855:
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1330:Crystal optics
1328:
1300:
1297:
1291:) response in
1289:delta function
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958:is given by a
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615:solutions for
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239:
236:
231:
228:
224:
220:
217:
214:
211:
208:
205:
202:
175:
172:
169:
166:
142:
139:
136:
133:
130:
127:
107:
104:
101:
98:
95:
92:
76:
73:
13:
10:
9:
6:
4:
3:
2:
2294:
2283:
2280:
2279:
2277:
2263:
2259:
2255:
2251:
2247:
2243:
2239:
2235:
2228:
2225:
2220:
2216:
2212:
2208:
2204:
2200:
2196:
2192:
2185:
2182:
2176:
2174:
2170:
2165:
2159:
2155:
2151:
2147:
2143:
2142:E.M. Lifshitz
2139:
2133:
2131:
2129:
2127:
2123:
2116:
2114:
2112:
2108:
2104:
2099:
2097:
2093:
2089:
2085:
2081:
2076:
2074:
2070:
2062:
2060:
2058:
2054:
2050:
2046:
2045:metamaterials
2042:
2035:
2032:
2028:
2023:
2021:
2017:
2012:
2010:
2006:
2005:magnetization
1990:
1983:
1979:
1973:
1970:
1967:
1964:
1953:
1952:polarizations
1949:
1938:
1935:December 2015
1929:
1925:
1922:This section
1920:
1917:
1913:
1912:
1906:
1904:
1894:
1891:December 2015
1885:
1881:
1878:This section
1876:
1873:
1869:
1868:
1864:
1856:
1854:
1852:
1847:
1844:
1838:
1829:
1826:December 2015
1820:
1816:
1813:This section
1811:
1808:
1804:
1803:
1797:
1795:
1791:
1774:
1771:
1762:
1753:
1730:
1727:
1718:
1703:
1697:
1688:
1679:
1670:
1655:
1649:
1640:
1631:
1622:
1613:
1610:
1601:
1592:
1587:
1583:
1577:
1573:
1565:
1564:
1563:
1543:
1532:
1528:
1523:
1521:
1502:
1496:
1493:
1484:
1473:
1469:
1460:
1457:
1453:
1447:
1443:
1439:
1433:
1430:
1421:
1410:
1407:
1403:
1394:
1390:
1386:
1380:
1377:
1368:
1357:
1353:
1345:
1344:
1343:
1341:
1337:
1329:
1327:
1325:
1322:
1318:
1317:metamaterials
1314:
1310:
1306:
1298:
1296:
1294:
1290:
1270:
1267:
1263:
1260:
1257:
1253:
1250:
1246:
1243:
1223:
1214:
1195:
1192:
1189:
1177:
1151:
1144:
1141:
1137:
1133:
1130:
1109:
1102:
1099:
1095:
1092:
1089:
1085:
1082:
1078:
1075:
1072:
1068:
1062:
1059:
1055:
1047:
1039:
1035:
1030:
1027:
1023:
1015:
1007:
1003:
999:
993:
990:
987:
975:
965:
964:
963:
961:
942:
939:
936:
924:
898:
893:
889:
882:
879:
876:
864:
858:
853:
849:
841:
840:
839:
817:
814:
811:
808:
805:
802:
799:
795:
789:
785:
778:
775:
772:
766:
763:
760:
754:
747:
728:
725:
722:
719:
716:
713:
710:
706:
700:
696:
689:
686:
683:
677:
674:
671:
665:
658:
657:
656:
642:
622:
614:
584:
559:
556:
552:
549:
546:
542:
539:
535:
532:
512:
508:
501:
498:
494:
491:
488:
484:
481:
477:
474:
468:
460:
455:
453:
436:
433:
429:
424:
421:
417:
410:
407:
403:
400:
397:
393:
390:
386:
383:
377:
354:
347:
344:
340:
336:
333:
326:
319:
316:
312:
309:
306:
302:
299:
295:
292:
286:
281:
278:
274:
266:
258:
254:
249:
246:
242:
234:
226:
222:
218:
212:
209:
206:
200:
193:
192:
191:
190:is given by:
189:
173:
170:
167:
164:
156:
137:
134:
131:
125:
102:
99:
96:
90:
81:
74:
72:
70:
66:
61:
59:
55:
51:
46:
41:
39:
35:
31:
27:
23:
19:
2237:
2233:
2227:
2197:(10): 1029.
2194:
2190:
2184:
2149:
2100:
2077:
2075:and above).
2066:
2052:
2040:
2033:
2026:
2024:
2019:
2015:
2013:
2009:polarization
1945:
1932:
1928:adding to it
1923:
1901:
1888:
1884:adding to it
1879:
1848:
1839:
1836:
1823:
1819:adding to it
1814:
1792:
1745:
1531:normal modes
1525:Considering
1524:
1517:
1340:polarization
1333:
1323:
1321:permittivity
1302:
1292:
1212:
1166:
913:
837:
456:
451:
369:
82:
78:
62:
42:
34:conductivity
30:permittivity
25:
15:
2138:L.D. Landau
2117:References
613:plane wave
65:dispersion
45:wavelength
38:wavevector
2262:0031-899X
2219:119408077
2073:gigahertz
2069:acoustics
1988:∂
1977:∂
1974:−
1965:×
1962:∇
1775:ω
1766:→
1754:ϵ
1722:→
1707:→
1698:⋅
1692:→
1674:→
1659:→
1650:⋅
1644:→
1632:−
1626:→
1614:ω
1605:→
1593:ϵ
1584:μ
1574:ω
1547:→
1497:ω
1488:→
1454:δ
1444:ϵ
1440:−
1434:ω
1425:→
1404:ϵ
1391:∑
1381:ω
1372:→
1264:−
1247:−
1224:σ
1196:ω
1181:~
1178:σ
1110:σ
1096:ω
1073:−
1051:∞
1048:−
1043:∞
1040:−
1036:∫
1019:∞
1016:−
1011:∞
1008:−
1004:∫
994:ω
979:~
976:σ
943:ω
928:~
925:σ
883:ω
868:~
865:σ
815:ω
809:−
779:
726:ω
720:−
690:
655:like so:
585:σ
553:−
536:−
513:σ
469:σ
378:σ
287:σ
270:∞
267:−
262:∞
259:−
255:∫
238:∞
235:−
230:∞
227:−
223:∫
171:σ
155:Ohm's law
2276:Category
2148:(1984).
1338:for the
1271:′
1254:′
1145:″
1134:″
1103:″
1086:″
1063:″
1031:″
560:′
543:′
502:′
485:′
452:nonlocal
437:′
425:′
411:′
394:′
348:′
337:′
320:′
303:′
282:′
250:′
2242:Bibcode
2199:Bibcode
2094:), the
2029:to the
1851:exciton
450:is the
58:plasmon
50:calcite
18:physics
16:In the
2260:
2217:
2160:
2047:where
2003:, the
1520:tensor
370:where
2215:S2CID
2101:Many
1293:x-x'
2258:ISSN
2158:ISBN
2082:and
2051:for
1311:and
635:and
2250:doi
2238:170
2207:doi
2067:In
1930:.
1886:.
1821:.
1303:In
32:or
20:of
2278::
2256:.
2248:.
2236:.
2213:.
2205:.
2195:49
2193:.
2172:^
2156:.
2144:;
2140:;
2125:^
2059:.
1731:0.
1562::
1522:.
1295:.
776:Re
687:Re
60:.
24:,
2264:.
2252::
2244::
2221:.
2209::
2201::
2166:.
2053:ÎĽ
2041:ε
2037:0
2034:ÎĽ
2027:ÎĽ
2020:ε
2016:ÎĽ
1991:t
1984:/
1980:B
1971:=
1968:E
1937:)
1933:(
1893:)
1889:(
1828:)
1824:(
1778:)
1772:,
1763:k
1757:(
1728:=
1719:k
1713:)
1704:E
1689:k
1683:(
1680:+
1671:E
1665:)
1656:k
1641:k
1635:(
1623:E
1617:)
1611:,
1602:k
1596:(
1588:0
1578:2
1544:E
1503:,
1500:)
1494:,
1485:k
1479:(
1474:j
1470:E
1466:)
1461:j
1458:i
1448:0
1437:)
1431:,
1422:k
1416:(
1411:j
1408:i
1400:(
1395:j
1387:=
1384:)
1378:,
1369:k
1363:(
1358:i
1354:P
1324:ε
1275:)
1268:t
1261:t
1258:,
1251:x
1244:x
1241:(
1235:m
1232:y
1229:s
1213:k
1199:)
1193:,
1190:k
1187:(
1152:.
1149:)
1142:t
1138:,
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1127:(
1121:m
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1100:t
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1090:+
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1056:d
1028:x
1024:d
1000:=
997:)
991:,
988:k
985:(
946:)
940:,
937:k
934:(
899:.
894:0
890:E
886:)
880:,
877:k
874:(
859:=
854:0
850:J
823:)
818:t
812:i
806:x
803:k
800:i
796:e
790:0
786:E
782:(
773:=
770:)
767:t
764:,
761:x
758:(
755:E
734:)
729:t
723:i
717:x
714:k
711:i
707:e
701:0
697:J
693:(
684:=
681:)
678:t
675:,
672:x
669:(
666:J
643:J
623:E
596:m
593:y
590:s
564:)
557:t
550:t
547:,
540:x
533:x
530:(
524:m
521:y
518:s
509:=
506:)
499:t
495:,
492:t
489:,
482:x
478:,
475:x
472:(
434:t
430:d
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418:d
415:)
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391:x
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355:,
352:)
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324:)
317:t
313:,
310:t
307:,
300:x
296:,
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290:(
279:t
275:d
247:x
243:d
219:=
216:)
213:t
210:,
207:x
204:(
201:J
174:E
168:=
165:J
141:)
138:t
135:,
132:x
129:(
126:E
106:)
103:t
100:,
97:x
94:(
91:J
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