1069:
1598:
805:
879:
1287:
1445:
1313:
varies are generally much longer than the spatial wavelength and temporal period of the carrier wave. A numerical solution of the envelope equation thus can use much larger space and time steps, resulting in significantly less computational effort.
1426:
679:
185:
1725:
433:
690:
245:
1180:
74:
The slowly varying envelope approximation is often used because the resulting equations are in many cases easier to solve than the original equations, reducing the order of—all or some of—the highest-order
1171:
864:
1780:
1654:
1064:{\displaystyle 2i\mathbf {k} _{0}\cdot \nabla E_{0}+{\frac {2i\omega _{0}}{c^{2}}}{\frac {\partial E_{0}}{\partial t}}-\left(k_{0}^{2}-{\frac {\omega _{0}^{2}}{c^{2}}}\right)E_{0}=0~.}
1841:
1812:
469:
96:
509:
1891:
1593:{\displaystyle k_{0}{\frac {\partial E_{0}}{\partial z}}+{\frac {\omega _{0}}{c^{2}}}{\frac {\partial E_{0}}{\partial t}}-{\frac {1}{2}}\,i\,\Delta _{\perp }E_{0}=0~.}
1342:
596:
297:
1861:
1295:, like the original wave equation, but now of first-order instead of second-order. It is valid for coherent forward-propagating waves in directions near the
1665:
1100:
1606:. This equation has enhanced validity as compared to the full SVEA: It represents waves propagating in directions significantly different from the
800:{\displaystyle \left|{\frac {\partial ^{2}E_{0}}{\partial t^{2}}}\right|\ll \left|\omega _{0}\,{\frac {\partial E_{0}}{\partial t}}\right|,}
192:
1292:
1603:
816:
1933:
1282:{\displaystyle \mathbf {k} _{0}\cdot \nabla E_{0}+{\frac {\omega _{0}}{c^{2}}}\,{\frac {\partial E_{0}}{\partial t}}=0~.}
1736:
1997:
Svelto, Orazio (1974). "Self-focussing, self-trapping, and self-phase modulation of laser beams". In Wolf, Emil (ed.).
1175:
This gives the following approximation to the wave equation, as a result of the slowly varying envelope approximation:
2012:
1981:
1624:
2004:
88:
2076:
2071:
1817:
1788:
440:
1973:
2028:
Bonifacio, R.; Caloi, R.M.; Maroli, C. (1993). "The slowly varying envelope approximation revisited".
478:
1869:
1335:
is taken in this direction. The SVEA is only applied to the second-order spatial derivatives in the
1421:{\displaystyle \Delta _{\perp }\equiv \partial ^{2}/\partial x^{2}+\partial ^{2}/\partial y^{2}}
674:{\displaystyle \left|\nabla ^{2}E_{0}\right|\ll \left|\mathbf {k} _{0}\cdot \nabla E_{0}\right|}
2030:
17:
180:{\displaystyle \nabla ^{2}E-{\frac {1}{c^{2}}}{\frac {\partial ^{2}E}{\partial t^{2}}}=0\,,}
2039:
1942:
8:
1894:
1092:
2043:
1946:
1999:
1846:
76:
2051:
2008:
1977:
1911:
518:
270:
1618:
In the one-dimensional case, another sufficient condition for the SVEA validity is
2047:
1950:
1906:
1429:
52:
44:
1720:{\displaystyle \ell _{\mathsf {p}}\gg \lambda \left(1-{\frac {v}{c}}\right)\,,}
2065:
1954:
428:{\displaystyle E(\mathbf {r} ,t)=\operatorname {\operatorname {Re} } \left,}
1866:
These conditions are much less restrictive in the relativistic limit where
590:, when taking derivatives, the highest-order derivatives may be neglected:
274:
79:. But the validity of the assumptions which are made need to be justified.
1931:
Arecchi, F.; Bonifacio, R. (1965). "Theory of optical maser amplifiers".
266:
64:
56:
472:
1897:, compared to the usual conditions required for the SVEA validity.
60:
240:{\displaystyle c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}~.}
24:
874:
Consequently, the wave equation is approximated in the SVEA as:
1166:{\displaystyle k_{0}^{2}-{\frac {\omega _{0}^{2}}{c^{2}}}=0~.}
1814:
is the length over which the radiation pulse is amplified,
563:
represents waves propagating forward, predominantly in the
48:
859:{\displaystyle k_{0}\equiv \left|\mathbf {k} _{0}\right|.}
1872:
1849:
1820:
1791:
1739:
1668:
1627:
1448:
1345:
1183:
1103:
882:
819:
693:
599:
481:
443:
300:
195:
99:
2027:
51:
pulse varies slowly in time and space compared to a
1885:
1855:
1835:
1806:
1775:{\displaystyle \lambda ={\frac {2\pi }{k_{0}}}\,,}
1774:
1719:
1648:
1592:
1420:
1281:
1165:
1063:
858:
799:
673:
503:
463:
427:
239:
179:
1304:-direction. The space and time scales over which
2063:
1930:
1613:
572:direction. As a result of the slow variation of
1863:is the group velocity of the radiating system.
1649:{\displaystyle \ell _{\mathsf {g}}\gg \lambda }
1967:
1322:Assume wave propagation is dominantly in the
67:—hence it is also referred to as the
16:"SVEA" redirects here. For other uses, see
1317:
292:, the following representation is useful:
1768:
1713:
1557:
1553:
1239:
761:
457:
453:
364:
173:
1968:Butcher, Paul N.; Cotter, David (1991).
1293:hyperbolic partial differential equation
1604:parabolic partial differential equation
37:slowly varying asymmetric approximation
2064:
1996:
1827:
1798:
1675:
1634:
475:of the quantity between brackets, and
869:
515:slowly varying envelope approximation
29:slowly varying envelope approximation
1934:IEEE Journal of Quantum Electronics
1836:{\displaystyle \ell _{\mathsf {p}}}
1807:{\displaystyle \ell _{\mathsf {g}}}
464:{\displaystyle \operatorname {Re} }
13:
1559:
1531:
1516:
1477:
1462:
1405:
1391:
1374:
1360:
1347:
1258:
1243:
1199:
970:
955:
904:
780:
765:
723:
702:
653:
606:
151:
137:
101:
14:
2088:
1970:The Elements of Nonlinear Optics
1186:
891:
839:
640:
392:
378:
351:
308:
548:. This inherently implies that
504:{\displaystyle i^{2}\equiv -1.}
2021:
1990:
1961:
1924:
1886:{\displaystyle {\frac {v}{c}}}
517:(SVEA) it is assumed that the
458:
450:
412:
373:
361:
347:
318:
304:
1:
1917:
1614:Alternative limit of validity
89:electromagnetic wave equation
43:) is the assumption that the
2052:10.1016/0030-4018(93)90363-A
7:
1900:
1091:such that they satisfy the
1073:It is convenient to choose
10:
2093:
1974:Cambridge University Press
87:For example, consider the
82:
15:
69:narrow-band approximation
1955:10.1109/JQE.1965.1072212
1339:-direction and time. If
538:only varies slowly with
273:of the (characteristic)
47:of a forward-travelling
35:, sometimes also called
1893:is close to 1, as in a
1843:is the pulse width and
1318:Parabolic approximation
1887:
1857:
1837:
1808:
1776:
1721:
1650:
1594:
1440:plane, the result is:
1422:
1283:
1167:
1065:
860:
801:
675:
505:
465:
429:
241:
181:
2031:Optics Communications
1888:
1858:
1838:
1809:
1777:
1722:
1651:
1595:
1423:
1284:
1168:
1066:
861:
802:
676:
506:
466:
430:
242:
182:
1972:(reprint ed.).
1870:
1847:
1818:
1789:
1737:
1666:
1625:
1446:
1343:
1181:
1101:
880:
817:
691:
597:
479:
441:
298:
193:
97:
63:of the signal to be
59:. This requires the
2077:Asymptotic analysis
2072:Theoretical physics
2044:1993OptCo.101..185B
1947:1965IJQE....1..169A
1895:free-electron laser
1138:
1118:
1093:dispersion relation
1021:
1001:
77:partial derivatives
2007:. pp. 23–25.
2000:Progress in Optics
1883:
1853:
1833:
1804:
1772:
1717:
1646:
1590:
1418:
1279:
1163:
1124:
1104:
1061:
1007:
987:
870:Full approximation
856:
797:
671:
501:
461:
425:
237:
177:
1912:WKB approximation
1881:
1856:{\displaystyle v}
1766:
1706:
1586:
1551:
1538:
1511:
1484:
1275:
1265:
1237:
1159:
1149:
1057:
1032:
977:
950:
787:
737:
519:complex amplitude
271:angular frequency
233:
229:
228:
165:
131:
2084:
2056:
2055:
2038:(3–4): 185–187.
2025:
2019:
2018:
2003:. Vol. 12.
1994:
1988:
1987:
1965:
1959:
1958:
1928:
1907:Ultrashort pulse
1892:
1890:
1889:
1884:
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1874:
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1490:
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1483:
1475:
1474:
1473:
1460:
1458:
1457:
1439:
1435:
1430:Laplace operator
1427:
1425:
1424:
1419:
1417:
1416:
1404:
1399:
1398:
1386:
1385:
1373:
1368:
1367:
1355:
1354:
1338:
1334:
1326:-direction, and
1325:
1312:
1303:
1288:
1286:
1285:
1280:
1273:
1266:
1264:
1256:
1255:
1254:
1241:
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1236:
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1226:
1225:
1216:
1211:
1210:
1195:
1194:
1189:
1172:
1170:
1169:
1164:
1157:
1150:
1148:
1147:
1137:
1132:
1123:
1117:
1112:
1090:
1081:
1070:
1068:
1067:
1062:
1055:
1048:
1047:
1038:
1034:
1033:
1031:
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1020:
1015:
1006:
1000:
995:
978:
976:
968:
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966:
953:
951:
949:
948:
939:
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921:
916:
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894:
865:
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852:
848:
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842:
829:
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742:
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666:
665:
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629:
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589:
571:
562:
547:
543:
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510:
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491:
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354:
346:
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328:
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291:
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186:
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166:
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149:
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117:
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2083:
2082:
2081:
2062:
2061:
2060:
2059:
2026:
2022:
2015:
1995:
1991:
1984:
1976:. p. 216.
1966:
1962:
1929:
1925:
1920:
1903:
1873:
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1005:
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377:
376:
369:
365:
350:
341:
337:
336:
332:
324:
307:
299:
296:
295:
278:
277:for the signal
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85:
21:
12:
11:
5:
2090:
2080:
2079:
2074:
2058:
2057:
2020:
2013:
1989:
1982:
1960:
1941:(4): 169–178.
1922:
1921:
1919:
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1914:
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84:
81:
9:
6:
4:
3:
2:
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2024:
2016:
2014:0-444-10571-9
2010:
2006:
2005:North Holland
2002:
2001:
1993:
1985:
1983:0-521-42424-0
1979:
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1971:
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930:
926:
923:
917:
912:
908:
901:
896:
886:
883:
875:
853:
849:
844:
834:
830:
825:
821:
794:
790:
783:
773:
769:
756:
752:
747:
743:
739:
731:
727:
716:
712:
706:
695:
667:
661:
657:
650:
645:
634:
630:
626:
620:
616:
610:
601:
593:
592:
591:
587:
583:
576:
567:
560:
556:
552:
542:
535:
531:
524:
520:
516:
511:
498:
495:
492:
487:
483:
474:
454:
447:
444:
435:
422:
418:
409:
404:
400:
396:
388:
383:
370:
366:
358:
355:
342:
338:
333:
329:
325:
321:
315:
312:
301:
293:
289:
285:
281:
276:
272:
268:
253:
247:
234:
223:
219:
213:
209:
204:
199:
196:
187:
174:
170:
167:
159:
155:
146:
141:
126:
122:
118:
113:
110:
105:
92:
90:
80:
78:
72:
70:
66:
65:narrow-banded
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
19:
2035:
2029:
2023:
1998:
1992:
1969:
1963:
1938:
1932:
1926:
1865:
1784:
1617:
1610:-direction.
1601:
1442:
1328:
1321:
1306:
1297:
1290:
1177:
1174:
1097:
1084:
1075:
1072:
876:
873:
585:
581:
574:
565:
558:
554:
550:
540:
533:
529:
522:
514:
512:
471:denotes the
436:
294:
287:
283:
279:
275:carrier wave
251:
248:
188:
93:
86:
73:
68:
40:
36:
32:
28:
22:
267:wave number
2066:Categories
1918:References
1602:This is a
1291:This is a
57:wavelength
1823:ℓ
1794:ℓ
1753:π
1741:λ
1696:−
1685:λ
1682:≫
1671:ℓ
1644:λ
1641:≫
1630:ℓ
1564:⊥
1560:Δ
1541:−
1532:∂
1517:∂
1493:ω
1478:∂
1463:∂
1406:∂
1392:∂
1375:∂
1361:∂
1357:≡
1352:⊥
1348:Δ
1259:∂
1244:∂
1219:ω
1200:∇
1197:⋅
1126:ω
1120:−
1009:ω
1003:−
980:−
971:∂
956:∂
931:ω
905:∇
902:⋅
831:≡
781:∂
766:∂
753:ω
744:≪
724:∂
703:∂
654:∇
651:⋅
631:≪
607:∇
496:−
493:≡
473:real part
455:⋅
448:
401:ω
397:−
389:⋅
330:
220:ε
210:μ
152:∂
138:∂
114:−
102:∇
1901:See also
265:are the
61:spectrum
45:envelope
2040:Bibcode
1943:Bibcode
1432:in the
1428:is the
513:In the
83:Example
25:physics
2011:
1980:
1785:where
1732:
1728:
1661:
1657:
1585:
1274:
1158:
1056:
812:
808:
686:
682:
437:where
232:
189:where
53:period
1730:with
810:with
2009:ISBN
1978:ISBN
1659:and
1082:and
684:and
544:and
269:and
258:and
49:wave
41:SVAA
33:SVEA
18:Svea
2048:doi
2036:101
1951:doi
249:If
55:or
39:or
23:In
2068::
2046:.
2034:.
1949:.
1937:.
1095::
584:,
557:,
532:,
499:1.
445:Re
326:Re
91::
71:.
27:,
2054:.
2050::
2042::
2017:.
1986:.
1957:.
1953::
1945::
1939:1
1879:c
1876:v
1851:v
1828:p
1799:g
1770:,
1762:0
1758:k
1750:2
1744:=
1715:,
1710:)
1704:c
1701:v
1693:1
1689:(
1676:p
1635:g
1608:z
1588:.
1582:0
1579:=
1574:0
1570:E
1555:i
1549:2
1546:1
1535:t
1525:0
1521:E
1507:2
1503:c
1497:0
1487:+
1481:z
1471:0
1467:E
1455:0
1451:k
1438:y
1436:×
1434:x
1414:2
1410:y
1402:/
1396:2
1388:+
1383:2
1379:x
1371:/
1365:2
1337:z
1332:0
1329:k
1324:z
1310:0
1307:E
1301:0
1298:k
1277:.
1271:0
1268:=
1262:t
1252:0
1248:E
1233:2
1229:c
1223:0
1213:+
1208:0
1204:E
1192:0
1187:k
1161:.
1155:0
1152:=
1145:2
1141:c
1135:2
1130:0
1115:2
1110:0
1106:k
1088:0
1085:ω
1079:0
1076:k
1059:.
1053:0
1050:=
1045:0
1041:E
1036:)
1028:2
1024:c
1018:2
1013:0
998:2
993:0
989:k
984:(
974:t
964:0
960:E
946:2
942:c
935:0
927:i
924:2
918:+
913:0
909:E
897:0
892:k
887:i
884:2
854:.
850:|
845:0
840:k
835:|
826:0
822:k
795:,
791:|
784:t
774:0
770:E
757:0
748:|
740:|
732:2
728:t
717:0
713:E
707:2
696:|
668:|
662:0
658:E
646:0
641:k
635:|
627:|
621:0
617:E
611:2
602:|
588:)
586:t
582:r
580:(
578:0
575:E
569:0
566:k
561:)
559:t
555:r
553:(
551:E
546:t
541:r
536:)
534:t
530:r
528:(
526:0
523:E
488:2
484:i
459:]
451:[
423:,
419:]
413:)
410:t
405:0
393:r
384:0
379:k
374:(
371:i
367:e
362:)
359:t
356:,
352:r
348:(
343:0
339:E
334:[
322:=
319:)
316:t
313:,
309:r
305:(
302:E
290:)
288:t
286:,
284:r
282:(
280:E
263:0
260:ω
255:0
252:k
235:.
224:0
214:0
205:1
200:=
197:c
175:,
171:0
168:=
160:2
156:t
147:E
142:2
127:2
123:c
119:1
111:E
106:2
31:(
20:.
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