1472:
1325:
Dynamic wavelength shift in semiconductor lasers occurs as a result of the change in refractive index in the active region during intensity modulation. It is possible to evaluate the shift in wavelength by determining the refractive index change of the active region as a result of carrier injection.
546:
In the photon density rate equation, the first term ΓGP is the rate at which photon density increases due to stimulated emission (the same term in carrier rate equation, with positive sign and multiplied for the confinement factor Γ), the second term is the rate at which photons leave the cavity, for
1243:
Spatial hole burning occurs as a result of the standing wave nature of the optical modes. Increased lasing power results in decreased carrier diffusion efficiency which means that the stimulated recombination time becomes shorter relative to the carrier diffusion time. Carriers are therefore
363:
229:
1250:
To account for gain compression due to the high power densities in semiconductor lasers, the gain equation is modified such that it becomes related to the inverse of the optical power. Hence, the following term in the denominator of the gain equation :
1108:
769:
1218:
1231:
The gain term, G, cannot be independent of the high power densities found in semiconductor laser diodes. There are several phenomena which cause the gain to 'compress' which are dependent upon optical power. The two main phenomena are
1326:
A complete analysis of spectral shift during direct modulation found that the refractive index of the active region varies proportionally to carrier density and hence the wavelength varies proportionally to injected current.
1404:
884:
505:
is the radiative recombination time constant, M is the number of modes modelled, μ is the mode number, and subscript μ has been added to G, Γ, and β to indicate these properties may vary for the different modes.
509:
The first term on the right side of the carrier rate equation is the injected electrons rate (I/eV), the second term is the carrier depletion rate due to all recombination processes (described by the decay time
976:
1315:
235:
99:
998:
594:
574:
537:
503:
452:
403:
1138:
474:
423:
1575:
1461:
77:
The laser diode rate equations can be formulated with more or less complexity to model different aspects of laser diode behavior with varying accuracy.
1335:
1667:
1247:
Spectral hole burning is related to the gain profile broadening mechanisms such as short intraband scattering which is related to power density.
1569:
788:
907:
1507:
1454:
358:{\displaystyle {\frac {dP_{\mu }}{dt}}=(\Gamma _{\mu }G_{\mu }-{\frac {1}{\tau _{p}}})P_{\mu }+\beta _{\mu }{\frac {N}{\tau _{r}}}}
1257:
1646:
576:
and the third term is the contribution of spontaneous emission from the carrier radiative recombination into the laser mode.
224:{\displaystyle {\frac {dN}{dt}}={\frac {I}{eV}}-{\frac {N}{\tau _{n}}}-\sum _{\mu =1}^{\mu =M}\Gamma _{\mu }G_{\mu }P_{\mu }}
24:
1672:
89:. This formulation requires one equation for the carrier density, and one equation for the photon density in each of the
1447:
1613:
1513:
48:
1592:
1103:{\displaystyle \beta _{\mu }={\frac {\beta _{0}}{1+(2(\lambda _{s}-\lambda _{\mu })/\delta \lambda _{s})^{2}}}}
1586:
1535:
1237:
1233:
1529:
55:
40:
588:, the gain of the μ mode, can be modelled by a parabolic dependence of gain on wavelength as follows:
1581:
1552:
71:
764:{\displaystyle G_{\mu }={\frac {\alpha N-\alpha N_{0}}{1+\epsilon \sum _{\mu =1}^{\mu =M}P_{\mu }}}}
1475:
540:
373:
1213:{\displaystyle \lambda _{\mu }=\lambda _{0}-\mu \delta \lambda +{\frac {(n-1)\delta \lambda }{2}}}
1519:
550:
513:
479:
428:
379:
67:
1524:
457:
369:
368:
where: N is the carrier density, P is the photon density, I is the applied current, e is the
408:
1640:
782:
is the full width at half maximum (FWHM) of the gain curve, the centre of which is given by
44:
1634:
1597:
90:
32:
1661:
85:
In the multimode formulation, the rate equations model a laser with multiple optical
70:
equations to help in further understanding the static and dynamic characteristics of
20:
774:
where: α is the gain coefficient and ε is the gain compression factor (see below). λ
63:
547:
internal absorption or exiting the mirrors, expressed via the decay time constant
1501:
1495:
1489:
86:
59:
17:
1433:
G. P. Agrawal, "Fiber-Optic
Communication Systems", Wiley Interscience, Chap. 3
1399:{\displaystyle \delta \lambda =k\left({\sqrt {\frac {I_{0}}{I_{th}}}}-1\right)}
1244:
depleted faster at the crest of the wave causing a decrease in the modal gain.
23:
model the electrical and optical performance of a laser diode. This system of
1439:
879:{\displaystyle \lambda (t)=\lambda _{0}+{\frac {k(N_{th}-N(t))}{N_{th}}}}
36:
1471:
28:
1329:
Experimentally, a good fit for the shift in wavelength is given by:
971:{\displaystyle N_{th}=N_{tr}+{\frac {1}{\alpha \tau _{p}\Gamma }}}
1443:
543:, which is proportional to the photon density and medium gain.
1310:{\displaystyle 1+\epsilon \sum _{\mu =1}^{\mu =M}P_{\mu }}
1124:
is the centre wavelength for spontaneous emission and δλ
405:
is the carrier lifetime, G is the gain coefficient (s),
1338:
1260:
1141:
1001:
910:
791:
597:
553:
539:) and the third term is the carrier depletion due to
516:
482:
460:
431:
411:
382:
238:
102:
901:
is the carrier density at threshold and is given by
1627:
1606:
1562:
1545:
1482:
897:
and k is the spectral shift constant (see below). N
1398:
1309:
1212:
1102:
970:
878:
763:
568:
531:
497:
468:
446:
417:
397:
357:
223:
1132:is the wavelength of the μ mode and is given by
1576:Vertical-external-cavity surface-emitting-laser
43:and to device and material parameters such as
1455:
8:
1128:is the spontaneous emission FWHM. Finally, λ
1502:Separate confinement heterostructure laser
1462:
1448:
1440:
1373:
1363:
1356:
1337:
1301:
1285:
1274:
1259:
1180:
1159:
1146:
1140:
1091:
1081:
1069:
1060:
1047:
1021:
1015:
1006:
1000:
956:
943:
931:
915:
909:
865:
833:
820:
811:
790:
752:
736:
725:
704:
685:
672:
657:
635:
611:
602:
596:
559:
554:
552:
522:
517:
515:
488:
483:
481:
461:
459:
437:
432:
430:
410:
388:
383:
381:
347:
338:
332:
319:
304:
295:
286:
276:
249:
239:
237:
215:
205:
195:
179:
168:
153:
144:
126:
103:
101:
985:is the carrier density at transparency.
1426:
1570:Vertical-cavity surface-emitting laser
1120:is the spontaneous emission factor, λ
62:solution, or used to derive a set of
7:
476:is the spontaneous emission factor,
54:The rate equations may be solved by
778:is the wavelength of the μ mode, δλ
962:
893:is the centre wavelength for N = N
412:
273:
192:
14:
1508:Distributed Bragg reflector laser
1417:is the lasing threshold current.
39:) in the device to the injection
27:relates the number or density of
1470:
1668:Ordinary differential equations
1647:List of semiconductor materials
25:ordinary differential equations
1223:where δλ is the mode spacing.
1195:
1183:
1088:
1066:
1040:
1034:
857:
854:
848:
826:
801:
795:
691:
682:
647:
641:
629:
620:
312:
269:
1:
1413:is the injected current and I
1496:Double heterostructure laser
569:{\displaystyle {\tau _{p}}}
532:{\displaystyle {\tau _{n}}}
498:{\displaystyle {\tau _{r}}}
447:{\displaystyle {\tau _{p}}}
425:is the confinement factor,
398:{\displaystyle {\tau _{n}}}
47:, photon lifetime, and the
1689:
1619:Laser diode rate equations
1614:Semiconductor laser theory
1514:Distributed-feedback laser
372:, V is the volume of the
1593:Semiconductor ring laser
541:stimulated recombination
469:{\displaystyle {\beta }}
454:is the photon lifetime,
81:Multimode rate equations
1587:Interband cascade laser
418:{\displaystyle \Gamma }
1400:
1311:
1296:
1214:
1104:
972:
880:
765:
747:
570:
533:
499:
470:
448:
419:
399:
359:
225:
190:
1536:External-cavity laser
1530:Quantum-cascade laser
1401:
1312:
1270:
1238:spectral hole burning
1215:
1105:
973:
881:
766:
721:
571:
534:
500:
471:
449:
420:
400:
360:
226:
164:
56:numerical integration
1673:Semiconductor lasers
1582:Hybrid silicon laser
1553:Volume Bragg grating
1476:Semiconductor lasers
1336:
1258:
1234:spatial hole burning
1139:
999:
908:
789:
595:
551:
514:
480:
458:
429:
409:
380:
236:
100:
72:semiconductor lasers
1520:Quantum well laser
1396:
1307:
1210:
1100:
968:
876:
761:
566:
529:
495:
466:
444:
415:
395:
355:
221:
1655:
1654:
1525:Quantum dot laser
1383:
1382:
1208:
1098:
966:
874:
759:
679:
370:elementary charge
353:
310:
264:
159:
139:
121:
1680:
1641:Gallium arsenide
1474:
1464:
1457:
1450:
1441:
1434:
1431:
1405:
1403:
1402:
1397:
1395:
1391:
1384:
1381:
1380:
1368:
1367:
1358:
1357:
1316:
1314:
1313:
1308:
1306:
1305:
1295:
1284:
1227:Gain Compression
1219:
1217:
1216:
1211:
1209:
1204:
1181:
1164:
1163:
1151:
1150:
1109:
1107:
1106:
1101:
1099:
1097:
1096:
1095:
1086:
1085:
1073:
1065:
1064:
1052:
1051:
1026:
1025:
1016:
1011:
1010:
977:
975:
974:
969:
967:
965:
961:
960:
944:
939:
938:
923:
922:
885:
883:
882:
877:
875:
873:
872:
860:
841:
840:
821:
816:
815:
770:
768:
767:
762:
760:
758:
757:
756:
746:
735:
710:
709:
708:
690:
689:
680:
678:
677:
676:
663:
662:
661:
636:
612:
607:
606:
575:
573:
572:
567:
565:
564:
563:
538:
536:
535:
530:
528:
527:
526:
504:
502:
501:
496:
494:
493:
492:
475:
473:
472:
467:
465:
453:
451:
450:
445:
443:
442:
441:
424:
422:
421:
416:
404:
402:
401:
396:
394:
393:
392:
364:
362:
361:
356:
354:
352:
351:
339:
337:
336:
324:
323:
311:
309:
308:
296:
291:
290:
281:
280:
265:
263:
255:
254:
253:
240:
230:
228:
227:
222:
220:
219:
210:
209:
200:
199:
189:
178:
160:
158:
157:
145:
140:
138:
127:
122:
120:
112:
104:
45:carrier lifetime
1688:
1687:
1683:
1682:
1681:
1679:
1678:
1677:
1658:
1657:
1656:
1651:
1635:Indium arsenide
1623:
1602:
1598:Polariton laser
1558:
1541:
1478:
1468:
1438:
1437:
1432:
1428:
1423:
1416:
1412:
1369:
1359:
1355:
1351:
1334:
1333:
1323:
1297:
1256:
1255:
1229:
1182:
1155:
1142:
1137:
1136:
1131:
1127:
1123:
1119:
1087:
1077:
1056:
1043:
1027:
1017:
1002:
997:
996:
991:
984:
952:
948:
927:
911:
906:
905:
900:
896:
892:
861:
829:
822:
807:
787:
786:
781:
777:
748:
711:
700:
681:
668:
664:
653:
637:
613:
598:
593:
592:
587:
582:
555:
549:
548:
518:
512:
511:
484:
478:
477:
456:
455:
433:
427:
426:
407:
406:
384:
378:
377:
343:
328:
315:
300:
282:
272:
256:
245:
241:
234:
233:
211:
201:
191:
149:
131:
113:
105:
98:
97:
83:
33:charge carriers
12:
11:
5:
1686:
1684:
1676:
1675:
1670:
1660:
1659:
1653:
1652:
1650:
1649:
1644:
1638:
1631:
1629:
1625:
1624:
1622:
1621:
1616:
1610:
1608:
1604:
1603:
1601:
1600:
1595:
1590:
1584:
1579:
1573:
1566:
1564:
1560:
1559:
1557:
1556:
1549:
1547:
1543:
1542:
1540:
1539:
1533:
1527:
1522:
1517:
1511:
1505:
1499:
1493:
1486:
1484:
1480:
1479:
1469:
1467:
1466:
1459:
1452:
1444:
1436:
1435:
1425:
1424:
1422:
1419:
1414:
1410:
1407:
1406:
1394:
1390:
1387:
1379:
1376:
1372:
1366:
1362:
1354:
1350:
1347:
1344:
1341:
1322:
1321:Spectral Shift
1319:
1318:
1317:
1304:
1300:
1294:
1291:
1288:
1283:
1280:
1277:
1273:
1269:
1266:
1263:
1228:
1225:
1221:
1220:
1207:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1179:
1176:
1173:
1170:
1167:
1162:
1158:
1154:
1149:
1145:
1129:
1125:
1121:
1117:
1111:
1110:
1094:
1090:
1084:
1080:
1076:
1072:
1068:
1063:
1059:
1055:
1050:
1046:
1042:
1039:
1036:
1033:
1030:
1024:
1020:
1014:
1009:
1005:
989:
982:
979:
978:
964:
959:
955:
951:
947:
942:
937:
934:
930:
926:
921:
918:
914:
898:
894:
890:
887:
886:
871:
868:
864:
859:
856:
853:
850:
847:
844:
839:
836:
832:
828:
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819:
814:
810:
806:
803:
800:
797:
794:
779:
775:
772:
771:
755:
751:
745:
742:
739:
734:
731:
728:
724:
720:
717:
714:
707:
703:
699:
696:
693:
688:
684:
675:
671:
667:
660:
656:
652:
649:
646:
643:
640:
634:
631:
628:
625:
622:
619:
616:
610:
605:
601:
585:
581:
580:The modal gain
578:
562:
558:
525:
521:
491:
487:
464:
440:
436:
414:
391:
387:
366:
365:
350:
346:
342:
335:
331:
327:
322:
318:
314:
307:
303:
299:
294:
289:
285:
279:
275:
271:
268:
262:
259:
252:
248:
244:
231:
218:
214:
208:
204:
198:
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188:
185:
182:
177:
174:
171:
167:
163:
156:
152:
148:
143:
137:
134:
130:
125:
119:
116:
111:
108:
91:optical cavity
82:
79:
21:rate equations
13:
10:
9:
6:
4:
3:
2:
1685:
1674:
1671:
1669:
1666:
1665:
1663:
1648:
1645:
1642:
1639:
1636:
1633:
1632:
1630:
1626:
1620:
1617:
1615:
1612:
1611:
1609:
1605:
1599:
1596:
1594:
1591:
1588:
1585:
1583:
1580:
1577:
1574:
1571:
1568:
1567:
1565:
1561:
1554:
1551:
1550:
1548:
1544:
1537:
1534:
1531:
1528:
1526:
1523:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1487:
1485:
1481:
1477:
1473:
1465:
1460:
1458:
1453:
1451:
1446:
1445:
1442:
1430:
1427:
1420:
1418:
1392:
1388:
1385:
1377:
1374:
1370:
1364:
1360:
1352:
1348:
1345:
1342:
1339:
1332:
1331:
1330:
1327:
1320:
1302:
1298:
1292:
1289:
1286:
1281:
1278:
1275:
1271:
1267:
1264:
1261:
1254:
1253:
1252:
1248:
1245:
1241:
1239:
1235:
1226:
1224:
1205:
1201:
1198:
1192:
1189:
1186:
1177:
1174:
1171:
1168:
1165:
1160:
1156:
1152:
1147:
1143:
1135:
1134:
1133:
1114:
1092:
1082:
1078:
1074:
1070:
1061:
1057:
1053:
1048:
1044:
1037:
1031:
1028:
1022:
1018:
1012:
1007:
1003:
995:
994:
993:
986:
957:
953:
949:
945:
940:
935:
932:
928:
924:
919:
916:
912:
904:
903:
902:
869:
866:
862:
851:
845:
842:
837:
834:
830:
823:
817:
812:
808:
804:
798:
792:
785:
784:
783:
753:
749:
743:
740:
737:
732:
729:
726:
722:
718:
715:
712:
705:
701:
697:
694:
686:
673:
669:
665:
658:
654:
650:
644:
638:
632:
626:
623:
617:
614:
608:
603:
599:
591:
590:
589:
579:
577:
560:
556:
544:
542:
523:
519:
507:
489:
485:
462:
438:
434:
389:
385:
375:
371:
348:
344:
340:
333:
329:
325:
320:
316:
305:
301:
297:
292:
287:
283:
277:
266:
260:
257:
250:
246:
242:
232:
216:
212:
206:
202:
196:
186:
183:
180:
175:
172:
169:
165:
161:
154:
150:
146:
141:
135:
132:
128:
123:
117:
114:
109:
106:
96:
95:
94:
92:
88:
80:
78:
75:
73:
69:
65:
61:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
19:
1618:
1546:Hybrid types
1429:
1408:
1328:
1324:
1249:
1246:
1242:
1230:
1222:
1115:
1112:
992:is given by
987:
980:
888:
773:
583:
545:
508:
367:
84:
76:
68:small signal
64:steady state
58:to obtain a
53:
49:optical gain
15:
1563:Other Types
1490:Laser diode
1483:Basic types
60:time-domain
18:laser diode
1662:Categories
1421:References
1628:Materials
1386:−
1343:λ
1340:δ
1303:μ
1287:μ
1276:μ
1272:∑
1268:ϵ
1202:λ
1199:δ
1190:−
1175:λ
1172:δ
1169:μ
1166:−
1157:λ
1148:μ
1144:λ
1079:λ
1075:δ
1062:μ
1058:λ
1054:−
1045:λ
1019:β
1008:μ
1004:β
963:Γ
954:τ
950:α
843:−
809:λ
793:λ
754:μ
738:μ
727:μ
723:∑
719:ϵ
698:α
695:−
670:λ
666:δ
659:μ
655:λ
651:−
639:λ
627:−
615:α
604:μ
557:τ
520:τ
486:τ
463:β
435:τ
413:Γ
386:τ
345:τ
334:μ
330:β
321:μ
302:τ
293:−
288:μ
278:μ
274:Γ
251:μ
217:μ
207:μ
197:μ
193:Γ
181:μ
170:μ
166:∑
162:−
151:τ
142:−
37:electrons
1578:(VECSEL)
376:region,
1572:(VCSEL)
1409:where I
981:where N
889:where λ
93:modes:
41:current
29:photons
1643:(GaAs)
1637:(InAs)
1607:Theory
1113:where
374:active
1589:(ICL)
1555:laser
1538:(ECL)
1532:(QCL)
1516:(DFB)
1510:(DBR)
1504:(SCH)
87:modes
1498:(DH)
1492:(LD)
1236:and
31:and
16:The
66:or
1664::
1415:th
1240:.
983:tr
899:th
895:th
74:.
51:.
1463:e
1456:t
1449:v
1411:0
1393:)
1389:1
1378:h
1375:t
1371:I
1365:0
1361:I
1353:(
1349:k
1346:=
1299:P
1293:M
1290:=
1282:1
1279:=
1265:+
1262:1
1206:2
1196:)
1193:1
1187:n
1184:(
1178:+
1161:0
1153:=
1130:μ
1126:s
1122:s
1118:0
1116:β
1093:2
1089:)
1083:s
1071:/
1067:)
1049:s
1041:(
1038:2
1035:(
1032:+
1029:1
1023:0
1013:=
990:μ
988:β
958:p
946:1
941:+
936:r
933:t
929:N
925:=
920:h
917:t
913:N
891:0
870:h
867:t
863:N
858:)
855:)
852:t
849:(
846:N
838:h
835:t
831:N
827:(
824:k
818:+
813:0
805:=
802:)
799:t
796:(
780:g
776:μ
750:P
744:M
741:=
733:1
730:=
716:+
713:1
706:0
702:N
692:]
687:2
683:)
674:g
648:)
645:t
642:(
633:2
630:(
624:1
621:[
618:N
609:=
600:G
586:μ
584:G
561:p
524:n
490:r
439:p
390:n
349:r
341:N
326:+
317:P
313:)
306:p
298:1
284:G
270:(
267:=
261:t
258:d
247:P
243:d
213:P
203:G
187:M
184:=
176:1
173:=
155:n
147:N
136:V
133:e
129:I
124:=
118:t
115:d
110:N
107:d
35:(
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