844:. In solid-state laser materials, the linewidth is several orders of magnitude larger so the spectra of emission and absorption are determined by distribution of excitation among sublevels rather than by the shape of the spectral line of each individual transition between sublevels. This distribution is determined by the effective temperature within each of laser levels. The McCumber hypothesis is that the distribution of excitation among sublevels is thermal. The effective temperature determines the spectra of emission and absorption ( The
3664:
363:
1706:
3258:
165:
1463:
1887:
3646:
No specific property of sublevels of active medium is required to keep the McCumber relation. It follows from the assumption about quick transfer of energy among excited laser levels and among lower laser levels. The McCumber relation (mc) has the same range of validity as the concept of the emission
2689:
3046:
861:
3638:
1475:
3429:
3767:
On the base of the second Law of thermodynamics, the experimental results were refuted in 2007. With the McCumber theory, the correction was suggested for the effective emission cross section (black thin curve). Then this correction was confirmed experimentally.
3058:
799:
544:
It is typical that the lasing properties of a medium are determined by the temperature and the population at the excited laser level, and are not sensitive to the method of excitation used to achieve it. In this case, the absorption cross-section
358:{\displaystyle {\frac {\sigma _{\rm {e}}(\omega )}{\sigma _{\rm {a}}(\omega )}}\exp \!\left({\frac {\hbar \omega }{k_{\rm {B}}T}}\right)=\left({\frac {N_{1}}{N_{2}}}\right)_{T}=\exp \!\left({\frac {\hbar \omega _{\rm {z}}}{k_{\rm {B}}T}}\right)}
1228:
2335:
the atom is excited.) The relation (D2) is a fundamental property of spontaneous and stimulated emission, and perhaps the only way to prohibit a spontaneous break of the thermal equilibrium in the thermal state of excitations and photons.
1718:
2440:
2073:
3763:
through a spectrally-selective window which is transparent in vicinity of 975 nm and reflective at other wavelengths. Due to the lack of emissivity at 975 nm the medium should warm, breaking the thermal equilibrium.
2834:
3727:
and reported in 3 independent journals. Typical behavior of the cross-sections reported is shown in Fig.2 with thick curves. The emission cross-section is practically zero at wavelength 975 nm; this property makes
1984:
1701:{\displaystyle ~~~D(\omega )={\frac {\frac {a(\omega )}{\sigma _{\rm {e}}(\omega )v(\omega )}}{{\frac {n_{1}}{n_{2}}}{\frac {\sigma _{\rm {a}}(\omega )}{\sigma _{\rm {e}}(\omega )}}-1}}~~~~~~~~~~~~~~{\rm {(D1)}}}
1176:
1094:
3467:
3297:
3655:
The McCumber relation is confirmed for various media. In particular, relation (1) makes it possible to approximate two functions of frequency, emission and absorption cross sections, with single fit .
868:
Consider the set of active centers (fig.1.). Assume fast transition between sublevels within each level, and slow transition between levels. According to the McCumber hypothesis, the cross-sections
422:
2205:
2256:
3714:
3253:{\displaystyle {\frac {n_{1}}{n_{2}}}{\frac {\sigma _{\rm {a}}(\omega )}{\sigma _{\rm {e}}(\omega )}}=\exp \!\left({\frac {\hbar \omega }{k_{\rm {B}}T}}\right)~~.~~~~~~~~{\rm {(n1n2)(mc1)}}}
2146:
619:
581:
110:
72:
2825:
2431:
2307:
836:
leads to equality of cross-section of absorption and emission. In the solid-state lasers the splitting of each of laser levels leads to the broadening which greatly exceeds the
924:
895:
451:
662:
1215:
1016:
3288:
1458:{\displaystyle ~~~n_{2}\sigma _{\rm {e}}(\omega )v(\omega )D(\omega )+n_{2}a(\omega )=n_{1}\sigma _{\rm {a}}(\omega )v(\omega )D(\omega )~~~~~~~~~~~~~~~{\rm {(balance)}}}
504:
2102:
1882:{\displaystyle ~~~D(\omega )~=~{\frac {{\frac {1}{\pi ^{2}}}{\frac {\omega ^{2}}{c^{3}}}}{\exp \!\left({\frac {\hbar \omega }{k_{\rm {B}}T}}\right)-1}}~~~~~{\rm {(D2)}}}
4193:
3824:
2764:
1916:
645:
534:
2725:
130:
471:
978:
951:
3458:
2790:
2363:
2333:
156:
2383:
804:
D.E.McCumber had postulated these properties and found that the emission and absorption cross-sections are not independent; they are related with
Equation (1).
2684:{\displaystyle ~~~a_{s,j}(\omega )=\sigma _{s,j}(\omega ){\frac {\omega ^{2}v(\omega )}{\pi ^{2}c^{3}}}~~.~~~~~~~~~~~~~~~~{\rm {comparison1}}~~{\rm {partial}}}
1989:
3995:
3958:
3041:{\displaystyle {\frac {a(\omega )}{\sigma _{\rm {e}}(\omega )}}={\frac {\omega ^{2}v(\omega )}{\pi ^{2}c^{3}}}~~~~~~~~~~~~~~~~~~({\rm {comparison)(av)}}}
3956:
R.M.Martin; R.S.Quimby (2006). "Experimental evidence of the validity of the McCumber theory relating emission and absorption for rare-earth glasses".
825:
1921:
3633:{\displaystyle \omega _{\rm {Z}}={\frac {k_{\rm {B}}T}{\hbar }}\ln \left({\frac {n_{1}}{n_{2}}}\right)_{T}~~~~~~~~~~~~~~~~.~~{({\rm {oz)}}}}
1103:
1021:
821:
4034:
3916:
3873:
3424:{\displaystyle ~\left({\frac {n_{1}}{n_{2}}}\right)_{\!T}=\exp \!\left({\frac {\hbar \omega _{\rm {Z}}}{k_{\rm {B}}T}}\right)~~~~,~~~}
20:(or McCumber theory) is a relationship between the effective cross-sections of absorption and emission of light in the physics of
841:
840:. In the case of an ideal two-level atom, the product of the linewidth and the lifetime is of order of unity, which obeys the
371:
2155:
2210:
3744:
3678:
3460:
indicates that the ratio of populations in evaluated in the thermal state. The zero-line frequency can be expressed as
3993:
D.Kouznetsov; J.-F.Bisson; K.Takaichi; K.Ueda (2005). "Single-mode solid-state laser with short wide unstable cavity".
4262:
1221:. (Note that in this approximation, there is no such thing as a spontaneous absorption) The balance of photons gives:
4157:
3921:
2107:
837:
586:
548:
77:
39:
2149:
4214:
4047:
3837:
3878:
2795:
794:{\displaystyle ~~~~~~~~~~~~~~~G(\omega )=N_{2}\sigma _{\rm {e}}(\omega )-N_{1}\sigma _{\rm {a}}(\omega )}
4257:
2388:
2261:
4147:
C.Yan; G.Zhao; L.Zhang; J.Xu; et al. (2006). "A new Yb-doped oxyorthosilicate laser crystal: Yb:Gd
900:
871:
427:
4223:
4166:
4108:
4056:
4004:
3967:
3930:
3846:
1218:
1181:
833:
813:
1097:
25:
986:
3266:
507:
480:
3751:
device would be possible. It would be sufficient to fill a box with reflecting walls with Yb:Gd
3263:
This relation is equivalent of the McCumber relation (mc), if we define the zero-line frequency
2078:
860:
2730:
1895:
624:
513:
4126:
3788:
3737:
21:
2697:
2694:
Neglecting the cooperative coherent effects, the emission is additive: for any concentration
115:
4231:
4174:
4116:
4064:
4012:
3975:
3938:
3887:
3854:
3748:
817:
456:
956:
929:
474:
3917:"Range of validity of McCumber theory in relating absorption and emission cross sections"
3437:
2769:
2342:
2312:
852:
by scientists even if the excited medium as whole is pretty far from the thermal state )
135:
4227:
4170:
4112:
4060:
4008:
3971:
3934:
3850:
4099:
3787:
D.E.McCumber. Einstein relations connecting broadband emission and absorption spectra.
2368:
2068:{\displaystyle ~n_{1}/n_{2}=\exp \!\left({\frac {\hbar \omega }{k_{\rm {B}}T}}\right)}
4251:
652:
648:
1711:
The thermal distribution of density of photons follows from blackbody radiation
3663:
2075:, which leads to the relation between the spectral rate of spontaneous emission
4178:
3760:
2207:, which is probability of emission of a photon within small spectral interval
829:
4016:
3979:
4130:
4121:
4086:
3743:
However, the property reported (thick curves) is not compatible with the
3719:
In 2006 the strong violation of McCumber relation was observed for Yb:Gd
1979:{\displaystyle ~\sigma _{\rm {a}}(\omega )=\sigma _{\rm {e}}(\omega )~}
4236:
4068:
3942:
3858:
1171:{\displaystyle ~n_{1}\sigma _{\rm {a}}(\omega )v(\omega )D(\omega )~}
1089:{\displaystyle ~n_{2}\sigma _{\rm {e}}(\omega )v(\omega )D(\omega )~}
980:. Therefore, we can deduce the relation, assuming the thermal state.
158:
be the effective temperature of the medium. The McCumber relation is
510:. Note that the right-hand side of Equation (1) does not depend on
3662:
2433:
can be expressed from consideration of idealized two-level atoms:
859:
3643:
Then (n1n2) becomes equivalent of the McCumber relation (mc).
4087:"Diode-pumped continuous-wave and passively mode-locked Yb:Gd
3051:
Then, the comparison of (D1) and (D2) gives the relation
424:
is thermal steady-state ratio of populations; frequency
3659:
Violation of the McCumber relation and perpetual motion
417:{\displaystyle \left({\frac {N_{1}}{N_{2}}}\right)_{T}}
3808:
Erbium-doped fiber amplifiers: fundamentals and theory
2200:{\displaystyle ~a(\omega ){\rm {d}}\omega {\rm {d}}t~}
1918:. For the case of idealized two-level active centers,
1018:
be group velocity of light in the medium, the product
4033:
W. Li; H. Pan; L. Ding; H. Zeng; et al. (2006).
3874:"Broadband laser materials and the McCumber relation"
3681:
3470:
3440:
3300:
3269:
3061:
2837:
2798:
2772:
2733:
2700:
2443:
2391:
2371:
2345:
2315:
2264:
2213:
2158:
2110:
2081:
1992:
1924:
1898:
1721:
1478:
1231:
1184:
1106:
1024:
989:
959:
932:
903:
874:
665:
627:
589:
551:
516:
483:
459:
430:
374:
168:
138:
118:
80:
42:
2251:{\displaystyle ~(\omega ,\omega +{\rm {d}}\omega )~}
3709:{\displaystyle \lambda ={\frac {2\pi c}{\omega }}}
3708:
3632:
3452:
3423:
3282:
3252:
3040:
2819:
2784:
2758:
2719:
2683:
2425:
2377:
2357:
2327:
2301:
2250:
2199:
2140:
2096:
2067:
1978:
1910:
1881:
1700:
1457:
1209:
1170:
1088:
1010:
972:
945:
918:
889:
793:
639:
613:
575:
528:
498:
465:
445:
416:
357:
150:
124:
112:be effective emission cross-sections at frequency
104:
66:
4085:W.Li; H.Pan; L.Ding; H.Zeng; et al. (2006).
3350:
3338:
3142:
2027:
1801:
308:
222:
655:at this frequency can be determined as follows:
2766:of sublevels, the same proportionality between
8:
2385:, the partial spectral emission probability
2141:{\displaystyle ~\sigma _{\rm {e}}(\omega )~}
3996:Journal of the Optical Society of America B
3959:Journal of the Optical Society of America B
1892:Both (4) and (5) hold for all frequencies
3759:and allow it to exchange radiation with a
614:{\displaystyle \sigma _{\rm {e}}(\omega )}
576:{\displaystyle \sigma _{\rm {a}}(\omega )}
105:{\displaystyle \sigma _{\rm {e}}(\omega )}
74:be the effective absorption cross-section
67:{\displaystyle \sigma _{\rm {a}}(\omega )}
4235:
4120:
3818:
3816:
3688:
3680:
3617:
3616:
3612:
3549:
3537:
3527:
3521:
3494:
3493:
3486:
3476:
3475:
3469:
3439:
3380:
3379:
3366:
3365:
3355:
3337:
3325:
3315:
3309:
3299:
3274:
3268:
3214:
3213:
3163:
3162:
3147:
3117:
3116:
3094:
3093:
3086:
3078:
3068:
3062:
3060:
2990:
2989:
2923:
2913:
2889:
2882:
2860:
2859:
2838:
2836:
2807:
2806:
2797:
2771:
2741:
2732:
2708:
2699:
2657:
2656:
2614:
2613:
2547:
2537:
2513:
2506:
2485:
2457:
2442:
2399:
2390:
2370:
2344:
2314:
2284:
2283:
2263:
2233:
2232:
2212:
2185:
2184:
2175:
2174:
2157:
2119:
2118:
2109:
2080:
2048:
2047:
2032:
2015:
2006:
2000:
1991:
1957:
1956:
1933:
1932:
1923:
1897:
1864:
1863:
1822:
1821:
1806:
1788:
1778:
1772:
1764:
1755:
1752:
1720:
1683:
1682:
1612:
1611:
1589:
1588:
1581:
1573:
1563:
1557:
1526:
1525:
1503:
1477:
1425:
1424:
1339:
1338:
1328:
1303:
1256:
1255:
1245:
1230:
1201:
1183:
1125:
1124:
1114:
1105:
1043:
1042:
1032:
1023:
988:
964:
958:
937:
931:
909:
908:
902:
880:
879:
873:
775:
774:
764:
741:
740:
730:
664:
626:
595:
594:
588:
557:
556:
550:
515:
489:
488:
482:
458:
436:
435:
429:
408:
396:
386:
380:
373:
338:
337:
324:
323:
313:
296:
284:
274:
268:
243:
242:
227:
200:
199:
177:
176:
169:
167:
137:
117:
86:
85:
79:
48:
47:
41:
28:, who proposed the relationship in 1964.
3825:"Comment on Efficient diode-pumped Yb:Gd
2827:holds for the effective cross-sections:
2727:of sites and for any partial population
4196:Comment on Efficient diode-pumped Yb:Gd
3833:laser (Appl.Phys.Lett.88,221117(2006))"
3777:
3505:
3358:
3150:
2035:
1809:
460:
316:
230:
4142:
4140:
4080:
4078:
4028:
4026:
3651:Confirmation of the McCumber relation
7:
4192:G.Zhao; L.Su; J.Xua; H.Zeng (2007).
3901:
3899:
3897:
3806:P.C.Becker, N.A.Olson, J.R.Simpson.
3783:
3781:
3736:an excellent material for efficient
3802:
3800:
2820:{\displaystyle ~\sigma _{\rm {e}}~}
3621:
3618:
3495:
3477:
3381:
3367:
3239:
3236:
3224:
3218:
3164:
3118:
3095:
3030:
3027:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2861:
2808:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2426:{\displaystyle ~a_{s,j}(\omega )~}
2302:{\displaystyle ~(t,t+{\rm {d}}t)~}
2285:
2234:
2186:
2176:
2120:
2049:
1958:
1934:
1868:
1823:
1687:
1613:
1590:
1527:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1340:
1257:
1126:
1044:
910:
881:
856:Deduction of the McCumber relation
776:
742:
596:
558:
490:
437:
339:
325:
244:
201:
178:
87:
49:
14:
926:do not depend on the populations
919:{\displaystyle \sigma _{\rm {e}}}
890:{\displaystyle \sigma _{\rm {a}}}
453:is called "zero-line" frequency;
446:{\displaystyle \omega _{\rm {z}}}
842:Heisenberg uncertainty principle
3667:Fig.2. Cross-sections for Yb:Gd
2104:and the emission cross-section
1210:{\displaystyle a(\omega )n_{2}}
583:and the emission cross-section
3624:
3613:
3245:
3233:
3230:
3215:
3130:
3124:
3107:
3101:
3033:
3024:
3021:
2986:
2904:
2898:
2873:
2867:
2850:
2844:
2528:
2522:
2503:
2497:
2475:
2469:
2417:
2411:
2293:
2268:
2242:
2217:
2171:
2165:
2132:
2126:
2091:
2085:
1970:
1964:
1946:
1940:
1874:
1865:
1740:
1734:
1693:
1684:
1625:
1619:
1602:
1596:
1551:
1545:
1539:
1533:
1516:
1510:
1497:
1491:
1450:
1426:
1376:
1370:
1364:
1358:
1352:
1346:
1318:
1312:
1293:
1287:
1281:
1275:
1269:
1263:
1194:
1188:
1162:
1156:
1150:
1144:
1138:
1132:
1080:
1074:
1068:
1062:
1056:
1050:
1002:
996:
788:
782:
754:
748:
720:
714:
608:
602:
570:
564:
213:
207:
190:
184:
99:
93:
61:
55:
1:
4035:"Efficient diode-pumped Yb:Gd
3747:. With such a material, the
2258:during a short time interval
647:can be related to the lasers
3745:second law of thermodynamics
1011:{\displaystyle ~v(\omega )~}
812:In the case of an idealized
3886:: S240–S242. Archived from
3283:{\displaystyle \omega _{Z}}
2365:, for each sublevel number
499:{\displaystyle k_{\rm {B}}}
4279:
4158:Solid State Communications
3922:Journal of Applied Physics
3794:(4A), 954–957 (1964)
2097:{\displaystyle a(\omega )}
1468:Which can be rewritten as
864:Fig.1. Sketch of sublevels
838:natural spectral linewidth
4179:10.1016/j.ssc.2005.12.023
2759:{\displaystyle ~n_{s,j}~}
1911:{\displaystyle ~\omega ~}
640:{\displaystyle ~\omega ~}
529:{\displaystyle ~\omega ~}
3290:as solution of equation
2309:, assuming that at time
651:in such a way, that the
4215:Applied Physics Letters
4048:Applied Physics Letters
4017:10.1364/JOSAB.22.001605
3980:10.1364/JOSAB.23.001770
3838:Applied Physics Letters
3647:cross-section itself.
2720:{\displaystyle ~q_{s}~}
2150:probability of emission
1178:is that of absorption;
125:{\displaystyle \omega }
4122:10.1364/OPEX.14.000686
3890:on September 28, 2007.
3879:Chinese Optics Letters
3823:D. Kouznetsov (2007).
3716:
3710:
3634:
3454:
3425:
3284:
3254:
3042:
2821:
2786:
2760:
2721:
2685:
2427:
2379:
2359:
2329:
2303:
2252:
2201:
2142:
2098:
2069:
1980:
1912:
1883:
1702:
1459:
1211:
1172:
1090:
1012:
974:
947:
920:
891:
865:
795:
641:
615:
577:
530:
500:
467:
466:{\displaystyle \hbar }
447:
418:
359:
152:
126:
106:
68:
3872:D.Kouznetsov (2007).
3711:
3666:
3635:
3455:
3426:
3285:
3255:
3043:
2822:
2787:
2761:
2722:
2686:
2428:
2380:
2360:
2339:For each site number
2330:
2304:
2253:
2202:
2143:
2099:
2070:
1981:
1913:
1884:
1703:
1460:
1212:
1173:
1096:is spectral rate of
1091:
1013:
975:
973:{\displaystyle N_{2}}
948:
946:{\displaystyle N_{1}}
921:
892:
863:
846:effective temperature
796:
642:
616:
578:
531:
501:
468:
448:
419:
360:
153:
127:
107:
69:
3679:
3468:
3438:
3298:
3267:
3059:
2835:
2796:
2770:
2731:
2698:
2441:
2389:
2369:
2343:
2313:
2262:
2211:
2156:
2148:. (We keep the term
2108:
2079:
1990:
1922:
1896:
1719:
1476:
1229:
1219:spontaneous emission
1217:is spectral rate of
1182:
1104:
1022:
987:
957:
930:
901:
872:
834:black-body radiation
828:which preserves the
663:
625:
587:
549:
514:
481:
457:
428:
372:
166:
136:
116:
78:
40:
24:. It is named after
4228:2007ApPhL..90f6103Z
4206:(Appl. Phys. Lett.
4171:2006SSCom.137..451Y
4113:2006OExpr..14..686L
4061:2006ApPhL..88v1117L
4009:2005JOSAB..22.1605K
3972:2006JOSAB..23.1770M
3935:2002JAP....92..180Q
3915:R.S.Quimby (2002).
3851:2007ApPhL..90f6101K
3453:{\displaystyle ~T~}
2785:{\displaystyle ~a~}
2358:{\displaystyle ~s~}
2328:{\displaystyle ~t~}
1098:stimulated emission
151:{\displaystyle ~T~}
4263:Solid-state lasers
3738:solid-state lasers
3717:
3706:
3630:
3450:
3421:
3280:
3250:
3038:
2817:
2782:
2756:
2717:
2681:
2423:
2375:
2355:
2325:
2299:
2248:
2197:
2138:
2094:
2065:
1976:
1908:
1879:
1698:
1455:
1207:
1168:
1086:
1008:
970:
943:
916:
887:
866:
791:
637:
611:
573:
526:
508:Boltzmann constant
496:
463:
443:
414:
355:
148:
122:
102:
64:
22:solid-state lasers
4237:10.1063/1.2435314
4069:10.1063/1.2206150
3943:10.1063/1.1485112
3859:10.1063/1.2435309
3810:(Academic, 1999).
3704:
3611:
3608:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3543:
3508:
3449:
3443:
3420:
3417:
3414:
3408:
3405:
3402:
3399:
3391:
3331:
3303:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3185:
3182:
3174:
3134:
3084:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2930:
2877:
2816:
2801:
2781:
2775:
2755:
2736:
2716:
2703:
2655:
2652:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2561:
2558:
2554:
2452:
2449:
2446:
2422:
2394:
2378:{\displaystyle j}
2354:
2348:
2324:
2318:
2298:
2267:
2247:
2216:
2196:
2161:
2152:for the quantity
2137:
2113:
2059:
1995:
1975:
1927:
1907:
1901:
1862:
1859:
1856:
1853:
1850:
1846:
1833:
1794:
1770:
1751:
1745:
1730:
1727:
1724:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1638:
1629:
1579:
1555:
1487:
1484:
1481:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1240:
1237:
1234:
1167:
1109:
1085:
1027:
1007:
992:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
636:
630:
525:
519:
402:
349:
290:
254:
217:
147:
141:
18:McCumber relation
4270:
4242:
4241:
4239:
4189:
4183:
4182:
4144:
4135:
4134:
4124:
4082:
4073:
4072:
4030:
4021:
4020:
4003:(8): 1605–1619.
3990:
3984:
3983:
3966:(9): 1770–1775.
3953:
3947:
3946:
3912:
3906:
3903:
3892:
3891:
3869:
3863:
3862:
3820:
3811:
3804:
3795:
3785:
3749:perpetual motion
3715:
3713:
3712:
3707:
3705:
3700:
3689:
3639:
3637:
3636:
3631:
3629:
3628:
3627:
3609:
3606:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3554:
3553:
3548:
3544:
3542:
3541:
3532:
3531:
3522:
3509:
3504:
3500:
3499:
3498:
3487:
3482:
3481:
3480:
3459:
3457:
3456:
3451:
3447:
3441:
3430:
3428:
3427:
3422:
3418:
3415:
3412:
3406:
3403:
3400:
3397:
3396:
3392:
3390:
3386:
3385:
3384:
3373:
3372:
3371:
3370:
3356:
3343:
3342:
3336:
3332:
3330:
3329:
3320:
3319:
3310:
3301:
3289:
3287:
3286:
3281:
3279:
3278:
3259:
3257:
3256:
3251:
3249:
3248:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3183:
3180:
3179:
3175:
3173:
3169:
3168:
3167:
3156:
3148:
3135:
3133:
3123:
3122:
3121:
3110:
3100:
3099:
3098:
3087:
3085:
3083:
3082:
3073:
3072:
3063:
3047:
3045:
3044:
3039:
3037:
3036:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2931:
2929:
2928:
2927:
2918:
2917:
2907:
2894:
2893:
2883:
2878:
2876:
2866:
2865:
2864:
2853:
2839:
2826:
2824:
2823:
2818:
2814:
2813:
2812:
2811:
2799:
2791:
2789:
2788:
2783:
2779:
2773:
2765:
2763:
2762:
2757:
2753:
2752:
2751:
2734:
2726:
2724:
2723:
2718:
2714:
2713:
2712:
2701:
2690:
2688:
2687:
2682:
2680:
2679:
2653:
2650:
2649:
2648:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2559:
2556:
2555:
2553:
2552:
2551:
2542:
2541:
2531:
2518:
2517:
2507:
2496:
2495:
2468:
2467:
2450:
2447:
2444:
2432:
2430:
2429:
2424:
2420:
2410:
2409:
2392:
2384:
2382:
2381:
2376:
2364:
2362:
2361:
2356:
2352:
2346:
2334:
2332:
2331:
2326:
2322:
2316:
2308:
2306:
2305:
2300:
2296:
2289:
2288:
2265:
2257:
2255:
2254:
2249:
2245:
2238:
2237:
2214:
2206:
2204:
2203:
2198:
2194:
2190:
2189:
2180:
2179:
2159:
2147:
2145:
2144:
2139:
2135:
2125:
2124:
2123:
2111:
2103:
2101:
2100:
2095:
2074:
2072:
2071:
2066:
2064:
2060:
2058:
2054:
2053:
2052:
2041:
2033:
2020:
2019:
2010:
2005:
2004:
1993:
1985:
1983:
1982:
1977:
1973:
1963:
1962:
1961:
1939:
1938:
1937:
1925:
1917:
1915:
1914:
1909:
1905:
1899:
1888:
1886:
1885:
1880:
1878:
1877:
1860:
1857:
1854:
1851:
1848:
1847:
1845:
1838:
1834:
1832:
1828:
1827:
1826:
1815:
1807:
1796:
1795:
1793:
1792:
1783:
1782:
1773:
1771:
1769:
1768:
1756:
1753:
1749:
1743:
1728:
1725:
1722:
1707:
1705:
1704:
1699:
1697:
1696:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1639:
1637:
1630:
1628:
1618:
1617:
1616:
1605:
1595:
1594:
1593:
1582:
1580:
1578:
1577:
1568:
1567:
1558:
1554:
1532:
1531:
1530:
1519:
1505:
1504:
1485:
1482:
1479:
1464:
1462:
1461:
1456:
1454:
1453:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1345:
1344:
1343:
1333:
1332:
1308:
1307:
1262:
1261:
1260:
1250:
1249:
1238:
1235:
1232:
1216:
1214:
1213:
1208:
1206:
1205:
1177:
1175:
1174:
1169:
1165:
1131:
1130:
1129:
1119:
1118:
1107:
1095:
1093:
1092:
1087:
1083:
1049:
1048:
1047:
1037:
1036:
1025:
1017:
1015:
1014:
1009:
1005:
990:
979:
977:
976:
971:
969:
968:
952:
950:
949:
944:
942:
941:
925:
923:
922:
917:
915:
914:
913:
896:
894:
893:
888:
886:
885:
884:
832:formula for the
818:detailed balance
800:
798:
797:
792:
781:
780:
779:
769:
768:
747:
746:
745:
735:
734:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
646:
644:
643:
638:
634:
628:
620:
618:
617:
612:
601:
600:
599:
582:
580:
579:
574:
563:
562:
561:
535:
533:
532:
527:
523:
517:
505:
503:
502:
497:
495:
494:
493:
472:
470:
469:
464:
452:
450:
449:
444:
442:
441:
440:
423:
421:
420:
415:
413:
412:
407:
403:
401:
400:
391:
390:
381:
364:
362:
361:
356:
354:
350:
348:
344:
343:
342:
331:
330:
329:
328:
314:
301:
300:
295:
291:
289:
288:
279:
278:
269:
259:
255:
253:
249:
248:
247:
236:
228:
218:
216:
206:
205:
204:
193:
183:
182:
181:
170:
157:
155:
154:
149:
145:
139:
131:
129:
128:
123:
111:
109:
108:
103:
92:
91:
90:
73:
71:
70:
65:
54:
53:
52:
4278:
4277:
4273:
4272:
4271:
4269:
4268:
4267:
4248:
4247:
4246:
4245:
4210:, 066101 2007)"
4203:
4199:
4191:
4190:
4186:
4154:
4150:
4146:
4145:
4138:
4094:
4090:
4084:
4083:
4076:
4042:
4038:
4032:
4031:
4024:
3992:
3991:
3987:
3955:
3954:
3950:
3914:
3913:
3909:
3904:
3895:
3871:
3870:
3866:
3832:
3828:
3822:
3821:
3814:
3805:
3798:
3786:
3779:
3774:
3758:
3754:
3735:
3731:
3726:
3722:
3690:
3677:
3676:
3674:
3670:
3661:
3653:
3533:
3523:
3517:
3516:
3489:
3488:
3471:
3466:
3465:
3436:
3435:
3375:
3374:
3361:
3357:
3351:
3321:
3311:
3305:
3304:
3296:
3295:
3270:
3265:
3264:
3158:
3157:
3149:
3143:
3112:
3111:
3089:
3088:
3074:
3064:
3057:
3056:
2919:
2909:
2908:
2885:
2884:
2855:
2854:
2840:
2833:
2832:
2802:
2794:
2793:
2768:
2767:
2737:
2729:
2728:
2704:
2696:
2695:
2543:
2533:
2532:
2509:
2508:
2481:
2453:
2439:
2438:
2395:
2387:
2386:
2367:
2366:
2341:
2340:
2311:
2310:
2260:
2259:
2209:
2208:
2154:
2153:
2114:
2106:
2105:
2077:
2076:
2043:
2042:
2034:
2028:
2011:
1996:
1988:
1987:
1952:
1928:
1920:
1919:
1894:
1893:
1817:
1816:
1808:
1802:
1797:
1784:
1774:
1760:
1754:
1717:
1716:
1607:
1606:
1584:
1583:
1569:
1559:
1556:
1521:
1520:
1506:
1474:
1473:
1334:
1324:
1299:
1251:
1241:
1227:
1226:
1197:
1180:
1179:
1120:
1110:
1102:
1101:
1038:
1028:
1020:
1019:
985:
984:
960:
955:
954:
933:
928:
927:
904:
899:
898:
875:
870:
869:
858:
810:
808:Idealized atoms
770:
760:
736:
726:
661:
660:
623:
622:
590:
585:
584:
552:
547:
546:
542:
512:
511:
484:
479:
478:
475:Planck constant
455:
454:
431:
426:
425:
392:
382:
376:
375:
370:
369:
333:
332:
319:
315:
309:
280:
270:
264:
263:
238:
237:
229:
223:
195:
194:
172:
171:
164:
163:
134:
133:
114:
113:
81:
76:
75:
43:
38:
37:
34:
12:
11:
5:
4276:
4274:
4266:
4265:
4260:
4250:
4249:
4244:
4243:
4201:
4197:
4184:
4165:(8): 451–455.
4152:
4148:
4136:
4107:(2): 686–695.
4100:Optics Express
4092:
4088:
4074:
4055:(22): 221117.
4040:
4036:
4022:
3985:
3948:
3929:(1): 180–187.
3907:
3893:
3864:
3830:
3826:
3812:
3796:
3776:
3775:
3773:
3770:
3756:
3752:
3733:
3729:
3724:
3720:
3703:
3699:
3696:
3693:
3687:
3684:
3672:
3668:
3660:
3657:
3652:
3649:
3641:
3640:
3626:
3623:
3620:
3615:
3605:
3552:
3547:
3540:
3536:
3530:
3526:
3520:
3515:
3512:
3507:
3503:
3497:
3492:
3485:
3479:
3474:
3446:
3434:the subscript
3432:
3431:
3411:
3395:
3389:
3383:
3378:
3369:
3364:
3360:
3354:
3349:
3346:
3341:
3335:
3328:
3324:
3318:
3314:
3308:
3277:
3273:
3261:
3260:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3188:
3178:
3172:
3166:
3161:
3155:
3152:
3146:
3141:
3138:
3132:
3129:
3126:
3120:
3115:
3109:
3106:
3103:
3097:
3092:
3081:
3077:
3071:
3067:
3049:
3048:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2988:
2926:
2922:
2916:
2912:
2906:
2903:
2900:
2897:
2892:
2888:
2881:
2875:
2872:
2869:
2863:
2858:
2852:
2849:
2846:
2843:
2810:
2805:
2778:
2750:
2747:
2744:
2740:
2711:
2707:
2692:
2691:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2564:
2550:
2546:
2540:
2536:
2530:
2527:
2524:
2521:
2516:
2512:
2505:
2502:
2499:
2494:
2491:
2488:
2484:
2480:
2477:
2474:
2471:
2466:
2463:
2460:
2456:
2419:
2416:
2413:
2408:
2405:
2402:
2398:
2374:
2351:
2321:
2295:
2292:
2287:
2282:
2279:
2276:
2273:
2270:
2244:
2241:
2236:
2231:
2228:
2225:
2222:
2219:
2193:
2188:
2183:
2178:
2173:
2170:
2167:
2164:
2134:
2131:
2128:
2122:
2117:
2093:
2090:
2087:
2084:
2063:
2057:
2051:
2046:
2040:
2037:
2031:
2026:
2023:
2018:
2014:
2009:
2003:
1999:
1972:
1969:
1966:
1960:
1955:
1951:
1948:
1945:
1942:
1936:
1931:
1904:
1890:
1889:
1876:
1873:
1870:
1867:
1844:
1841:
1837:
1831:
1825:
1820:
1814:
1811:
1805:
1800:
1791:
1787:
1781:
1777:
1767:
1763:
1759:
1748:
1742:
1739:
1736:
1733:
1709:
1708:
1695:
1692:
1689:
1686:
1636:
1633:
1627:
1624:
1621:
1615:
1610:
1604:
1601:
1598:
1592:
1587:
1576:
1572:
1566:
1562:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1529:
1524:
1518:
1515:
1512:
1509:
1502:
1499:
1496:
1493:
1490:
1466:
1465:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1342:
1337:
1331:
1327:
1323:
1320:
1317:
1314:
1311:
1306:
1302:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1259:
1254:
1248:
1244:
1204:
1200:
1196:
1193:
1190:
1187:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1128:
1123:
1117:
1113:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1046:
1041:
1035:
1031:
1004:
1001:
998:
995:
967:
963:
940:
936:
912:
907:
883:
878:
857:
854:
814:two-level atom
809:
806:
802:
801:
790:
787:
784:
778:
773:
767:
763:
759:
756:
753:
750:
744:
739:
733:
729:
725:
722:
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4258:Spectroscopy
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850:temperature
4252:Categories
3772:References
3761:black body
826:absorption
132:, and let
32:Definition
3702:ω
3695:π
3683:λ
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2036:ℏ
1968:ω
1954:σ
1944:ω
1930:σ
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1840:−
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758:−
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231:ℏ
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188:ω
174:σ
120:ω
97:ω
83:σ
59:ω
45:σ
4131:19503386
822:emission
820:for the
4224:Bibcode
4167:Bibcode
4109:Bibcode
4057:Bibcode
4005:Bibcode
3968:Bibcode
3931:Bibcode
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3675:versus
506:is the
473:is the
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953:and
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3671:SiO
3348:exp
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306:exp
220:exp
4254::
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