Broadband laser materials and the McCumber relation
Broadband laser materials and the McCumber relation by 
Copyleft 2007 by Author. Published in Chinese Optics Letters , Vol.05 , Issue S1 , PP.240242(2007)
http://col.org.cn/abstract.aspx?id=COL05S1S2403 
The McCumber relation can be deduced without assuming that all active centers have the same structure of sublevels. The range of validity of the McCumber relation is the same as that of the effective emission crosssection.
Contents
Introduction[edit]
The concept of effective crosssections allow to treat the laser medium as a twolevel system. Such a concept is widely used in physics of solidstate lasers; often, these effective crosssections are called simply crosssections. The McCumber relation ^{[1]}^{[2]} expresses the emission crossseccion in terms of the absorption crosssection :
where is temperature, is Boltzmann constant and is zeroline frequency, at which the emission and absorption crosssections are equal. The relation (mc) is validated for various media ^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}.
The original deduction of the McCumber relation ^{[1]}, as well as the adaptation in the textbook ^{[2]} assume, that all active centers are equal. It cannot be applied as is to the broadband laser materials with different sites of the active centers.
This allowed the interpretation of results for Yb:Gd_{2}SiO_{5} by cites ^{[6]}^{[7]}^{[8]} as an indication, that the effective crosssections of broadband composite materials have no need to satisfy the McCumber relation: the peak of at wavelength 950 nm corresponds to the gap of (See Fig.0.)
However, a medium with such effective crosssections ^{[6]}^{[7]}^{[8]} would be good not only for an efficient laser, but also for a Perpetual Motion of Second Kind; the correction of the emission crosssection ^{[9]}^{[10]} was suggested (thin black curve in Fig.0) and confirmed^{[11]}. In order to avoid such confusions, the deduction of the McCumber relation should be generalized.
After the presentation ^{[9]}, I was asked for the general deduction of the McCumber relation as a substitute of the speculation ^{[10]}about the w:gedanken experiment with perpetual motion. Below, such a deduction is suggested.
In this paper, the generalization of the deduction of the McCumber relation is suggested. I show, that the McCumber relation follows from the fundamental properties of the Einstein coefficients ^{[12]}^{[13]}^{[14]}^{[15]}, and applies to any material with fast transitions within each of two sets of levels and relatively slow transitions between these two sets.
Active centers[edit]
The sketch of sublevels of active centers is shown in fig.1. Consider two subsets of quantum states: level 1 and level 2. Assume slow optical transitions from level 1 to level 2.
(This property makes the medium suitable for a laser action.)
Assume quick transfer of energy between neighbors, which leads to the fast thermalization within each of laser levels.
Then, the refractive index ^{[16]} and gain ^{[5]} are determined by the populations and of the the laser levels. In this case, and only in this case, the effective crosssections and of absorption and emission have sense.
Thermalization[edit]
Use of effective crosssections assumes the thermalization of quantum states within each of laser levels. However, the population of the laser levels can be far from a thermal state, allowing the lasing. The gain can be expressed as
where and are population of lower and upper laser levels.
Keeping the consideration phenomenological, the spontaneous emission can be characterized with the Einstein coefficients ^{[12]}^{[17]}^{[18]}^{[19]}; the rate of emission of spontaneous photons at frequency can be expressed as
where is probability of spontaneous emission by a random active center per time per frequency, assuming that it is excited. is equivalent of the Einstein coefficient . Notation is used here to avoid confusion with the Einstein coefficient , which has no established expression (see notes at Table 7.7 of ^{[13]}); not only value, but even dimensions of the Einstein coefficients depend on scale we use: frequencies or wavelengths.
Decay[edit]
The decay rate of the excited level can be expressed in terms of the coefficient :
The crosssection and and the coefficient do not depend on the populations and of the active medium and the density of photons of frequency . In this approximation, the properties of the medium are determined by 3 functions , and , and we have no need to consider noninear processes \cite{desu} which produced a given population; as gain, as refraction index as function of frequency are determined by the populations and . The functions , and are equivalent of the Einstein coefficients, but have an advantage: their values do not depend on system of notations. In the following, the consideration of relations between Einstein coefficients ^{[12]}^{[13]}^{[14]}^{[15]} is rewritten, taking into account sublevels (Fig.1).
Detailed balance[edit]
Functions , and of frequency are related, as the Einstein coefficients are. These relations can be found from the principle of detailed balance.
Although the expression (g) is good for a nonequilibrium medium, it is valid also at the thermal equilibrium, when the spectral rate of emission (both spontaneous and stimulates) of photons at any frequency is equal to that of absorption.
Consider a thermal state. Let be croup velocity of light in the medium.
The product is spectral rate of stimulated emission, and is that of absorption; is spectral rate of spontaneous emission. (Note that in this approximation, there is no such thing as a spontaneous absorption.)
The balance of photons gives:
Rewrite it as
The thermal distribution of density of photons follows from blackbody radiation ^{[13]}
Both (D1) and (D2)) hold for all frequencies . For the case of idealized twolevel active centers, , and , which leads to the relation between the spectral rate of spontaneous emission and the emission crosssection ^{[13]}. (We keep the term w:probability of emission for the quantity , which is probability of emission of a photon within small spectral interval during a short time interval , assuming that at time the atom is excited.) The relation (D2) is 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.
For each site number , for each sublevel number , the partial spectral emission probability can be expressed from consideration of idealized twolevel atoms ^{[13]}:
Neglecting the cooperative coherent effects, the emission is additive: for any concentration of sites and for any partial population of sublevels, the same proportionality between and holds for the effective crosssections:
Then, the comparison of (D1) and (D2) gives the relation
This relation is equivalent of the McCumber relation (mc), if we define the zeroline frequency as solution of equation
the subscript indicates that the ratio of populations in evaluated in the thermal state. The zeroline frequency can be expressed as
Then, (n1n2) becomes equivalent of the McCumber relation (mc).
We see, 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 crosssection itself.
Thermal ratio of populations[edit]
The zeroline frequency is determined by (oz) in terms of ratio of populations of levels at given thermal state with temperature . In general, depends on the temperature. This dependence can be expressed explicitly in terms of energies of sublevels.
Consider first the homogeneous medium, and numerate the sublevels as it is shown in Fig.2. Let be total number of sublevels in the system. Let the variable numerate these sublevels. Let first sublevels be in the lower level, they correspond to values . The following sublevels belong to the upper level; they correspond to . Let be energy of th sublevel. Then, the thermalequilibrium ratio of populations can be expressed as follows:
Sites[edit]
For a medium with different active sites (Fig.1), let numerate the kinds of a site. Let be concentration of th site, and be energy of the th sublevel at th site. Then,
At small temperatures, and the only zeroth term is important in the summation. It is typical case for the Ybdoped laser materials, when the zeroline frequency corresponds to the transition between the lowest sublevels.
The use of the formal expression (mono) and, especially, (multi) requires the knowledge of the energy of sublevels. It may be practical, to determine the emission crosssection from the spectrum of the spontaneous emission, (which is easier to measure), using Eq.(comparison). Then,
The integral of can be checked using Eq.(tau), while the lifetime is known. Then, the zeroline can be determined, comparing the ratio of the crosssections to the exponential in the righthand side of Eq.(n1n2). The deviation of the righthand side of the expression
from a constant is a measure of the error of a description of a process in terms of the effective emission crosssection. The strong deviation ^{[6]}^{[7]}^{[8]}^{[10]}^{[9]} may indicate, that the effective emission crosssection has no sense, and more detailed kinetic of excitations of various sites (or may be even subleveles) should be taken into account. Until now, there is no evidence that the concept of the effective crosssections does not apply to Yb:Gd_{2}SiO_{5} . The strong violation of the McCumber relation in graphics presented by ^{[6]}^{[7]}^{[8]} can be attributed also to the errors at the measurement of caused by the reabsorption in vicinity of the zeroline.
Conclusion[edit]
The McCumber relation (mc) follows from the assumption of fast redistribution of energy among laser sublevels. Only in this case, the effective crosssections can be used to characterize the laser medium.
The deduction suggested applies to broadband materials with different sites. This approximation will be broken at low concentration of the active centers, as well as at the excitation with very strong and short pulses. In both cases, the different sites interact with electromagnetic field faster than they exchange the energy. In any of these cases, the medium cannot be characterized with the singlevalued emission crosssection function ; the effective crosssections should be defined for each site, and the kinetic of the transfer of the excitations should be considered.
The effective emission crosssection and the McCumber relation have the same range of validity. The deviation from a constant of the righthand side of the estimate (check) for the steadystate ratio of populations characterizes the error of measurement of the effective crosssections.
Acknowledgment[edit]
Author is grateful to JeanFrançois Bisson, Susanne T. FredrichThornton, Kenichi Ueda, Akira Shirakawa and Alexander Kaminskii for the important discussion.
References[edit]
 ↑ ^{1.0} ^{1.1} ^{1.2} D.E.McCumber. Einstein relations connecting broadband emission and absorption spectra. PRB 136 (4A), 954957 (1964)
 ↑ ^{2.0} ^{2.1} ^{2.2} P.C.Becker, N.A.Olson, J.R.Simpson. Erbiumdoped fiber amplifiers: fundamentals and theory (Academic, 1999).
 ↑ R.S.Quimby. Range of validity of McCumber theory in relating absorption and emission cross sections,J. Appl. Phys.92, 180187 (2002) doi:10.1063/1.1485112 ; http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JAPIAU000092000001000180000001
 ↑ R.M.Martin, R.S.Quimby. Experimental evidence of the validity of the McCumber theory relating emission and absorption for rareearth glasses", JOSAB, 23, (9): 17701775 (2006)
 ↑ ^{5.0} ^{5.1} D. Kouznetsov, J. F. Bisson, K. Takaichi, K. Ueda. Highpower single mode solid state laser with short unstable cavity. JOSAB 22, 16051619 (2005), http://josab.osa.org/abstract.cfm?id=84730
 ↑ ^{6.0} ^{6.1} ^{6.2} ^{6.3} W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, J. Xu. Efficient diodepumped Yb:Gd_{2}SiO_{5} laser. APL 88, 221117 (2006), http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APPLAB000088000022221117000001
 ↑ ^{7.0} ^{7.1} ^{7.2} ^{7.3} W. Li, H. Pan, L. Ding, H. Zeng, G. Zhao, C. Yan, L. Su, J. Xu. Diodepumped continuouswave and passively modelocked Yb:Gd_{2}SiO_{5}laser. Optics Express 14, 686695 (2006) http://www.opticsexpress.org/search.cfm
 ↑ ^{8.0} ^{8.1} ^{8.2} ^{8.3} C. Yan, G. Zhao, L. Zhang, J. Xu, X. Liang, D. Juan, W. Li, H. Pan, L. Ding, H. Zeng. A new Ybdoped oxyorthosilicate laser crystal: Yb:Gd_{2}SiO_{5}. Solid State Communications 137, 451455 (2006), doi:10.1016/j.ssc.2005.12.023, http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVW4HYTYFW3&_user=10&_coverDate=02%2F28%2F2006&_alid=615430684&_rdoc=1&_fmt=summary&_orig=search&_cdi=5545&_sort=d&_docanchor=&view=c&_ct=1&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=b6673cac47762a0184b56a206d31ccd6
 ↑ ^{9.0} ^{9.1} ^{9.2} D. Kouznetsov. Broadband laser materials and the McCumber relation. 5th Asia Pacific Laser Symposium, presentation VI12 (Guillin, China, Nov.2327, 2006);
 ↑ ^{10.0} ^{10.1} ^{10.2} D. Kouznetsov. Comment on Efficient diodepumped Yb:Gd_{2}SiO_{5} laser. APL 90, 066101 (2007), http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APPLAB000090000006066101000001&idtype=cvips&gifs=yes
 ↑ G.Zhao, L.Su, J.Xua, and H.Zeng. Response to Comment on Efficient diodepumped Yb:Gd_{2}SiO_{5} laser (Appl. Phys. Lett. 90, 066101 2007).  APL 90, 066103 (2007), http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APPLAB000090000006066103000001&idtype=cvips&gifs=yes
 ↑ ^{12.0} ^{12.1} ^{12.2} R. C. Hilborn. ``Einstein coefficients, cross sections, values, dipole moments, and all that". Am. J. Phys. 50, 982946 (1982), http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=AJPIAS&CURRENT=NO&ONLINE=YES&smode=strresults&sort=rel&maxdisp=25&threshold=0&pjournals=AJPIAS&pyears=2001%2C2000%2C1999&possible1=982&possible1zone=fpage&fromvolume=50&SMODE=strsearch&OUTLOG=NO&viewabs=AJPIAS&key=DISPLAY&docID=1&page=1&chapter=0
 ↑ ^{13.0} ^{13.1} ^{13.2} ^{13.3} ^{13.4} ^{13.5} \bibitem{e2} A. Thorne, U. Litzen, S. Johansson. {\em Spectrophysics: principles and applications.} (SpringerVerlag, 1999)
 ↑ ^{14.0} ^{14.1} H. Haken. {\em Laser Theory}. (SpringerVerlag, 1984)
 ↑ ^{15.0} ^{15.1} P. W. Millobi, J. H. Eberly. {\em Lasers}. (John Wiley & Sons, 1988)
 ↑ J.F. Bisson and D. Kouznetsov. Saturation Broadening and KramersKronig Relations in QuasiThreeLevel Lasers". J. of Lightwave Techn.
 ↑ A.Thorne, U.Litzen, S.Johansson. Spectrophysics: principles and applications. (SpringerVerlag, 1999)
 ↑ H.Haken. {\em Laser Theory}. (SpringerVerlag, 1984)
 ↑ P.W.Millobi, J.H.Eberly. {\em Lasers}. (John Wiley \& Sons, 1988)
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