Broadband laser materials and the McCumber relation

Broadband laser materials and the McCumber relation
by Dmitrii Kouznetsov
130078Broadband laser materials and the McCumber relationDmitrii Kouznetsov


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 cross-section.

Introduction edit

The concept of effective cross-sections allow to treat the laser medium as a two-level system. Such a concept is widely used in physics of solid-state lasers; often, these effective cross-sections are called simply cross-sections. The McCumber relation [1][2] expresses the emission cross-seccion   in terms of the absorption cross-section  :

 

where   is temperature,   is Boltzmann constant and   is zero-line frequency, at which the emission and absorption cross-sections 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.

 
Fig.0. Cross-sections for Yb:Gd2SiO5

This allowed the interpretation of results for Yb:Gd2SiO5 by cites [6][7][8] as an indication, that the effective cross-sections of broad-band 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 cross-sections [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 cross-section [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.

 
Fig.1. Sketch of sublevels

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 cross-sections   and   of absorption and emission have sense.

Thermalization edit

Use of effective cross-sections 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][13][14][15]; 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 cross-section   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 non-equilibrium 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 two-level active centers,  , and  , which leads to the relation between the spectral rate of spontaneous emission   and the emission cross-section   [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 two-level 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 cross-sections:

 

Then, the comparison of (D1) and (D2) gives the relation

 

This relation is equivalent of the McCumber relation (mc), if we define the zero-line frequency   as solution of equation

 

the subscript   indicates that the ratio of populations in evaluated in the thermal state. The zero-line 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 cross-section itself.

Thermal ratio of populations edit

The zero-line 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.

 
Fig.2. Numeration 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 thermal-equilibrium 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 Yb-doped laser materials, when the zero-line 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 cross-section 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 zero-line can be determined, comparing the ratio of the cross-sections to the exponential in the right-hand side of Eq.(n1n2). The deviation of the right-hand 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 cross-section. The strong deviation [6][7][8][10][9] may indicate, that the effective emission cross-section   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 cross-sections does not apply to Yb:Gd2SiO5 . 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 zero-line.

Conclusion edit

The McCumber relation (mc) follows from the assumption of fast redistribution of energy among laser sublevels. Only in this case, the effective cross-sections 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 single-valued emission cross-section function  ; the effective cross-sections should be defined for each site, and the kinetic of the transfer of the excitations should be considered.

The effective emission cross-section and the McCumber relation have the same range of validity. The deviation from a constant of the right-hand side of the estimate (check) for the steady-state ratio of populations characterizes the error of measurement of the effective cross-sections.

Acknowledgment edit

Author is grateful to Jean-François Bisson, Susanne T. Fredrich-Thornton, Ken-ichi Ueda, Akira Shirakawa and Alexander Kaminskii for the important discussion.

References edit

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