Broadband laser materials and the McCumber relation

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
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 expresses the emission cross-seccion $$~\sigma_{\rm e}(\omega)~$$ in terms of the absorption cross-section $$~\sigma_{\rm a}(\omega)~$$: $$ \sigma_{\rm e}(\omega)~=~\sigma_{\rm a}(\omega)\!~\exp\!\left(\hbar\frac {\omega_{\rm Z}\!-\!\omega}{k_{\rm B}T} \right), {(\rm mc)} $$ where $$~T~$$ is temperature, $$~k_{\rm B}~$$ is Boltzmann constant and $$~\omega_{\rm Z}~$$ is zero-line frequency, at which the emission and absorption cross-sections are equal. The relation (mc) is validated for various media.

The original deduction of the McCumber relation , as well as the adaptation in the textbook 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   as an indication, that the effective cross-sections of broad-band composite materials have no need to satisfy the McCumber relation: the peak of $$~\sigma_{\rm a}~$$ at wavelength 950 nm corresponds to the gap of $$~\sigma_{\rm e}~$$ (See Fig.0.)

However, a medium with such effective cross-sections 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 was suggested (thin black curve in Fig.0) and confirmed. In order to avoid such confusions, the deduction of the McCumber relation should be generalized.

After the presentation , I was asked for the general deduction of the McCumber relation as a substitute of the speculation about the 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  , 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
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 and gain are determined by the populations $$~n_1$$ and $$~n_2~$$ of the the laser levels. In this case, and only in this case, the effective cross-sections $$~\sigma_{\rm a}(\omega)~$$ and $$~\sigma_{\rm e}(\omega)~$$ of absorption and emission have sense.

Thermalization
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 $$ g(\omega) = n_2\!~\sigma_{\rm e}(\omega) - n_1\!~ \sigma_{\rm a}(\omega),~ {\rm  (g)} $$ where $$~n_1~$$ and $$~n_2~$$ are population of lower and upper laser levels.

Keeping the consideration phenomenological, the spontaneous emission can be characterized with the Einstein coefficients   ; the rate of emission of spontaneous photons at frequency $$\omega$$ can be expressed as $$ a(\omega)n_2, {\rm (a)} $$ where $$~a(\omega)~$$ is probability of spontaneous emission by a random active center per time per frequency, assuming that it is excited. $$~a(\omega)~$$ is equivalent of the Einstein coefficient $$~A_{21}~$$. Notation $$~a~$$ is used here to avoid confusion with the Einstein coefficient $$~A_{\rm 21}~$$, which has no established expression (see notes at Table 7.7 of ); not only value, but even dimensions of the Einstein coefficients depend on scale we use: frequencies or wavelengths.

Decay
The decay rate $$~1/\tau~$$ of the excited level can be expressed in terms of the coefficient $$~a~$$: $$ \frac{1}{\tau}=\int_{0}^{\infty} a(\omega)~ {\rm d} \omega~ {\rm (tau)} $$

The cross-section $$~\sigma_{\rm a}(\omega)~$$ and $$~\sigma_{\rm e}(\omega)~$$ and the coefficient $$~a(\omega)~$$ do not depend on the populations $$~n_1~$$ and $$~n_2~$$ of the active medium and the density $$~D(\omega)~$$ of photons of frequency $$~\omega~$$. In this approximation, the properties of the medium are determined by 3 functions $$~\sigma_{\rm a}(\omega)~$$,$$~\sigma_{\rm e}(\omega)~$$ and $$~a(\omega)~$$, 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 $$~n_1~$$ and $$~n_2~$$. The functions $$~a(\omega)~$$, $$~\sigma_{\rm a}(\omega)~$$ and $$~\sigma_{\rm e}(\omega)~$$ 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   is rewritten, taking into account sublevels (Fig.1).

Detailed balance
Functions $$~\sigma_{\rm a}(\omega)~$$,$$~\sigma_{\rm e}(\omega)~$$ and $$~a(\omega)~$$ of frequency $$~\omega~$$ 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 $$~\omega~$$ is equal to that of absorption.

Consider a thermal state. Let $$~v(\omega)~$$ be croup velocity of light in the medium.

The product $$~n_2\sigma_{\rm e}(\omega) v(\omega)D(\omega)~$$ is spectral rate of stimulated emission, and $$~n_1\sigma_{\rm a}(\omega) v(\omega)D(\omega)~$$ is that of absorption; $$a(\omega)n_2$$ 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: $$ n_2\sigma_{\rm e}(\omega) v(\omega)D(\omega)+n_2 a(\omega)= n_1\sigma_{\rm a}(\omega) v(\omega)D(\omega) {\rm (balance)} $$

Rewrite it as $$ D(\omega)= \frac{\frac{a(\omega)}{\sigma_{\rm e}(\omega) v(\omega)} } {\frac{n_1}{n_2} \frac{\sigma_{\rm a}(\omega)}{\sigma_{\rm e}(\omega)}-1} {\rm (D1)} $$

The thermal distribution of density of photons follows from blackbody radiation $$ 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)} $$

Both (D1) and (D2)) hold for all frequencies $$~\omega~$$. For the case of idealized two-level active centers, $$~\sigma_{\rm a}(\omega)=\sigma_{\rm e}(\omega)~$$, and $$~n_1/n_2=\exp\!\left( \frac{\hbar\omega}{k_{\rm B}T} \right)$$, which leads to the relation between the spectral rate of spontaneous emission $$a(\omega)$$ and the emission cross-section $$~\sigma_{\rm e}(\omega)~$$  . (We keep the term probability of emission for the quantity $$~a(\omega){\rm d}\omega{\rm d}t~$$, which is probability of emission of a photon within small spectral interval $$~(\omega,\omega+{\rm d}\omega)~$$ during a short time interval $$~(t,t+{\rm d}t)~$$, assuming that at time $$~t~$$ 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 $$~s~$$, for each sublevel number $$j$$, the partial spectral emission probability $$~a_{s,j}(\omega)~$$ can be expressed from consideration of idealized two-level atoms : $$ a_{s,j}(\omega)=\sigma_{s,j}(\omega) \frac{\omega^2 v(\omega)}{\pi^2c^3}. ~{\rm comparison1} {\rm partial} $$

Neglecting the cooperative coherent effects, the emission is additive: for any concentration $$~q_{s}~$$ of sites and for any partial population $$~n_{s,j}~$$ of sublevels, the same proportionality between $$~a~$$ and $$~\sigma_{\rm e}~$$ holds for the effective cross-sections: $$ \frac{a(\omega)}{\sigma_{\rm e}(\omega)}= \frac{\omega^2 v(\omega)}{\pi^2c^3} (\rm comparison)(av) $$

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

$$ \frac{n_1}{n_2} \frac{\sigma_{\rm a}(\omega)} {\sigma_{\rm e}(\omega)}= \exp\!\left( \frac{\hbar\omega}{k_{\rm B}T}\right). {\rm (n1n2) (mc1)} $$

This relation is equivalent of the McCumber relation (mc), if we define the zero-line frequency $$\omega_{Z}$$ as solution of equation

$$~\left(\frac{n_2}{n_1}\right)_{\!T}= \exp\!\left(\frac{\hbar \omega_{\rm Z}}{k_{\rm B}T}\right)~, $$

the subscript $$~T~$$ indicates that the ratio of populations in evaluated in the thermal state. The zero-line frequency can be expressed as $$ \omega_{\rm Z}=\frac{k_{\rm B}T}{\hbar} \ln \left(\frac{n_1}{n_2}\right)_{T} ~.{(\rm oz)} $$ 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
The zero-line frequency is determined by (oz) in terms of ratio $$~(n_2/n_1)_{T}~$$ of populations of levels at given thermal state with temperature $$T$$. In general, $$~\omega_{\rm Z}~$$ 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 $$~U~$$ be total number of sublevels in the system. Let the variable $$~j~$$ numerate these sublevels. Let first $$~L~$$ sublevels be in the lower level, they correspond to values $$~0\le j\le L\!-\!1~$$. The following $$~U\!-\!L~$$ sublevels belong to the upper level; they correspond to $$~L\le j\le U\!-\!1~$$. Let $$~\varepsilon_j~$$ be energy of $$~j~$$th sublevel. Then, the thermal-equilibrium ratio of populations can be expressed as follows: $$ \left(\frac{n_2}{n_1}\right)_{\!T}= \frac {\sum_{j=L}^{U-1}\exp\!\left(-\frac{\hbar\varepsilon_j}{K_{\rm B} T } \right)} {\sum_{j=0}^{L-1}\exp\!\left(-\frac{\hbar\varepsilon_j}{K_{\rm B} T } \right)} \approx \exp\!\left(-\frac{\hbar\varepsilon_L}{K_{\rm B} T } \right), {\rm (mono)} $$

Sites
For a medium with different active sites (Fig.1), let $$~s~$$ numerate the kinds of a site. Let $$~q_{s}~$$ be concentration of $$~s~$$th site, and $$~\varepsilon_{q,j}~$$ be energy of the $$~j$$-th sublevel at $$~s~$$th site. Then, $$ \left(\frac{n_1}{n_0}\right)_{\!T}= \frac {~\sum_{s} ~q_{s}~ \frac {\sum_{j=L}^{U-1}\exp\!\left(-\frac{\hbar\varepsilon_{s,j}}{K_{\rm B} T } \right)} {\sum_{j=0}^{U-1}\exp\!\left(-\frac{\hbar\varepsilon_{s,j}}{K_{\rm B} T } \right)} ~} {~\sum_{s} ~q_{s}~ \frac {\sum_{j=0}^{L-1}\exp\!\left(-\frac{\hbar\varepsilon_{s,j}}{K_{\rm B} T } \right)} {\sum_{j=0}^{U-1}\exp\!\left(-\frac{\hbar\varepsilon_{s,j}}{K_{\rm B} T } \right)} ~} (\rm multi) $$ At small temperatures, $$~ \frac{\hbar\varepsilon_{q,L+1}-\hbar\varepsilon_{q,L}}{k_{\rm B}T}\gg 1,$$ $$~ \frac{\hbar\varepsilon_{q,1}-\hbar\varepsilon_{q,0}}{k_{\rm B}T}\gg 1,~$$ 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, $$ \sigma_{\rm e}(\omega)=\frac{\pi^2c^3}{\omega^2 v(\omega)}a(\omega). $$ The integral of $$~\sigma_{\rm e}(\omega)~$$ can be checked using Eq.(tau), while the lifetime $$~\tau~$$ 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 $$ \left(\frac{n_1}{n_2}\right)_T =\frac{\sigma_{\rm e}(\omega)}{\sigma_{\rm a}(\omega)}\exp\!\left(\frac{\hbar\omega} {k_{\rm B}T}\right) {\rm (check)} $$ 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    may indicate, that the effective emission cross-section $$~\sigma_{\rm e}(\omega)~$$ 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:Gd$2$SiO$5$. The strong violation of the McCumber relation in graphics presented by  can be attributed also to the errors at the measurement of $$\sigma_{\rm e}$$ caused by the reabsorption in vicinity of the zero-line.

Conclusion
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 $$~\sigma_{\rm e}(\omega)~$$; 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
Author is grateful to Jean-François Bisson, Susanne T. Fredrich-Thornton, Ken-ichi Ueda, Akira Shirakawa and Alexander Kaminskii for the important discussion.