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McCumber relation

The McCumber relation (or McCumber theory) refers to the effective cross-sections of absorption and emission of light in the physics of solid-state lasers ^[1]^[2].

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1 Definition
2 Gain
3 Idealized atoms
4 Deduction of the McCumber relation
- 4.1 Confirmation of the McCumber relation
5 Violation of the McCumber relation and perpetual motion
6 References

Definition

Let $σ a (ω)$ and the emission cross-section $σ e (ω)$ be effective absorption and emisison cross-sections at frequency $ω$ , and let $~T~$ be the effective temperature of the medium. The McCumbner relation is

(1) $\frac{\sigma_{\rm e}(\omega)}{\sigma_{\rm a}(\omega)}\exp\!\left( \frac{\hbar \omega}{k_{\rm B} T}\right) =\left(\frac{N_1}{N_2}\right)_T =\exp\!\left( \frac{\hbar \omega_{\rm z}}{k_{\rm B} T}\right)$

where $\left(\frac{N_1}{N_2}\right)_T$ is thermal steady-state ratio of populations; frequensy $ω z$ is called "zero-line" frequency ^[3]^[4]; $\hbar$ is the Planck constant and $k B$ is the Boltzmann constant. Note that the right-hand side of Equation (1) does not depend on $~\omega~$ .

Gain

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 way of excitation, used to achieve a given population at the upper laser level. In this case, the absorption cross-section $σ a (ω)$ and the emission cross-section $σ e (ω)$ at frequency $~\omega~$ can be determined in such a way, that the gain at this frequency can be determined as follows:

(2) $~~~~~~~~~~~~~~~G(\omega)=N_2 \sigma_{\rm e}(\omega)-N_1 \sigma_{\rm a}(\omega)$

D.E.McCumber had postulated this properties, and found that the emisison and absorption cross-sections are not independent ^[1]^[2]; they are related with Equation (1).

Idealized atoms

In the case of an idealized two-level atom, the consideration of the detailed balance for the emission and absorption, that preserves the Max Planck formula for the black body radiation 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 natural spectral linewidth. In the case of an ideal two-level atom, the product of the linewidth and the lifetime is of order of unity, which follows to the Heisenberg uncertainty principle. In the solid-state laser materials, the linewidth is several orders of magnitude larger. Therefore, the spectra of emission and absorption are determined with distribution of excigtation among sublevels rather than by the shape of the spectral line of each individual transition between sublevels. This distribution is determined by the effectie temperature within each of laser levels. The hypothesis of McCumber is that the distribution of excitation among sublevels is termal. Then, the effectie temperature^[5] determines the spectra of emission and absorption.

Deduction of the McCumber relation

Consider the set of active centers (fig.1.). Assime fast transition between sublevels within each level, and slow transition between levels. According to the McCumber hypothesis, the corss-sections $σ a$ and $σ e$ do not depend on the populations $N 1$ and $N 2$ . Therefore, we can deduce the relation, assumnig the 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; $a (ω) n 2$ 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:

(3) $~~~ n_2\sigma_{\rm e}(\omega) v(\omega)D(\omega)+n_2 a(\omega)= n_1\sigma_{\rm a}(\omega) v(\omega)D(\omega) ~~~~~~~~~~~~~~~{\rm (balance)}$

Rewrite it as

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

The thermal distribution of density of photons follows from blackbody radiation ^[6]

(5) $~~~ D(\omega)~=~ \frac{\frac{1}{\pi^2} \frac{\omega^2}{c^3}} {\exp\!\left(\frac{\hbar\omega}{k_{\rm B}T}\right)-1} ~~~~~ {\rm (D2)}$

Both (4) and (5) 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 $a (ω)$ and the emission cross-section $~\sigma_{\rm e}(\omega)~$ ^[6]. (We keep the term w: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 ^[6]:

(6) $~~~ a_{s,j}(\omega)=\sigma_{s,j}(\omega) \frac{\omega^2 v(\omega)}{\pi^2c^3}~~. ~~~~~~~~~~~~~~~~{\rm comparison1} ~~{\rm partial}$

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:

(7) $\frac{a(\omega)}{\sigma_{\rm e}(\omega)}= \frac{\omega^2 v(\omega)}{\pi^2c^3} ~~~~~~~~~~~~~~~~~~(\rm comparison)(av)$

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

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

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

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

the subscript $~T~$ indicates that the ratio of populations in evaluated in the thermal state. The zero-line frequency can be expressed as

(10) $\omega_{\rm Z}=\frac{k_{\rm B}T}{\hbar} \ln \left(\frac{n_1}{n_2}\right)_{T} ~~~~~~~~~~~~~~~~.~~{(\rm oz)}$

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.

Confirmation of the McCumber relation

The McCumber relation is confirmed for various media ^[7]^[8]. In particular, relation (1) allows to approximate two functions of frequency, emission and absorption cross sections, with single fit ^[9].

Violation of the McCumber relation and perpetual motion

Y.2006, the strong violation of McCumber relation was observed for Yb:Gd₂SiO₅ and reported in 3 independent journals^[10]^[11]^[12]. Typical behaivor of the cross-sections is shown in FIg.2 with thick curves. The emission cross-section is practically zero at wavelength 975nm; this property makes Yb:Gd₂SiO₅ an excellent material for efficient solid-state lasers.

The project of a Perpetual motion was suggested basing on these prpoperties. It is sufficient to fill a box with reflecting walls with Yb:Gd₂SiO₅, and allow it to exchange radiation with a blackbody through the spectrally-selective window, which is transparent in vicinity of 975nm, and reflector at other wavelengths. Due to the lack of emissivity at 975nm, the medium should warm, breaking the thermal equilibrium.

Unfortunately, this brilliant opportunity was closed in 2007; the correction of the effective emission cross section (black thin cirve) was suggested ^[3] and confirmed^[13].

References

^ ^a ^b D.E.McCumber. Einstein relations connecting broadband emission and absorption spectra. PRB 136 (4A), 954-957 (1964)
^ ^a ^b P.C.Becker, N.A.Olson, J.R.Simpson. Erbium-doped fiber amplifiers: fundamentals and theory (Academic, 1999).
^ ^a ^b D. Kouznetsov (2007). "Comment on Efficient diode-pumped Yb:Gd₂SiO₅ laser (Appl.Phys.Lett.88,221117(2006))". APL 90: 066101.
^ D.Kouznetsov (2007). "Broadband laser materials and the McCumber relation". Chinese Optics Letters 5: S240-S242.
^ Effective temperature: the scientific slang it is called temperatire even if the excited medium as whole is pretty far from the thermal state)
^ ^a ^b ^c e2
^ R.S.Quimby (2002). "Range of validity of McCumber theory in relating absorption and emission cross sections". J. Appl. Phys. 92: 180-187. doi:10.1063/1.1485112.
^ R.M.Martin; R.S.Quimby (2006). "Experimental evidence of the validity of the McCumber theory relating emission and absorption for rare-earth glasses". JOSAB 23 (9): 1770-1775.
^ D.Kouznetsov; J.-F.Bisson, K.Takaichi, K.Ueda (2005). "Single-mode solid-state laser with short wide unstable cavity". JOSAB 22 (8): 1605-1619.
^ W. Li; H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, J. Xu. (2006). "Efficient diode-pumped Yb:Gd₂SiO₅ laser.". APL 88: 221117.
^ W.Li; H.Pan, L.Ding, H.Zeng, G.Zhao, C.Yan, L.Su, J.Xu. "[http://www.opticsexpress.org/search.cfm year=2006 Diode-pumped continuous-wave and passively mode-locked Yb:Gd₂SiO₅laser.]". Optics Express 14.
^ C.Yan; G.Zhao, L.Zhang, J.Xu, X.Liang, D.Juan, W.Li, H.Pan, L.Ding, H.Zeng. (2006). "A new Yb-doped oxyorthosilicate laser crystal: Yb:Gd₂SiO₅.". Solid State Communications 137: 451-455.
^ G.Zhao (2007). "Response to Comment on Efficient diode-pumped Yb:Gd₂SiO₅ laser (Appl. Phys. Lett. 90, 066101 2007)". APL 90: 066103.

Categories: Spectroscopy | Solid-state lasers

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