To use all functions of this page, please activate cookies in your browser.
my.chemeurope.com
With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.
 My watch list
 My saved searches
 My saved topics
 My newsletter
McCumber relation
Additional recommended knowledge
DefinitionLet σ_{a}(ω) and the emission crosssection σ_{e}(ω) be effective absorption and emisison crosssections at frequency ω, and let be the effective temperature of the medium. The McCumbner relation is
where is thermal steadystate ratio of populations; frequensy ω_{z} is called "zeroline" frequency ^{[3]}^{[4]}; is the Planck constant and k_{B} is the Boltzmann constant. Note that the righthand side of Equation (1) does not depend on . GainIt 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 crosssection σ_{a}(ω) and the emission crosssection σ_{e}(ω) at frequency can be determined in such a way, that the gain at this frequency can be determined as follows:
D.E.McCumber had postulated this properties, and found that the emisison and absorption crosssections are not independent ^{[1]}^{[2]}; they are related with Equation (1). Idealized atomsIn the case of an idealized twolevel 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 crosssection of absorption and emission. In the solidstate 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 twolevel atom, the product of the linewidth and the lifetime is of order of unity, which follows to the Heisenberg uncertainty principle. In the solidstate 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 relationConsider 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 corsssections 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 be croup velocity of light in the medium. The product is spectral rate of stimulated emission, and 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:
Rewrite it as
The thermal distribution of density of photons follows from blackbody radiation ^{[6]}
Both (4) and (5) 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 a(ω) and the emission crosssection ^{[6]}. (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 j, the partial spectral emission probability can be expressed from consideration of idealized twolevel atoms ^{[6]}:
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 ω_{Z} 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. Confirmation of the McCumber relationThe 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:Gd_{2}SiO_{5} and reported in 3 independent journals^{[10]}^{[11]}^{[12]}. Typical behaivor of the crosssections is shown in FIg.2 with thick curves. The emission crosssection is practically zero at wavelength 975nm; this property makes Yb:Gd_{2}SiO_{5} an excellent material for efficient solidstate 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:Gd_{2}SiO_{5}, and allow it to exchange radiation with a blackbody through the spectrallyselective 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
Categories: Spectroscopy  Solidstate lasers 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "McCumber_relation". A list of authors is available in Wikipedia. 