# Reciprocity Method

Definition: a spectroscopic method which is often used for obtaining the scaling of emission spectra of laser gain media

German: Reziprozitätsmethode

Categories: methods, physical foundations

Author: Dr. Rüdiger Paschotta

The principle of reciprocity in the context of absorption and emission cross-sections goes back to Albert Einstein. Considering the simplest possible case – a transition between just two non-degenerate electronic energy levels – that principle says that the absorption and emission cross-sections for this transition must be identical.

Einstein already considered an important generalization for the situation that the upper and lower level may have degeneracies, i.e. that they actually consist of multiple levels, each having the same energy. This situation often occurs for the electronic states of isolated atoms or ions, as long as they are not exposed to electric or magnetic fields. Here, effective transition cross-sections can be used which describe the likelihood of transitions between *any* of the levels involved, and the ratio of effective emission to absorption cross-sections is no longer unity, but rather equals the degeneracy factor of the lower level divided by that of the upper level. This is easy to understand: for example, emission (but not absorption) is favored by a large degeneracy of the lower level, i.e. by a large “choice” of final states in the lower level manifold.

In solid-state gain media, the situation is more complicated because the interaction of laser-active ions with the crystal field partly removes the degeneracies. There are therefore Stark level manifolds with a spread of level energies. As this splitting can be comparable to (or larger than) the thermal energy <$k_\textrm{B} T$>, the average population fractions for the sublevels differ according to a Boltzmann distribution. As a result, emission from the highest lying sublevel of the upper level manifold becomes weaker, as does absorption from the highest lying sublevel of the lower level manifold. Even in this regime, however, the principle of reciprocity can still be used in a convenient form, which was published by McCumber in 1964 [1] in the context of his spectroscopic theory, now called McCumber theory. The corresponding article quotes the McCumber relation

$${\sigma _{{\rm{abs}}}}(\nu ) = {\sigma _{{\rm{em}}}}(\nu )\;\exp \left( {\frac{{h\nu - {E_0}}}{{{k_{\rm{B}}}T}}} \right)$$which is often used to process spectroscopic data. The constant <$E_0$> can be calculated using the reciprocity principle, if the Stark level positions within the manifolds are known.

Note that the reciprocity relation is not always fulfilled with high precision, since vibronic interactions can lead to deviations [2].

### Bibliography

[1] | D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra”, Phys. Rev. 136 (4A), A954 (1964), DOI:10.1103/PhysRev.136.A954 |

[2] | B. F. Aull and H. Jenssen, “Vibronic interactions in Nd:YAG resulting in nonreciprocity of absorption and stimulated emission cross-sections”, IEEE J. Quantum Electron. 18 (5), 925 (1982), DOI:10.1109/JQE.1982.1071611 |

See also: effective transition cross-sections, McCumber theory, Füchtbauer–Ladenburg equation, fluorescence

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