Reciprocity Method

In the simplest possible case — a transition between just two non-degenerate electronic energy levels — the reciprocity principle says that the absorption and emission cross-sections for this transition must be identical:

Here, ($\sigma_{\textrm{abs}}(\nu)$) is the absorption cross-section and ($\sigma_{\textrm{em}}(\nu)$) is the stimulated emission cross-section at optical frequency ($\nu$). This equality results form the consideration that for a two-level system in thermal equilibrium, the net rate of upward and downward transitions must balance when the radiation field has the spectral distribution of black-body radiation.

Einstein already considered an important generalization for the situation where the upper and/or lower levels may be degenerate, i.e., that they actually consist of multiple sublevels, each with 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.

In this case, effective transition cross-sections can be used, which describe the likelihood of transitions between any of the levels involved. The ratio of effective emission to absorption cross-sections is then no longer unity, but rather equals the degeneracy factor of the lower level ($g_\textrm{l}$) divided by that of the upper level ($g_\textrm{u}$):

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 partially removes the degeneracies. The electronic multiplets split into 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 population fractions of the sublevels differ according to a Boltzmann distribution. As a result:

• Emission from the highest-lying sublevels of the upper manifold is reduced, as these sublevels are less populated.
• Absorption from the highest-lying sublevels of the lower manifold is also reduced due to low occupancy.

Even in this regime, 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

which is often used to process spectroscopic data of laser gain media. The constant ($E_0$) (sometimes called the McCumber energy or zero-line energy) can be calculated using the reciprocity principle, if the Stark level positions within the manifolds are known, or it can be obtained from suitable integrals over measured spectra.

Note that the reciprocity relation is not always fulfilled with high precision, since vibronic interactions and other effects (e.g., strong inhomogeneous broadening or non-thermal level populations) can lead to deviations [2].

In many gain media — for example, rare-earth-doped fibers or bulk laser crystals — high-quality absorption spectra are easier to measure than fluorescence-based emission spectra. Using the McCumber relation and a known (or fitted) value of ($E_0$), one can proceed as follows:

• Measure ($\sigma_{\textrm{abs}}(\nu)$) over a sufficiently broad spectral range.
• Apply the McCumber formula to calculate ($\sigma_{\textrm{em}}(\nu)$) with reasonable accuracy.

This is particularly common for designing and modeling fiber amplifiers and fiber lasers (e.g., Er³⁺, Yb³⁺, or Nd³⁺-doped systems).

Because the McCumber relationship contains the factor ($\exp[(h\nu - E_0)/(k_\textrm{B}T)]$), it explicitly predicts how the ratio of emission to absorption cross-section changes with temperature. This is used to estimate how the gain spectrum shifts with temperature, adjust models for devices operating at cryogenic or elevated temperatures, and interpret measurements taken at different environmental conditions.

Experimental data on absorption and emission can be checked for internal consistency:

• If independently measured ($\sigma_{\textrm{abs}}(\nu)$) and ($\sigma_{\textrm{em}}(\nu)$) satisfy the McCumber relation within expected accuracy, this supports the assumptions behind McCumber theory and validates the extracted cross-sections.
• Conversely, a lack of agreement may indicate non-thermal population distributions, strong vibronic coupling, or experimental artifacts (e.g., calibration errors).

When the positions of the Stark sublevels are known (e.g., from high-resolution spectroscopy), one can compute the McCumber energy ($E_0$) from the Boltzmann-weighted averages of upper and lower manifolds. Conversely, by fitting the McCumber relation to experimental spectra, one can infer effective level parameters and gain insight into the structure of the Stark manifolds.