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Effective Transition Cross Sections

Definition: a modified type of transition cross sections which apply to optical transitions between Stark level manifolds

German: effektive Wirkungsquerschnitte

Formula symbol: σeff

Units: m

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energy levels of ytterbium ions in Yb:YAG
Figure 1: Energy levels of Yb3+ ions in Yb:YAG, and the usual pump and laser transitions. The optical transitions take place between the 2F7/2 and the 2F5/2 Stark level manifolds.

Optical transition cross sections are defined so that the rate of optical transitions (per active ion) starting from a certain electronic level is the transition cross section σ times the photon flux (i.e., the optical intensity divided by the photon energy). This concept often cannot be directly applied to optical transitions, e.g. in solid-state laser gain media (e.g. rare-earth-doped laser crystals and active fibers), because such media exhibit Stark level manifolds containing multiple electronic sublevels with slightly different energies. Here, optical transitions can occur between different combinations of sublevels in the two involved Stark level manifolds. A frequently encountered difficulty is that neither the exact energetic positions of the sublevels are known, nor the transition cross sections for all the combinations of different sublevels. These quantities can be difficult to measure, essentially because the different optical transitions are spectrally broadened by phonon-induced transitions within the Stark level manifolds, so that their contributions to the absorption and emission spectra overlap. Particularly in glass materials, the spectral broadening hides essentially all information on the different sublevels, whereas in some crystals (such as Yb:YAG) the absorption and emission spectra reveal clearly the contributions from different sublevels.

The concept of effective transition cross sections is very useful, particularly for media with strong spectral broadening. Effective cross sections incorporate both the occupation probabilities for different sublevels of both involved Stark level manifolds and the transition cross sections for all pairs of sublevels. Their use is simple: the rate of optical transitions starting from a certain Stark level manifold is the effective transition cross section σ times the photon flux (i.e., the optical intensity divided by the photon energy). Effective cross sections are usually directly obtained from absorption and emission measurements, and the knowledge of sublevel positions and cross sections for the contributing transitions is not required. For example, the measured absorption spectrum of an electronically non-excited sample reveals the effective absorption cross sections for transitions from the ground-state manifold to higher-lying Stark level manifolds.

effective transition cross sections of Yb-doped glass
Figure 2: Effective absorption and emission cross sections of ytterbium-doped germanosilicate glass, as used in the cores of ytterbium-doped fibers, at room temperature. (Data from spectroscopic measurements by R. Paschotta)

As an example, Figure 2 shows effective cross sections of an ytterbium-doped fiber. Here, the strongest transition is that between the lowest-lying energy levels in both manifolds; it is seen as the “zero-phonon line” at ≈ 975 nm. The (weaker) absorption at shorter wavelengths (e.g. 920 nm) is due to transitions to higher-lying sublevel in the upper manifold, which involves the emission of one or several phonons during subsequent thermalization. Similarly, the emission at longer wavelength (e.g. 1040 nm) is related to transitions to higher-lying levels of the ground-state manifold, and involves phonon emission during thermalization in the ground-state manifold.

Although effective transition cross sections are in principle very simple to use, some important aspects must be considered:

Effective transition cross sections are often used for rate equation modeling. The dynamic variables are then the population densities of the different Stark level manifolds, not distinguishing the sublevels.

See also: transition cross sections, rate equation modeling

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