Definition: material parameters for quantifying the likelihood or rate of optical transition events
More specific term: effective transition cross-sections
Categories: laser devices and laser physics, physical foundations
Formula symbol: σ
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Author: Dr. Rüdiger Paschotta
In laser physics, transition cross-sections are used to quantify the likelihood of optically induced transition events, e.g. of absorption or stimulated emission. For a laser ion in a certain electronic state, the rate of transitions (in events per second) is given as the corresponding cross-section times the photon flux density (in photons per square meter and second):
where R is the transition rate (in units of s−1), σ is the transition cross-section, I is the optical intensity, and h ν is the photon energy. Such transition rates are used in rate equation modeling.
Transition cross-sections depend on the optical frequency (or wavelength). In classical models, such transitions involve resonances, which can lead to strongly peaked cross-section spectra. In a photon picture, those peaks correspond to wavelengths where the photon energy fits to the difference of energies of two involved levels.
Absorption and Gain Coefficient
If absorbing atoms or ions with an absorption cross-section σabs are distributed with a number density N in a medium, this leads to an intensity absorption coefficient α of the medium which is the product of N and σabs. (For fibers or other waveguides with an undoped cladding, it may be necessary to include also an overlap factor Γ.) In an analogous fashion, the gain coefficient for atoms or ions with a given emission cross-section σem can be calculated.
Transition Cross-sections of Laser Gain Media
For a laser gain medium, the most relevant cross-sections are the absorption and emission cross-sections at the pump and laser wavelengths. In many cases, there is only an absorption cross-section for the pump wavelength and an emission cross-section (also called laser cross-section) for the laser wavelengths. The other transition cross-sections are effectively zero, because the lower laser level is quickly depopulated. For such a four-level gain medium, the threshold pump power is inversely proportional to the product of emission cross-section (at the laser wavelength) and upper-state lifetime. The cross-sections influence not only the achievable pump absorption and gain, but also the saturation behavior and the rates of spontaneous transition processes. If there is a non-zero absorption cross-section at the laser wavelength, this causes reabsorption (→ quasi-three-level laser gain media).
In solid-state gain media, laser transitions normally involve different Stark levels of the upper and lower electronic levels (or more precisely, upper and lower Stark level manifolds). These sublevel-transitions are spectrally overlapping, so that they are hard to distinguish (except perhaps at very low temperatures). Depending on the exact wavelength, transitions between different Stark levels contribute with varying degrees. A convenient technique for laser modeling is therefore to use so-called effective transition cross-sections, which are a kind of averages of the cross-sections of different sublevel transitions, with weight factors depending on population densities in thermal equilibrium (which implies that they are intrinsically temperature-dependent). Effective transition cross-sections can directly be obtained from spectroscopic measurements without resolving the different sublevel transitions (which would be difficult e.g. in the case of doped glasses). Also, they can be directly used in laser models.
Typical values of absorption and emission cross-sections of laser crystals are in the range 10−20–10−18 cm2. Single ions in glasses exhibit similar values, but due to inhomogeneous broadening, the average cross-sections of the ions in a glass are often substantially smaller – often several times or even by an order of magnitude. For gain media operating on allowed transitions (rather than forbidden transitions), such as laser dyes or semiconductors, the cross-sections are much higher.
Laser gain media with high emission cross-sections and/or broad gain bandwidth tend to have a low upper-state lifetime because of strong spontaneous emission (→ radiative lifetime). Even though quantum noise processes determine the rate of spontaneous emission, that rate is also proportional to the emission cross-section and bandwidth. See the article on radiative lifetime for more details.
Measurement of Transition Cross-sections
Absorption cross-sections for transitions starting at the ground-state manifold are often obtained from absorption spectra, measured e.g. with a white light source. The number density (concentration) of the ions must be known for that calculation.
The spectral profile of emission cross-sections (usually without absolute scaling) can be obtained from fluorescence spectra, provided that no reabsorption modifies the spectral shape and there is no spectral overlap of different fluorescent transitions. Absolute values of emission cross-sections can be calculated with the reciprocity method, e.g. via the Füchtbauer–Ladenburg equation. The McCumber relation  can also be useful. Emission cross-sections e.g. for rare earth ions can also be estimated from absorption cross-sections using Judd–Ofelt theory.
Excited-state absorption (e.g. from the upper laser level) is also characterized with cross-sections. These are more difficult to measure than those for ground-state absorption, because (a) it is necessary to populate at least partially the corresponding starting level during the measurement, (b) it can be difficult to determine the fraction of ions in that level, and (c) the measured absorption may be modified by absorption or stimulated emission from other levels, which are also populated.
For non-isotropic media, transition cross-sections can strongly depend on the direction of the polarization of light. For example, the laser gain can be strongly polarization-dependent.
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See also: stimulated emission, spontaneous emission, laser transitions, radiative lifetime, effective transition cross-sections, Giles parameters, McCumber theory, doping concentration
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