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Stark Level Manifolds

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

Atoms and ions as used in laser gain media exhibit not just a few energy levels, but rather Stark level manifolds, each one consisting of some number of levels with identical or at least similar energies. They are labeled to indicate certain quantum states. For example, the lowest three energy levels of the sodium atom are ²S_1/2, ²P_1/2 and ²P_3/2 in the common notation for the case of LS coupling. All have the multiplicity (<$2s +1$>) of 2, orbital states S or P (<$l = 0$> and <$l = 1$>, respectively) and the quantum number <$j = 1/2$> or <$j = 3/2$>. The exact energy values can be modified by the influence of electric fields (Stark effect) and magnetic fields (Zeeman effect).

In the case of gas lasers, the atoms or ions are usually not subject to significant electric and magnetic fields, so that a Stark level splitting does not occur. One may then still not completely ignore the presence of multiple degenerate energy levels, but use rather simple physical models where only the number of degenerate sub-levels is taken into account. (In some cases, however, the Doppler effect due to movement of atoms needs to be taken into account.)

In solid-state laser gain media, where atoms are placed in close vicinity to each other, the atoms are subject to electric fields which depend on their neighborhood. This leads to some significant amount of Stark level splitting, i.e., to a lifting of the otherwise existing level degeneracy, which can have substantial effects on laser operation. Depending on the circumstances, different kinds of laser models can be appropriate:

• In the case of rare-earth-doped laser crystals, it is often the case that all relevant atoms (or ions) experience the same kind of neighborhood and thus have essentially the same level structure. Typical examples are Nd³⁺:YAG, Nd³⁺:YLF and Er³⁺:YAG. In spectroscopic measurements, one can then easily identify the different Stark levels. In physical models (e.g. rate equation models), one may include all relevant levels and the transitions between those.
• In the case of rare-earth-doped laser glasses, and also for various disordered laser crystals, the exact level energies vary so much that the different levels cannot easily be distinguished by spectroscopy. The transition energies strongly overlap, and physical models usually need to work with each Stark level manifold as a whole, and with effective transition cross-sections.
• In transition-metal-doped laser gain media, the strong interactions of the electrons with phonons result in strong spectral broadening, so that the sub-levels can again not be resolved.

Population Distribution Within Stark Level Manifolds

For isolated atoms or ions, each sub-level within a manifold can be expected to be a stationary state, as long as there are no additional effects (e.g. optical fields) inducing transitions between those levels. When a specific sub-level is populated, for example by bracket optical pumping], that population may stay for a substantial time.

The situation is entirely different in solid media, where the electrons interact not only with other parts of their atom, but also with other atoms. In particular, lattice vibrations, in quantum mechanics described with phonons, lead to very rapid transitions between the sub-levels within any manifold. As a result of that, there is very rapid thermalization (typically on a picosecond timescale) within each manifold, although the total population in a certain manifold may have a lifetime which is many orders of magnitude longer. The relative population numbers of the sub-levels are then simply determined by a Boltzmann distribution.

For that reason, many laser models for solid-state lasers work with population numbers (or densities) only of the Stark level manifolds, not of individual levels. A four-level laser gain medium, for example, is then understood to be one where four Stark level manifolds are relevant. Assuming quasi-instantaneous thermalization within the manifolds (which is in most cases a very reasonable approximation), one achieves a substantial simplification of the models. In particular, the spectral shapes of absorption and emission transitions (e.g. for pump absorption, fluorescence and stimulated emission on laser transitions) are then fixed. Deviations from that simple behavior are sometimes encountered, for example when an optical amplifier is applied to intense ultrashort pulses with pulse durations below the thermalization time.

Based on certain further assumptions, McCumber theory can further be used to relate to spectral shapes of absorption and emission transitions between the same pair of level manifolds. This is often a useful tool in the spectroscopy of solid-state laser gain media, e.g. for estimating the strength of reabsorption in spectral regions where it is difficult to be measured directly.

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