effective transition cross-sections (original) (raw)

Author: the photonics expert (RP)

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

Category: laser devices and laser physics

• properties of laser gain media
  • transition cross-sections
    * effective transition cross-sections

Related: transition cross-sectionsoptical transitionsStark level manifoldsrate equation modeling

Units: m2

Formula symbol: ($\sigma_\textrm{eff}$)

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DOI: 10.61835/u7u Cite the article: BibTex BibLaTex plain textHTML Link to this page! LinkedIn

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Contents

What are Effective Transition Cross-sections?

Effective transition cross-sections are a concept which is often applied in the context of rate equation modeling of solid-state laser gain media. This concept is very useful for simplifying the modeling related to sublevels of Stark level manifolds. The dynamic variables are then the population densities of the different Stark level manifolds, not distinguishing the sublevels.

Problem with Sublevels of Stark Level Manifolds

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 ($\sigma$) times the photon flux (i.e., the optical intensity divided by the photon energy). However, in the context of solid-state laser gain media (e.g. rare-earth-doped laser crystals and active fibers) one faces the problem that optical transitions (related to absorption or stimulated emission) can occur between different combinations of sublevels in the two involved Stark level manifolds. Unfortunately, in many cases 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.

Solution with Effective Cross-sections

A convenient solution is provided by the concept of effective transition cross-sections. These 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: By definition, the rate of optical transitions starting from a certain Stark level manifold is the product of the following factors:

• the effective transition cross-section ($\sigma$)
• the photon flux (i.e., the optical intensity divided by the photon energy)
• the number of ions in the initial Stark level manifold

where for the latter number we do not care in which sublevel the ion might be. So instead of dealing with individual sublevels and “ordinary” transition cross-sections, one uses populations of whole Stark level manifolds in conjunction with effective transition cross-sections.

Obtaining Effective Cross-sections

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.

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 sublevels in the upper manifold, which involves the emission of one or several phonons during subsequent thermalization. Similarly, the emission at longer wavelengths (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.

Important Considerations

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

• Effective cross-sections are intrinsically temperature-dependent: Temperature changes can affect not only the electronic sublevel positions, but also the relative occupation probabilities within the sublevels. The latter effect is often dominant. The lower the temperature, the more are the level populations concentrated on the lowest-lying sublevels. Higher temperatures allow for absorption at longer wavelengths (starting from higher-lying sublevels of the lower manifold) and for emission at shorter wavelengths (starting from higher-lying sublevels of the higher manifold). In the example of Figure 2, higher temperatures would increase the absorption tail around 1000–1050 nm as well as the emission tail at 900–950 nm, while somewhat reducing the height of the main peak at 975 nm.
• Albert Einstein found that the transition cross-section for some absorption process should be the same as that for stimulated emission on the same transition. That rule does not apply to effective cross-sections in Stark level manifolds. There is a modified and more complicated rule resulting from McCumber theory.
• Effective cross-sections can be used only when the level population within each Stark level manifold can be assumed to be in thermal equilibrium. In solid-state media, thermalization is provided by phonons on a time scale of picoseconds, so that this condition is in most practical cases very well fulfilled. Deviations can occur e.g. in regenerative amplifiers for ultrashort pulses, where an intense pulse can extract a significant amount of energy from the gain medium in a time which is too short to allow for thermalization within the pulse duration.

Frequently Asked Questions

What are effective transition cross-sections?

Effective transition cross-sections are parameters used in rate equation modeling of solid-state laser gain media. They represent the averaged effect of all possible optical transitions between the sublevels of two Stark level manifolds, simplifying calculations.

Why are effective cross-sections necessary for solid-state laser media?

In materials like rare-earth-doped crystals, electronic levels split into many sublevels (Stark manifolds). The properties of transitions between all these individual sublevels are often unknown and masked by spectral broadening, making a simplified model necessary.

How are effective cross-sections obtained?

They are typically derived directly from measured absorption and emission spectra of the gain medium, without needing detailed knowledge of the energies and transition strengths of the individual sublevels.

Are effective transition cross-sections dependent on temperature?

Yes, they are substantially temperature-dependent. A change in temperature alters the statistical occupation of the various sublevels within a Stark manifold, which in turn changes the absorption and emission spectra and thus the effective cross-sections.

Does the simple Einstein relation apply to effective cross-sections?

No. The rule that absorption and stimulated emission cross-sections are equal for a given transition does not apply to effective cross-sections. Instead, a more complex relationship described by McCumber theory is required.

When can the concept of effective cross-sections be used?

It can be used when the population within each Stark level manifold is in thermal equilibrium. This condition is usually met because thermalization via phonons is extremely fast (picoseconds), but it can fail for intense, ultrashort pulses.

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