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Multi-phonon Transitions

Definition: transitions between electronic levels of atoms or ions in solid media, involving the emission of multiple phonons

German: Multiphononen-Übergänge

Category: physical foundations

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Ions in solids such as laser gain media (for example) have different energy levels, and transitions between those can be caused by absorption of light, by spontaneous or stimulated emission, and by various non-radiative mechanisms. An example for the latter possibility are multi-phonon transitions, where multiple phonons (quanta of lattice vibrations) are simultaneously emitted and carry away the difference of the excitation energies of the two levels. That way, transitions with a larger energy difference are possible, compared with the emission of a single phonon.

Transition rates of multi-phonon transitions decreases rapidly (exponentially) with increasing order (i.e., number of emitted phonons). With only two or three phonons required, transitions can be very rapid, while for five phonons, for example, they may be negligibly slow.

Importance of Maximum Phonon Energy

An important parameter is the maximum phonon energy of a material, because it determines the minimum number of required phonons for a multi-phonon transition between two levels with a given difference in excitation energy. There are cases where very strong multi-phonon transition rates occur for ions in a high phonon energy glass such as fused silica, while that rate is negligible for a low phonon energy glass, e.g. ZBLAN, a fluoride glass containing heavy metals.

Temperature Dependence

Multiphonon emission can already occur at low temperatures, where phonon modes of the material are hardly populated. However, the multi-phonon transition rate can increase with temperature due to stimulated emission of phonons, involving thermally populated phonon modes [1]. This happens when kBT is not much smaller than the energy of the involved phonons. As a consequence, metastable level lifetimes can be reduced at increasing temperatures.

Relevance of Multi-phonon Processes in Laser Gain Media

Multi-phonon processes are relevant for solid-state laser gain media, and this in different ways:

  • They can play an essential role in the function of a laser by providing useful transitions. For example, in neodymium-doped laser gain media, one typically pumps ions from their ground state to an excited state above the upper laser level. From there, multi-phonon transitions quickly take them to the upper laser level (4F3/2). Also, the lower laser level of most four-level solid-state gain media is rapidly depopulated by multiphonon emission; that is important because otherwise one could have reabsorption losses on the laser transition.
  • On the other hand, it can be very detrimental if the laser transition itself is bypassed by multi-phonon processes, or if other important metastable states are depopulated. Therefore, many many mid-infrared laser sources, e.g. 3-μm erbium-doped fiber lasers, need to be realized with low phonon energy glasses. (Low phonon energies are also important for transparency in the mid-infrared wavelength region.) Also, some upconversion lasers based on erbium-doped or thulium-doped glass require heavy metal glasses (e.g. fluoride glasses) as host media, since high phonon energy glasses such as silica would lead to too low lifetimes of required metastable levels.

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[1]L. A. Riseberg and H. W. Moos, “Multiphonon orbit–lattice relaxation of excited states of rare earth ions in crystals”, Phys. Rev. 174 (2), 429 (1968), doi:10.1103/PhysRev.174.429
[2]C. B. Layne et al., “Multiphonon relaxation of rare earth ions in oxide glasses”, Phys. Rev. B 16 (1), 10 (1977), doi:10.1103/PhysRevB.16.10
[3]Y. V. Orlovskii et al., “Multiple-phonon nonradiative relaxation: experimental rates in fluoride crystals doped with Er”, Phys. Rev. B 49 (6), 3821 (1994), doi:10.1103/PhysRevB.49.3821
[4]Y. V. Orlovskii et al., “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 μm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium”, Opt. Materials 18, 355 (2002), doi:10.1016/S0925-3467(01)00174-4
[5]Z. Burshtein, “Radiative, nonradiative, and mixed-decay transitions of rare-earth ions in dielectric media”, Opt. Eng. 49, 091005 (2010), doi:10.1117/1.3483907

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See also: laser gain media, rare-earth-doped laser gain media, upconversion lasers, fluoride fibers, quenching, non-radiative transitions, multiphonon absorption
and other articles in the category physical foundations


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