In a laser gain medium, amplification is associated with the population of an excited state, from which stimulated emission can occur. Even without stimulated emission, the lifetime of this upper-level population is finite due to spontaneous emission and possibly due to additional quenching effects. Typically, the upper-state population decays exponentially with a certain decay time (the upper-state lifetime), assuming the absence of pumping and stimulated emission. More precisely, the decay time is the time after which this population has decayed to 1/e (≈ 37%) of the initial value.
Spontaneous emission leads to fluorescence, the lifetime of which (fluorescence lifetime) is of course identical to the upper-state lifetime.
The decay rate is enhanced if there are additional transitions to other lower-lying energy levels. In addition, the upper-state lifetime can be reduced by quenching processes, involving e.g. the deexcitation at impurities or crystal defects, or energy transfers between different laser ions. The decay of the upper-state population is then not necessarily of exponential nature; non-exponential decay is frequently observed (particularly for measurements with sufficiently short pump pulses). For example, some quenching processes lead to a fast decay, as long as the upper-level population is high, but have little influence on later stages of the decay.
For some gain media such as Cr:forsterite and Cr:YAG, the upper-state lifetime is strongly temperature-dependent. The reason can be phonon-assisted non-radiative relaxation, which becomes stronger at higher temperatures.
An effective upper-state lifetime can be defined under lasing conditions, which includes the effect of stimulated emission. For a four-level laser medium, the effective upper-state lifetime is reduced e.g. by a factor of 2 when the laser is pumped twice above the threshold pump power.
Note that the lower laser level can also have a finite lifetime, the so-called lower-state lifetime.
The upper-state lifetimes of different kinds of laser gain media differ considerably:
- Assuming the existence of dipole-allowed transitions, excited levels of atoms or ions typically have lifetimes of the order of nanoseconds.
- The lifetime of carriers in the conduction band of a direct band gap semiconductor (as used for semiconductor lasers) is also typically a few nanoseconds.
- Rare-earth-doped gain media typically operate on weakly allowed transitions, leading to much longer lifetimes between a few microseconds (e.g. for titanium–sapphire) and ≈ 8–10 milliseconds (e.g. for erbium-doped fiber amplifiers). Their upper laser levels are called metastable.
Note that the threshold powers for different gain media vary much less than the upper-state lifetimes do, since long upper-state lifetimes imply low emission cross sections, and the threshold power depends on the σ−τ product (see below).
Importance for Lasers
A long upper-state lifetime in a laser gain medium means that a significant population inversion can be maintained with a relatively low pump power. The gain efficiency and thus also the threshold pump power of a laser also depend on the emission cross section (apart from other factors); the threshold pump power is inversely proportional to the product of upper-state lifetime and emission cross section (called the σ−τ product), at least for four-level lasers.
A long upper-state lifetime is desirable for continuously pumped Q-switched lasers, because it makes it possible to store large amounts of energy.
Measurement of the Upper-state Lifetime
The upper-state lifetime can be measured e.g. by populating the upper laser level with a short laser pulse and monitoring the decay of the fluorescence. Alternatively, one may use an optical chopper (typically, with a rotation disc) in conjunction with a continuous-wave laser beam, but the switching is then much slower; it may still be sufficient for lifetimes of the order of milliseconds are hundreds of microseconds.
Note that in highly doped media the measured upper-state lifetime may be increased by reabsorption of the fluorescence, particularly if radiation trapping due to total internal reflection at the surfaces of the medium enhances this effect [1, 2]. Reabsorption effects can be suppressed by using a lightly doped powder of the substance immersed in a liquid with comparable refractive index, or by dominantly recording fluorescence from some edge of a sample, using a pinhole (pinhole method).
Note that the fluorescence decay becomes non-exponential in situations with significant reabsorption. Non-exponential decay can also result from various other conditions.
|||D. S. Sumida and T. Y. Fan, “Effect of radiation trapping on fluorescence lifetime and emission cross section measurements in solid-state laser media”, Opt. Lett. 19 (17), 1343 (1994)|
|||H. Kühn et al., “Model for the calculation of radiation trapping and description of the pinhole method”, Opt. Lett. 32 (13), 1908 (2007)|
|||I. G. Kisialiou, “Free of reabsorption upper-state lifetime measurements by the method of transient gratings”, Appl. Opt. 51 (22), 5458 (2012)|