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Upper-state Lifetime

Definition: the lifetime of the population in the upper laser level

Opposite term: lower-state lifetime

German: Lebensdauer des oberen Zustands

Category: physical foundationsphysical foundations


Cite the article using its DOI: https://doi.org/10.61835/hur

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In a laser gain medium, amplification is associated with the population in 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.

Ideally, the decay of the upper-state population is caused only by the unavoidable spontaneous emission. Under these conditions, the inverse lifetime (then called radiative lifetime) is

$$\frac{1}{{{\tau _{{\rm{rad}}}}}} = \frac{{8\pi \;{n^2}}}{{{c^2}}}\;\int {{\nu ^2}{\sigma _{{\rm{em}}}}(\nu )\;{\rm{d}}\nu } = 8\pi \;{n^2}c\;\int {\frac{{{\sigma _{{\rm{em}}}}(\lambda )}}{{{\lambda ^4}}}\;{\rm{d}}\lambda } $$

This shows that the decay is fast for large laser cross-sections and a large gain bandwidth.

The decay rate is enhanced if there are additional radiative or non-radiative transitions to other lower-lying energy levels. In particular, there can be 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 later on.

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.

In the case of semiconductor devices (semiconductor optical amplifiers and lasers), the upper-state lifetime is usually called the carrier lifetime.

Typical Values

The upper-state lifetimes of different kinds of laser gain media differ considerably:

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 <$\sigma -\tau$> 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 <$\sigma -\tau$> product), at least for four-level lasers.

The upper-state lifetime also influences the laser dynamics, e.g. the relaxation oscillations and the tendency for spiking.

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 or 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, as mentioned above.

More to Learn

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[1]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); https://doi.org/10.1364/OL.19.001343
[2]H. Kühn et al., “Model for the calculation of radiation trapping and description of the pinhole method”, Opt. Lett. 32 (13), 1908 (2007); https://doi.org/10.1364/OL.32.001908
[3]I. G. Kisialiou, “Free of reabsorption upper-state lifetime measurements by the method of transient gratings”, Appl. Opt. 51 (22), 5458 (2012); https://doi.org/10.1364/AO.51.005458

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