Upper-state lifetime

The upper-state lifetime $\tau$ of a laser medium is the time constant with which the population of the upper laser level decays when pumping stops and no stimulated emission is present. After a pump pulse the population falls as $e^{-t/\tau}$, to 37 % after one lifetime and 5 % after three. Values span about seven orders of magnitude across common gain media:

The figures for doped crystals and glasses depend somewhat on dopant concentration, host composition and temperature. In a semiconductor the corresponding quantity is the carrier lifetime at threshold, which falls as the carrier density rises.

Measurement

The lifetime is measured as a fluorescence decay: a short pump pulse excites the sample, a fast detector records the emission at the laser wavelength, and an exponential fit gives $\tau$. For the long lifetimes of rare-earth ions a chopped CW diode and an ordinary photodiode suffice; nanosecond semiconductor lifetimes need time-correlated single-photon counting or differential carrier-lifetime measurements from the electrical impedance. The measured fluorescence lifetime equals the upper-state lifetime when the emission comes from that level alone and the sample is thin enough that reabsorption and amplified emission do not distort the decay.

Radiative and nonradiative decay

The total decay rate is the sum of a radiative part and a nonradiative part,

$$\frac{1}{\tau} = \frac{1}{\tau_\text{rad}} + \frac{1}{\tau_\text{nr}},$$

and the fluorescence quantum efficiency is $\tau/\tau_\text{rad}$. Nonradiative recombination through multiphonon emission, energy transfer between neighboring ions (concentration quenching) or defects shortens the lifetime and turns the stored energy into heat. In Nd:YAG, quenching is one reason doping is kept near 1 %. In semiconductors, Auger and defect recombination play the same role.

Energy storage and Q-switching

A long upper-state lifetime lets a medium accumulate inversion while the cavity is held off and release it in a single pulse, the basis of Q-switching. Under constant pumping, the inversion approaches its steady state with time constant $\tau$; the fraction of the maximum available after a pumping interval $T$ is $1 - e^{-T/\tau}$. For Nd:YAG at a 1 kHz repetition rate this is 0.99; at $1/\tau = 4.3$ kHz it is 0.63; at 20 kHz it is 0.20. This is why the pulse energy of a Q-switched Nd:YAG laser falls above a few kilohertz. Ti:sapphire, with a lifetime 72 times shorter, stores little energy and is usually pumped by a pulsed laser when used as an amplifier. Diode lasers store energy for only about a nanosecond, which limits gain switching to picosecond pulses of small energy and gives them relaxation oscillations in the gigahertz range.

The long lifetime of erbium has the opposite consequence in an EDFA: the gain cannot follow modulation at gigabit rates and responds only to the average signal power, so data channels do not cross-modulate at the bit rate; adding or dropping whole channels still produces gain transients over tens to hundreds of microseconds.

Saturation intensity

The lifetime sets the intensity at which a four-level medium's gain is halved, the saturation intensity of gain saturation:

$$I_\text{sat} = \frac{h\nu}{\sigma\,\tau}.$$

For Nd:YAG at 1064 nm, $h\nu = 1.165$ eV $= 1.87 \times 10^{-19}$ J. With an emission cross section $\sigma = 2.8 \times 10^{-19}$ cm² (quoted values range up to about $6.5 \times 10^{-19}$ cm² depending on how the line structure is treated) and $\tau = 230$ µs,

$$I_\text{sat} \approx 2.9\ \text{kW/cm}^2.$$

The corresponding saturation fluence $h\nu/\sigma$, independent of $\tau$, is 0.67 J/cm² and governs pulse amplification. Ti:sapphire, with $\sigma \approx 3 \times 10^{-19}$ cm² at 800 nm and $\tau = 3.2$ µs, has $I_\text{sat} \approx 260$ kW/cm², which is why it needs tightly focused pumping. The product $\sigma\tau$ is a figure of merit for CW lasers: the threshold pump power of a four-level laser scales inversely with it.

Pitfalls

Radiation trapping, in which emitted photons are reabsorbed and re-emitted, lengthens the apparent lifetime in thick or highly doped samples, a known problem for Yb-doped and Er-doped media. Lifetimes of many hosts fall with temperature as nonradiative paths open. In a laser running above threshold, the effective lifetime of the upper level is shortened by stimulated emission; the tabulated value refers to spontaneous decay only.

Common questions

What is the difference between upper-state lifetime and fluorescence lifetime?

For most laser media they are the same measured quantity: the decay time of spontaneous emission from the upper laser level. They differ only when reabsorption or emission from other levels affects the measurement.

Why does a long upper-state lifetime help Q-switching?

The medium can integrate pump power over a time of order $\tau$ before spontaneous decay drains it, so more energy is stored when the switch opens.

Is a long lifetime always better?

No. It lowers the saturation intensity and the pump threshold, but it lowers the relaxation-oscillation frequency and makes a laser more prone to spiking and slow transients after disturbances.

References: A. E. Siegman, Lasers (University Science Books, 1986); W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006); O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994).