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Gain Efficiency

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

The gain efficiency of an amplifier or a laser gain medium can be defined as the small-signal gain divided by the pump power required to achieve this gain in the steady state. For pulsed pumping, it may also be defined as the small-signal gain divided by the pump pulse energy, or alternatively by the stored energy in the gain medium (which is lower due to non-perfect pumping efficiency). The latter definition can also make sense in the context of continuous pumping (even for continuous-wave operation); note that the stored energy is related to the applied continuous pump power.

For three-level gain media, where the gain without pumping is negative, it is more sensible to use the differential gain efficiency, i.e. the derivative of the small-signal gain with respect to the pump power.

The gain efficiency should not be confused with the power conversion efficiency. For example, an optical amplifier based on a laser crystal may have a low gain efficiency (due to a large mode area) but nevertheless a high power conversion efficiency.

Relation to Stored Energy

Generally, the gain increases with increasing stored energy, but the relation between those quantities may generally be non-trivial.

A simple relation results if we assume a flat-top profile of the laser (or signal) beam in the gain medium with a mode area ($A$). Here, the dependence of the gain coefficient ($g$) on the stored energy can be simply calculated as

for a laser gain medium with emission and absorption cross-sections ($\sigma_\rm{em}$) and ($\sigma_\rm{abs}$) and photon energy ($h\nu$) at the signal wavelength. (For a four-level gain medium, ($\sigma_\rm{abs}$) = 0.) The power amplification factor is ($\exp(g)$).

The equation shows that the gain efficiency in terms of stored energy is directly and inversely related to the saturation energy. High laser cross-sections lead to a high gain efficiency, but also to a low saturation energy.

In practice, one often has a smoothly varying intensity profile. Those laser-active ions sitting in places with high beam intensity can contribute more to the gain. Therefore, the achieved gain is no longer strictly related to the stored energy or applied pump power. For accurate simulations, this needs to be taken into account.

Continuous Pumping

The pump power required for achieving a certain stored energy in the steady state depends on the upper-state lifetime of the laser transition: the shorter this lifetime, the higher is the rate with which ions needs to be pumped into the upper laser level. For the differential gain efficiency in terms of pump power, this leads to the equation

where ($\eta_\textrm{p}$) is the pump efficiency, including the pump absorption efficiency, the quantum efficiency of the pumping process, and the quantum defect. ($P_\rm{sat}$) is the saturation power, which is the saturation energy divided by the upper-state lifetime.

Fiber Amplifiers vs. Bulk Devices

Fiber amplifiers typically have a small effective mode area and can thus easily reach differential gain efficiencies of several dB/mW, with special optimization even more than 10 dB/mW.

In contrast, bulk amplifiers and lasers can be operated with far larger mode areas, implying correspondingly lower gain efficiencies, although that may in part be compensated by higher emission cross-sections of crystalline gain media.

Is a High Gain Efficiency Desirable?

A high gain efficiency can be desirable for an amplifier when a high gain is wanted. However, it can be preferable to have a not too large gain efficiency in cases where a high energy needs to be stored in a gain medium – for example for Q switching of a laser, or if pulses need to be amplified to high energies. Therefore, when they need to operate devices with relatively large beam areas.

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