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Definition: the phenomenon that the gain of an amplifier is reduced for high input signal powers
An amplifier device such as a laser gain medium cannot maintain a fixed gain for arbitrarily high input powers, because this would require adding arbitrary amounts of power to the amplified signal. Therefore, the gain must be reduced for high input powers; this phenomenon is called gain saturation (or gain compression).
In the case of a laser gain medium, the gain does not instantly adjust to the level according to the optical input power, because the gain medium stores some amount of energy, and the stored energy determines the gain. For example, a sudden increase in the input power of a laser gain medium will reduce the gain only within a certain time, because the population of excited laser ions is only reduced with a certain finite rate. This has important consequences for the laser dynamics.
The general dynamic equation for the gain is
where g is the gain coefficient (assumed to be small), gss the small-signal gain (for a given pump intensity), 
\g the gain relaxation time, P the power of the amplified beam, and Esat the saturation energy of the gain medium.
Note that the power amplification factor is exp(g), and must not be confused with g.
Gain Saturation in the Steady State
In the steady state (i.e., for long time scales with constant pump power and resonator losses), the gain is
where Psat is the saturation power. Note that it has been implicitly assumed that the pump rate is constant, i.e. there are no effects of pump saturation. This assumption is well justified in most, but not all cases.
Figure 1: Saturated gain versus signal power in the steady state.
For example, the gain is reduced to half the small-signal gain if the signal power equals the saturation power.
Calculations for large gain are more sophisticated, essentially because the optical intensity varies significantly within the amplifier. A straightforward approach is to divide a high-gain amplifier into a sequence of low-gain amplifiers, which can all be treated with the low-gain approximation. However, there are numerically more efficient techniques which do not require such a subdivision.
The transverse variation of optical intensity of a laser beam can modify the saturation characteristics: laser-active ions in the outer parts of the beam require a higher optical power to be saturated. This effect somewhat modifies the saturation curves as shown e.g. in Figures 1 and 2.
The presented equations can also be used for high repetition rate pulse trains with sufficiently low pulse energy. Gain saturation may then be determined only by the average power.
Gain Saturation by an Optical Pulse
For optical amplification of a short pulse (with a duration well below the upper-state lifetime), spontaneous emission during the time of pulse amplification is not important. Also, the influence of pump light can usually be neglected (except for pulsed pumping with high intensity). The gain after the pulse is then reduced by a factor exp(−Ep / Esat), where Ep is the pulse energy. For example, the gain is reduced to 1 / e ≈ 37% of the initial value, if Ep = Esat.
Figure 2: Reduction in gain by a short pulse.
The effective gain as experienced by the pulse is some averaged value, as the gain decreases during pulse amplification. This average gain can be calculated by considering the reduction in the stored energy in the gain medium. In the simpler case, where the gain is small, so that the intensity is approximately constant within the amplifier, the result is
where g0 is the initial gain coefficient. In the more general case with arbitrarily high gain, one may use the Frantz-Nodvick equation
and obtain
Note that an implicit underlying assumption is that there is always a thermal equilibrium within the involved Stark level manifolds. That assumption may be violated for an intense femtosecond pulse. In such cases, the effective amplification and energy extraction may be smaller than estimated with the presented equations.
Homogeneous and Inhomogeneous Saturation
An important issue is the homogeneous or inhomogeneous nature of gain saturation. Homogeneous gain saturation means that the spectral shape of the gain is not affected by the saturation. This is the case e.g. when all laser-active ions have the same emission spectrum. In some gain media, particularly in disordered media such as glasses, the laser ions can occupy different sites in the lattice, and the differing local electric fields affect the wavelengths and strength of the different transitions. This can lead to inhomogeneous saturation (Figure 3), where the spectral shape changes; typically, the gain around the laser wavelength is saturated more than the gain at other wavelengths.
Figure 3: Demonstration of inhomogeneous gain saturation. A laser at 1064 nm saturates the gain around 1064 nm more than the gain at other wavelengths. For comparison, the unsaturated gain (without laser power) is shown as a dotted curve.
Another cause of inhomogeneous gain medium can be spatial hole burning in linear laser resonators, caused by the wavelength-dependent standing-wave pattern in the gain medium.
In any case, inhomogeneous gain saturation can make it difficult to achieve single-frequency operation, since non-lasing resonator modes are favored in terms of gain. The homogeneous or inhomogeneous nature of gain saturation also has important effects on the mode-locking behavior and particularly for Q switching and for amplifiers.
Saturation Characteristics of Different Gain Media
Different kinds of gain media differ very much in their gain saturation characteristics:
- Solid-state gain media based on ion-doped crystals or glasses typically operate on so-called weakly allowed transitions, leading to small laser cross sections, large saturation fluences and intensities, and long upper-state lifetimes (microseconds to milliseconds). Such lasers are suitable for Q switching but also have a tendency for spiking behavior. For passive mode locking, they tend to exhibit Q-switching instabilities.
- Semiconductors and laser dyes exhibit a small saturation fluence and intensity, in addition to a short upper-state lifetime of typically a few nanoseconds. They are thus not suitable for Q switching, but for passive mode locking without Q-switching instabilities, and they react very quickly to changes in the pump power. The latter has implications on the intensity noise.
- Optical parametric amplifiers instantaneously adapt the gain to the amplified signal power level, since they do not store energy in the gain medium.
See also: saturable absorbers, laser dynamics, inhomogeneous saturation, spatial hole burning
Categories: amplifiers, lasers, physical foundations
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