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Gain Saturation

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Definition: the phenomenon that the gain of an amplifier is reduced for high input signal powers

German: Verstärkungssättigung

Categories: optical amplifiers, lasers, physical foundations

How to cite the article; suggest additional literature

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

evolution of gain

where g is the logarithmic gain coefficient, i.e., the natural logarithm of the ratio of output power to input power; it is assumed to be small. (Otherwise, the optical power would vary substantially within the gain medium.) gss is the small-signal gain (for a given pump intensity), τg 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

steady-state gain

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.

gain saturation in steady state

Figure 1: Saturated gain versus signal power (for a constant pump 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.

gain saturation by a pulse

Figure 2: Reduction in gain by a short pulse. The solid curve shows the gain after the pulse, and the dashed curve the average gain experienced by the 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

gain for a short pulse

where g0 is the initial gain coefficient. (See the dashed curve in Figure 2.) In the more general case with arbitrarily high gain, one may use the Frantz-Nodvick equation

Frantz-Nodvick equation

and obtain

effective gain of a high-gain saturated amplifier

(Note that as in the equations above, g0 and gp are dimensionless gain coefficients, not the gain per unit length.)

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.

Figure 3 shows a numerically simulated example where an optical pulse is amplified in a fiber amplifier The shape of the amplified pulse is substantially distorted, since the amplifier gain drops substantially during the pulse due to gain saturation.

pulse amplification in a fiber amplifier

Figure 3: Amplification of an optical pulse in an ytterbium-doped fiber amplifier. The solid blue curve shows the output power versus time; it substantially deviates from the output power for neglected gain saturation (dot-dashed curve), since the gain is substantially saturated. The black dashed curve shows how the upper-level population drops during the pulse. The diagram has been taken from a case study done with the RP Fiber Power software.

Such strong saturation effects are common for cases where one tries to extract much of the stored energy from a high-gain amplifier with a single pulse. Numerical simulations are often needed for analyzing such devices. Note that the saturation conditions can be very different between the input and output ends of the amplifier, and nonlinear effects can additionally complicate the behavior.

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 4), where the spectral shape changes; typically, the gain around the laser wavelength is saturated more than the gain at other wavelengths.

inhomogeneous gain saturation

Figure 4: 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:

See also: saturable absorbers, laser dynamics, inhomogeneous saturation, spatial hole burning
and other articles in the categories optical amplifiers, lasers, physical foundations

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