Gain saturation

The small-signal gain $g_0$ of an active medium describes the gain at signal intensities low enough that the upper-level population is set entirely by pumping, with negligible depletion by stimulated emission. As signal intensity increases, stimulated emission depletes the upper-level population faster than pumping can refill it, and the gain drops.

For homogeneously-broadened gain media, the saturation behavior follows:

where $I_{\text{sat}}$ is the saturation intensity, the intensity at which gain drops to half the small-signal value. For inhomogeneously-broadened systems (Doppler-broadened atomic vapors, spectrally-broadened solid-state laser materials), the more general form includes a square root:

In this form the gain falls to half only at $I=3I_{\text{sat}}$. For semiconductor gain media, the saturation behavior depends on carrier dynamics (recombination time, free-carrier lifetime, hot-carrier effects) and on whether the saturation is dominated by interband or intraband processes.

Saturation power $P_{\text{sat}}$ is the power at which the gain drops to half its small-signal value. For a uniform beam in a waveguide of area $A$: $P_{\text{sat}}=I_{\text{sat}}\cdot A$.

Datasheets quote the saturation output power, the output at which an amplifier's gain has fallen 3 dB:

The two figures have different origins. An EDFA's output, like that of a Raman amplifier, is limited mainly by the pump power it converts, while an SOA's is set by its carrier lifetime and mode area; the saturation power entry works through both.

Why saturation matters.

• Output power scaling. The output power at which a high-gain amplifier's gain has compressed by 3 dB is about $\ln 2\cdot P_{\text{sat}}$, nearly independent of the small-signal gain. Higher $P_{\text{sat}}$ allows higher output for the same gain.
• Cross-talk in SOAs. SOA recovery time ($\sim 100$ ps) is comparable to bit periods of $10-40$ Gb/s signals, producing pattern-dependent gain that couples between WDM channels. EDFAs have much longer recovery time ($\sim$ ms) so gain is essentially constant across modulation patterns. Channel cross-talk is dramatically lower.
• Noise behavior. Gain reduction at high signal also reduces effective ASE buildup; partially-saturated amplifier chains have lower OSNR penalty than fully-linear ones.
• Pulse amplification. Saturable gain combined with saturable absorber gives the gain-loss balance required for mode-locked pulse formation. Q-switching holds the cavity loss high while the inversion builds far above the normal threshold, then drops it so the stored energy leaves as one giant pulse; gain switching is a separate technique that modulates the pump instead.

Self-saturation in lasers. The lasing intracavity intensity drives the gain medium into saturation; the steady-state output occurs at the intensity where gain equals total cavity loss. Solving the rate equations gives the standard output-power-vs-pump relation: $P_{\text{out}}={\eta }_{\text{sl}}(I-I_{\text{th}})$, with slope efficiency ${\eta }_{\text{sl}}$.

In an inhomogeneously broadened medium saturation is local in frequency and burns a hole rather than lowering the whole line; see spectral hole burning. In a standing-wave cavity it is also local in space; see spatial hole burning.