Saturation power (P_sat)

Saturation power is the power at which an amplifier's gain has dropped to half (–3 dB below) its small-signal value. For a homogeneously-broadened gain medium:

$$G(P) \;=\; \frac{G_0}{1 + P / P_\text{sat}},$$

where $G_0$ is the unsaturated (small-signal) gain.

A more useful definition for distributed amplifiers (fiber amplifiers): saturation output power $P_\text{sat,out}$ is the output power at which the gain has dropped to half $G_0$, related by

$$P_\text{sat,out} \;\approx\; \ln(2) \cdot P_\text{sat,intrinsic} \cdot \frac{G_0 - 1}{G_0 \ln G_0}.$$

Datasheets generally quote $P_\text{sat,out}$ in dBm — the readable, system-relevant figure.

Typical saturation output powers.

Amplifier	$P_\text{sat,out}$	Small-signal gain
Telecom EDFA (low-noise preamp)	$+10$ to $+15$ dBm	25 – 30 dB
Telecom EDFA (booster amp)	$+20$ to $+27$ dBm	15 – 25 dB
High-power industrial Yb fiber amp	$+30$ to $+50$ dBm (1 – 100 W)	varies
Telecom SOA	$-5$ to $+10$ dBm	15 – 25 dB
Raman amplifier (1 W pump)	$+20$ to $+25$ dBm	5 – 25 dB

Why EDFA $\gg$ SOA in $P_\text{sat}$. The saturation power scales with the upper-state lifetime:

$$P_\text{sat} \;\propto\; \frac{h\nu \, A_\text{eff}}{\sigma \, \tau},$$

where $\sigma$ is the stimulated emission cross-section, $\tau$ is the upper-state lifetime, and $A_\text{eff}$ is the effective mode area.

• Erbium upper-state lifetime: $\sim 10$ ms ($^4I_{13/2}$ metastable level)
• Semiconductor upper-state lifetime: $\sim 0.1 - 1$ ns (radiative + non-radiative)

The $10^7$ longer lifetime in erbium gives correspondingly higher saturation power and corresponds to the very different applications: EDFAs as essentially-linear high-output amplifiers, SOAs as fast-switching elements where saturation is sometimes desired (wavelength conversion, regeneration, nonlinear signal processing).

Operating point selection.

For a target output power $P_\text{out}$ from an amplifier with small-signal gain $G_0$ and saturation output power $P_\text{sat,out}$:

• $P_\text{out} \ll P_\text{sat,out}$: amplifier operates linearly, gain ≈ $G_0$
• $P_\text{out} \approx P_\text{sat,out}$: gain compressed by ~3 dB
• $P_\text{out} \gg P_\text{sat,out}$: gain $\propto P_\text{pump} / P_\text{out}$; saturation-clamped output

Most telecom WDM systems operate EDFAs at $P_\text{out}$ near $P_\text{sat,out}$ to maximize energy efficiency while keeping the gain spectrum nearly flat across the channel comb. Operating well into saturation reduces ASE buildup (and OSNR penalty) per amplifier stage at the cost of pattern-dependent effects in some signal formats.