Saturation power (P_sat)

Saturation power is the optical power at which an amplifier's gain has fallen to half its small-signal value. Two related figures go by the name. The saturation power of the medium, $P_{\text{sat}}$, is the power at which the local gain coefficient is halved:

$$g=\frac{g_0}{1+P\mathrm{/}{P}_{\text{sat}}},P_{\text{sat}}=\frac{h\nu A}{\sigma \tau },$$

where $h\nu$ is the photon energy, $A$ the area of the light in the gain medium, $\sigma$ the stimulated-emission cross-section and $\tau$ the lifetime of the upper state or of the carriers. The figure on a datasheet is the saturation output power $P_{\text{sat,out}}$: the output power at which the gain of the whole amplifier has dropped 3 dB below its small-signal value $G_0$. For an amplifier that saturates homogeneously and has high gain, $P_{\text{sat,out}}\approx \ln 2\cdot P_{\text{sat}}=0.69P_{\text{sat}}$.

From the medium to the amplifier

With internal loss neglected, the power grows along the amplifier as $dP\mathrm{/}dz=g_{0}P\mathrm{/}(1+P\mathrm{/}{P}_{\text{sat}})$. Integrating over the length, with $G_0=e^{g_{0}L}$ and $G=P_{\text{out}}\mathrm{/}{P}_{\text{in}}$, gives the saturated gain implicitly:

$$G=G_0\exp \text{\hspace{0.17em}⁣}[-\frac{(G-1)P_{\text{in}}}{P_{\text{sat}}}]\mathrm{.}$$

Setting $G=G_0\mathrm{/}2$ gives the 3 dB saturation output power,

$$P_{\text{sat,out}}=\frac{G_0\ln 2}{G_0-2}{P}_{\text{sat}},$$

which approaches $0.69P_{\text{sat}}$ at high gain and does not depend on $G_0$ there. At high gain, the output at $x$ dB of compression is $(x\ln 10\mathrm{/}10)P_{\text{sat}}$:

For an amplifier with $G_0=25$ dB and $P_{\text{sat,out}}=+10$ dBm, the model gives $P_{\text{sat}}=14.3$ mW, a gain of 24 dB at an output of +5.2 dBm (input −18.8 dBm) and of 22 dB at +10.0 dBm (input −12.0 dBm). Datasheets quote the 3 dB point, measured by sweeping the input power and recording the gain against the output power.

What sets the saturation power

A larger mode area, a smaller cross-section and a shorter lifetime all raise $P_{\text{sat}}$. In a semiconductor optical amplifier, the area is the active region's cross-section divided by the confinement factor, the differential gain $a=dg\mathrm{/}dN$ takes the place of the cross-section, and $\tau$ is the carrier lifetime. For an active region 1.5 µm wide and 0.1 µm thick with a confinement factor of 0.3, $a=3\times {10}^{-16}$ cm² and $\tau =0.3$ ns, at 1550 nm,

$$P_{\text{sat}}=\frac{h\nu (wd\mathrm{/}\mathrm{\Gamma })}{a\tau }=7.1\text{ mW }(+8.5\text{ dBm}),$$

and the saturation output power is about 4.9 mW (+6.9 dBm), a little below the +8 to +13 dBm typical of commercial SOAs. High-power SOAs raise it with a lower confinement factor, a waveguide that flares toward the output, and a higher current, which shortens the carrier lifetime.

The erbium-doped fiber amplifier is the opposite case. The upper-state lifetime of erbium, about 10 ms, is some seven orders of magnitude longer than a semiconductor carrier lifetime, and its cross-sections are small, so by the formula above the erbium medium itself saturates at low power. An EDFA nevertheless delivers much more output than an SOA because it runs deep in saturation, where the output is limited by the pump power it converts rather than by $P_{\text{sat}}$: each pump photon adds at most one signal photon, so the power added to the signal cannot exceed $({\lambda }_p\mathrm{/}{\lambda }_s)P_{\text{pump}}$, 63 mW from 100 mW of 980 nm pump at 1550 nm. The saturation output power of an EDFA therefore scales with its pump power, and with 200 mW of pump the limit is +21 dBm. The EDFA and SOA entries give typical datasheet values.

Speed of saturation

The lifetime also sets how fast the gain responds. Erbium's gain changes over milliseconds, so an EDFA saturates on the average power of the signals passing through it and imposes no pattern on individual bits or on neighboring WDM channels; only adding or dropping channels moves its gain, which is why line amplifiers use automatic gain control. An SOA's gain follows the carrier lifetime, a fraction of a nanosecond, so a saturated SOA distorts fast data and transfers the modulation of one channel onto others (gain saturation describes these effects). Linear SOA applications stay well below $P_{\text{sat,out}}$, while wavelength converters and regenerators drive the SOA into saturation deliberately.

For pulses shorter than the lifetime, the relevant quantity is the saturation energy $E_{\text{sat}}=P_{\text{sat}}\tau =h\nu A\mathrm{/}\sigma$, the pulse energy that depletes the gain substantially. The SOA above has $E_{\text{sat}}=2.1$ pJ, which is why SOAs distort short pulses of even modest energy, while the long lifetime of rare-earth-doped fibers lets pulsed fiber amplifiers store and release far larger energies.

Common questions

What is the saturation output power of an amplifier?

The output power at which the gain has fallen 3 dB below its small-signal value. For a homogeneously saturating amplifier with high gain it is $0.69P_{\text{sat}}$, and the 1 dB compression point lies 4.8 dB below it.

What is the difference between saturation power and saturation output power?

The saturation power $P_{\text{sat}}$ is a property of the gain medium: the local power at which the gain coefficient is halved. The saturation output power is a property of the whole amplifier: the output at which its total gain is 3 dB compressed. The two are related by the formula above.

Why do EDFAs have higher output power than SOAs?

Not because the erbium medium has a higher intrinsic saturation power; with its long lifetime it has a low one. An EDFA's output is limited by the pump power it converts, at most ${\lambda }_p\mathrm{/}{\lambda }_s$ signal watts per pump watt, and pump lasers of hundreds of milliwatts are available. An SOA is limited by its carrier dynamics, which fix $P_{\text{sat}}$ at milliwatts to tens of milliwatts.

References: E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994). M. J. Connelly, Semiconductor Optical Amplifiers (Kluwer, 2002). L. M. Frantz and J. S. Nodvik, "Theory of pulse propagation in a laser amplifier," Journal of Applied Physics 34, 2346 (1963). G. P. Agrawal, Fiber-Optic Communication Systems, 5th ed. (Wiley, 2021).