Small-signal gain

Small-signal gain is the gain of an amplifying medium when the light passing through it is too weak to change the population inversion that the pump has set up. It is quoted in two forms. The small-signal gain coefficient $g_0$, in cm⁻¹, is the local exponential growth rate of the optical gain before any saturation; the small-signal gain of a whole amplifier, $G_0$, is the ratio of output to input power in that regime, usually given in decibels. A C-band erbium-doped fiber amplifier typically has a small-signal gain of 20–35 dB, with some designs approaching 40 dB, and a telecom semiconductor optical amplifier 15–30 dB fiber to fiber.

Gain coefficient and amplifier gain

For a uniform medium of length $L$ with internal loss neglected, the power grows exponentially and

$$G_0 = \exp(g_0 L).$$

In decibels, $10\log_{10} G_0 = 4.343\, g_0 L$. An amplifier with 30 dB of small-signal gain therefore has $g_0 L = 6.91$, a factor of 1000 in power. In a waveguide amplifier the relevant coefficient is the net modal gain, $\Gamma g - \alpha_i$: a 1 mm SOA chip with 25 dB of gain has a net modal gain of 57.6 cm⁻¹. In a doped fiber, $g_0$ varies along the length because the pump is absorbed as it travels, so $G_0$ is the exponential of the integrated gain coefficient.

Inside a laser, $g_0$ is the gain the medium would have if the cavity did not lase. Above threshold the intracavity power saturates the gain down to the threshold value, so the ratio of $g_0$ to the threshold gain sets how far above threshold the laser runs and, together with the loss, the optimum output coupler transmission.

Saturation

As the signal grows, stimulated emission removes excited ions or carriers faster than the pump replaces them. For a homogeneously broadened medium the local gain coefficient falls as

$$g(P) = \frac{g_0}{1 + P/P_\text{sat}},$$

where $P_\text{sat}$ is the saturation power of the medium. Integrating this along the amplifier gives an implicit relation for the large-signal gain $G$:

$$G = G_0 \exp\!\left[-\frac{(G-1)P_\text{in}}{P_\text{sat}}\right].$$

It has no closed-form solution and is solved numerically. For an amplifier with $G_0 = 25$ dB and a 3 dB saturation output power of +10 dBm, which corresponds to $P_\text{sat} = 14.3$ mW, the gain is 24.0 dB at an input of −18.8 dBm and 22.0 dB at −12.0 dBm. The gain saturation entry covers inhomogeneous media and the semiconductor case, and the gain compression factor entry the fast intraband saturation in diode lasers.

Measurement

Small-signal gain is measured with the pump on and a probe weak enough that the gain no longer depends on it. A tunable laser or a set of fixed lasers is attenuated to a low input power, and output and input are read with a power meter or an optical spectrum analyzer. With the 25 dB amplifier above, an input of −30 dBm already compresses the gain by 0.09 dB, −35 dBm by 0.03 dB and −40 dBm by 0.01 dB, so the input is lowered until a further reduction leaves the gain unchanged within the measurement uncertainty. Two corrections apply. The amplified spontaneous emission that reaches the detector inside its bandwidth adds to the measured output, which matters most at the lowest inputs; an OSA measurement subtracts the ASE level interpolated under the signal. And for fiber-coupled devices the quoted figure includes the input and output coupling losses, which for an SOA are 2–6 dB per facet.

Practical significance

Small-signal gain is the figure of merit for preamplifiers, which receive weak signals and never approach saturation, and it is the regime in which SOA and preamplifier noise figures are usually specified, although EDFA datasheets often give the noise figure at a stated, saturating input power. Booster and in-line amplifiers run in saturation, where the output power is set by the pump power (the drive current in an SOA) and the small-signal gain matters mainly as headroom. In an SOA a high small-signal gain also raises the sensitivity to residual facet reflections, since the gain ripple grows with $G\sqrt{R_1R_2}$.

Common questions

What is the difference between small-signal gain and saturated gain?

Small-signal gain is the gain at input powers far below the saturation power, where it does not depend on the input. Saturated gain is the lower gain observed when the signal itself depletes the inversion, and it falls as the input rises.

How large is the small-signal gain of an EDFA?

Typically 20–35 dB for a single C-band amplifier with tens to a few hundred milliwatts of 980 nm or 1480 nm pump, and close to 40 dB in some high-gain designs. It depends on the erbium length, pump power and wavelength.

References: A. E. Siegman, Lasers (University Science Books, 1986); E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994); M. J. Connelly, Semiconductor Optical Amplifiers (Kluwer, 2002); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).