Noise figure (NF)

Noise figure in decibels is

$$\text{NF}=10{\log }_{10}\text{\hspace{0.17em}⁣}\left(\frac{{\text{SNR}}_{\text{in}}}{{\text{SNR}}_{\text{out}}}\right)=10{\log }_{10}F,$$

where $F$ is the linear noise factor and SNR is the signal-to-noise ratio at the photodetected output.

For an optical amplifier with gain $G$ and population-inversion factor $n_{\text{sp}}$ (which amplifier texts also call the spontaneous emission factor, a different quantity from the $\beta$ of a laser):

$$F=2n_{\text{sp}}(1-\frac{1}{G})+\frac{1}{G}\mathrm{.}$$

In the high-gain limit ($G\gg 1$):

$$F\to 2n_{\text{sp}},$$

with a fundamental quantum-mechanical lower bound of $n_{\text{sp}}=1$ (full population inversion), giving NF $\ge 3$ dB. Real amplifiers exceed this due to partial inversion and additional loss mechanisms.

Typical values:

Raman amplifiers achieve effective NF below 0 dB because gain is distributed along the transmission fiber. The signal is amplified before it is fully attenuated, effectively reducing the equivalent input loss compared to a discrete amplifier at the span end.

Cascade. For $N$ amplifiers in series with gains $G_i$ and noise factors $F_i$ (Friis formula):

$$F_{\text{total}}=F_1+\frac{F_2-1}{G_1}+\frac{F_3-1}{G_1{G}_2}+\dots$$

The first amplifier in a chain dominates the overall noise figure if its gain is high. This is why the first stage of an amplifier chain, and a pre-amplifier immediately preceding the receiver, are specified for low NF, while later amplifiers can tolerate higher NF.

NF directly limits the achievable OSNR of an amplified link: each amplifier in a span adds noise proportional to $F_i\mathrm{/}{G}_i$. Minimizing per-amplifier NF and selecting span lengths to maintain adequate per-stage gain is the central design problem for long-haul telecom.

The OSNR and Noise Figure Calculator applies the Friis formula to up to five amplifiers and losses and shows how much of the total each stage contributes.