Quantum-limited noise figure of networks of linear optical elements

The quantum-limited noise figure (NF) for a general combination of linear optical elements is obtained in a compact form. The entries in this expression are obtained from the classical input–output transmission coefficient of the signal, and of the various transmission coefficients from the noise inputs to the output. The same result is also obtained by means of a semiclassical derivation. The linear elements can be: population-inversion amplifiers, parametric amplifiers and/or wavelength converters, beam splitters, lossy fibers, etc. They form a network with a simple directed graph (no loops). We identify a class of networks for which Friis’ formula for concatenated elements holds. We verify the general formula for a variety of well-known cases. We also verify that the NF for an optical communication link periodically amplified by nondegenerate phase-sensitive parametric amplifiers (PSAs) is only about half of that for a similar system using degenerate PSAs.