Generalised expression of the noise figure of phase sensitive amplifiers for an arbitrary number of modes

Phase sensitive amplifiers (PSA), contrary to usual phase insensitive (PIA), are in principle capable achieve noiseless amplification, i.e. exhibit a quantum-limited noise figure (NF) of 0 dB. When implemented using four-wave mixing (FWM) nonlinear fiber, extra waves can be generated by undesired FWM processes, which may introduce input ports for vacuum fluctuations, thus potentially degrading the NF. In this situation, we give here general analytical quantum derivation PSA NF, valid an arbitrary number nonlinearly coupled modes. This expression is usable as soon linear input-output relation found annihilation and creation operators involved It predicts that level depends on interacting waves. We illustrate usefulness case six waves, corresponding four example signal NF degraded 0.4 dB, compared 10 dB obtained PIA operation same scheme.