The Generalized Droop Formula for Low Signal to Noise Ratio Optical Links

We present a theoretical model that fully supports the recently disclosed generalized droop formula (GDF) for calculating the signal-to-noise ratio (SNR) of constant-output power (COP) amplified coherent links operated at a very low SNR. For single-mode nonlinear COP links we compare the GDF-SNR to the better known generalized SNR (GSNR) that uses the Gaussian noise (GN) model for constant-gain (CG) amplifiers. We find that at all medium to large SNRs the GSNR well matches with the GDF, while at GSNR below 6 dB the GSNR over-estimates the correct GDF-SNR by more than 0.5 dB. Fortunately, the GDF-SNR turns out to be approximately a simple function of the GSNR, which allows adaptation of the widespread GSNR also to very low SNR links. A key finding of this article is that the end-to-end model underlying the GDF is a concatenation of per-span first-order regular perturbation (RP1) models with end-span power renormalization. This fact allows the GDF to well reproduce the SNR of highly nonlinear systems, well beyond the RP1 limit underlying the GN model. The GDF is successfully extended to the case where the bandwidth/modes of the COP amplifiers are not entirely filled by the transmitted multiplex. Finally, the GDF is extended to CG amplified links and is shown to improve on known GN models of highly nonlinear propagation with CG amplifiers.