Revisiting the nonlinear Gaussian noise model for hybrid fiber spans

We rederive from first principles and generalize the theoretical framework of the nonlinear Gaussian noise model to the case of coherent optical systems with multiple fiber types per span and ideal Nyquist spectra. We focus on the accurate numerical evaluation of the integral for the nonlinear noise variance for hybrid fiber spans. This task consists in addressing four computational aspects: (1) Adopting a novel transformation of variables (other than using hyperbolic coordinates) that changes the integrand to a more appropriate form for numerical quadrature; (2) Evaluating analytically the integral at its lower limit, where the integrand presents a singularity; (3) Dividing the interval of integration into subintervals of size $\text{π}$ and approximating the integral over each subinterval by using various algorithms; and (4) Deriving an upper bound for the relative error when the interval of integration is truncated in order to accelerate computation. We apply the proposed analytical model to the performance evaluation of coherent optical communications systems with hybrid fiber spans composed of quasi-single-mode and single-mode fiber segments. More specifically, the model is used to optimize the lengths of the optical fiber segments that compose each span in order to maximize the system performance. We check the validity of the optimal fiber segment lengths per span provided by the analytical model by using Monte Carlo simulation, where the Manakov equation is solved numerically using the split-step Fourier method. We show that the analytical model predicts the lengths of the optical fiber segments per span with satisfactory accuracy so that the system performance, in terms of the Q-factor, is within 0.1 dB from the maximum given by Monte Carlo simulation.