Revisiting the nonlinear Gaussian noise model: The case of 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: (i) Adopting a novel transformation of variables (other than using hyperbolic coordinates) that changes the integrand to a more appropriate form for numerical quadrature; (ii) Evaluating analytically the integral at its lower limit, where the integrand presents a singularity; (iii) Dividing the interval of integration into subintervals of size pi and approximating the integral in each subinterval by using various algorithms; and (iv) Deriving an upper bound for the relative error when the interval of integration is truncated in order to accelerate computation. We apply the proposed model to coherent optical communications systems with hybrid fiber spans composed of quasi-singlemode fiber and single-mode fiber segments. The accuracy of the final analytical relationship for the nonlinear noise variance in long-haul coherent optical communications systems with hybrid fiber spans is checked using the split-step Fourier method and Monte Carlo simulation. It is shown to be adequate to within 0.1 dBQ for the determination of the optimal fiber segment lengths per span that maximize system performance.