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An accurate,closed-form expression to evaluate the nonlinear interference (NLI) noise power in Nyquist-spaced, coherent optical communication systems using backward-pumped Raman amplification is presented. This enables rapid estimation of the signal-to-noise ratio (SNR) and avoids the need of integral evaluations and split-step simulations.The accuracy of the proposed formula is compared to numerical integration of the Gaussian noise (GN) model and split-step simulations over a wide range of parameters, including three different fiber types. Additionally, the impact of pump depletion on the NLI noise power is studied and the formula is applied to a second-order Raman-amplified system. In the case of first-order amplification and negligible pump depletion, a maximum deviation of 0.34 dB in NLI coefficient between the GN model and the closed-form formula is found, which corresponds to a maximum deviation of ~0.1 dB in optimal SNR or similar figures of merit (e.g., maximum reach). When pump depletion is considered,it is shown that the NLI coefficient becomes a function of launch power and as a result the cubic power dependence of the NLI noise power is no longer valid in such regimes. Finally, for the second-order Raman-amplified system, a maximum deviation of 0.39 dB in NLI coefficient is reported.

A Closed-Form Expression to Evaluate Nonlinear Interference in Raman-Amplified Links