  Fiber nonlinearities, that lead to nonlinear signal interference (NLI), are
typically regarded as an instantaneous material response with respect to the
optical field. However, in addition to an instantaneous part, the nonlinear
fiber response consists of a delayed contribution, referred to as the Raman
response. The imaginary part of its Fourier transform, referred to as the Raman
gain spectrum, leads to inter-channel stimulated Raman scattering (ISRS). ISRS
is a nonlinear effect that redistributes optical power from high to lower
frequencies during propagation. However, as the nonlinear fiber response is
causal, the Raman spectrum obeys the Kramers-Kronig relations resulting in the
real part of the complex valued Raman spectrum. While the impact of the
imaginary part (i.e. ISRS) is well studied, the direct implications of its
associated real part on the NLI are unexplored.
  In this work, a theory is proposed to analytically quantify the impact of the
real Raman spectrum on the nonlinear interference power. Starting from a
generalized Manakov equation, an extension of the ISRS Gaussian Noise (GN)
model is derived to include the real Raman spectrum and, thus, to account for
the complete nonlinear Raman response. Accurate integral expressions are
derived and approximations in closed-form are proposed. Different formulations
for the case of single -and dual polarized signals are derived and novel
analytical approximations of the real Raman spectrum are proposed. Moreover, it
is analytically shown that the real Raman spectrum scales the strength of the
instantaneous nonlinear distortions depending on the frequency separation of
the interacting frequencies. A simple functional form is derived to assess the
scaling of the NLI strength. The proposed theory is validated by numerical
simulations over C-and C+L band, using experimentally measured fiber data.
