  Coherent optical transmission systems naturally lead to a four dimensional
(4D) signal space, i.e., two polarizations each with two quadratures. In this
paper we derive an analytical model to quantify the impact of Kerr nonlinearity
on such 4Dspaces, taking the interpolarization dependency into account. This is
in contrast to previous models such as the GN and EGN models, which are valid
for polarization multiplexed (PM)formats, where the two polarizations are seen
as independent channels on which data is multiplexed. The proposed model agrees
with the EGN model in the special case of independent two-dimensional
modulation in each polarization. The model accounts for the predominant
nonlinear terms in a WDM system, namely self-phase modulation and and
cross-phase modulation. Numerical results show that the EGN model may
inaccurately estimate the nonlinear interference of 4D formats. This nonlinear
interference discrepancy between the results of the proposed model and the EGN
model could be up to 2.8 dB for a system with 80 WDM channels. The derived
model is validated by split-step Fourier simulations, and it is shown to follow
simulations very closely.
