In optical communications, four-dimensional (4D) modulation formats encode information onto the quadrature components of two arbitrary orthogonal states of polarisation of the optical field. Many analytical models available in the optical communication literature allow, within a first-order perturbation framework, the computation of the average power of the nonlinear interference (NLI) accumulated in coherent fibre-optic transmission systems. However, all such models only operate under the assumption of transmitted polarisation-multiplexed two-dimensional (PM-2D) modulation formats, which only represent a limited subset of the possible dual-polarisation 4D (DP-4D) formats. Namely, only those where data transmitted on each polarisation channel are mutually independent and identically distributed. This paper presents a step-by-step mathematical derivation of the extension of existing NLI models to the class of arbitrary DP-4D modulation formats. In particular, the methodology adopted follows the one of the popular enhanced Gaussian noise model, albeit dropping most assumptions on the geometry and statistic of the transmitted 4D modulation format. The resulting expressions show that, whilst in the PM-2D case the NLI power depends only on different statistical high-order moments of each polarisation component, for a general DP-4D constellation, several other cross-polarisation correlations also need to be taken into account.

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 4D spaces, 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 splitstep Fourier simulations, and it is shown to follow simulations very closely.

Optical transmission systems intrinsically enjoy a four-dimensional (4D) constellation space, corresponding to two quadratures in two polarization states. In this paper, we introduce a general nonlinear model that is valid for 4D symmetric modulation formats. We take the inter-polarization dependency into account to derive this model. The model accounts for all perturbative nonlinear interference (NLI) terms, including selfchannel, cross-channel and multi-channel interferences. Split step Fourier simulations show that the proposed model has the ability to predict the NLI with high levels of accuracy for both low and high fiber dispersion regimes. The simulation results further show that previous models, including the EGN model, inaccurately predict the NLI in certain scenarios.

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