Abstract
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.
Highlights
With the resurgence of polarisation-diverse, optical coherent detection, transmission of information over an optical fibre is typically performed exploiting four degrees of freedom of the optical field: two quadrature components over two orthogonal states of polarisation
Power depends only on different statistical high-order moments of each polarisation component, for a general dual-polarisation 4D (DP-4D) constellation, several other cross-polarisation correlations need to be taken into account
Despite the popularity of polarisation-multiplexed 2D (PM-2D) formats, a substantial amount of research work in the literature has been devoted to more general 4D formats, i.e., 4D constellations which are not necessarily generated as Cartesian products of a component 2D constellation [1,2]
Summary
With the resurgence of polarisation-diverse, optical coherent detection, transmission of information over an optical fibre is typically performed exploiting four degrees of freedom of the optical field: two quadrature components over two orthogonal states of polarisation. Accurately predicting the amount of nonlinear interference generated by transmission of a given constellation in an optical fibre is key to optimising its shape in N dimensions. [10,11] introduced for the first time the possibility of predicting the dependency of nonlinear interference power as a function of the modulation format features, i.e., geometrical shape and statistical properties. One underlying key point of all previous models is the transmission of PM-2D modulation formats, where data on the two polarisation channels are assumed to be independent and identically distributed. Under this constraint, one can predict the NLI power using the statistical properties of the 2D component modulation format. A reliable optimisation of both geometry and symbol probability of occurrence of such 4D formats is enabled for the optical fibre channel
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