  The first-order (FO) perturbation theory-based nonlinearity compensation
(PB-NLC) technique has been widely investigated to combat the detrimental
effects of the intra-channel Kerr nonlinearity in polarization-multiplexed
(Pol-Mux) optical fiber communication systems. However, the NLC performance of
the FO-PB-NLC technique is significantly limited in highly nonlinear regimes of
the Pol-Mux long-haul optical transmission systems. In this paper, we extend
the FO theory to second-order (SO) to improve the NLC performance. This
technique is referred to as the SO-PB-NLC. A detailed theoretical analysis is
performed to derive the SO perturbative field for a Pol-Mux optical
transmission system. Following that, we investigate a few simplifying
assumptions to reduce the implementation complexity of the SO-PB-NLC technique.
The numerical simulations for a single-channel system show that the SO-PB-NLC
technique provides an improved bit-error-rate performance and increases the
transmission reach, in comparison with the FO-PB-NLC technique. The complexity
analysis demonstrates that the proposed SO-PB-NLC technique has a reduced
computational complexity when compared to the digital back-propagation with one
step per span.
