Voronoi constellations (VCs) are considered as an effective geometric shaping method due to their high power efficiencies and low complexity. In this paper, the performance of 16-and 32-dimensional VCs with a variety of spectral efficiencies transmitted in the nonlinear fiber channel are investigated. Both single-channel and wavelength-division multiplexing systems are considered for the transmission of the VCs, as well as different signal processing schemes, including chromatic dispersion compensation and digital backpropagation. Multiple performance metrics including the uncoded bit error rate, mutual information (MI), and generalized mutual information (GMI) of VCs are evaluated. Compared with quadrature amplitude modulation (QAM) formats, the VCs provide 1.0-2.4 dB launch power gains, up to 0.50 bits/symbol/dimension-pair MI gains, up to around 30% potential reach increase at the same MI, and up to 0.30 bits/symbol/dimension-pair GMI gains in a limited launch power range. The observed performance gains over QAM are found higher than in the back-to-back case. Moreover, a general GMI estimation method for very large constellations using importance sampling is proposed for the first time.

Multidimensional geometric shaping has been shown to outperform uniform quadrature amplitude modulation (QAM) in optical communication systems but the complexity of symbol decision and bit mapping can often be significant as dimensionality increases. In this paper, a low-complexity geometric shaping method based on multidimensional lattices is investigated both in experiments and simulations. The modulation formats designed based on this method are called Voronoi constellations (VCs) and we study them in 8, 16, and 32 dimensions. We obtain transmission reach improvements of up to 22 and 70% for VCs compared to 4QAM and 16QAM, respectively, in nonlinear long-haul fiber transmission. Moreover, we compare different physical realizations of multidimensional VCs over wavelengths, polarizations, and time slots in both the Gaussian and nonlinear fiber channels. We demonstrate that different physical realizations perform similarly in the fiber-optic back-to-back channel. However, in long-haul transmission systems, spreading the dimensions over time slots can increase the transmission reach up to 4% compared to wavelengths and polarizations. Furthermore, the mutual information and generalized mutual information are estimated and compared to QAM formats at the same spectral efficiencies.

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