Sphere Shaping with Dispersion-Aware Sequence Selection for Long-Haul Transmission Systems

We introduce a nonlinear-tolerant design approach to Enumerative Sphere Shaping (ESS) for $M$-QAM signaling, utilizing sequence selection techniques based on a sign-independent metric: the Energy Dispersion Index, EDI (a scheme denoted E-ESS) and a novel sign-dependent metric: EDI of a dispersed sequence, D-EDI (a scheme denoted D-E-ESS). These approaches are designed to minimize rate loss and enhance transmission performance in nonlinear optical fiber transmission systems, catering to both short-distance and long-haul scenarios. Our simulation results reveal significant performance gains over conventional ESS, with improvements up to $0.4$~bits/4D-symbol. These improvements were observed over a 205 km single-span standard single mode fiber link in WDM transmission, with five dual-polarization channels, each operating at a net rate of $400$~Gbit/s. Furthermore, we demonstrate that D-E-ESS surpasses conventional ESS by $0.03$~bits/4D-symbol in achievable information rate over a $30$~span link with $80$~km span in a single-wavelength 8 discrete multi-bands (DMB) transmission with an $880$~Gbit/s net rate per channel, achieving similar performance to sequence selection based on a full split-step Fourier method (SSFM) simulation. We also demonstrate that D-EDI exhibits a more accurate negative correlation with the transmission performance compared to EDI. Notably, our proposed D-E-ESS scheme maintains robust performance improvements even when we consider reduced-complexity versions of it, consistently delivering throughput enhancements across various block lengths and selected sequence lengths.

Multi-dimensional Energy Limitation in Sphere Shaping for Nonlinear Interference Noise Mitigation

We propose Four-Dimensional (4D) energy limit enumerative sphere shaping (ESS) of $M$-QAM signaling to minimize rate loss and improve the transmission performance over non-linear WDM optical-fiber systems. Simulation results show that the proposed scheme outperforms the conventional ESS by $0.19$~bit/4D-symbol in achievable information rate over a $205$-km single-span link and a WDM transmission of five polarization-division-multiplexed channels with $400$-Gbit/s net rate per channel. We also study the achieved performance over several shaping block lengths and show that the achieved gains do not scale well over multi-span systems.

Fault-Tolerant Four-Dimensional Constellation for Coherent Optical Transmission Systems

We propose a 4-dimensional 2-ary amplitude ring-switched modulation format with 64 symbols, which is denoted as 4D-2A-RS64 encoded over two polarization tributaries to improve the transmission performance over long-haul optical fibers in the presence of the non-linear Kerr effect. At a spectral efficiency of 6 bits per 4D, simulation results show that this format outperforms the polarization division multiplexed (PDM) 8QAM-star modulation as well as the 4D-2A-8PSK over links without inline dispersion management. We evaluate the performance for a WDM transmission of $11\times90~\mathrm{Gbaud}$ channels over a multi-span SSMF link. For an achievable information rate of $4.8\mathrm{bit/s/Hz}$, the maximum transmission distance is improved by $10.6\%$ (400 km) and $4\%$ (160 km) compared to PDM-8QAM-star and 4D-2A-8PSK respectively. The achieved gains are composed of a linear part and a non-linear part, respectively from the improved Euclidean-distance distribution and the constant power property of the 4D modulation. The geometric shaping of the proposed scheme is easy to implement and is robust to Mach-Zehnder modulator (MZM) imbalances and quantization errors stemming from the finite digital-to-analog converter (DAC) resolution. This robustness is compared to the one of other geometric-shaped non-linearity tolerant 4D schemes such as the 4D-2A-8PSK and the 4D-64PRS that can be both outperformed by our scheme in severe conditions.