# Two-Dimensional Constellation Shaping in Fiber-Optic Communications
## 1 Introduction  
The advances in photonics integrated circuit [1; 2], software-defined networking [3; 4], application-oriented fibers [5; 6], and optical amplifiers [7; 8], have been greatly pushing forward development and commercialization of optical communications. In the modern optical transport systems, especially in terrestrial and transoceanic fiber-optic communications, advanced modulation formats have been overwhelmingly implemented in optical transceivers [9; 10; 11; 12; 13; 14], thanks to the ever-cheaper optical frontend and powerful digital signal processing (DSP) chips with smaller size and lower power consumption. Traditional quadrature amplitude modulation (QAM) formats have been applied extensively to realize high-capacity and long-reach optical communications, and we have witnessed numerous two-dimensional QAM (2D-QAM)-based hero experiments in recent years to explore the highest possible system capacity and longest transmission distance [15; 16; 17]. Due to the loss profile of the standard single mode fiber (SSMF) and gain profile of commercial Erbium-doped fiber amplifiers (EDFAs), C-band window (1530-1565 nm) is mostly used for data loading [18]. However, the capacity bottleneck becomes more visible for the traditional QAM-based C-band optical transmissions [19]. In order to meet the increasing bandwidth requirements, in particular the upcoming 5G infrastructure, more advanced solutions are expected to be introduced to optical infrastructures. We can roughly divide the promising solutions into two categories: coded modulation [20; 21; 22; 23; 24; 25; 26; 27] and extended multiplexing [17; 28; 29; 30; 31]. The idea of coded modulation is increasing information bits per channel use, including higher-order 2D modulation formats, like 1024 QAM [22], multidimensional QAM formats, like 4D optimized constellations [23], and constellation shaping techniques [24; 25; 26], like probabilistic shaping (PS)-64 QAM [25]. Alternatively, extended multiplexing is a more straightforward solution, which mainly contains C+L band wavelength multiplexing [17; 28] and space-division multiplexing [29; 30; 31]. All the solutions come with the trade-off choice between optical complexity and electrical complexity. In order to implement C+L band wavelength multiplexing, C-band EDFAs and L-band EDFAs should be used together to simultaneously amplify the channel loss per span [17]. Raman amplifier may be another option [32], but the telecom industry is not infavor of it because of its high cost. Space-division multiplexing based on few-mode fiber or multi-core fiber is an efficient way to directly boost the aggregate capacity. However, it seems to have been put at the bottom of the to-do list, because (i) it will be quite challenging to replace and rebuild the current fiber links, (ii) space-division-multiplexing-based optical amplifiers are too expensive for commercial applications, and (iii) complex multi-input multi-output (MIMO) channel equalization may be required to compensate channel crosstalk [33]. The telecom industry always chooses the most cost-efficient way to upgrade their communication systems. Therefore, extended multiplexing solution is barely found in the roadmap of optical communications, especially in the field of long-reach optical communications. On the contrary, coded modulation seems to be a more attracting solution. Higher modulation formats are more common now, but they have more stringent requirements on optical signal-to-noise ratio (OSNR), digital-to-analog converter/analog-to-digital converter (DAC/ADC), and DSP recovery. Multidimensional QAM solution can further enlarge the minimum Euclidean distance between constellation points, but it requires powerful DSP technology to recover signals. What is more, such performance optimization can come at a price of less information bits per channel use [34]. In traditional QAM formats, the constellation points are located on a uniform grid.