# Novel Equalization Techniques for Space Division Multiplexing Based on Stokes Space Update Rule
## 1 Introduction  
The achievable data transmission capacity over optical fiber has been growing continuously at exponential rate during the last 30 years. One of the main enablers for this capacity increase is the development of coherent detection, which efficiently maps the full optical field into the electrical domain, presents better sensitivity than direct detection, and allows to overcome signal impairments by means of digital signal processing (DSP) [1]. DSP also relaxes the receiver specifications in favor of equalization, allowing the use of polarization division multiplexing (PDM) without a tight control of the state of polarization of light [2].  
On the other hand, the improvement of physical components, such as optical fibers with lower signal impairments and attenuation, allows longer transmission distances while lowering the DSP requirements. In addition, the development of advanced Erbium-Doped Fiber Amplifiers (EDFAs) providing lower noise figures (NF) and wider amplification ranges, enables the allocation of more optical channels on larger optical spectrums by means of dense wavelength-division multiplexing (DWDM), efficiently increases the transmitted throughput on the optical fiber.  
By the exploitation of high spectral efficient coding [3], we are finally approaching the fundamental capacity limit of the standard single mode fiber (SSMF) [4; 5], where each research effort to increase capacity results in diminishing marginal returns. This scenario, commonly termed as the capacity crunch of optical communications, drives industry and academia to look for alternatives to overcome this limitation.  
From a product point of view, we live in the information era where enormous amount of data is generated, transported, and consumed every day. The optical transport network accounts for most of the responsibility in providing the requested data to the end users. Due to the proliferation of bandwidth demanding services such as video streaming, cloud services, or social networks, optical networks have been experimenting a continuous growth of their traffic. This growth is forecasted to continuously increase due to the mainstream of very high definition screens, virtual reality (VR), advance cloud services, or internet-of-things (IoT), with even higher requirements in terms of bandwidth and latency. Fueled by this demand, operators and telecommunications companies are compelled to increase the provided bandwidth, facing important trade-offs concerning: cost-per-bit, form factor, or power consumption of the optical transponders. If the current capacity demand continues to increase, the commercial telecommunication equipment will finally reach the fundamental physical limitation currently experimented in the telecommunication research.  
The research community has shown an increasing interest in Space Division Multiplexing (SDM) as the feasible way to increase capacity. SDM adds the spatial dimension to overcome the fundamental limitation by considering multiple parallel spatial propagation paths. Different SDM variants are currently being investigated. In this paper, we provide a discussion of the different SDM approaches and the main DSP techniques needed for each case. We focus on different equalization methods including novel equalization techniques based on Stokes Space recently introduced in [6; 7; 8].  
The presented algorithms are thoroughly studied and evaluated by simulations. The Stokes algorithm implemented in this paper is insensitive to frequency offsets, which eliminates the need of its pre-compensation, enabling to simplify the future SDM receivers at the expense of increasing the convergence speed.