# Novel Equalization Techniques for Space Division Multiplexing Based on Stokes Space Update Rule
## 2 Space Division Multiplexing  
SDM has acquired great popularity enabling the transmission of previously unthinkable baudrates. Many experiments have validated the high throughput achievable by SDM. Some remarkable cases are the 2.15 Pb/s transmission over 31 km employing a 22 multi-core single mode fiber [9] or the 57.6 Tb/s over 119 km employing a few-mode fiber (FMF) [10]. Further reading and a thorough review of different SDM transmission records can be found at [11]. Since SDM applies to any communication system transmitting different spatial-optical signals, it encompasses a wide range of categories. We review the different SDM technologies [3], with a clear focus on the complexity of the DSP needed. Depending on the transmission medium, two types of equalizers are considered. A single-mode 2 \(\times\) 2 equalizer, capable of compensating Chromatic Dispersion (CD), Polarization Modal Dispersion (PMD), and State-of-Polarization (SoP) rotation, which is the conventional coherent receiver for SMF and valid for SDM systems exhibiting low or negligible crosstalk, and a multi-mode \(2M\times 2M\) equalizer, where \(M\) is the number of modes that are mixing, capable of compensating all the impairments of the single-mode equalizer, and in addition differential modal group delay (DMGD) and mode mixing between the different spatial modes. The simpler approach to implement SDM is packaging multiple fiber together in fiber-bundles. Since the fibers involved maintain their properties, no crosstalk is expected between the different fibers, requiring parallel single-mode equalizers. A similar approach consists of placing different cores inside the same fiber resulting in multi-core fibers (MCF). As the core spacing may be in the range of the decay of the single cores' evanescent waves, crosstalk between the different cores may arise [12]. Fiber design of MCF (core number, core spacing, etc.) is critical for minimizing the crosstalk between modes and achieving optimum propagation settings for each scenario [13]. Depending on the fiber parameters and the distance of the link, the accumulated crosstalk may allow signal recovery by means of parallel single-mode equalization for each transmitted mode. Alternatively, if the crosstalk between the modes is high enough, multi-mode MIMO equalization \(2M\times 2M\) may be needed. Another approach is to increase the numerical aperture of the fiber, allowing the propagation of higher order modes inside the same fiber. Depending on the number of supported modes, the respective fibers are multi-mode fiber (MMF) or few-mode fiber (FMF). The transmission over MMF or FMF generally results in mixing between different modes and needs complex DSP for its compensation [14] with multi-mode equalizers. Mode multiplexing in MMF and FMF is commonly focused on LP modes [3]. Alternatively, another set of orthogonal modes may be excited, like orbital angular momentum (OAM) modes, that exhibit less degeneracy of propagation constant and thus less severe (resonant) crosstalk between the modes. The generation of OAM modes can be done by means of ring resonators or its conversion by free space optics employing Spatial Light Modulators (SLM) [15]. OAM modes also require the design of specific optical fibers that are capable of removing aforementioned degeneracy and thus of suppressing mode coupling such as the inverse-parabolic graded-index fiber (IPGIF) [16]. Even though these fibers can prevent mode coupling in theory, the transmission of OAM modes with tolerable crosstalk is only possible for short distances (in the order of a few km [17]) due to the imperfections of the fiber, fiber coupling, and bending. Another type of fiber merging previous approaches is the hybrid fiber, which combines few-mode fiber and multicore fiber (FMF-MCF), resulting in higher density of modes and enabling the transmission of multiple modes per core [18]. A variant of MCF is the coupled-core fibers involving a higher granularity of cores in the fiber making dense multi-mode equalization indispensable due to strong crosstalk.