Sercan Ömer Arik

MIMO Signal Processing for Mode-Division Multiplexing: An overview of channel models and signal processing architectures

We present the fundamentals of multiple-input, multiple-output (MIMO) signal processing for mode-division multiplexing (MDM) in multimode fiber (MMF). As an introduction, we review current long-haul optical transmission systems and how continued traffic growth motivates study of new methods to increase transmission capacity per fiber. We describe the key characteristics of MIMO channels in MMF, contrasting these with wireless MIMO channels. We review MMF channel models, the statistics derived from them, and their implications for MDM system performance and complexity. We show that optimizing performance and complexity requires management of channel parameters-particularly group delay (GD) spread and mode-dependent loss and gain-by design of transmission fibers and optical amplifiers, and by control of mode coupling along the link. We describe a family of fibers optimized for low GD spread, which decreases with an increasing number of modes. We compare the performance and complexity of candidate MIMO signal processing architectures in a representative long-haul system design, and show that programmable frequency-domain equalization (FDE) of chromatic dispersion (CD) and adaptive FDE of modal dispersion (MD) is an attractive combination. We review two major algorithms for adaptive FDE of MD-least mean squares (LMS) and recursive least squares (RLS)-and analyze their complexity, throughput efficiency, and convergence time. We demonstrate that, with careful physical link design and judicious choice of signal processing architectures, it is possible to overcome MIMO signal processing challenges in MDM systems.

Effect of Mode Coupling on Signal Processing Complexity in Mode-Division Multiplexing

Mode-division multiplexing systems employ multi-input multi-output (MIMO) equalization to compensate for chromatic dispersion (CD), modal dispersion (MD) and modal crosstalk. The computational complexity of MIMO equalization depends on the number of modes and on the group delay (GD) spread arising from CD and MD. Assuming the strong-coupling regime, in which the total system length far exceeds the correlation length of modal fields, we quantify the GD spread arising from MD, showing that it can be reduced significantly by mode coupling. We evaluate the computational complexity of various MIMO single-carrier equalizers, considering separate or combined equalization of CD and MD, in the time or frequency domain. We present numerical examples for the optimally designed graded-index depressed-cladding fibers supporting D=6, 12, 20 or 30 modes in two polarizations. Assuming a 2000-km system length, a 1-km correlation length, and a combined CD+MD frequency-domain equalizer, the complexity (in complex multiplications per two-dimensional symbol) is a factor 1.4, 1.7, 2.2, 2.8 times higher for D=6, 12, 20, 30 than for polarization-multiplexed systems in standard single-mode fiber (D=2).

Adaptive Frequency-Domain Equalization in Mode-Division Multiplexing Systems

Long-haul mode-division multiplexing (MDM) employs adaptive multi-input multi-output (MIMO) equalization to compensate for modal crosstalk and modal dispersion. MDM systems must typically use MIMO frequency-domain equalization (FDE) to minimize computational complexity, in contrast to polarization-division-multiplexed systems in single-mode fiber, where time-domain equalization (TDE) has low complexity and is often employed to compensate for polarization effects. We study two adaptive algorithms for MIMO FDE: least mean squares (LMS) and recursive least squares (RLS). We analyze tradeoffs between computational complexity, cyclic prefix efficiency, adaptation time and output symbol-error ratio (SER), and the impact of channel group delay spread and fast Fourier transform (FFT) block length on these. Using FDE, computational complexity increases sublinearly with the number of modes, in contrast to TDE. Adaptation to an initially unknown fiber can be achieved in ~3-5 μs using RLS or ~15-25 μs using LMS in fibers supporting 6-30 modes. As compared to LMS, RLS achieves faster adaptation, higher cyclic prefix efficiency, lower SER, and greater tolerance to mode-dependent loss, but at the cost of higher complexity per FFT block. To ensure low computational complexity and fast adaptation in an MDM system, a low overall group delay spread is required. This is achieved here by a family of graded-index graded depressed-cladding fibers in which the uncoupled group delay spread decreases with an increasing number of modes, in concert with strong mode coupling.

Coupled-Core Multi-Core Fibers for Spatial Multiplexing

Coupled-core multi-core fibers (MCFs) offer characteristics that are beneficial for long-haul spatially multiplexed transmission. Using full-vector solution of the wave equation, we study the number of propagating modes and several modal characteristics, including intermodal beat lengths, group delay spread, mode-dependent chromatic dispersion, and intramodal and intermodal effective areas. We identify a range of design parameters that simultaneously optimizes these characteristics. Our results demonstrate the limited accuracy of perturbation-based analyses in characterizing MCFs with closely spaced cores.

Optical network scaling: roles of spectral and spatial aggregation.

As the bit rates of routed data streams exceed the throughput of single wavelength-division multiplexing channels, spectral and spatial traffic aggregation become essential for optical network scaling. These aggregation techniques reduce network routing complexity by increasing spectral efficiency to decrease the number of fibers, and by increasing switching granularity to decrease the number of switching components. Spectral aggregation yields a modest decrease in the number of fibers but a substantial decrease in the number of switching components. Spatial aggregation yields a substantial decrease in both the number of fibers and the number of switching components. To quantify routing complexity reduction, we analyze the number of multi-cast and wavelength-selective switches required in a colorless, directionless and contentionless reconfigurable optical add-drop multiplexer architecture. Traffic aggregation has two potential drawbacks: reduced routing power and increased switching component size.

Group Delay Management and Multiinput Multioutput Signal Processing in Mode-Division Multiplexing Systems

Multi-input multi-output (MIMO) digital signal processing (DSP) for mode-division multiplexing (MDM) may have high complexity, owing to a plurality of modes and a potentially long group delay (GD) spread in multimode fiber (MMF). This paper addresses the management of GD in MMF and its implications for the complexity and performance of MIMO DSP. First, we review the generalized Jones and Stokes representations for modeling propagation in MMF, and describe key GD properties derived using the two representations. Then, we describe three approaches for GD management: 1) optimized fiber design, 2) mode coupling, and 3) GD compensation. For approach 1), we explain design principles for minimizing the GD spread. We review experimental results to date, showing that fabrication nonidealities significantly increase the GD spread, and this approach alone may not achieve sufficiently low GD spread. For approach 2), we describe mechanisms for inducing intragroup and intergroup coupling. We describe mode scrambler designs based on photonic lanterns or long-period fiber gratings, both of which can ensure strong intergroup coupling. For approach 3), we review GD-compensated system design principles and show that GD compensation is only partially effective in the presence of random intragroup or intergroup coupling. Finally, we provide an overview of adaptive MIMO frequency-domain equalization algorithms. Considering tradeoffs between complexity, performance, and adaptation time, we show that the GD spread is a key factor determining the feasibility of MIMO DSP, and its feasibility requires judicious GD management.

Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform

Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. We show that by judiciously choosing sample points on these curved reference surfaces, it is possible to represent the diffracted signals in a nonredundant manner. The change in sample spacing with distance reflects the structure of Fresnel diffraction. This sampling grid also provides a simple and robust basis for accurate and efficient computation, which naturally handles the challenges of sampling chirplike kernels.

Delay Spread Reduction in Mode-Division Multiplexing: Mode Coupling Versus Delay Compensation

Reduction of the group delay (GD) spread is crucial for minimizing signal processing complexity in mode-division multiplexing. Strong mode coupling and GD compensation (concatenating different fibers with opposing GD ordering) are two approaches for reducing the end-to-end GD spread. In this paper, we study the GD behavior in systems where mode coupling and GD compensation are both present. Using a propagation model in generalized Stokes space, we describe the evolution of the GD variance by coupled differential equations. By integration of these equations, we evaluate the GD variance in GD compensated systems with different mode coupling lengths and GD compensation lengths. When the mode coupling length is much longer than the GD compensation length, a low GD variance can be obtained as a result of GD compensation. By contrast, when the mode coupling length is much shorter than the GD compensation length, GD compensation becomes ineffective, but a low GD variance can be obtained as a result of strong mode coupling. The largest GD variance is obtained when the mode coupling length is comparable to the GD compensation length.

Diversity-multiplexing tradeoff in mode-division multiplexing

The capacity of mode-division multiplexing (MDM) systems is limited, for a given outage probability, by mode-dependent loss (MDL) and gain. Modal degrees of freedom may be exploited to increase transmission rate (multiplexing gain) or lower outage probability (diversity gain), but there is a fundamental tradeoff between the achievable multiplexing and diversity gains. In this Letter, we present the diversity-multiplexing tradeoff in MDM systems for the first time, studying the impact of signal-to-noise ratio, MDL, and frequency diversity order on the tradeoff in the strong-mode-coupling regime.

Fundamental structure of Fresnel diffraction: longitudinal uniformity with respect to fractional Fourier order

Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. Transverse samples can be taken on these surfaces with separation that increases with propagation distance. Here, we are concerned with the separation of the spherical reference surfaces along the longitudinal direction. We show that these surfaces should be equally spaced with respect to the fractional Fourier transform order, rather than being equally spaced with respect to the distance of propagation along the optical axis. The spacing should be of the order of the reciprocal of the space-bandwidth product of the signals. The space-dependent longitudinal and transverse spacings define a grid that reflects the structure of Fresnel diffraction.