We study the statistics of optical data transmission in a noisy nonlinear fiber channel with a weak dispersion management and zero average dispersion. Applying analytical expressions for the output probability density functions both for a nonlinear channel and for a linear channel with additive and multiplicative noise we calculate in a closed form a lower bound estimate on the Shannon capacity for an arbitrary signal-to-noise ratio.

We scrutinize the concept of integrable nonlinear communication channels, resurrecting and extending the idea of eigenvalue communications in a novel context of nonsoliton coherent optical communications. Using the integrable nonlinear Schrödinger equation as a channel model, we introduce a new approach-the nonlinear inverse synthesis method-for digital signal processing based on encoding the information directly onto the nonlinear signal spectrum. The latter evolves trivially and linearly along the transmission line, thus, providing an effective eigenvalue division multiplexing with no nonlinear channel cross talk. The general approach is illustrated with a coherent optical orthogonal frequency division multiplexing transmission format. We show how the strategy based upon the inverse scattering transform method can be geared for the creation of new efficient coding and modulation standards for the nonlinear channel.

What is the maximum rate at which information can be transmitted error-free in fibre–optic communication systems? For linear channels, this was established in classic works of Nyquist and Shannon. However, despite the immense practical importance of fibre–optic communications providing for >99% of global data traffic, the channel capacity of optical links remains unknown due to the complexity introduced by fibre nonlinearity. Recently, there has been a flurry of studies examining an expected cap that nonlinearity puts on the information-carrying capacity of fibre–optic systems. Mastering the nonlinear channels requires paradigm shift from current modulation, coding and transmission techniques originally developed for linear communication systems. Here we demonstrate that using the integrability of the master model and the nonlinear Fourier transform, the lower bound on the capacity per symbol can be estimated as 10.7 bits per symbol with 500 GHz bandwidth over 2,000 km.

Fiber-optic communication systems are nowadays facing serious challenges due to fast growing demand on capacity from various new applications and services. It is now well recognised that nonlinear effects limit the spectral efficiency and transmission reach of modern fiber-optic communications. Nonlinearity compensation is therefore widely believed to be of paramount importance for increasing the capacity of future optical networks. Recently, there has been a steadily growing interest in the application of a powerful mathematical tool -the nonlinear Fourier transform (NFT) -in the development of fundamentally novel nonlinearity mitigation tools for fiber-optic channels. It has been recognized that, within this paradigm, the nonlinear crosstalk due to the Kerr effect is effectively absent, and fiber nonlinearity due to Kerr effect can enter as a constructive element rather than a degrading factor. The novelty and the mathematical complexity of the NFT, the versatility of the proposed system designs, and the lack of a unified vision of an optimal NFT-type communication system however constitute significant difficulties for communication researchers. In this paper, we therefore survey the existing approaches in a common framework and review the progress in this area with a focus on practical implementation aspects. First, an overview of existing key algorithms for the efficacious computation of the direct and inverse NFT is given, and the issues of accuracy and numerical complexity are elucidated. We then describe different approaches for the utilization of the NFT in practical transmission schemes. After that we discuss the differences, advantages and challenges of various recently emerged system designs employing the NFT, and the efficiency estimation available up-to-date. With many practical implementation aspects still being open, our minireview is aimed at helping researchers to assess the perspectives, understand the bottle-necks, and envision the development paths in the upcoming of NFT-based transmission technologies.

Using the integrable nonlinear Schrödinger equation (NLSE) as a channel model, we describe the application of nonlinear spectral management for effective mitigation of all nonlinear distortions induced by the fiber Kerr effect. Our approach is a modification and substantial development of the so-called "eigenvalue communication" idea first presented in A. Hasegawa, T. Nyu, J. Lightwave Technol. 11, 395 (1993). The key feature of the nonlinear Fourier transform (inverse scattering transform) method is that for the NLSE, any input signal can be decomposed into the so-called scattering data (nonlinear spectrum), which evolve in a trivial manner, similar to the evolution of Fourier components in linear equations. We consider here a practically important weakly nonlinear transmission regime and propose a general method of the effective encoding/modulation of the nonlinear spectrum: The machinery of our approach is based on the recursive Fourier-type integration of the input profile and, thus, can be considered for electronic or all-optical implementations. We also present a novel concept of nonlinear spectral pre-compensation, or in other terms, an effective nonlinear spectral pre-equalization. The proposed general technique is then illustrated through particular analytical results available for the transmission of a segment of the orthogonal frequency division multiplexing (OFDM) formatted pattern, and through WDM input based on Gaussian pulses. Finally, the robustness of the method against the amplifier spontaneous emission is demonstrated, and the general numerical complexity of the nonlinear spectrum usage is discussed.

Transmission through a complex network of nonlinear one-dimensional leads is discussed by extending the stationary scattering theory on quantum graphs to the nonlinear regime. We show that the existence of cycles inside the graph leads to a large number of sharp resonances that dominate scattering. The latter resonances are then shown to be extremely sensitive to the nonlinearity and display multi-stability and hysteresis. This work provides a framework for the study of light propagation in complex optical networks.The study of quantum graphs has gained its popularity in recent years [1] not only because graphs emulate successfully complex mesoscopic and optical networks, but also because they manage to reproduce universal properties (such as level statistics, transmission fluctuations and others) observed in generic quantum chaotic systems. Here we generalize quantum graph theory to the nonlinear domain. The theory will be applied in particular to show the effect of nonlinearity on transmission through networks of nonlinear fibers. Our model may also be used as a simple yet non-trivial model where the universal properties derived from detailed numerical computations of Bose-Einstein condensates in non-regular traps [2][3][4][5][6][7] could be further investigated.Scattering is studied as a stationary process. The main finding is that the sharp resonances which dominate scattering in networks with complex connectivity lead to a dramatic amplification of the nonlinear effects: while non-resonant scattering hardly deviates from the predictions of the linear theory, tuning the parameters to a nearby resonance (without changing the incoming field intensity) brings the system into the nonlinear regime which is signaled by multi-stability and hysteresis. For this reason we revisit the theory of scattering in the linear regime and demonstrate that sharp resonances with large amplification of the incoming wave inside the system are very frequent for graphs compared to other complex (chaotic) scattering systems. The origin of this effect can be related to the topology of the graph (existence of cycles) and leads to a power-law distribution for the amplification. I. THE NONLINEAR SCHRÖDINGER EQUATION ON GRAPHSConsider a general metric graph which consists of V vertices connected by B internal bonds and N leads to infinity as illustrated in Fig. 1. The bonds and leads will be collectively referred to as edges. have the coordinate x l ∈ [0, ∞) and x l = 0 is at the vertex where the lead is attached. The wave function on the graph is a bounded piecewise continuous and differentiable function on the edges written collectively as(1)where ψ e (x e ) is the wave functions on the edge e. While the model can describe far more general settings we restrict ourselves in this exploratory work to the discussion of stationary scattering. The wave function on edge e satisfies the stationary nonlinear Schrödinger equationHere, g e is real nonlinear coupling parameter which we assume constant on each edge (but it may take different values ...

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