Nonlinear Fourier Spectrum Research Articles

Channel model for the dual-polarization b-modulated nonlinear frequency-division multiplexing optical transmission systems.

We consider optical transmission systems based on the nonlinear frequency division multiplexing (NFDM) concept, i.e., the systems employing the nonlinear Fourier transform (NFT) for signal processing and data modulation. Our work specifically addresses the double-polarization (DP) NFDM setup that utilizes the so-called b-modulation, the most efficient NFDM method proposed up-to-date. We extend the previously-developed analytical approach based on the adiabatic perturbation theory for the continuous nonlinear Fourier spectrum (b-coefficient) onto the DP case to obtain the leading order of continuous input-output signal relation, i.e., the asymptotic channel model, for an arbitrary b-modulated DP-NFDM optical communication system. Our main result is in deriving the relatively simple analytical expressions for the power spectral density of the components of effective conditionally Gaussian input-dependent noise emerging inside the nonlinear Fourier domain. We also demonstrate that our analytical expressions are in remarkable agreement with direct numerical results if one extracts the "processing noise" arising due to the imprecision of numerical NFT operations.

Role of Homoclinic Breathers in the Interpretation of Experimental Measurements, With Emphasis on the Peregrine Breather

A class of generalized homoclinic solutions of the nonlinear Schrödinger (NLS) equation in 1+1 dimensions is studied. These are homoclinic breathers that are shown to be derivable from the ratio of Riemann theta functions for the genus-2 solutions of the nonlinear Schrödinger equation. We discuss how these solutions behave in the homoclinic limit for which a fundamental parameter ε goes to zero, ε→0 (such that two points of simple spectrum converge to double points at some particular lambda-plane eigenvalue). The homoclinic solutions cover the entire lambda plane (the Riemann surface of the NLS equation) and are given in terms of simple trigonometric functions. When the spectral eigenvalues converge to the carrier amplitude in the lambda plane we have the Peregrine breather. While the Peregrine solution is often called a soliton, it is in reality a breather, albeit occurring at the “singular point” corresponding to the carrier eigenvalue in the lambda plane and consequently “breathes” only once in its lifetime. The Peregrine breather separates small-amplitude modulations below the carrier from large amplitude modulation above the carrier. This fact means that the Peregrine breather has a “central” role in the lambda plane characterization of the NLS nonlinear spectrum. The Akhmediev breather occurs somewhat below the carrier (and is therefore a small-amplitude modulation) and the Kuznetsov-Ma breather occurs above the carrier (and is therefore a large-amplitude modulation). The general homoclinic solutions can be constructed everywhere in the lambda plane and are shown to be a useful tool to interpret the nonlinear Fourier spectrum of space and time series recorded in the laboratory and ocean environment. Nonlinear filtering is suggested as a way to extract breather trains from experimental time series. The generalized homoclinic breathers can be thought of as “extreme wave packets” or “rogue wave” solutions of water waves for scientific and engineering applications in various fields of physics including physical oceanography and nonlinear optics.

Neural networks for computing and denoising the continuous nonlinear Fourier spectrum in focusing nonlinear Schr\xf6dinger equation

We combine the nonlinear Fourier transform (NFT) signal processing with machine learning methods for solving the direct spectral problem associated with the nonlinear Schrödinger equation. The latter is one of the core nonlinear science models emerging in a range of applications. Our focus is on the unexplored problem of computing the continuous nonlinear Fourier spectrum associated with decaying profiles, using a specially-structured deep neural network which we coined NFT-Net. The Bayesian optimisation is utilised to find the optimal neural network architecture. The benefits of using the NFT-Net as compared to the conventional numerical NFT methods becomes evident when we deal with noise-corrupted signals, where the neural networks-based processing results in effective noise suppression. This advantage becomes more pronounced when the noise level is sufficiently high, and we train the neural network on the noise-corrupted field profiles. The maximum restoration quality corresponds to the case where the signal-to-noise ratio of the training data coincides with that of the validation signals. Finally, we also demonstrate that the NFT b-coefficient important for optical communication applications can be recovered with high accuracy and denoised by the neural network with the same architecture.

Convolutional long short-term memory neural network equalizer for nonlinear Fourier transform-based optical transmission systems

We evaluate improvement in the performance of the optical transmission systems operating with the continuous nonlinear Fourier spectrum by the artificial neural network equalisers installed at the receiver end. We propose here a novel equaliser designs based on bidirectional long short-term memory (BLSTM) gated recurrent neural network and compare their performance with the equaliser based on several fully connected layers. The proposed approach accounts for the correlations between different nonlinear spectral components. The application of BLSTM equaliser leads to a 16x improvement in terms of bit-error rate (BER) compared to the non-equalised case. The proposed equaliser makes it possible to reach the data rate of 170 Gbit/s for one polarisation conventional nonlinear Fourier transform (NFT) based system at 1000 km distance. We show that our new BLSTM equalisers significantly outperform the previously proposed scheme based on a feed-forward fully connected neural network. Moreover, we demonstrate that by adding a 1D convolutional layer for the data pre-processing before BLSTM recurrent layers, we can further enhance the performance of the BLSTM equaliser, reaching 23x BER improvement for the 170 Gbit/s system over 1000 km, staying below the 7% forward error correction hard decision threshold (HD-FEC).

Nonlinear Fourier Spectrum Characterization of Time-Limited Signals

Addressing the optical communication systems employing the nonlinear Fourier transform (NFT) for the data modulation/demodulation, we provide an explicit proof for the properties of the signals emerging in the so-called $b$ -modulation method, the nonlinear signal modulation technique that provides explicit control over the signal extent. We present details of the procedure and related rigorous mathematical proofs addressing the case where the time-domain profile corresponding to the $b$ -modulated data has a limited duration, and when the bound states corresponding to specifically chosen discrete solitonic eigenvalues and norming constants, are also present. We also prove that the number of solitary modes that we can embed without violating the exact localisation of the time-domain profile, is actually infinite. Our theoretical findings are illustrated with numerical examples, where simple example waveforms are used for the $b$ -coefficient, demonstrating the validity of the developed approach. We also demonstrate the influence of the bound states on the noise tolerance of the b-modulated system.

Dispersion and Nonlinearity Identification for Single-Mode Fibers Using the Nonlinear Fourier Transform

Efficient fiber-optic communication requires precise knowledge of the fiber coefficients, but these often change over time due to factors such as aging or bending. We propose a novel algorithm that identifies the average second-order dispersion and Kerr nonlinearity coefficient of a fiber, without employing any special training signals. Instead, ordinary input and output data recorded during normal operation is used. To the best of our knowledge, this is the first such algorithm. The algorithm is based on the nonlinear Fourier spectrum of the signal, which is known to evolve trivially as the signal propagates through an idealized model of the fiber. The algorithm varies the values of the fiber coefficients until the corresponding nonlinear Fourier spectrum at transmitter and receiver match optimally. We test the algorithm on simulated transmission data over a 1600 km link, and accurately identify the fiber coefficients. The identification algorithm is in particular well suited for providing a fiber model for nonlinear Fourier transform-based communication.

Submicrometer Nanoparticle Systems – Effective Nonlinear Filters for Real-Time Image Processing

We realize real-time image processing due to nonlinear Fourier-spectrum filtering by an emulsion of submicrometer particles of oil in water. We show that the emulsion of submicrometer particles of oil in water can be used as an effective amplitude–phase filter that nonlinearly changes the amplitude and phase of coherent radiation.

Nonlinear Fourier transform of time-limited and one-sided signals

In this article, we study the properties of the nonlinear Fourier spectrum in order to gain better control of the temporal support of the signals synthesized using the inverse nonlinear Fourier transform. In particular, we provide necessary and sufficient conditions satisfied by the nonlinear Fourier spectrum such that the generated signal has a prescribed support. In our exposition, we assume that the support is a simply connected domain that is either a bounded interval or the half-line, which amounts to studying the class of signals which are either time-limited or one-sided, respectively. Further, it is shown that the analyticity properties of the scattering coefficients of the aforementioned classes of signals can be exploited to improve the numerical conditioning of the differential approach of inverse scattering. Here, we also revisit the integral approach of inverse scattering and provide the correct derivation of the so called Töplitz inner-bordering algorithm. Finally, we conduct extensive numerical tests in order to verify the analytical results presented in the article. These tests also provide us an opportunity to compare the performance of the two aforementioned numerical approaches in terms of accuracy and complexity of computations.

Contour integrals for numerical computation of discrete eigenvalues in the Zakharov-Shabat problem.

We propose a novel algorithm for the numerical computation of discrete eigenvalues in the Zakharov-Shabat problem. Our approach is based on contour integrals of the nonlinear Fourier spectrum function in the complex plane of the spectral parameter. The reliability and performance of the new approach are examined in application to a single eigenvalue, multiple eigenvalues, and the degenerate breather's multiple eigenvalue. We also study the impact of additive white Gaussian noise on the stability of numerical eigenvalues computation.

Modulation Over Nonlinear Fourier Spectrum: Continuous and Discrete Spectrum

Nonlinear frequency division multiplexed (NFDM) systems are considered when data are modulated in both parts of the nonlinear Fourier spectrum: Continuous spectrum and discrete spectrum. An efficient algorithm is introduced to generate a time-domain signal from a given nonlinear spectrum. The transmission of such NFDM symbols is experimentally demonstrated over 1460 km standard single-mode fiber with EDFA-only amplification. In each NFDM symbol, the continuous spectrum is modulated by 64 $\times$ 0.5 Gbaud orthogonal frequency-division multiplexing (OFDM) symbols with 32-QAM format whereas the discrete spectrum contains four eigenvalues with the same imaginary part, each one is modulated by 8-PSK format, resulting a line rate of 55.3 Gb/s. The cross-talk between different nonlinear modes is quantified in terms of cross-correlation and the performance loss is computed in terms of mutual information when each nonlinear modes is detected individually.

Some results on discrete eigenvalues for the Stochastic Nonlinear Schrödinger Equation in fiber optics

Abstract We study a stochastic Nonlinear Schrödinger Equation (NLSE), with additive white Gaussian noise, by means of the Nonlinear Fourier Transform (NFT). In particular, we focus on the propagation of discrete eigenvalues along a focusing fiber. Since the stochastic NLSE is not exactly integrable by means of the NFT, then we use a perturbation approach, where we assume that the signal-to-noise ratio is high. The zeroth-order perturbation leads to the deterministic NLSE while the first-order perturbation allows to describe the statistics of the discrete eigenvalues. This is important to understand the properties of the channel for recently devised optical transmission techniques, where the information is encoded in the nonlinear Fourier spectrum.

Exact solution of the Zakharov–Shabat scattering problem for doubly-truncated multisoliton potentials

Recent studies have revealed that multisoliton solutions of the nonlinear Schrödinger equation, as carriers of information, offer a promising solution to the problem of nonlinear signal distortions in fiber optic channels. In any nonlinear Fourier transform based transmission methodology seeking to modulate the discrete spectrum of the multisolitons, choice of an appropriate windowing function is an important design issue on account of the unbounded support of such signals. Here, we consider the rectangle function as the windowing function for the multisolitonic signal and provide a recipe for computing the exact solution of the associated Zakharov–Shabat (ZS) scattering problem for the windowed/doubly-truncated multisoliton potential. The idea consists in expressing the Jost solution of the doubly-truncated multisoliton potential in terms of the Jost solution of the original potential. The proposed method allows us to avoid prohibitive numerical computations normally required in order to accurately quantify the effect of time-domain windowing on the nonlinear Fourier spectrum of the multisolitonic signals. Further, the method devised in this work also applies to general type of signals admissible as ZS scattering potential, and, may prove to be a useful tool in the theoretical analysis of such systems.

Nonlinear signal multiplexing for communication beyond the Kerr nonlinearity limit

Current optical fibre transmission systems rely on modulation, coding and multiplexing techniques that were originally developed for linear communication channels. However, linear transmission techniques are not fully compatible with a transmission medium with a nonlinear response, which is the case for an optical fibre. As a consequence, the Kerr nonlinearity in fibre imposes a limit on the performance and the achievable transmission rate of the conventional optical fibre communication systems. Here we show that a transmission performance beyond the conventional Kerr nonlinearity limit can be achieved by encoding all the available degrees of freedom and nonlinearly multiplexing signals in the so-called nonlinear Fourier spectrum, which evolves linearly along the fibre link. This result strongly motivates a fundamental paradigm shift in modulation, coding and signal-processing techniques for optical fibre transmission technology. The Kerr nonlinearity limit for optical fibre communications is surpassed by using nonlinear multiplexing.

64 × 0.5 Gbaud Nonlinear Frequency Division Multiplexed Transmissions With High Order Modulation Formats

Nonlinear frequency division multiplexing (NFDM) with the modulation of the nonlinear Fourier spectrum (both discrete and continuous parts) has been recently considered as a promising transmission method to combat fiber nonlinearity impairments. In this paper, we experimentally demonstrate the generation, detection, and transmission performance of NFDM systems where the transmitted information is encoded directly onto the continuous part of the nonlinear spectrum using 64 overlapped 0.5 Gbaud channels with high order QAM formats (e.g., 16 QAM, 32 QAM, and 64 QAM). In comparison with the conventional orthogonal frequency division multiplexing transmission, which is the linear counterpart of NFDM transmission, clear performance advantage of NFDM systems, up to 1.3 dB, has been observed. This result indicates for the first time the performance benefit of NFDM transmission over its linear counterpart, showing the high potential of this emerging technology.

Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform

The periodic, inverse scattering transform (PIST) is a powerful analytical tool in the theory of integrable, nonlinear evolution equations. Osborne pioneered the use of the PIST in the analysis of data form inherently nonlinear physical processes. In particular, Osborne's so-called nonlinear Fourier analysis has been successfully used in the study of waves whose dynamics are (to a good approximation) governed by the Korteweg–de Vries equation. In this paper, the mathematical details and a new application of the PIST are discussed. The numerical aspects of and difficulties in obtaining the nonlinear Fourier (i.e., PIST) spectrum of a physical data set are also addressed. In particular, an improved bracketing of the “spectral eigenvalues” (i.e., the ±1 crossings of the Floquet discriminant) and a new root-finding algorithm for computing the latter are proposed. Finally, it is shown how the PIST can be used to gain insightful information about the phenomenon of soliton-induced acoustic resonances, by computing the nonlinear Fourier spectrum of a data set from a simulation of internal solitary wave generation and propagation in the Yellow Sea.

Theory of Pump Depletion and Spike Formation in Stimulated Raman Scattering.

By using the inverse spectral transform, the SRS equations are solved and the explicit output data is given for arbitrary laser pump and Stokes seed profiles injected on a vacuum of optical phonons. For long duration laser pulses, this solution is modified such as to take into account the damping rate of the optical phonon wave. This model is used to interprete the experiments of Druhl, Wenzel and Carlsten (Phys. Rev. Lett., (1983) vol. 51, p. 1171), in particular the creation of a spike of (anomalous) pump radiation. The related nonlinear Fourier spectrum does not contain discrete eigenvalue, hence this Raman spike is not a soliton.

Computation of the direct scattering transform for the nonlinear Schroedinger equation

The cubic nonlinear Schroedinger equation (NLS) describes the space-time evolution of narrow-banded wave trains in one space and one time (1 + 1) dimensions. The richness of nonlinear wave motions described by NLS is exemplified by the fully nonlinear envelope soliton and “breather” solutions, which are fully understood only in terms of the general solution of the equation as described by the inverse scattering transform (1ST); the method may be viewed as a nonlinear generalization of the linear Fourier transform. Herein we develop a numerical algorithm for determining the scattering transform spectrum of a nonlinear wave train described by the NLS equation. The analysis of space or time series data obtained from computer simulations of nonlinear, narrow-band wave trains or from experimental measurements is thus a central point of discussion. In particular we develop a numerical algorithm for computing the direct scattering transform (DST) which may be interpreted as the nonlinear Fourier spectrum of the complex envelope function of a wave train; the fact that the nonlinear Fourier modes are constants of the motion for all time provides a physical basis for the analysis of data. While the nonlinear Fourier method is specifically applied to the NLS equation, the approach is easily generalized to include the class of spectral problems due to Ablowitz, Kaup, Newell, and Segur (AKNS). This class includes several other nonlinear wave equations of physical interest in (1 + 1), including the Kortewegde Vries (KdV), modified KdV, sine-Gordon, and sinh-Gordon equations.