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- 摘要
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- 实验方案
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Transmission of Higher Order Solitons created by Optical Multiplexing
摘要： The nonlinear Fourier transform (NFT) is a promising tool to linearize the inherently nonlinear optical fiber channel. The NFT transforms a time domain signal into the continuous and the discrete spectrum. The discrete spectrum is composed of an arbitrary number of complex valued discrete eigenvalues and their associated amplitudes. These discrete eigenvalues relate to solitons, which maintain their shape or return to it in an oscillating manner, while passing through the optical channel. Higher order solitons consisting of multiple eigenvalues are complex pulses, which are created and demodulated by sophisticated digital signal processing (DSP) leading to demanding hardware requirements. This paper shows a way to work with higher order solitons in a WDM like fashion by using optical-electrical signal processing and presents boundaries of this method. Optical-electrical signal processing decreases the required electrical and electro-optical hardware specifications substantially and enables to use a simplified DSP. The proposed creation method is subsequently employed to transmit higher order solitons consisting of five QPSK modulated eigenvalues. Furthermore, the optical-electrical processing is benchmarked against the Darboux transformation, which creates higher order solitons purely numerically. The results show that for a 5th order soliton transmission the proposed method can significantly reduce the hardware requirements and DSP complexity.
关键词： solitons,nonlinear Fourier transform,optical signal processing,nonlinear optics,optical fiber communication,Darboux transform
更新于2025-09-23 15:21:01
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Analysis of laser radiation using the Nonlinear Fourier transform
摘要： Modern high-power lasers exhibit a rich diversity of nonlinear dynamics, often featuring nontrivial co-existence of linear dispersive waves and coherent structures. While the classical Fourier method adequately describes extended dispersive waves, the analysis of time-localised and/or non-stationary signals call for more nuanced approaches. Yet, mathematical methods that can be used for simultaneous characterisation of localized and extended fields are not yet well developed. Here, we demonstrate how the Nonlinear Fourier transform (NFT) based on the Zakharov-Shabat spectral problem can be applied as a signal processing tool for representation and analysis of coherent structures embedded into dispersive radiation. We use full-field, real-time experimental measurements of mode-locked pulses to compute the nonlinear pulse spectra. For the classification of lasing regimes, we present the concept of eigenvalue probability distributions. We present two field normalisation approaches, and show the NFT can yield an effective model of the laser radiation under appropriate signal normalisation conditions.
关键词： eigenvalue probability distributions,signal normalisation,Zakharov-Shabat spectral problem,mode-locked pulses,Nonlinear Fourier transform
更新于2025-09-12 10:27:22
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[IEEE 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Munich, Germany (2019.6.23-2019.6.27)] 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems
摘要： The conventional Fourier transform is widely used mathematical methods in science and technology. It allows representing the signal/field under study as a set of spectral harmonics, that it many situations simplify understanding of such signal/field. In some linear equations, where spectral harmonics evolve independently of each other, the Fourier transform provides a straightforward description of otherwise complex dynamics. Something similar is available for certain classes of nonlinear equations that are integrable using the inverse scattering transform [1, 2], also known as the nonlinear Fourier transform (NFT). Here we discuss potential of its application in dissipative, non-integrable systems to characterize coherent structures. We present a new approach for describing the evolution of a nonlinear system considering the cubic Ginzburg-Landau Equation (CGLE) as a particularly important example in the context of laser system modeling.
关键词： Coherent Structures,Ginzburg-Landau Equation,Dissipative Systems,Nonlinear Fourier Transform
更新于2025-09-11 14:15:04
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[IEEE 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Munich, Germany (2019.6.23-2019.6.27)] 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Artificial Neural Network-Based Equaliser in the Nonlinear Fourier Domain for Fibre-Optic Communication Applications
摘要： Nonlinear Fourier transform (NFT) has shown its potential to overcome some challenges of nonlinear signal distortions in fibre-optic communications systems. However, there is yet so much unknown about fundamental properties and traits of a communication based on NFT. One of the most important aspects of an optical communication system is its robustness against the inevitable amplified spontaneous emission (ASE) noise coming from optical amplifiers. The ASE noise not only contaminates the signal as it does for all systems, but it also has a special detrimental effect on NFT-based systems; destroying the very basic concept of integrability of the nonlinear Schrodinger equation (NLSE). Moreover, this noise, undergoing a nonlinear transformation, i.e. “direct NFT” is dependent on the signal power. There are initial studies to model the noise in the nonlinear Fourier domain and some approximations for particular cases of modulation such as discrete spectrum-only or continuous spectrum-only modulations are available. The limited extent of the mentioned studies makes it difficult to use them in order to design an optimum receiver - the primary engineering purpose of analysing the noise properties. On the other hand, machine learning-based techniques to study the noise characteristics in fibre-optic communication have been tried and promising results are already obtained. Machine learning (ML) can also be used to find the impact of noise and alleviate the extent of its perturbation. In particular, in an NFT-based system, where data is mapped on the discrete spectrum (DS) of a signal, at the receiver, the calculated DS contains some spurious elements which are the result of the noise. These points are usually filtered out considered to be void of useful information about the transmitted signal. However, the correlation between these points and the main elements of the DS points that there is mutual information which can be used in order to improve the detection performance at the receiver. In this work, we use a simple Neural Network (NN) to back-propagate the received DS of a signal in a periodic NFT-based communication system. As shown in Fig1. a, drawn from a 64-QAM constellation, each symbol is mapped on the DS of a periodic signal as described in [5,6]. Performing the exact inverse transformation, a signal with the given DS is constructed and sent to a noisy link with 11 spans of 80 km length standard SMF. At the receiver, the DS is calculated and even the out-of-band components are passed to the NN-based equaliser. All data from the calculated DS is used as the input feature vector to our 2-layer (10 neurons each) NN. Raman amplification is used and assumed to provide perfect power loss compensation along the span.
关键词： equaliser,amplified spontaneous emission noise,Neural Network,fibre-optic communications,Nonlinear Fourier transform
更新于2025-09-11 14:15:04
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[IEEE 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Munich, Germany (2019.6.23-2019.6.27)] 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Statistical Properties of Phase and Eigenvalues of Nonlinear Fourier Transform of Second Order Solitons
摘要： Due to the inherently nonlinear nature of optical fibres, the increased demand for transmission capacity means that fibre optic communication systems will reach a limit, known as the Linear Capacity Limit. A radically new solution has received significant attention in the past few years, which is based on Nonlinear Fourier Transform (NFT). Under NFT a signal q(t) in time domain transforms into a continuous, qc(λ), and a discrete, qd(λk), complex spectral part, with continuous and discrete eigenvalues λ, and λk, respectively. Considering only multisolitons (a class of optical signals that have discrete NFT eigenvalues), it is well-known that 1) complex eigenvalues λk are invariant, and 2) the spectral part propagates as: (cid:3031)((cid:1878)) = |(cid:1869)(cid:3038) (cid:1869)(cid:3038) (cid:3031)(0)|(cid:1857)(cid:3037)(cid:2957)((cid:3053)); Φ((cid:1852)) = ∠(cid:1869)(cid:3038) (cid:3031)(0) ? 4(cid:2019)(cid:3038) (cid:2870) (cid:1878), where |(cid:1869)(cid:3038) (cid:3031)(0)| and ∠(cid:1869)(cid:3038) (cid:3031)(0) are the initial spectral amplitude and phase associated with the eigenvalue λk, respectively. In the discrete NFT domain, QPSK modulation has been proposed to encode data on the phase of the NFT spectral components qd(λk). For efficient encoding/decoding of the proposed modulation schemes, it is crucial to have knowledge of NFT noise perturbation on spectral phases and eigenvalues. There are limited reports on correlations among eigenvalues in multi-soliton systems and correlation between eigenvalues and their corresponding phases. However, there is no report with detailed analysis of correlation properties among NFT parameters of multi-soliton signals as a function of initial signal parameters. Here, we investigate correlations of the discrete spectra components in a two-eigenvalue system related to their relative NFT phase differences and distinguish a correlation pattern, which reaches extreme values at a NFT phase difference of π. We show this pattern to be unrelated to the choice of eigenvalues nor their spectral amplitude or constant phase offsets. We show that not only the eigenvalues but also the phases of signals in NFT domain are correlated, approaching a maxima at a NFT phase differences of π. We consider second-order solitons with different eigenvalue pairs, where for each of them, 129 symbols have been defined in the NFT domain with varying initial NFT phase differences Φ1,2=∠qd 2 between 0 and 2π, representing Φ1,2 on discrete positions of propagation over z. For every symbol, 1500 copies have been produced and exposed to white Gaussian noise. Using a combination of the fast inverse NFT algorithm and bidirectional NFT algorithm, each signal is translated into time domain, introduced to Gaussian noise, and translated back into NFT domain. The resulting scattering data has been used to estimate the eigenvalues λ1,2 and corresponding spectral components qd 1,2. The ellipticity (defined as the relative difference of principal axes of the scattering cloud) as a function of phase difference for a range of eigenvalue pairs, Fig.1 a,b, shows almost an ellipse at a phase difference of π and a circle at π/2. While both (Im((cid:79)1),Im((cid:79)2)) and (∠qd 2) are highly correlated at the phase difference of π, |Correlation coefficient|≈1, they are negative of each other, where its reason is under investigation. These correlation properties help to gain deeper physical insight of noise properties of signals in NFT domain, which is crucial for developing efficient NFT communication schemes.
关键词： phase,eigenvalues,Nonlinear Fourier Transform,correlation,solitons
更新于2025-09-11 14:15:04