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Gui T, Lu C, Lau APT, Wai PKA. High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform. OPTICS EXPRESS 2017; 25:20286-20297. [PMID: 29041711 DOI: 10.1364/oe.25.020286] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2017] [Accepted: 08/08/2017] [Indexed: 06/07/2023]
Abstract
In this paper, we experimentally investigate high-order modulation over a single discrete eigenvalue under the nonlinear Fourier transform (NFT) framework and exploit all degrees of freedom for encoding information. For a fixed eigenvalue, we compare different 4 bit/symbol modulation formats on the spectral amplitude and show that a 2-ring 16-APSK constellation achieves optimal performance. We then study joint spectral phase, spectral magnitude and eigenvalue modulation and found that while modulation on the real part of the eigenvalue induces pulse timing drift and leads to neighboring pulse interactions and nonlinear inter-symbol interference (ISI), it is more bandwidth efficient than modulation on the imaginary part of the eigenvalue in practical settings. We propose a spectral amplitude scaling method to mitigate such nonlinear ISI and demonstrate a record 4 GBaud 16-APSK on the spectral amplitude plus 2-bit eigenvalue modulation (total 6 bit/symbol at 24 Gb/s) transmission over 1000 km.
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Terekhov IS, Vergeles SS, Turitsyn SK. Conditional probability calculations for the nonlinear Schrödinger equation with additive noise. PHYSICAL REVIEW LETTERS 2014; 113:230602. [PMID: 25526108 DOI: 10.1103/physrevlett.113.230602] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2014] [Indexed: 06/04/2023]
Abstract
The method for the computation of the conditional probability density function for the nonlinear Schrödinger equation with additive noise is developed. We present in a constructive form the conditional probability density function in the limit of small noise and analytically derive it in a weakly nonlinear case. The general theory results are illustrated using fiber-optic communications as a particular, albeit practically very important, example.
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Affiliation(s)
- I S Terekhov
- Budker Institute of Nuclear Physics of SB RAS, Novosibirsk 630090, Russia and Novosibirsk State University, Novosibirsk 630090, Russia
| | - S S Vergeles
- Landau Institute for Theoretical Physics, Moscow 119334, Russia
| | - S K Turitsyn
- Novosibirsk State University, Novosibirsk 630090, Russia and Aston Institute of Photonic Technologies, Aston University, Aston Triangle, Birmingham B4 7ET, United Kingdom
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Kuratsuji H. Stochastic theory of an optical vortex in nonlinear media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:013202. [PMID: 23944571 DOI: 10.1103/physreve.88.013202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2013] [Indexed: 06/02/2023]
Abstract
A stochastic theory is given of an optical vortex occurring in nonlinear Kerr media. This is carried out by starting from the nonlinear Schrödinger type equation which accommodates vortex solution. By using the action functional method, the evolution equation of vortex center is derived. Then the Langevin equation is introduced in the presence of random fluctuations, which leads to the Fokker-Planck equation for the distribution function of the vortex center coordinate by using a functional integral. The Fokker-Planck equation is analyzed for a specific form of pinning potential by taking into account an interplay between the strength of the pinning potential and the random parameters, diffusion and dissipation constants. This procedure is performed by several approximate schemes.
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Affiliation(s)
- Hiroshi Kuratsuji
- Faculty of Science and Engineering, Ritsumeikan University-BKC, Noji-Hill, Kusatsu City, 525-8577, Japan.
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4
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Gautier É. Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications. ESAIM-PROBAB STAT 2005. [DOI: 10.1051/ps:2005005] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
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Peleg A, Dohnal T, Chung Y. Effects of dissipative disorder on front formation in pattern forming systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:027203. [PMID: 16196757 DOI: 10.1103/physreve.72.027203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2005] [Revised: 06/21/2005] [Indexed: 05/04/2023]
Abstract
We study the effects of weak disorder in the linear gain coefficient on front formation in pattern forming systems described by the cubic-quintic nonlinear Schrödinger equation. We calculate the statistics of the front amplitude and position. We show that the distribution of the front amplitude has a loglognormal diverging form at the maximum possible amplitude and that the distribution of the front position has a lognormal tail. The theory is in good agreement with our numerical simulations. We show that these results are valid for other types of dissipative disorder and relate the loglognormal divergence of the amplitude distribution to the form of the emerging front tail.
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Affiliation(s)
- Avner Peleg
- Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA
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6
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Debussche A, de Bouard A. Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise. ANN PROBAB 2005. [DOI: 10.1214/009117904000000964] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Mayol C, Toral R, Mirasso CR. Derivation of amplitude equations for nonlinear oscillators subject to arbitrary forcing. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:066141. [PMID: 15244701 DOI: 10.1103/physreve.69.066141] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2003] [Indexed: 05/24/2023]
Abstract
By using a generalization of the multiple scales technique we develop a method to derive amplitude equations for zero-dimensional forced systems. The method allows to consider either additive or multiplicative forcing terms and can be straightforwardly applied to the case that the forcing is white noise. We give examples of the use of this method to the case of the van der Pol-Duffing oscillator. The writing of the amplitude equations in terms of a Lyapunov potential allow us to obtain an analytical expression for the probability distribution function which reproduces reasonably well the numerical simulation results.
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Affiliation(s)
- Catalina Mayol
- Department de Física, Universitat de les Illes Balears, and Instituto Mediterráneo de Estudios Avanzados, CSIC-UIB, Ed. Mateu Orfila, Campus UIB, 07122 Palma de Mallorca, Spain
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8
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Ho KP. Non-Gaussian statistics of soliton timing jitter induced by amplifier noise. OPTICS LETTERS 2003; 28:2165-2167. [PMID: 14649929 DOI: 10.1364/ol.28.002165] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Based on first-order perturbation theory of the soliton, the Gordon-Haus timing jitter induced by amplifier noise is found to be non-Gaussian distributed. Both frequency and timing jitter have larger tail probabilities than Gaussian distribution given by the linearized perturbation theory. The timing jitter has a larger discrepancy from Gaussian distribution than does the frequency jitter.
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Affiliation(s)
- Keang-Po Ho
- Graduate Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan.
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9
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Derevyanko SA, Turitsyn SK, Yakushev DA. Non-Gaussian statistics of an optical soliton in the presence of amplified spontaneous emission. OPTICS LETTERS 2003; 28:2097-2099. [PMID: 14587827 DOI: 10.1364/ol.28.002097] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We apply an approach based on the Fokker-Planck equation to study the statistics of optical soliton parameters in the presence of additive noise. This rigorous method not only allows us to reproduce and justify the classical Gordon-Haus formula but also leads to new exact results.
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Affiliation(s)
- S A Derevyanko
- Photonics Research Group, Aston University, Birmingham B4 7ET, UK.
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Chertkov M, Chung Y, Dyachenko A, Gabitov I, Kolokolov I, Lebedev V. Shedding and interaction of solitons in weakly disordered optical fibers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:036615. [PMID: 12689187 DOI: 10.1103/physreve.67.036615] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2002] [Indexed: 05/24/2023]
Abstract
The propagation of the soliton pattern through optical fiber with weakly disordered dispersion coefficient is considered. Solitons perturbed by this disorder radiate and, as a consequence, decay. The average radiation profile is found. Emergence of a long-range intrachannel interaction between the solitons (mediated by this radiation) is reported. We show that soliton in a multisoliton pattern experiences a random jitter: intersoliton separation is zero mean Gaussian random field. Fluctuations of this separation are estimated by deltay approximately Dz(2)square root mu, where D measures the disorder strength, z is the propagation distance, and mu stands for the transmission rate (number of solitons per unit length of the fiber). Direct numerical simulations are used to validate theoretical predictions for single soliton decay and two-soliton interaction. Relevance of these results to fiber optics communication technology is discussed.
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Affiliation(s)
- M Chertkov
- Theoretical Division, LANL, Los Alamos, New Mexico 87545, USA
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Moore RO, Biondini G, Kath WL. Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems. OPTICS LETTERS 2003; 28:105-107. [PMID: 12656499 DOI: 10.1364/ol.28.000105] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We discuss the application of importance-sampling techniques to the numerical simulation of transmission impairments induced by amplified spontaneous emission noise in soliton-based optical transmission systems. The method allows one to concentrate numerical simulations on the noise realizations that are most likely to result in transmission errors, thus leading to increases in speed of several orders of magnitude over standard Monte Carlo methods. We demonstrate the technique by calculating the probability distribution function of amplitude and timing fluctuations.
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Affiliation(s)
- R O Moore
- Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA.
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Green AG, Littlewood PB, Mitra PP, Wegener LGL. Schrödinger equation with a spatially and temporally random potential: effects of cross-phase modulation in optical communication. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:046627. [PMID: 12443369 DOI: 10.1103/physreve.66.046627] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2002] [Revised: 08/02/2002] [Indexed: 05/24/2023]
Abstract
We model the effects of cross-phase modulation in frequency (or wavelength) division multiplexed optical communications systems, using a Schrödinger equation with a spatially and temporally random potential. Green's functions for the propagation of light in this system are calculated using Feynman path-integral and diagrammatic techniques. This propagation leads to a non-Gaussian joint distribution of the input and output optical fields. We use these results to determine the amplitude and timing jitter of a signal pulse and to estimate the system capacity in analog communication.
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Affiliation(s)
- A G Green
- Theoretical Physics, Oxford University, United Kingdom
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Falkovich G, Stepanov MG. Role of interaction in causing errors in optical soliton transmission. OPTICS LETTERS 2002; 27:13-15. [PMID: 18007700 DOI: 10.1364/ol.27.000013] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
We consider two solitons propagating under a filter-control scheme and describe the timing jitter that is caused by spontaneous-emission noise and enhanced by attraction between solitons. We find the bit-error rate as a function of system parameters (filtering and noise level), timing, initial distance, and the phase difference between solitons.
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Falkovich GE, Stepanov MG, Turitsyn SK. Statistics of interacting optical solitons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:067602. [PMID: 11736321 DOI: 10.1103/physreve.64.067602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2001] [Indexed: 05/23/2023]
Abstract
We examine statistics of two interacting optical solitons and describe timing jitter caused by spontaneous emission noise and enhanced by pulse interaction. Dynamics of phase difference is shown to be of crucial importance in determining the probability distribution function (PDF) of the distance between solitons. We find analytically the non-Gaussian tail of the PDF to be exponential. The propagation distance that corresponds to a given bit-error rate is described as a function of system parameters (filtering and noise level), initial distance, and initial phase difference between solitons. We find the interval of parameters where a larger propagation distance can be achieved for higher density of information.
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Affiliation(s)
- G E Falkovich
- Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
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