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Wang Z, He M, Ling X, Zhang L, Zhao C. Numerical investigation of the fractional-soliton mode-locked fiber laser. OPTICS LETTERS 2024; 49:5499-5502. [PMID: 39352991 DOI: 10.1364/ol.534793] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/04/2024] [Accepted: 09/01/2024] [Indexed: 10/04/2024]
Abstract
We propose and numerically investigate a fractional-soliton mode-locked fiber laser by utilizing an intracavity spectral pulse shaper (SPS). The fiber laser can generate stable fractional-soliton pulses for three different Lévy index α (1 < α < 2), whose profiles are all close to the sech shape. We find that the positions of Kelly sidebands, pulse energy, and peak power of the emitted fractional pulses conform to three theoretical expressions, respectively. The numerical results are in good agreement with the theoretical analyses. In addition, the intracavity dynamics of the fractional pulses have been discussed. Our findings not only deepen the fundamental understanding of temporal fractional soliton but also provide a novel, to the best of our knowledge, approach to generating stable ultrashort fractional pulses.
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Malomed BA. Basic fractional nonlinear-wave models and solitons. CHAOS (WOODBURY, N.Y.) 2024; 34:022102. [PMID: 38341765 DOI: 10.1063/5.0190039] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/02/2023] [Accepted: 01/09/2024] [Indexed: 02/13/2024]
Abstract
This review article provides a concise summary of one- and two-dimensional models for the propagation of linear and nonlinear waves in fractional media. The basic models, which originate from Laskin's fractional quantum mechanics and more experimentally relevant setups emulating fractional diffraction in optics, are based on the Riesz definition of fractional derivatives, which are characterized by the respective Lévy indices. Basic species of one-dimensional solitons, produced by the fractional models which include cubic or quadratic nonlinear terms, are outlined too. In particular, it is demonstrated that the variational approximation is relevant in many cases. A summary of the recently demonstrated experimental realization of the fractional group-velocity dispersion in fiber lasers is also presented.
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Affiliation(s)
- Boris A Malomed
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
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Chatterjee P, Modak R. One-dimensional Lévy quasicrystal. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2023; 35:505602. [PMID: 37708897 DOI: 10.1088/1361-648x/acf9d4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2023] [Accepted: 09/14/2023] [Indexed: 09/16/2023]
Abstract
Space-fractional quantum mechanics (SFQM) is a generalization of the standard quantum mechanics when the Brownian trajectories in Feynman path integrals are replaced by Lévy flights. We introduce Lévy quasicrystal by discretizing the space-fractional Schrödinger equation using the Grünwald-Letnikov derivatives and adding on-site quasiperiodic potential. The discretized version of the usual Schrödinger equation maps to the Aubry-André (AA) Hamiltonian, which supports localization-delocalization transition even in one dimension. We find the similarities between Lévy quasicrystal and the AA model with power-law hopping, and show that the Lévy quasicrystal supports a delocalization-localization transition as one tunes the quasiperiodic potential strength and shows the coexistence of localized and delocalized states separated by mobility edge. Hence, a possible realization of SFQM in optical experiments should be a new experimental platform to test the predictions of AA models in the presence of power-law hopping.
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Affiliation(s)
- Pallabi Chatterjee
- Department of Physics, Indian Institute of Technology Tirupati, Tirupati 517619, India
| | - Ranjan Modak
- Department of Physics, Indian Institute of Technology Tirupati, Tirupati 517619, India
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Strunin DV, Malomed BA. Symmetry-breaking transitions in quiescent and moving solitons in fractional couplers. Phys Rev E 2023; 107:064203. [PMID: 37464702 DOI: 10.1103/physreve.107.064203] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2023] [Accepted: 05/08/2023] [Indexed: 07/20/2023]
Abstract
We consider phase transitions, in the form of spontaneous symmetry breaking (SSB) bifurcations of solitons, in dual-core couplers with fractional diffraction and cubic self-focusing acting in each core, characterized by Lévy index α. The system represents linearly coupled optical waveguides with the fractional paraxial diffraction or group-velocity dispersion (the latter system was used in a recent experiment [Nat. Commun. 14, 222 (2023)10.1038/s41467-023-35892-8], which demonstrated the first observation of the wave propagation in an effectively fractional setup). By dint of numerical computations and variational approximation, we identify the SSB in the fractional coupler as the bifurcation of the subcritical type (i.e., the symmetry-breaking phase transition of the first kind), whose subcriticality becomes stronger with the increase of fractionality 2-α, in comparison with very weak subcriticality in the case of the nonfractional diffraction, α=2. In the Cauchy limit of α→1, it carries over into the extreme subcritical bifurcation, manifesting backward-going branches of asymmetric solitons which never turn forward. The analysis of the SSB bifurcation is extended for moving (tilted) solitons, which is a nontrivial problem because the fractional diffraction does not admit Galilean invariance. Collisions between moving solitons are studied too, featuring a two-soliton symmetry-breaking effect and merger of the solitons.
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Affiliation(s)
- Dmitry V Strunin
- School of Mathematics, Physics and Computing, University of Southern Queensland, Toowoomba, Queensland 4350, Australia
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
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Zhao W, Ge L, Shen M. Interactions of Airy beams in nonlinear media with fourth-order diffraction. OPTICS EXPRESS 2023; 31:14715-14725. [PMID: 37157330 DOI: 10.1364/oe.488852] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/10/2023]
Abstract
We investigate to the best of our knowledge the first time the interactions of in-phase and out-of-phase Airy beams in Kerr, saturable and nonlocal nonlinear media with fourth-order diffraction using split-step Fourier transform method. Directly numerical simulations show that normal and anomalous fourth-order diffractions have profound effects on the interactions of the Airy beams in Kerr and saturable nonlinear media. We demonstrate the dynamics of the interactions in detail. In nonlocal media with fourth-order diffraction, nonlocality induces a long-range attractive force between Airy beams, leading to the formation of stable bound states of both in-phase and out-of-phase breathing Airy soliton pairs which are always repulsive in local media. Our results have potential applications in all-optical devices for communication and optical interconnects, etc.
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Stabilization of Axisymmetric Airy Beams by Means of Diffraction and Nonlinearity Management in Two-Dimensional Fractional Nonlinear Schrödinger Equations. Symmetry (Basel) 2022. [DOI: 10.3390/sym14122664] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
The propagation dynamics of two-dimensional (2D) ring-Airy beams is studied in the framework of the fractional Schrödinger equation, which includes saturable or cubic self-focusing or defocusing nonlinearity and Lévy index ((LI) alias for the fractionality) taking values 1≤α≤2. The model applies to light propagation in a chain of optical cavities emulating fractional diffraction. Management is included by making the diffraction and/or nonlinearity coefficients periodic functions of the propagation distance, ζ. The management format with the nonlinearity coefficient decaying as 1/ζ is considered too. These management schemes maintain stable propagation of the ring-Airy beams, which maintain their axial symmetry, in contrast to the symmetry-breaking splitting instability of ring-shaped patterns in 2D Kerr media. The instability driven by supercritical collapse at all values α<2 in the presence of the self-focusing cubic term is eliminated, too, by the means of management.
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Kumar S, Li P, Malomed BA. Domain walls in fractional media. Phys Rev E 2022; 106:054207. [PMID: 36559477 DOI: 10.1103/physreve.106.054207] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2022] [Accepted: 10/17/2022] [Indexed: 06/17/2023]
Abstract
Currently, much interest is drawn to the analysis of optical and matter-wave modes supported by the fractional diffraction in nonlinear media. We predict a new type of such states in the form of domain walls (DWs) in the two-component system of immiscible fields. Numerical study of the underlying system of fractional nonlinear Schrödinger equations demonstrates the existence and stability of DWs at all values of the respective Lévy index (α<2), which determines the fractional diffraction, and at all values of the XPM/SPM ratio β in the two-component system above the immiscibility threshold. The same conclusion is obtained for DWs in the system which includes the linear coupling, alongside the XPM interaction between the immiscible components. Analytical results are produced for the scaling of the DW's width. The DW solutions are essentially simplified in the special case of β=3, as well as close to the immiscibility threshold. In addition to symmetric DWs, asymmetric ones are constructed too, in the system with unequal diffraction coefficients and/or different Lévy indices of the two components.
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Affiliation(s)
- Shatrughna Kumar
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
| | - Pengfei Li
- Department of Physics, Taiyuan Normal University, Jinzhong 030619, China
- Institute of Computational and Applied Physics, Taiyuan Normal University, Jinzhong 030619, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
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Zhang X, Zhang J, Chen C, Zhang L. Controllable focusing behavior of chirped Pearcey-Gaussian pulses under time-dependent potentials. OPTICS EXPRESS 2022; 30:34835-34847. [PMID: 36242487 DOI: 10.1364/oe.471329] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2022] [Accepted: 08/26/2022] [Indexed: 06/16/2023]
Abstract
We investigate the propagation dynamics of the Pearcey-Gaussian (PG) pulses in the presence of time-dependent potentials in a linear medium both theoretically and numerically. We demonstrate that the combination of the linear potential and the initial chirp of PG pulses can flexibly control the propagation trajectory and inherent focusing properties of the PG pulses. When the parabolic potential is taken into account, the chirped PG pulses are periodically focused and reversed. By adjusting the parabolic potential and the pulse chirp, the characteristics of the focal points, such as position, intensity, and spacing between focal points, can be manipulated effectively. The interaction of two temporally separated PG pulses still shows a periodic evolution with controllable focusing characteristics. These results can broaden the application range of PG pulses and provide some inspiration for the control of PG pulses under nonlinear conditions.
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Zhou W, Liu A, Huang X, Bai Y, Fu X. Propagation dynamics of Laguerre-Gaussian beams in the fractional Schrödinger equation with noise disturbance. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2022; 39:736-743. [PMID: 35471400 DOI: 10.1364/josaa.455164] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2022] [Accepted: 03/06/2022] [Indexed: 06/14/2023]
Abstract
The evolution of Laguerre-Gaussian (LG) beams in the fractional Schrödinger equation (FSE) with Gaussian noise disturbance is numerically investigated. Without noise disturbance, the peak intensity of LG beams increases with the increment of radial or azimuthal indices, and the turning point of the peak intensity between different radial indices exists. As propagation distance gets longer, the intensity of the outermost sub-lobe exceeds that of the main lobe. When Gaussian noise is added, for a given noise level, the stability of peak intensity is enhanced as the Lévy index increases, while the center of gravity shows the opposite phenomenon. Moreover, the increment of the radial index can weaken the stability of the center of gravity. We also investigate the stability of the peak intensity of Airy beams in the FSE, and generally, the stability of LG beams is better than that of Airy beams. All these properties show that LG beams modeled by the FSE have potential applications in optical manipulation and communications.
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Wang J, Jin Y, Gong X, Yang L, Chen J, Xue P. Generation of random soliton-like beams in a nonlinear fractional Schrödinger equation. OPTICS EXPRESS 2022; 30:8199-8211. [PMID: 35299566 DOI: 10.1364/oe.448972] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2021] [Accepted: 02/07/2022] [Indexed: 06/14/2023]
Abstract
We investigate the generation of random soliton-like beams based on the Kuznetsov-Ma solitons in a nonlinear fractional Schrödinger equation (NLFSE). For Lévy index α = 1, the Kuznetsov-Ma solitons split into two nondiffracting beams during propagation in linear regime. According to the different input positions of the Kuznetsov-Ma solitons, the diffraction-free beams can be divided into three different types: bright-dark, dark-bright and bright-bright beams. In the nonlinear regime, the Kuznetsov-Ma solitons can be evolved into random soliton-like beams due to the collapse. The number of soliton-like beams is related to the nonlinear coefficient and the Lévy index. The bigger the nonlinear coefficient, the more beams generated. Moreover, the peak intensity of soliton-like beams presents a Gaussian distribution under the large nonlinear effect. In practice, the evolution of KM soliton can be realized by a plane wave with a Gaussian perturbation, which can be confirmed that they have the similar dynamics of propagation. In two dimensions, the plane wave with a Gaussian perturbation can be evolved into a bright-dark axisymmetric ring beam in the linear regime. Under the nonlinear modulation, the energy accumulates to the center and finally breaks apart into random beam filaments.
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Liu X, Malomed BA, Zeng J. Localized Modes in Nonlinear Fractional Systems with Deep Lattices. ADVANCED THEORY AND SIMULATIONS 2022. [DOI: 10.1002/adts.202100482] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Affiliation(s)
- Xiuye Liu
- State Key Laboratory of Transient Optics and Photonics Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences Xi'an 710119 China
- University of Chinese Academy of Sciences Beijing 100049 China
| | - Boris A. Malomed
- Department of Physical Electronics School of Electrical Engineering Faculty of Engineering, and the Center for Light‐Matter Interaction Tel Aviv University Ramat Aviv Tel Aviv P.O.B. 39040 Israel
- Instituto de Alta Investigación Universidad de Tarapacá Casilla 7D Arica Chile
| | - Jianhua Zeng
- State Key Laboratory of Transient Optics and Photonics Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences Xi'an 710119 China
- University of Chinese Academy of Sciences Beijing 100049 China
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Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results. PHOTONICS 2021. [DOI: 10.3390/photonics8090353] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The article produces a brief review of some recent results which predict stable propagation of solitons and solitary vortices in models based on the nonlinear Schrödinger equation (NLSE) including fractional one-dimensional or two-dimensional diffraction and cubic or cubic-quintic nonlinear terms, as well as linear potentials. The fractional diffraction is represented by fractional-order spatial derivatives of the Riesz type, defined in terms of the direct and inverse Fourier transform. In this form, it can be realized by spatial-domain light propagation in optical setups with a specially devised combination of mirrors, lenses, and phase masks. The results presented in the article were chiefly obtained in a numerical form. Some analytical findings are included too, in particular, for fast moving solitons and the results produced by the variational approximation. Moreover, dissipative solitons are briefly considered, which are governed by the fractional complex Ginzburg–Landau equation.
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Li P, Li R, Dai C. Existence, symmetry breaking bifurcation and stability of two-dimensional optical solitons supported by fractional diffraction. OPTICS EXPRESS 2021; 29:3193-3210. [PMID: 33770923 DOI: 10.1364/oe.415028] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2020] [Accepted: 12/25/2020] [Indexed: 06/12/2023]
Abstract
We study existence, bifurcation and stability of two-dimensional optical solitons in the framework of fractional nonlinear Schrödinger equation, characterized by its Lévy index, with self-focusing and self-defocusing saturable nonlinearities. We demonstrate that the fractional diffraction system with different Lévy indexes, combined with saturable nonlinearity, supports two-dimensional symmetric, antisymmetric and asymmetric solitons, where the asymmetric solitons emerge by way of symmetry breaking bifurcation. Different scenarios of bifurcations emerge with the change of stability: the branches of asymmetric solitons split off the branches of unstable symmetric solitons with the increase of soliton power and form a supercritical type bifurcation for self-focusing saturable nonlinearity; the branches of asymmetric solitons bifurcates from the branches of unstable antisymmetric solitons for self-defocusing saturable nonlinearity, featuring a convex shape of the bifurcation loops: an antisymmetric soliton loses its stability via a supercritical bifurcation, which is followed by a reverse bifurcation that restores the stability of the symmetric soliton. Furthermore, we found a scheme of restoration or destruction the symmetry of the antisymmetric solitons by controlling the fractional diffraction in the case of self-defocusing saturable nonlinearity.
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Chen J, Zeng J. Spontaneous symmetry breaking in purely nonlinear fractional systems. CHAOS (WOODBURY, N.Y.) 2020; 30:063131. [PMID: 32611086 DOI: 10.1063/5.0006050] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2020] [Accepted: 05/26/2020] [Indexed: 06/11/2023]
Abstract
Spontaneous symmetry breaking, a spontaneous course of breaking the spatial symmetry (parity) of the system, is known to exist in many branches of physics, including condensed-matter physics, high-energy physics, nonlinear optics, and Bose-Einstein condensates. In recent years, the spontaneous symmetry breaking of solitons in nonlinear wave systems is broadly studied; understanding such a phenomenon in nonlinear fractional quantum mechanics with space fractional derivatives (the purely nonlinear fractional systems whose fundamental properties are governed by a nonlinear fractional Schrödinger equation), however, remains pending. Here, we survey symmetry breaking of solitons in fractional systems (with the fractional diffraction order being formulated by the Lévy index α) of a nonlinear double-well structure and find several kinds of soliton families in the forms of symmetric and anti-symmetric soliton states as well as asymmetric states. Linear stability and dynamical properties of these soliton states are explored relying on linear-stability analysis and direct perturbed simulations, with which the existence and stability regions of all the soliton families in the respective physical parameter space are identified.
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Affiliation(s)
- Junbo Chen
- State Key Laboratory of Transient Optics and Photonics, Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an 710119, China
| | - Jianhua Zeng
- State Key Laboratory of Transient Optics and Photonics, Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an 710119, China
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Huang C, Dong L. Dissipative surface solitons in a nonlinear fractional Schrödinger equation. OPTICS LETTERS 2019; 44:5438-5441. [PMID: 31730077 DOI: 10.1364/ol.44.005438] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2019] [Accepted: 10/16/2019] [Indexed: 06/10/2023]
Abstract
We study the existence and stability of dissipative surface solitons supported by the nonlinear fractional Schrödinger equation (NLFSE) with an interface between a semi-infinite chirped lattice and a uniform Kerr medium. In such a system, the existence domain of dissipative surface solitons depends on an upper cutoff value of the linear gain coefficient at a fixed nonlinear loss. The results of the linear stability analysis are in good agreement with that of the propagation simulation in a fractional dimension. Stable dissipative surface solitons generally feature low energy and small propagation constants and adapt to a wide range of two-photon absorption. The instability of solitons can be suppressed by increasing the chirp rate of the lattice. Robust nonlinear dissipative surface states can be easily excited by a Gaussian input beam. Similar characteristics of the two-dimensional dissipative surface solitons are also addressed.
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