Vector Solitons Research Articles

Generation of vector soliton mode-locked Er-doped fiber laser using ZrTe2 as saturable absorber

We demonstrate a passively mode-locked fiber laser employing zirconium telluride (ZrTe2) as saturable absorber and illustrate its application for vector soliton generation in erbium-doped fiber lasers. By tuning the pump power and polarization controller, stable mode-locked operations in different operations are obtained with different cavity lengths. In addition, we also further research on characteristics of the vector solitons formed in fiber laser. The vector features of the obtained solitons are experimentally explored, and these solitons are proved to be group velocity locked vector solitons. Our research results indicate that ZrTe2 nanosheets could be available saturable absorber for ultrashort pulse fiber lasers, especially for generating vector soliton pulses in fiber lasers.

Electrostatic wave interaction via asymmetric vector solitons as precursor to rogue wave formation in non-Maxwellian plasmas

An asymmetric pair of coupled nonlinear Schrödinger (CNLS) equations has been derived through a multiscale perturbation method applied to a plasma fluid model, in which two wavepackets of distinct (carrier) wavenumbers (k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_1$$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_2$$\\end{document}) and amplitudes (Ψ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi _1$$\\end{document} and Ψ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi _2$$\\end{document}) are allowed to co-propagate and interact. The original fluid model was set up for a non-magnetized plasma consisting of cold inertial ions evolving against a κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}-distributed electron background in one dimension. The reduction procedure resulting in the CNLS equations has provided analytical expressions for the dispersion, self-modulation and cross-coupling coefficients in terms of the two carrier wavenumbers. These coefficients present no symmetry whatsoever, in the general case (of different wavenumbers). The possibility for coupled envelope (vector soliton) solutions to occur has been investigated. Although the CNLS equations are asymmetric and non-integrable, in principle, the system admits various types of vector soliton solutions, physically representing nonlinear, localized electrostatic plasma modes, whose areas of existence is calculated on the wavenumbers’ parameter plane. The possibility for either bright (B) or dark (D) type excitations for either of the (2) waves provides four (4) combinations for the envelope pair (BB, BD, DB, DD), if a set of explicit criteria is satisfied. Moreover, the soliton parameters (maximum amplitude, width) are also calculated for each type of vector soliton solution, in its respective area of existence. The dependence of the vector soliton characteristics on the (two) carrier wavenumbers and on the spectral index κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} characterizing the electron distribution has been explored. In certain cases, the (envelope) amplitude of one component may exceed its counterpart (second amplitude) by a factor 2.5 or higher, indicating that extremely asymmetric waves may be formed due to modulational interactions among copropagating wavepackets. As κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} decreases from large values, modulational instability occurs in larger areas of the parameter plane(s) and with higher growth rates. The distribution of different types of vector solitons on the parameter plane(s) also varies significantly with decreasing κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}, and in fact dramatically for κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} between 3 and 2. Deviation from the Maxwell-Boltzmann picture therefore seems to favor modulational instability as a precursor to the formation of bright (predominantly) type envelope excitations and freak waves.

Controllable vector soliton in (2+1)-dimensional coupled nonlinear Schrödinger equations with varying coefficients

Controllable vector soliton in (2+1)-dimensional coupled nonlinear Schrödinger equations with varying coefficients

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z(N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian–Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase difference in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

Phase transitions in 2D multistable mechanical metamaterials via collisions of soliton-like pulses

In recent years, mechanical metamaterials have been developed that support the propagation of an intriguing variety of nonlinear waves, including transition waves and vector solitons (solitons with coupling between multiple degrees of freedom). Here we report observations of phase transitions in 2D multistable mechanical metamaterials that are initiated by collisions of soliton-like pulses in the metamaterial. Analogous to first-order phase transitions in crystalline solids, we observe that the multistable metamaterials support phase transitions if the new phase meets or exceeds a critical nucleus size. If this criterion is met, the new phase subsequently propagates in the form of transition waves, converting the rest of the metamaterial to the new phase. More interestingly, we numerically show, using an experimentally validated model, that the critical nucleus can be formed via collisions of soliton-like pulses. Moreover, the rich direction-dependent behavior of the nonlinear pulses enables control of the location of nucleation and the spatio-temporal shape of the growing phase.

Almost *-Ricci solitons on contact strongly pseudo-convex integrable CR-manifolds

We prove that if contact strongly pseudo-convex integrable CR-manifold admits a *-Ricci soliton where the soliton vector Z is contact, then the Reeb vector field ? is an eigenvector of the Ricci operator at each point if and only if ? is constant. Then we study contact strongly pseudo-convex integrable CR-manifold such that 1 is a almost *-Ricci soliton with potential vector field Z collinear with ?. To this end, we prove that if a 3-dimensional contact metric manifoldMwith Q? = ?Q which admits a gradient almost *-Ricci soliton, then either M is flat or f is constant.

Yamabe solitons in contact geometry

It is shown that the scalar curvature of a Yamabe soliton as a Sasakian manifold is constant and the soliton vector field is Killing. The same conclusion is shown to hold for a Yamabe soliton as a $K$-contact manifold $M^{2n+1}$ if any one of the following conditions hold: (i) its scalar curvature is constant along the soliton vector field $V$, (ii) $V$ is an eigenvector of the Ricci operator with eigenvalue $2n$, (iii) $V$ is gradient.

Optical Nonlinearity of Violet Phosphorus and Applications in Fiber Lasers

A D-shaped fiber is coated with a new two-dimensional nanomaterial, violet phosphorus (VP), to create a saturable absorber (SA) with a modulation depth of 3.68%. Subsequently, the SA is inserted into a fiber laser, enabling successful generation of dark solitons and bright–dark soliton pairs through adjustment of the polarization state within the cavity. Through further study, mode-locked pulses are achieved, proving the existence of polarization-locked vector solitons. The results indicate that VP can be used as a polarization-independent SA.

Pulse energy exchange of vector solitons in a fiber laser

The energy exchange between orthogonal polarization components is crucial for the build-up of vector solitons (VSs). Unlike previous observations of energy exchange in the frequency domain, our experiments analyzed pulse energy flows in the time domain. We provide evidence to demonstrate the influence of the four-wave mixing (FWM) and cross-phase modulation (XPM) effect on VSs build-up in passive mode-locked fiber lasers through a perspective of pulse energy exchange. The results indicate that the energy exchange of PRVS caused by FWM and XPM is stronger than that of PLVS. The liner energy exchange caused by the birefringence of fiber and PC influences the period of energy exchange. After stabilization, the intra-cavity energy evolution period is one roundtrip for PLVS while serval roundtrips for PRVS. In the future, we can achieve PLVS by adjusting the linear energy exchange through cavity birefringence, thereby meeting the industrial demand for stable and uniform pulse trains.

Higher-order matrix nonlinear Schrödinger equation with the negative coherent coupling: binary Darboux transformation, vector solitons, breathers and rogue waves

Optical fiber communication plays a crucial role in modern communication. In this work, we focus on the higher-order matrix nonlinear Schrödinger equation with negative coherent coupling in a birefringent fiber. For the slowly varying envelopes of two interacting optical modes, we construct a binary Darboux transformation using the corresponding Lax pair. With vanishing seed solutions and the binary Darboux transformation, we investigate vector degenerate soliton and exponential soliton solutions. By utilizing these soliton solutions, we demonstrate three types of degenerate solitons and double-hump bright solitons. Furthermore, considering non-vanishing seed solutions and applying the binary Darboux transformation, we obtain vector breather solutions, and present the vector single-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, double-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, and vector degenerate beak-type breathers. Additionally, we take the limit in the breather solutions and derive vector rogue wave solutions. We illustrate the beak-type rogue waves and bright-dark rogue waves. Humps of these vector double-hump waves can separate into two individual humps. The results obtained in this work may potentially provide valuable insights for experimentally manipulating the separation of two-hump solitons, breathers, and rogue waves in optical fibers.

Inverse scattering transform algorithm for the Manakov system

A numerical algorithm is described for solving the inverse spectral scattering problem associated with the Manakov model of the vector nonlinear Schrödinger equation. This model of wave processes simultaneously considers dispersion, nonlinearity and polarization effects. It is in demand in nonlinear physical optics and is especially perspective for describing optical radiation propagation through the fiber communication lines. In the presented algorithm, the solution to the inverse scattering problem based on the inversion of a set of nested matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations, using a block version of the Levinson-type Toeplitz bordering algorithm. Numerical tests carried out by comparing calculations with known exact analytical solutions confirm the stability and second order of accuracy of the proposed algorithm. We also give an example of the algorithm application to simulate the collision of a differently polarized pair of Manakov optical vector solitons.

Ricci solitons on anisotropic fluid spacetime in f(R, T)-gravity

In this paper we study the behaviour of anisotropic fluid spacetimes in f(R, T)-gravity for admitting Ricci Soliton. We obtain the form of tangential pressure of the fluid for steady, shrinking and expanding Ricci soliton when Ricci soliton vector field becomes a Killing vector field. Modified Poisson and Liouville equations are derived from the Ricci soliton equation in f(R, T)-gravity having anisotropic fluid. Also harmonic significance of Ricci soliton on anisotropic fluid spacetime in f(R, T)-gravity with a harmonic potential function is derived. We derive field equations for perfect fluid plane symmetric static spacetime and obtain their solutions. For each solution we then obtain Ricci soliton vector fields. We observe that such spacetimes admit expanding, shrinking or steady Ricci solitons. All the obtained solitons are trivial in nature. When these Ricci soliton vector fields become Killing vector fields then our results are in full agreement with the theorems deduced from our study.

Nonlinearity managed vector solitons

The evolution of vector solitons under nonlinearity management is studied. The averaged over strong and rapid modulations in time of the inter-species interactions vector Gross-Pitaevskii equation (GPE) is derived. The averaging gives the appearance of the effective nonlinear quantum pressure depending on the population of the other component. Using this system of equations, the existence and stability of the vector solitons under the action of the strong nonlinearity management (NM) is investigated. Using a variational approach the parameters of NM vector solitons are found. The numerical simulations of the full time-dependent coupled GPE confirm the theoretical predictions.

A binary Darboux transformation for multi-component nonlinear Schrödinger equations and dark vector soliton solutions

By taking the plane wave potentials as the seed solutions, we harness a binary Darboux transformation to generate dark vector soliton solutions for multi-component nonlinear Schrödinger equations. We introduce a generalized Darboux matrix such that the eigenvalues could equal the adjoint eigenvalues. The method which is purely algebraic could be useful and convenient, particularly in the construction of dark soliton solutions of integrable systems.

Modulating dual-wavelength optical fiber vector solitons

Optical fiber vector solitons have potential applications in the field of high-capacity optical fiber communications and have been widely explored in recent years. Here, we theoretically modulate dual-wavelength optical fiber vector solitons in an optical fiber system at a wavelength regime of 1 μm while considering the influence of group velocity dispersion. When the input dual-wavelength optical fiber vector solitons have the same two central wavelengths of 1057 nm and 1063 nm in orthogonal directions, the output modulated optical fiber vector solitons’ pulse shapes and optical spectra will maintain their peak intensities upon the change of the projection angle. When the two orthogonal central wavelengths of the input dual-wavelength optical fiber vector solitons are slightly different (1056 nm and 1062 nm in one polarization direction, 1058 nm and 1064 nm in the other direction), dual-peak pulse shapes appear and are accompanied by different wavelength peak intensities when the propagation distance increases. Our simulation results examine the out-cavity modulation of dual-wavelength optical fiber vector solitons and can be expanded to multi-wavelength optical fiber vector solitons’ modulation.

Suppression of polarization instability in ultrashort Fabry–Pérot fiber laser

Due to the weak birefringence from the intracavity fiber, vector solitons are easily generated in GHz-fundamental-repetition-rate mode-locked fiber lasers, which can exhibit diverse time-varying polarization dynamics. These soliton dynamics can lead to polarization instability of the pulse train, which prevents industrial and scientific applications that require stable and uniform pulse trains. However, it is hard to suppress the polarization instability due to insufficient space for inserting traditional polarizers and difficulties in enhancing the fiber birefringence in the ultrashort fiber laser cavity. To this end, here we propose an alternative method that can effectively transform the vector solitons into scalar solitons by exploring gold nanorod (GNR) film as a polarizer in the laser cavity. First, the theoretical studies of polarization dynamics in ultrashort Fabry–Pérot (FP) fiber lasers with and without intracavity GNR film are conducted. The results indicate that the use of GNR film can significantly suppress the polarization instability and generate scalar solitons. Then, the large-scale preparation of GNRs with longitudinal surface plasmon resonance absorption peak of >1380 nm is realized by a two-step seed-mediated method, and the GNR film with an operation wavelength range covering the C + L band is fabricated by electrospinning. Finally, we apply the GNR film to a 2-cm-long FP fiber laser, and convert polarization rotation vector solitons to linearly polarized solitons (LPSs) at 1.5 μm. The polarization extinction ratio of the improved LPS pulse train is up to 33 dB.

Existence of Ricci soliton vector fields on Vaidya spacetime

In this paper, we investigate the behaviour of the Vaidya spacetime admitting a Ricci soliton vector field, where we have found the expressions for the four components of the vector field, and we have shown, under this condition, that the spacetime reduces to the Schwarzschild spacetime. Moreover, we have found the desired expression for the potential function h to satisfy in order for a gradient Ricci soliton vector field to exist on the Vaidya spacetime.

Linear coupling-related pulse splitting in fiber lasers.

We demonstrate a unique pulse-splitting mechanism dominated by the linear coupling between two vector modes in a mode-locked fiber laser using polarization-maintaining fiber. As the linear coupling strength increases, the pulse experiences larger perturbations and manifests as stronger spectral sidebands. Correspondingly, the temporal pedestals possessing a higher intensity become untrapped and eventually evolve into a stable pulse. Such linear coupling-related pulse splitting is ubiquitous both in normal- and anomalous-dispersion regimes, fundamentally differing from that induced by the excessive nonlinear phase shift. Experimental observations fully sustain numerical results and provide a flexible approach to managing the number and energy of vector solitons.

Emergent Phenomena of Vector Solitons Induced by the Linear Coupling (Laser Photonics Rev. 17(9)/2023)

Emergent Phenomena of Vector Solitons Induced by the Linear Coupling (Laser Photonics Rev. 17(9)/2023)

Characterization of the ultrafast fiber laser mode-locked by nonlinear multimode interference

A passively mode-locked fiber laser with switchable output states is demonstrated by using an all-fiber saturable absorber (SA) based on a hybrid single mode fiber-graded index multimode fiber-single mode fiber-graded index multimode fiber-single mode fiber (SMSMS) incorporated into a ring cavity. We investigate the vectorial characteristics of the output soliton in order to analyze the influence of polarization dependence of the SMSMS structure on solitons generation. Due to the appearance of vector solitons, it is proved that nonlinear multimode interference (NL-MMI) plays a dominant role in solitons generation but not nonlinear polarization rotation (NPR). Meanwhile, abundant mode-locked states are achieved in the laser, such as fundamental soliton, higher-order harmonics, bound twin-pulse solitons, and quasi-harmonic bound states. This configuration provides a useful platform for investigating ultrafast soliton dynamics.