Higher-order Solitons Research Articles

Emergence of breathers in non-linear pulse compression

We report on the universality of the emergence of Akhmediev breathers in soliton effect pulse compression dynamics by using explicit analytic solutions of breathers and solitons in the non-linear Schrödinger equation. We show that at maximum compression point, a higher-order soliton and a breather solution are equivalent with a nearly identical full-width at half maximum duration. From this, we demonstrate numerically how breathers that develop in the initial stage of continuous wave supercontinuum generation transform into an ensemble of solitons in the presence of system perturbation.

Soliton formation and stability under the interplay between parity-time-symmetric generalized Scarf-II potentials and Kerr nonlinearity

We present an alternative type of parity-time (PT)-symmetric generalized Scarf-II potentials, which makes possible for non-Hermitian Hamiltonians in the classical linear Schrödinger system to possess fully real spectra with unique features such as the multiple PT-symmetric breaking behaviors and to support one-dimensional (1D) stable PT-symmetric solitons of power-law waveform, namely power-law solitons, in focusing Kerr-type nonlinear media. Moreover, PT-symmetric high-order solitons are also derived numerically in 1D and 2D settings. Around the exactly obtained nonlinear propagation constants, families of 1D and 2D localized nonlinear modes are also found numerically. The majority of fundamental nonlinear modes can still keep steady in general, whereas the 1D multipeak solitons and 2D vortex solitons are usually susceptible to suffering from instability. Likewise, similar results occur in the defocusing Kerr-nonlinear media. The obtained results will be useful for understanding the complex dynamics of nonlinear waves that form in PT-symmetric nonlinear media in other physical contexts.

Pulse compression and pedestal suppression by self-similar propagation in nonlinear optical loop mirror

We propose and investigate a unique nonlinear pulse compression mechanism which provides low-pedestal, few-cycle ultrashort pulses at 800 nm. In order to accomplish this aim, we propose a fiber compressor based on self similar design and nonlinear optical loop mirror (NOLM) configuration. The proposed NOLM consists of chloroform infiltrated polarization maintaining tapered photonic crystal fiber (TPCF) which has been designed on the basis of self-similar technique. The coupler of the NOLM plays a key role in the suppression of pedestal and formation of high-quality compressed pulses. The dynamics of higher-order chirped solitons in a designed pulse compressor has been simulated numerically, to rigorously incorporate self-switching characteristics of NOLM and the self-similar property of the fiber. For a 140-fs input pulse with 100 W, quality analysis at the output of NOLM reveals a compressed pulse with a FWHM of 6.98 fs, compression factor of 20.05, and pedestal energy of 2.5% in a 4.9 cm long fiber.

Bright soliton solutions to a nonlocal nonlinear Schrödinger equation of reverse-time type

Two types of multiple bright soliton solutions for a focusing nonlocal reverse-time nonlinear Schrodinger equation are derived by using the Hirota’s bilinear method and the Kadomtsev–Petviashvili hierarchy reduction. Compared with the case of the local Manakov system, both types of soliton solutions are restricted to ones with even numbers. For the first type of solution, the fundamental paired soliton exhibits two paralleled line solitons along with the time axis and the special breathing soliton. For the second type of solution, the fundamental paired soliton only allows head-on collisions with the same velocity, in which the collision with spatial interferences and the degenerate soliton with position shifts can be realized with certain parameters. General higher-order soliton solutions of two types describe the superposition of the fundamental paired soliton, which show a few interesting dynamical behaviors such as the superposed breathing solitons, the interaction of two paired soliton bound states, the interaction of two degenerate solitons with position shifts and the breathing soliton with position shifts.

Effects of third-order dispersion on temporal soliton compression in dispersion-engineered silicon photonic crystal waveguides

High-order temporal soliton compression in dispersion-engineered silicon photonic crystal waveguides will play an important role in future integrated photonic circuits compatible with complementary metal–oxide–semiconductors. Here, we report the physical mechanisms of high-order temporal soliton compression affected by third-order dispersion (TOD) combined with free carrier dispersion (FCD) in a dispersion engineered silicon photonic crystal waveguide with wideband low anomalous dispersion. Through numerical temporal soliton evolution analysis, we report what we believe is the first demonstration of the dual opposite effects of TOD on temporal soliton compression, which are strengthening or weakening through two different physical mechanisms, not only depending on the sign of TOD but also the relative magnitude of TOD-induced equivalent group velocity dispersion (GVD) β 2 , equ to the original GVD β 2 . We further find that FCD counteracts the effects of negative TOD on the soliton compression, while it reinforces the effects of positive TOD on the soliton compression. These results will help to design suitable dispersion-engineered silicon waveguides for superior on-chip temporal pulse compression in optical communications and processing application fields.

Multiple bright soliton solutions of a reverse-space nonlocal nonlinear Schrödinger equation

Multiple bright soliton solutions in determinant form for a focusing nonlocal nonlinear Schrödinger (NLS) equation are derived via the bilinear method and the KP hierarchy reduction. It is shown that only multi-soliton solutions with even numbers are allowed and each paired soliton exhibits the head-on collision with the same velocity in the interaction process. One paired soliton describes the different collision patterns including the centering-hump, the centering-valley, and the spatial interference as well as a degenerate soliton with position shifts. Higher-order soliton solutions depict the interactions among different types of the paired soliton, in which a reduced four-soliton solution exhibits an interesting breathing soliton with position shifts.

Cascaded single-mode fibers for higher-order soliton compression at 2 µm.

We present the design of single-mode fibers for two-stage higher-order soliton compression at 2 µm wavelength and achieve high-degree pulse compression in cascaded single-mode fibers. The compression performance for the initial input pulse width from 1 to 50 ps is also investigated. For the initial third-order soliton of 10 ps, a compression factor of 75.7 has been achieved, and the pedestal energy is only 46.66%.

Effect of an input beam’s shape and curvature on the nonlinear effects in graded-index fibers

We present a general framework capable of describing the nonlinear propagation of pulsed optical beams of arbitrary shapes and phase fronts inside a graded-index (GRIN) fiber. The main assumption made is that the spatial self-imaging features of the beam are not affected by the temporal evolution of optical pulses. A propagation kernel known from the work done in the 1970s is used to obtain a distance-dependent nonlinear coefficient that captures all spatial effects within an effective nonlinear Schrödinger equation. We consider three specific beam shapes (Gaussian, circular, and square) to study the impact of the shape, position, and curvature of optical beams on the complex spatiotemporal dynamics specific to GRIN fibers. In particular, we focus on the impact of an input beam’s shape on the modulation-instability sidebands and the generation of multiple dispersive waves when higher-order solitons form inside a GRIN fiber. The results of our numerical analysis indicate that for beam widths chosen to yield the same value of the effective mode area at the input end of the fiber, the nonlinear effects are pronounced considerably when a Gaussian beam is launched into the fiber. We also found that even though the self-imaging period is doubled when an off-centered Gaussian beam is launched into a GRIN fiber, it does not affect the nonlinear evolution because the effective beam area still maintains the same periodicity, as long as the shift in the beam’s center is not so large that it does not remain confined to the fiber’s core.

General high-order localized waves to the Bogoyavlenskii–Kadomtsev–Petviashvili equation

In this work, the Bogoyavlenskii–Kadomtsev–Petviashvili equation is investigated. By means of the Hirota bilinear system and Pfaffian, we demonstrate that the Bogoyavlenskii–Kadomtsev–Petviashvili equation has the Grammian determinant solution. On the basis of the Grammian determinant solution, we derive a class of exponentially localized wave solutions. It is shown that the exponentially localized wave solutions describe the fission and fusion phenomena between kink-type soliton and breather-type soliton. Further, general high-order localized waves consisting of kink-type, lump-type and breather-type solitons are derived by means of the general differential operators. These general high-order localized waves contain more plentiful dynamical behaviors. It is shown that the mixture of multiple wave solutions exhibit the fission and fusion phenomena among kink-type, lump-type and breather-type solitons. In addition, by choosing appropriate parameters, we construct general high-order lump-type soliton solutions. The results show that high-order lump-type solitons propagate with the variable speed on the curves. After collision, the lump-type solitons do not pass through each other, but rather are reflected back. The propagation paths of the lump-type solitons are changed completely.

Various kinds of high-order solitons to the Bogoyavlenskii–Kadomtsev–Petviashvili equation

In this work, the Bogoyavlenskii–Kadomtsev–Petviashvili equation which is used to describe the wave phenomenon in fluid mechanics is investigated. Based on the bilinear representation, perturbation method and Taylor expansion approach, we derive various kinds of high-order solitons including the N-kink soliton, n-order lump-type soliton and mixture solution of kink soliton and lump-type soliton. First, N-kink soliton solution is obtained by the bilinear representation and perturbation method. Second, by using the Taylor expansion approach for the 2n-kink soliton solution, n-order lump-type soliton is obtained. Third, by mean of the Taylor expansion approach for 2n-kink soliton solution in the N-kink soliton solution (1 < 2n < N), we construct the mixture solution consisting of (N − 2n)-kink soliton and n-order lump-type soliton. Interestingly, the collision between kink soliton and lump-type soliton can give rise to a high-order lump-type soliton.

Preventing critical collapse of higher-order solitons by tailoring unconventional optical diffraction and nonlinearities

Self-trapped modes suffer critical collapse in two-dimensional cubic systems. To overcome such a collapse, linear periodic potentials or competing nonlinearities between self-focusing cubic and self-defocusing quintic nonlinear terms are often introduced. Here, we combine both schemes in the context of an unconventional and nonlinear fractional Schrödinger equation with attractive-repulsive cubic–quintic nonlinearity and an optical lattice. We report theoretical results for various two-dimensional trapped solitons, including fundamental gap and vortical solitons as well as the gap-type soliton clusters. The latter soliton family resembles the recently-found gap waves. We uncover that, unlike the conventional case, the fractional model exhibiting fractional diffraction order strongly influences the formation of higher band gaps. Hence, a new route for the study of self-trapped modes in these newly emergent higher band gaps is suggested. Regimes of stability and instability of all the soliton families are obtained with the help of linear-stability analysis and direct simulations.

A Chaotic State in the Transition From Single Toward Multiple Pulse for Soliton Fiber Lasers

High-order soliton in anomalous-dispersion ultrafast fiber lasers is rarely researched due to its intrinsic instability in dissipative systems. For the first time, we numerically demonstrate the ultrafast pulse in fiber lasers will deviate from the fundamental soliton under strong pulse energy before splitting into multiple pulses. High-order soliton effect results in the single-pulse splitting into the dual-pulse. Sustaining switching between the single- and dual-pulse induced by the high-order soliton evolution and cross-gain-modulation was observed and analyzed. The results offer significant and novel insights for the soliton shaping dynamics and instability in anomalous-dispersion fiber lasers.

Pump-Probe Study of Plasma Dynamics in Gas-Filled Photonic Crystal Fiber Using Counterpropagating Solitons

We present a pump-probe technique for monitoring ultrafast polarizability changes. In particular, we use it to measure the plasma density created at the temporal focus of a self-compressing higher-order pump soliton in gas-filled hollow-core photonic crystal fiber. This is done by monitoring the wavelength of the dispersive wave emission from a counter-propagating probe soliton. By varying the relative delay between pump and probe, the plasma density distribution along the fiber can be mapped out. Compared to the recently introduced interferometric side-probing for monitoring the plasma density, our new technique is relatively immune to instabilities caused by air turbulence and mechanical vibration. The results of two experiments on argon- and krypton-filled fiber are presented, and compared to numerical simulations. The technique provides an important new tool for probing photoionization in many different gases and gas mixtures as well as ultrafast changes in dispersion in many other contexts.

Transforming higher order bright and dark solitons to the first order solitons in Kerr medium: A review

A higher order soliton is a soliton where the initial energy is higher than the fundamental one by a specific coefficient. The value of the coefficient is square of an integer number. The intensity profile of higher order soliton (spatial or temporal) is not constant but varies periodically during propagation. The period of their evolution, called soliton-period. However, when the coefficient is not the square of an integer, the complicated intensity profile by propagation occurs. In this work, it is demonstrated and discussed the reason why higher order soliton is made, and how it modifies a factor in Nonlinear Schrödinger Equation (NLS), in positive and negative third order nonlinear (Kerr) medium. In this modified NLS equation, the exact analytical fundamental soliton solution which doesn’t show the soliton-period is obtained. Followed by numerical simulation by MATLAB, with Split-Step method, and comparison between modified initial beam-width which produces fundamental soliton and nonmodified initial beam-width that gives the other order of solitons for bright and dark solitons.

Propagation Characteristics of Higher-Order Mixed-Pattern Solitons in Nonlinear Media

In this paper, we study the propagation and evolution of higher-order mixed-pattern solitons in nonlinear media with a strong nonlocality. The propagation analytic formula of higher-order mixed-pattern solitons is derived and discussed in detail. Taking the fourth-order mixed-pattern solitons as an example and based on numerical calculations, we present the variations of second-order moment beam width, the light intensity, and the wavefront curvature. We show that the propagation evolution exhibits unique characteristics depending on different relative factors of higher-order mixed-pattern solitons. The influence of the beam order on the propagation evolution is also discussed.

Soliton-effect optical pulse compression in CMOS-compatible ultra-silicon-rich nitride waveguides

The formation of optical solitons arises from the simultaneous presence of dispersive and nonlinear properties within a propagation medium. Chip-scale devices that support optical solitons harness high field confinement and flexibility in dispersion engineering for significantly smaller footprints and lower operating powers compared to fiber-based equivalents. High-order solitons evolve periodically as they propagate and experience a temporal narrowing at the start of each soliton period. This phenomenon allows strong temporal compression of optical pulses to be achieved. In this paper, soliton-effect temporal compression of optical pulses is demonstrated on a CMOS-compatible ultra-silicon-rich nitride (USRN) waveguide. We achieve 8.7× compression of 2 ps optical pulses using a low pulse energy of ∼16 pJ, representing the largest demonstrated compression on an integrated photonic waveguide to date. The strong temporal compression is confirmed by numerical calculations of the nonlinear Schrödinger equation to be attributed to the USRN waveguide’s large nonlinearity and negligible two-photon absorption at 1550 nm.

Excitation Modes of Bright Matter-Wave Solitons.

We experimentally study the excitation modes of bright matter-wave solitons in a quasi-one-dimensional geometry. The solitons are created by quenching the interactions of a Bose-Einstein condensate of cesium atoms from repulsive to attractive in combination with a rapid reduction of the longitudinal confinement. Adeliberate mismatch of quench parameters allows for the excitation of breathing modes of the emerging soliton and for the determination of its breathing frequency as a function of atom number and confinement. In addition, we observe signatures of higher-order solitons and the splitting of the wave packet after the quench. Our experimental results are compared to analytical predictions and to numerical simulations of the one-dimensional Gross-Pitaevskii equation.

Investigations of energy and an exchange of energies between electrostatic solitons: higher-order corrections and efficiency improvement of semiconductor plasmas

ABSTRACT Investigations of the nonlinear excitation and collisions of electrostatic solitons in a dense semiconductor plasma composed of electrons and holes are improved by using the higher-order corrections. Applying the extended Poincaré-Lighthill-Kuo (EPLK) method to obtain the Korteweg–de Vries (KdV) equations, which govern the nonlinear excitation of electrostatic solitons. Furthermore, the phase shift equations due to the collisions between electrostatic solitons are obtained. A theoretical analysis is improved by employing the KdV equations with the effects of the fifth – order dispersion terms. The numerical illustrations demonstrate that the higher-order soliton energy depends significantly on the quantum semiconductor plasma number density. On the other hand, the density of the semiconductor plasma has a weak effect on the lowest-order soliton energy. Therefore, one has to be careful about the choosing semiconductor plasma parameters to avoid any deficiency of the modern semiconductor devices.

Nonlocal Pair Correlations in a Higher-Order Bose Gas Soliton.

The truncated Wigner and positive-P phase-space representations are used to study the dynamics of a one-dimensional Bose gas. This allows calculations of the breathing quantum dynamics of higher-order solitons with 10^{3}-10^{5} particles, as in realistic Bose-Einstein condensation experiments. Although classically stable, these decay quantum mechanically. Our calculations show that there are large nonlocal correlations and nonclassical quantum entanglement.

Invite paper: Self-imaging in multimode graded-index fibers and its impact on the nonlinear phenomena

Abstract The phenomenon of periodic self-imaging of optical beams, occurring inside any graded-index (GRIN) medium, was studied during the decade of the 1970s and was exploited to commercialize the GRIN lens. It has been found in recent years that the periodic self-imaging also affects the nonlinear propagation of optical pulses inside multimode GRIN fibers. In this paper, we first present the theory of self-imaging in linear GRIN fibers using a modal expansion approach. It is shown that the optical field at any point inside the fiber can be written without any reference to the fiber modes as a two-dimensional integration over the input field using a propagation kernel that is similar to that found in diffraction theory. However, this kernel has a specific property that reproduces the input field precisely in a periodic fashion along the length of a GRIN fiber (self-imaging). We apply this kernel to study the propagation of a Gaussian beam and discuss how self-imaging is modified by self-focusing produced by the Kerr nonlinearity. We then consider propagation of the continuous and pulsed Gaussian beams inside a GRIN fiber and discuss how self-imaging affects the modulation instability, leads to the formation of GRIN solitons, and produces novel temporal and spectral features when short optical pulses are launched that are intense enough to form high-order solitons.