Annihilation of photorefractive solitons

Collision-induced Hopf-type bifurcation reversible transitions in a dual-wavelength femtosecond fiber laser

Collisions refer to a striking nonlinear interaction process in dissipative systems, revealing the particle-like properties of solitons. In dual-wavelength mode-locked fiber lasers, collisions are inherent and periodic. However, how collisions influence the dynamical transitions in the dual-wavelength mode-locked state has not yet been explored. In our work, dispersion management triggers the complex interactions between solitons in the cavity. We reveal the smooth or Hopf-type bifurcation reversible transitions of dual-color soliton molecules (SMs) during the collision by the real-time spectral measurement technique of time-stretch Fourier transform. The reversible transitions between stationary SMs and vibrating SMs, reveal that the cavity parameters pass through a bifurcation point in the collision process without active external intervention. The numerical results confirm the universality of collision-induced bifurcation behavior. These findings provide new insights into collision dynamics in dual-wavelength ultrafast fiber lasers. Furthermore, the study of inter-molecular collisions is of great significance for other branches of nonlinear science.

Intense Wave Formation from Multiple Soliton Fusion and the Role of Extra Dimensions

We experimentally and numerically explore the role of dimensionality in multiple (three or more) soliton fusion supported by nonreciprocal energy exchange. Three-soliton fusion into an intense wave is found when an extra dimension, with no broken inversion symmetry, is involved. The phenomenon is observed for 2+1D spatial waves in photorefractive crystals, where solitons are supported by a spatially local saturated Kerr-like self-focusing and fusion is driven by the leading nonlocal correction, the spatial analog of the nonlinear Raman effect.

Evidence of Chaotic Dynamics in Three-Soliton Collisions

We observe chaotic optical wave dynamics characterized by erratic energy transfer and soliton annihilation and creation in the aftermath of a three-soliton collision in a photorefractive crystal. Irregular dynamics are found to be mediated by the nonlinear Raman effect, a coherent interaction that leads to nonreciprocal soliton energy exchange. Results extend the analogy between solitons and particles to the emergence of chaos in three-body physics and provide new insight into the origin of the irregular dynamics that accompany extreme and rogue waves.

Recent advances in real-time spectrum measurement of soliton dynamics by dispersive Fourier transformation

Mode-locking lasers have not only produced huge economic benefits in industrial fields and scientific research, but also provided an excellent platform to study diverse soliton phenomena. However, the real-time characterization of the ultrafast soliton dynamics remains challenging for traditional electronic instruments due to their relatively low response bandwidth and slow scan rate. Consequently, it is urgent for researchers to directly observe these ultrafast evolution processes, rather than just indirectly understand them from numerical simulations or averaged measurement data. Fortunately, dispersive Fourier transformation (DFT) provides a powerful real-time measurement technique to overcome the speed limitations of traditional electronic measurement devices by mapping the frequency spectrum onto the temporal waveform. In this review, the operation principle of DFT is discussed and the recent progress in characterizing the ultrafast transient soliton dynamics of mode-locking lasers is summarized, including soliton explosions, soliton molecules, noise-like pulses, rogue waves, and mode-locking buildup processes.

Opto-chemo-mechanical transduction in photoresponsive gels elicits switchable self-trapped beams with remote interactions

Next-generation photonics envisions circuitry-free, rapidly reconfigurable systems powered by solitonic beams of self-trapped light and their particlelike interactions. Progress, however, has been limited by the need for reversibly responsive materials that host such nonlinear optical waves. We find that repeatedly switchable self-trapped visible laser beams, which exhibit strong pairwise interactions, can be generated in a photoresponsive hydrogel. Through comprehensive experiments and simulations, we show that the unique nonlinear conditions arise when photoisomerization of spiropyran substituents in pH-responsive poly(acrylamide-co-acrylic acid) hydrogel transduces optical energy into mechanical deformation of the 3D cross-linked hydrogel matrix. A Gaussian beam self-traps when localized isomerization-induced contraction of the hydrogel and expulsion of water generates a transient waveguide, which entraps the optical field and suppresses divergence. The waveguide is erased and reformed within seconds when the optical field is sequentially removed and reintroduced, allowing the self-trapped beam to be rapidly and repeatedly switched on and off at remarkably low powers in the milliwatt regime. Furthermore, this opto-chemo-mechanical transduction of energy mediated by the 3D cross-linked hydrogel network facilitates pairwise interactions between self-trapped beams both in the short range where there is significant overlap of their optical fields, and even in the long range--over separation distances of up to 10 times the beam width--where such overlap is negligible.

3-D Spiraling Self-Trapped Light Beams in Photochemical Systems

A pair of visible laser beams self-trap and spiral about each other as they propagate through polymer gels undergoing two different photochemical reactions. When launched into gels that undergo photopolymerization of methacrylate substituents or photo-oxidation of iodide anion, two non-coplanar (skewed) Gaussian beams collide and spiral about each other as they advance through the evolving medium. In the absence of chemical reactions, the linearly polarized beams broaden naturally and propagate along their original, straight-pathed trajectories. By contrast, refractive index gradients generated by the photochemical reactions elicit self-trapping and introduce an attractive interaction between the self-trapped beams. The self-trapped beams spiral about each other when this mutual attraction perfectly counterbalances their original tendency to diverge away from each other. These findings show that the photochemically mediated interactions of incident optical fields within the gel medium impart a helical trajectory and angular velocity to the self-trapped beam pair.

Soliton annihilation into a polychromatic dispersive wave

We investigate the propagation of a soliton in an axially varying optical fiber with a progressive change from anomalous to normal dispersion regimes. Spectral and temporal measurements provide evidence for a complete annihilation of the soliton, which explodes into a polychromatic dispersive wave. This interpretation is confirmed by numerical solution of the generalized nonlinear Schrödinger equation.

Stable radiating gap solitons and their resonant interactions with dispersive waves in systems with a parametric pump

We study the formation of gap solitons in the presence of a parametric pump. It is shown that a parametric pump can stabilize stationary solitons continuously emitting dispersive waves. The resonant interactions of the radiation and the solitons are studied and it is shown that the solitons can be effectively controlled by the radiation. In particular it is shown that the solitons can collide or get pinned to inhomogeneities due to the interactions mediated by the resonant radiation.

Universal fractal structures in the weak interaction of solitary waves in generalized nonlinear Schrödinger equations

Weak interactions of solitary waves in the generalized nonlinear Schrödinger equations are studied. It is first shown that these interactions exhibit similar fractal dependence on initial conditions for different nonlinearities. Then by using the Karpman-Solov'ev method, a universal system of dynamical equations is derived for the velocities, amplitudes, positions, and phases of interacting solitary waves. These dynamical equations contain a single parameter, which accounts for the different forms of nonlinearity. When this parameter is zero, these dynamical equations are integrable, and the exact analytical solutions are derived. When this parameter is nonzero, the dynamical equations exhibit fractal structures which match those in the original wave equations both qualitatively and quantitatively. Thus the universal nature of fractal structures in the weak interaction of solitary waves is analytically established. The origin of these fractal structures is also explored. It is shown that these structures bifurcate from the initial conditions where the solutions of the integrable dynamical equations develop finite-time singularities. Based on this observation, an analytical criterion for the existence and locations of fractal structures is obtained. Lastly, these analytical results are applied to the generalized nonlinear Schrödinger equations with various nonlinearities such as the saturable nonlinearity, and predictions on their weak interactions of solitary waves are made.

Mechanisms supporting long propagation regimes of photorefractive solitons

We extend investigation of one-dimensional solitons in biased photorefractive crystals to long propagation regimes, where self-trapping over a large number of linear diffraction lengths combines with the progressive growth of generally distortive spatially nonlocal components. Results indicate that saturation halts the radiative misshaping of the soliton, which follows that specific bending trajectory along which its evolution is governed by the same local screening nonlinearity that intervenes in short propagation conditions, where spatial nonlocality has a negligible effect. This finding not only allows the prediction of the curvature and of the relative role of charge displacement and diffusion, but implies a set of interesting observable effects, such as boomerangs, counterpropagating and cavity geometries, quasirectilinear and anomalous collisions, along with specific consequences on soliton arrays and on coupling to bulk gratings.

Interactive dynamics of two copropagating laser beams in underdense plasmas

The interaction of two copropagating laser beams with crossed polarization in the underdense plasmas has been investigated analytically with the variational approach and numerically. The coupled envelope equations of the two beams include both the relativistic mass correction and the ponderomotive force effect. It is found that the relativistic effect always plays the role of beam attraction, while the ponderomotive force can play both the beam attraction and beam repulsion, depending upon the beam diameters and their transverse separation. In certain conditions, the two beam centers oscillate transversely around a propagation axis. In this case, the ponderomotive effect can lead to a higher oscillation frequency than that accounting for the relativistic effect only. The interaction of two beams decreases the threshold power for self-focusing of the single beam. A strong self-trapping beam can channel a weak one.

Collisions between optical spatial solitons propagating in opposite directions

We formulate the theory describing the evolution and interactions between optical spatial solitons that propagate in opposite directions. We show that coherent collisions between counterpropagating solitons give rise to a new focusing mechanism resulting from the interference between the beams, and that interactions between such solitons are insensitive to the relative phase between the beams.

Self-focusing and merging of two copropagating laser beams in underdense plasma

The propagation of two laser beams copropagating in underdense plasma has been studied numerically by solving their coupled envelope equations. It shows that two beams can merge each other, or split into three beams, or propagate with unstable trajectories, depending upon their power and initial beam separation. During the merging process, strong emission of radiation is observed. It also shows that the density cavitation channels due to the transverse ponderomotive force of the beams tend to trap them inside and prevent them from merging each other.

Complexity and regularity of vector-soliton collisions

In this paper, we extensively investigate the collision of vector solitons in the coupled nonlinear Schrödinger equations. First, we show that for collisions of orthogonally polarized and equal-amplitude vector solitons, when the cross-phase modulational coefficient beta is small, a sequence of reflection windows similar to that in the phi(4) model arises. When beta increases, a fractal structure unlike phi(4)'s gradually emerges. But when beta is greater than one, this fractal structure disappears. Analytically, we explain these collision behaviors by a variational model that qualitatively reproduces the main features of these collisions. This variational model helps to establish that these window sequences and fractal structures are caused entirely or partially by a resonance mechanism between the translational motion and width oscillations of vector solitons. Next, we investigate collision dependence on initial polarizations of vector solitons. We discovered a sequence of reflection windows that is phase induced rather than resonance induced. Analytically, we have derived a simple formula for the locations of these phase-induced windows, and this formula agrees well with the numerical data. Last, we discuss collision dependence on relative amplitudes of initial vector solitons. We show that when vector solitons have different amplitudes, the collision structure simplifies. Feasibility of experimental observation of these results is also discussed at the end of the paper.

Interactions between two-dimensional composite vector solitons carrying topological charges

We present a comprehensive study of interactions (collisions) between two-dimensional composite vector solitons carrying topological charges in isotropic saturable nonlinear media. We numerically study interactions between such composite solitons for different regimes of collision angle and report numerous effects which are caused solely by the "spin" (topological charge) carried by the second excited mode. The most intriguing phenomenon we find is the delayed-action interaction between interacting composite solitons carrying opposite spins. In this case, two colliding solitons undergo a fusion process and form a metastable bound state that decays after long propagation distances into two or three new solitons. Another noticeable effect is spin-orbit coupling in which angular momentum is being transferred from "spin" to orbital angular momentum. This phenomenon occurs at angles below the critical angle, including the case when the initial soliton trajectories are in parallel to one another and lie in the same plane. Finally, we report on shape transformation of vortex component into a rotating dipole-mode solitons that occurs at large collision angles, i.e., at angles for which scalar solitons of all types simply go through one another unaffected.

Delayed-action interaction and spin-orbit coupling between solitons

We report on new fundamental phenomena in soliton interactions: delayed-action interaction and "spin"-orbit coupling upon collision between two-dimensional composite solitons carrying topological charges.

Self-bending of the coupled spatial soliton pairs in a photorefractive medium with drift and diffusion nonlinearity

The propagation of two incoherently coupled laser beams (coupled soliton pairs) in the photorefractive crystal with drift and diffusion components of nonlinear response is investigated. By the effective particles method we have shown that not only the well-known Manakov's soliton pairs but also asymmetric pairs can propagate undistorted in photorefractive crystal with diffusion nonlinearity along the parabolic trajectory for the definite relations of propagation constants. We numerically found the exact profiles of the specific multihump soliton solutions that are possible only in the photorefractive medium with nonlocal diffusion response. The stability properties and specific features of pair collisions are analyzed.

Analytical solution for photorefractive screening solitons

We study formation and interaction of one-dimensional screening solitons in a photorefractive medium with sublinear dependence of the photoconductivity on light intensity. We find an exact analytical solution to the corresponding nonlinear Schrodinger equation. We show that these solitons are stable in propagation and their interaction is generic for solitons of saturable nonlinearity. In particular, they may fuse or "give birth" to new solitons upon collision.

Spatial screening solitons as particles

Photorefractive spatial screening solitons are treated as rays using geometrical optics. The ray picture is transformed into a classical mechanics picture, in which solitons move self-consistently as particles in a potential created by the induced change in the refractive index. The Hamiltonian equations of motion are integrated to yield trajectories that agree with the optical center-of-mass trajectories. The motion in the transverse plane is found to be not central and the orbits are not closed, preventing the spiraling of solitons.

Optical Spatial Solitons and Their Interactions: Universality and Diversity

Spatial solitons, beams that do not spread owing to diffraction when they propagate, have been demonstrated to exist by virtue of a variety of nonlinear self-trapping mechanisms. Despite the diversity of these mechanisms, many of the features of soliton interactions and collisions are universal. Spatial solitons exhibit a richness of phenomena not found with temporal solitons in fibers, including effects such as fusion, fission, annihilation, and stable orbiting in three dimensions. Here the current state of knowledge on spatial soliton interactions is reviewed.

Dynamics of formation and interaction of photorefractive screening solitons

An experimental and numerical investigation of the dynamical, time-dependent effects accompanying the formation and interaction of two-dimensional spatial screening solitons in a photorefractive strontium barium niobate crystal is performed. These effects include initial diffraction, collapse to the soliton shape, the oscillation of beam diameters, beam bending, and the rotation, twisting, and turning of soliton pairs. The dynamics of complex spiraling of two incoherent solitons is considered in more detail.