Conservative and dissipative fiber Bragg solitons (a review)

We present a comparative review of two classes of optical solitons—conservative and dissipative solitons—propagating in single-mode optical fibers in which refractive-index gratings are induced such that their period is comparable with the radiation wavelength. Fibers that have the Kerr nonlinearity and negligibly small losses and that do not gain radiation (conservative system) are described by traditional equations of the approximation of slowly varying amplitudes, and effects caused by the nonlinearity of the medium, such as nonlinear switching, optical bistability, and formation of conservative Bragg solitons are considered. It is shown that the passage beyond the scope of the approximation of slowly varying amplitudes makes it possible to describe new important effects, including localization of soliton centers near maxima of the refractive-index grating. Bright and dark conservative solitons are demonstrated, which are formed when the Kerr nonlinearity is replaced by the nonlinearity of two-level atomic systems. The properties of conservative solitons in resonance semiconductor Bragg structures with quantum wells are considered. Results of experimental studies of nonlinear effects in fibers with Bragg gratings are presented. For an active single-mode fiber with a Bragg refractive-index grating and nonlinear gain and absorption, dissipative solitons are described using the approximation of slowly varying amplitudes and inertialess nonlinearity. It is shown that the dissipative factors qualitatively change the properties of solitons compared to the conservative case. Using the Maxwell-Bloch equations, it is demonstrated that the ratio between the gain and absorption relaxation times significantly affects the stability of localized structures. The interaction of dissipative optical Bragg solitons is described. It is shown that, beyond the scope of the approximation of slowly varying amplitudes, the average velocity of propagating dissipative Bragg solitons acquires only discrete values, and formation of pairs of solitons with two values of the phase difference becomes possible. For a birefringent fiber, dissipative vector optical Bragg solitons are demonstrated.     

Regular and stochastic motion of dissipative optical solitons

Investigations of the motion of dissipative optical solitons and their complexes in wide-aperture nonlinearly optical (with coherent pump radiation) and laser (with incoherent pump radiation) systems have been reviewed. An important characteristic of dissipative solitons is the topology of the energy fluxes, which determines the internal structure of individual solitons and makes it possible to certainly separate the cases of the weak and strong interactions between the solitons. It has been shown that the character of the regular motion of dissipative soliton structures in a homogeneous system is determined by the symmetry of the transverse distributions of the intensity and energy flux; the motion of asymmetric structures is curvilinear. This is also valid for complexes of three-dimensional dissipative optical solitons, “laser bullets.” The extreme possibilities of localization of solitons are determined by quantum noises. The corresponding Brownian motion of the center of the dissipative optical soliton is characterized by a much lower level of the statistic dispersion of the coordinates of its center and velocity than that in the case of conservative solitons.     

Midband dissipative spatial solitons

We show dissipative spatial solitons in nonlinear optical microresonators in which the refractive index is laterally modulated. In addition to "normal" and "staggered" dissipative solitons, similar to those in spatially modulated conservative systems, a narrow "midband" soliton is shown, having no counterparts in conservative systems.     

Bragg optical solitons beyond the coupled mode approximation

Properties of conservative and dissipative Bragg solitons formed in single-mode optical fibers with induced longitudinal modulation of the refractive index are analyzed beyond the standard approximation of coupled modes, or of slowly varying amplitudes. It is shown that, if the initial velocity of a Bragg soliton is smaller than a critical value, the soliton stops in the process of propagation.     

Gradient induced motion control of drifting solitary structures in a nonlinear optical single feedback experiment

We realize an absolute position control of drifting dissipative optical solitons by injecting an incoherent amplitude parameter gradient onto the nonlinear optical system. This allows for two-dimensional, arbitrary control patterns. The control of the soliton drift velocity is studied applying a periodic, hexagonally shaped modulation. The guiding of dissipative solitons by one- and two-dimensional parameter modulations is demonstrated. Furthermore, one-dimensional, line-shaped parameter modulations are designed to act as barriers for dissipative solitons, allowing implementations of position selectors for solitons. The interaction of dissipative optical solitons with barriers is studied for different barrier parameters.     

Dissipative discrete spatial optical solitons in a system of coupled optical fibers with the Kerr and resonance nonlinearities

The propagation of monochromatic radiation in a system of weakly coupled single-mode optical fibers with saturable amplification and absorption and Kerr nonlinearity of the refractive index is analyzed. Conditions of stability and bistability of plane-wave regimes are determined. Discrete dissipative optical solitons are found and their stability is studied. The hysteresis dependences of the peak intensity of the discrete solitons on the value of the Kerr nonlinearity and the input beam intensity are demonstrated. The numerical estimates of the parameters of the spatial dissipative discrete solitons are presented.     

Dissipative optical solitons in the absence of bistability and modulation instability

Transversely one-dimensional localized field structures in a wide-aperture interferometer with inertialess nonlinearity of a threshold form are analyzed theoretically for the case when modulation instability of the transversely homogeneous field distributions is absent. It is shown that dissipative solitons exist even in the case of monostability (in the absence of bistability of the homogeneous distributions). In this case, the number of types of solitons turns out to be finite. The possibility of existence of pulsating solitons in an interferometer with inertialess nonlinearity is indicated.     

Observation of a discrete family of dissipative solitons in a nonlinear optical system

We report on the observation of a discrete family of spatial dissipative solitons in a simple optical pattern forming system, which is based on a modified single-mirror feedback arrangement. After a pitchfork bifurcation the system possesses two (nearly) equivalent coexisting states of different polarizations. The spatial solitons correspond to excursions from one of the two states serving as a background state towards the other one. The members of the soliton family differ in the number of high-amplitude radial oscillations. The observations are in good agreement with numerical simulations and general expectations for dissipative solitons.     

Discrete solitons in inhomogeneous waveguide arrays

The existence and dynamical properties of discrete solitons in inhomogeneous waveguide arrays with a Kerr nonlinearity are studied in two different configurations. First we investigate the effect of a longitudinal periodic modulation of the coupling strength on the dynamics of discrete solitons. It is shown that resonances of internal modes of the soliton with the longitudinal structure may lead to soliton oscillations and decay. Second we study the existence and stability of discrete solitons in arrays exhibiting a linear variation of the waveguide effective index in the transverse direction. We find that resonant coupling between conventional discrete solitons and linear Wannier-Stark states leads to the formation of so-called hybrid discrete solitons.     

Dissipative photonic lattice solitons

We show that discrete dissipative optical lattice solitons are possible in waveguide array configurations that involve periodically patterned semiconductor optical amplifiers and saturable absorbers. The characteristics of these low-power soliton states are investigated, and their propagation constant eigenvalues are mapped on Floquet-Bloch band diagrams. The prospect of observing such low-power dissipative lattice solitons is discussed in detail.     

Three-dimensional dissipative optical solitons

A brief overview of recent theoretical results in the area of three-dimensional dissipative optical solitons is given. A systematic analysis demonstrates the existence and stability of both fundamental (spinless) and spinning three-dimensional dissipative solitons in both normal and anomalous group-velocity regimes. Direct numerical simulations of the evolution of stationary solitons of the three-dimensional cubic-quintic Ginzburg-Landau equation show full agreement with the predictions based on computation of the instability eigenvalues from the linearized equations for small perturbations. It is shown that the diffusivity in the transverse plane is necessary for the stability of vortex solitons against azimuthal perturbations, while fundamental (zero-vorticity) solitons may be stable in the absence of diffusivity. It has also been found that, at values of the nonlinear gain above the upper border of the soliton existence domain, the three-dimensional dissipative solitons either develop intrinsic pulsations or start to expand in the temporal (longitudinal) direction keeping their structure in the transverse spatial plane. Presented at 9-th International Workshop on Nonlinear Optics Applications, NOA 2007, May 17–20, 2007, Świnoujście, Poland     

Curvilinear motion of laser soliton complexes

Conditions under which the center of inertia of stable complexes of dissipative optical solitons moves curvilinearly have been determined. Such a character of the motion of dissipative structures is caused by asymmetry in the distribution of the intensity and energy fluxes, and it is pronounced for laser solitons with strong interaction. The results of the numerical simulation of these complexes in the model of surface emitting lasers or laser amplifiers with saturated amplification and absorption are presented. Such complexes may be observed in various spatially distributed nonlinear dissipative systems, in particular, in the form of discrete solitons.     

All-fiber erbium-doped dissipative soliton laser with multimode interference based on saturable-reserve saturable hybrid optical switch

Reverse saturable absorption is essential for the realization of dissipative solitons. In this paper, we introduce reverse saturable absorption by using nonlinear multimode interference (NL-MMI), for the first time, to the best of our knowledge, and obtain a stable dissipative soliton operation. By adjusting the coupling efficiency from multimode fiber to single mode fiber, the absorption properties of NL-MMI can be switched between saturation and reverse saturation. The dissipative soliton can be obtained with pulse width of 975 fs in the experiment, the 3-dB bandwidth at 1555 nm is 16 nm, and the maximum output power is 11.48 mW. The nonlinear absorption optical modulation and high damage threshold characteristics of the NL-MMI based ultrafast optical switch provide a new idea for realizing dissipative solitons.     

Dissipative vector Bragg solitons in a birefringent optical fiber

Motionless dissipative vector solitons are found numerically in a birefringent polarization-conserving optical fiber with a longitudinal Bragg grating and nonlinear amplification and absorption in the polarization capture regime. It is shown that dissipative vector Bragg solitons exist and are stable even if the birefringence is comparatively strong.     

Collision Dynamics of Dissipative Matter-Wave Solitons in a Perturbed Optical Lattice

We investigate the stability and collision dynamics of dissipative matter-wave solitons formed in a quasi-onedimensional Bose-Einstein condensate with linear gain and three-body recombination loss perturbed by a weak optical lattice.It is shown that the linear gain can modify the stability of the single dissipative soliton moving in the optical lattice.The collision dynamics of two individual dissipative matter-wave solitons explicitly depend on the linear gain parameter,and they display different dynamical behaviors in both the in-phase and out-of-phase interaction regimes.     

Asymmetric moving localized structures in a wide-aperture nonlinear interferometer

The existence of a branch of asymmetric solitary localized radiation structures moving in the transverse direction with a constant velocity is demonstrated for the slit (transversely one-dimensional) scheme of a wide-aperture interferometer with a threshold nonlinearity which is excited by external coherent radiation. This branch is shown to split off the branch of symmetric motionless structures at a certain bifurcation value of the external-radiation intensity. The generalization to other schemes, for which the existence of dissipative optical solitons is known, is discussed.     

Dissipative solitons of the Bose-Einstein condensate of excitons

A mirrorless scheme for forming dissipative solitons has been proposed. It consists of a thin layer of a nonlinear medium excited by coherent laser radiation. The existence of dark dissipative solitons of the Bose-Einstein condensate of excitons in such a semiconductor film has been numerically demonstrated. The sizes of an excition soliton and required laser-radiation level are two orders of magnitude smaller than the respective quantities for the case of optical dissipative solitons.     

Complex Dynamics of Optical Solitons Interacting with Nanoparticles

JETP Letters - The mutual interactions of the dissipative optical solitons with trapped nanoparticles are studied. It is shown that the attraction of the nanoparticles to the solitons can result in...     

The evolution of two-frequency solitons in an optical fiber with a longitudinally nonuniform nonlinearity

The evolution of two-frequency solitons in an optical fiber, as well as the practically important special case of absence of the second-harmonic wave, in the presence of a longitudinal nonuniformity of the coefficients characterizing the propagation nonlinearity are considered. The solitons found for media with constant values of the nonlinearity coefficients are used as initial distributions for media with a periodic dependence of the nonlinearity coefficients on the longitudinal coordinate. Modulation of the coefficient of cubic or quadratic nonlinearity is shown to result in oscillations of the peak intensity of the solitons (in both their components if two-color solitons are considered). In the case of a weak modulation of the nonlinearity coefficients, oscillations of the peak intensity occur at the frequency coinciding with the frequency of modulation of the nonlinearity coefficients. Under the weak influence of a periodically modulated cubic nonlinearity, parameters of quadratic solitons also oscillate upon the propagation. Regions of stability of solitons in the space of the modulation parameters are established.     

Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling

We reveal the existence of stable dissipative soliton complexes with curvilinear motion of their center of mass. This type of motion results from the field distribution asymmetry and is well pronounced for asymmetric complexes of laser solitons with strong coupling. We present results of numerical simulations of such complexes in a model of wide-aperture lasers or laser amplifiers with saturable gain and absorption. The complex consists of a pair of strongly coupled vortex solitons weakly coupled with a number of other vortex solitons. Similar complexes are expected to exist in different spatially distributed nonlinear dissipative systems, including schemes with discrete dissipative solitons.