Families of Bragg-grating solitons in a cubic–quintic medium

We investigate the existence and stability of solitons in an optical waveguide equipped with a Bragg grating (BG) in which nonlinearity contains both cubic and quintic terms. The model has straightforward realizations in both temporal and spatial domains, the latter being most realistic. Two different families of zero-velocity solitons, which are separated by a border at which solitons do not exist, are found in an exact analytical form. One family may be regarded as a generalization of the usual BG solitons supported by the cubic nonlinearity, while the other family, dominated by the quintic nonlinearity, includes novel “two-tier” solitons with a sharp (but nonsingular) peak. These soliton families also differ in the parities of their real and imaginary parts. A stability region is identified within each family by means of direct numerical simulations. The addition of the quintic term to the model makes the solitons very robust: simulating evolution of a strongly deformed pulse, we find that a larger part of its energy is retained in the process of its evolution into a soliton shape, only a small share of the energy being lost into radiation, which is opposite to what occurs in the usual BG model with cubic nonlinearity.     

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.     

Numerical analysis of the stability of optical bullets (2 + 1) in a planar waveguide with cubic–quintic nonlinearity

In this paper, we have presented a numerical analysis of the stability of optical bullets (2 + 1), or spatiotemporal solitons (2 + 1), in a planar waveguide with cubic–quintic nonlinearity. The optical spatiotemporal solitons are the result of the balance between the nonlinear parameters, of dispersion (dispersion length, L _D) and diffraction (diffraction length, L _d) with temporal and spatial auto-focusing behavior, respectively. With the objective of ensure the stability and preventing the collapse or the spreading of pulses, in this study we explore the cubic–quintic nonlinearity with the optical fields coupled by cross-phase modulation and considering several values for the non linear parameter α We have shown the existence of stable light bullets in planar waveguide with cubic–quintic nonlinearity through the study of spatiotemporal collisions of the light bullets.     

Dissipative solitons in active Bragg gratings

Motionless and moving bright dissipative solitons in an optical fiber with a Bragg grating and non-linear amplification and absorption are found numerically and studied. These solitons form a one-parameter family whose parameter—the soliton velocity—can continuously change in a certain range. The presence of saturation of the nonlinearity is necessary for stability of such solitons. Neglect of saturation of the cubic-in-field polarization of the medium results in the instability of the possible localized structures.     

Optical solitons in fiber Bragg gratings having Kerr law of refractive index with extended Kudryashov’s method and new extended auxiliary equation approach

This paper reveals optical solitons and other solutions to fiber Bragg gratings with dispersive reflectivity having Kerr law of nonlinear refractive index. Bragg gratings are indeed a technological marvel that supplements chromatic dispersion when its count runs low. The extended Kudryashov’s method and new extended auxiliary equation method have been implemented. Chirped and chirp–free bright, dark and singular solitons, with dispersive reflectivity, are presented.     

Exact vortex solitons in a quasi-two-dimensional Bose–Einstein condensate with spatially inhomogeneous cubic–quintic nonlinearity

The exact vortex soliton solutions of the quasi-two-dimensional cubic–quintic Gross–Pitaevskii equation with spatially inhomogeneous nonlinearities are constructed by similarity transformation. It is demonstrated that spatially inhomogeneous cubic–quintic nonlinearity can support exact vortex solitons in which there are two quantum numbers S and m. The radius structures and density distributions of these vortex solitons are studied, and it is shown that the number of ring structure of the vortex solitons increases by one with increasing the “radial quantum number” m by one.     

Stable high-dimensional solitons in nonlocal competing cubic-quintic nonlinear media

We find and stabilize high-dimensional dipole and quadrupole solitons in nonlocal competing cubic-quintic nonlinear media. By adjusting the propagation constant, cubic, and quintic nonlinear coefficients, the stable intervals for dipole and quadrupole solitons that are parallel to the x-axis and those after rotating 45° counterclockwise around the origin of coordinate are found. For the dipole solitons and those after rotation, their stability is controlled by the propagation constant, the coefficients of cubic and quintic nonlinearity. The stability of quadrupole solitons is controlled by the propagation constant and the coefficient of cubic nonlinearity, rather than the coefficient of quintic nonlinearity, though there is a small effect of the quintic nonlinear coefficient on the stability. Our proposal may provide a way to generate and stabilize some novel high-dimensional nonlinear modes in a nonlocal system.     

Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices

The nonlinear lattice — a new and nonlinear class of periodic potentials — was recently introduced to generate various nonlinear localized modes. Several attempts failed to stabilize two-dimensional (2D) solitons against their intrinsic critical collapse in Kerr media. Here, we provide a possibility for supporting 2D matter-wave solitons and vortices in an extended setting — the cubic and quintic model — by introducing another nonlinear lattice whose period is controllable and can be different from its cubic counterpart, to its quintic nonlinearity, therefore making a fully “nonlinear quasi-crystal”.A variational approximation based on Gaussian ansatz is developed for the fundamental solitons and in particular, their stability exactly follows the inverted Vakhitov–Kolokolov stability criterion, whereas the vortex solitons are only studied by means of numerical methods. Stability regions for two types of localized mode — the fundamental and vortex solitons — are provided. A noteworthy feature of the localized solutions is that the vortex solitons are stable only when the period of the quintic nonlinear lattice is the same as the cubic one or when the quintic nonlinearity is constant, while the stable fundamental solitons can be created under looser conditions. Our physical setting (cubic-quintic model) is in the framework of the Gross–Pitaevskii equation or nonlinear Schrödinger equation, the predicted localized modes thus may be implemented in Bose–Einstein condensates and nonlinear optical media with tunable cubic and quintic nonlinearities.     

Stable two-dimensional solitons supported by radially inhomogeneous self-focusing nonlinearity

We demonstrate that modulation of the local strength of the cubic self-focusing (SF) nonlinearity in the two-dimensional geometry, in the form of a circle with contrast Δg of the SF coefficient relative to the ambient medium with a weaker nonlinearity, stabilizes a family of fundamental solitons against the critical collapse. The result is obtained in an analytical form, using the variational approximation and Vakhitov-Kolokolov stability criterion, and corroborated by numerical computations. For the small contrast, the stability interval of the soliton's norm scales as ΔN~Δg (the replacement of the circle by an annulus leads to a reduction of the stability region by perturbations breaking the axial symmetry). To further illustrate this mechanism, we demonstrate, in an exact form, the stabilization of one-dimensional solitons against the critical collapse under the action of a locally enhanced quintic SF nonlinearity.     

Influence of uniaxial crystal material cladding on reflectivity and dispersion of uniform fiber Bragg grating

The fiber Bragg grating with cladding made of uniaxial crystal material whose optical axis is parallel to the axis of fiber Bragg grating proposed in the paper published in 2003 was investigated again and an error was corrected in the calculation, its effective index, reflectivity and dispersion were examined using coupled-mode theory and numeric solution. The calculated results indicate that no low frequency cutoff phenomenon exists in the HE₁₁ mode, more power is transmitted by the core of the fiber with cladding made of isotropic material, the reflectivity of the fiber Bragg grating with cladding made of uniaxial crystal material is much higher than that with cladding made of isotropic material, the parameter K_cl, i.e., the ratio of the extraordinary to the ordinary ray refractive index, has a stronger impact on the reflectivity, Bragg wavelength and the dispersion of this kind of fiber Bragg grating when it varied from 1.00 to 1.01 than in other regions. This means that the characteristics of the fiber Bragg grating with uniaxial crystal cladding can be changed through adjusting K_cl while keeping its length, periodicity and the other parameters as constants.     

Stable helical solitons in optical media

We present a review of new results which suggest the existence of fully stable spinning solitons (self-supporting localised objects with an internal vorticity) in optical fibres with self-focusing Kerr (cubic) nonlinearity, and in bulk media featuring a combination of the cubic self-defocusing and quadratic nonlinearities. Their distinctive difference from other optical solitons with an internal vorticity, which were recently studied in various optical media, theoretically and also experimentally, is that all the spinning solitons considered thus far have been found to be unstable against azimuthal perturbations. In the first part of the paper, we consider solitons in a nonlinear optical fibre in a region of parameters where the fibre carries exactly two distinct modes, viz., the fundamental one and the first-order helical mode. From the viewpoint of application to communication systems, this opens the way to doubling the number of channels carried by a fibre. Besides that, these solitons are objects of fundamental interest. To fully examine their stability, it is crucially important to consider collisions between them, and their collisions with fundamental solitons, in (ordinary or hollow) optical fibres. We introduce a system of coupled nonlinear Schrödinger equations for the fundamental and helical modes with nonstandard values of the cross-phase-modulation coupling constants, and show, in analytical and numerical forms, results of collisions between solitons carried by the two modes. In the second part of the paper, we demonstrate that the interaction of the fundamental beam with its second harmonic in bulk media, in the presence of self-defocusing Kerr nonlinearity, gives rise to the first ever example of completely stable spatial ring-shaped solitons with intrinsic vorticity. The stability is demonstrated both by direct simulations and by analysis of linearized equations.     

光纤布拉格光栅中隙孤子的EPP方法分析

提出了利用EPP方法分析光纤布拉格光栅中隙孤子的解。基于非线性耦合模式方程(NLCME)定性地分析了无微扰条件下的隙孤子参数与孤子的其它特性的关系。利用EPP方法分析了隙孤子的能量特性。证明了隙孤子的速度影响形态特性和能量分布。从理论上解释了已观察到的一系列隙孤子的试验现象,对光纤布拉格光栅中产生隙孤子的应用具有理论意义。     

Ultraslow solitons due to large quintic nonlinearity in coupled quantum well structures driven by two control laser beams

In this paper, we have shown the existence of large amounts of quintic nonlinearity in asymmetric three-coupled quantum wells, which arise due to a probe pulse and two controlling laser beams. The possibilities of generation and propagation of ultraslow bright optical solitons in these systems have been examined in situations of both Kerr and quintic nonlinearities. We have also demonstrated numerically that these solitons are stable. The modulation instability of a continuous or quasi-continuous wave probe beam has been also investigated and the role of quintic nonlinearity in suppressing this instability has been addressed.     

光纤布拉格光栅中的隙孤子存在条件

提出光纤布拉格光栅中产生隙孤子的条件和参量制约关系。利用非线性耦合模式方程建立光纤布拉格光栅中孤子的传播方程,通过扰动方法建立了参量的微分方程,计算得到参量近似解。以周期非线性光学介质中隙孤子存在的条件为依据,数学计算分析得到两组参量关系不等式。最终通过数值计算说明了这些参量之间存在制约关系和物理意义。从而理论上说明了在光纤布拉格光栅中隙孤子存在需要选择适当参量。为光纤布拉格光栅中产生隙孤子的实验和进一步的工程应用提供了理论基础。     

暗孤子在高斯变迹光纤光栅中的演化

运用快速傅里叶方法数值模拟了暗孤子在高斯变迹光纤光栅中的演化以及暗孤子之间的相互作用。结果表明,在一定条件下光纤光栅的正常色散区可以维持暗孤子的稳定传输,并且孤子的稳定性与f的取值(孤子离禁带的位置)和非线性系数有关。输入两个暗孤子时,孤子之间的相互作用随f的增大而加强,当f继续增大时孤子中心和背景处出现震荡现象并演化产生灰孤子;f越大,两个孤子的稳定性对非线性效应越敏感。     

Counterpropagating spatial solitons in reflection gratings with a longitudinally modulated Kerr nonlinearity

We investigate quasi-Bragg-matched counterpropagating spatial solitons in a reflection grating in the presence of a longitudinally modulated Kerr nonlinearity. The physical interplay of linear reflection and Kerr self-focusing with the modulation in the nonlinearity yields a variety of elaborate self-action mechanisms. We first analytically predict the existence of symmetric soliton pairs supported by a pure Kerr-like effective nonlinearity. We then analytically derive two families of solitons, associated with the linear grating eigenmodes, supported by an effective "incoherent" Kerr-like coupling arising from the exact balance between the modulation in the nonlinearity and the Kerr interaction due to beam interference.     

Effect of finite relaxation rates on the stability of dissipative fiber Bragg solitons

We present a study of the effect of finite relaxation rates of media on the stability of dissipative solitons in single-mode fibers with longitudinal Bragg grating and nonlinear gain created by one active medium, and absorption created by one passive medium that gives the stability to solitons. The system of Maxwell–Bloch equations for this model is introduced. Based on the analysis of this system, it is shown that dissipative Bragg solitons are stable in the case when the active medium has greater relaxation rates than the passive medium.     

A novel dual-wavelength fiber Bragg grating and its application in fiber ring laser

A novel method for fabricating dual-wavelength fiber Bragg gratings (FBGs) by using one phase mask is developed. The method is based on a double-exposure technique. Our technique lends itself to writing gratings with controllable reflectivity and separation of two Bragg wavelengths. A grating with two equal transmission peaks of 20.25 dB is obtained by this method and the separation of the two Bragg wavelengths is about 0.8 nm. With the grating, we demonstrate a dual-wavelength erbium-doped fiber ring laser whose interval of the two peaks is 0.8 nm. The laser's peak powers can get 3.1 mW above and have a good stability.