Chirped and chirpfree soliton solutions of generalized nonlinear Schrödinger equation with distributed coefficients

Using the F-expansion method, we systematically present exact solutions of the generalized nonlinear nonlinear Schrödinger equation with varying intermodal dispersion and nonlinear gain or loss. This approach allows us to obtain large variety of solutions in terms of Jacobi-elliptical and Weierstrass-elliptical functions. The chirped and unchirped spatiotemporal soliton solutions and trigonometric-function solutions have been also obtained as limiting cases. The dynamics of these spatiotemporal soliton is discussed in context of optical fiber communication. To visualize the propagation characteristics of chirp and unchirped dark-bright soliton solutions, few numerical simulations are given. It is found that wave profile of solitons depend on the group velocity dispersion and the gain or loss functions.     

Stability of chirped bright and dark soliton-like solutions of the cubic complex Ginzburg-Landau equation with variable coefficients

We consider an inhomogeneous optical fiber system described by the generalized cubic complex Ginzburg-Landau (CGL) equation with varying dispersion, nonlinearity, gain (loss), nonlinear gain (absorption) and the effect of spectral limitation. Exact chirped bright and dark soliton-like solutions of the CGL equation were found by using a suitable ansatz. Furthermore, we analyze the features of the solitons and consider the problem of stability of these soliton-like solutions under finite initial perturbations. It is shown by extensive numerical simulations that both bright and dark soliton-like solutions are stable in an inhomogeneous fiber system. Finally, the interaction between two chirped bright and dark soliton-like pulses is investigated numerically.     

Exact chirped gray soliton solutions of the nonlinear Schrödinger equation with variable coefficients

In this paper, we consider the nonlinear Schrödinger equation with variable coefficients, and by using direct transformation of variables and functions, the explicit chirped gray one- and two-soliton solutions are presented. Based on the exact solutions, we in detail analyze the propagation characteristics of the chirped gray soliton, including the stability against either the deviation from integrable condition or the initial perturbation, and interaction between the chirped gray solitons. The results show that the gray soliton can be compressed by choosing the appropriate initial chirp, and the chirped gray pulses can stably propagate along optical fibers remaining the character of solitons.     

光孤子脉冲在光纤放大器中的传播

本文建立了包括增益色散,受激喇曼散射,双光子吸收效应的理论模型,讨论了光孤子脉冲在光纤放大器中的传播.数值计算结果表明:在反常色散范围内,光孤子的放大是不稳定的.增益色散导致光脉冲对称分裂,而受激喇曼散射则导致不对称的分裂.在增益色散和受激喇曼散射的共同作用下可获得新的时域和频域特征.有限带宽的放大能抑制受激喇曼散射引起的自频移.本文对有啁啾的光孤子脉冲的放大也进行了分析.     

Chirped soliton-like solutions for nonlinear Schrödinger equation with variable coefficients

The exact chirped bright and dark soliton-like solutions of generalized nonlinear Schrödinger equation including linear and nonlinear gain(loss) with variable coefficients describing dispersion-management or soliton control is obtained detailedly in this paper. To begin our numerical studies of the stability of the solutions, we present a periodically distributed dispersion management or soliton control system as an example. It is found that both the bright and dark soliton-like solutions are stable during propagation in the given system. The numerical results are well in accordance with those obtained by analytical methods.     

有限宽度背景中的啁啾灰孤子

基于在正常色散区的变系数非线性薛定谔方程,考虑一个带有微扰的参数渐减光纤系统,并利用数值模拟方法,对超高斯型有限宽度背景波和有限宽度背景中啁啾灰孤子的传输进行详细地研究.结果表明,超高斯背景波可以在带有微扰的参数渐减光纤系统中不受负载啁啾灰孤子的影响而稳定传输.当取背景波脉宽与啁啾孤子的初始脉宽比例大于或等于50时,有限宽度背景中啁啾灰孤子的数值结果基本与精确解相吻合.即使选取的背景波脉宽不宽,有限宽度背景中的啁啾灰脉冲仍可以很好的保持其孤子性质.     

光纤中变系数非线性Schr(o)dinger方程的孤子解及其应用

基于推广的立方非线性Klein—Gordon方程对一般形式的变系数非线性Schrodinger方程进行研究，讨论了无啁啾情形的孤子解，发现了包括亮、暗孤子解和类孤子解在内的一些新的精确解．同时对基本孤子的色散控制方法进行了简单讨论．作为特例，常系数非线性Schrodinger方程和两类特殊的变系数非线性Schrodinger方程的结果和已知的形式一致．此外，还研究了一个周期增益或损耗的光纤系统，得到了有意义的结果．     

Propagation characteristics of chirped soliton in periodic distributed amplification systems with variable coefficients

Propagation characteristics of chirped soliton in periodic distributed amplification systems with variable coefficients are investigated by using the split-step Fourier method. The soliton pulse still maintains soliton characteristics after propagation in the system. The shape of output spectrum is still with a single peak. The temporal FWHM of an unchirped soliton performs a slight oscillation with the increase of propagation distance. The oscillation amplitude is slightly increased with the increase of the gain or loss parameter σ. The chirped soliton is obviously and periodically compressed and broadened with the increase of propagation distance in the system. The lager the value of chirp parameter |C| and σ is, the greater the variation of temporal width with the distance is. The effect of chirp on temporal width of soliton in a amplification system is greater than that in a loss system. Three types of numerical experiments show that the soliton with different perturbations can stably propagate along the fiber. In practical applications, one can obtain a suitable soliton pulse by controlling the parameters of the fiber system.     

Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients

A broad class of exact self-similar solutions to the nonlinear Schr?dinger equation (NLSE) with distributed dispersion, nonlinearity, and gain or loss has been found. Appropriate solitary wave solutions applying to propagation in optical fibers and optical fiber amplifiers with these distributed parameters have also been studied. These solutions exist for physically realistic dispersion and nonlinearity profiles in a fiber with anomalous group velocity dispersion. They correspond either to compressing or spreading solitary pulses which maintain a linear chirp or to chirped oscillatory solutions. The stability of these solutions has been confirmed by numerical simulations of the NLSE.     

Dark solitons in optical fiber with inhomogeneous effects

We study the nonlinear wave propagation in an inhomogeneous optical fiber core in the normal dispersive regime. In order to include the inhomogeneous physical effects, the nonlinear Schrödinger equation (NLSE), which governs the solitary pulse propagation in optical fiber, is modified by adding terms for phase modulation and power gain or loss. The modified NLSEs are bilinearized and exact dark soliton solutions are obtained. The results are discussed.     

亚皮秒啁啾孤子在凸形色散与线性色散平坦光纤中的传输特性比较

提出采用凸形色散平坦光纤(DFF-CVDP)传输亚皮秒啁啾孤子,利用分步傅里叶方法数值研究了亚皮秒啁啾孤子在DFF-CVDP中的传输特性,并与在线性色散平坦光纤(DFF-LDP)中的传输特性做了比较。结果表明,孤子脉冲在上述两种光纤中传输时,光纤损耗导致了孤子脉冲宽度随传输距离增加稍有展宽,脉冲方均根谱宽随传输距离增加逐渐减小。在DFF-LDP中,脉冲展宽更快,正啁啾对脉冲展宽的影响比负啁啾的影响更大,正啁啾对应的方均根光谱比其他情况下的宽得多。啁啾脉冲时域宽度随传输距离增加出现了衰减振荡,振荡周期和振幅随啁啾参量|C|的增加而增大。光孤子脉冲在两种光纤中传输时域波形保持不变,仍具有孤子特性。在DFF-LDP中,3β使得脉冲光谱红移从而诱导了脉冲时延,负啁啾减弱了3β的作用,正啁啾加强了β3的影响。在DFF-CVDP中传输可以忽略3β的影响。     

Impact of slow gain dynamics on soliton molecules in mode-locked fiber lasers

We theoretically demonstrate and experimentally confirm the major influence of gain dynamics on soliton molecules that self-assemble in mode-locked lasers. Both slow gain recovery and depletion play a pivotal role in the formation of chirped soliton molecules characterized by an increasing separation from leading to trailing pulses. These chirped molecules actually consist of many pulses and may be termed macromolecules. They are experimentally observed in a fiber laser and numerically modeled by an approach that properly includes the slow gain dynamics. Furthermore, it is shown that these processes stabilize soliton trains in fiber lasers by inhibiting internal oscillations.     

拉曼增益对孤子传输特性的影响

利用考虑拉曼增益效应的非线性薛定谔方程, 在忽略光纤损耗的情况下, 采用基于MATLAB的分步傅里叶数值算法, 得出线性算符和非线性算符具体的表达式, 分步作用于光孤子脉冲传输方程, 仿真模拟了光孤子在光纤中传输时的演变. 与不考虑拉曼增益的光孤子在光纤中传输相对比, 探析了拉曼增益对孤子传输特性的影响.拉曼增益会破坏孤子的传输周期, 导致孤子在光纤中传输时快速衰减, 并且影响程度和输入孤子的脉冲峰值功率大小有关, 拉曼增益对基态孤子和高阶孤子的影响也不相同. 关键词： 拉曼增益 孤子 对称分步傅里叶法 非线性薛定谔方程     

用推广的李群约化法求解非线性薛定谔方程

用推广的经典李群约化法，得到了色散系数、非线性系数、补偿（或损失）系数为时、空变量函数时的非线性薛定谔方程的精确解.深入研究了非线性薛定谔模型的一般孤波解与线性调频孤波解在光纤通讯与光纤放大器中的潜在应用. 关键词： 李群约化 非线性薛定谔方程 光纤通讯     

Optical parametric chirped pulse amplification based on photonic crystal fibre

A compact two-stage optical parametric chirped pulse amplifier based on photonic crystal fibre is demonstrated.A 1064-nm soliton pulse is obtained in a home-made photonic crystal fibre (PCF) with femtosecond pulse pumping and then amplified to 2 mJ in an Nd:YAG regenerative amplifier.After the amplified pulses pass through the LBO crystal,the 532-nm double-frequency light with an energy of 0.8 mJ and a duration of over 100 ps at 10-Hz repetition rate is generated as a pump source in the following two-stage optical parametric amplification (OPA).The 850-nm chirped signal light gain from the stretcher is 1.5×10 4 in the first-stage OPA while it is 120 in the second-stage OPA.The total signal gain of optical parametric chirped pulse amplification (OPCPA) can reach 1.8×10 6.     

Abundant exact solutions for a strong dispersion-managed system equation

The generalized nonlinear Schr?dinger equation (NLSE), which governs the dynamics of dispersion-managed (DM) solitons, is considered. A novel transformation is constructed such that the DM fibre system equation with optical loss (gain) is transformed to the standard NLSE under a restricted condition. Abundant new soliton and periodic wave solutions are obtained by using the transformation and the solutions of standard NLSE. Further, we discuss their main properties and the interaction scenario between two neighbouring solitons by using direct computer simulation.     

拉曼增益对双折射光纤中孤子传输特性的影响

本文采用考虑拉曼增益的耦合非线性薛定谔方程,利用分步傅里叶方法求解并仿真模拟了光孤子脉冲在不同性质的双折射光纤中传输时的演化过程.结果表明,拉曼增益可以有效抑制非线性耦合导致的孤子漂移,同时会导致光孤子脉冲峰值在传输时不断增大,产生拉曼放大效应.拉曼增益也可以有效抑制双折射光纤中传输的相邻光孤子之间的相互作用.     

Nonlinear compression of optical solitons

In this paper, we consider nonlinear Schrödinger (NLS) equations, both in the anomalous and normal dispersive regimes, which govern the propagation of a single field in a fiber medium with phase modulation and fibre gain (or loss). The integrability conditions are arrived from linear eigen value problem. The variable transformations which connect the integrable form of modified NLS equations are presented. We succeed in Hirota bilinearzing the equations and on solving, exact bright and dark soliton solutions are obtained. From the results, we show that the soliton is alive, i.e. pulse area can be conserved by the inclusion of gain (or loss) and phase modulation effects.     

复变系数Ginzburg-Landau方程的啁啾组合孤波解

基于描述非均匀光纤系统的复系数Ginzburg-Landau方程,通过拟解法得到了该方程的精确啁啾组合孤波解,并分析了该解的特性.通过大量的数值模拟,发现在有限的初始扰动下这些组合孤波解是在非均匀光纤系统中稳定的.最后,为了进一步研究组合孤波解的稳定性,我们还探讨了组合孤波的相互作用.     

高斯脉冲在半导体光放大器中传输的解析表征

采用解析方法,在考虑材料损耗和色散的情况下,详细研究了无啁啾高斯脉冲和啁啾高斯脉冲在半导体光放大器中传输的物理过程,分析了强度增益、脉冲宽度和频率啁啾与线宽增强因子、色散系数、小信号增益特征参数及初始啁啾之间的关系。结果表明:当输入变换极限的高斯脉冲时,色散会引起增益压缩,脉冲展宽和频率啁啾;同样情况下,线宽增强因子越大,脉宽加宽越明显,输出脉冲啁啾越大,且随着线宽增强因子的增大,输出脉冲啁啾极大值向特征参数值较小的一边移动。当输入啁啾高斯脉冲时,初始脉冲啁啾越大,增益压缩越明显,啁啾系数为正时,脉冲单纯展宽,输出啁啾随特征参数的增大而逐渐减小,啁啾系数为负时,初始啁啾与群速度色散导致的啁啾相互竞争,致使脉冲先被压缩后被展宽;脉冲最窄处对应的特征参数随线宽增强因子的增大而先增大后减小,输出啁啾随特征参数的增大而经历振荡后趋于平稳。