光纤孤子传播的自相似性和通讯限制的研究
|
摘要
大千世界,孤子万千。自从1834年J.S. Russell首次在浅水波中观察到孤子现象以来,人们相继在物理、化学、生物领域上发现大量的孤子,它们均传播不发散、不破裂。即使是绝对真空,人们也证实其存在电磁波孤子。但是,最能体现孤子多样性的领域是光学。绝大多数研究学者都赞成光孤子研究是所有孤子科学中的前沿课题。为什么会这样?一个重要的原因是过去十多年里光通讯技术的快速发展。这段时间中,来自高容量光通讯和全光信息处理的急速膨胀需求导致了非线性光学研究上的大规模投资。另外一个原因在于目前许多必要的技术如强大但价廉的激光器、方便获得的超快光学检测和取样技术、快速发展的材料制造科学均己经成熟。并且,光学之美在于它允许我们去直接研究繁多的高非线性效应、洞彻所涉及物理的每一个细节、分离潜在的基本效应。显然,这些也是我们选择光孤子特别是光纤孤子作为我们研究课题的主要原因。
广义上讲,光孤子是自捕获的局域波包(光束或脉冲),它们在色散介质中传播时不展宽(注意,当代非线性光学术语把所有的自捕获光束或脉冲统称为“孤子”)。它们的存在是由于倾向展宽波包的色散与倾向压缩波包的非线性之间的精确平衡。对于光纤孤子,这种平衡体现在光纤波导折射率的波长相关和强度相关之间的均衡。光纤孤子的粒子特性或强稳定性使它们不仅具有基础研究兴趣,而且可用于长距离光纤通讯。这两个领域均吸引大量的研究人员。最近,基础研究兴趣已经转向变化色散、非线性、和增益光纤中的自相似光波传播(不限于孤子)。同时在应用方面,人们致力于研究放大自发辐射噪声或其他噪声所产生的、影响光通讯性能的物理限制。因此,在本论文中,我们的主要关切是研究参数变化系统中的自相似光波传播、超短光孤子的定时抖动(利用我们发展的孤子微扰模型)、和随意非线性和色散管理孤子的相抖动(利用变分法或矩值法)。这些工作概括如下。
1.自相似光波传播和线性稳定性分析
首先,我们系统地研究参数变化系统中孤子、连续波、cnoidal波、抛物自相似子、厄密高斯(HG)脉冲、和混合脉冲的自相似传播。通过对系数可变NLS方程的解析研究和数值模拟,我们发现,这些自相似孤子或脉冲具有许多值得关注的演化特性。举几个例子,在这些参数变化系统中,抛物自相似子的脉冲演化只与脉冲输入能量有关,与输入脉冲形状无关,但是其他脉冲的自相似演化均受它们的初始形状影响。此外,由于连续增强的线性啁啾,抛物自相似子是单调展宽的,但是孤子、HG、和混合脉冲的情形则不然。在合适的初始条件下,这三种脉冲会遭受到初始的脉冲压缩过程,只要GVD感应的啁啾与初始啁啾符号相反。我们也在论文中详细讨论了孤子、HG、和混合脉冲之间的区别。尽管这些,上述自相似脉冲也具有许多共同特征,如自相似的脉冲外形和严格线性的脉冲啁啾。利用这些特性和合适的参数条件,人们可以很容易地从光纤、放大器、或者吸收介质中获取严格确定的线性啁啾输出脉冲。
我们接着对这些结果进行推广,在原来的NLS方程模型中增加非线性增益效应。利用相同的对称约化技术,我们可以获得该模型的精确啁啾自相似孤子解。通过添加高斯白噪声和输入高斯脉冲,我们数值证明了这类孤子的稳定性。我们同时演示,这些啁啾孤子的脉冲位置可以通过裁剪色散轮廓图来精确控制,并且,只要非线性啁啾参数选择适当,sech型孤子可以稳定存在于反常、正常、和其他管理的色散区。
在结束这个话题时,我们利用线性稳定性分析方法研究了自相似孤子的稳定性问题,从而揭示了孤子稳定性增强的可能性(利用色散管理技术)。此外,自相似脉冲(特别是抛物自相似子)的实验实现也得到了详细的回顾。
2.新颖孤子微扰模型和定时抖动的计算
众所周知,除了光学损耗或耗散,定时抖动是限制强度调制孤子通讯总传输距离的关键因素。为了找出定时抖动的解析表达式,许多模型如绝热微扰理论、线性化方法、变分法等相继提出。本质上,它们均是基于对标准NLS方程的小微扰,在足够精度范围内,它们适合用于阐述一些表象效应。但是,在亚皮秒━飞秒区,由于高阶效应的加入,这些模型在一定程度上不能高效、准确地描述孤子噪声效应。
因此,我们提出一个扩展的孤子微扰模型,其实质是基于高阶NLS方程的小微扰。利用该模型,我们相继发展了绝热微扰理论和线性化方法,从而得到孤子参数的动力学方程。根据这些演化方程并充分考虑三阶色散、自陡、和受激Raman散射所产生的自频移效应,我们解析计算飞秒亮暗孤子的定时抖动,所得到的结果与数值模拟完全吻合。我们指出,暗孤子的定时抖动大约只有亮孤子的一半,并且,在亚皮秒━飞秒区,Raman抖动通常远大于Gordon-Haus抖动。更有趣的是,我们发现在一定参数条件下,高阶效应对定时抖动的影响可以相互抵消,从而理论上允许获得较小的定时抖动。可以期望,这些解析结果将比基于NLS方程的著名微扰理论所得到的结果具有更广泛的应用。
3.非线性和色散管理孤子的相抖动
鉴于相调制光纤通讯上的实际应用,随意非线性和色散管理孤子的相波动也在本文中得到详细的解析和数值探讨。利用变分法或矩值法,我们首次得到该复杂孤子系统相抖动的精确解析表达式。值得注意的是,在我们的推导过程中,啁啾波动效应连同色散和非线性效应均得到充分的考虑。我们指出,在非线性相噪声的控制上,不管输入孤子啁啾化与否,啁啾波动效应均通过纤维色散扮演着重要的角色。更重要地,我们从该表达式看出,在给定噪声强度和输入孤子参数情况下,相方差可由系统累积色散、色散━非线性之比完全确定。很显然,该成果会对最近非线性相噪声控制的有趣话题产生积极的影响,同时能提供有效途径来减轻系统的非线性和色散破坏。
接着,我们数值例证了以上解析结果。在第一个演示中,我们考虑熟悉的常色散和非线性系统。它不仅准确再现一些著名的结论,如相方差的立方增长,相方差与脉宽的三次方成反比等等,而且更新我们的观点,即残余频移除了产生大的定时抖动外,同时会产生不可忽略的非线性相噪声。我们的解析结果也被应用到另一个实验可实现的孤子系统,其色散随距离指数递增(递减),但非线性保持常数。显然,通过选择合适的啁啾参数,这个特殊的孤子系统允许快速的脉冲增宽(压缩),而且不辐射任何色散波。我们看出,非线性相噪声在脉冲展宽过程中(不考虑初始频移效应时)会得到极大的抑制,但是在脉冲压缩时则会急剧的放大。在最后的演示中,我们考虑理论上感兴趣的余弦变化孤子系统。我们可清晰地看出,在该管理系统中,不管孤子的初始频移存在与否,非线性相噪声均表现为近线性增长。总之,我们的解析结果适用广义NLS方程描述的随意管理孤子系统,也必将在光纤通讯和光信息处理等领域上找到潜在的应用。
4.随机分步Fourier方法
在本文最后部分,我们发展了一个强大、高效的随机广义NLS方程的模拟算法。该算法是基于常规分步Fourier方法之上,同时综合考虑了乘法和加法噪声的作用。利用该算法,我们已经数值计算孤子传播的定时抖动和相抖动,而且所获得的数值结果与理论预测完全一致。微扰作用下的脉冲演化也可以通过该算法进行数值模拟。
Solitons are ubiquitous. Since the first observation of soliton by J.S. Russell in the water of a shallow canal in 1834, numerous examples of solitons have been found in various physical, chemical, and biological environments, all propagating without spreading out or breaking up. Solitons of electromagnetic waves were even identified in an absolute vacuum. However, the large variety of soliton manifestations is in optics. Most researchers agree that optical solitons are at the forefront of soliton research in all the branches of science in which solitons are studied. One important reason why did this happen is the optical-communication technology boom of the past decade. During that period, the burgeoning demand for high-capacity optical communications and all-optical information processing has led to large investments of resources in nonlinear optics research. Another reason lies in the fact that the necessary technologies such as powerful but less expensive lasers, readily available monitoring and sampling techniques for ultrafast optics, and rapidly developed material science for fabricating complex photonic structures, have matured. Moreover, the beauty of optics is that it allows one to study a variety of highly nonlinear effects directly, visualizing every detail of the physics involved, and isolating the underlying effects. As such, we would like to choose the study of optical solitons, especially temporal solitons propagating in optical fibers, as our research topic that will be discussed heavily in this thesis.
In a broad sense, optical solitons are self-trapped, localized, wave-packets (beams or pulses) that do not broaden while propagating in a dispersive medium (Noting that modern nonlinear optics nomenclature now identifies all self-trapped optical beams or pulses as“solitons”even though this is a terminology reserved for integrable sets). They exist by virtue of the balance between the dispersion (or diffraction) that tends to expand the wave packet, and the nonlinear effect that tends to localize it. In the case of solitons in optical fibers, the balance is between the wavelength dependence and the intensity dependence of refractive index of the fiber waveguide. The particlelike nature or the robustness of optical solitons in fibers makes them interesting not only for fundamental research, but also for long-haul optical communications, both fields fascinating many people. Recently, the issue of fundamental interest has involved the self-similarity of optical wave propagation (not restricted to solitons) in a fiber waveguide with varying dispersion, nonlinearity, and gain. As respects the issue involving applications in optical telecommunications, the physical limits on communication performance caused by amplified spontaneous emission (ASE) noise or otherwise were also extensively pursued over the years. Hence in this thesis, our primary concern is to study the self-similar optical wave propagations in optical fibers with varying parameters, the timing jitter of ultrashort solitons (of fs order) by virtue of our formulated soliton perturbation model, and the phase fluctuations of arbitrarily nonlinearity- and dispersion-managed solitons using the variational approach or moment method. These works are summarized as follows.
1. Self-Similar Optical Wave Propagations and Linear Stability Analysis
To begin with, we present a systematical study of the self-similar propagations of solitons, continuous waves, cnoidal waves, parabolic similaritons, Hermite-Gaussian (HG) pulses, and hybrid pulses in parametrically varying systems. By analytical exploration and numerical simulations of the generalized NLS equation with varying coefficients, many remarkable properties of these self-similar solitons or pulses are found. To mention a few, the pulse evolutions of parabolic similaritons in such arbitrarily managed systems are uniquely determined by the input energy, not by the initial pulse shapes, but the other pulse evolutions are greatly affected by their input pulse shapes. In addition, owing to the continuously enhanced linear chirp, parabolic similaritons are monotonically broadened during propagation, but it is not always the case for solitons, HG and hybrid pulses. Under appropriate initial conditions, the latter three will go through initial pulse narrowing stage when GVD induces a chirp in opposition to the initial chirp. The discrepancies among solitons, HG pulses, and hybrid pulses are also discussed in our thesis. Despite all this, these self-similar pulses are clearly shown to share many universal features such as self-similarity in pulse shape and enhanced linearity in pulse chirp. By virtue of these features one can achieve directly the well-defined linearly chirped output pulses from an optical fiber, an amplifier, or an absorption medium under favorable conditions.
We then make a substantial extension to these results by considering the nonlinear gain into the above-mentioned NLS equation model. The exact chirped self-similar soliton solutions are obtained by using the same symmetry reduction technique and their stabilities are verified numerically by adding Gaussian white noise and by evolving from an initial chirped Gaussian pulse, respectively. It is demonstrated that the pulse position of these chirped solitons can be precisely piloted by tailoring the dispersion profile, and that the sech-shaped solitons can propagate stably in the anomalous, normal, or other managed dispersion regime, according to the magnitude of the nonlinear chirp parameter.
Before ending this issue, we study the stability problem of self-similar solitons with a linear stability analysis, and thereby reveal the availability of an enhanced stability against perturbations via dispersion management techniques. The experimental realizations of these self-similar pulses, especially parabolic similaritons, are also reviewed in detail.
2. A Novel Soliton Perturbation Model and Calculation of Timing Jitter
It is well known that apart from the optical losses or dissipation, timing jitter is the key factor which limits the total transmission distance of the intensity-modulated optical communication link. To find the analytical expression for timing jitter, many models such as the adiabatic perturbation theory, linearized method, and variational approach had been proposed. In essence, they are all based on the small perturbations to the well-analyzed NLS equation and can be applicable for elucidating some phenomenological effects with sufficient accuracy. In the subpicosecond-femtosecond regime, however, these models fail to a certain extent in the efficient and accurate description of the noise effects on soliton evolutions because of the introduction of higher-order effects.
Accordingly, we propose an extended soliton perturbation model which only considers the small perturbations to the higher-order nonlinear Schrodinger equation. Based on this model, we develop the adiabatic perturbation theory and linearized method by formulation of dynamic equations for soliton parameters. In terms of our formulated equations, we calculate the timing jitter of fs solitons (bright or dark) analytically with the third-order dispersion, self-steepening, and self-frequency shift arising from stimulated Raman scattering all taken into account. A good agreement between our analytical results and numerical simulations is obtained. We show that the timing jitter for dark solitons is nearly one half of that for bright solitons and the Raman jitter always dominates the Gordon-Haus one in femtosecond regime. More interestingly, we find that the higher-order effects on timing jitter can be cancelled out under a certain parametric condition and thus a significant reduction of timing jitter is allowed. As expected, these analytical results would have more extensive applications than those obtained by use of the well-known perturbation theory of the NLS equation.
3. Phase Jitter of Nonlinearity- and Dispersion-Managed Solitons
In view of the practical relevance to phase-modulated optical communications, the phase fluctuations of arbitrarily nonlinearity- and dispersion-managed solitons are investigated in this thesis both analytically and numerically. By the aid of either variational approach or moment method, we obtain for the first time an exact closed-form expression for the variances of phase jitter in these complicated soliton systems. Notably, in our derivations, the effect of chirp fluctuations has been critically taken into account as well as the dispersive and nonlinear effects. It is shown that the chirp fluctuations effect plays an important role in the control of nonlinear phase noise via fiber dispersion, independently of whether the input solitons are initially chirped or not. Significantly, we find from this expression that the phase variance can be uniquely determined by the accumulated dispersion and the ratio of local dispersion to nonlinearity for given noise intensity and some input parameters. This achievement bears on the recent intriguing issues about the control of nonlinear phase noise and may offer effective ways of mitigating the nonlinearity and dispersion penalties.
We then corroborate our analytical results with numerical simulations by way of several interesting examples. The first demonstration involves the familiar soliton system with constant dispersion and nonlinearity. It not only exactly reproduces some well-known results such as a cubic growth of phase variance with distance, an inverse phase variance dependence on the cube of pulse width, to name a few, but also renews our view by finding that the residual frequency shift can produce nonnegligible nonlinear phase noise apart from causing large timing jitter. We also apply our results to another experimentally accessible soliton system in which the dispersion increases (decreases) exponentially but the nonlinearity remains constant. By choosing appropriate chirp parameter, this special soliton system allows for a rapid pulse broadening (compression) without radiating dispersive waves. We show that the nonlinear phase noise can be greatly suppressed, if the frequency shift is eliminated initially, in the process of pulse broadening but drastically amplified as solitons get compressed. We finish our demonstrations with a theoretically intrigued soliton system which has a cosinoidally varying dispersion and nonlinearity. It is clearly shown that the nonlinear phase noise in such a system can grow linearly on an average, independently of whether the initial frequency shift vanishes or not. Taken altogether, our analytical results can apply to arbitrarily managed systems within the framework of generalized NLS equation, and may find potential applications in areas such as optical communications and optical information processing.
4. Stochastic Split-Step Fourier Method
In the final part of thesis, we develop a robust and efficient algorithm for simulating the stochastic generalized NLS equation. This algorithm is based on the conventional split-step Fourier method, with an incorporation of both the multiplicative and additive noises. Using this algorithm, we have numerically calculated the timing or phase jitter of solitons when propagating and obtained a good agreement with analytical predictions. The pulse evolutions under perturbations also can be simulated with this algorithm.
广义上讲,光孤子是自捕获的局域波包(光束或脉冲),它们在色散介质中传播时不展宽(注意,当代非线性光学术语把所有的自捕获光束或脉冲统称为“孤子”)。它们的存在是由于倾向展宽波包的色散与倾向压缩波包的非线性之间的精确平衡。对于光纤孤子,这种平衡体现在光纤波导折射率的波长相关和强度相关之间的均衡。光纤孤子的粒子特性或强稳定性使它们不仅具有基础研究兴趣,而且可用于长距离光纤通讯。这两个领域均吸引大量的研究人员。最近,基础研究兴趣已经转向变化色散、非线性、和增益光纤中的自相似光波传播(不限于孤子)。同时在应用方面,人们致力于研究放大自发辐射噪声或其他噪声所产生的、影响光通讯性能的物理限制。因此,在本论文中,我们的主要关切是研究参数变化系统中的自相似光波传播、超短光孤子的定时抖动(利用我们发展的孤子微扰模型)、和随意非线性和色散管理孤子的相抖动(利用变分法或矩值法)。这些工作概括如下。
1.自相似光波传播和线性稳定性分析
首先,我们系统地研究参数变化系统中孤子、连续波、cnoidal波、抛物自相似子、厄密高斯(HG)脉冲、和混合脉冲的自相似传播。通过对系数可变NLS方程的解析研究和数值模拟,我们发现,这些自相似孤子或脉冲具有许多值得关注的演化特性。举几个例子,在这些参数变化系统中,抛物自相似子的脉冲演化只与脉冲输入能量有关,与输入脉冲形状无关,但是其他脉冲的自相似演化均受它们的初始形状影响。此外,由于连续增强的线性啁啾,抛物自相似子是单调展宽的,但是孤子、HG、和混合脉冲的情形则不然。在合适的初始条件下,这三种脉冲会遭受到初始的脉冲压缩过程,只要GVD感应的啁啾与初始啁啾符号相反。我们也在论文中详细讨论了孤子、HG、和混合脉冲之间的区别。尽管这些,上述自相似脉冲也具有许多共同特征,如自相似的脉冲外形和严格线性的脉冲啁啾。利用这些特性和合适的参数条件,人们可以很容易地从光纤、放大器、或者吸收介质中获取严格确定的线性啁啾输出脉冲。
我们接着对这些结果进行推广,在原来的NLS方程模型中增加非线性增益效应。利用相同的对称约化技术,我们可以获得该模型的精确啁啾自相似孤子解。通过添加高斯白噪声和输入高斯脉冲,我们数值证明了这类孤子的稳定性。我们同时演示,这些啁啾孤子的脉冲位置可以通过裁剪色散轮廓图来精确控制,并且,只要非线性啁啾参数选择适当,sech型孤子可以稳定存在于反常、正常、和其他管理的色散区。
在结束这个话题时,我们利用线性稳定性分析方法研究了自相似孤子的稳定性问题,从而揭示了孤子稳定性增强的可能性(利用色散管理技术)。此外,自相似脉冲(特别是抛物自相似子)的实验实现也得到了详细的回顾。
2.新颖孤子微扰模型和定时抖动的计算
众所周知,除了光学损耗或耗散,定时抖动是限制强度调制孤子通讯总传输距离的关键因素。为了找出定时抖动的解析表达式,许多模型如绝热微扰理论、线性化方法、变分法等相继提出。本质上,它们均是基于对标准NLS方程的小微扰,在足够精度范围内,它们适合用于阐述一些表象效应。但是,在亚皮秒━飞秒区,由于高阶效应的加入,这些模型在一定程度上不能高效、准确地描述孤子噪声效应。
因此,我们提出一个扩展的孤子微扰模型,其实质是基于高阶NLS方程的小微扰。利用该模型,我们相继发展了绝热微扰理论和线性化方法,从而得到孤子参数的动力学方程。根据这些演化方程并充分考虑三阶色散、自陡、和受激Raman散射所产生的自频移效应,我们解析计算飞秒亮暗孤子的定时抖动,所得到的结果与数值模拟完全吻合。我们指出,暗孤子的定时抖动大约只有亮孤子的一半,并且,在亚皮秒━飞秒区,Raman抖动通常远大于Gordon-Haus抖动。更有趣的是,我们发现在一定参数条件下,高阶效应对定时抖动的影响可以相互抵消,从而理论上允许获得较小的定时抖动。可以期望,这些解析结果将比基于NLS方程的著名微扰理论所得到的结果具有更广泛的应用。
3.非线性和色散管理孤子的相抖动
鉴于相调制光纤通讯上的实际应用,随意非线性和色散管理孤子的相波动也在本文中得到详细的解析和数值探讨。利用变分法或矩值法,我们首次得到该复杂孤子系统相抖动的精确解析表达式。值得注意的是,在我们的推导过程中,啁啾波动效应连同色散和非线性效应均得到充分的考虑。我们指出,在非线性相噪声的控制上,不管输入孤子啁啾化与否,啁啾波动效应均通过纤维色散扮演着重要的角色。更重要地,我们从该表达式看出,在给定噪声强度和输入孤子参数情况下,相方差可由系统累积色散、色散━非线性之比完全确定。很显然,该成果会对最近非线性相噪声控制的有趣话题产生积极的影响,同时能提供有效途径来减轻系统的非线性和色散破坏。
接着,我们数值例证了以上解析结果。在第一个演示中,我们考虑熟悉的常色散和非线性系统。它不仅准确再现一些著名的结论,如相方差的立方增长,相方差与脉宽的三次方成反比等等,而且更新我们的观点,即残余频移除了产生大的定时抖动外,同时会产生不可忽略的非线性相噪声。我们的解析结果也被应用到另一个实验可实现的孤子系统,其色散随距离指数递增(递减),但非线性保持常数。显然,通过选择合适的啁啾参数,这个特殊的孤子系统允许快速的脉冲增宽(压缩),而且不辐射任何色散波。我们看出,非线性相噪声在脉冲展宽过程中(不考虑初始频移效应时)会得到极大的抑制,但是在脉冲压缩时则会急剧的放大。在最后的演示中,我们考虑理论上感兴趣的余弦变化孤子系统。我们可清晰地看出,在该管理系统中,不管孤子的初始频移存在与否,非线性相噪声均表现为近线性增长。总之,我们的解析结果适用广义NLS方程描述的随意管理孤子系统,也必将在光纤通讯和光信息处理等领域上找到潜在的应用。
4.随机分步Fourier方法
在本文最后部分,我们发展了一个强大、高效的随机广义NLS方程的模拟算法。该算法是基于常规分步Fourier方法之上,同时综合考虑了乘法和加法噪声的作用。利用该算法,我们已经数值计算孤子传播的定时抖动和相抖动,而且所获得的数值结果与理论预测完全一致。微扰作用下的脉冲演化也可以通过该算法进行数值模拟。
Solitons are ubiquitous. Since the first observation of soliton by J.S. Russell in the water of a shallow canal in 1834, numerous examples of solitons have been found in various physical, chemical, and biological environments, all propagating without spreading out or breaking up. Solitons of electromagnetic waves were even identified in an absolute vacuum. However, the large variety of soliton manifestations is in optics. Most researchers agree that optical solitons are at the forefront of soliton research in all the branches of science in which solitons are studied. One important reason why did this happen is the optical-communication technology boom of the past decade. During that period, the burgeoning demand for high-capacity optical communications and all-optical information processing has led to large investments of resources in nonlinear optics research. Another reason lies in the fact that the necessary technologies such as powerful but less expensive lasers, readily available monitoring and sampling techniques for ultrafast optics, and rapidly developed material science for fabricating complex photonic structures, have matured. Moreover, the beauty of optics is that it allows one to study a variety of highly nonlinear effects directly, visualizing every detail of the physics involved, and isolating the underlying effects. As such, we would like to choose the study of optical solitons, especially temporal solitons propagating in optical fibers, as our research topic that will be discussed heavily in this thesis.
In a broad sense, optical solitons are self-trapped, localized, wave-packets (beams or pulses) that do not broaden while propagating in a dispersive medium (Noting that modern nonlinear optics nomenclature now identifies all self-trapped optical beams or pulses as“solitons”even though this is a terminology reserved for integrable sets). They exist by virtue of the balance between the dispersion (or diffraction) that tends to expand the wave packet, and the nonlinear effect that tends to localize it. In the case of solitons in optical fibers, the balance is between the wavelength dependence and the intensity dependence of refractive index of the fiber waveguide. The particlelike nature or the robustness of optical solitons in fibers makes them interesting not only for fundamental research, but also for long-haul optical communications, both fields fascinating many people. Recently, the issue of fundamental interest has involved the self-similarity of optical wave propagation (not restricted to solitons) in a fiber waveguide with varying dispersion, nonlinearity, and gain. As respects the issue involving applications in optical telecommunications, the physical limits on communication performance caused by amplified spontaneous emission (ASE) noise or otherwise were also extensively pursued over the years. Hence in this thesis, our primary concern is to study the self-similar optical wave propagations in optical fibers with varying parameters, the timing jitter of ultrashort solitons (of fs order) by virtue of our formulated soliton perturbation model, and the phase fluctuations of arbitrarily nonlinearity- and dispersion-managed solitons using the variational approach or moment method. These works are summarized as follows.
1. Self-Similar Optical Wave Propagations and Linear Stability Analysis
To begin with, we present a systematical study of the self-similar propagations of solitons, continuous waves, cnoidal waves, parabolic similaritons, Hermite-Gaussian (HG) pulses, and hybrid pulses in parametrically varying systems. By analytical exploration and numerical simulations of the generalized NLS equation with varying coefficients, many remarkable properties of these self-similar solitons or pulses are found. To mention a few, the pulse evolutions of parabolic similaritons in such arbitrarily managed systems are uniquely determined by the input energy, not by the initial pulse shapes, but the other pulse evolutions are greatly affected by their input pulse shapes. In addition, owing to the continuously enhanced linear chirp, parabolic similaritons are monotonically broadened during propagation, but it is not always the case for solitons, HG and hybrid pulses. Under appropriate initial conditions, the latter three will go through initial pulse narrowing stage when GVD induces a chirp in opposition to the initial chirp. The discrepancies among solitons, HG pulses, and hybrid pulses are also discussed in our thesis. Despite all this, these self-similar pulses are clearly shown to share many universal features such as self-similarity in pulse shape and enhanced linearity in pulse chirp. By virtue of these features one can achieve directly the well-defined linearly chirped output pulses from an optical fiber, an amplifier, or an absorption medium under favorable conditions.
We then make a substantial extension to these results by considering the nonlinear gain into the above-mentioned NLS equation model. The exact chirped self-similar soliton solutions are obtained by using the same symmetry reduction technique and their stabilities are verified numerically by adding Gaussian white noise and by evolving from an initial chirped Gaussian pulse, respectively. It is demonstrated that the pulse position of these chirped solitons can be precisely piloted by tailoring the dispersion profile, and that the sech-shaped solitons can propagate stably in the anomalous, normal, or other managed dispersion regime, according to the magnitude of the nonlinear chirp parameter.
Before ending this issue, we study the stability problem of self-similar solitons with a linear stability analysis, and thereby reveal the availability of an enhanced stability against perturbations via dispersion management techniques. The experimental realizations of these self-similar pulses, especially parabolic similaritons, are also reviewed in detail.
2. A Novel Soliton Perturbation Model and Calculation of Timing Jitter
It is well known that apart from the optical losses or dissipation, timing jitter is the key factor which limits the total transmission distance of the intensity-modulated optical communication link. To find the analytical expression for timing jitter, many models such as the adiabatic perturbation theory, linearized method, and variational approach had been proposed. In essence, they are all based on the small perturbations to the well-analyzed NLS equation and can be applicable for elucidating some phenomenological effects with sufficient accuracy. In the subpicosecond-femtosecond regime, however, these models fail to a certain extent in the efficient and accurate description of the noise effects on soliton evolutions because of the introduction of higher-order effects.
Accordingly, we propose an extended soliton perturbation model which only considers the small perturbations to the higher-order nonlinear Schrodinger equation. Based on this model, we develop the adiabatic perturbation theory and linearized method by formulation of dynamic equations for soliton parameters. In terms of our formulated equations, we calculate the timing jitter of fs solitons (bright or dark) analytically with the third-order dispersion, self-steepening, and self-frequency shift arising from stimulated Raman scattering all taken into account. A good agreement between our analytical results and numerical simulations is obtained. We show that the timing jitter for dark solitons is nearly one half of that for bright solitons and the Raman jitter always dominates the Gordon-Haus one in femtosecond regime. More interestingly, we find that the higher-order effects on timing jitter can be cancelled out under a certain parametric condition and thus a significant reduction of timing jitter is allowed. As expected, these analytical results would have more extensive applications than those obtained by use of the well-known perturbation theory of the NLS equation.
3. Phase Jitter of Nonlinearity- and Dispersion-Managed Solitons
In view of the practical relevance to phase-modulated optical communications, the phase fluctuations of arbitrarily nonlinearity- and dispersion-managed solitons are investigated in this thesis both analytically and numerically. By the aid of either variational approach or moment method, we obtain for the first time an exact closed-form expression for the variances of phase jitter in these complicated soliton systems. Notably, in our derivations, the effect of chirp fluctuations has been critically taken into account as well as the dispersive and nonlinear effects. It is shown that the chirp fluctuations effect plays an important role in the control of nonlinear phase noise via fiber dispersion, independently of whether the input solitons are initially chirped or not. Significantly, we find from this expression that the phase variance can be uniquely determined by the accumulated dispersion and the ratio of local dispersion to nonlinearity for given noise intensity and some input parameters. This achievement bears on the recent intriguing issues about the control of nonlinear phase noise and may offer effective ways of mitigating the nonlinearity and dispersion penalties.
We then corroborate our analytical results with numerical simulations by way of several interesting examples. The first demonstration involves the familiar soliton system with constant dispersion and nonlinearity. It not only exactly reproduces some well-known results such as a cubic growth of phase variance with distance, an inverse phase variance dependence on the cube of pulse width, to name a few, but also renews our view by finding that the residual frequency shift can produce nonnegligible nonlinear phase noise apart from causing large timing jitter. We also apply our results to another experimentally accessible soliton system in which the dispersion increases (decreases) exponentially but the nonlinearity remains constant. By choosing appropriate chirp parameter, this special soliton system allows for a rapid pulse broadening (compression) without radiating dispersive waves. We show that the nonlinear phase noise can be greatly suppressed, if the frequency shift is eliminated initially, in the process of pulse broadening but drastically amplified as solitons get compressed. We finish our demonstrations with a theoretically intrigued soliton system which has a cosinoidally varying dispersion and nonlinearity. It is clearly shown that the nonlinear phase noise in such a system can grow linearly on an average, independently of whether the initial frequency shift vanishes or not. Taken altogether, our analytical results can apply to arbitrarily managed systems within the framework of generalized NLS equation, and may find potential applications in areas such as optical communications and optical information processing.
4. Stochastic Split-Step Fourier Method
In the final part of thesis, we develop a robust and efficient algorithm for simulating the stochastic generalized NLS equation. This algorithm is based on the conventional split-step Fourier method, with an incorporation of both the multiplicative and additive noises. Using this algorithm, we have numerically calculated the timing or phase jitter of solitons when propagating and obtained a good agreement with analytical predictions. The pulse evolutions under perturbations also can be simulated with this algorithm.
引文
[1] J.S. Russell, in Reports of the Meetings of the British Association for the Advancement of Science, 1844, Liverpool Meeting, 1838.
[2] M. Segev, Solitons: A Universal Phenomenon of Self-Trapped Wave Packets, Opt. Photon. News, 2002(2), 13: 27.
[3] R.W. Boyd, Nonlinear Optics (Academic Press, Boston, 2003).
[4] Y.R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
[5] T.H. Maiman, Stimulated Optical Radiation in Ruby, Nature 1960, (187): 493–494.
[6] J. Hecht, Beam: The Race to Make the Laser, Opt. Photon. News, 2005(7/8), 16: 24-29.
[7] P.A. Franken, A.E. Hill, C.W. Peters et al., Generation of Optical Harmonics, Phys. Rev. Lett. 1961, (7): 118-119.
[8] A. Hasegawa and F.D.Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys. Lett. 1973, (23): 142-144.
[9] L.F. Mollenauer, R.H. Stolen, and J.P. Gordon, Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers, Phys. Rev. Lett. 1980, (45): 1095-1098.
[10] G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).
[11] L.F. Mollenauer, J.P. Gordon, and S.G. Evangelides, Laser Focus World 159 (November 1991).
[12] Y.N. Karamzin and A.P. Sukhorukov, Mutual focusing of high-power light beams in media with quadratic nonlinearity, Sov. Phys. JETP 1975, (41): 414-420.
[13] G.I. Stegeman, M. Sheik-Bahae, E.V. Stryland et al., Large nonlinear phase shifts in second-order nonlinear-optical processes, Opt. Lett. 1993, (18): 13-15.
[14] L. Torner and A.P. Sukhorukov, Quadratic Solitons, Opt. Photon. News, 2002(2), 13: 42-47.
[15] A.V. Buryaka, P.D. Trapanib, D.V. Skryabinc et al., Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep. 2002, (370): 63-235.
[16] S.L. McCall and E.L. Hahn, Self-Induced Transparency by Pulsed Coherent Light, Phys. Rev. Lett. 1967, (18): 908-911.
[17] H.M. Gibbs and R.E. Slusher, Peak Amplification and Breakup of a Coherent Optical Pulse in a Simple Atomic Absorber, Phys. Rev. Lett. 1970, (24): 638–641.
[18] L.V. Hau, S.E. Harris, Z. Dutton et al., Light speed reduction to 17 metres per second in an ultracold atomic gas, Nature 1999, (397): 594–598.
[19] C. Liu, Z. Dutton, C. H. Behroozi et al., Observation of coherent optical information storage in an atomic medium using halted light pulses, Nature 2001, (409): 490–493.
[20] M. Fleischhauer, A. Imamoglu, and J.P. Marangos, Electromagnetically induced transparency: Optics in coherent media, Rev. Mod. Phys., 2005, (77): 633-673.
[21] Y. Wu and L. Deng, Ultraslow Optical Solitons in a Cold Four-State Medium, Phys. Rev. Lett. 2004, (93): 143904 1-4.
[22] T. Hong, Spatial Weak-Light Solitons in an Electromagnetically Induced Nonlinear Waveguide, Phys. Rev. Lett. 2003, (90): 183901 1-4.
[23] G.I. Stegeman and M. Segev, Optical Spatial Solitons and Their Interactions: Universality and Diversity, Science 1999, (286):1518-1523.
[24] Y.S. Kivshar and G.I. Stegeman, Spatial Optical Solitons: Guiding Light for Future Technologies, Opt. Photon. News, 2002(2), 13: 59-63.
[25] S. Trillo and W. Torruellas, Spatial Solitons (Springer, New York, 2001).
[26] Y.S. Kivshar and B. Luther-Davies, Dark optical solitons: physics and applications, Phys. Rep. 1998, (298), 81-197.
[27] R.Y. Chiao, E. Garmire, C.H. Townes, Self-Trapping of Optical Beams, Phys. Rev. Lett. 1964, (13): 479-482.
[28] M. Segev, B. Crosignani, A. Yariv, et al., Spatial Solitons in Photorefractive Media, Phys. Rev. Lett. 1992, (68): 923-926.
[29] M. Segev and D.N. Christodoulides, Incoherent Solitons: Self-Trapping of Weakly Correlated Wave Packets, Opt. Photon. News, 2002(2), 13: 70-76.
[30] J.W. Fleischer, M. Segev, N.K. Efremidis et al., Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature 2003, (422): 147-150.
[31] J.W. Fleischer, T. Carmon, M. Segev et al., Observation of Discrete Solitons in Optically-Induced Real-Time Waveguide Arrays, Phys. Rev. Lett. 2003, (90): 023902.
[32] A. Barthelemy, S. Maneuf, C. Froehly, Soliton propagation and self-trapping of laser beams by the optical Kerr effect, Opt. Commun. 1985, (55): 201-206.
[33] B. Crosignani and G. Salamo, Photorefractive Solitons, Opt. Photon. News, 2002(2), 13: 38-41.
[34] M. Mitchell, Z. Chen, M. Shih et al., Self-Trapping of Partially Spatially Incoherent Light, Phys. Rev. Lett. 1996, (77): 490-493.
[35] M. Mitchell and M. Segev, Self-trapping of incoherent white light, Nature 1997, (387): 880-883.
[36] Z. Chen, M. Mitchell, M. Segev et al., Self-Trapping of Dark Incoherent Light Beams, Science 1998, (280): 889-892.
[37] A.W. Snyder and A. P. Sheppard, Collisions, steering, and guidance with spatial solitons, Opt. Lett. 1993, (18): 482-484.
[38] M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, Cambridge, 1991).
[39] M. Segev and G.I. Stegeman, Self-Trapping of Optical Beams: Spatial Solitons, Phys. Today 1998, (51): 42-48.
[40] F. Lederer and Y. Silberberg, Discrete Solitons, Opt. Photon. News, 2002(2), 13: 48-53.
[41] S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys. JETP 1974, (38): 248-253.
[42] J.U. Kang, G.I. Stegeman, J.S. Aitchison et al., Observation of Manakov Spatial Solitons in AlGaAs Planar Waveguides, Phys. Rev. Lett. 1996, (76): 3699-3702.
[43] W.E. Torruellas, Z. Wang, D.J. Hagan et al., Observation of Two-Dimensional Spatial Solitary Waves in a Quadratic Medium, Phys. Rev. Lett. 1995, (74): 5036-5039.
[44] F. Wise and P. Di Trapani, The Hunt for Light Bullets: Spatiotemporal Solitons, Opt. Photon. News, 2002(2), 13: 28-32.
[45] X. Liu, L.J. Qian, and F.W. Wise, Generation of Optical Spatiotemporal Solitons, Phys. Rev. Lett. 1999, (82): 4631-4634.
[46] N.J. Zabusky and M.D. Kruskal, Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States, Phys. Rev. Lett. 1965, (15): 240-243.
[47] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 1972, (34): 62-69.
[48] J.P. Gordon, Interaction forces among solitons in optical fibers, Opt. Lett. 1983, (8): 596-598.
[49] T. Kanna and M. Lakshmanan, Exact Soliton Solutions, Shape Changing Collisions, and Partially Coherent Solitons in Coupled Nonlinear Schrodinger Equations, Phys. Rev. Lett. 2001, (86): 5043-5046.
[50] K. Porsezian and K. Nakkeeran, Optical Solitons in Presence of Kerr Dispersion and Self-Frequency Shift, Phys. Rev. Lett. 1996, (76): 3955-3958.
[51] M. Gedalin, T.C. Scott, and Y.B. Band, Optical Solitary Waves in the Higher Order Nonlinear Schrodinger Equation, Phys. Rev. Lett. 1997, (78): 448-451.
[52] H.A. Haus and W.S. Wong, Solitons in optical communications, Rev. Mod. Phys. 1996, (68): 423-444.
[53] G.P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 2002).
[54] M. Peccianti, C. Conti, G. Assanto et al., All-optical switching and logic gating with spatial solitons in liquid crystals, Appl. Phys. Lett. 2002, (81): 3335-3337.
[55] M. Peccianti, C. Conti, G. Assanto et al., Routing of anisotropic spatial solitons and modulational instability in liquid crystals, Nature (London) 2004, (432): 733-737.
[56] R. McLeod, K. Wagner, and S. Blair, (3+1)-dimensional optical soliton dragging logic, Phys. Rev. A 1995, (52): 3254-3278.
[57] N.N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).
[58] N.N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, New York, 2005).
[59] Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE J. Quantum Electron. 1987, (23): 510-524.
[60] Z. Li, L. Li, H. Tian et al., Chirped Femtosecond Solitonlike Laser Pulse Form with Self-Frequency Shift, Phys. Rev. Lett. 2002, (89):263901 1-4.
[61] W.H. Reeves, D.V. Skryabin, F. Biancalana et al., Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres, Nature 2003, (424): 511-515.
[62] R. Conte and M. Musette, Linearity inside nonlinearity: exact solutions to the complex Ginzburg-Landau equation, Physica D 1993, (69): 1–17.
[63] N. Akhmediev and V.V. Afanasjev, Novel Arbitrary-Amplitude Soliton Solutions of the Cubic-Quintic Complex Ginzburg-Landau Equation, Phys. Rev. Lett. 1995, (75): 2320- 2323.
[64] J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Pulsating, Creeping, and Erupting Solitons in Dissipative Systems, Phys. Rev. Lett. 2000, (85):2937-2940.
[65] S.T. Cundiff, J.M. Soto-Crespo, and N. Akhmediev, Experimental Evidence for Soliton Explosions, Phys. Rev. Lett. 2002, (88):073903 1-4.
[66] J.M. Soto-Crespo and N. Akhmediev, Soliton as Strange Attractor: Nonlinear Synchronization and Chaos, Phys. Rev. Lett. 2005, (95): 024101 1-4.
[67] R.J. Deissler and H.R. Brand, Periodic, Quasiperiodic, and Chaotic Localized Solutions of the Quintic Complex Ginzburg-Landau Equation, Phys. Rev. Lett.1994,(72): 478-481.
[68] R.J. Deissler and H.R. Brand, Interaction of Breathing Localized Solutions for Subcritical Bifurcations, Phys. Rev. Lett. 1995, (74): 4847-4850.
[69] R.J. Deissler and H.R. Brand, Effect of Nonlinear Gradient Terms on Breathing Localized Solutions in the Quintic Complex Ginzburg-Landau Equation, Phys. Rev. Lett. 1998, (81): 3856-3859.
[70] A. Hasegawa, Optical Solitons in Fibers for Communication systems, Opt. Photon. News, 2002(2), 13: 33-37.
[71] C. Lin, H. Kogelnik, and L.G. Cohen, Optical-pulse equalization of low-dispersion transmission in single-mode fibers in the 1.3-1.7-μm spectral region, Opt. Lett. 1980, (5): 476-478.
[72] N.J. Smith, F.M. Knox, N.J. Doran et al., Enhanced power solitons in optical fibres with periodic dispersion management, Electron. Lett. 1996, (32): 54-55.
[73] Q. Quraishi, S.T. Cundiff, B. Ilan et al., Dynamics of Nonlinear and Dispersion Managed Solitons, Phys. Rev. Lett. 2005, (94): 243904 1-4.
[74] H.S. Eisenberg, Y. Silberberg, R. Morandotti et al., Diffraction Management, Phys. Rev. Lett. 2000, (85): 1863-1866.
[75] M.J. Ablowitz and Z.H. Musslimani, Discrete Diffraction Managed Spatial Solitons, Phys. Rev. Lett. 2001, (87): 254102 1-4.
[76] L.F. Mollenauer, P.V. Mamyshev, J. Gripp et al., Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersion- managed solitons, Opt. Lett. 2000, (25): 704-706.
[77] D.W. Prather, Photonic Crystals: An Engineering Perspective, Opt. Photon. News, 2002(6), 13: 16-19.
[78] J. Knight, T. Birks, B. Mangan et al., Photonic Crystal Fibers: New Solutions in Fiber Optics, Opt. Photon. News, 2002(3), 13: 26-30.
[79] V.N. Serkin and A. Hasegawa, Novel Soliton Solutions of the Nonlinear Schrodinger Equation Model, Phys. Rev. Lett. 2000, (85): 4502-4505.
[80] V.I. Kruglov, A.C. Peacock, and J.D. Harvey, Exact Self-Similar Solutions of the Generalized Nonlinear Schrodinger Equation with Distributed Coefficients, Phys. Rev. Lett. 2003, (90): 113902 1-4.
[81] S. Chen and L. Yi, Chirped self-similar solutions of a generalized nonlinear Schrodinger equation model, Phys. Rev. E 2005, (71): 016606 1-4.
[82] T. S. Raju, P.K. Panigrahi, and K. Porsezian, Self-similar propagation and compression of chirped self-similar waves in asymmetric twin-core fibers with nonlinear gain, Phys. Rev. E 2005, (72): 046612 1-6.
[83] L. Wang, L. Li, Z. Li et al., Generation, compression, and propagation of pulse trains in the nonlinear Schrodinger equation with distributed coefficients, Phys. Rev. E 2005, (72): 036614 1-7.
[84] W.J. Tomlinson, R.H. Stolen, and A.M. Johnson, Optical wave breaking of pulses in nonlinear optical fibers, Opt. Lett. 1985, (10): 457-459.
[85] D. Anderson, M. Desaix, M. Lisak et al., Wave breaking in nonlinear-optical fibers, J. Opt. Soc. Am. B 1992, (9): 1358-1361.
[86] T. Brabec and F. Krausz, Intense few-cycle laser fields: Frontiers of nonlinear optics, Rev. Mod. Phys. 2000, (72): 545-591.
[87] G. Steinmeyer, D.H. Sutter, L. Gallmann et al., Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics, Science 1999, (286):1507-1512.
[88] D. Méchin, S.-H. Im, V. I. Kruglov et al., Experimental demonstration of similariton pulse compression in a comblike dispersion-decreasing fiber amplifier, Opt. Lett. 2006, (31): 2106-2108.
[89] V.N. Serkin, A. Hasegawa, and T.L. Belyaeva, Comment on “Exact Self-Similar Solutions of the Generalized Nonlinear Schrodinger Equation with Distributed Coefficients”, Phys. Rev. Lett. 2004, (92): 199401 1-1.
[90] G.I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge University Press, Cambridge, 1996).
[91] S. Chen, L. Yi, D.-S. Guo et al., Self-similar evolutions of parabolic, Hermite- Gaussian, and hybrid optical pulses: Universality and diversity, Phys. Rev. E 2005, (72): 016622 1-5.
[92] P.L. Kelley, Self-Focusing of Optical Beams, Phys. Rev. Lett. 1965, (15): 1005- 1008.
[93] I.V. Barashenkov, Stability Criterion for Dark Solitons, Phys. Rev. Lett. 1996, (77): 1193-1197.
[94] A.V. Buryak, Y.S. Kivshar, and S. Trillo, Stability of Three-Wave Parametric Solitons in Diffractive Quadratic Media, Phys. Rev. Lett. 1996, (77): 5210-5213.
[95] D.E. Pelinovsky, A.V. Buryak, and Y.S. Kivshar, Instability of Solitons Governed by Quadratic Nonlinearities, Phys. Rev. Lett. 1995, (75): 591-595.
[96] E.A. Ostrovskaya, Y.S. Kivshar, D.V. Skryabin et al., Stability of Multihump Optical Solitons, Phys. Rev. Lett. 1999, (83): 296-299.
[97] D. Mihalache, D. Mazilu, and L. Torner, Stability of Walking Vector Solitons, Phys. Rev. Lett. 1998, (81): 4353-4356.
[98] V. Skarka and N. B. Aleksia, Stability Criterion for Dissipative Soliton Solutions of the One-, Two-, and Three-Dimensional Complex Cubic-Quintic Ginzburg-Landau Equations, Phys. Rev. Lett. 2006, (96): 013903 1-4.
[99] N.N. Akhmediev and A. Ankiewicz,Solitons Around Us: Integrable, Hamiltonian and Dissipative Systems,In: Optical Solitons, Theoretical and Experimental Challenges, Lecture Notes in Physics, vol. 613, Editors: K. Porsezian and V.C. Kurakose (Springer, Berlin, 2003), p. 105-126.
[100] E.A. Kuznetsov, A.M. Rubenchik, and V.E. Zakharov, Soliton stability in plasmas and hydrodynamics, Phys. Rep. 1986, (142): 103-165.
[101] A. Hasegawa, Amplification and reshaping of optical solitons in a glass fiber - IV:Use of stimulated Raman process, Opt. Lett. 1983, (8): 650-652.
[102] L.F. Mollenauer and K. Smith, Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain, Opt. Lett. 1988, (13): 675-677.
[103] A. Hasegawa, Optical Soliton Theory and Its Applications in Communication, in: Optical Solitons, Theoretical and Experimental Challenges, eds. K. Porsezian and V.C. Kuriakose (Springer, Berlin, 2002), 9-31.
[104] A. Hasegawa, Massive WDM and TDM Soliton Transmission Systems (Kluwer Academic, Dordrecht, The Netherlands, 2000).
[105] M. Nakazawa, H. Kubota, K. Suzuki et al., Ultrahigh-speed long-distance TDM and WDM soliton transmission technologies, IEEE J. Sel. Top. Quantum Electron. 2000, (6): 363-396.
[106] K. Fukuchi, M. Kakui, A. Sasaki et al., 1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely-dispersion-managed transmission line employing 1.55/1.58-μm hybrid repeaters, in European Conference on Optical Communication (Institute of Electrical Engineers, London, 1999), paper PD42.
[107] K. Suzuki, H. Kubota, and M. Nakazawa, 1 Tb/s (40 Gb/s×25 channel) DWDM quasi-DM soliton transmission over 1,500 km using dispersion-managed single-mode fiber and conventional C-band EDFAs, Optical Fiber Communication Conference and Exhibit, 2001, (2): TuN7-1-3.
[108] M. Nakazawa, K. Suzuki, and H. Kubota, 160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation, Electron. Lett. 1999, (35): 1358-1359.
[109] G. Zhu, L. Mollenauer, and C. Xu, Experimental Demonstration of Postnonlinearity Compensation in a Multispan DPSK Transmission, IEEE Photon. Technol. Lett. 2006, (18): 1007-1009.
[110] R.-J. Essiambre and G.P. Agrawal, Soliton communication systems, in: Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1997), Vol. 37, p.185-256.
[111] E. Iannone, F. Matera, A. Mecozzi et al., Nonlinear Optical Communications Networks (Wiley, New York,1998).
[112] K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, New York, 2005).
[113] W. Schottky, über spontane Stromschwankungen in verschiedenen Elektrizitatsleitern. Ann. Phys. (Leipz.) 1918, (57): 541-568.
[114] H.A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, New York, 2000).
[115] P.D. Drummond, R.M. Shelby, S.R. Friberg et al., Quantum solitons in optical fibers, Nature (London) 1993, (365): 307-313.
[116] J.P. Gordon and H.A. Haus, Random walk of coherently amplified solitons in optical fiber transmission, Opt. Lett. 1986, (11): 665-667.
[117] J.P. Gordon and L.F. Mollenauer, Phase noise in photonic communications systems using linear amplifiers, Opt. Lett. 1990, (15): 1351-1353.
[118] P.D. Drummond and J.F. Corney, Quantum noise in optical fibers. I. Stochastic equations, J. Opt. Soc. Am. B 2001, (18): 139-152.
[119] J.F. Corney and P.D. Drummond, Quantum noise in optical fibers. II. Raman jitter in soliton communications, J. Opt. Soc. Am. B 2001, (18): 153-161.
[120] S. Chen, S. Zhang, and D. Shi, Quantum Theory of Femtosecond Soliton Propagation in Single-Mode Optical Fibers, Commun. Theor. Phys. 2004, (41): 943-948.
[121] S. Chen, D. Shi, and L. Yi, Timing jitter of femtosecond solitons in single-mode optical fibers: A perturbation model, Phys. Rev. E 2004, (69): 046602 1-12.
[122] L.F. Mollenauer and J.P. Gordon, Birefringence-mediated timing jitter in soliton transmission, Opt. Lett. 1994, (19): 375-377.
[123] L.F. Mollenauer, P.V. Mamyshev, and M.J. Neubelt, Measurement of timing jitter in filter-guided soliton transmission at 10 Gbits/s and achievement of 375 Gbits/s-Mm, error free, at 12.5 and 15 Gbits/s, Opt. Lett. 1994, (19): 704-706.
[124] P.V. Mamyshev and L.F. Mollenauer, Soliton collisions in wavelength-division- multiplexed dispersion-managed systems, Opt. Lett. 1999, (24): 448-450.
[125] A. Mecozzi, J.D. Moores, H.A. Haus et al., Soliton transmission control, Opt. Lett. 1991, (16):1841-1843.
[126] G.M. Carter, J.M. Jacob, C.R. Menyuk et al., Timing-jitter reduction for a dispersion-managed soliton system: experimental evidence, Opt. Lett. 1997, (22): 513-515.
[127] R.-M. Mu, V.S. Grigoryan, C.R. Menyuk et al., Timing-jitter reduction in a dispersion-managed soliton system, Opt. Lett. 1998, (23): 930-932.
[128] R.J. Essiambre and G.P. Agrawal, Timing jitter of ultrashort solitons in high-speed communication systems. II. Control of jitter by periodic optical phase conjugation, J. Opt. Soc. Am. B 1997, (14): 323-330.
[129] A.H. Gnauck and P. J. Winzer, Optical Phase-Shift-Keyed Transmission, J. Lightwave Technol. 2005, (23): 115-130.
[130] C. J. McKinstrie, C. Xie, and C. Xu, Effects of cross-phase modulation on phase jitter in soliton systems with constant dispersion, Opt. Lett. 2003, (28): 604-606.
[131] S. Kumar, Analysis of intrachannel impairments in differential phase-shift keying transmission systems, Opt. Lett. 2005, (30): 2053-2055.
[132] A.G. Green, P.P. Mitra, and L.G.L. Wegener, Effect of chromatic dispersion onnonlinear phase noise, Opt. Lett. 2003, (28): 2455-2457.
[133] S. Kumar, Effect of dispersion on nonlinear phase noise in optical transmission systems, Opt. Lett. 2005, (30): 3278-3280.
[134] K.-P. Ho and H.-C. Wang, Effect of dispersion on nonlinear phase noise, Opt. Lett. 2006, (31): 2109-2111.
[135] J.N. Kutz and P.K.A. Wai, Noise-induced amplitude and chirp jitter in dispersion- managed soliton communications, Opt. Lett. 1998, (23): 1022-1024.
[136] X. Liu, X. Wei, R.E. Slusher et al., Improving transmission performance in differential phase-shift-keyed systems by use of lumped nonlinear phase-shift compensation, Opt. Lett. 2002, (27): 1616-1618.
[137] C. Xu and X. Liu, Postnonlinearity compensation with data-driven phase modulators in phase-shift keying transmission, Opt. Lett. 2002, (27): 1619-1621.
[138] S.A. Derevyanko and S.K. Turitsyn, Reduction of the phase jitter in differential phase-shift-keying soliton transmission systems by in-line Butterworth filters, Opt. Lett. 2004, (29): 35-37.
[139] D. Boivin, M. Hanna, P.-A. Lacourt et al., Reduction of phase jitter in dispersion- managed systems by in-line filtering, Opt. Lett. 2004, (29): 688-690.
[140] C.J. McKinstrie, S. Radic, and C. Xie, Reduction of soliton phase jitter by in-line phase conjugation, Opt. Lett. 2003, (28): 1519-1521.
[141] H.A. Haus and Y. Lai, Quantum theory of soliton squeezing: a linearized approach, J. Opt. Soc. Am. B 1990, (7): 386-392.
[142] D.J. Kaup, Perturbation theory of solitons in optical fibers, Phys. Rev. A 1990, (42): 5689-5694.
[143] Y.S. Kivshar and B.A. Malomed, Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys. 1989, (61): 763-915.
[144] S.N. Vlasov, V.A. Petrishchev, and V.I. Talanov, Averaged description of wave beams in linear and nonlinear media (the method of moments), Radiophys. Quantum Electron. 1971, (14): 1062–1070.
[145] D. Anderson, Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A 1983, (27): 3135-3145.
[146] J. Santhanam and G.P.Agrawal, Reduced timing jitter in dispersion-managed lightwave systems through parametric amplification, J. Opt. Soc. Am. B 2003, (20): 284-291.
[147] S. Chen et al., Phase jitter of nonlinearity- and dispersion-managed solitons: Moment-method description taking into account the chirp fluctuations effect, Phys. Rev. E (prepare).
[148] M. Suzuki, I. Morita, N. Edagawa et al., Reduction of Gordon-Haus timing jitter byperiodic dispersion compensation in soliton transmission, Electron. Lett. 1995, (31): 2027-2029.
[149] P.V. Mamyshev and L.F. Mollenauer, Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission, Opt. Lett. 1996, (21): 396-398.
[150] A.M. Niculae, W. Forysiak, A.J. Gloag et al., Soliton collisions with wavelength- division multiplexed systems with strong dispersion management, Opt. Lett. 1998, (23): 1354-1356.
[151] C. Paré, A. Villeneuve, P.-A. Bélanger et al., Compensating for dispersion and the nonlinear Kerr effect without phase conjugation, Opt. Lett. 1996, (21): 459-461.
[152] C. Paré, A. Villeneuve, S. LaRochelle, Split compensation of dispersion and self-phase modulation in optical communication systems, Opt. Commun. 1999, (160): 130-138.
[153] F.O. Ilday and F.W. Wise, Nonlinearity management: a route to high-energy soliton fiber lasers, J. Opt. Soc. Am. B 2002, (19): 470-476.
[154] D.E. Pelinovsky, P.G. Kevrekidis, and D.J. Frantzeskakis, Averaging for Solitons with Nonlinearity Management, Phys. Rev. Lett. 2003, (91): 240201 1-4.
[155] G.L. Lamb, JR., Analytical Descriptions of Ultrashort Optical Pulse Propagation in a Resonant Medium, Rev. Mod. Phys. 1971, (43): 99-124.
[156] I.R. Gabitov and S.V. Manakov, Propagation of Ultrashort Optical Pulses in Degenerate Laser Amplifier, Phys. Rev. Lett. 1983, (50): 495-498.
[157] S. An and J.E. Sipe, Universality in the dynamics of phase grating formation in optical fibers, Opt. Lett. 1991, (16): 1478-1480.
[158] C.R. Menyuk, D. Levi, and P. Winternitz, Self-Similarity in Transient Stimulated Raman Scattering, Phys. Rev. Lett. 1992, (69): 3048-3051.
[159] D. Anderson, M. Desaix, M. Karlsson et al., Wave-breaking-free pulses in nonlinear-optical fibers, J. Opt. Soc. Am. B 1993, (10): 1185-1190.
[160] T.M. Monro, P.D. Millar, L. Poladian et al., Self-similar evolution of self-written waveguides, Opt. Lett. 1998, (23): 268-270.
[161] T.M. Monro, D. Moss, M. Bazylenko et al., Observation of Self-Trapping of Light in a Self-Written Channel in a Photosensitive Glass, Phys. Rev. Lett. 1998, (80): 4072-4075.
[162] M. Soljacic, M. Segev, and C.R. Menyuk, Self-similarity and fractals in soliton- supporting systems, Phys. Rev. E 2000, (61): R1048-R1051.
[163] S. Sears, M. Soljacic, M. Segev et al., Cantor Set Fractals from Solitons, Phys. Rev. Lett. 2000, (84): 1902-1905.
[164] M.E. Fermann, V.I. Kruglov, B.C. Thomsen et al., Self-Similar Propagation andAmplification of Parabolic Pulses in Optical Fibers, Phys. Rev. Lett. 2000, (84): 6010- 6013.
[165] V.I. Kruglov, A.C. Peacock, J.D. Harvey et al., Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers, J. Opt. Soc. Am. B 2002, (19): 461-469.
[166] C. Finot and G. Millot, Synthesis of optical pulses by use of similaritons, Opt. Express 2004, (12): 5104-5109.
[167] C. Finot and G. Millot, Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers, Opt. Lett. 2004, (29): 2533-2535.
[168] V.I. Kruglov, A.C. Peacock, J.M. Dudley et al., Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers, Opt. Lett. 2000, (25): 1753-1755.
[169] C. Finot and G. Millot, Collisions between similaritons in optical fiber amplifiers, Opt. Express 2005, (13): 7653-7665.
[170] G. Chang, H.G. Winful, A. Galvanauskas et al., Self-similar parabolic beam generation and propagation, Phys. Rev. E 2005, (72): 016609 1-4.
[171] P.D. Drummond and K.V. Kheruntsyan, Asymptotic solutions to the Gross- Pitaevskii gain equation: Growth of a Bose-Einstein condensate, Phys. Rev. E 2000, (63): 013605 1-5.
[172] J.P. Limpert, T. Schreiber, T. Clausnitzer et al., High-power femtosecond Yb-doped fiber amplifier, Opt. Express 2002, (10): 628-638.
[173] C. Finot, G. Millot, C. Billet et al., Experimental generation of parabolic pulses via Raman amplification in optical fiber, Opt. Express 2003, (11): 1547-1552.
[174] F.O. Ilday, J. R. Buckley, W.G. Clark et al., Self-Similar Evolution of Parabolic Pulses in a Laser, Phys. Rev. Lett. 2004, (92): 213902 1-4.
[175] C. Billet, J.M. Dudley, N. Joly et al., Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm, Opt. Express 2005, (13): 3236-3241.
[176] C. Finot, F. Parmigiani, P. Petropoulos et al., Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime, Opt. Express 2006, (14): 3161-3170.
[177] C. Finot, S. Pitois, and G. Millot, Regenerative 40 Gbit/s wavelength converter based on similariton generation, Opt. Lett. 2005, (30): 1776-1778.
[178] T. Hirooka and M. Nakazawa, Parabolic pulse generation by use of a dispersion- decreasing fiber with normal group-velocity dispersion, Opt. Lett. 2004, (29): 498-500.
[179] V.N. Serkin and A. Hasegawa, Exactly Integrable Nonlinear Schrodinger Equation Models With Varying Dispersion, Nonlinearity and Gain: Application for SolitonDispersion Managements, IEEE J. Sel. Top. Quantum Electron. 2002, (8): 418-431.
[180] V.I. Kruglov, A.C. Peacock, and J.D. Harvey, Exact solutions of the generalized nonlinear Schrodinger equation with distributed coefficients, Phys. Rev. E 2005, (71): 056619 1-11.
[181] V.N. Serkin, M. Matsumoto, and T.L. Belyaeva, Nonlinear Bloch Waves, JETP Lett. 2001, (73): 59-62.
[182] M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan et al., Dispersion of bound electron nonlinear refraction in solids, IEEE J. Quantum Electron. 1991, (27): 1296-1309.
[183] C. Liberale, V. Degiorgio, M. Marangoni et al., Measurement of the nonlinear phase shift induced by cascaded interactions in a periodically poled lithium niobate waveguide, Opt. Lett. 2005, (30): 2448-2450.
[184] R. DeSalvo, D.J. Hagan, M. Sheik-Bahae et al., Self-focusing and self-defocusing by cascaded second-order effects in KTP, Opt. Lett. 1992, (17): 28-30.
[185] E.T. Copson, Asymptotic Expansions (Cambridge University press, 1965).
[186] D.B.S. Soh, J. Nilsson, and A.B. Grudinin, Efficient femtosecond pulse generation using a parabolic amplifier combined with a pulse compressor. II. Finite gain-bandwidth effect, J. Opt. Soc. Am. B 2006, (23): 10-19.
[187] S.A. Ponomarenko and G.P. Agrawal, Do Solitonlike Self-Similar Waves Exist in Nonlinear Optical Media? Phys. Rev. Lett. 2006, (97): 013901 1-4.
[188] K. Tai, A. Hasegawa, and A. Tomita, Observation of Modulational Instability in Optical Fibers, Phys. Rev. Lett. 1986, (56): 135-138.
[189] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1972).
[190] Y.V. Kartashov, V.A. Aleshkevich, V.A. Vysloukh et al., Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity, Phys. Rev. E 2003, (67): 036613 1-11.
[191] H.J. Shin, Multisoliton complexes moving on a cnoidal wave background, Phys. Rev. E 2005, (71): 036628 1-10.
[192] S. Chen et al., Soliton complexes on a cnoidal wave background described by the generalized nonlinear schrodinger equation with varying coefficients, Phys. Rev. E (prepare).
[193] S. Chen, H. Liu, S. Zhang et al., Compression of Hermite–Gaussian pulses in an engineered optical fiber absorber with varying dispersion and nonlinearity, Phys. Lett. A 2006, (253): 493-496.
[194] H.J.A. da Silva, J.J. O’Reilly, Optical pulse modeling with Hermite-Gaussian functions, Opt. Lett. 1989, (14): 526-528
[195] A. Hook, D. Anderson, M. Lisak et al., Soliton-supported pulses at normaldispersion in optical fibers, J. Opt. Soc. Am. B 1993, (10): 2313-2320.
[196] S.-P. Gorza, N. Roig, Ph. Emplit et al., Snake Instability of a Spatiotemporal Bright Soliton Stripe, Phys. Rev. Lett. 2004, (92): 084101 1-4.
[197] S. Chen, H. Liu, and Y. Yi, Numerically simulating stochastic generalized nonlinear Schrodinger equations: Robust algorithm, Computational Physics, Proceedings of the joint conference of ICCP6 and CCP2003, edited by X.-G. Zhao, S. Jiang and X.-J. Yu (Rinton Press, New Jersey, 2005).
[198] L. Bergé, Wave collapse in physics: principles and applications to light and plasma waves, Phys. Rep. 1998, (303): 259-370.
[199] D.E. Pelinovsky, Y.S. Kivshar, and V.V. Afanasjev, Internal modes of envelope solitons, Physica D 1998, (116): 121-142.
[200] M. Facao and D.F. Parker, Stability of screening solitons in photorefractive media, Phys. Rev. E 2003, (68): 016610 1-7.
[201] T. Kapitula, Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrodinger equation, Physica D 1998, (116): 95-120.
[202] S. Chen et al., Phase fuctuations of linearly chirped solitons in a noisy optical fiber channel with varying dispersion, nonlinearity, and gain, Phys. Rev. A (prepare).
[203] K. Tamura and M. Nakazawa, Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers, Opt. Lett. 1996, (21): 68-70.
[204] T.Schreiber, C.K.Nielsen, B.Ortac et al., Microjoule-level all-polarization- maintaining femtosecond fiber source, Opt. Lett. 2006, (31): 574-576.
[205] C. Finot, S. Pitois, G. Millot et al., Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers, IEEE J. Sel. Top. Quantum Electron. 2004, (10): 1211-1218.
[206] C. Finot and G. Millot, Interaction between optical parabolic pulses in a Raman fiber amplifier, Opt. Express 2005, (13): 5825-5830.
[207] E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, New York, 1994).
[208] W. Cao and P.K.A. Wai, Amplification and compression of ultrashort fundamental solitons in an erbium-doped nonlinear amplifying fiber loop mirror, Opt. Lett. 2003, (28): 284-286.
[209] S.K. Varshney, T. Fujisawa, K. Saitoh et al., Novel design of inherently gain- flattened discrete highly nonlinear photonic crystal fiber Raman amplifier and dispersion compensation using a single pump in C-band, Opt. Express 2005, (13): 9516-9526.
[210] R. Trebino, K.W. DeLong, D.N. Fittinghoff et al., Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating, Rev. Sci.Instrum. 1997, (68): 3277-3295.
[211] F.W. Kerfoot and P.K. Runge, Future directions for undersea communications, AT&T Tech. J. 1995,(74): 93-102.
[212] N.S. Bergano et al., 40 Gb/s WDM transmission of eight 5 Gb/s data channels over transoceanic distances using the conventional NRZ modulation format, in OFC’95, optical fiber communication: summaries of papers presented at the Conference on Optical Fiber Communication, San Diego, California, 1995 Technical Digest Series, Vol. 8 (Optical Society of America, Washington, D.C.), p. PD19.
[213] N. Savage, Too much fiber? Opt. Photon. News, 2002(3), 13: 32-37.
[214] S. Chen, D. Shi, S. Zhang, and L. Yi, General formulation for parametrically controlled dark-soliton jitter in high-speed optical transmission systems, Opt. Commun. 2004, (242), 503-510.
[215] S. Chen, Novel Strategy for Numerically Evaluating Dark-Soliton Jitter in Optical Communications, Commun. Theor. Phys. 2005, (43): 847-849.
[216] D. Anderson and M. Lisak, Nonlinear asymmetric self-phase modulation and self- steepening of pulses in long optical waveguides, Phys. Rev. A 1983, (27): 1393- 1398.
[217] E.J. Woodbury, and W.K. Ng, Ruby laser operation in the near IR, Proc. IREE 1962, (50): 2347.
[218] R.H. Stolen and E. P. Ippen, Raman gain in glass optical waveguides, Appl. Phys. Lett. 1973, (22): 276-278.
[219] R.H. Stolen, Nonlinearity in fiber transmission, Proc. IEEE 1980, (68): 1232-1236.
[220] J.P. Gordon, Theory of the soliton self-frequency shift, Opt. Lett. 1986, (11): 662-664.
[221] F.M. Mitschke and L.F. Mollenauer, Discovery of the solitons self-frequency shift, Opt. Lett. 1986, (11): 659-661.
[222] V.P. Kalosha and J. Herrmann, Formation of Optical Subcycle Pulses and Full Maxwell-Bloch Solitary Waves by Coherent Propagation Effects, Phys. Rev. Lett. 1999, (83): 544-547.
[223] H.A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).
[224] D. Mihalache, N. Truta, and L.-C. Crasovan, Painlevé analysis and bright solitary waves of the higher-order nonlinear Schrodinger equation containing third-order dispersion and self-steepening term, Phys. Rev. E 1997, (56): 1064-1070.
[225] Y. Kodama and A. Hasegawa, Generation of asymptotically stable optical solitons and suppression of the Gordon-Haus effect, Opt. Lett. 1992, (17): 31-33.
[226] Y. S. Kivshar, M. Haelterman, P. Emplit et al., Gordon-Haus effect on dark solitons,Opt. Lett. 1994, (19): 19-21.
[227] V.I. Karpman, Perturbation theory for solitons, Sov. Phys. JETP, 1977, (46): 281-291.
[228] D.J. Kaup, A perturbation expansion for the Zakharov-Shabat inverse scattering transform, SIAM J. Appl. Math. 1976, (31): 121-133.
[229] A.I. Maimistov and M. Basharov, Nonlinear Optical Waves (Kluwer, Dordrecht, The Netherlands, 1999).
[230] S.L. Palacios, A. Guinea, J.M. Fernández-Díaz et al., Dark solitary waves in the nonlinear Schrodinger equation with third order dispersion, self-steepening, and self-frequency shift, Phys. Rev. E 1999, (60): R45-R47.
[231] A. Mahalingam and K. Porsezian, Propagation of dark solitons with higher-order effects in optical fibers, Phys. Rev. E 1999, (64): 046608 1-9.
[232] L. Li, Z. Li, Z. Xu et al., Gray optical dips in the subpicosecond regime, Phys. Rev. E 2002, (66): 046616 1-8.
[233] R. Hao, L. Li, Z. Li et al., Exact multisoliton solutions of the higher-order nonlinear Schrodinger equation with variable coefficients, Phys. Rev. E 2004, (70): 066603 1-6.
[234] Z. Li, L. Li, H. Tian et al., New Types of Solitary Wave Solutions for the Higher Order Nonlinear Schrodinger Equation, Phys. Rev. Lett. 2000, (84): 4096-4099.
[235] R. Yang, L. Li, R. Hao et al., Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrodinger equation, Phys. Rev. E 2005, (71): 036616 1-8.
[236] K. Nakkeeran, K. Porsezian, P. S. Sundaram et al., Optical Solitons in N-Coupled Higher Order Nonlinear Schrodinger Equations, Phys. Rev. Lett. 1998, (80): 1425-1428.
[237] R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys. (N.Y.) 1973, (14): 805-809.
[238] N. Sasa and J. Satsuma, New-type of soliton solutions for a higher-order nonlinear Schrodinger equation, J. Phys. Soc. Jpn. 1991, (60): 409-417.
[239] L.F. Mollenauer, J.P. Gordon, and S.G. Evangelides, The sliding-frequency guiding filter: an improved form of soliton jitter control, Opt. Lett. 1992, (17): 1575-1577.
[240] W.H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
[241] J.P. Hamaide, P. Emplit and M. Haelterman, Dark-soliton jitter in amplified optical transmission systems, Opt. Lett. 1991, (16): 1578-1580.
[242] M. Nakazawa and K. Suzuki, 10 Gbit/s pseudorandom dark soliton data transmission over 1200 km, Electron. Lett. 1995, (31): 1076-1077.
[243] P.V.Mamyshev, P.G.J. Wigley, J. Wilson et al., Adiabatic compression of Schrodinger solitons due to the combined perturbations of higher-order dispersion and delayed nonlinear response, Phys. Rev. Lett. 1993, (71): 73-76.
[244] M. Nakazawa, H.Kubota, K. Kurokawa et al., Femtosecond optical soliton propagation over long distance using adiabatic trapping and soliton standardization, J. Opt. Soc. Am. B 1991, (8): 1811-1817.
[245] K. Qiu, Soliton position jitter caused by periodic amplification and self-frequency shift, Electron. Lett. 1994, (30): 439-440.
[246] K.C. Kao and G.A. Hockham, Dielectric fiber surface waveguides for optical frequencies, Proc. IEE, 1966, (133): 1151-1158.
[247] J. G. Proakis, Digital Communications, (McGraw Hill, New York, 2000).
[248] A.H. Gnauck, G. Raybon, S. Chandrasekhar et al., 2.5 Tb/s (64×42.7 Gb/s) Transmission Over 40×100 km NZDSF Using RZ-DPSK Format and All-Raman- Amplified Spans, Post Deadline paper, OFC, Anaheim, CA, March 17-22, 2002.
[249] C.J. McKinstrie and C. Xie, Phase Jitter in Single-Channel Soliton Systems with Constant Dispersion, IEEE J. Sel. Top. Quantum Electron. 2002, (8): 616-625.
[250] C. Xu, X. Liu, L.F. Mollenauer et al., Comparison of Return-to-Zero Differential Phase-Shift Keying and ON–OFF Keying in Long-Haul Dispersion Managed Transmission, IEEE Photon. Technol. Lett., 2003, (15): 617-619.
[251] M. Hanna, H. Porte, J.-P. Goedgebuer et al., Performance assessment of DPSK soliton transmission system, Electon. Lett. 2001, (37): 644-646.
[252] H. Tsuchida, Correlation between amplitude and phase noise in a mode-locked Cr:LiSAF laser, Opt. Lett. 1998, (23): 1686-1688.
[253] W.K. Marshall, B. Crosignani, and A. Yariv, Laser phase noise to intensity noise conversion by lowest-order group-velocity dispersion in optical fiber: exact theory, Opt. Lett. 2000, (25): 165-167.
[254] H. Kim and A.H. Gnauck, Experimental Investigation of the Performance Limitation of DPSK Systems Due to Nonlinear Phase Noise, IEEE Photon. Technol. Lett., 2003, (15): 320-322.
[255] K.-P. Ho, Probability density of nonlinear phase noise, J. Opt. Soc. Am. B 2003, (20): 1875-1879.
[256] A. Mecozzi, Probability density functions of the nonlinear phase noise, Opt. Lett. 2004, (29): 673-675.
[257] X. Wei and X. Liu, Analysis of intrachannel four-wave mixing in differential phase- shift keying transmission with large dispersion, Opt. Lett. 2003, (28): 2300-2302.
[258] X. Wei, X. Liu, S.H. Simon, and C. J. McKinstrie, Intrachannel four-wave mixing in highly dispersed return-to-zero differential-phase-shift-keyed transmission with a nonsymmetric dispersion map, Opt. Lett. 2006, (31): 29-31.
[259] P.P. Mitra and J.B. Stark, Nonlinear limits to the information capacity of optical fibre communications, Nature 2001, (411): 1027-1030.
[260] J.M. Kahn and K.-P. Ho, Communications technology: A bottleneck for optical fibres, Nature 2001, (411): 1007-1010.
[261] X. Wei, A.H. Gnauck, D.M. Gill et al., Optical π/2-DPSK and its Tolerance to Filtering and Polarization-Mode Dispersion, IEEE Photon. Technol. Lett., 2003, (15): 1639-1641.
[262] U.-V. Koc and X. Wei, Combined Effect of Polarization-Mode Dispersion and Chromatic Dispersion on Strongly Filtered π/2-DPSK and Conventional DPSK, IEEE Photon. Technol. Lett., 2004, (16): 1588-1590.
[263] C. Xie, L. Muller, H. Haunstein et al., Comparison of System Tolerance to Polarization-Mode Dispersion Between Different Modulation Formats, IEEE Photon. Technol. Lett., 2003, (15): 1168-1170.
[264] M. Hanna, D. Boivin, P.-A. Lacourt et al., Calculation of optical phase jitter in dispersion managed systems by use of the moment method, J. Opt. Soc. Am. B 2004, (21): 24-28.
[265] C.J. McKinstrie and T.I. Lakoba, Probability-density function for energy perturbations of isolated optical pulses, Opt. Express 2003, (11): 3628-3648.
[266] S.A. Derevyanko, S.K. Turitsyn, and D.A.Yakushev, Fokker-Planck equation approach to the description of soliton statistics in optical fiber transmission systems, J. Opt. Soc. Am. B 2005, (22): 743-752.
[267] S. Chen et al., Study of chirp statistics in nonlinearity- and dispersion-managed soliton systems: a Fokker-Planck equation method, Opt. Lett. (prepare).
[268] S.K. Turitsyn, T. Schaefer, and V.K. Mezentsev, Generalized momentum method to describe high-frequency solitary wave propagation in systems with varying dispersion, Phys. Rev. E 1998, (58): R5264-R5267.
[269] K.J. Blow, N.J. Doran, and S.J.D. Phoenix, The soliton phase, Opt. Commun. 1992, (88): 137-140.
[270] K. Ito, Lectures on Stochastic Processes (Tata Institute of Fundamental Research, Bombay, 1960).
[271] R.L. Stratanovich, Topics in the Theory of Random Noise (Gordon & Breach, New York, 1963), Vols. I and II.
[272] C.W. Gardiner, Handbook of Stochastic Methods, 2nd Ed. (Springer, Berlin, 1996).
[273] P.E. Kloeden and E. Platen, The Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992).
[274] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, England, 1995).
[275] P.D. Drummond and I.K. Mortimer, Computer simulations of multiplicative stochastic differential equations, J. Comput. Phys. 1991, (93): 144-170.
[276] M.J. Werner and P.D. Drummond, Robust algorithms for solving stochastic partial differential equations, J. Comput. Phys. 1997, (132): 312-326.
[277] B. Khubchandani, P.N. Guzdar, and R. Roy, Influence of stochasticity on multiple four-wave-mixing processes in an optical fiber, Phys. Rev. E 2002, (66): 066609 1-8.
[278] W.H. Press, B.P. Flannery, S.A. Teukolsky et al., Numerical recipes: the art of scientific computing (Cambridge University Press, Cambridge, England, 1986).
[279] J. Garcia-Ojalvo and J.M. Sancho, Noise in spatially extended systems (Springer, New York, 1999).
[280] J.W. Cooley, P.A.W. Lewis, and P.D. Welch, The fast Fourier transform and its applications, IEEE Trans. Education, 1969, (12): 27-34.
[2] M. Segev, Solitons: A Universal Phenomenon of Self-Trapped Wave Packets, Opt. Photon. News, 2002(2), 13: 27.
[3] R.W. Boyd, Nonlinear Optics (Academic Press, Boston, 2003).
[4] Y.R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
[5] T.H. Maiman, Stimulated Optical Radiation in Ruby, Nature 1960, (187): 493–494.
[6] J. Hecht, Beam: The Race to Make the Laser, Opt. Photon. News, 2005(7/8), 16: 24-29.
[7] P.A. Franken, A.E. Hill, C.W. Peters et al., Generation of Optical Harmonics, Phys. Rev. Lett. 1961, (7): 118-119.
[8] A. Hasegawa and F.D.Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys. Lett. 1973, (23): 142-144.
[9] L.F. Mollenauer, R.H. Stolen, and J.P. Gordon, Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers, Phys. Rev. Lett. 1980, (45): 1095-1098.
[10] G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).
[11] L.F. Mollenauer, J.P. Gordon, and S.G. Evangelides, Laser Focus World 159 (November 1991).
[12] Y.N. Karamzin and A.P. Sukhorukov, Mutual focusing of high-power light beams in media with quadratic nonlinearity, Sov. Phys. JETP 1975, (41): 414-420.
[13] G.I. Stegeman, M. Sheik-Bahae, E.V. Stryland et al., Large nonlinear phase shifts in second-order nonlinear-optical processes, Opt. Lett. 1993, (18): 13-15.
[14] L. Torner and A.P. Sukhorukov, Quadratic Solitons, Opt. Photon. News, 2002(2), 13: 42-47.
[15] A.V. Buryaka, P.D. Trapanib, D.V. Skryabinc et al., Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep. 2002, (370): 63-235.
[16] S.L. McCall and E.L. Hahn, Self-Induced Transparency by Pulsed Coherent Light, Phys. Rev. Lett. 1967, (18): 908-911.
[17] H.M. Gibbs and R.E. Slusher, Peak Amplification and Breakup of a Coherent Optical Pulse in a Simple Atomic Absorber, Phys. Rev. Lett. 1970, (24): 638–641.
[18] L.V. Hau, S.E. Harris, Z. Dutton et al., Light speed reduction to 17 metres per second in an ultracold atomic gas, Nature 1999, (397): 594–598.
[19] C. Liu, Z. Dutton, C. H. Behroozi et al., Observation of coherent optical information storage in an atomic medium using halted light pulses, Nature 2001, (409): 490–493.
[20] M. Fleischhauer, A. Imamoglu, and J.P. Marangos, Electromagnetically induced transparency: Optics in coherent media, Rev. Mod. Phys., 2005, (77): 633-673.
[21] Y. Wu and L. Deng, Ultraslow Optical Solitons in a Cold Four-State Medium, Phys. Rev. Lett. 2004, (93): 143904 1-4.
[22] T. Hong, Spatial Weak-Light Solitons in an Electromagnetically Induced Nonlinear Waveguide, Phys. Rev. Lett. 2003, (90): 183901 1-4.
[23] G.I. Stegeman and M. Segev, Optical Spatial Solitons and Their Interactions: Universality and Diversity, Science 1999, (286):1518-1523.
[24] Y.S. Kivshar and G.I. Stegeman, Spatial Optical Solitons: Guiding Light for Future Technologies, Opt. Photon. News, 2002(2), 13: 59-63.
[25] S. Trillo and W. Torruellas, Spatial Solitons (Springer, New York, 2001).
[26] Y.S. Kivshar and B. Luther-Davies, Dark optical solitons: physics and applications, Phys. Rep. 1998, (298), 81-197.
[27] R.Y. Chiao, E. Garmire, C.H. Townes, Self-Trapping of Optical Beams, Phys. Rev. Lett. 1964, (13): 479-482.
[28] M. Segev, B. Crosignani, A. Yariv, et al., Spatial Solitons in Photorefractive Media, Phys. Rev. Lett. 1992, (68): 923-926.
[29] M. Segev and D.N. Christodoulides, Incoherent Solitons: Self-Trapping of Weakly Correlated Wave Packets, Opt. Photon. News, 2002(2), 13: 70-76.
[30] J.W. Fleischer, M. Segev, N.K. Efremidis et al., Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature 2003, (422): 147-150.
[31] J.W. Fleischer, T. Carmon, M. Segev et al., Observation of Discrete Solitons in Optically-Induced Real-Time Waveguide Arrays, Phys. Rev. Lett. 2003, (90): 023902.
[32] A. Barthelemy, S. Maneuf, C. Froehly, Soliton propagation and self-trapping of laser beams by the optical Kerr effect, Opt. Commun. 1985, (55): 201-206.
[33] B. Crosignani and G. Salamo, Photorefractive Solitons, Opt. Photon. News, 2002(2), 13: 38-41.
[34] M. Mitchell, Z. Chen, M. Shih et al., Self-Trapping of Partially Spatially Incoherent Light, Phys. Rev. Lett. 1996, (77): 490-493.
[35] M. Mitchell and M. Segev, Self-trapping of incoherent white light, Nature 1997, (387): 880-883.
[36] Z. Chen, M. Mitchell, M. Segev et al., Self-Trapping of Dark Incoherent Light Beams, Science 1998, (280): 889-892.
[37] A.W. Snyder and A. P. Sheppard, Collisions, steering, and guidance with spatial solitons, Opt. Lett. 1993, (18): 482-484.
[38] M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, Cambridge, 1991).
[39] M. Segev and G.I. Stegeman, Self-Trapping of Optical Beams: Spatial Solitons, Phys. Today 1998, (51): 42-48.
[40] F. Lederer and Y. Silberberg, Discrete Solitons, Opt. Photon. News, 2002(2), 13: 48-53.
[41] S.V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys. JETP 1974, (38): 248-253.
[42] J.U. Kang, G.I. Stegeman, J.S. Aitchison et al., Observation of Manakov Spatial Solitons in AlGaAs Planar Waveguides, Phys. Rev. Lett. 1996, (76): 3699-3702.
[43] W.E. Torruellas, Z. Wang, D.J. Hagan et al., Observation of Two-Dimensional Spatial Solitary Waves in a Quadratic Medium, Phys. Rev. Lett. 1995, (74): 5036-5039.
[44] F. Wise and P. Di Trapani, The Hunt for Light Bullets: Spatiotemporal Solitons, Opt. Photon. News, 2002(2), 13: 28-32.
[45] X. Liu, L.J. Qian, and F.W. Wise, Generation of Optical Spatiotemporal Solitons, Phys. Rev. Lett. 1999, (82): 4631-4634.
[46] N.J. Zabusky and M.D. Kruskal, Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States, Phys. Rev. Lett. 1965, (15): 240-243.
[47] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 1972, (34): 62-69.
[48] J.P. Gordon, Interaction forces among solitons in optical fibers, Opt. Lett. 1983, (8): 596-598.
[49] T. Kanna and M. Lakshmanan, Exact Soliton Solutions, Shape Changing Collisions, and Partially Coherent Solitons in Coupled Nonlinear Schrodinger Equations, Phys. Rev. Lett. 2001, (86): 5043-5046.
[50] K. Porsezian and K. Nakkeeran, Optical Solitons in Presence of Kerr Dispersion and Self-Frequency Shift, Phys. Rev. Lett. 1996, (76): 3955-3958.
[51] M. Gedalin, T.C. Scott, and Y.B. Band, Optical Solitary Waves in the Higher Order Nonlinear Schrodinger Equation, Phys. Rev. Lett. 1997, (78): 448-451.
[52] H.A. Haus and W.S. Wong, Solitons in optical communications, Rev. Mod. Phys. 1996, (68): 423-444.
[53] G.P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 2002).
[54] M. Peccianti, C. Conti, G. Assanto et al., All-optical switching and logic gating with spatial solitons in liquid crystals, Appl. Phys. Lett. 2002, (81): 3335-3337.
[55] M. Peccianti, C. Conti, G. Assanto et al., Routing of anisotropic spatial solitons and modulational instability in liquid crystals, Nature (London) 2004, (432): 733-737.
[56] R. McLeod, K. Wagner, and S. Blair, (3+1)-dimensional optical soliton dragging logic, Phys. Rev. A 1995, (52): 3254-3278.
[57] N.N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).
[58] N.N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, New York, 2005).
[59] Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE J. Quantum Electron. 1987, (23): 510-524.
[60] Z. Li, L. Li, H. Tian et al., Chirped Femtosecond Solitonlike Laser Pulse Form with Self-Frequency Shift, Phys. Rev. Lett. 2002, (89):263901 1-4.
[61] W.H. Reeves, D.V. Skryabin, F. Biancalana et al., Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibres, Nature 2003, (424): 511-515.
[62] R. Conte and M. Musette, Linearity inside nonlinearity: exact solutions to the complex Ginzburg-Landau equation, Physica D 1993, (69): 1–17.
[63] N. Akhmediev and V.V. Afanasjev, Novel Arbitrary-Amplitude Soliton Solutions of the Cubic-Quintic Complex Ginzburg-Landau Equation, Phys. Rev. Lett. 1995, (75): 2320- 2323.
[64] J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Pulsating, Creeping, and Erupting Solitons in Dissipative Systems, Phys. Rev. Lett. 2000, (85):2937-2940.
[65] S.T. Cundiff, J.M. Soto-Crespo, and N. Akhmediev, Experimental Evidence for Soliton Explosions, Phys. Rev. Lett. 2002, (88):073903 1-4.
[66] J.M. Soto-Crespo and N. Akhmediev, Soliton as Strange Attractor: Nonlinear Synchronization and Chaos, Phys. Rev. Lett. 2005, (95): 024101 1-4.
[67] R.J. Deissler and H.R. Brand, Periodic, Quasiperiodic, and Chaotic Localized Solutions of the Quintic Complex Ginzburg-Landau Equation, Phys. Rev. Lett.1994,(72): 478-481.
[68] R.J. Deissler and H.R. Brand, Interaction of Breathing Localized Solutions for Subcritical Bifurcations, Phys. Rev. Lett. 1995, (74): 4847-4850.
[69] R.J. Deissler and H.R. Brand, Effect of Nonlinear Gradient Terms on Breathing Localized Solutions in the Quintic Complex Ginzburg-Landau Equation, Phys. Rev. Lett. 1998, (81): 3856-3859.
[70] A. Hasegawa, Optical Solitons in Fibers for Communication systems, Opt. Photon. News, 2002(2), 13: 33-37.
[71] C. Lin, H. Kogelnik, and L.G. Cohen, Optical-pulse equalization of low-dispersion transmission in single-mode fibers in the 1.3-1.7-μm spectral region, Opt. Lett. 1980, (5): 476-478.
[72] N.J. Smith, F.M. Knox, N.J. Doran et al., Enhanced power solitons in optical fibres with periodic dispersion management, Electron. Lett. 1996, (32): 54-55.
[73] Q. Quraishi, S.T. Cundiff, B. Ilan et al., Dynamics of Nonlinear and Dispersion Managed Solitons, Phys. Rev. Lett. 2005, (94): 243904 1-4.
[74] H.S. Eisenberg, Y. Silberberg, R. Morandotti et al., Diffraction Management, Phys. Rev. Lett. 2000, (85): 1863-1866.
[75] M.J. Ablowitz and Z.H. Musslimani, Discrete Diffraction Managed Spatial Solitons, Phys. Rev. Lett. 2001, (87): 254102 1-4.
[76] L.F. Mollenauer, P.V. Mamyshev, J. Gripp et al., Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersion- managed solitons, Opt. Lett. 2000, (25): 704-706.
[77] D.W. Prather, Photonic Crystals: An Engineering Perspective, Opt. Photon. News, 2002(6), 13: 16-19.
[78] J. Knight, T. Birks, B. Mangan et al., Photonic Crystal Fibers: New Solutions in Fiber Optics, Opt. Photon. News, 2002(3), 13: 26-30.
[79] V.N. Serkin and A. Hasegawa, Novel Soliton Solutions of the Nonlinear Schrodinger Equation Model, Phys. Rev. Lett. 2000, (85): 4502-4505.
[80] V.I. Kruglov, A.C. Peacock, and J.D. Harvey, Exact Self-Similar Solutions of the Generalized Nonlinear Schrodinger Equation with Distributed Coefficients, Phys. Rev. Lett. 2003, (90): 113902 1-4.
[81] S. Chen and L. Yi, Chirped self-similar solutions of a generalized nonlinear Schrodinger equation model, Phys. Rev. E 2005, (71): 016606 1-4.
[82] T. S. Raju, P.K. Panigrahi, and K. Porsezian, Self-similar propagation and compression of chirped self-similar waves in asymmetric twin-core fibers with nonlinear gain, Phys. Rev. E 2005, (72): 046612 1-6.
[83] L. Wang, L. Li, Z. Li et al., Generation, compression, and propagation of pulse trains in the nonlinear Schrodinger equation with distributed coefficients, Phys. Rev. E 2005, (72): 036614 1-7.
[84] W.J. Tomlinson, R.H. Stolen, and A.M. Johnson, Optical wave breaking of pulses in nonlinear optical fibers, Opt. Lett. 1985, (10): 457-459.
[85] D. Anderson, M. Desaix, M. Lisak et al., Wave breaking in nonlinear-optical fibers, J. Opt. Soc. Am. B 1992, (9): 1358-1361.
[86] T. Brabec and F. Krausz, Intense few-cycle laser fields: Frontiers of nonlinear optics, Rev. Mod. Phys. 2000, (72): 545-591.
[87] G. Steinmeyer, D.H. Sutter, L. Gallmann et al., Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics, Science 1999, (286):1507-1512.
[88] D. Méchin, S.-H. Im, V. I. Kruglov et al., Experimental demonstration of similariton pulse compression in a comblike dispersion-decreasing fiber amplifier, Opt. Lett. 2006, (31): 2106-2108.
[89] V.N. Serkin, A. Hasegawa, and T.L. Belyaeva, Comment on “Exact Self-Similar Solutions of the Generalized Nonlinear Schrodinger Equation with Distributed Coefficients”, Phys. Rev. Lett. 2004, (92): 199401 1-1.
[90] G.I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge University Press, Cambridge, 1996).
[91] S. Chen, L. Yi, D.-S. Guo et al., Self-similar evolutions of parabolic, Hermite- Gaussian, and hybrid optical pulses: Universality and diversity, Phys. Rev. E 2005, (72): 016622 1-5.
[92] P.L. Kelley, Self-Focusing of Optical Beams, Phys. Rev. Lett. 1965, (15): 1005- 1008.
[93] I.V. Barashenkov, Stability Criterion for Dark Solitons, Phys. Rev. Lett. 1996, (77): 1193-1197.
[94] A.V. Buryak, Y.S. Kivshar, and S. Trillo, Stability of Three-Wave Parametric Solitons in Diffractive Quadratic Media, Phys. Rev. Lett. 1996, (77): 5210-5213.
[95] D.E. Pelinovsky, A.V. Buryak, and Y.S. Kivshar, Instability of Solitons Governed by Quadratic Nonlinearities, Phys. Rev. Lett. 1995, (75): 591-595.
[96] E.A. Ostrovskaya, Y.S. Kivshar, D.V. Skryabin et al., Stability of Multihump Optical Solitons, Phys. Rev. Lett. 1999, (83): 296-299.
[97] D. Mihalache, D. Mazilu, and L. Torner, Stability of Walking Vector Solitons, Phys. Rev. Lett. 1998, (81): 4353-4356.
[98] V. Skarka and N. B. Aleksia, Stability Criterion for Dissipative Soliton Solutions of the One-, Two-, and Three-Dimensional Complex Cubic-Quintic Ginzburg-Landau Equations, Phys. Rev. Lett. 2006, (96): 013903 1-4.
[99] N.N. Akhmediev and A. Ankiewicz,Solitons Around Us: Integrable, Hamiltonian and Dissipative Systems,In: Optical Solitons, Theoretical and Experimental Challenges, Lecture Notes in Physics, vol. 613, Editors: K. Porsezian and V.C. Kurakose (Springer, Berlin, 2003), p. 105-126.
[100] E.A. Kuznetsov, A.M. Rubenchik, and V.E. Zakharov, Soliton stability in plasmas and hydrodynamics, Phys. Rep. 1986, (142): 103-165.
[101] A. Hasegawa, Amplification and reshaping of optical solitons in a glass fiber - IV:Use of stimulated Raman process, Opt. Lett. 1983, (8): 650-652.
[102] L.F. Mollenauer and K. Smith, Demonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gain, Opt. Lett. 1988, (13): 675-677.
[103] A. Hasegawa, Optical Soliton Theory and Its Applications in Communication, in: Optical Solitons, Theoretical and Experimental Challenges, eds. K. Porsezian and V.C. Kuriakose (Springer, Berlin, 2002), 9-31.
[104] A. Hasegawa, Massive WDM and TDM Soliton Transmission Systems (Kluwer Academic, Dordrecht, The Netherlands, 2000).
[105] M. Nakazawa, H. Kubota, K. Suzuki et al., Ultrahigh-speed long-distance TDM and WDM soliton transmission technologies, IEEE J. Sel. Top. Quantum Electron. 2000, (6): 363-396.
[106] K. Fukuchi, M. Kakui, A. Sasaki et al., 1.1-Tb/s (55×20-Gb/s) dense WDM soliton transmission over 3,020-km widely-dispersion-managed transmission line employing 1.55/1.58-μm hybrid repeaters, in European Conference on Optical Communication (Institute of Electrical Engineers, London, 1999), paper PD42.
[107] K. Suzuki, H. Kubota, and M. Nakazawa, 1 Tb/s (40 Gb/s×25 channel) DWDM quasi-DM soliton transmission over 1,500 km using dispersion-managed single-mode fiber and conventional C-band EDFAs, Optical Fiber Communication Conference and Exhibit, 2001, (2): TuN7-1-3.
[108] M. Nakazawa, K. Suzuki, and H. Kubota, 160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation, Electron. Lett. 1999, (35): 1358-1359.
[109] G. Zhu, L. Mollenauer, and C. Xu, Experimental Demonstration of Postnonlinearity Compensation in a Multispan DPSK Transmission, IEEE Photon. Technol. Lett. 2006, (18): 1007-1009.
[110] R.-J. Essiambre and G.P. Agrawal, Soliton communication systems, in: Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1997), Vol. 37, p.185-256.
[111] E. Iannone, F. Matera, A. Mecozzi et al., Nonlinear Optical Communications Networks (Wiley, New York,1998).
[112] K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, New York, 2005).
[113] W. Schottky, über spontane Stromschwankungen in verschiedenen Elektrizitatsleitern. Ann. Phys. (Leipz.) 1918, (57): 541-568.
[114] H.A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, New York, 2000).
[115] P.D. Drummond, R.M. Shelby, S.R. Friberg et al., Quantum solitons in optical fibers, Nature (London) 1993, (365): 307-313.
[116] J.P. Gordon and H.A. Haus, Random walk of coherently amplified solitons in optical fiber transmission, Opt. Lett. 1986, (11): 665-667.
[117] J.P. Gordon and L.F. Mollenauer, Phase noise in photonic communications systems using linear amplifiers, Opt. Lett. 1990, (15): 1351-1353.
[118] P.D. Drummond and J.F. Corney, Quantum noise in optical fibers. I. Stochastic equations, J. Opt. Soc. Am. B 2001, (18): 139-152.
[119] J.F. Corney and P.D. Drummond, Quantum noise in optical fibers. II. Raman jitter in soliton communications, J. Opt. Soc. Am. B 2001, (18): 153-161.
[120] S. Chen, S. Zhang, and D. Shi, Quantum Theory of Femtosecond Soliton Propagation in Single-Mode Optical Fibers, Commun. Theor. Phys. 2004, (41): 943-948.
[121] S. Chen, D. Shi, and L. Yi, Timing jitter of femtosecond solitons in single-mode optical fibers: A perturbation model, Phys. Rev. E 2004, (69): 046602 1-12.
[122] L.F. Mollenauer and J.P. Gordon, Birefringence-mediated timing jitter in soliton transmission, Opt. Lett. 1994, (19): 375-377.
[123] L.F. Mollenauer, P.V. Mamyshev, and M.J. Neubelt, Measurement of timing jitter in filter-guided soliton transmission at 10 Gbits/s and achievement of 375 Gbits/s-Mm, error free, at 12.5 and 15 Gbits/s, Opt. Lett. 1994, (19): 704-706.
[124] P.V. Mamyshev and L.F. Mollenauer, Soliton collisions in wavelength-division- multiplexed dispersion-managed systems, Opt. Lett. 1999, (24): 448-450.
[125] A. Mecozzi, J.D. Moores, H.A. Haus et al., Soliton transmission control, Opt. Lett. 1991, (16):1841-1843.
[126] G.M. Carter, J.M. Jacob, C.R. Menyuk et al., Timing-jitter reduction for a dispersion-managed soliton system: experimental evidence, Opt. Lett. 1997, (22): 513-515.
[127] R.-M. Mu, V.S. Grigoryan, C.R. Menyuk et al., Timing-jitter reduction in a dispersion-managed soliton system, Opt. Lett. 1998, (23): 930-932.
[128] R.J. Essiambre and G.P. Agrawal, Timing jitter of ultrashort solitons in high-speed communication systems. II. Control of jitter by periodic optical phase conjugation, J. Opt. Soc. Am. B 1997, (14): 323-330.
[129] A.H. Gnauck and P. J. Winzer, Optical Phase-Shift-Keyed Transmission, J. Lightwave Technol. 2005, (23): 115-130.
[130] C. J. McKinstrie, C. Xie, and C. Xu, Effects of cross-phase modulation on phase jitter in soliton systems with constant dispersion, Opt. Lett. 2003, (28): 604-606.
[131] S. Kumar, Analysis of intrachannel impairments in differential phase-shift keying transmission systems, Opt. Lett. 2005, (30): 2053-2055.
[132] A.G. Green, P.P. Mitra, and L.G.L. Wegener, Effect of chromatic dispersion onnonlinear phase noise, Opt. Lett. 2003, (28): 2455-2457.
[133] S. Kumar, Effect of dispersion on nonlinear phase noise in optical transmission systems, Opt. Lett. 2005, (30): 3278-3280.
[134] K.-P. Ho and H.-C. Wang, Effect of dispersion on nonlinear phase noise, Opt. Lett. 2006, (31): 2109-2111.
[135] J.N. Kutz and P.K.A. Wai, Noise-induced amplitude and chirp jitter in dispersion- managed soliton communications, Opt. Lett. 1998, (23): 1022-1024.
[136] X. Liu, X. Wei, R.E. Slusher et al., Improving transmission performance in differential phase-shift-keyed systems by use of lumped nonlinear phase-shift compensation, Opt. Lett. 2002, (27): 1616-1618.
[137] C. Xu and X. Liu, Postnonlinearity compensation with data-driven phase modulators in phase-shift keying transmission, Opt. Lett. 2002, (27): 1619-1621.
[138] S.A. Derevyanko and S.K. Turitsyn, Reduction of the phase jitter in differential phase-shift-keying soliton transmission systems by in-line Butterworth filters, Opt. Lett. 2004, (29): 35-37.
[139] D. Boivin, M. Hanna, P.-A. Lacourt et al., Reduction of phase jitter in dispersion- managed systems by in-line filtering, Opt. Lett. 2004, (29): 688-690.
[140] C.J. McKinstrie, S. Radic, and C. Xie, Reduction of soliton phase jitter by in-line phase conjugation, Opt. Lett. 2003, (28): 1519-1521.
[141] H.A. Haus and Y. Lai, Quantum theory of soliton squeezing: a linearized approach, J. Opt. Soc. Am. B 1990, (7): 386-392.
[142] D.J. Kaup, Perturbation theory of solitons in optical fibers, Phys. Rev. A 1990, (42): 5689-5694.
[143] Y.S. Kivshar and B.A. Malomed, Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys. 1989, (61): 763-915.
[144] S.N. Vlasov, V.A. Petrishchev, and V.I. Talanov, Averaged description of wave beams in linear and nonlinear media (the method of moments), Radiophys. Quantum Electron. 1971, (14): 1062–1070.
[145] D. Anderson, Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A 1983, (27): 3135-3145.
[146] J. Santhanam and G.P.Agrawal, Reduced timing jitter in dispersion-managed lightwave systems through parametric amplification, J. Opt. Soc. Am. B 2003, (20): 284-291.
[147] S. Chen et al., Phase jitter of nonlinearity- and dispersion-managed solitons: Moment-method description taking into account the chirp fluctuations effect, Phys. Rev. E (prepare).
[148] M. Suzuki, I. Morita, N. Edagawa et al., Reduction of Gordon-Haus timing jitter byperiodic dispersion compensation in soliton transmission, Electron. Lett. 1995, (31): 2027-2029.
[149] P.V. Mamyshev and L.F. Mollenauer, Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission, Opt. Lett. 1996, (21): 396-398.
[150] A.M. Niculae, W. Forysiak, A.J. Gloag et al., Soliton collisions with wavelength- division multiplexed systems with strong dispersion management, Opt. Lett. 1998, (23): 1354-1356.
[151] C. Paré, A. Villeneuve, P.-A. Bélanger et al., Compensating for dispersion and the nonlinear Kerr effect without phase conjugation, Opt. Lett. 1996, (21): 459-461.
[152] C. Paré, A. Villeneuve, S. LaRochelle, Split compensation of dispersion and self-phase modulation in optical communication systems, Opt. Commun. 1999, (160): 130-138.
[153] F.O. Ilday and F.W. Wise, Nonlinearity management: a route to high-energy soliton fiber lasers, J. Opt. Soc. Am. B 2002, (19): 470-476.
[154] D.E. Pelinovsky, P.G. Kevrekidis, and D.J. Frantzeskakis, Averaging for Solitons with Nonlinearity Management, Phys. Rev. Lett. 2003, (91): 240201 1-4.
[155] G.L. Lamb, JR., Analytical Descriptions of Ultrashort Optical Pulse Propagation in a Resonant Medium, Rev. Mod. Phys. 1971, (43): 99-124.
[156] I.R. Gabitov and S.V. Manakov, Propagation of Ultrashort Optical Pulses in Degenerate Laser Amplifier, Phys. Rev. Lett. 1983, (50): 495-498.
[157] S. An and J.E. Sipe, Universality in the dynamics of phase grating formation in optical fibers, Opt. Lett. 1991, (16): 1478-1480.
[158] C.R. Menyuk, D. Levi, and P. Winternitz, Self-Similarity in Transient Stimulated Raman Scattering, Phys. Rev. Lett. 1992, (69): 3048-3051.
[159] D. Anderson, M. Desaix, M. Karlsson et al., Wave-breaking-free pulses in nonlinear-optical fibers, J. Opt. Soc. Am. B 1993, (10): 1185-1190.
[160] T.M. Monro, P.D. Millar, L. Poladian et al., Self-similar evolution of self-written waveguides, Opt. Lett. 1998, (23): 268-270.
[161] T.M. Monro, D. Moss, M. Bazylenko et al., Observation of Self-Trapping of Light in a Self-Written Channel in a Photosensitive Glass, Phys. Rev. Lett. 1998, (80): 4072-4075.
[162] M. Soljacic, M. Segev, and C.R. Menyuk, Self-similarity and fractals in soliton- supporting systems, Phys. Rev. E 2000, (61): R1048-R1051.
[163] S. Sears, M. Soljacic, M. Segev et al., Cantor Set Fractals from Solitons, Phys. Rev. Lett. 2000, (84): 1902-1905.
[164] M.E. Fermann, V.I. Kruglov, B.C. Thomsen et al., Self-Similar Propagation andAmplification of Parabolic Pulses in Optical Fibers, Phys. Rev. Lett. 2000, (84): 6010- 6013.
[165] V.I. Kruglov, A.C. Peacock, J.D. Harvey et al., Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers, J. Opt. Soc. Am. B 2002, (19): 461-469.
[166] C. Finot and G. Millot, Synthesis of optical pulses by use of similaritons, Opt. Express 2004, (12): 5104-5109.
[167] C. Finot and G. Millot, Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers, Opt. Lett. 2004, (29): 2533-2535.
[168] V.I. Kruglov, A.C. Peacock, J.M. Dudley et al., Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers, Opt. Lett. 2000, (25): 1753-1755.
[169] C. Finot and G. Millot, Collisions between similaritons in optical fiber amplifiers, Opt. Express 2005, (13): 7653-7665.
[170] G. Chang, H.G. Winful, A. Galvanauskas et al., Self-similar parabolic beam generation and propagation, Phys. Rev. E 2005, (72): 016609 1-4.
[171] P.D. Drummond and K.V. Kheruntsyan, Asymptotic solutions to the Gross- Pitaevskii gain equation: Growth of a Bose-Einstein condensate, Phys. Rev. E 2000, (63): 013605 1-5.
[172] J.P. Limpert, T. Schreiber, T. Clausnitzer et al., High-power femtosecond Yb-doped fiber amplifier, Opt. Express 2002, (10): 628-638.
[173] C. Finot, G. Millot, C. Billet et al., Experimental generation of parabolic pulses via Raman amplification in optical fiber, Opt. Express 2003, (11): 1547-1552.
[174] F.O. Ilday, J. R. Buckley, W.G. Clark et al., Self-Similar Evolution of Parabolic Pulses in a Laser, Phys. Rev. Lett. 2004, (92): 213902 1-4.
[175] C. Billet, J.M. Dudley, N. Joly et al., Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm, Opt. Express 2005, (13): 3236-3241.
[176] C. Finot, F. Parmigiani, P. Petropoulos et al., Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime, Opt. Express 2006, (14): 3161-3170.
[177] C. Finot, S. Pitois, and G. Millot, Regenerative 40 Gbit/s wavelength converter based on similariton generation, Opt. Lett. 2005, (30): 1776-1778.
[178] T. Hirooka and M. Nakazawa, Parabolic pulse generation by use of a dispersion- decreasing fiber with normal group-velocity dispersion, Opt. Lett. 2004, (29): 498-500.
[179] V.N. Serkin and A. Hasegawa, Exactly Integrable Nonlinear Schrodinger Equation Models With Varying Dispersion, Nonlinearity and Gain: Application for SolitonDispersion Managements, IEEE J. Sel. Top. Quantum Electron. 2002, (8): 418-431.
[180] V.I. Kruglov, A.C. Peacock, and J.D. Harvey, Exact solutions of the generalized nonlinear Schrodinger equation with distributed coefficients, Phys. Rev. E 2005, (71): 056619 1-11.
[181] V.N. Serkin, M. Matsumoto, and T.L. Belyaeva, Nonlinear Bloch Waves, JETP Lett. 2001, (73): 59-62.
[182] M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan et al., Dispersion of bound electron nonlinear refraction in solids, IEEE J. Quantum Electron. 1991, (27): 1296-1309.
[183] C. Liberale, V. Degiorgio, M. Marangoni et al., Measurement of the nonlinear phase shift induced by cascaded interactions in a periodically poled lithium niobate waveguide, Opt. Lett. 2005, (30): 2448-2450.
[184] R. DeSalvo, D.J. Hagan, M. Sheik-Bahae et al., Self-focusing and self-defocusing by cascaded second-order effects in KTP, Opt. Lett. 1992, (17): 28-30.
[185] E.T. Copson, Asymptotic Expansions (Cambridge University press, 1965).
[186] D.B.S. Soh, J. Nilsson, and A.B. Grudinin, Efficient femtosecond pulse generation using a parabolic amplifier combined with a pulse compressor. II. Finite gain-bandwidth effect, J. Opt. Soc. Am. B 2006, (23): 10-19.
[187] S.A. Ponomarenko and G.P. Agrawal, Do Solitonlike Self-Similar Waves Exist in Nonlinear Optical Media? Phys. Rev. Lett. 2006, (97): 013901 1-4.
[188] K. Tai, A. Hasegawa, and A. Tomita, Observation of Modulational Instability in Optical Fibers, Phys. Rev. Lett. 1986, (56): 135-138.
[189] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1972).
[190] Y.V. Kartashov, V.A. Aleshkevich, V.A. Vysloukh et al., Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity, Phys. Rev. E 2003, (67): 036613 1-11.
[191] H.J. Shin, Multisoliton complexes moving on a cnoidal wave background, Phys. Rev. E 2005, (71): 036628 1-10.
[192] S. Chen et al., Soliton complexes on a cnoidal wave background described by the generalized nonlinear schrodinger equation with varying coefficients, Phys. Rev. E (prepare).
[193] S. Chen, H. Liu, S. Zhang et al., Compression of Hermite–Gaussian pulses in an engineered optical fiber absorber with varying dispersion and nonlinearity, Phys. Lett. A 2006, (253): 493-496.
[194] H.J.A. da Silva, J.J. O’Reilly, Optical pulse modeling with Hermite-Gaussian functions, Opt. Lett. 1989, (14): 526-528
[195] A. Hook, D. Anderson, M. Lisak et al., Soliton-supported pulses at normaldispersion in optical fibers, J. Opt. Soc. Am. B 1993, (10): 2313-2320.
[196] S.-P. Gorza, N. Roig, Ph. Emplit et al., Snake Instability of a Spatiotemporal Bright Soliton Stripe, Phys. Rev. Lett. 2004, (92): 084101 1-4.
[197] S. Chen, H. Liu, and Y. Yi, Numerically simulating stochastic generalized nonlinear Schrodinger equations: Robust algorithm, Computational Physics, Proceedings of the joint conference of ICCP6 and CCP2003, edited by X.-G. Zhao, S. Jiang and X.-J. Yu (Rinton Press, New Jersey, 2005).
[198] L. Bergé, Wave collapse in physics: principles and applications to light and plasma waves, Phys. Rep. 1998, (303): 259-370.
[199] D.E. Pelinovsky, Y.S. Kivshar, and V.V. Afanasjev, Internal modes of envelope solitons, Physica D 1998, (116): 121-142.
[200] M. Facao and D.F. Parker, Stability of screening solitons in photorefractive media, Phys. Rev. E 2003, (68): 016610 1-7.
[201] T. Kapitula, Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrodinger equation, Physica D 1998, (116): 95-120.
[202] S. Chen et al., Phase fuctuations of linearly chirped solitons in a noisy optical fiber channel with varying dispersion, nonlinearity, and gain, Phys. Rev. A (prepare).
[203] K. Tamura and M. Nakazawa, Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers, Opt. Lett. 1996, (21): 68-70.
[204] T.Schreiber, C.K.Nielsen, B.Ortac et al., Microjoule-level all-polarization- maintaining femtosecond fiber source, Opt. Lett. 2006, (31): 574-576.
[205] C. Finot, S. Pitois, G. Millot et al., Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers, IEEE J. Sel. Top. Quantum Electron. 2004, (10): 1211-1218.
[206] C. Finot and G. Millot, Interaction between optical parabolic pulses in a Raman fiber amplifier, Opt. Express 2005, (13): 5825-5830.
[207] E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, New York, 1994).
[208] W. Cao and P.K.A. Wai, Amplification and compression of ultrashort fundamental solitons in an erbium-doped nonlinear amplifying fiber loop mirror, Opt. Lett. 2003, (28): 284-286.
[209] S.K. Varshney, T. Fujisawa, K. Saitoh et al., Novel design of inherently gain- flattened discrete highly nonlinear photonic crystal fiber Raman amplifier and dispersion compensation using a single pump in C-band, Opt. Express 2005, (13): 9516-9526.
[210] R. Trebino, K.W. DeLong, D.N. Fittinghoff et al., Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating, Rev. Sci.Instrum. 1997, (68): 3277-3295.
[211] F.W. Kerfoot and P.K. Runge, Future directions for undersea communications, AT&T Tech. J. 1995,(74): 93-102.
[212] N.S. Bergano et al., 40 Gb/s WDM transmission of eight 5 Gb/s data channels over transoceanic distances using the conventional NRZ modulation format, in OFC’95, optical fiber communication: summaries of papers presented at the Conference on Optical Fiber Communication, San Diego, California, 1995 Technical Digest Series, Vol. 8 (Optical Society of America, Washington, D.C.), p. PD19.
[213] N. Savage, Too much fiber? Opt. Photon. News, 2002(3), 13: 32-37.
[214] S. Chen, D. Shi, S. Zhang, and L. Yi, General formulation for parametrically controlled dark-soliton jitter in high-speed optical transmission systems, Opt. Commun. 2004, (242), 503-510.
[215] S. Chen, Novel Strategy for Numerically Evaluating Dark-Soliton Jitter in Optical Communications, Commun. Theor. Phys. 2005, (43): 847-849.
[216] D. Anderson and M. Lisak, Nonlinear asymmetric self-phase modulation and self- steepening of pulses in long optical waveguides, Phys. Rev. A 1983, (27): 1393- 1398.
[217] E.J. Woodbury, and W.K. Ng, Ruby laser operation in the near IR, Proc. IREE 1962, (50): 2347.
[218] R.H. Stolen and E. P. Ippen, Raman gain in glass optical waveguides, Appl. Phys. Lett. 1973, (22): 276-278.
[219] R.H. Stolen, Nonlinearity in fiber transmission, Proc. IEEE 1980, (68): 1232-1236.
[220] J.P. Gordon, Theory of the soliton self-frequency shift, Opt. Lett. 1986, (11): 662-664.
[221] F.M. Mitschke and L.F. Mollenauer, Discovery of the solitons self-frequency shift, Opt. Lett. 1986, (11): 659-661.
[222] V.P. Kalosha and J. Herrmann, Formation of Optical Subcycle Pulses and Full Maxwell-Bloch Solitary Waves by Coherent Propagation Effects, Phys. Rev. Lett. 1999, (83): 544-547.
[223] H.A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).
[224] D. Mihalache, N. Truta, and L.-C. Crasovan, Painlevé analysis and bright solitary waves of the higher-order nonlinear Schrodinger equation containing third-order dispersion and self-steepening term, Phys. Rev. E 1997, (56): 1064-1070.
[225] Y. Kodama and A. Hasegawa, Generation of asymptotically stable optical solitons and suppression of the Gordon-Haus effect, Opt. Lett. 1992, (17): 31-33.
[226] Y. S. Kivshar, M. Haelterman, P. Emplit et al., Gordon-Haus effect on dark solitons,Opt. Lett. 1994, (19): 19-21.
[227] V.I. Karpman, Perturbation theory for solitons, Sov. Phys. JETP, 1977, (46): 281-291.
[228] D.J. Kaup, A perturbation expansion for the Zakharov-Shabat inverse scattering transform, SIAM J. Appl. Math. 1976, (31): 121-133.
[229] A.I. Maimistov and M. Basharov, Nonlinear Optical Waves (Kluwer, Dordrecht, The Netherlands, 1999).
[230] S.L. Palacios, A. Guinea, J.M. Fernández-Díaz et al., Dark solitary waves in the nonlinear Schrodinger equation with third order dispersion, self-steepening, and self-frequency shift, Phys. Rev. E 1999, (60): R45-R47.
[231] A. Mahalingam and K. Porsezian, Propagation of dark solitons with higher-order effects in optical fibers, Phys. Rev. E 1999, (64): 046608 1-9.
[232] L. Li, Z. Li, Z. Xu et al., Gray optical dips in the subpicosecond regime, Phys. Rev. E 2002, (66): 046616 1-8.
[233] R. Hao, L. Li, Z. Li et al., Exact multisoliton solutions of the higher-order nonlinear Schrodinger equation with variable coefficients, Phys. Rev. E 2004, (70): 066603 1-6.
[234] Z. Li, L. Li, H. Tian et al., New Types of Solitary Wave Solutions for the Higher Order Nonlinear Schrodinger Equation, Phys. Rev. Lett. 2000, (84): 4096-4099.
[235] R. Yang, L. Li, R. Hao et al., Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrodinger equation, Phys. Rev. E 2005, (71): 036616 1-8.
[236] K. Nakkeeran, K. Porsezian, P. S. Sundaram et al., Optical Solitons in N-Coupled Higher Order Nonlinear Schrodinger Equations, Phys. Rev. Lett. 1998, (80): 1425-1428.
[237] R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys. (N.Y.) 1973, (14): 805-809.
[238] N. Sasa and J. Satsuma, New-type of soliton solutions for a higher-order nonlinear Schrodinger equation, J. Phys. Soc. Jpn. 1991, (60): 409-417.
[239] L.F. Mollenauer, J.P. Gordon, and S.G. Evangelides, The sliding-frequency guiding filter: an improved form of soliton jitter control, Opt. Lett. 1992, (17): 1575-1577.
[240] W.H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
[241] J.P. Hamaide, P. Emplit and M. Haelterman, Dark-soliton jitter in amplified optical transmission systems, Opt. Lett. 1991, (16): 1578-1580.
[242] M. Nakazawa and K. Suzuki, 10 Gbit/s pseudorandom dark soliton data transmission over 1200 km, Electron. Lett. 1995, (31): 1076-1077.
[243] P.V.Mamyshev, P.G.J. Wigley, J. Wilson et al., Adiabatic compression of Schrodinger solitons due to the combined perturbations of higher-order dispersion and delayed nonlinear response, Phys. Rev. Lett. 1993, (71): 73-76.
[244] M. Nakazawa, H.Kubota, K. Kurokawa et al., Femtosecond optical soliton propagation over long distance using adiabatic trapping and soliton standardization, J. Opt. Soc. Am. B 1991, (8): 1811-1817.
[245] K. Qiu, Soliton position jitter caused by periodic amplification and self-frequency shift, Electron. Lett. 1994, (30): 439-440.
[246] K.C. Kao and G.A. Hockham, Dielectric fiber surface waveguides for optical frequencies, Proc. IEE, 1966, (133): 1151-1158.
[247] J. G. Proakis, Digital Communications, (McGraw Hill, New York, 2000).
[248] A.H. Gnauck, G. Raybon, S. Chandrasekhar et al., 2.5 Tb/s (64×42.7 Gb/s) Transmission Over 40×100 km NZDSF Using RZ-DPSK Format and All-Raman- Amplified Spans, Post Deadline paper, OFC, Anaheim, CA, March 17-22, 2002.
[249] C.J. McKinstrie and C. Xie, Phase Jitter in Single-Channel Soliton Systems with Constant Dispersion, IEEE J. Sel. Top. Quantum Electron. 2002, (8): 616-625.
[250] C. Xu, X. Liu, L.F. Mollenauer et al., Comparison of Return-to-Zero Differential Phase-Shift Keying and ON–OFF Keying in Long-Haul Dispersion Managed Transmission, IEEE Photon. Technol. Lett., 2003, (15): 617-619.
[251] M. Hanna, H. Porte, J.-P. Goedgebuer et al., Performance assessment of DPSK soliton transmission system, Electon. Lett. 2001, (37): 644-646.
[252] H. Tsuchida, Correlation between amplitude and phase noise in a mode-locked Cr:LiSAF laser, Opt. Lett. 1998, (23): 1686-1688.
[253] W.K. Marshall, B. Crosignani, and A. Yariv, Laser phase noise to intensity noise conversion by lowest-order group-velocity dispersion in optical fiber: exact theory, Opt. Lett. 2000, (25): 165-167.
[254] H. Kim and A.H. Gnauck, Experimental Investigation of the Performance Limitation of DPSK Systems Due to Nonlinear Phase Noise, IEEE Photon. Technol. Lett., 2003, (15): 320-322.
[255] K.-P. Ho, Probability density of nonlinear phase noise, J. Opt. Soc. Am. B 2003, (20): 1875-1879.
[256] A. Mecozzi, Probability density functions of the nonlinear phase noise, Opt. Lett. 2004, (29): 673-675.
[257] X. Wei and X. Liu, Analysis of intrachannel four-wave mixing in differential phase- shift keying transmission with large dispersion, Opt. Lett. 2003, (28): 2300-2302.
[258] X. Wei, X. Liu, S.H. Simon, and C. J. McKinstrie, Intrachannel four-wave mixing in highly dispersed return-to-zero differential-phase-shift-keyed transmission with a nonsymmetric dispersion map, Opt. Lett. 2006, (31): 29-31.
[259] P.P. Mitra and J.B. Stark, Nonlinear limits to the information capacity of optical fibre communications, Nature 2001, (411): 1027-1030.
[260] J.M. Kahn and K.-P. Ho, Communications technology: A bottleneck for optical fibres, Nature 2001, (411): 1007-1010.
[261] X. Wei, A.H. Gnauck, D.M. Gill et al., Optical π/2-DPSK and its Tolerance to Filtering and Polarization-Mode Dispersion, IEEE Photon. Technol. Lett., 2003, (15): 1639-1641.
[262] U.-V. Koc and X. Wei, Combined Effect of Polarization-Mode Dispersion and Chromatic Dispersion on Strongly Filtered π/2-DPSK and Conventional DPSK, IEEE Photon. Technol. Lett., 2004, (16): 1588-1590.
[263] C. Xie, L. Muller, H. Haunstein et al., Comparison of System Tolerance to Polarization-Mode Dispersion Between Different Modulation Formats, IEEE Photon. Technol. Lett., 2003, (15): 1168-1170.
[264] M. Hanna, D. Boivin, P.-A. Lacourt et al., Calculation of optical phase jitter in dispersion managed systems by use of the moment method, J. Opt. Soc. Am. B 2004, (21): 24-28.
[265] C.J. McKinstrie and T.I. Lakoba, Probability-density function for energy perturbations of isolated optical pulses, Opt. Express 2003, (11): 3628-3648.
[266] S.A. Derevyanko, S.K. Turitsyn, and D.A.Yakushev, Fokker-Planck equation approach to the description of soliton statistics in optical fiber transmission systems, J. Opt. Soc. Am. B 2005, (22): 743-752.
[267] S. Chen et al., Study of chirp statistics in nonlinearity- and dispersion-managed soliton systems: a Fokker-Planck equation method, Opt. Lett. (prepare).
[268] S.K. Turitsyn, T. Schaefer, and V.K. Mezentsev, Generalized momentum method to describe high-frequency solitary wave propagation in systems with varying dispersion, Phys. Rev. E 1998, (58): R5264-R5267.
[269] K.J. Blow, N.J. Doran, and S.J.D. Phoenix, The soliton phase, Opt. Commun. 1992, (88): 137-140.
[270] K. Ito, Lectures on Stochastic Processes (Tata Institute of Fundamental Research, Bombay, 1960).
[271] R.L. Stratanovich, Topics in the Theory of Random Noise (Gordon & Breach, New York, 1963), Vols. I and II.
[272] C.W. Gardiner, Handbook of Stochastic Methods, 2nd Ed. (Springer, Berlin, 1996).
[273] P.E. Kloeden and E. Platen, The Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992).
[274] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, England, 1995).
[275] P.D. Drummond and I.K. Mortimer, Computer simulations of multiplicative stochastic differential equations, J. Comput. Phys. 1991, (93): 144-170.
[276] M.J. Werner and P.D. Drummond, Robust algorithms for solving stochastic partial differential equations, J. Comput. Phys. 1997, (132): 312-326.
[277] B. Khubchandani, P.N. Guzdar, and R. Roy, Influence of stochasticity on multiple four-wave-mixing processes in an optical fiber, Phys. Rev. E 2002, (66): 066609 1-8.
[278] W.H. Press, B.P. Flannery, S.A. Teukolsky et al., Numerical recipes: the art of scientific computing (Cambridge University Press, Cambridge, England, 1986).
[279] J. Garcia-Ojalvo and J.M. Sancho, Noise in spatially extended systems (Springer, New York, 1999).
[280] J.W. Cooley, P.A.W. Lewis, and P.D. Welch, The fast Fourier transform and its applications, IEEE Trans. Education, 1969, (12): 27-34.