On The Self-Similarities And Communication Limits Of Optical Solitons Propagating In Fibers

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 AnalysisTo 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 JitterIt 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 SolitonsIn 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 MethodIn 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.