Thesis advisor: Krzysztof Kempa / In this work, we first examine nonlocal behavior in plasmonic systems and develop or expand upon models that enable calculation of higher-order, nonlocal responses for systems with novel geometries. The effects of nonlocality, i.e., spatial dispersion, are prominent in nanostuctures with small feature sizes, and accurate calculations of the nonlocal response of nanostructures are increasingly important for the study of novel physics at the nanoscale. Next, we consider a specific biological system, double-stranded DNA, and investigate the nonlocal and nonlinear model that describes its dynamics. We consider the regime of strong driving with THz radiation and study the parameter-space where molecular damage occurs, motivated by the prospect of using selective damage for potential novel therapies. In a related study, we also consider the possibility of generating THz radiation through the nonlinear, difference-frequency response of a plasmonic system. Plasmonic difference-frequency generation could enable deep penetration of THz signals into the body and, therefore, these projects are intimately connected. Ultimately, these two regimes of behavior, nonlocality and nonlinearity, represent rich areas for applicable research. / Thesis (PhD) — Boston College, 2018. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Lasers are an essential technology enabling countless fields of optics, however, their operation wavelengths are limited to isolated regions across the optical spectrum due to the need for suitable gain media. Parametric frequency conversion (PFC) is an attractive means to convert existing lasers to new colors using nonlinear optical interactions rather than the material properties of the host medium, allowing for the development of high power laser sources across the entire optical spectrum. PFC in bulk χ(2) crystals has led to the development of the optical parametric oscillator, which is currently the standard source for high power light at non-traditional wavelengths in the laboratory setting. Ideally, however, one could implement PFC in an optical fiber, thus leveraging the crucial benefits of a guided-wave geometry: alignment-free, compact, and robust operation. Four-wave mixing (FWM) is a nonlinear effect in optical fibers that can be used to convert frequencies, the major challenge being conservation of momentum, or phase matching, between the interacting light waves. Phase matching can be satisfied through the interaction of different spatial modes in a multi-mode fiber, however, previous demonstrations have been limited by mode stability and narrow-band FWM gain. Alternatively, phase matching within the fundamental mode can be realized in high-confinement waveguides (such as photonic crystal fibers), but achieving the anomalous waveguide dispersion necessary for phase matching at pump wavelengths near ∼1 μm (where the highest power fiber lasers emit) comes at the cost of reducing the effective area of the mode, thus limiting power-handling. Here, we specifically consider the class of Bessel-like LP0,m modes in step-index fibers. It has been shown that these modes can be selectively excited and guided stably for long lengths of fiber, and mode stability increases with mode order ‘m’. The effective area of modes in these fibers can be very large (>6000 μm2 demonstrated) and is decoupled from dispersion, allowing for phase matching within a single mode in a power-scalable platform. Furthermore, step-index fibers can guide many different LP0,m modes, allowing access to a highly multi-moded basis set with which to study FWM interactions between different modes. In this thesis we develop techniques to excite, propagate, and characterize LP0,m modes in order to demonstrate FWM in two regimes: monomode interactions comprising waves all belonging to the same mode, and intermodal interactions between different modes. In the monomode regime we demonstrate parametric sources which operate at near-infrared wavelengths under-served by conventional fiber lasers, including 880, 974, 1173, and 1347 nm. The output pulses for these systems are ∼300 ps in duration and reach peak powers of ∼10 kW, representing, to the best our knowledge, the highest peak power fiber laser sources demonstrated at these wavelengths to date. In the intermodal regime, we demonstrate a cascade of FWM processes between different modes that lead to a series of discrete peaks in the visible portion of the spectrum, increasing monotonically in mode order from LP0,7 at 678 nm to LP0,16 at 443 nm. This cascade underscores the huge number of potential FWM interactions between different LP0,m modes available in a highly multi-mode fiber, which scale as N4 for N guided modes. Finally, we demonstrate a novel intermodal FWM process pumped between the LP0,4 and LP0,5 modes of a step-index fiber, which provides broadband FWM gain (63 nm at 1550 nm) while maintaining wavelength separations of nearly an octave (762 nm) – a result that cannot be replicated in the single-mode regime. We seed this process to generate a ∼10 kW, ∼300-ps pulsed fiber laser wavelength-tunable from 786-795 nm; representing a fiber analogue of the ubiquitous Ti:Sapphire laser.

This thesis presents several advances in the technology and applications of photonic crystal fibres achieved over the last three years. Chapters 1 and 2 give the background material important to understand the results presented in chapters 3, 4 and 5. In chapter 1, linear properties of optical fibres are described. This chapter focuses particularly on how the engineering of the cladding structure of solid core photonic crystal fibres can be used to vary the fibre properties, most importantly the group index and dispersion. Propagation in all-solid photonic bandgap fibres is also discussed in terms of the anti-resonant reflecting optical waveguide model. Chapter 2 introduces the nonlinear optical effects that are important to understand the work presented in chapters 4 and 5. In chapter 3, a method to reduce bend losses in all-solid photonic bandgap fibres is outlined. The reduction of these losses is achieved by redesigning the high-index inclusions in the cladding structure to suppress cladding modes that strongly couple to the fundamental core-guided mode when the fibre is bent. In chapter 4, a method of tapering photonic crystal fibres in order to decrease the dispersion along their length is described. The tapers are used to compress solitons via adiabatic soliton compression and a combination of adiabatic soliton compression and soliton effect compression, achieving a factor of 15 compression of a transform-limited pulse to below 50 fs. Chapter 5 describes how engineering the cladding structure of photonic crystal fibres can be used to generate shorter frequencies in supercontinuum generation. The method by which this achieved is experimentally verified and then exploited to generate a continuum incorporating the entire visible spectrum using low cost, low maintenance pump sources.

In Chapter 2 we study the uniqueness problem of sign-changing solutions for a nonlinear scalar equation. It is well-known that positive solution is radially symmetric and unique up to a translation. Recently, there are many works on the existence and multiplicity of sign-changing solutions. However much less is known for uniqueness, even in the radially symmetric class. In Chapter 2, we solve this problem for nearly critical nonlinearity by Lyaponov-Schmidt reduction. Moreover, we can also prove the non-degeneracy. / In Chapter 3 we are concerned with the uniqueness problem for coupled nonlinear Schrodinger equations. The problem is to classify all positive solutions. In Chapter 3, some sufficient conditions are given. In particular, we have a sufficient and necessary condition in one dimension. The proof is elementary because only the implicit function theorem, integration by parts, and the uniqueness for scalar equation are needed. / In Chapter 4 we go back to the nonlinear scalar equation and consider the traveling wave solutions. Using an infinite dimensional Lyaponov-Schmidt reduction, new examples of traveling wave solutions are constructed. Our approach explains the difference between two dimension and higher dimensions, and also explores a connection between moving fronts and the mean curvature flow. This is the first such traveling waves connecting the same states. / This thesis is devoted to the study of nonlinear elliptic equations and systems. It is divided into two parts. In the first part, we study the uniqueness problem, and in the second part, we are concerns with traveling wave solutions. / Yao, Wei. / Adviser: Jun Cheng Wei. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 132-142). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

In this thesis, we investigate the global robust output regulation problem for nonlinear systems subject to time-varying or nonlinear exosystems. / One of the crucial issues in output regulation problem is the design of the appropriate internal model. Internal model is a dynamical compensator which possesses an essential ability of generating all possible steady-state input information asymptotically, and it should not only lead to a well-defined augmented system but also ensure the stabilizability of the augmented system. Besides, stabilization techniques for the augmented system should also be carefully chosen to meet the needs in different scenarios, e.g. the time-varying settings. Efforts are put on both sides throughout the thesis. / Output regulation problem, also known as servomechanism problem, is one of the central topics in control theory. The control objective is to design a feedback control law for the given plant so as to achieve asymptotic tracking for a class of reference signals and asymptotic rejection for a class of disturbance signals while maintaining the stability of closed-loop system. The reference or the disturbance signals are assumed to be generated from a dynamical system called the exosystem. Normally, the exosystem is a linear autonomous system, e.g. a harmonic oscillator, and the exogenous signals represent step or ramp signals, or sinusoidal signals contains finite number of harmonics. The extensions of the exosystem, from linear to nonlinear, autonomous to non-autonomous, significantly enlarge the categories of the exogenous signals, and more importantly, such extensions motivate the development of the output regulation theory in both scientific research and practical application. / Paying special attention to the appearance of time-varying or nonlinear exosystems, our research is mainly conducted under the general framework for tackling the output regulation problem. In general, first we convert the output regulation problem of the original plant into the stabilization problem of the augmented system which is composed of the plant and the designed internal model. Second, we achieve the global stabilization of the augmented system by robust and adaptive control approaches, according to both parameter uncertainty and dynamic uncertainty in either plant or the exosystem. / The main contributions of the thesis are outlined as follows. 1. A framework for handling the robust output regulation problem for general timevarying nonlinear systems subject to time-varying exosystem is proposed. Especially, certain existence conditions of a time-varying internal model is given, and problem conversion can be achieved. As an application of this framework, we give the solvability conditions of the output regulation problem for the time-varying nonlinear systems in output feedback form. Further, when parameter uncertainties occurred in the time-varying exosystem, we solve the corresponding adaptive robust output regulation problem resorting to some adaptive control methods. These results can also be applied to the time-varying nonlinear systems in lower triangular form. 2. The global robust output regulation problem for nonlinear systems subject to nonlinear exosystem is considered. A new class of internal models is introduced which relaxes the existence conditions of the former one. Also, this class of internal models has the merit that it is zero input globally asymptotically stable which greatly facilitates the global stabilization of the augmented system. Compared with the existing results, the new method solves the global robust output regulation problem without restrictions on the initial conditions or trajectory bounds of the exosystems, and the bound of the parameter uncertainties of the plant is not necessarily known. Moreover, utilizing the Nussbaum gain technique, the unknown control direction case can also be handled by modifying the control law. 3. The theoretical results have been applied to several practical control problems, such as the global disturbance rejection problem for FitzHugh-Nagumo model with Mathieu equation, the synchoniztion of periodically-forced pendulum with Rayleigh equation, etc .. / Yang, Xi. / Adviser: Jie Huang. / Source: Dissertation Abstracts International, Volume: 73-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 123-134). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Lei, Kin Pang. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Among various non-smooth dynamic systems, the periodically forced oscillation system with impact is perhaps the most common in engineering applications. Usually it has an oscillator with fixed or unfixed stops. The dynamics becomes complicate due to the impact against the stops. Sometimes it leads to bifurcation and even turns to chaos. Its present applications include MEMS switch device, escapement in watch movement and so on. / As a branch of mechanics, the multi-body dynamic system is well-studied. In particular, the non-smooth dynamical system attracts many researchers because of its importance and diversity. The main behaviours of such a system include contact (slip-stick motion), friction and impact. Although various models have been developed for these behaviours and their results are often satisfactory, the truth is that they are still far from completion. In the past twenty some years, various new methods have been developed. However, none of them is universally applicable. One of the difficulties is that there are a number of explicit discontinuities, such as: (a) Coulomb friction gives a discontinuous law for the forces as a function of velocities, and (b) The contact conditions give forces that are not only discontinuous in position, but also unbounded and give rise to discontinuities in the velocities. / This thesis presents a systematic study on the periodically forced oscillation system with impact. Various existing methods are discussed and compared. In particular, impulsive differential equation, Poincare map and perturbation theory are applied. Two practical cases are included: a first-order system and the Swiss lever escapement mechanism. The latter has significant engineering value as the Swiss level escapement is the key component of mechanical watch movement. The precision dynamic model has very high numerical accuracy in describing/predicting their dynamics. The research helps to optimize the design of a commercial product. The model is validated by means of experiment. / Fu, Yu. / Adviser: Du Ruxu. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3745. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 137-142). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

by Zheng Yu-kun. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Includes bibliographies.

by Yu Wai Kuen. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 71-76). / Preface --- p.1 / PART I / Chapter Chapter 1 --- Inverse Scattering Method / Chapter §1 --- Introduction --- p.5 / Chapter §2 --- Rapidly decreasing solutions of the GNLSE --- p.6 / Chapter Chapter 2 --- Modified Inverse Scattering Method / Chapter §1 --- Introduction --- p.25 / Chapter §2 --- Singular solutions of the KdV equation --- p.25 / PART II / Chapter Chapter 3 --- Backlund Transformation Method / Chapter §1 --- Introduction --- p.37 / Chapter §2 --- Solution by Backlund transformation --- p.37 / Chapter §3 --- Clairin's method for finding Backlund transformations --- p.46 / Chapter §4 --- Construction of multi-soliton solutions --- p.48 / Chapter Chapter 4 --- Dressing Method And Hirota Direct Method / Chapter §1 --- Introduction --- p.51 / Chapter §2 --- Zakharov-Shabat's dressing method --- p.52 / Chapter §3 --- Hirota direct method --- p.57 / Chapter Chapter 5 --- Group Reduction Method / Chapter §1 --- Introduction --- p.61 / Chapter §2 --- Method of group reduction --- p.61 / Bibliography --- p.71

Chen Zhiyong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 54-57). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Nonlinear Control --- p.1 / Chapter 1.2 --- Global Stabilization --- p.2 / Chapter 1.3 --- Output Regulation --- p.3 / Chapter 1.4 --- Contributions of the Thesis --- p.4 / Chapter 2 --- Global Robust Stabilization of Cascaded Polynomial Systems --- p.5 / Chapter 2.1 --- Introduction --- p.5 / Chapter 2.2 --- Preliminaries --- p.6 / Chapter 2.3 --- Basic Results --- p.8 / Chapter 2.4 --- The Algorithm --- p.11 / Chapter 2.5 --- An Example --- p.14 / Chapter 2.6 --- Concluding Remarks --- p.16 / Chapter 3 --- Output Regulation of Singular Nonlinear Systems by Normal Output Feedback --- p.18 / Chapter 3.1 --- Introduction --- p.18 / Chapter 3.2 --- Preliminaries --- p.20 / Chapter 3.3 --- Main Result --- p.24 / Chapter 3.4 --- An Example --- p.34 / Chapter 3.5 --- Concluding Remarks --- p.35 / Chapter 4 --- Robust Output Regulation of Singular Nonlinear Systems --- p.37 / Chapter 4.1 --- Introduction --- p.37 / Chapter 4.2 --- Problem Description and Standard Assumptions --- p.38 / Chapter 4.3 --- A Preliminary Result --- p.40 / Chapter 4.4 --- Solvability of the Problem --- p.48 / Chapter 4.5 --- Concluding Remarks --- p.51 / Chapter 5 --- Conclusions --- p.52 / Bibliography --- p.54 / Biography --- p.58