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Nonlinear optical phenomena associated with high-order soliton breakup in photonic crystal fibres and squeezed state generation in three dimensional photonic crystal microcavities are investigated. In both cases, the properties of periodically patterned, high-index contrast dielectric structures are engineered to control the dispersion and local field enhancements of the electromagnetic field. Ultra-short pulse propagation in a polarization-maintaining microstructured fibre (with 1 um core diameter and 1.1 m length) is investigated experimentally and theoretically. For an 80 MHz train of 130 fs pulses with average propagating powers in the fibre up to 13.8 mW, the output spectra consist of multiple discrete solitons that shift continuously to lower energies as they propagate in the lowest transverse mode of the fibre. The number of solitons and the amount that they shift both increase with the launched power. All of the data is quantitatively consistent with solutions of the nonlinear Schrodinger equation, but only when the Raman nonlinearity is treated without approximation, and self-steepening is included. The feasibility of using a parametric down-conversion process to generate squeezed electromagnetic states in 3D photonic crystal microcavity structures is investigated for the first time. The spectrum of the squeezed light is theoretically calculated by using an open cavity quantum mechanical formalism. The cavity communicates with two main channels, which model vertical radiation losses and coupling into a single-mode waveguide respectively. The amount of squeezing is determined by the correlation functions relating the field quadratures of light coupled into the waveguide. All of the relevant model parameters are realistically estimated using 3D finite-difference time-domain (FDTD) simulations. Squeezing up to ~20% below the shot noise level is predicted for reasonable optical excitation levels. To preserve the squeezed nature of the light generated in the microcavity, a unidirectional coupling geometry from the microcavity to a ridge waveguide in a slab photonic crystal structure is studied. The structure was successfully fabricated in a silicon membrane, and experimental measurements of the efficiency for the signal coupled out of the structure are in good agreement with the result of FDTD simulations. The coupling efficiency of the cavity mode to the output channel is ~60%. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

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This thesis proposes and develops a novel optic configuration, Kerr phase-interrogator, which investigates the phase-shift between two sinusoidally modulated optical signals (SMOS) utilizing Kerr effect. The Kerr phase-interrogator gives birth to an entirely new technique for measuring the phase-shift between two light-waves. Taking advantage of all-optical signal processing, ultrafast responses, and being free from the coherent properties of a laser source, the Kerr phase-interrogator based technique for measuring the phase-shift is a promising novel approach for monitoring and sensing applications. The thesis begins with theoretically demonstrating the operation of Kerr phase-interrogator. As the core of optical process occurs in Kerr phase-interrogator, nonlinear interactions between two SMOSs in the Kerr medium are theoretically analyzed utilizing the models of nonlinear phase-modulation and four-wave mixing (FWM). The phase-modulation-based model is intuitive and allows for conceptual understanding of the operation of the Kerr phase-interrogator. However, this model does not account for the impact of chromatic-dispersion (CD) of the Kerr medium on the operation of the Kerr phase-interrogator. Compared with the former model, the FWM-based model is essential for acquiring insight into Kerr phase-interrogator, and can explain the CD impact of the Kerr medium. The analytical solution of the power of the first order sideband as a result of the nonlinear interaction is obtained in both theoretical models. The obtained solution shows sinusoidal dependence of the power on the phase-shift of the SMOSs. Utilizing this sinusoidal dependence, the phase-shift of two SMOSs can be acquired by measuring the power of the first-order sideband. Birefringence and CD are critical factors that affect the nonlinear interactions and thus impact the operation of Kerr phase-interrogator. In this work, vector analysis is performed on the nonlinear interaction between two SMOSs in a Kerr medium with randomly varying birefringence, and the effect of polarization-states of SMOSs on the operation of Kerr phase-interrogator is investigated. Impact of CD of Kerr medium on the operation of Kerr phase-interrogator is theoretically investigated using theory of FWM and is experimentally verified. Four typical applications, which comprehensively reflect the advantages of Kerr phase-interrogator, are proposed and experimentally demonstrated in this thesis. First, we present a novel approach for measurements of CD in long optical fibers using a Kerr phase-interrogator. The Kerr phase-interrogator measures the phase variation of a SMOS induced by CD in a fiber under test as the laser carrier wavelength is varied. This approach takes advantage of all-optical signal-processing based on Kerr effect to acquire the phase variation, and consequently removes the requirement of complex electrical signal-processors in existing techniques of CD measurement. CD measurement for several fibers is experimentally demonstrated. Second, a novel temperature sensor that utilizes temperature dependence of reflection group-delay in a linearly chirped fiber Bragg grating is presented. The reflection group-delay of chirped grating changes with temperature leading to a variation in the phase of a SMOS reflected from the grating. A Kerr phase-interrogator converts the phase-variation into power variation allowing for temperature sensing with a resolution of 0.0089 oC and a sensitivity of 1.122 rad/oC. Third, a Kerr phase-interrogator is applied for implementation of real-time CD monitoring. CD induces a phase-shift between two SMOSs carried by two different wavelengths. A Kerr phase-interrogator converts the phase-shift into power variation and CD monitoring is achieved by measurement of the power variation in real time with a resolution of 0.196 ps/nm. This application takes advantages of ultrafast response of Kerr phase-interrogator and achieves the real-time monitoring. Lastly, a novel approach for incoherent optical frequency-domain reflectometry based on a Kerr phase-interrogator is presented. The novel approach eliminates the limitation of finite coherent length of the light source, and achieves measurement of long-range distance beyond the coherent length of the light source. Long-range detection of reflection points as far as 151 km at a spatial-resolution of 11.2 cm is experimentally demonstrated.

It is assumed that processes to be identified or controlled can be described by linear or non-linear differential equations with unknown coefficients aᵢ. For on-line identifications a model is constructed to have the same form of differential equation as the process, but with adjustable parameters αᵢ replacing the aᵢ. The parameters αᵢ are adjusted in a steepest descent fashion; they are shown to converge to the aᵢ as long as an adjustment gain K(t) is non-negative, and not identically zero. An approximate analysis is carried out to determine the best constant K which gives the fastest identification. Optimal control theory is introduced to find the best piecewise continuous K(t) in the interval 0 ≤ K(t) ≤ Kmax . From the exact solution in a special case, a switched suboptimal K which always gives fast identification is determined. Identification schemes are developed for processes which include an unknown non-linearity that, can be assumed to be piecewise linear. An adaptive control, system is developed to control processes with unknown time-varying coefficients. The system is shown to be stable as long as the process inverse is stable; the process need not be stable. Systems to control linear and non-linear unstable processes are designed and simulated. The limitations of the adaptive system are determined, and compared with the limitations of conventional feedback systems. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

Sufficient conditions for the absolute stability of a class of nonlinear sampled-data systems are derived using the techniques of system transformations due to Aizerman and Gantmacher. These criteria are based on different forms used to approximate the area under the nonlinear characteristic. It is also shown that the stability criterion can be improved by relaxing a restriction on the slope of nonlinear element. In multiple nonlinearily continuous systems, by the application of numerical techniques, it was shown that some improvement over previous stability bounds can be made. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

This thesis is in two parts, both considering the absolute stability of nonlinear systems. In the first two chapters the stability of certain classes of nonlinear time invariant systems involving several nonlinearities is considered. A number of graphical methods are given for testing the stability of these systems. The graphical tests are equivalent to a weakened form of the Popov criterion. The third chapter derives a stability condition for nonlinear systems involving a linear time-varying gain. The time-varying gain is assumed to satisfy conditions on its magnitude and rate of change. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

This dissertation is concerned with aspects of the newly-proposed approach to nonlinear acoustics in which the Lagrangian description of gas motion is followed. It contains a systematic survey of the approach which leads to the so-called dynamic piston problem. Then new situations regarding the piston problem are studied. These situations cover cases of varying applied pressure and results concerning the formation of shock discontinuities are presented. / Dissertation (MSc)--University of Pretoria, 2013. / Mathematics and Applied Mathematics / unrestricted

The nonlinear dynamics of three physical systems has been investigated. Diode resonator systems are experimentally shown to display a period doubling route to chaos, quasiperiodic states, periodic locking states, and Hopf bifurcation to chaos. Particularly, the transition from quasiperiodic states to chaos in line-coupled systems agrees well with the Curry-Yorke model. The SPICE program has been modified to give realistic models for the diode resonator systems.

An airship with a moving gondola is investigated with the goal of achieving a large pitch angle and of minimizing the total energy consumption required to goal position. The airship in this study is equipped with different actuation tools such as a moving gondola and a vectoring thrust, which can be used for various flight modes. The efficiency of the actuation methods employed is studied and compared in various flight scenarios, based on the airship’s ability to reach the desired position while consuming the least amount of energy. The nonlinear dynamic model is derived using Newton-Euler equations. Backstepping and incremental nonlinear dynamic inversion (INDI) controllers are designed to track a desired trajectory by controlling the position of the gondola and the thrust. The dynamic models are then implemented and simulated in the Matlab/Simulink to evaluate the effectiveness of the controllers in different environmental conditions. The simulation results show the effectiveness of the controllers used, and a larger pitch angle of -89o, can be reached thanks to the movement of the gondola to the front of the airship on the curved keel. The airship prototype was used for the experimental test to evaluate the pitch tracking performance of each of the controllers. The experimental results show that the prototype used can generate a -90o pitch angle, thereby improving manoeuvrability and allowing for rapid changes in altitude. The energy model is developed to evaluate and compare the energy required by the airship for ascent, cruise, and descent flights, using different actuation methods. The effectiveness of the composite control strategy is demonstrated by completing the flight mission with the least amount of energy consumed. An optimization method is then developed to find the optimum design configuration to reduce the cost function, based on energy consumption, of the different flight scenarios while always respecting the design constraints. The Heuristic technique is used to obtain the optimal flight trajectory based on the platform’s ability to complete the desired mission while minimising energy consumption. The results show that for pitch tracking, the vectored thrust has a rapid response, and the required thrust is high. Therefore, this configuration requires more energy than the moving gondola control configuration, in all cases studied. The composite configuration is found to be the most efficient method for completing the flight trajectory with the least amount of energy. The total energy consumption of the entire flight is reduced by about 17% by using the optimization algorithm to select the best actuation method for each flight mode.