In recent years, second harmonic generation (SHG) microscopy has revolutionised the field of biological imaging by offering a new means of visualising the fine structures of collagen tissues with excellent image penetration while minimising photodamage. This project involves the design and construction of a SHG microscope that is built around a compact femtosecond fibre laser for collagen imaging. Operating at 1032 nm, the microscope has demonstrated a penetration depth of beyond 320 microns in collagen, which is considerably superior to depths of 250 to 300 microns achievable with a conventional SHG microscope coupled to a Ti:sapphire excitation laser. The imaging characteristics of the microscope have been tested with a modified sample of bovine pericardium. The results indicate the microscope is polarisation-sensitive to the tissue structure and is capable to detecting signal changes at 10 μm resolution. This thesis will describe in detail, to our best knowledge, the first SHG microscope equipped with a compact and robust all-fibre femtosecond 1032 nm laser source.

Short optical pulses and engineered nonlinear media is a powerful combination. Mode locked pulses exhibit high peak powers and short pulse duration and the engineered ferro-electric KTiOPO4 facilitates several different nonlinear processes. In this work we investigate the use of structured, second-order materials for generation, characterization and frequency conversion of short optical pulses. By cascading second harmonic generation and difference frequency generation the optical Kerr effect was emulated and two different Nd-based laser cavities were mode locked by the cascaded Kerr lensing effect. In one of the cavities 2.8 ps short pulses were generated and a strong pulse shortening took place through the interplay of the cavity design and the group velocity mismatch in the nonlinear crystal. The other laser had a hybrid mode locking scheme with active electro-optic modulation and passive cascaded Kerr lensing incorporated in a single partially poled KTP crystal. The long pulses from the active modulation were shortened when the passive mode locking started and 6.9 ps short pulses were generated. High-efficiency frequency conversion is not a trivial task in periodically poled materials for short pulses due to the large group velocity mismatch. Optimization of parameters such as the focussing condition and the crystal temperature allowed us to demonstrate 64% conversion efficiency by frequency doubling the fs pulses from a Yb:KYW laser in a single pass configuration. Quasi phase matching also offers new possibilities for nonlinear interactions. We demonstrated that it is possible to simultaneously utilize several phase matched second harmonic interactions, resulting in a dual-polarization second harmonic beam. Short pulse duration of the fundamental wave is a key parameter in the novel method that we demonstrated for characterization of the nonlinearity of periodically poled crystals. The method utilizes the group velocity mismatch between the two polarizations in a type II second harmonic generation configuration. The domain walls of PPKTP exhibit second order nonlinearities that are forbidden in the bulk material. This we used in a single shot frequency resolved optical gating arrangement. The spectral resolution came from Čerenkov phase matching, a non-collinear phase matching scheme that exhibits a substantial angular dispersion. The second harmonic light was imaged upon a CCD camera and with the spectral distribution on one axis and the temporal autocorrelation on the other. From this image we retrieved the full temporal profile of the fundamental pulse, as well as the phase. The spectral dispersion provided by the Čerenkov phase matching was large enough to characterize optical pulses as long as ~200 fs in a compact setup. The Čerenkov frequency resolved optical gating method samples a thin stripe of the beam, i.e. the area close to the domain wall. This provides the means for high spatial resolution measurements of the spectral-temporal characteristics of ultrafast optical fields. / QC 20100831

Optical spatial solitons in nematic liquid crystals, termed nematicons, have become an excellent test bed for nonlinear optics, ranging from fundamental effects to potential uses, such as designing and demonstrating all-optical switching and routing circuits in reconfigurable settings and guided-wave formats. Following their demonstration in planar voltage-assisted nematic liquid crystal cells, the spatial routing of nematicons and associated waveguides have been successfully pursued by exploiting birefringent walkoff, interactions between solitons, electro-optic controlling, lensing effects, boundary effects, solitons in twisted arrangements, refraction and total internal reflection and dark solitons. Refraction and total internal reflection, relying on an interface between two dielectric regions in nematic liquid crystals, provides the most striking results in terms of angular steering. In this thesis, the refraction and total internal reflection of self-trapped optical beams in nematic liquid crystals in the case of a planar cell with two separate regions defined by independently applied bias voltages have been investigated with the aim of achieving a broader understanding of the nematicons and their control. The study of the refraction of nematicons is then extended to the equivalent refraction of optical vortices. The equations governing nonlinear optical beam propagation in nematic liquid crystals are a system consisting of a nonlinear Schr¨odinger-type equation for the optical beam and an elliptic Poisson equation for the medium response. This system of equations has no exact solitary wave solution or any other exact solutions. Although numerical solutions of the governing equations can be found, it has been found that modulation theories give insight into the mechanisms behind nonlinear optical beam evolution, while giving approximate solutions in good to excellent agreement with full numerical solutions and experimental results. The modulation theory reduces the infinite-dimensional partial differential equation problem to a finite dynamical system of comparatively simple ordinary differential equations which are, then easily solved numerically. The modulation theory results on the refraction and total internal reflection of nematicons are in excellent agreement with experimental data and numerical simulations, even when accounting for the birefringent walkoff. The modulation theory also gives excellent results for the refraction of optical vortices of +1 topological charge. The modulation theory predicts that the vortices can become unstable on interaction with the nematic interface, which is verified in quantitative detail by full numerical solutions. This prediction of their azimuthal instability and their break-up into bright beams still awaits an experimental demonstration, but the previously obtained agreement of modulation theory models with the behaviour of actual nematicons leads us to expect the forthcoming observation of the predicted effects with vortices as well.

Many physics and engineering problems are modeled by differential equations. In many instances these equations are nonlinear and exact solutions are difficult to obtain. Numerical schemes are often used to find approximate solutions. However, numerical solutions do not describe the qualitative behaviour of mechanical systems and are insufficient in determining the general properties of certain systems of equations. The need for analytical methods is self-evident and major developments were seen in the 1990’s. With the aid of faster processing equipment today, we are able to compute analytical solutions to highly nonlinear equations that are more accurate than numerical solutions. In this study we discuss solutions to nonlinear partial differential equations with focus on non-perturbation analytical methods. The non-perturbation methods of choice are the homotopy analysis method (HAM) developed by Shijun Liao and the variational iteration method (VIM) developed by Ji-Huan He. The aim is to compare the solutions obtained by these modern day analytical methods against each other focusing on accuracy, convergence and computational efficiency. The methods were applied to three test problems, namely, the heat equation, Burgers equation and the Bratu equation. The solutions were compared against both the exact results as well as solutions generated using the finite difference method, in some cases. The results obtained show that the HAM successfully produces solutions which are accurate, faster converging and requires less computational resources than the VIM. However, the VIM still provides accurate solutions that are also in good agreement with the closed form solutions of the test problems. The FDM also produced good results which were used as a further comparison to the analytical solutions. The findings of this study is in agreement with those published in the literature. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

Quantum dots and nanorods are becoming increasingly important structures due to their potential applications that range from photovoltaic devices to medicine. The majority of the research on carrier dynamics in these structures has been in the optical regime, with little work performed at Terahertz frequencies where excitonic dynamics can be more directly probed. In this work, we examine theoretically the interaction of Terahertz radiation with colloidal CdSe nanorods to determine the dynamics of excitons generated via a short optical pulse. We calculate the energies and wavefunctions for the excitons within the envelope function approximation in the low density limit where there is at most one exciton per nanorod. The linear Terahertz transmittance and absorbance is found for nanorods that are approximately 70 nm in length and 7 nm in diameter and are compared with experimental results that have shown the first observation of intra-excitonic transitions in nanorods. We find absorbance peaks at 8.5 THz and 11 THz that result from polarizations in the longitudinal (rod axis) and transverse directions respectively. Our theoretical results show that the 8.5 THz and 11 Thz peaks are due to 1s-2pz and 1s-2px transitions respectively. The theoretical absorbance spectra is in good agreement with the experimental one and show that only the ground state is significantly populated 1 ps after optical excitation. This provides strong evidence of rapid trapping of excited holes into the ligand used to passivate the nanorods. A full set of dynamical equations were then constructed from Heisenberg's equation of motion, and were used to model the excitonic correlations as a function of time. Transmittance and absorbance were calculated for different nanorod orientations and electric field strengths in both the linear and nonlinear regime. These results were then averaged over nanorod orientation in order to more accurately reflect experimental conditions. Nonlinearity was found to become significant at peak pulse field strengths of 7 kV/cm and greater. Due to two-photon processes, we predict the 2pz-3dz transition that is not observed in the linear regime will be clearly seen in the nonlinear absorbance spectrum. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2014-05-14 23:37:58.604

<p>If electronics are ever to be completely replaced by optics, a significant possibility in the wake of the fiber revolution, it is likely that nonlinear materials will play a central and enabling role. Indeed, nonlinear optics is the study of the mechanisms through which light can change the nature and properties of matter and, as a corollary, how one beam or color of light can manipulate another or even itself within such a material. However, of the many barriers preventing such a lofty goal, the narrow and limited range of properties supported by nonlinear materials, and natural materials in general, stands at the forefront. Many industries have turned instead to artificial and composite materials, with homogenizable metamaterials representing a recent extension of such composites into the electromagnetic domain. In particular, the inclusion of nonlinear elements has caused metamaterials research to spill over into the field of nonlinear optics. Through careful design of their constituent elements, nonlinear metamaterials are capable of supporting an unprecedented range of interactions, promising nonlinear devices of novel design and scale. In this context, I cast the basic properties of nonlinear metamaterials in the conventional formalism of nonlinear optics. Using alternately transfer matrices and coupled mode theory, I develop two complementary methods for characterizing and designing metamaterials with arbitrary nonlinear properties. Subsequently, I apply these methods in numerical studies of several canonical metamaterials, demonstrating enhanced electric and magnetic nonlinearities, as well as predicting the existence of nonlinear magnetoelectric and off-diagonal nonlinear tensors. I then introduce simultaneous design of the linear and nonlinear properties in the context of phase matching, outlining five different metamaterial phase matching methods, with special emphasis on the phase matching of counter propagating waves in mirrorless parametric amplifiers and oscillators. By applying this set of tools and knowledge to microwave metamaterials, I experimentally confirm several novel nonlinear phenomena. Most notably, I construct a backward wave nonlinear medium from varactor-loaded split ring resonators loaded in a rectangular waveguide, capable of generating second-harmonic opposite to conventional nonlinear materials with a conversion efficiency as high as 1.5\%. In addition, I confirm nonlinear magnetoelectric coupling in two dual gap varactor-loaded split ring resonator metamaterials through measurement of the amplitude and phase of the second-harmonic generated in the forward and backward directions from a thin slab. I then use the presence of simultaneous nonlinearities in such metamaterials to observe nonlinear interference, manifest as unidirectional difference frequency generation with contrasts of 6 and 12 dB in the forward and backward directions, respectively. Finally, I apply these principles and intuition to several plasmonic platforms with the goal of achieving similar enhancements and configurations at optical frequencies. Using the example of fluorescence enhancement in optical patch antennas, I develop a semi-classical numerical model for the calculation of field-induced enhancements to both excitation and spontaneous emission rates of an embedded fluorophore, showing qualitative agreement with experimental results, with enhancement factors of more than 30,000. Throughout these series of works, I emphasize the indispensability of effective design and retrieval tools in understanding and optimizing both metamaterials and plasmonic systems. Ultimately, when weighed against the disadvantages in fabrication and optical losses, the results presented here provide a context for the application of nonlinear metamaterials within three distinct areas where a competitive advantage over conventional materials might be obtained: fundamental science demonstrations, linear and nonlinear anisotropy engineering, and extremely compact resonant all-optical devices.</p> / Dissertation

Non-negative matrices arise naturally in population models. In this thesis, we first study Perron- Frobenius theory of non-negative irreducible matrices. We use this theory to investigate the asymptotic behaviour of discrete time linear autonomous models. Then we discuss an application for this in age structured population. Furthermore, we study Liapunov stability of a general non-linear autonomous model. We consider a general nonlinear autonomous model that arises in structured population. We assume that the associated nonlinear matrix of this model is non-increasing at all density levels. Then, we show the existence of global extinction. In addition, we show the stability condition of the extinction equilibrium of the this model in the Liapunov sense. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2013.

Large amplitude interfacial waves are an important feature of annular gas-liquid two-phase flow. They act as a source of entrainment for liquid droplets. They occur for liquid flow rates above a critical value which depends on the gas flow rate. This thesis examines the formulation of a mathematical model to describe the behaviour of these nonlinear waves. Attention is focussed on the case of vertical upwards flow with reference to the experimental conditions for the rig at AERE Harwell. A comprehensive account is given of the limitations and similarities of mathematical models proposed by earlier research workers and their applicability to vertical two-phase flow. The most suitable approaches are found to be kinematic wave theory and an integral method. Experiments have been carried out at AERE Harwell to determine the relationship between liquid flux and film thickness required by kinematic wave theory and also to test some of the theory's predictions. There is a discussion of the difficulties involved in modelling the stresses exerted by the turbulent gas core on disturbance waves. The applicability of Benjamin's 'quasi-laminar' theory is considered. A linear stability analysis indicates that the interface is always unstable. The linear theory cannot provide a criterion for disturbance wave inception. Alternative explanations for wave inception are suggested. The SMAC (Simplified Marker And Cell) numerical method has been developed to model the time dependent behaviour of large amplitude waves in vertical annular two-phase flow. Finally, it is proposed that any realistic mathematical model for disturbance waves should be based upon kinematic wave theory and should take account of wave-breaking.

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