Orientador: Prof. Dr. Marcos Roberto da Rocha Gesualdi / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2015. / Os feixes de Airy tem atraido um grande interesse em fotonica, uma vez que tem propriedades vantajosas como a propagacao nao-difrativa e aceleracao sem acao de forcas externas. Por outro lado, a holografia e uma tecnica interferometrica que permite o registro e a reconstrucao de frentes de onda de objetos e feixes opticos, pois um holograma carrega as informacoes de intensidade e fase de um objeto ou feixe eholografadof. Recentemente, o aumento da capacidade de processamento e armazenagem dos microcomputadores; o desenvolvimento de novos dispositivos optoeletronicos, como os moduladores espaciais de luz (SLMs), as cameras CCDs (Coupled Charge Device) de alta resolucao; e, novos materiais fotossensiveis (materiais eletro-opticos e fotorrefrativos, PRMs); vem possibilitando a viabilidade dos hologramas gerados por computador (CGH, Computer Generate Hologram) e a implementacao experimental de sistemas holograficos de registro e reconstrucao numericas e opticas de objetos tridimensionais e geracao de ondas e feixes opticos especiais. Neste trabalho, nos apresentamos um estudo dos feixes de Airy e suas caracteristicas nao-difrativas e de aceleracao durante sua propagacao. Numa primeira etapa, montamos um sistema experimental que permitiu o registro (construcao) numerico e a reconstrucao de hologramas de feixes de Airy usando dispositivos LC-SLM. Numa segunda etapa, montamos um sistema experimental que permitiu o registro e a reconstrucao optica de hologramas de feixes de Airy usando o metodo de holografia fotorrefrativa com cristais ....12......20 (BTO) para a geracao optica de feixes de Airy. Na messma linha, a partir da obtencao numerica dos campos, mostramos a possibilidade de criar matrizes de feixes de Airy em numa so reconstruccao holografica. Paralelamente, fizemos um analise dos resultados de reconstrucao numerica optica dos hologramas e dos resultados de reconstrucao optica destes hologramas, bem como, suas potencialidades para aplicacoes tecnologicas. / The Airy beams have attracted great interest in photonics, once it has advantageous properties as the non-diffractive propagation and acceleration without action of external forces. On the other hand, holography is an interferometric technique that allows the recording and reconstruction of wavefront objects and optical beams, due to hologram carries the intensity and phase information of an object or beam. Recently, increased processing capacity and storage microcomputers; the development of novel optoelectronic devices such as spatial light modulators (SLMs), the CCD camera (Charge Coupled Device) with high resolution; and new photosensitive materials (electro-optic and photorefractive materials, DRP); has enabled the viability of computer-generated holograms (CGH, Computer Generate Hologram) and the experimental implementation of holographic systems of numerical and optical recording-reconstruction of three-dimensional objects and generation of special optical beams. In this work, we present a study of the beam Airy their non-diffractive characteristics and acceleration during its propagation. In a first step, we developed a experimental system that allowed me the record (construction) numerical and optical reconstruction holograms of Airy beams using LC-SLM devices. In a second step, developed a experimental system that allowed me the record and the optical reconstruction Airy beam holograms using photorefractive holography method with ....12......20 (BTO) crystal for optical generation of beams Airy. Similarly, upon the obtaining numerical fields, we show the possibility to create arrays of Airy beams in a single holographic recontruccao. In parallel, we did an analysis of the results of the numerical optical reconstruction of hologrmas and the resultlts of optical reconstruction of such holograms, as well as their potential for technological applications.

Over the years, non-spreading or non-diffracting wave configurations have been systematically investigated in optics. Perhaps the best known example of a diffraction-free optical wave is the so-called Bessel beam, first suggested and observed by Durnin et al. This work sparked considerable theoretical and experimental activity and paved the way toward the discovery of other interesting non-diffracting solutions. In 1979 Berry and Balazs made an important observation within the context of quantum mechanics: they theoretically demonstrated that the Schrodinger equation describing a free particle can exhibit a non-spreading Airy wavepacket solution. This work remained largely unnoticed in the literature-partly because such wavepackets cannot be readily synthesized in quantum mechanics. In this dissertation we investigate both theoretically and experimentally the acceleration dynamics of non-spreading optical Airy beams in both one- and two-dimensional configurations. We show that this class of finite energy waves can retain their intensity features over several diffraction lengths. The possibility of other physical realizations involving spatio-temporal Airy wavepackets is also considered. As demonstrated in our experiments, these Airy beams can exhibit unusual features such as the ability to remain quasi-diffraction-free over long distances while their intensity features tend to freely accelerate during propagation. We have demonstrated experimentally that optical Airy beams propagating in free space can perform ballistic dynamics akin to those of projectiles moving under the action of gravity. The parabolic trajectories of these beams as well as the motion of their center of gravity were observed in good agreement with theory. Another remarkable property of optical Airy beams is their resilience in amplitude and phase perturbations. We show that this class of waves tends to reform during propagation in spite of the severity of the imposed perturbations. In all occasions the reconstruction of these beams is interpreted through their internal transverse power flow. The robustness of these optical beams in scattering and turbulent environments was also studied. The experimental observation of self-trapped Airy beams in unbiased nonlinear photorefractive media is also reported. This new class of non-local self-localized beams owes its existence to carrier diffusion effects as opposed to self-focusing. These finite energy Airy states exhibit a highly asymmetric intensity profile that is determined by the inherent properties of the nonlinear crystal. In addition, these wavepackets self-bend during propagation at an acceleration rate that is independent of the thermal energy associated with two-wave mixing diffusion photorefractive nonlinearity.

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Modal analysis on slabs, thin section petrography, microprobe analyses, and geological mapping have been performed on the Mount Airy granite and enclosing country rock. The pluton.is unzoned, simply intruded, and composed of a medium-grained, white granite, containing 33-41% plagioclase (An₁₀₋₂₀), 27-35% quartz, 20-28% alkali feldspar (Or₉₀), and 3-13% color index minerals, primarily biotite, muscovite, and epidote. A more granodiorite aplite phase intrudes the granite and is composed of An₉₋₃ plagioclase, Or₉₅ alkali feldspar, quartz, apatite, muscovite, biotite, and traces of epidote and zircon, and garnet. Aplite is a late segregation of granite and is associated with pegmatite. The enclosing country rock is amphibolite, pelitic schist, and pin-striped gneiss of the Alligator Back Formation. The pelites in contact with the granite on the southeast side have the assemblage garnet + biotite + staurolite consistent with the albite-epidote amphibolite facies metamorphism of the region. The granite has deformed the prior Ordovician foliation of the country rock and is itself cut by shears and folds close to the Brevard Zone. These are the result of Devonian-Mississippian movement on the Brevard. Some of the muscovite is primary in the granite, and epidote is not. Most of the muscovite, sphene, and low-An plagioclase are subsolidus. Alteration of biotite and plagioclase show an early, water-rich fluid evolving into a more CO₂-rich fluid as the rocks cooled. An injection temperature of 700-750°C and pressure of at least 5.5-7 Kbar is consistent with the mineral assemblages seen in the granite, xenoliths, and contact country rocks. The notable lack of hydrothermal activity at the granite margins is attributed to low H₂O content of the original magma and hydration reactions within the granite below the solidus. / Master of Science

This thesis consists of two parts. In the first part, we study a class of special functions Aik (y), k = 2, 4, 6, ··· generalising the classical Airy function Ai(y) to higher orders and in the second part, we apply expressions and properties of Ai4(y) to spectral problem of a specific operator. The first part is however motivated by latter part. We establish regularity properties of Aik (y) and particularly show that Aik (y) is smooth, bounded, and extends to the complex plane as an entire function, and obtain pointwise bounds on Aik (y) for all k. Some analytic properties of Aik (y) are also derived allowing one to express Aik (y) as a finite sum of certain generalised hypergeometric functions. We further obtain full asymptotic expansions of Aik (y) and their first derivative Ai'(y) both for y > 0 and for y < 0. Using these expansions, we derive expressions for the negative real zeroes of Aik (y) and Ai'(y). Using expressions and properties of Ai4(y), we extensively study spectral properties of a non-local operator H whose physical interpretation is the massless relativistic quartic anharmonic oscillator in one dimension. Various spectral results for H are derived including estimates of eigenvalues, spectral gaps and trace formula, and a Weyl-type asymptotic relation. We study asymptotic behaviour, analyticity, and uniform boundedness properties of the eigenfunctions Ψn(x) of H. The Fourier transforms of these eigenfunctions are expressed in two terms, one involving Ai4(y) and another term derived from Ai4(y) denoted by Āi4(y). By investigating the small effect generated by Āi4(y) this work shows that eigenvalues λn of H are exponentially close, with increasing n Ε N, to the negative real zeroes of Ai4(y) and those of its first derivative Ai'4(y) arranged in alternating and increasing order of magnitude. The eigenfunctions Ψ(x) are also shown to be exponentially well-approximated by the inverse Fourier transform of Ai4(|y| - λn) in its normalised form.

M.V. Cerrillo, W.H. Kautz. / "November 15, 1951." / Bibliography: p. 11. / Army Signal Corps Contract DA36-039 SC-64637 Project 102B. Dept. of the Army Projects 3-99-10-022 and DA3-99-10-000.

The main purpose of this thesis is to use modern goal-oriented adaptive methods of Lie group analysis to construct the optimal sys- tem of Black-Scholes equation. We will show in this thesis how to obtain all invariant solutions by constructing what has now become so popular, optimal system of sub-algebras, the main Lie algebra admit- ted by the Black-Scholes equation. First, we obtain the commutator table of already calculated symmetries of the Black-Scholes equation. We then followed with the calculations of transformation of the gen- erators with the Lie algebra L6 which provides one-parameter group of linear transformations for the operators. Here we make use of the method of Lie equations to solve the partial di®erential equations. Next, we consider the construction of optimal systems of the Black- Scholes equation where the method requires a simpli¯cation of a vector to a general form to each of the transformations of the generators. Further, we construct the invariant solutions for each of the op- timal system. This study is motivated by the analysis of Lie groups which is being taken to another level by ALGA here in Blekinge In- stitute Technology, Sweden. We give a practical and in-depth steps and explanation of how to construct the commutator table, the calcu- lation of the transformation of the generators and the construction of the optimal system as well as their invariant solutions. Keywords: Black-Scholes Equation, commutators, commutator table, Lie equa- tions, invariant solution, optimal system, generators, Airy equation, structure constant, / It was an accolade for us to work with Professor Nail.H. Ibrgimov. +46762600953

La thèse présente l’étude de la propagation et de l’auto-focalisation de faisceaux d’Airy dans un milieu photoréfractif. Le faisceau d’Airy est un faisceau dit accélérant qui, dans l’espace libre, présente une trajectoire curviligne, ne se déforme pas et est capable de se régénérer après un obstacle. L’auto-focalisation de faisceaux conventionnels, tels les faisceaux gaussiens, a été étudiée dans des milieux nonlinéaires en particulier pour des applications de routage tout-optique. En propageant des faisceaux optiques à travers de tels milieux photosensibles, il est possible de graver optiquement des guides d’onde retraçant la trajectoire de ces faisceaux. C’est dans ce contexte que le faisceau d’Airy suscite beaucoup d’intérêt, grâce à sa forme et sa trajectoire uniques. Dans ce mémoire nous étudions expérimentalement comme théoriquement les mécanismes d’auto-focalisation du faisceau d’Airy. Durant le régime transitoire de l’effet d’auto-focalisation, nous montrons des dynamiques spatiotemporelles singulières qui suggèrent une analogie avec les interactions gravitationnelles entre un objet massique et une onde se propageant dans l’espace-temps courbe. Dans un second temps, nous ajoutons un faisceau d’Airy se propageant dans la direction opposée au premier afin d’analyser leurs interactions. Ensuite, nous testons ces structures guidantes photoinduites par un ou deux faisceaux d’Airy, qui révèlent des possibilités de guidage uniques, non accessibles avec deux faisceaux conventionnels. Ces faisceaux optiques peuvent permettre de réaliser des fonctions de couplage, routage et multiplexage optique. Par ailleurs, nous étudions les limites de la force de guidage en augmentant la nonlinéarité d’autofocalisation du système. Les dynamiques spatiotemporelles qui en dérivent présentent des comportements et une évolution particuliers suggérant des applications dans le routage tout-optique stationnaire tout comme dynamique. Pour conclure, cette thèse nous permet de démontrer les alternatives prometteuses que nous offre le faisceau d’Airy dans la physique générale et plus particulièrement dans la photonique pour le routage tout-optique / In this thesis we study the propagation and the self-focusing of Airy beams in a photorefractive crystal. The Airy beam is a so-called accelerating beam which propagates in free space along a curved trajectory and with a shape-preserving and self-healing nature. The self-focusing of conventional beams, such as Gaussian beams, has been studied in nonlinear media in particular for all-optical routing solutions. By propagating optical beams in such photosensitive media, one can induce waveguides with the shape of the optical beams’ trajectories. The unique shape and trajectory of the Airy beam however suggest innovative waveguide possibilities. In this manuscript we theoretically and experimentally study the self-focusing mechanisms of the Airy beam. In particular during the transient self-focusing effect, we enlighten peculiar spatiotemporal dynamics suggesting an analogy with the gravitational interactions between a mass and a wave propagating in a curved spacetime. In a second step we add an Airy beam propagating in the opposite direction to analyze their cross-coupling interactions. The guiding structures induced by one or two counterpropagating Airy beams are then tested and show peculiar guiding possibilities that are not achievable using two conventional beams: optical beams can be guided along curved trajectories and eventually split into multiple beams. Furthermore the limits of the waveguiding strength are studied by increasing the self-focusing nonlinearity of the system. The resulting spatiotemporal dynamics present a peculiar behavior and evolution with possible applications in static and dynamical all-optical routing as well as optical computing such as random number generation. Finally with this thesis we demonstrate that the Airy beam offers promising alternatives in general physics and more specifically in photonics for all-optical routing

Light scattering by small particles is the physical phenomenon that produces the natural rainbow in the sky. The same phenomenon can be reproduced in laboratory with monochromatic light giving rise to non-intrusive laser based techniques for the measurement of size and refractive index of particles suspended in a medium possessing lower refractive index. These techniques are commonly called "Rainbow Thermometry" techniques and appear at the beginning of the nineties. The measurement of size and refractive index of the particles is obtained through the analysis of the interference fringe image generated in the far field by means of the interaction between the laser light and the particles themselves. The work presented in this thesis has as objective the improvement of the Rainbow Techniques both on the theoretical aspects and the application.<p>In this thesis the Airy theory, which is used as theoretical model for the Standard Rainbow Thermometry, has been improved to minimize the discrepancies it was presenting respect to more complex and complete theories and it has been generalized to spherical particles presenting a spherically symmetric refractive index gradient. This generalized model has been used to evaluate the size and temperature of a n-octane burning droplet in standard atmosphere with good results. The generalization of the Standard Rainbow Thermometry to multiple particles, Global Rainbow Thermometry, is presented both theoretically and experimentally and the role of the particle asphericity in the light scattered intensity is evaluated.<p>Two experimental application of the Global Rainbow Thermometry are reported. The first one concerns the measurement of size and refractive index of silicon oil droplets suspended in an aqueous bulk. This experiment allows the Global Rainbow Thermometry validation through the comparison with measurements simultaneously obtained with well known experimental techniques as Back-Light scattering and thermocouple measurement with satisfactory results.<p>The second experiment concerns Global Rainbow Thermometry measurement performed on an evaporating flat-fan water spray. The results obtained, both for temperature and for size, are in satisfactory agreement with predictions already done by other authors and with numerical simulations performed by the engineering code CASIMIRE.<p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished

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