Dissertations / Theses on the topic 'Linear wave scattering'

Using the method of matched asymptotic expansions the reflection and transmission coefficients are calculated for scattering of oblique water waves by a vertical barrier. Here an assumption is made that the barrier is small compared to the wavelength and the depth of water. A number of sloshing problems are considered. The eigenfrequencies are calculated when a body is placed in a rectangular tank. Here the bodies considered are a vertical surface-piercing or bottom-mounted barrier, and circular and elliptic cylinders. When the body is a vertical barrier, the eigenfunction expansion method is applied. When the body is either a circular or elliptic cylinder, and the motion is two-dimensional, the boundary element method is applied to calculate the eigenfrequencies. For comparison, two approximations, "a wide-spacing", and "a small-body" are used for a vertical barrier and circular cylinder. In the wide-spacing approximation, the assumption is made that the wavelength is small compared with the distance between the body and walls. The small-body approximation means that a typical dimension of the body is much larger than the cross-sectional length scale of the fluid motion. For an elliptic cylinder, the method of matched asymptotic expansions is used and compared with the result of the boundary- element method. Also a higher-order solution is obtained using the method of matched asymptotic expansions, and it is compared with the exact solution for a surface-piercing barrier. Again the assumption is made that the length scale of the motion is much larger than a typical body dimension. Finally, the drift force on multiple bodies is considered the ratio of horizontal drift force in the direction of wave advance on two cylinders to that on an isolated cylinder is calculated. The method of matched asymptotic expansions is used under the assumption that the wavelength is much greater than the cylinder spacing.

This thesis describes the experimental investigation of three different strongly correlated transition-metal oxide systems. The magnetic behaviour of each has been probed using inelastic neutron spectroscopy. A distinctive hour-glass excitation spectrum has been observed in the layered cobaltate La_1.75Sr_0.25CoO₄. This spectrum is similar to that measured in a related cobaltate La_1.67Sr_0.33CoO₄, although it appears broader. The spectrum has been reproduced using a spin wave model derived from a disordered cluster spin glass ground state. Signatures of spin glass behaviour have also been observed in bulk magnetisation measurements of La_1.75Sr_0.25CoO₄. These findings, once more, demonstrate the emergence of an hour-glass spectrum from a ground state that combines quasi-one dimensional magnetic correlations and disorder. Additionally, this study shows that charge and magnetic stripe order persists to lower dopings in La_2-xSr_xCoO₄ than previously thought. The complete magnetic excitation spectrum of the multiferroic compound CuO has been measured for the first time. A high energy, one-dimensional magnetic spectrum is observed and modelled using the Muller ansatz derived for the S=1/2 Heisenberg antiferromagnetic chain. At lower energies, a three-dimension spectrum is observed. The measured spectrum is inconsistent with all previous theoretical estimates of the dominant inter-chain exchange interactions in CuO. The inter-chain dispersion is successfully described by a phenomenological model based on linear spin wave theory. The third material investigated, LuFe₂O₄ demonstrates complex charge and magnetic order, the precise nature of which is still under debate. The full spectrum of in-plane excitations in LuFe₂O₄ has been measured and a complicated dispersion consistent with six magnetic modes is observed. These findings are compatible with structures described by a magnetic unit cell containing six spins. The dispersion can be described by a spin wave model derived from a bilayer structure comprised of charge-rich and charge-poor monolayers. This structure is consistent with the original site-specific model for the 3D magnetic ordering in LuFe₂O₄.

Lorsque les deux ondes se couplent en basse frequence, on obtient un effet de pression de radiation ou lieu de force ponderomotrice. Ces notions sont etudiees dans un plasma magnetise ou non, ainsi que le cas ou la force ponderomotrice depend lentement du temps. On envisage deux applications : on etudie, a l'aide d'un double blanc microonde la generation d'une resonance par battement d'ondes, avec une detection sensible au transfert d'action. On considere la possibilite de diagnostic du plasma magnetise par optique non lineaire. L'analyse de la polarisation d'une onde diffusee par des fluctuations electromagnetiques est enfin traitee

Nous étudions l’analyse asymptotique en relativité générale sous deux aspects: le peeling et le scattering (diffusion) conforme. Le peeling est construit pour les champs scalaires linéaire et non-linéaires et pour les champs de Dirac en espace-temps de Kerr (qui est non-stationnaire et à symétrie simplement axiale), généralisant les travaux de L. Mason et J-P. Nicolas (2009, 2012). La méthode des champs de vecteurs (estimations d’énergie géométriques) et la technique de compactification conforme sont développées. Elles nous permettent de formuler les définitions du peeling à tous ordres et d’obtenir les données initiales optimales qui assurent ces comportements. Une théorie de la diffusion conforme pour les équations de champs sans masse de spîn n/2 dans l’espace-temps de Minkowski est construite.En effectuant les compactifications conformes (complète et partielle), l’espace-temps est complété en ajoutant une frontière constituée de deux hypersurfaces isotropes représentant respectivement les points limites passés et futurs des géodésiques de type lumière. Le comportement asymptotique des champs s’obtient en résolvant le problème de Cauchy pour l’équation rééchelonnée et en considérant les traces des solutions sur ces bords. L’inversibilité des opérateurs de trace, qui associent le comportement asymptotique passé ou futur aux données initiales, s’obtient en résolvant le problème de Goursat sur le bord conforme. L’opérateur de diffusion conforme est alors obtenu par composition de l’opérateur de trace futur avec l’inverse de l’opérateur de trace passé
This work explores two aspects of asymptotic analysis in general relativity: peeling and conformal scattering.On the one hand, the peeling is constructed for linear and nonlinear scalar fields as well as Dirac fields on Kerr spacetime, which is non-stationary and merely axially symmetric. This generalizes the work of L. Mason and J-P. Nicolas (2009, 2012). The vector field method (geometric energy estimates) and the conformal technique are developed. They allow us to formulate the definition of the peeling at all orders and to obtain the optimal space of initial data which guarantees these behaviours. On the other hand, a conformal scattering theory for the spin-n/2 zero rest-mass equations on Minkowski spacetime is constructed. Using the conformal compactifications (full and partial), the spacetime is completed with two null hypersurfaces representing respectively the past and future end points of null geodesics. The asymptotic behaviour of fields is then obtained by solving the Cauchy problem for the rescaled equation and considering the traces of the solutions on these hypersurfaces. The invertibility of the trace operators, that to the initial data associate the future or past asymptotic behaviours, is obtained by solving the Goursat problem on the conformal boundary. The conformal scattering operator is then obtained by composing the future trace operator with the inverse of the past trace operator

Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface. We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem.

L'objet de cette thèse est l'étude des équations de Schrödinger et des ondes, à la fois linéaires et non linéaires, dans des domaines extérieurs. Nous nous intéressons en particulier aux inégalités dites de Strichartz, qui sont une famille d'estimations dispersives mesurant la décroissance du flot linéaire, particulièrement utiles à l'étude des problèmes non linéaires correspondants. Dans des géométries dites non-captantes, c'est à dire où tous les rayons de l'optique géométrique partent à l'infini, de nombreux résultats montrent que de telles estimations sont aussi bonnes que dans l'espace libre. D'autre part, la présence d'une trajectoire captive induit nécessairement une perte au niveau d'une autre famille d'estimations à priori, les estimations d'effet régularisant et de décroissance locale de l'énergie, respectivement pour Schrödinger et pour les ondes. En contraste de quoi, nous montrons des estimations de Strichartz sans perte dans une géométrie captante instable : l'extérieur de plusieurs obstacles strictement convexes vérifiant la condition d'Ikawa. La seconde partie de cette thèse est dédiée à l'étude du comportement en temps long des équations non-linéaires sous-jacentes. Lorsque le domaine dans lequel elles vivent n'induit pas trop de concentration de l'énergie, on s'attend à ce qu'elles diffusent, c'est à dire se comportent de manière linéaire asymptotiquement en temps. Nous montrons un tel résultat pour les ondes non linéaires critiques à l'extérieur d'une classe d'obstacles généralisant la notion d'étoilé. A l'extérieur de deux obstacles strictement convexes, nous obtenons un résultat de rigidité concernant les solutions à flot compact, premier pas vers un résultat général. Enfin, nous nous intéressons à l'équation de Schrödinger non linéaire, dans l'espace libre, mais avec un potentiel. Nous montrons que les solutions diffusent si l'on prend un potentiel répulsif, ainsi qu'une somme de deux potentiels répulsifs ayant des surfaces de niveau convexes, ce qui fournit un exemple de diffusion dans une géométrie captante analogue à l'extérieur de deux convexes stricts
We are concerned with Schrödinger and wave equations, both linear and non linear, in exterior domains. In particular, we are interested in the so-called Strichartz estimates, which are a family of dispersive estimates measuring decay for the linear flow. They turn out to be particularly useful in order to study the corresponding non linear equations. In non-captive geometries, where all the rays of geometrical optics go to infinity, many results show that Strichartz estimates hold with no loss with respect to the flat case. Moreover, the local smoothing estimates for the Schrödinger equation, respectively the local energy decay for the wave equation, which are another family of dispersive estimates, are known to fail in any captive geometry. In contrast, we show Strichartz estimates without loss in an unstable captive geometry: the exterior of many strictly convex obstacles verifying Ikawa's condition. The second part of this thesis is dedicated to the study of the long time asymptotics of the corresponding non linear equations. We expect that they behave linearly in large times, or scatter, when the domain they live in does not induce too much concentration effect. We show such a result for the non linear critical wave equation in the exterior of a class of obstacles generalizing star-shaped bodies. In the exterior of two strictly convex obstacles, we obtain a rigidity result concerning compact flow solutions, which is a first step toward a general result. Finally, we consider the non linear Schrödinger equation in the free space but with a potential. We prove that solutions scatter for a repulsive potential, and for a sum of two repulsive potentials with strictly convex level surfaces. This provides a scattering result in a framework similar to the exterior of two strictly convex obstacles

Orientador: Hugo Luis Fragnito
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin
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Resumo: Esta tese apresenta um estudo teórico e experimental sobre Amplificadores Paramétricos a Fibra Óptica. Mostramos primeiramente a potencialidade do amplificador em termos de largura de banda plana, necessária para aplicações em sistemas WDM. Foram desenvolvidas equações do ripple para os diferentes tipos de espectro de alto ganho possíveis. Usando fibras de dispersão deslocada e fibras altamente não-lineares curtas, construímos amplificadores de dois bombeios com uma banda plana de amplificação de 52 nm e 84 nm, respectivamente. Em ambos os casos, a banda obtida foi comparável à obtida com outras tecnologias como Érbio ou Raman. Pesquisamos também o crosstalk entre canais em amplificadores paramétricos. Mostramos como as variações de ë 0 ao longo da fibra aumentam o crosstalk, pois se incrementam os processos de mistura de quatro ondas espúrios. Em amplificadores de dois bombeios, obtivemos os melhores resultados para aplicações WDM com fibras curtas (L ~ 0.8 km). Finalmente, a partir da análise da penalidade do fator de mérito Q do sinal, estudamos como os efeitos da modulação de fase do/s laser/s de bombeio, necessária para suprimir o espalhamento Brillouin estimulado, são prejudiciais para a performance destes amplificadores. Com base nesse resultado, procuramos eliminar o espalhamento Brillouin com um método alternativo que consiste na aplicação de distribuições de deformação mecânica ao longo da fibra. Obtivemos um aumento de 10.7 dB em 285 m de fibra altamente não-linear, o que é da ordem dos valores obtidos modulando a fase do/s bombeio/s. Aplicamos a técnica em dispositivos paramétricos obtendo um conversor de comprimento de onda com um sinal convertido de largura de banda estreito, sintonizável ao longo de 70 nm de comprimento de onda do sinal de entrada
Abstract: This thesis presents a theoretical and experimental study on fiber optics parametric amplifiers. For the double-pumped case, we show the performance of the amplifier in terms of spectral bandwidth and gain uniformity, both fundamental properties for WDM applications. By using short dispersion shifted fibers and highly non-linear fibers, we made parametric amplifiers with 52 and 84 nm of flat bandwidth, respectively. In both cases, the bandwidth is similar to the one obtained with optical amplifiers based on other technologies such as doped fibers, Raman gain or semiconductors. We also deduced equations of the gain ripple for all the possible types of high gain spectra. We also investigated the inter-channel crosstalk in parametric amplifiers. We showed that the variations of the zero dispersion wavelength along the fiber increase the crosstalk because those variations enhances the spurious four wave mixing tones. Finally, the degradation of the signal quality factor (Q) was studied for both, single and double-pumped parametric amplifiers. We showed that the phase modulation applied to the pump lasers to suppress the stimulated Brillouin backscattering can strongly degrade the performance of the amplifier. One alternative technique that we developed was based on applying different mechanical stress distributions along the fiber length. We experimentally demonstrated, using 285 m of highly non-linear fibers, that the backscattering threshold can be increased by 10.7 dB. This value is similar to that obtained using the phase modulation schema. The method was successfully applied to parametric devices. In particular we made a narrow linewidth wavelength converter with 70 nm of signal tuning band
Doutorado
Física
Doutor em Ciências

Par l'utilisation de lasers de puissance de type picoseconde (longueur d'onde 532 nm, durée 25ps) dans des milieux liquides et cristallins, nous avons apporté notre contribution a : - l'aspect faisabilité de la conjugaison de phase en picoseconde, -l'évaluation du comportement transitoire ou pas des phénomènes, - des applications particulières a ce domaine. Nous avons caractérisé et étudie par voie expérimentale et théorique différents paramètres entrant en jeu dans les deux types d'expériences susceptibles de produire la conjugaison de phase- mélange quatre ondes et rétrodiffusions stimulées- en vue d'applications de type temporel ou imagerie, dans les liquides (principalement sulfure de carbone et benzène). Le comportement en excitation picoseconde des matériaux photoréfractifs (oxyde de bismuth silicium et niobate de lithium) a été étudié par des expériences de mélange quatre ondes et mélange deux ondes. Il ressort des principaux résultats -expérimentaux et théoriques- que certains sont propre au régime d'excitation picoseconde alors que d'autres sont d'ordre général et indépendants du régime temporel d'excitation.

Nous etudions la reponse optique non-lineaire d'un cristal de chlorure de cuivre en interaction avec un faisceau laser intense et de grande finesse spectrale. Un calcul non perturbatif utilisant le formalisme de la matrice densite permet d'obtenir la fonction dielectrique du cristal, represente par un modele a trois ou quatre niveaux (niveau fondamental, excitonique, biexcitonique) dans ces conditions. Les renormalisations des courbes de dispersion, du coefficient d'absorption et de reflexion normale sont calculees. L'utilisation de lasers a colorants a incidence rasante pompes par un laser a excimere permet l'etude experimentale de ces renormalisations par diffusion hyper-raman, melange de quatre ondes et reflexion normale.

Nous avons construit des lasers qui ont servi a des experiences de melange a quatres ondes (reseau transitoire). Nous avons d'abord etudie la reponse non lineaire du troisieme ordre lorsque le milieu presente un continuum large d'etats (colorants). Le signal obtenu sur le cresyl-violet en film mince de colle cellulosique a ete analyse. La technique du reseau transitoire a ete ensuite utilisee pour caracteriser la dynamique de molecules dans un milieu poreux (vycor) et etudier la geometrie des connexions entre pores. Nous avons etudie la rotation de l'azobenzene en solution soit libre soit contenue a l'interieur de l'espace poreux. Ces experiences peuvent etre realisees sur des echelles allant de la femtoseconde a la seconde, permettant ainsi l'etude tant du temps de coherence que des proprietes de diffusion des materiaux

Les résonateurs à modes de galerie sont des cavités diélectriques qui supportent des modes à très haut facteur de qualité et à faible volume qui demeurent confinés à l'interface air-diélectrique pour des durées pouvant atteindre voire dépasser la microseconde. L'intérêt de ce fort confinement des modes pour de longues durées est l'accentuation de l'interaction lumière-matière. Par conséquent, de nombreuses interactions non-linéaires telles que l'effet Kerr ou encore l'effet Brillouin à des puissances seuil inversement proportionnelles au carré voire au cube du facteur de qualité du résonateur ont lieu en son sein. Ces propriétés donnent accès à de nombreuses applications dans des domaines divers et variés tels que la spectroscopie, les télécommunications ou encore les micro-ondes. Les travaux de cette thèse ont pour but d'exploiter les non-linéarités Kerr et Brillouin dans les résonateurs à mode galerie à la longueur d'onde de 1550 nm afin de générer des micro-ondes ultra-stables à des fréquences comprises entre 5 et 30 GHz. Le premier chapitre introduit la théorie, la fabrication, le couplage et la caractérisation de résonateurs à des modes de galerie. Le second chapitre concerne la génération de micro-ondes. Nous y présentons nos résultats expérimentaux, la modélisation numérique de peignes de Kerr, ainsi que l'analyse d'instabilités oscillatoires d'origine thermique observées lors de nos travaux expérimentaux, puis nous concluons. Le dernier chapitre traite de l'interaction photons-phonons via le processus de diffusion Brillouin stimulée dans ces mêmes expérimentaux ainsi que le modèle temporel que nous avons développé pour suivre la dynamique de l'onde transmise et de celle rétrodiffusée. Le dernier chapitre conclue nos travaux. Les travaux présentés dans ce manuscrit ont été financé par l'European Research Council dans le cadre du projet NextPhase
Whispering galery mode resonators are dielectric cavities that support modes with ultra-high quality factor and small volume that remain confined in their inner periphery for time duratioons that can be as long as few microseconds. The strong confinement of these modes for such long durations strongly enhances nonlinear effect suchs as Kerr effect or Brillouin effect. These resonators can therefore be used for several applications such as spectroscopy, telecommunications or microwave generation. The objective of this thesis is to use Kerr and Brillouin nonlinearities in these resonators at the laser wavelength of 1550 nm, in order to generate high spectral purity microwave signals with frequencies rangong fros 5 to 30 GHz. The first chapter oh the thesis intriduces the theory, fabrication, coupling and characterisation of whispering gallery mode resonators. The second chapter is about the generation of Kerr optical frequency combs in these resonators and their application to the generation of microwave signals. We present our experimental resuktsdn the numerical modelling of Kerr combs, the analysis of oscillatory instabilities (due to thermal effect) observed during our experiments, and conclue. The third chapter concerns photon-phonon interactions via stimulated Brillouin scattering in these resonators and their application to the generation of microwave signals. We present our experimental results and the temporal model that we developed to track the dynamics of the forward and backscattered fields. The last chapter conclude the thesis. The research presented in this thesis has benne funded by the European Research Council through the project Nextphase

The reflection and transmission of an electromagnetic TE-polarized plane wave at a dielectric film between two linear semi-infinite media (substrate and cladding) is considered. All media are assumed to be homogeneous in x- and z- direction, isotropic, and non-magnetic. The permittivity of the film is assumed to be characterized by a continuously differentiable function of the transverse coordinate and the field. To obtain solutions of Maxwell´s equations that satisfy the boundary conditions the problem is reduced to a Helmholtz equation, which is transformed to a Volterra integral equation for the field intensity inside the film. The Volterra equation is solved by iteration subject to the appropriate boundary conditions. The (iteration) solutions for the linear case and for the nonlinear case are expressed in terms of a uniformly convergent series and a uniformly convergent sequence, respectively. The uniform convergence is proved using the Banach Fixed-Point Theorem. The condition for its applicability leads to a condition for the parameters of the problem. By iterating the Volterra equation an approximate solution for the intensity inside the film is presented. The mathematical basis of the procedure is outlined in detail. Using an approximate solution, the phase function,the phase shifts on reflection and transmission, the reflectivity and the absorptance are determined.Further iterations of the Volterra equation are possible.Semianalytical and numerical examples illustrate the main features of the approach. The method is succesfully applied to different permittivity functions (real, complex, Kerr-like and saturable nonlinear). The agreement between the approximate analytical solutions and numerical solutions is satisfactory. It seems that the method proposed can serve as a means to optimize certain parameters of the problem (material and/or geometrical) for particular purposes.

碩士
國立清華大學
光電工程研究所
101
There have been numerical investigations for electron and electromagnetic waves in the one-dimensional disordered lattice. The electron eigenmodes of one-dimensional disordered lattice are investigated for three different models. The electromagnetic eigenmodes of one-dimensional disordered lattice are also investigated. The degree of localization of the wavelength at the band, gap edge and gap are studied. Finally, the disordered lattice solitons are presented.