La diffraction des rayons X a été développée depuis près d'un siècle pour la détermination de structures cristallographiques. Mais la détermination des structures les plus complexes (protéines, structures incommensurables...) nécessite l'utilisation de la diffraction anomale, i.e. la mesure des intensités de diffraction à plusieurs longueurs d'onde au voisinage du seuil d'absorption d'un élément du cristal. Cette technique permet d'obtenir une information sur la phase du facteur de structure, ainsi que sur les positions des atomes anomaux. Dans cette thèse, nous présentons la Diffraction Anomale Dispersive (DAD), qui permet de mesurer simultanément les intensités diffractées à plusieurs longueurs d'onde, pour de nombreuses réflexions. Nous présentons deux modes de collecte, continu (DDAFS-Dispersive Diffraction Anomalous fine Structure), et discret (SMAD-Simultaneous Multiwavelength Anomalous Diffraction). Nous avons développé une procédure et un programme (DAD) pour l'analyse quantitative des images de diffraction dispersive. Ce programme permet également l'analyse d'images de diffraction monochromatique présentant des réflexions satellites proches des pics principaux. Nous présentons les deux premières expériences quantitatives de diffraction dispersive sur des cristaux biologiques. Nos résultats montrent que la détermination de structure par la méthode SMAD est possible. Des améliorations aux protocoles de collecte et d'analyse sont encore nécessaires pour ces cristaux. Une partie importante de cette thèse a été consacrée à l'étude de (TaSe4)2I : ce cristal quasi-1D présente une transition de Peierls, la condensation des atomes de tantale étant recherchée depuis 15 ans. Notre étude a d'abord caractérisé la structure en domaines de ce matériau, et la diffraction anomale a mis en évidence de manière spécifique la tétramérisation des atomes de tantale, accompagnant la modulation acoustique déjà connue.

Theoretical and experimental studies of real-time interrogation of fiber Bragg grating (FBG) sensors based on chirped pulse compression with increased interrogation resolution and signal-to-noise ratio are presented. Two interrogation systems are proposed in this thesis. In the first interrogation system, a linearly chirped FBG (LCFBG) is employed as the sensing element. By incorporating the LCFBG in an optical interferometer as the sensor encoding system, employing wavelength-to-time mapping and chirped pulse compression technique, the correlation of output microwave waveform with a chirped reference waveform would provide an interrogation result with high speed and high resolution. The proposed system can provide an interrogation resolution as high as 0.25 μ at a speed of 48.6 MHz. The second interrogation system is designed to achieve simultaneous measurement of strain and temperature. In this system, a high-birefringence LCFBG (Hi-Bi LCFBG) is employed as a sensing element.

This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem? The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions. The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported on H^s for s<1/2, and local solutions to the quartic KdV cannot be obtained below H^{1/2} by using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}. Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}. The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.

Theoretical and experimental studies of real-time interrogation of fiber Bragg grating (FBG) sensors based on chirped pulse compression with increased interrogation resolution and signal-to-noise ratio are presented. Two interrogation systems are proposed in this thesis. In the first interrogation system, a linearly chirped FBG (LCFBG) is employed as the sensing element. By incorporating the LCFBG in an optical interferometer as the sensor encoding system, employing wavelength-to-time mapping and chirped pulse compression technique, the correlation of output microwave waveform with a chirped reference waveform would provide an interrogation result with high speed and high resolution. The proposed system can provide an interrogation resolution as high as 0.25 μ at a speed of 48.6 MHz. The second interrogation system is designed to achieve simultaneous measurement of strain and temperature. In this system, a high-birefringence LCFBG (Hi-Bi LCFBG) is employed as a sensing element.

This thesis presents a Hamiltonian formulation of the electromagnetic fields in structured (inhomogeneous) media of arbitrary dimensionality, with arbitrary material dispersion and absorption consistent with causality. The method is based on an identification of the photonic component of the polariton modes of the system. Although the medium degrees of freedom are introduced in an oscillator model, only the macroscopic response of the medium appears in the derived eigenvalue equation for the polaritons. For both the discrete transparent-regime spectrum and the continuous absorptive-regime spectrum, standard codes for photonic modes in nonabsorptive systems can easily be leveraged to calculate polariton modes. Two applications of the theory are presented: pulse propagation and spontaneous parametric down-conversion (SPDC). In the propagation study, the dynamics of the nonfluctuating part of a classical-like pulse are expressed in terms of a Schr\"{o}dinger equation for a polariton effective field. The complex propagation parameters of that equation can be obtained from the same generalized dispersion surfaces typically used while neglecting absorption, without incurring additional computational complexity. As an example I characterize optical pulse propagation in an Au/MgF$_2$ metallodielectric stack, using the empirical response function, and elucidate the various roles of Bragg scattering, interband absorption and field expulsion. Further, I derive the Beer coefficient in causal structured media. The SPDC calculation is rigorous, captures the full 3D physics, and properly incorporates linear dispersion. I obtain an expression for the down-converted state, quantify pair-production properties, and characterize the scaling behavior of the SPDC energy. Dispersion affects the normalization of the polariton modes, and calculations of the down-conversion efficiency that neglect this can be off by 100$\%$ or more for common media regardless of geometry if the pump is near the band edge. Furthermore, I derive a 3D three-wave group velocity walkoff factor; due to the interplay of a topological property with a symmetry property, I show that even if down-conversion is into a narrow forward cone, neglect of the transverse walkoff can lead to an overestimate of the SPDC energy by orders of magnitude.

This thesis presents a Hamiltonian formulation of the electromagnetic fields in structured (inhomogeneous) media of arbitrary dimensionality, with arbitrary material dispersion and absorption consistent with causality. The method is based on an identification of the photonic component of the polariton modes of the system. Although the medium degrees of freedom are introduced in an oscillator model, only the macroscopic response of the medium appears in the derived eigenvalue equation for the polaritons. For both the discrete transparent-regime spectrum and the continuous absorptive-regime spectrum, standard codes for photonic modes in nonabsorptive systems can easily be leveraged to calculate polariton modes. Two applications of the theory are presented: pulse propagation and spontaneous parametric down-conversion (SPDC). In the propagation study, the dynamics of the nonfluctuating part of a classical-like pulse are expressed in terms of a Schr\"{o}dinger equation for a polariton effective field. The complex propagation parameters of that equation can be obtained from the same generalized dispersion surfaces typically used while neglecting absorption, without incurring additional computational complexity. As an example I characterize optical pulse propagation in an Au/MgF$_2$ metallodielectric stack, using the empirical response function, and elucidate the various roles of Bragg scattering, interband absorption and field expulsion. Further, I derive the Beer coefficient in causal structured media. The SPDC calculation is rigorous, captures the full 3D physics, and properly incorporates linear dispersion. I obtain an expression for the down-converted state, quantify pair-production properties, and characterize the scaling behavior of the SPDC energy. Dispersion affects the normalization of the polariton modes, and calculations of the down-conversion efficiency that neglect this can be off by 100$\%$ or more for common media regardless of geometry if the pump is near the band edge. Furthermore, I derive a 3D three-wave group velocity walkoff factor; due to the interplay of a topological property with a symmetry property, I show that even if down-conversion is into a narrow forward cone, neglect of the transverse walkoff can lead to an overestimate of the SPDC energy by orders of magnitude.

The aim of this work was to investigate and apply new DLLME systems coupled with LA-ICP-MS and ultra-performance liquid chromatography (UPLC) techniques for the preconcentration and determination of Cr(VI), Co(II), Cu(II) ir Ni(II) ions. / Šioje daktaro disertacijoje apibendrintų mokslinių tyrimų tikslas – ištirti ir pritaikyti naujus Cr(VI), Co(II), Cu(II) ir Ni(II) koncentravimo ir nustatymo metodus apjungiant dispersinę skystafazę mikroekstrakciją (DSME) su lazerinio išgarinimo induktyviai sužadintos plazmos masių spektrometrijos (LA-ICP-MS) bei ultraefektyviosios skysčių chromatografijos (UESCh) metodais.

Šioje daktaro disertacijoje apibendrintų mokslinių tyrimų tikslas – ištirti ir pritaikyti naujus Cr(VI), Co(II), Cu(II) ir Ni(II) koncentravimo ir nustatymo metodus apjungiant dispersinę skystafazę mikroekstrakciją (DSME) su lazerinio išgarinimo induktyviai sužadintos plazmos masių spektrometrijos (LA-ICP-MS) bei ultraefektyviosios skysčių chromatografijos (UESCh) metodais. / The aim of this work was to investigate and apply new DLLME systems coupled with LA-ICP-MS and ultra-performance liquid chromatography (UPLC) techniques for the preconcentration and determination of Cr(VI), Co(II), Cu(II) ir Ni(II) ions.

The most commonly used method for the time-domain Maxwell equations is the Finite-Difference Time-Domain method (FDTD). This is an explicit, second-order accurate method, which is used on a staggered Cartesian grid. The main drawback with the FDTD method is its inability to accurately model curved objects and small geometrical features. This is due to the Cartesian grid, which leads to a staircase approximation of the geometry and small details are not resolved at all. This thesis presents different ways to circumvent this drawback, but still take advantage of the benefits of the FDTD method. An approach to avoid staircasing errors but still retain the efficiency of the FDTD method is to use a hybrid grid. A few layers of unstructured cells are used close to curved objects and a Cartesian grid is used for the rest of the domain. For the choice of solver on the unstructured grid two different alternatives are compared: an explicit Finite-Volume Time-Domain (FVTD) solver and an implicit Finite-Element Time-Domain (FETD) solver. The hybrid solvers calculate the scattering from complex objects much more efficiently compared to using FDTD on highly resolved Cartesian grids. For the same accuracy in the solution roughly a factor of 10 in memory requirements and a factor of 20 in execution time are gained. The ability to model features that are small relative to the cell size is often important in electromagnetic simulations. In this thesis a technique to generalize a well-known subcell model for thin wires, in order to take arbitrarily oriented wires in FETD and FDTD into account, is proposed. The method gives considerable modeling flexibility compared to earlier methods and is proven stable. The results show excellent consistency and very good accuracy on different antenna configurations. The recursive convolution method is often used to model frequency dispersive materials in FDTD. This method is used to enable modeling of such materials in the unstructured FVTD and FETD solvers. The stability of both solvers is analyzed and their accuracy is demonstrated by computing the radar cross section for homogeneous as well as layered spheres with frequency dependent permittivity.

This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem? The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions. The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported on H^s for s<1/2, and local solutions to the quartic KdV cannot be obtained below H^{1/2} by using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}. Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}. The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.