This dissertation presents the contributions and the research conducted in developing and implementing Microwave Tomography (MWT) systems. MWT is an imaging modality which aims to interrogate an object of interest by microwave energy, and quantitatively “find” the interior spatial distribution of its dielectric properties using field measurements taken outside the object. Due to the inherent non-linearity of the MWT problem, a substantial amount of electromagnetic scattering data is required to ensure a robust inversion and quantitatively accurate imaging results. This research benefits a variety of applications including biomedical imaging, industrial non-destructive testing, and security applications. Developing a MWT system, requires many critical components including the bandwidth and polarization purity of the collected fields as well as calibration of the fields scattered by the object of interest. Two generations of MWT systems were designed, implemented, calibrated and tested at the University of Manitoba (UM). These systems aim different approaches for near-field measurements which are referred to as the direct and indirect methods. With regard to the antenna design, a novel methodology applicable to broadband planar antennas is introduced. This technique is based on a combination of field modelling, herein, the finite element method and transmission line modelling. In the first generation of the UM MWT systems, a suitable antenna system was utilized. The system under study was a prototype, where twenty-four co-resident antennas encircle the object of interest to directly measure the fields. In the second generation of the UM MWT systems, the feasibility of using a novel technique to indirectly measure the fields by a secondary array of near-field scatterer probes was studied. The technique is based on the Modulated Scatterer Technique (MST). In this system, antennas are called ``collectors", since the role of antennas are changed to collecting probes' scattered fields. A number of PIN diodes were utilized to activate the probes. Finally, the capability of the probe system was investigated and its performance with the previously constructed tomography systems was compared. Various dielectric phantoms were utilized to test the accuracy of the systems.

This dissertation presents the contributions and the research conducted in developing and implementing Microwave Tomography (MWT) systems. MWT is an imaging modality which aims to interrogate an object of interest by microwave energy, and quantitatively “find” the interior spatial distribution of its dielectric properties using field measurements taken outside the object. Due to the inherent non-linearity of the MWT problem, a substantial amount of electromagnetic scattering data is required to ensure a robust inversion and quantitatively accurate imaging results. This research benefits a variety of applications including biomedical imaging, industrial non-destructive testing, and security applications. Developing a MWT system, requires many critical components including the bandwidth and polarization purity of the collected fields as well as calibration of the fields scattered by the object of interest. Two generations of MWT systems were designed, implemented, calibrated and tested at the University of Manitoba (UM). These systems aim different approaches for near-field measurements which are referred to as the direct and indirect methods. With regard to the antenna design, a novel methodology applicable to broadband planar antennas is introduced. This technique is based on a combination of field modelling, herein, the finite element method and transmission line modelling. In the first generation of the UM MWT systems, a suitable antenna system was utilized. The system under study was a prototype, where twenty-four co-resident antennas encircle the object of interest to directly measure the fields. In the second generation of the UM MWT systems, the feasibility of using a novel technique to indirectly measure the fields by a secondary array of near-field scatterer probes was studied. The technique is based on the Modulated Scatterer Technique (MST). In this system, antennas are called ``collectors", since the role of antennas are changed to collecting probes' scattered fields. A number of PIN diodes were utilized to activate the probes. Finally, the capability of the probe system was investigated and its performance with the previously constructed tomography systems was compared. Various dielectric phantoms were utilized to test the accuracy of the systems.

Temporal pulse shaping based on integrated Bragg gratings is investigated in this work to achieve arbitrary output waveforms. The grating structure is simulated based on the sidewall-etching geometry in an AlGaAs platform. The inverse scattering employin the Gel'fan-Levithan-Marchenko theorem and the layer peeling method provides a tool to determine grating structures from a desired spectral reflection response. Simulations of pulse shaping considered flat-top and triangular pulses as well as one-to-one and one-to-many pulse shaping. The suggested grating profiles revealed a compromise between performance and grating length. The integrated grating, a few hundred microns in length, could generate flat-top pulses with pulse durations as short as 500 fs with rise/fall times of 200 fs; the results are comparable to previous work in free-space optics and fiber optics. The theories and the devised algorithms could serve as a design station for advanced grating devices for, but not restricted to, optical pulse shaping.

Temporal pulse shaping based on integrated Bragg gratings is investigated in this work to achieve arbitrary output waveforms. The grating structure is simulated based on the sidewall-etching geometry in an AlGaAs platform. The inverse scattering employin the Gel'fan-Levithan-Marchenko theorem and the layer peeling method provides a tool to determine grating structures from a desired spectral reflection response. Simulations of pulse shaping considered flat-top and triangular pulses as well as one-to-one and one-to-many pulse shaping. The suggested grating profiles revealed a compromise between performance and grating length. The integrated grating, a few hundred microns in length, could generate flat-top pulses with pulse durations as short as 500 fs with rise/fall times of 200 fs; the results are comparable to previous work in free-space optics and fiber optics. The theories and the devised algorithms could serve as a design station for advanced grating devices for, but not restricted to, optical pulse shaping.

L'objectif de ce travail fut le développement de nouvelles méthodes pour aborder certainsproblèmes inverses en élasticité, en tirant parti de la présence d'un petit paramètre dans ces problèmespour construire des approximation asymptotiques d'ordre élevé.La première partie est consacrée à l'identification de la taille et la position d'une inhomogénéité$BTrue$ enfouie dans un domaine élastique tridimensionnel. Nous nous concentrons sur l'étude defonctions-co^uts $Jbb(Br)$ quantifiant l'écart entre $BTrue$ et une hétérogénéité ``test'' $Br$. Unetelle fonction-co^ut peut en effet être minimisée par rapport à tout ou partie des caractéristiques del'inclusion ``test'' $Br$ (position, taille, propriétés mécaniques ...) pour établir la meilleurecorrespondance possible entre $Br$ et $BTrue$. A cet effet, nous produisons un développement asymptotique de $Jbb$en la taille $incsize$ de $Br$, qui en constitue une approximation polynomiale plus aisée à minimiser. Cedéveloppement, établi jusqu'à l'ordre $O(incsize^6)$, est justifié par une estimation du résidu. Uneméthode d'identification adaptée est ensuite présentée et illustrée par des exemples numériques portant surdes obstacles de formes simples dans l'espace libre $Rbb^3$.L'objet de la seconde partie est de caractériser une inclusion microstructurée de longueur $ltot$, modéliséeen une dimension, composée de couches de deux matériaux alternés périodiquement, en supposant que les plusbasses de ses fréquences propres de transmission (TEs) sont connues. Ces fréquences sont les valeurs propres d'unproblème dit de transmission intérieur (ITP). Afin de disposer d'un modèle propiceà l'inversion, tout en prenant en compte les effets de la microstructure, nous nous reposons sur des approximationsde l'ITP exact obtenues par homogénéisation. A partir du modèle homogénéisé d'ordre 0, nous établissonstout d'abord une méthode simple pour déterminer les paramètres macroscopiques ($ltot$ et contrastes matériaux)d'une telle inclusion. Pour avoir accès à la période de la microstructure, nous nous intéressons ensuite àdes modèles homogénéisés d'ordre élevé, pour lesquels nous soulignons le besoin de conditions aux limitesadaptées. / The purpose of this work was to develop new methods to address inverse problems in elasticity,taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptoticexpansions.The first part is dedicated to the localization and size identification of a buried inhomogeneity $BTrue$ in a 3Delastic domain. In this goal, we focused on the study of functionals $Jbb(Br)$ quantifying the misfit between $BTrue$and a trial homogeneity $Br$. Such functionals are to be minimized w.r.t. some or all the characteristics of the trialinclusion $Br$ (location, size, mechanical properties ...) to find the best agreement with $BTrue$. To this end, weproduced an expansion of $Jbb$ with respect to the size $incsize$ of $Br$, providing a polynomial approximationeasier to minimize. This expansion, established up to $O(incsize^6)$ in a volume integral equations framework, isjustified by an estimate of the residual. A suited identification procedure is then given and supported by numericalillustrations for simple obstacles in full-space $Rbb^3$.The main purpose of this second part is to characterize a microstructured two-phases layered1D inclusion of length $ltot$, supposing we already know its low-frequency transmission eigenvalues (TEs). Thoseare computed as the eigenvalues of the so-called interior transmission problem (ITP). To provide a convenient invertiblemodel, while accounting for the microstructure effects, we then relied on homogenized approximations of the exact ITPfor the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method torecover the macroscopic parameters ($ltot$ and material contrast) of such inclusion. To access to the period of themicrostructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundaryconditions.

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The main topic of the thesis are inverse scattering problems of electromagnetic waves from periodic structures. We study first the direct problem and its numerical resolution using volume integral equation methods with a focus on the case of strongly singular integral operators and discontinuous coefficients. In a second investigation of the direct problem we study conditions on the material parameters under which well-posedness is ensured for all positive wave numbers. Such conditions exclude the existence of guided waves. The considered inverse scattering problem is related to shape identification. To treat this class of inverse problems, we investigate the so-called Factorization method as a tool to identify periodic patterns from measured scattered waves. In this thesis, these measurements are always related to plane incident waves. The outline of the thesis is the following: The first chapter is the introduction where we give the state of the art and new results of the topics studied in the thesis. The main content consists of five chapters, divided into two parts. The first part deals with the scalar case where the TM electromagnetic polarization is considered. In the second chapter we present the volume integral equation method with new results on Garding inequalities, convergence theory and numerical validation. The third chapter is devoted to the analysis of the Factorization method for the inverse scalar problem as well as some numerical experiments. The second part is dedicated to the study of 3-D Maxwell's equations. The fourth and fifth chapters are respectively generalizations of the results of the second and third ones to the case of Maxwell's equations. The sixth chapter contains the analysis of uniqueness conditions for the direct scattering problem, that is, absence of guided modes.

This dissertation presents a contribution to quantitative microwave imaging for breast tumor detection. The study made in the frame of a joint supervision Ph.D. thesis between University Paris-SUD 11 (France) and Mälardalen University (Sweden), has been conducted through two experimental microwave imaging setups, the existing 2.45 GHz planar camera (France) and the multi-frequency flexible robotic system, (Sweden), under development. In this context a 2D scalar flexible numerical tool based on a Newton-Kantorovich (NK) scheme, has been developed. Quantitative microwave imaging is a three dimensional vectorial nonlinear inverse scattering problem, where the complex permittivity of an object is reconstructed from the measured scattered field, produced by the object. The NK scheme is used in order to deal with the nonlinearity and the ill-posed nature of this problem. A TM polarization and a two dimensional medium configuration have been considered in order to avoid its vectorial aspect. The solution is found iteratively by minimizing the square norm of the error with respect to the scattered field data. Consequently, the convergence of such iterative process requires, at least two conditions. First, an efficient calibration of the experimental system has to be associated to the minimization of model errors. Second, the mean square difference of the scattered field introduced by the presence of the tumor has to be large enough, according to the sensitivity of the imaging system. The existing planar camera associated to a flexible 2D scalar NK code, are considered as an experimental platform for quantitative breast imaging. A preliminary numerical study shows that the multi-view planar system is quite efficient for realistic breast tumor phantoms, according to its characteristics (frequency, planar geometry and water as a coupling medium), as long as realistic noisy data are considered. Furthermore, a multi-incidence planar system, more appropriate in term of antenna-array arrangement, is proposed and its concept is numerically validated. On the other hand, an experimental work which includes a new fluid-mixture for the realization of a narrow band cylindrical breast phantom, a deep investigation in the calibration process and model error minimization, is presented. This conducts to the first quantitative reconstruction of a realistic breast phantom by using multi-view data from the planar camera. Next, both the qualitative and quantitative reconstruction of 3D inclusions into the cylindrical breast phantom, by using data from all the retina, are shown and discussed. Finally, the extended work towards the flexible robotic system is presented. / A dissertation prepared through an international convention for a joint supervision thesis with Université Paris-SUD 11, France / Microwaves in biomedicine

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