Ce travail de thèse constitue une contribution originale et importante à la compréhension de la transformation d’espace et ouvre la voie au design de nouvelles structures éléctromagnétiques. Le couplage entre cette technique innovante et les métamatériaux a permis la réalisation de prototypes aux propriétés uniques. C’est ainsi que nous avons pu concevoir une cape d’invisibilité polygonale, un adapteur de modes ou encore une antenne directive ou isotrope. La fabrication de notre antenne très directive par cette méthode est le seul prototype dans la littérature qui allie le contrôle de la permittivité et la perméabilité à partir de résonnateurs électriques et magnétiques. Ce contrôle ultime de la lumière à partir d’une ingénierie de l’espace trouvera son utilité dans la recherche fondamentale mais aussi pour les ingénieurs et dévellopeurs recherchant plus de précision dans leur conception de dispositifs électromagnétiques. / This phD work is an original and important contribution to the understanding of transformation optics and paves the way to the design of new electromagnetic structures. The coupling between this innovative technique and metamaterials has led to prototypes with unique properties. We have thus developed an invisibility polygonal cloak, an electromagnetic taper, a directional antenna and isotropic source. The realization of our high-directive antenna with this method is the only prototype in the literature that combines controlled variations of the permittivity and permeability from electric and magnetic resonators. The ultimate control of light from an engineering space will find its usefulness in fundamental research but also for engineers and developers who are looking for more precision in the design of electromagnetic devices.

The first part of this study is focused on numerical studies of light scattering from a single microscopic particle using the Discrete Dipole Approximation (DDA) method. The conventional DDA formalism is generalized to two cases: (a) inelastic light scattering from a dielectric particle and (b) light scattering from a particle with magnetic permeability u /= 1. The first generalization is applied to simulations of Raman scattering from bioaerosol particles, and the second generalization is applied to confi rmation of irregular invisibility cloaks made from metamaterials. In the second part, radiative transfer in a coupled atmosphere-ocean system is solved to study the asymptotic nature of the polarized light in deep oceans. The rate at which the radiance and the polarization approach their asymptotic forms in an ideal homogeneous water body are studied. Effects of the single scattering albedo and the volume scattering function are studied. A more realistic water body with vertical pro files for oceanic optical properties determined by a Case 1 water model is then assumed to study the e ffects of wavelength, Raman scattering, and surface waves. Simulated Raman scattering patterns computed from the generalized DDA formalism are found to be sensitive to the distribution of Raman active molecules in the host particle. Therefore one can infer how the Raman active molecules are distributed from a measured Raman scattering pattern. Material properties of invisibility cloaks with a few irregular geometries are given, and field distributions in the vicinity of the cloaked particles computed from the generalized DDA formalism con rm that the designated material properties lead to invisibility. The radiative transfer model calculation in deep oceans suggest that the underwater radiance approaches its asymptotic form more quickly than the polarization does. Therefore, a vector radiative transfer solution is necessary for asymptotic light field studies. For a typical homogeneous water body whose scattering property is characterized by the Petzold phase function, a single scattering albedo of w0 > 0:8 is required in order that the asymptotic regime can be reached before there are too few photons to be detected.

Research on metamaterials has been growing ever since the first experimental realization of double negative media. The theory of transformation optics provides people with a perfect tool to make use of vast possibilities of the constitutive parameters for metamaterials. A lot of fascinating designs have been brought to us by transformation optics, with invisibility cloaks being the most intensely studied. The present thesis aims to develop the basic theory of transformation optics, and utilize it to design invisibility cloaks for various applications. After the background description of this field, the theory of transformation optics is first introduced. Formulas of transformation medium parameters and transformed fields are derived with every detail explained, so that the working knowledge of transformation optics can be grasped with minimal prerequisite mathematics. Proof of form invariance of full Maxwell’s equations with sources is presented. Design procedure of transformation optics is then demonstrated by creating perfect invisibility cloaks. The introduction to basic theory is followed by discussions on our works included in our published papers. As our first application, a method of designing two-dimensional reduced cloaks of complex shapes is proposed to relieve the difficulty of singularity occurring in perfect cloaks. The simple and intuitive method is the first way to design two-dimensional reduced cloaks of shapes other than cylindrical. Elliptical and bowtie shaped reduced cloaks are presented to verify the effectiveness of the method. Prominent scattering reduction is observed for both examples. Considering the practical realization, transformations continuous in the whole space must be the identity operation outside certain volume, and thus they can only manipulate fields locally. Discontinuous transformations are naturally considered to break the limitation. We study the possible reflections from such a transformation medium due to a discontinuous transformation by a new concept of inverse transformation. This way, the reflection falls into the framework of transformation optics as well. A necessary and sufficient condition for no reflection is derived as a special case. Unlike the invisibility realized by perfect cloaks, cloaking an object over a dielectric half-space has advantages in some particular applications. Starting from a perfect cloak, a half-space cloak is designed to achieve this. In our design, two matching strips embedded in the dielectric ground are used to induce proper reflection in the upper air space, so that the reflected field is the same as that from the bare dielectric ground. Cloaks obtained from singular transformations and even reduced models all have null principal value in their material parameters, making invisibility inherently very narrowband. In contrast, a carpet cloak designed by only coordinate deformation does not have the narrowband issue, and can perform well in a broad spectrum. The invisibility accomplished by the carpet cloak is also for the half-space case as our previous design. In this part, we extend the original version of a carpet cloak above a PEC sheet to a general dielectric ground. / QC 20110415

The first part of this Thesis deals with the analysis of piecewise-smooth mechanical systems and the definition of special stability criteria in presence of non-conservative follower forces. To illustrate the peculiar stability properties of this kind of dynamical system, a reference 2 d.o.f. structure has been considered, composed of a rigid bar, with one and constrained to slide, without friction, along a curved profile, whereas the other and is subject to a follower force. In particular, the curved constraint is assumed to be composed of two circular profiles, with different and opposite curvatures, defining two separated subsystems. Due to this jump in the curvature, located at the junction point between the curved profiles, the entire mechanical structure can be modelled by discontinuous equations of motion, the differential equations valid in each subsystem can be combined, leading to the definition of a piecewise-smooth dynamical system. When a follower force acts on the structure, an unexpected and counterintuitive behaviour may occur: although the two subsystems are stable when analysed separately, the composed structure is unstable and exhibits flutter-like exponentially-growing oscillations. This special form of instability, previously known only from a mathematical point of view, has been analysed in depth from an engineering perspective, thus finding a mechanical interpretation based on the concept of non-conservative follower load. Moreover, the goal of this work is also the definition of some stability criteria that may help the design of these mechanical piecewise-smooth systems, since classical theorems cannot be used for the investigation of equilibrium configurations located at the discontinuity. In the literature, this unusual behaviour has been explained, from a mathematical perspective, through the existence of a discontinuous invariant cone in the phase space. For this reason, starting from the mechanical system described above, the existence of invariant cones in 2 d.o.f. mechanical systems is investigated through Poincaré maps. A complete theoretical analysis on piecewise-smooth dynamical systems is presented and special mathematical properties have been discovered, valid for generic 2~d.o.f. piecewise-smooth mechanical systems, which are useful for the characterisation of the stability of the equilibrium configurations. Numerical tools are implemented for the analysis of a 2~d.o.f. piecewise-smooth mechanical system, valid for piecewise-linear cases and extendible to the nonlinear ones. A numerical code has been developed, with the aim of predicting the stability of a piecewise-linear dynamical system a priori, varying the mechanical parameters. Moreover, “design maps” are produced for a given subset of the parameters space, so that a system with a desired stable or unstable behaviour can easily be designed. The aforementioned results can find applications in soft actuation or energy harvesting. In particular, in systems devoted to exploiting the flutter-like instability, the range of design parameters can be extended by using piecewise-smooth instead of smooth structures, since unstable flutter-like behaviour is possible also when each subsystem is actually stable. The second part of this Thesis deals with the numerical analysis of an elastic cloak for transient flexural waves in Kirchhoff-Love plates and the design of special metamaterials for this goal. In the literature, relevant applications of transformation elastodynamics have revealed that flexural waves in thin elastic plates can be diverted and channelled, with the aim of shielding a given region of the ambient space. However, the theoretical transformations which define the elastic properties of this “invisibility cloak” lead to the presence of a strong compressive prestress, which may be unfeasible for real applications. Moreover, this theoretical cloak must present, at the same time, high bending stiffness and a null twisting rigidity. In this Thesis, an orthotropic meta-structural plate is proposed as an approximated elastic cloak and the presence of the prestress has been neglected in order to be closer to a realistic design. With the aim of estimating the performance of this approximated cloak, a Finite Element code is implemented, based on a sub-parametric technique. The tool allows the investigation of the sensitivity of specific stiffness parameters that may be difficult to match in a real cloak design. Moreover, the Finite Element code is extended to investigate a meta-plate interacting with a Winkler foundation, to analyse how the substrate modulus transforms in the cloak region. This second topic of the Thesis may find applications in the realization of approximated invisibility cloaks, which can be employed to reduce the destructive effects of earthquakes on civil structures or to shield mechanical components from unwanted vibrations.