A la fin des années 80, ont été introduits les cristaux photoniques et les métamatériaux, permettant de concevoir des dispositifs avec des propriétés optiques très intéressantes. La découverte de ces propriétés artificielles est considérée comme l’une des avancées les plus spectaculaires de la photonique moderne, avec des applications importantes telles que la création de matériaux à réfraction négative, ou la conception de lentilles plates parfaites. Ces structures présentent néanmoins des pertes optiques dues aux métaux qui rendent plus difficile l’obtention de lentilles super-résolues. Une nouvelle approche, basée sur l’anisotropie du métamatériau, a alors été proposée comme une alternative très intéressante pour la conception de lentilles plates à super-résolution. Parallèlement les limites du modèle de Drude pour la description de la réponse optique des métaux ont été mises en évidence expérimentalement. Ce manuscrit de thèse présente une étude théorique et numérique d’empilements de couches minces métallo-diélectriques se comportant dans certains domaines de fonctionnement comme des milieux hyperboliques. Après avoir étudié le lien existant entre la réfraction négative et les décalages géants, le manuscrit se concentre sur la conception de lentilles plates permettant d’obtenir de la super-résolution : des images offrant une meilleure résolution que celle permise par les lois de la diffraction classique. Pour répondre à ces objectifs, nous avons développé une théorie basée sur l’approximation du milieu hyperbolique (obtenue avec un empilement métallo-diélectrique) par un milieu isotrope effectif à l’aide d’un développement parabolique du vecteur d’onde de Bloch. Les outils nécessaires pour toute étude de l’influence de la non-localité intrinsèque des métaux sur les propriétés optiques des structures sont ensuite développés et appliqués aux métallo-diélectriques. / In the early 80’s, planar or periodic photonics crystals have been introduced in order to control light and to obtain entirely new optical properties. The unrivalled properties of these metamaterials are of tremendous interest for advanced photonic systems, with some important applications such as materials with negative refraction or flat lenses. However, these structures present optical losses induced by metals defects and experimental fabrication at nanometric scales that prevent them to reach the expected performances. A new approach based on describing metallo-dielectric as anisotropic materials has then been proposed as an alternative description. In parallel, the limits of the Drude model have been experimentally highligthed. In this context this manuscript presents a theoretical and numerical study of metallo-dielectric multilayers that can be considered as homogeneous media with a hyperbolic dispersion relation. We first present the link between negative refraction and large negative lateral shifts, and then focus on the design of flat lenses with subwavelength resolution : structures allowing a better resolution than the classical diffraction limit. We thus developed a theory based on the approximation of the hyperbolic medium, by a homogeneous and isotropic medium with a parabolic development of the vector of wave of Bloch. Finally, the tools required to study the influence of the intrinsic nonlocality of metals on the optical properties of multilayers are developped and the formalism is applied to metallo-dielectric lenses.

The static and dynamic response of lattice materials is investigated to disclose and control the connection between microstructure and effective behavior. The analytical methods developed in the thesis aim at providing a new understanding of material instabilities and strain localizations as well as effective tools for controlling wave propagation in lattice structures. The time-harmonic dynamics of arbitrary beam lattices, deforming flexurally and axially in a plane, is formulated analytically to analyze the influence of the mechanical parameters on the dispersion properties of the spectrum of Floquet-Bloch waves. Several forms of dynamic localizations are shown to occur for in-plane wave propagation of grid-like elastic lattices. It is demonstrated that lattices of rods, despite being `simple' structures, can exhibit a completely different channeled response depending on the characteristics of the forcing source (i.e. frequency and direction) as well as on the slenderness of the elastic links. It is also shown how the lattice parameters can be tuned to attain specific dispersion properties, such as flat bands and sharp Dirac cones. In the research field of material instabilities, a key result proposed in this thesis is the development of both static and dynamic homogenization methods capable of accounting for second-order effects in the macroscopic response of prestressed lattices. These methods, the former based on an incremental strain-energy equivalence and the latter based on the asymptotic analysis of lattice waves, allow the identification of the incremental constitutive operator capturing the macroscopic incremental response of arbitrary lattice configurations. The homogenization framework has allowed the systematic analysis of prestress-induced phenomena on the incremental response of both the lattice structure and its `effective' elastic solid, which in turn has enabled the identification of the complex interplay between microstructure, prestress, loss of ellipticity (shear band formation) and short-wavelength bifurcations. Potential new applications for the control of wave propagation are also shown to be possible by leveraging the inclusion of second-order terms in the incremental dynamics. In particular, the tunability of the prestress state in a square lattice structure has been exploited to obtain dynamic interfaces with designable transmission properties. The interface can be introduced in a material domain by selectively prestressing the desired set of ligaments and the prestress level can be tuned to achieve total reflection, negative refraction, and wave channeling. The obtained results open new possibilities for the realization of engineered materials endowed with a desired constitutive response, as well as to enable the identification of novel dynamic material instabilities.