This dissertation focuses on basic science and application of structured light in nonlinear metamaterials and engineered media. The emergence of metamaterials has a strong potential to enable a plethora of novel nonlinear light-matter interactions and even new nonlinear materials. In particular, nonlinear focusing and defocusing effects are of paramount importance for manipulation of the minimum focusing spot size of structured light beams necessary for nanoscale trapping, manipulation, and fundamental spectroscopic studies. However, until now the majority of studies of nonlinear light-matter interactions focused on solid-state metamaterials. In this dissertation, I investigated the possibility of light manipulation in soft-matter, gaseous as well as negative-index engineered media. In particular, colloidal suspensions offer a promising platform for engineering polarizibilities and realization of large and tunable nonlinearities. I performed detailed theoretical, numerical and initial experimental studies of the phenomenon of spatial modulational instability leading to light beam filamentation in an engineered soft-matter nonlinear media. Modulational instability is a phenomenon that reveals itself as the exponential growth of weak perturbations in the presence of an intense pump beam propagating in a nonlinear medium. It plays a key role in such nonlinear optical processes as supercontinuum generation, light filamentation, rogue waves, and ring (or necklace) beam formation. Moreover, I show that propagation of light carrying an orbital angular momentum in nonlinear engineered media leads to formation of complex photonic structures, including optical necklace beams and chiral filaments.;I show that the modulational instability in colloidal suspensions with exponential nonlinearity results into fast collapse, while in the case of saturable nonlinearity the modulation instability results into enhanced transmission and the break up of the input beam into filaments. In negative-index metamaterial, by deriving an analytical expression for the spatial modulation-instability gain for the Kerr-nonlinearity case and show that a specific condition relating the diffraction and the nonlinear lengths must be fulfilled for the azimuthal modulation instability to occur. I predicted the rotation of the necklace beams due to the transfer of orbital angular momentum of the original vortex beam to the rotational motion of the necklace beams. I predict and demonstrate that the direction of rotation is opposite in positive- and negative-index materials.;Finally, I investigated the structured light propagation in air and numerically demonstrated the possibility of formation chiral photonic structures and large arrays of chiral filaments. These results pave new ways for many fundamental physical studies of engineered photonic structures and applications, including remote sensing, lightning protection, and microwave-radiation manipulation.
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