Wave transport in complex prime arrays and hyperuniform systems
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Abstract
Complex dielectric structures play an important role in a wide range of optical applications such as, for instance, novel light sources and random lasers, photonic filters and waveguides, lensless imaging systems, broadband sensors and spectroscopic devices. Due to the crucial role played by wave interference effects in the multiple scattering regime, the optics of disordered dielectric structures shows profound analogies with the transport of electrons in disordered metallic alloys and semiconductors. As a result, various mesoscopic phenomena known for the electron wave transport in disordered materials, such as Anderson localization, have found their counterparts in disordered optical materials as well. However, despite a continued research effort, Anderson localization of optical waves does not occur in open-scattering random media when the vector nature of light is fully considered. This is credited to the detrimental effects of the near-field coupling of electromagnetic waves confined at the sub-wavelength scale between adjacent scatterers in dense scattering systems. Moreover, it is difficult to create simple design rules for the optimization of uniform random media, limiting their applicability to optical device engineering. As a result, there is currently a compelling need to create optical media that are structurally complex, yet deterministic/controllable, thus providing an alternative route to achieve stronger light-matter coupling and localization effects compared to traditional random systems. In this thesis, I will follow a novel approach to transport and wave localization that leverages the aperiodic structural complexity of prime number distributions in complex quadratic fields and in random media with stealthy hyperuniform order. Specifically, by using multidisciplinary methods from spectral graph theory, correlated random walks, and quantum diffusion I will uncover structure-property relationships, optical transport and localization properties of new scattering media with engineering applications to novel light sources, spectroscopy, and imaging. Our theoretical analysis combines the rigorous Green's matrix solution of the multiple scattering problem with the methods of spatial statistics of point patterns to explore the geometrical properties (i.e., hyperuniformity, statistical isotropy, etc.) that give rise to localization effects in complex scattering systems. This study provides access to diffusion modes, geometrical modes, and asymptotic scaling laws for anomalous diffusion processes on complex environments at a modest computational cost, thus enabling a systematic study of novel photonic environments driven by prime number distributions.