Abstract
In this paper we propose a novel approach to aperiodic order in optical science and technology that leverages the intrinsic structural complexity of certain non-polynomial (hard) problems in number theory and cryptography for the engineering of optical media with novel transport and wave localization properties. In particular, we address structure-property relationships in a large number (900) of light scattering systems that physically manifest the distinctive aperiodic order of elliptic curves and the associated discrete logarithm problem over finite fields. Besides defining an extremely rich subject with profound connections to diverse mathematical areas, elliptic curves offer unprecedented opportunities to engineer light scattering phenomena in aperiodic environments beyond the limitations of traditional random media. Our theoretical analysis combines the interdisciplinary methods of point patterns spatial statistics with the rigorous Green’s matrix solution of the multiple wave scattering problem for electric and magnetic dipoles and provides access to the spectral and light scattering properties of novel deterministic aperiodic structures with enhanced light-matter coupling for nanophotonics and metamaterials applications to imaging and spectroscopy.
Highlights
Our results demonstrate that the light scattering properties of particle arrays designed according to the proposed elliptic curve approach are distinctively different from the ones of uniform random systems, despite close similarities are observed in both point pattern and spectral statistics
In this paper we introduce a novel class of deterministic aperiodic photonic systems that physically implement the distinctive aperiodic order of elliptic curves and their associated discrete logarithm problem
(900) of aperiodic photonic systems that manifest an extremely rich spectrum of scattering and localization properties that can be engineered to outperform the performances of traditional uniform random media in terms of optical confinement and directional light scattering
Summary
Deterministic structures with aperiodic though long-range ordered distributions of scattering potentials have a long history in the electronics and optics communities due to significant advantages in design and compatibility with standard fabrication technologies compared to random systems [22,23] These structures manifest unique spectral characteristics that lead to physical properties that cannot be found in either periodic or uniform random media, such as multifractal density of eigenstates with varying degrees of spatial localization, known as critical modes [24,25,26,27,28], anomalous photon transport regimes [29,30], and distinctive wave localization transitions [31,32]. Our findings underline the importance of structural correlations in elliptic curve-based structures for the improvement of photonic systems and show that the solution of the associated wave scattering problem reveals remarkable differences in the scattering and localization properties that may become important for the optical identification of vulnerabilities in elliptic-curve cryptosystems
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