Aperiodic Photonics of Elliptic Curves

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. * dalnegro@bu.edu arXiv:1912.08871v1 [cond-mat.dis-nn]

Wave Transport and Localization in Prime Number Landscapes

“…The decay dynamics of a light emitter embedded in a complex dielectric environment can be rigorously understood by computing its local density of states (LDOS) and the corresponding Purcell enhancement factor as a function of frequency. The knowledge of the LDOS spectra allows one to accurately determine the spectral locations of the resonant modes with the highest quality factors, such as those that are located near the bandgap regions [16,25,26,54,[70][71][72]87]. In order to characterize the emission dynamics in these complex aperiodic environments, we have computed the enhancement of the LDOS with respect to its free space value, or the Purcell spectrum, for all the CPAs consisting of approximately 1,000 electric dipoles with the electric polarizability α(ω) −4Γ 0 c 2 /[ω 0 (ω 2 − ω 2 0 + iΓ 0 ω 2 /ω 0 )] [66].…”

“…In particular, we apply the interdisciplinary methods of spatial statistics [21], spectral graph theory [22], and multifractal scaling analysis [23,24] to gain information on the geometrical and connectivity properties of the investigated structures that drive their characteristic localization behavior. Using the Green's matrix method, which has been extensively utilized in the study of the scattering resonances of open aperiodic media [13,[25][26][27][28][29][30][31], we investigate the spectral statistics of scattering resonances by means of extensive numerical calculations of large-scale aperiodic arrays that cannot otherwise be accessed via traditional numerical methods such as Finite Difference Time Domain (FDTD) or Finite Elements (FEM). Importantly, the Green's matrix method allows one to obtain full spectral information and access spatial and temporal localization properties from the frequency positions and lifetimes (i.e., the inverse of the resonance width) of all the scattering resonances supported by the investigated systems.…”

Sub-diffusive wave transport and weak localization transition in three-dimensional stealthy hyperuniform disordered systems

“…The predictions of Green's matrix spectral method captures the physics of the multiple scattering problem in the limit of small electric dipole scatterers [33,38,[70][71][72]. This approximation is valid for particles with a small size parameter x = ka << 1 (k is the wavenumber and a is the particle radius) [58,73,74]. This approach has been extensively utilized to study the scattering properties of large-scale aperiodic media, providing access to robust transport and spectral information that cannot otherwise be accessed using traditional numerical methods such as Finite Difference Time Domain (FDTD) or Finite Elements (FEM) [70,[74][75][76].…”

Cavity-enhanced light–matter interaction in Vogel-spiral devices as a platform for quantum photonics

“…long-range ordered systems that lack translational symmetry, to achieve light confinement in two and three dimensions. Beyond quasicrystals, aperiodic order can be used to engineer photonic devices with well-defined deterministic mathematical rules 18,22,23 , providing compatibility with planar nano-fabrication technologies 24 , as well as dis-tinctive spectral and optical properties [25][26][27][28] , including optical angular momentum 29 , characteristics that make them appealing for quantum photonics applications [30][31][32][33] .…”

“…This makes golden-angle Vogel spirals a very appealing photonic platform, due to more robust fabrication tolerances than traditional photonic crystals 42 . The presence of band-gaps despite the relatively low index contrast between silicon nitride and air is related to the long-range order in a nearly-isotropic geometry 22,28,43,49 . Isotropic gaps also imply reduced group velocity modes and therefore increased light-matter interaction, thus making these devices interesting for non-linear optics applications and for the realisation of low-threshold lasers 42,47 .…”