Abstract
In this paper, we study the wave transport and localization properties of novel aperiodic structures that manifest the intrinsic complexity of prime number distributions in imaginary quadratic fields. In particular, we address structure-property relationships and wave scattering through the prime elements of the nine imaginary quadratic fields (i.e., of their associated rings of integers) with class number one, which are unique factorization domains (UFDs). Our theoretical analysis combines the rigorous Green’s matrix solution of the multiple scattering problem with the interdisciplinary methods of spatial statistics and graph theory analysis of point patterns to unveil the relevant structural properties that produce wave localization effects. The onset of a Delocalization-Localization Transition (DLT) is demonstrated by a comprehensive study of the spectral properties of the Green’s matrix and the Thouless number as a function of their optical density. Furthermore, we employ Multifractal Detrended Fluctuation Analysis (MDFA) to establish the multifractal scaling of the local density of states in these complex structures and we discover a direct connection between localization, multifractality, and graph connectivity properties. Finally, we use a semi-classical approach to demonstrate and characterize the strong coupling regime of quantum emitters embedded in these novel aperiodic environments. Our study provides access to engineering design rules for the fabrication of novel and more efficient classical and quantum sources as well as photonic devices with enhanced light-matter interaction based on the intrinsic structural complexity of prime numbers in algebraic fields.
Highlights
The spatial localization of light in open scattering environments with a refractive index that randomly fluctuates over the wavelength scale has attracted intense research activities in the last decades thanks to its wealth of mesoscopic physical effects with potential applications to advanced photonics technology [1, 2]
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)
We have presented a comprehensive analysis of the structural, spectral, and localization properties of novel aperiodic arrays of scattering dipoles that inherit the intrinsic complexity of prime numbers in imaginary quadratic fields
Summary
The spatial localization of light in open scattering environments with a refractive index that randomly fluctuates over the wavelength scale has attracted intense research activities in the last decades thanks to its wealth of mesoscopic physical effects with potential applications to advanced photonics technology [1, 2]. Despite a continued research effort, Anderson localization of optical waves remains an evasive phenomenon since it does not occur in open-scattering threedimensional random media when the vector nature of light is considered [12]. This is credited to the detrimental effects of the evanescent-field coupling of randomly distributed vector dipole scatterers that confine electromagnetic waves at the subwavelength (near-field) scale in dense scattering media [13, 14].
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