Photonic implementation of algebraic number theory

We study the unique spectral and optical features of a new class of aperiodic arrays generated from the distribution of prime numbers in complex quadratic fields as well as quaternion primes. By using a multiple scattering spectral method, we have discovered several unique spectral properties, such as light localization, critical level statistics, and the existence of critical modes, i.e. extended fractal modes with long lifetime that cannot be supported in traditional systems. A systematic analysis based on LDOS calculations unveil the new functionalities of these complex aperiodic platforms for lasing applications. Our results unveil the importance of aperiodic structures characterized by a coexistence of singular and continuous spectral components for the engineering of new photonic architectures based on algebraic number theory.