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Abstract Many natural patterns and shapes, such as meandering coastlines, clouds, or turbulent flows, exhibit a characteristic complexity that is mathematically described by fractal geometry. Here, we extend the reach of fractal concepts in photonics by experimentally demonstrating multifractality of light in arrays of dielectric nanoparticles that are based on fundamental structures of algebraic number theory. Specifically, we engineered novel deterministic photonic platforms based on the aperiodic distributions of primes and irreducible elements in complex quadratic and quaternions rings. Our findings stimulate fundamental questions on the nature of transport and localization of wave excitations in deterministic media with multi-scale fluctuations beyond what is possible in traditional fractal systems. Moreover, our approach establishes structure–property relationships that can readily be transferred to planar semiconductor electronics and to artificial atomic lattices, enabling the exploration of novel quantum phases and many-body effects. 

Abstract This paper presents the design, fabrication, and characterization of dual‐band multi‐focal diffractive microlenses with sub‐wavelength thickness and the capability to simultaneously focus visible and near‐infrared spectral bands at two different focal positions. This technology utilizes high‐index and low‐loss sputtered hydrogenated amorphous Si, enabling a sub‐wavelength thickness of only 235 nm. Moreover, the proposed flat lens concept is polarization insensitive and can be readily designed to operate across any desired wavelength regime. Imaging under unpolarized broadband illumination with independent focal planes for two targeted spectral bands is experimentally demonstrated, enabling the encoding of the depth information of a sample into different spectral images. In addition, with a small footprint of only 100 µm and a minimum feature size of 400 nm, the proposed dual‐band multi‐focal diffractive microlenses can be readily integrated with vertical detector arrays to simultaneously concentrate and spectrally select electromagnetic radiation. This approach provides novel opportunities for spectroscopic and multispectral imaging systems with advanced detector architectures.