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We propose a rigorous approach for the inverse design of functional photonic structures by coupling the adjoint optimization method and the two-dimensional generalized Mie theory (2D-GMT) for the multiple scattering problem of finite-size arrays of dielectric nanocylinders optimized to display desired functions. We refer to these functional scattering structures as "photonic patches". We briefly introduce the formalism of 2D-GMT and the critical steps necessary to implement the adjoint optimization algorithm to photonic patches with designed radiation properties. In particular, we showcase several examples of periodic and aperiodic photonic patches with optimal nanocylinder radii and arrangements for radiation shaping, wavefront focusing in the Fresnel zone, and for the enhancement of the local density of states (LDOS) at multiple wavelengths over micron-size areas. Moreover, we systematically compare the performances of periodic and aperiodic patches with different sizes and find that optimized aperiodic Vogel spiral geometries feature significant advantages in achromatic focusing compared to their periodic counterparts. Our results show that adjoint optimization coupled to 2D-GMT is a robust methodology for the inverse design of compact photonic devices that operate in the multiple scattering regime with optimal desired functionalities. Without the need of spatial meshing, our approach provides efficient solutions at strongly reduced computational burden compared to standard numerical optimization techniques and suggests compact device geometries for on-chip photonics and metamaterials technologies.

We propose the inverse design of ultracompact, broadband focusing spectrometers based on adaptive deep diffractive neural networks (a-D$^2$NNs). Specifically, we introduce and characterize two-layer diffractive devices with engineered angular dispersion that focus and steer broadband incident radiation along predefined focal trajectories with desired bandwidth and $5$ nm spectral resolution. Moreover, we systematically study the focusing efficiency of two-layer devices with side length $L=100~\mu\mathrm{m}$ and focal length $f=300~\,\mu\mathrm{m}$ across the visible spectrum and we demonstrate accurate reconstruction of the emission spectrum from a commercial superluminescent diode. The proposed a-D$^2$NNs design method extends the capabilities of efficient multi-focal diffractive optical devices to include single-shot focusing spectrometers with customized focal trajectories for applications to ultracompact multispectral imaging and lensless microscopy.

We investigate the localization of waves in aperiodic structures that manifest the characteristic multiscale complexity of certain arithmetic functions with a central role in number theory. In particular, we study the eigenspectra and wave localization properties of tight-binding Schr\"{o}dinger equation models with on-site potentials distributed according to the Liouville function $\lambda(n)$, the M\"{o}bius function $\mu(n)$, and the Legendre sequence of quadratic residues modulo a prime (QRs). We employ Multifractal Detrended Fluctuation Analysis (MDFA) and establish the multifractal scaling properties of the energy spectra in these systems. Moreover, by systematically analyzing the spatial eigenmodes and their level spacing distributions, we show the absence of level repulsion with broadband localization across the entire energy spectra. Our study introduces deterministic aperiodic systems whose eigenmodes are all strongly localized in realistic finite one-dimensional systems and provides opportunities for novel quantum and classical devices of particular importance to cold-atom experiments in engineered speckle potentials and enhanced light-matter interactions.

In this paper we study the structural, scattering, and wave localization properties of multifractal arrays of electric point dipoles generated from multiplicative random fields with different degrees of multiscale correlations. Specifically, using the rigorous Green's matrix method, we investigate the scattering resonances and wave localization behavior of systems with $N=10^{4}$ dipoles and demonstrate an enhanced localization behavior in highly inhomogeneous multifractal structures compared to homogeneous fractals, or monofractals. We show distinctive spectral properties, such as the absence of level repulsion in the strong multiple scattering regime and power-law statistics of level spacings, which indicate a clear localization transition enhanced in non-homogeneous multifractals. Our findings unveil the importance of multifractal structural correlations in the multiple scattering regime of electric dipole arrays and provide an efficient model for the design of multiscale nanophotonic systems with enhanced light-matter coupling and localization phenomena beyond what is possible with traditional fractal systems.

We propose an efficient inverse design approach for multifunctional optical elements based on adaptive deep diffractive neural networks (a-D$^2$NNs). Specifically, we introduce a-D$^2$NNs and design two-layer diffractive devices that can selectively focus incident radiation over two well-separated spectral bands at desired distances. We investigate focusing efficiencies at two wavelengths and achieve targeted spectral lineshapes and spatial point-spread functions (PSFs) with optimal focusing efficiency. In particular, we demonstrate control of the spectral bandwidths at separate focal positions beyond the theoretical limit of single-lens devices with the same aperture size. Finally, we demonstrate devices that produce super-oscillatory focal spots at desired wavelengths. The proposed method is compatible with current diffractive optics and doublet metasurface technology for ultracompact multispectral imaging and lensless microscopy applications.

In this paper, we develop a deep learning approach for the accurate solution of challenging problems of near-field microscopy that leverages the powerful framework of physics-informed neural networks (PINNs) for the inversion of the complex optical parameters of nanostructured environments. Specifically, we show that PINNs can be flexibly designed based on the full-vector Maxwell's equations to inversely retrieve the spatial distributions of the complex electric permittivity and magnetic permeability of unknown scattering objects in the resonance regime from near-field data. Moreover, we demonstrate that PINNs achieve excellent convergence to the true material parameters under both plane wave and point source (localized) excitations, enabling parameter retrieval in scanning near-field optical microscopy (SNOM). Our method is computationally efficient compared to traditional data-driven deep learning approaches as it requires only a single dataset for training. Furthermore, we develop and successfully demonstrate adaptive PINNs with trainable loss weights that largely improve the accuracy of the inverse reconstruction for high-index materials compared to standard PINNs. Finally, we demonstrate the full potential of our approach by retrieving the space-dependent permittivity of a three-dimensional (3D) unknown object from near-field data. The presented framework paves the way to the development of a computationally-driven, accurate, and non-invasive platform for the simultaneous retrieval of the electric and magnetic parameters of resonant nanostructures from measured optical images, with applications to biomedical imaging, optical remote sensing, and characterization of metamaterial devices.

We propose a novel framework for the systematic design of lensless imaging systems based on the hyperuniform random field solutions of nonlinear reaction-diffusion equations from pattern formation theory. Specifically, we introduce a new class of imaging point-spread-functions (PSFs) with enhanced isotropic behavior and controllable sparsity. We investigate the PSFs and the modulated transfer functions (MTFs) for a number of nonlinear models and demonstrate that two-phase isotropic random fields with hyperuniform disorder are ideally suited to construct imaging PSFs with improved performances compared to PSFs based on the Perlin noise. Additionally, we introduce a phase retrieval algorithm based on the non-paraxial Rayleigh-Sommerfeld diffraction theory and introduce diffractive phase plates with PSFs designed from hyperuniform random fields, called hyperuniform phase plates (HPPs). Finally, using high-fidelity object reconstruction, we demonstrate improved image quality using engineered HPPs across the visible range. The proposed framework is suitable for high-performance lensless imaging systems for on-chip microscopy and spectroscopy applications., Comment: Optics Letters in review

We investigate the spatial decay and temporal localization properties of quasimodes (i.e., scattering resonances) of two-dimensional Vogel spirals, composed of deterministic, aperiodic arrays of electric dipoles. By determining the structural entropy and localization maps of Vogel spirals using the Green's matrix method, we show that three distinctive decay types of quasimodes coexist in Vogel spirals: exponential, power-law, and Gaussian. While the exponential and the power-law decays typically occur in disordered media and multifractal systems, respectively, the Gaussian decay is demonstrated to characterize, on average, the most localized quasimodes of Vogel spirals, both spatially (smallest participation ratios) and temporarily (longest lifetimes). These decay forms are demonstrated by a no-fitting analysis of the localization maps, independently corroborated by calculating the electric field in real space, which also provides a direct evidence of the algebraic spatial decay of critical quasimodes. Altogether our findings unveil a rich spectrum of both long-lived and spatially localized quasimodes that coexist in Vogel spirals and can be of direct relevance to novel optical functionalities for applications to light sources and sensing devices., Comment: 8 pages, 3 figures

In this paper we propose a novel approach to enhance the efficiency of the two-photon spontaneous emission process that is driven by the multifractal optical mode density of photonic structures based on the aperiodic distributions of Eisenstein and Gaussian primes. In particular, using the accurate Mie-Lorenz multipolar theory in combination with multifractal detrended fluctuation analysis, we compute the local density of states of periodic and aperiodic systems and demonstrate the formation of complete bandgaps with distinctive fractal scaling behavior for scattering arrays of dielectric nanocylinders. Moreover, we systematically study the Purcell enhancement and the most localized optical mode resonances in these novel aperiodic photonic systems and compute their two-photon spontaneous emission rates based on the general Green's tensor approach. Our results demonstrate that the excitation of the highly-resonant critical states of Eisenstein and Gaussian photonic arrays across broadband multifractal spectra gives rise to significantly enhanced emission rates compared to what is possible at the band-edges of periodic structures with comparable size. Besides defining a novel approach for enhanced quantum two-photon sources on the chip, the engineering of aperiodic bandgap structures with multifractal mode density may provide access to novel electromagnetic resonant phenomena in a multiscale-invariant vacuum for quantum nanophotonics applications., Comment: 14 pages, 9 Figures

Optical rogue waves are demonstrated in the far-field scattered radiation from photonic arrays designed according to the aperiodic distributions of prime elements in complex quadratic fields. Specifically, by studying light diffraction from Eisenstein and Gaussian prime arrays we establish a connection between the formation of optical rogue waves and multifractality in the visible single-scattering regime. We link strong multifractality with the heavy-tail probability distributions that describe the fluctuations of scattered radiation from the fabricated arrays. Our findings pave the way to control high-intensity rogue waves using deterministic arrays of dielectric nanostructures for enhanced sensing and lithographic applications., Comment: 9 pages, 7 figures

Refractory materials exhibit high damage tolerance, which is attractive for the creation of nanoscale field-emission electronics and optoelectronics applications that require operation at high peak current densities and optical intensities. Recent results have demonstrated that the optical properties of titanium nitride, a refractory and CMOS-compatible plasmonic material, can be tuned by adding silicon and oxygen dopants. However, to fully leverage the potential of titanium (silicon oxy)nitride, a reliable and scalable fabrication process with few-nm precision is needed. In this work, we developed a fabrication process for producing engineered nanostructures with gaps between 10 and 15 nm, aspect ratios larger than 5 with almost 90{\deg} steep sidewalls. Using this process, we fabricated large-scale arrays of electrically-connected bow-tie nanoantennas with few-nm free-space gaps. We measured a typical variation of 4 nm in the average gap size. Using applied DC voltages and optical illumination, we tested the electronic and optoelectronic response of the devices, demonstrating sub-10-V tunneling operation across the free-space gaps, and quantum efficiency of up to 1E-3 at 1.2 {\mu}m, which is comparable to a bulk silicon photodiode at the same wavelength. Tests demonstrated that the titanium silicon oxynitride nanostructures did not significantly degrade, exhibiting less than 5 nm of shrinking of the average gap dimensions over few-{\mu}m^2 areas after roughly 6 hours of operation. Our results will be useful for developing the next generation of robust and CMOS-compatible nanoscale devices for high-speed and low-power field-emission electronics and optoelectronics applications.

The polarization- and angle- resolved optical response of the anapole mode in silicon nano-disks array have been experimentally and theoretically investigated. The good agreement betweenmeasured data and simulations yields to a consistent description of the anapole mode behaviorthat exhibits different features for TE or TM polarization excitation. Scattering matrix calculationallows us to disentangle scattered and diffused light contributions and to provide a quantitativeestimation of the absorbance enhancement associated to 2D excitation of the anapole mode. Weperformed the multipolar decomposition of the far-field scattered radiation for both TE and TMpolarizations and unambiguously identified the anapole resonant condition in excellent agreementwith the experimental results over a large range of incident angles. Our findings demonstrate thecontrolled excitation of electromagnetic anapole modes in engineered arrays of silicon nano-disksfor the development of optical nanostructures with enhanced light-matter interaction, Comment: 10 pages, 4 figures and 1 table, submitted to Phys.Rev.B

We design, fabricate, and characterize multifunctional and compact diffractive microlenses with sub-wavelength thickness and the capability to simultaneously focus visible and near-infrared spectral bands at two different focal positions with 24% and 15% measured focusing efficiency, respectively. Our technology utilizes high-index and low-loss sputtered hydrogenated amorphous Si, enabling a sub-wavelength thickness of only 235nm. Moreover, the proposed flat lens concept is polarization insensitive and can be readily designed to operate across any desired wavelength regime. Imaging under 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$\mu$m and a minimum feature size of 400nm, the proposed multifunctional and compact diffractive microlenses can be readily integrated with vertical detector arrays to simultaneously concentrate and spectrally select electromagnetic radiation. This provides novel opportunities for spectroscopic and multispectral imaging systems with advanced detector architectures.

We design and characterize phase-modulated, ultra-compact, silicon-based axilens devices that combine efficient point focusing and grating selectivity within scalable 4-level phase mask configurations. The proposed designs are polarization insensitive and maintain a large focusing efficiency over a broad spectral band. Specifically, we select and systematically characterize structures designed for visible and near-infrared (NIR) operation in the $750nm-950nm$ wavelength range. These devices are ideally suited for monolithic integration atop the substrate layers of focal plane arrays (FPAs) for use in multi-band photo-detection and imaging. We demonstrate linear control of multi-wavelength focusing on a single achromatic plane and provide an application consisting of a proof-of-concept ultra-compact single-lens spectrometer with only 300 nm thickness and 70${\mu}m$ diameter, achieving a minimum distinguishable wavelength $\Delta\lambda=40nm$ at $\lambda_0=850nm$. The proposed devices add fundamental spectroscopic capabilities to compact imaging devices for a number of applications ranging from spectral sorting to visible and NIR multispectral imaging and detection.

We design and characterize a novel axilens-based diffractive optics platform that flexibly combines efficient point focusing and grating selectivity and is compatible with scalable top-down fabrication based on a 4-level phase mask configuration. This is achieved using phase-modulated compact axilens devices that simultaneously focus incident radiation of selected wavelengths at predefined locations with larger focal depths compared to traditional Fresnel lenses. In addition, the proposed devices are polarization insensitive and maintain a large focusing efficiency over a broad spectral band. Specifically, here we discuss and characterize modulated axilens configurations designed for long-wavelength infrared (LWIR) in the $6~\mu$m--12~$\mu$m wavelength range and in the $4~\mu$m--6~$\mu$m mid-wavelength infrared (MWIR) range. These devices are ideally suited for monolithic integration atop the substrate layers of infrared focal plane arrays (IR-FPAs) and for use as compact microspectrometers. We systematically study their focusing efficiency, spectral response, and cross talk ratio, and we demonstrate linear control of multi-wavelength focusing on a single plane. Our design method leverages Rayleigh-Sommerfeld (RS) diffraction theory and is validated numerically using the Finite Element Method (FEM). Finally, we demonstrate the application of spatially modulated axilenses to the realization of compact, single-lens spectrometer. By optimizing our devices, we achieve a minimum distinguishable wavelength interval of $\Delta\lambda=240nm$ at $\lambda_0=8{\mu}m$ and $\Delta\lambda=165nm$ at $\lambda_0=5{\mu}m$. The proposed devices add fundamental spectroscopic capabilities to compact imaging devices for a number of applications ranging from spectral sorting to LWIR and MWIR phase contrast imaging and detection.

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.

In this paper we employ the emerging paradigm of physics-informed neural networks (PINNs) for the solution of representative inverse scattering problems in photonic metamaterials and nano-optics technologies. In particular, we successfully apply mesh-free PINNs to the difficult task of retrieving the effective permittivity parameters of a number of finite-size scattering systems that involve many interacting nanostructures as well as multi-component nanoparticles. Our methodology is fully validated by numerical simulations based on the Finite Element Method (FEM). The development of physics-informed deep learning techniques for inverse scattering can enable the design of novel functional nanostructures and significantly broaden the design space of metamaterials by naturally accounting for radiation and finite-size effects beyond the limitations of traditional effective medium theories.