We propose an efficient inverse design approach for multifunctional optical elements based on adaptive deep diffractive neural networks (aD^{2}NNs). Specifically, we introduce aD^{2}NNs and design twolayer diffractive devices that can selectively focus incident radiation over two wellseparated spectral bands at desired distances. We investigate focusing efficiencies at two wavelengths and achieve targeted spectral line shapes and spatial pointspread functions (PSFs) with optimal focusing efficiency. In particular, we demonstrate control of the spectral bandwidths at separate focal positions beyond the theoretical limit of singlelens devices with the same aperture size. Finally, we demonstrate devices that produce superoscillatory 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.
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We propose a novel framework for the systematic design of lensless imaging systems based on the hyperuniform random field solutions of nonlinear reactiondiffusion equations from pattern formation theory. Specifically, we introduce a new class of imaging pointspread functions (PSFs) with enhanced isotropic behavior and controllable sparsity. We investigate PSFs and modulated transfer functions for a number of nonlinear models and demonstrate that twophase isotropic random fields with hyperuniform disorder are ideally suited to construct imaging PSFs with improved performances compared to PSFs based on Perlin noise. Additionally, we introduce a phase retrieval algorithm based on nonparaxial Rayleigh–Sommerfeld diffraction theory and introduce diffractive phase plates with PSFs designed from hyperuniform random fields, called hyperuniform phase plates (HPPs). Finally, using highfidelity object reconstruction, we demonstrate improved image quality using engineered HPPs across the visible range. The proposed framework is suitable for highperformance lensless imaging systems for onchip microscopy and spectroscopy applications.

We design and characterize a novel axilensbased diffractive optics platform that flexibly combines efficient point focusing and grating selectivity and is compatible with scalable topdown fabrication based on a fourlevel phase mask configuration. This is achieved using phasemodulated compact axilens devices that simultaneously focus incident radiation of selected wavelengths at predefined locations with larger focal depths compared with traditional Fresnel lenses. In addition, the proposed devices are polarizationinsensitive and maintain a large focusing efficiency over a broad spectral band. Specifically, here we discuss and characterize modulated axilens configurations designed for longwavelength infrared (LWIR) in the 6 µm–12 µm wavelength range and in the 4 µm–6 µm midwavelength infrared (MWIR) range. These devices are ideally suited for monolithic integration atop the substrate layers of infrared focal plane arrays and for use as compact microspectrometers. We systematically study their focusing efficiency, spectral response, and crosstalk ratio; further, we demonstrate linear control of multiwavelength focusing on a single plane. Our design method leverages Rayleigh–Sommerfeld diffraction theory and is validated numerically using the finite element method. Finally, we demonstrate the application of spatially modulated axilenses to the realization of a compact, singlelens spectrometer. By optimizing our devices, we achieve a minimum distinguishable wavelength interval of
$\mathrm{\Delta <\#comment/>}\mathrm{\lambda <\#comment/>}=240\phantom{\rule{thickmathspace}{0ex}}\mathrm{n}\mathrm{m}$ at${\mathrm{\lambda <\#comment/>}}_{c}=8\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\xb5<\#comment/>}\mathrm{m}$ and$\mathrm{\Delta <\#comment/>}\mathrm{\lambda <\#comment/>}=165\phantom{\rule{thickmathspace}{0ex}}\mathrm{n}\mathrm{m}$ at${\mathrm{\lambda <\#comment/>}}_{c}=5\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\xb5<\#comment/>}\mathrm{m}$ . 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. 
We design and characterize compact phasemodulated axilens devices that combine efficient point focusing and grating selectivity within fourlevel phase mask configurations. Specifically, we select and characterize in detail two device configurations designed for longwavelength infrared (LWIR) operation in the
$6\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\xb5<\#comment/>}\mathrm{m}\phantom{\rule{negativethinmathspace}{0ex}}<\#comment/>\phantom{\rule{negativethinmathspace}{0ex}}12\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\xb5<\#comment/>}\mathrm{m}$ wavelength range. These devices are ideally suited for monolithic integration atop the substrate layers of infrared focal plane arrays (IRFPAs) for use in multiband LWIR photodetection. We systematically study their focusing efficiency, spectral response, and crosstalk ratio, and we demonstrate a singlecomponent microspectrometer. Our design method leverages the Rayleigh–Sommerfeld (RS) diffraction theory that is validated numerically using the finite element method (FEM). The proposed devices are broadband and polarization insensitive and add fundamental spectroscopic capabilities to miniaturized optical components for a number of applications in LWIR detection and spectroscopy.