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Arrays of wavelength scale scatterers are a promising platform for designingoptical elements with a compact footprint. The large number of degrees offreedom in this system allows for unique and plentiful functionalities.However, the many variables also create a complex design problem. Whileintuitive forward design methods work for simple optical elements, they oftenfail to produce complicated elements, especially those involving multipleelements. We present an inverse design methodology for large arrays ofwavelength scale spheres based on both adjoint optimization or sensitivityanalysis and generalized multi-sphere Mie theory as a solution to the designproblem. We validate our methodology by designing two sets of optical elementswith scatterers on sub-wavelength and super-wavelength periodicity grids. Bothsets consist of a singlet and a doublet lens, with one and two layers ofspheres respectively designed for 1550 nm. The designed NA is ~0.33 (~0.5) forthe sub-wavelength (super-wavelength) periodic structure. We find that with thesub-wavelength periodicity, the full width at half maximum of the focal spotproduced by the singlet and doublet is smaller than that produced by an ideallens with the same geometric parameters. Finally, we simulate a realisticexperimental scenario for the doublet where the spheres are placed on asubstrate with the same refractive index. We find the performance is similar,but with lower intensity at the focal spot and larger spot size. The methoddescribed here will simplify the design procedure for complicatedmultifunctional optical elements and or scatterer array-based volume opticsbased on a specified figure of merit.

Inverse design of optical elements based on arrays of dielectric spheres