  Arrays of wavelength scale scatterers are a promising platform for designing
optical elements with a compact footprint. The large number of degrees of
freedom in this system allows for unique and plentiful functionalities.
However, the many variables also create a complex design problem. While
intuitive forward design methods work for simple optical elements, they often
fail to produce complicated elements, especially those involving multiple
elements. We present an inverse design methodology for large arrays of
wavelength scale spheres based on both adjoint optimization or sensitivity
analysis and generalized multi-sphere Mie theory as a solution to the design
problem. We validate our methodology by designing two sets of optical elements
with scatterers on sub-wavelength and super-wavelength periodicity grids. Both
sets consist of a singlet and a doublet lens, with one and two layers of
spheres respectively designed for 1550 nm. The designed NA is ~0.33 (~0.5) for
the sub-wavelength (super-wavelength) periodic structure. We find that with the
sub-wavelength periodicity, the full width at half maximum of the focal spot
produced by the singlet and doublet is smaller than that produced by an ideal
lens with the same geometric parameters. Finally, we simulate a realistic
experimental scenario for the doublet where the spheres are placed on a
substrate with the same refractive index. We find the performance is similar,
but with lower intensity at the focal spot and larger spot size. The method
described here will simplify the design procedure for complicated
multifunctional optical elements and or scatterer array-based volume optics
based on a specified figure of merit.
