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We propose a rigorous approach for the inverse design of functional photonicstructures by coupling the adjoint optimization method and the two-dimensionalgeneralized Mie theory (2D-GMT) for the multiple scattering problem offinite-size arrays of dielectric nanocylinders optimized to display desiredfunctions. We refer to these functional scattering structures as "photonicpatches". We briefly introduce the formalism of 2D-GMT and the critical stepsnecessary to implement the adjoint optimization algorithm to photonic patcheswith designed radiation properties. In particular, we showcase several examplesof periodic and aperiodic photonic patches with optimal nanocylinder radii andarrangements for radiation shaping, wavefront focusing in the Fresnel zone, andfor the enhancement of the local density of states (LDOS) at multiplewavelengths over micron-size areas. Moreover, we systematically compare theperformances of periodic and aperiodic patches with different sizes and findthat optimized aperiodic Vogel spiral geometries feature significant advantagesin achromatic focusing compared to their periodic counterparts. Our resultsshow that adjoint optimization coupled to 2D-GMT is a robust methodology forthe inverse design of compact photonic devices that operate in the multiplescattering regime with optimal desired functionalities. Without the need ofspatial meshing, our approach provides efficient solutions at strongly reducedcomputational burden compared to standard numerical optimization techniques andsuggests compact device geometries for on-chip photonics and metamaterialstechnologies.

Inverse design of functional photonic patches by adjoint optimization coupled to the generalized Mie theory