Multipolar FFT-Based Adjoint Optimization for Designing Goal-Oriented Materials with Metallic Insertions

In this paper, we present a fast and efficient algorithm for optimizing the orientation of dipoles centered in a 3D equispaced array. By using appropriate goal functions, the method enables the design of lenses with focal points at varying distances. The approach combines several numerical tools, including the Fast Fourier Transform (FFT), multipolar expansion, the quasi-Newton method (BFGS), and the adjoint method, to achieve the optimization objectives. The algorithm is capable of handling problems with a large number of degrees of freedom and diverse goal functions. In each iteration, it performs a full-wave electromagnetic simulation to compute the cost function for the current dipole orientations and calculates the gradient to guide the optimization toward improvement.