Accelerated Optimization of Metasurfaces with the Woodbury Matrix Identity

Direct optimization of the impedances of a metasurface is difficult due to the large number of subwavelength elements comprising a metasurface. Gradient-based optimization techniques numerically approximate the gradient using finite differences requiring at least one cost evaluation for each element in the metasurface. If the cost evaluation is expensive or the number of elements is large, then the calculation of the gradient becomes time-consuming and impractical. When metasurfaces are modeled using integral equations, the cost function evaluation involves matrix inversions of the linear system constructed with the method of moments. Since each component of the gradient perturbs the linear system along a few diagonal elements of the impedance matrix, the matrix inversions can be accelerated using the Woodbury Matrix Identity. This accelerated solution of the linear system allows the gradient to be calculated efficiently permitting the direct optimization of metasurface impedances. In this paper, an acceleration technique for the gradient descent optimization of metasurfaces modeled with integral equations is presented. Numerical results are provided and show up to a 26.5 times improvement in computation time for the cases presented.