Fast and Accurate Optimization of Metasurfaces With Gradient Descent and the Woodbury Matrix Identity

A fast metasurface optimization strategy for finite-size metasurfaces modeled using integral equations is presented. The metasurfaces considered are constructed from finite, patterned metallic claddings represented by homogenized impedance sheets supported by grounded dielectric spacers. Integral equations are used to model the response of the metasurface to a known excitation and solved by the method of moments. An accelerated gradient descent optimization algorithm is presented that enables the direct optimization of such metasurfaces. The gradient is normally calculated by solving the method of moments problem
$N+1$
times, where
$N$
is the number of homogenized elements in the metasurface. Since the calculation of each component of the
$N$
-dimensional gradient involves perturbing the moment method impedance matrix along one element of its diagonal and inverting the result, this numerical gradient calculation can be accelerated using the Woodbury matrix identity. The Woodbury matrix identity allows the inverse of the perturbed impedance matrix to be computed at a low cost by forming a rank-r correction to the inverse of the unperturbed impedance matrix. Timing diagrams show up to a 26.5 times improvement in algorithm times when the acceleration technique is applied.