This paper discusses computation of shape derivatives of electromagnetic fields produced by complex 2-periodic structures. A dual set of forward and adjoint problems for Maxwell's equations are solved with the method of moments (MoM) to calculate the full gradient of the object function by the adjoint variable method (AVM). The periodic fast multipole method (pFMM) is used to accelerate the solution of integral equations for electromagnetic scattering problems with periodic boundary conditions (PBC). This technique is applied to shape optimization problems for negative-index metamaterials (NIM) with a double-fishnet structure. Numerical results demonstrate that the figure of merit (FOM) of metamaterials can reach a maximum value when the shape parameters are optimized iteratively by a gradient-based optimization method.
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