Topology and shape optimization of plasmonic nano-antennas

Metallic nano-antennas are devices used to concentrate the energy in light into regions that are much smaller than the wavelength. These structures are currently used to develop new measurement and printing techniques, such as optical microscopy with sub-wavelength resolution, and high-resolution lithography. Here, we analyze and design a nano-antenna in a two-dimensional setting with the source being a planar TE-polarized wave. The design problem is to place silver and air in a pre-specified design region to maximize the electric energy in a small given target region. At optical frequencies silver exhibits extreme dielectric properties, having permittivity with a negative real part. We prove existence and uniqueness of solutions to the governing non-standard Helmholtz equation with absorbing boundary conditions. To solve the design optimization problem, we develop a two-stage procedure. The first stage uses a material distribution parameterization and aims at finding a conceptual design without imposing any a priori information about the number of shapes of components comprising the nano-antenna. The second design stage uses a domain variation approach and aims at finding a precise shape. Both of the above design problems are formulated as non-linear mathematical programming problems that are solved using the method of moving asymptotes. The final designs perform very well and the electric energy in the target region is several orders of magnitude larger than when there is only air in the design region. The performance of the optimized designs is verified with a high order interior penalty method.