Topology optimization of hyperbolic metamaterials for an optical hyperlens

This paper presents a level set-based topology optimization method for the microstructural design of an optical hyperlens. The resolution of conventional optics is generally diffraction limited, but the diffraction limit can be overcome by a hyperlens that converts evanescent waves to propagation waves, utilizing a cylindrical geometry to magnify the subwavelength features of imaged objects, which allows such features to be resolved beyond the diffraction limit at the hyperlens output. To support electromagnetic waves that contain information of subwavelength features, the hyperlens must have material properties that include a positive permittivity in the angular direction and a negative permittivity in the radial direction. Here, an energy-based homogenization method is used to obtain the effective permittivity of a metamaterial hyperlens unit cell. A level set-based topology optimization is applied so that the boundaries of the structure are clearly expressed. Moreover, a finite element mesh regeneration scheme is used to precisely capture the boundaries of the metal and dielectric materials of the unit cell that will ultimately form the hyperlens. The optimization algorithm uses the finite element method (FEM) to solve the equilibrium and adjoint equations. Optimum design examples for the design of a hyperlens microstructure are provided to confirm the utility and validity of the presented method.