Creating Materials with Negative Refraction Index using Topology Optimization

We apply topology optimization along with full modeling of the electromagnetic (acoustic) field to create metamaterials with negative refraction index. We believe that our approach can be used in the design of metamaterials with specific effective permittivity and permeability e.g. by adapting the approach presented in [1].
We model the problem in 2D in the frequency domain using the Helmholtz equation and discretize the model using the hybrid WBM-FEM method [2]. We consider a modulated plane wave incident at an angle on a slab consisting of a periodic array of identical design cells whose size is on the order of the wavelength. We seek a distribution of solid and air in the design cell yielding a prescribed negative refraction index for the slab.
Our objective is to minimize the difference in amplitude between the solution to the model problemand a prescribed modulated plane wave behind the slab. The direction of propagation for the prescribed wave is chosen to match the angle of incidence of the incoming plane wave and its position isused to select the refraction index for the slab. We introducing a continuous design field and apply The Method of Moving Asymptotes to perform the optimization. A filter is used for regularizationand a projection step applied to obtain clean 0/1 designs. A continuation scheme is used to avoidstagnation in the optimization.
Metamaterials with negative refraction index designed using this method are presented. The angular dependence of the refraction index and of the reflection and transmission coefficients are investigated.