This paper describes a class of shape optimization problems for optical metamaterials comprised of periodic microscale inclusions composed of a dielectric, low-dimensional material suspended in a non-magnetic bulk dielectric. The shape optimization approach is based on a homogenization theory for time-harmonic Maxwell's equations that describes effective material parameters for the propagation of electromagnetic waves through the metamaterial. The control parameter of the optimization is a deformation field representing the deviation of the microscale geometry from a reference configuration of the cell problem. This allows for describing the homogenized effective permittivity tensor as a function of the deformation field. We show that the underlying deformed cell problem is well-posed and regular. This, in turn, proves that the shape optimization problem is well-posed. In addition, a numerical scheme is formulated that utilizes an adjoint formulation with either gradient descent or BFGS as optimization algorithms. The developed algorithm is tested numerically on a number of prototypical shape optimization problems with a prescribed effective permittivity tensor as the target.

翻译：本文描述了一类由介电低维材料构成、悬浮于非磁性块体电介质中的周期性微尺度内含物的光学超材料形状优化问题。该形状优化方法基于时谐麦克斯韦方程组的均质化理论，该理论描述了电磁波在超材料中传播的有效材料参数。优化的控制参数是代表微尺度几何形状偏离胞元问题参考构型的变形场。这使得均质化有效介电常数张量可以表示为变形场的函数。我们证明了底层变形胞元问题的适定性与正则性，进而证明了形状优化问题的适定性。此外，我们制定了基于伴随公式的数值方案，采用梯度下降法或BFGS算法作为优化算法。该算法在若干以指定有效介电常数张量为目标的原型形状优化问题上进行了数值测试。