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：本文研究一类由介电低维材料制成的周期性微尺度夹杂物构成的光学超材料的形状优化问题，该夹杂物悬浮于非磁性体介电材料中。形状优化方法基于时谐麦克斯韦方程组均质化理论，该理论描述了电磁波在超材料中传播的有效材料参数。优化控制参数为反映微尺度几何偏离胞体问题参考构型的形变场，据此可将均质化有效介电张量描述为形变场的函数。我们证明了所述的变形胞体问题具有适定性和正则性，进而推导出形状优化问题的适定性。此外，本文提出一种数值方案，该方案采用伴随公式并结合梯度下降法或BFGS算法作为优化算法。通过以指定有效介电张量为目标的若干典型形状优化问题，对所开发的算法进行了数值测试。