  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.
