  In this paper we consider the free-form optimization of eigenvalues in
electromagnetic systems by means of shape-variations with respect to small
deformations. The objective is to optimize a particular eigenvalue to a target
value. We introduce the mixed variational formulation of the Maxwell eigenvalue
problem introduced by Kikuchi (1987) in function spaces of
(H(\operatorname{curl}; \Omega)) and (H^1(\Omega)). To handle this formulation,
suitable transformations of these spaces are utilized, e.g., of Piola-type for
the space of (H(\operatorname{curl}; \Omega)). This allows for a formulation of
the problem on a fixed reference domain together with a domain mapping. Local
uniqueness of the solution is obtained by a normalization of the the
eigenfunctions. This allows us to derive adjoint formulas for the derivatives
of the eigenvalues with respect to domain variations. For the solution of the
resulting optimization problem, we develop a particular damped inverse BFGS
method that allows for an easy line search procedure while retaining positive
definiteness of the inverse Hessian approximation. The infinite dimensional
problem is discretized by mixed finite elements and a numerical example shows
the efficiency of the proposed approach.
