  In this paper, we suggest a new Heterogeneous Multiscale Method (HMM) for the
(time-harmonic) Maxwell scattering problem with high contrast. The method is
constructed for a setting as in Bouchitt\'e, Bourel and Felbacq (C.R. Math.
Acad. Sci. Paris 347(9-10):571--576, 2009), where the high contrast in the
parameter leads to unusual effective parameters in the homogenized equation. We
present a new homogenization result for this special setting, compare it to
existing homogenization approaches and analyze the stability of the two-scale
solution with respect to the wavenumber and the data. This includes a new
stability result for solutions to time-harmonic Maxwell's equations with
matrix-valued, spatially dependent coefficients. The HMM is defined as direct
discretization of the two-scale limit equation. With this approach we are able
to show quasi-optimality and a priori error estimates in energy and dual norms
under a resolution condition that inherits its dependence on the wavenumber
from the stability constant for the analytical problem. This is the first
wavenumber-explicit resolution condition for time-harmonic Maxwell's equations.
Numerical experiments confirm our theoretical convergence results.
