  In this work, we apply the finite element heterogeneous multiscale method to
a class of dispersive first-order time-dependent Maxwell systems. For this
purpose, we use an analytic homogenization result, which shows that the
effective system contains additional dispersive effects. We provide a careful
study of the (time-dependent) micro problems, including $\mathrm{H}^2$ and
micro errors estimates. Eventually, we prove a semi-discrete error estimate for
the method.
