  We develop an essentially optimal numerical method for solving multiscale
Maxwell wave equations in a domain $D\subset{\mathbb R}^d$. The problems depend
on $n+1$ scales: one macroscopic scale and $n$ microscopic scales. Solving the
macroscopic multiscale homogenized problem, we obtain the desired macroscopic
and microscopic information. This problem depends on $n+1$ variables in
${\mathbb R}^d$, one for each scale that the original multiscale equation
depends on, and is thus posed in a high dimensional tensorized domain. The
straightforward full tensor product finite element (FE) method is exceedingly
expensive. We develop the sparse tensor product FEs that solve this multiscale
homogenized problem with essentially optimal number of degrees of freedom, that
is essentially equal to that required for solving a macroscopic problem in a
domain in ${\mathbb R}^d$ only, for obtaining a required level of accuracy.
Numerical correctors are constructed from the FE solution. For two scale
problems, we derive a rate of convergence for the numerical corrector in terms
of the microscopic scale and the FE mesh width. Numerical examples confirm our
analysis.
