  This paper is concerned with uniqueness for reconstructing a periodic
inhomogeneous medium covered on a perfectly conducting plate. We deal with the
problem in the frame of time-harmonic Maxwell systems without TE or TM
polarization. An orthogonal relation for two refractive indices is obtained,
and then inspired by Kirsch's idea, the refractive index can be identified by
utilizing the eigenvalues and eigenfunctions of a quasi-periodic
Sturm-Liouville eigenvalue problem.
