  In this paper, we consider a numerical method to solve scattering problems
with multi-periodic layers with different periodicities. The main tool applied
in this paper is the Bloch transform. With this method, the problem is written
into an equivalent coupled family of quasi-periodic problems. As the Bloch
transform is only defined for one fixed period, the inhomogeneous layer with
another period is simply treated as a non-periodic one. First, we approximate
the refractive index by a periodic one where its period is an integer multiple
of the fixed period, and it is decomposed by finite number of quasi-periodic
functions. Then the coupled system is reduced into a simplified formulation. A
convergent finite element method is proposed for the numerical solution, and
the numerical method has been applied to several numerical experiments. At the
end of this paper, relative errors of the numerical solutions will be shown to
illustrate the convergence of the numerical algorithm.
