  This paper concerns the inverse scattering problem to reconstruct a local
perturbation in a periodic structure. Unlike the periodic problems, the
periodicity for the scattered field no longer holds, thus classical methods,
which reduce quasi-periodic fields in one periodic cell, are no longer
available. Based on the Floquet-Bloch transform, a numerical method has been
developed to solve the direct problem, that leads to a possibility to design an
algorithm for the inverse problem. The numerical method introduced in this
paper contains two steps. The first step is initialization, that is to locate
the support of the perturbation by a simple method. This step reduces the
inverse problem in an infinite domain into one periodic cell. The second step
is to apply Newton-CG method to solve the associated optimization problem. The
perturbation is then approximated by a finite spline basis. Numerical examples
are given at the end of this paper, shows the efficiency of the numerical
method.
