We consider the direct and inverse scattering problem of electromagnetic and acoustic wave phenomena for an unbounded, inhomogeneous and bi-periodic media, which includes a local defect. At first, we show the unique solvability of the vector valued scattering problem formulated in terms of the Maxwell's equations, as well as the scalar valued scattering problem modeled by the Helmholtz equation, by assuming some reasonable presumptions on the regularity of the parameter. Moreover, we show some regularity results for the Bloch-Floquet transformed solution w.r.t. the quasiperiodicy and derive a numerical method for the approximation of the solution based on the finite-elements method. We continue with the inverse problem of reconstructing the perturbation. For that, we consider different measurement operators, prove the injectivity and apply a Newton method. Furthermore, we introduce the Factorization method as a fast imaging method to localize the support of the perturbation
