This paper discusses the well-posedness and the multiscale algorithm for the heterogeneous scattering of Maxwell's equations in dispersive media with a periodic microstructure or with many subdivided periodic microstructures. An exact transparent boundary condition is developed to reduce the scattering problem into an initial-boundary value problem in heterogeneous materials. The well-posedness and the stability analysis for the reduced problem are derived. The multiscale asymptotic expansions of the solution for the reduced problem are presented. The convergence results of the multiscale asymptotic method are proved for the dispersive media with a periodic microstructure. A multiscale Crank-Nicolson mixed finite element method (FEM) is proposed where the perfectly matched layer (PML) is utilized to truncate infinite domain problems. Numerical test studies are then carried out to validate the theoretical results.