  In this paper, we consider the scattering of a time-dependent electromagnetic
wave by an elastic body immersed in the lower half-space of a two-layered
background medium which is separated by an unbounded rough surface. By
proposing two exact transparent boundary conditions (TBCs) on the artificial
planes, we reformulate the unbounded scattering problem into an equivalent
initial-boundary value problem in a strip domain with the well-posedness and
stability proved using the Laplace transform, variational method and energy
method. A perfectly matched layer (PML) is then introduced to truncate the
interaction problem with two finite layers containing the elastic body, leading
to a PML problem in a finite strip domain. We further verify the existence,
uniqueness and stability estimate of solution for the PML problem. Finally, we
establish the exponential convergence in terms of the thickness and parameters
of the PML layers via an error estimate on the electric-to-magnetic (EtM)
capacity operators between the original problem and the PML problem.
