  Electromagnetic waves in a system with a space and time dependent boundary
experience both diffraction and Doppler-like frequency conversion. In order to
analyse such situations, conventional methods call for either the eigenmodes or
the dyadic Green's function in space and time dependent media. Here, we propose
a dynamical differential method which does not require either of them. Our
method utilises a dynamical coordinate transformation in order to simplify the
calculation of the optical response of the space and time dependent system. We
reveal that the diffraction symmetry is broken in the presence of
traveling-wave type spatiotemporal modulation.
