  We investigate deviations from the plane wave model in the interaction of
charged particles with strong electromagnetic fields. A general result is that
integrability of the dynamics is lost when going from lightlike to timelike or
spacelike field dependence. For a special scenario in the classical regime we
show how the radiation spectrum in the spacelike (undulator) case becomes
well-approximated by the plane wave model in the high energy limit, despite the
two systems being Lorentz inequivalent. In the quantum problem, there is no
analogue of the WKB-exact Volkov solution. Nevertheless, WKB and uniform-WKB
approaches give good approximations in all cases considered. Other approaches
that reduce the underlying differential equations from second to first order
are found to miss the correct physics for situations corresponding to barrier
transmission and wide-angle scattering.
