  We consider radiation emitted by colour-charged and massive particles
crossing strong plane wave backgrounds in gauge theory and gravity. These
backgrounds are treated exactly and non-perturbatively throughout. We compute
the back-reaction on these fields from the radiation emitted by the probe
particles: classically through background-coupled worldline theories, and at
tree-level in the quantum theory through three-point amplitudes. Consistency of
these two methods is established explicitly. We show that the gauge theory and
gravity amplitudes are related by the double copy for amplitudes on plane wave
backgrounds. Finally, we demonstrate that in four-dimensions these calculations
can be carried out with a background-dressed version of the massive
spinor-helicity formalism.
