  Nonlinear Compton scattering is calculated for the collision of an electron
with a plane wave pulse. A mid infra-red (IR) peak arises in the photon
spectrum due to long-range interference associated with the pulse envelope. The
case of a flat-top pulse is studied as a toy model for pulse envelope effects
and reduced to two final-state momentum integrations; the case of a
sine-squared pulse is studied numerically. A perturbative expansion in
charge-field coupling reveals that already at intermediate intensities, many
orders are required to correctly capture the structure of the mid-IR peak. By
regularising the classical result, it is shown that the mid-IR peak is due to
plane-wave ponderomotive effects on the pulse envelope. Finally, it is shown
that the mid-IR peak can be isolated using energy, angle and polarisation
filters.
