  Some non linear, second order QED processes in the presence of intense plane
electromagnetic waves are investigated. Analytic expressions with general
kinematics are derived for Compton scattering and $e^+e^-$ pair production in a
circularly polarised external electromagnetic field. The differential cross
sections were investigated numerically for photon energies up to 50 MeV,
external field intensity parameter $\nu^2$ to value 2, and all scattering
angles. The variation of full cross sections with respect to external field
intensity was also established. The presence of the external field led to
resonances in the Compton scattering and pair production differential cross
sections. These resonances were investigated by calculating the electron self
energy in the presence of the external field. At resonance the differential
cross sections were enhanced by several orders of magnitude. The resonances
occurred for values of external field intensity parameter $\nu^2 <1$, lowering
the limit of $\nu^2 \sim 1$ at which point non linear effects in first order
external field QED processes become important. Generally, full cross sections
increased with increasing external field intensity, though peaking sharply for
Compton scattering and levelling off for pair production. An application was
made to non linear background studies at $e^+e^-$ linear colliders. The pair
production process and electron self energy were studied for the case of a
constant crossed electromagnetic field. It was found that previous analytic
expressions required the external field to be azimuthally symmetric. New
analytic expressions for the more general non azimuthally symmetric case were
developed and a numerical parameter range equivalent to that proposed for
future linear collider designs was considered.
