  The new scheme of the energy measurement of the extremely high energy
electron beam with the inverse Compton scattering between electrons and
microwave photons requires the precise calculation of the interaction cross
section of electrons and microwave photons in a resonant cavity. In the local
space of the cavity, the electromagnetic field is expressed by Bessel
functions. Although Bessel functions can form a complete set of orthogonal
basis, it is difficult to quantify them directly as fundamental wave functions.
Fortunately, with the Fourier expansion of Bessel functions, the local
electromagnetic field can be considered as the superposition of a series of
plane waves. Therefore, with corresponding corrections of the cross section
formula of the classical Compton scattering, the cross section of the linear or
nonlinear microwave Compton scattering in the local space can be described
accurately. As an important application of our results in astrophysics,
corresponding ground verification devices can be designed to perform
experimental verifications on the prediction of the Sunyaev-Zeldovich (SZ)
effect of the cosmic microwave background radiation. Our results could also
provide a new way to generate wave sources with strong practical value, such as
the terahertz waves, the ultra-violet (EUV) waves, or the mid-infrared beams.
