  Narrowband x- and gamma-ray sources based on the inverse Compton scattering
of laser pulses suffer from a limitation of the allowed laser intensity due to
the onset of nonlinear effects that increase their bandwidth. It has been
suggested that laser pulses with a suitable frequency modulation could
compensate this ponderomotive broadening and reduce the bandwidth of the
spectral lines, which would allow to operate narrowband Compton sources in the
high-intensity regime. In this paper we, therefore, present the theory of
nonlinear Compton scattering in a frequency modulated intense laser pulse. We
systematically derive the optimal frequency modulation of the laser pulse from
the scattering matrix element of nonlinear Compton scattering, taking into
account the electron spin and recoil. We show that, for some particular
scattering angle, an optimized frequency modulation completely cancels the
ponderomotive broadening for all harmonics of the backscattered light. We also
explore how sensitive this compensation depends on the electron beam energy
spread and emittance, as well as the laser focusing.
