  We present a novel and quite general analysis of the interaction of a
high-field chirped laser pulse and a relativistic electron, in which exquisite
control of the spectral brilliance of the upshifted Thomson-scattered photon is
shown to be possible. Normally, when Thomson scattering occurs at high field
strengths, there is ponderomotive line broadening in the scattered radiation.
This effect makes the bandwidth too large for some applications and reduces the
spectral brilliance. We show that such broadening can be corrected and
eliminated by suitable frequency modulation of the incident laser pulse.
Further, we suggest a practical realization of this compensation idea in terms
of a chirped-beam driven free electron laser oscillator configuration and show
that significant compensation can occur, even with the imperfect matching to be
expected in these conditions.
