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. PACS numbers: 29.20.Ej, 29.25.Bx, 29.27.Bd, 07.85.Fv Sources of electromagnetic radiation relying upon Thomson scattering are increasingly being applied in fundamental physics research [1], and compact acceleratorbased sources specifically designed for potential user facilities have been built [2]. One remarkable feature of the radiation emerging from such sources, compared to bremsstrahlung sources, is the narrowband nature of the radiation produced. For example, applications to Xray structure determination [3], dark-field imaging [4,5], phase contrast imaging [6], and computed tomography [7] have been demonstrated experimentally and take full advantage of the narrow bandwidth of the Thomson source.Given that narrow bandwidth is desired, it is important to know and understand the sources of bandwidth of the scattered radiation and the limitations imposed on the performance of Thomson sources. For applications where the normalized vector potential of the incident laser pulse is much less than one (the low-field regime), the line width of the radiation from a scattering event reproduces the line width of the incident laser pulse. Unfortunately, when the normalized vector potential increases, as is desired for stronger sources, a detuning red-shift arises during the scattering events that tends to spread out the spectrum [8][9][10]. Physically, the scattering electron slows down, by a varying amount, as the incident pulse is traversed.In a recent paper, Ghebregziabher, Shadwick, and Umstadter (GSU) observed that frequency modulation (FM), or "chirping", of the scattering laser pulse can compensate for such ponderomotive line broadening, and suggested a form for this modulation [11]. Motivated by their observation, we present the exact analytic solution for optimal FM, recovering the low-field linewidth even in the high-field regime. The narrowing of the scattered pulse is Fourier-limited only by the duration of the incident pulse.The essence of laser pulse chirping is analogous to free electron laser (FEL) undulator tapering [12][13][14][15][16]. In tapering, as deceleration occurs due to the FEL emission, the field strength is adjusted to preserve the...