Effects of nonlinearity in Thomson scattering of a high intensity laser pulse from electrons are analyzed. Analytic expressions for laser pulse shaping in amplitude and frequency are obtained which control spectrum broadening for arbitrarily high laser pulse intensities. These analytic solutions allow prediction of the spectral form and required laser parameters to avoid broadening. The predictions are validated by numerical calculations. This control over the scattered radiation bandwidth allows of narrow bandwidth sources to be produced using high scattering intensities, which in turn greatly improves scattering yield for future x- and gamma-ray sources.
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