Relative narrow bandwidth-high energy radiation can be produced through Thomson scattering, where highly relativistic electrons collide with a laser pulse. The bandwidth of such a source is determined, among others factors, by the bandwidth of the laser pulse and the energy spread of the electrons. Here we investigate how the bandwidth of such a source can be minimized, with a particular emphasis on electron bunches with a correlated energy spread of several percent, that are typical for plasma based accelerator schemes. We show that by introducing a chirp on the laser pulse it is possible to compensate the broadening effect due to the energy spread of the electrons, and obtain the same bandwidth as a quasi-monochromatic plane wave laser pulse colliding with a monoenergetic electron bunch. Ultimately, the bandwidth of a Thomson source is limited by the acceptance angle and the initial transverse momentum of electrons (emittance).
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