The nonlinear Thomson backscattering of electrons interacting with a circularly polarized laser pulse is studied by using the single-electron model. By means of theoretical analysis and computer simulation, the influence of the central collision place and the beam waist radius on the spectral characteristics of the scattered light are discussed. It is indicated that the spectrum is nonlinear broadened as a result of redshifting of the Thomson scattered frequency. Only when the central collision point is set at a proper place under the tight focusing situation, can quasi-monochromatic x-rays be generated. Taking into account the requirements for the radiation intensity of scattered light, the optimum situation to generate quasi-monochromatic x-rays is explored. In addition, we find that the spectral and angular distribution of the backscattered light do not vary with azimuth angle.
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