We study incoherent Thomson scattering between an ultrashort laser pulse and an electron beam accelerated from a laser wakefield. The energy chirp effects of the accelerated electron beam on the final radiation spectrum bandwidth are investigated. It is found that the scattered X-ray radiation has the minimum spectrum width and highest intensity as electrons are accelerated up to around the dephasing point. Furthermore, it is proposed the electron acceleration process inside the wakefield can be studied by use of 90• Thomson scattering. The dephasing position and beam energy chirp can be deduced from the intensity and bandwidth of the scattered radiation.PACS numbers: 52.38. Kd, 41.75.Jv, 41.60.Ap Nowadays high bright X-ray sources based on intense laser pulses have attracted a lot of attentions due to their wide applications, relative low cost, easy operation, and unique characters [1]. Compared with traditional accelerator based synchrotron radiation sources these new sources have shorter duration and easier synchronism with lasers, which makes them more flexible for pump-probe technique. Many mechanisms are proposed to generate X-ray radiations. For incoherent sources, radiation based on inner shell electron excitation through laser-solid interaction are extensively studied [2-4]; electron betatron radiation inside a laser wakefield or radiation from laser Thomson scattering through laser gas interactions are also studied [5][6][7][8]. For coherent sources, radiation based on high harmonics generation in laser-solid interaction or coherent Thomson scattering from laser nanometer electron sheet interaction are studied [9][10][11]. The latter has attracted more and more interests recently due to the feasibility of electron sheet generation resulting from laser plasma interactions [12][13][14].In this paper, we focus on incoherent radiation from laser electron Thomson scattering. We use the electron beam accelerated from a laser wakefield accelerator [15,16], which may allow easy synchronization between the electron beams and laser pulses. We show that, due to the energy chirp of the electrons accelerated inside a wakefield, the final spectrum of the scattered radiation can be narrowed compared with a normal un-chirped electron beam. In the meanwhile, we propose to deduce the acceleration process inside a wakefield by diagnosing the radiation spectrum. This provides a new possible approach to detect the wakefield.For simplicity, here we use a one-dimensional (1D) model for the electron beam accelerated in a wakefield. This gives a proof-of-principle description for Thomson scattering with chirped electron beams. As one knows for the electrons accelerated along the same phase space trajectory in a wakefield * Electronic address: minchen@sjtu.edu.cn † Electronic address: zmsheng@sjtu.edu.cn whose normalized phase velocity is β p = v p /c, the relation between its longitudinal momentum (p z ) and phase position ψ satisfies:, where H is the Hamiltonian along the specific trajectory in the phase space set by the w...