Thomson backscattering in combined fields with a general elliptical polarization of laser field and an applied constant magnetic field is studied. The monotonous and non-monotonous dependence of emission spectra on the ellipticity and the initial phase of laser field are found. We have also revealed the scaling law of the radiation spectra about the initial axial momentum in the general elliptic polarized case that is similar to the circularly polarized one. However, the scaling law about the laser intensity exists conditionally and the scale invariance about the magnetic resonance parameter exists, too. Another interesting feature is the feasibility to obtain the THz radiation by this scheme.

With the saddle point analysis method for the Bessel function structure and property, the convergence problem and the scaling laws of Thomson backscattering spectra are solved and studied in both cases that are for the plane wave laser field without and with applied external constant magnetic field. Some unclear points appeared in previous work are clarified. The extension of the method to a general situation for the laser field with an arbitrary polarization is discussed. We also make a simple analysis and discussion about the optimal spectra dependence of field parameters and its implication to practical applications.