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 extemal 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.