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.