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The Thomson backscattering spectra in combined uniform magnetic andcosine-envelope circularly-polarized laser fields are studied in detail. Withan introduction of the envelope modulation, the radiation spectra exhibit highcomplexity attributed to the strong nonlinear interactions. On the other hand,four fundamental laws related to the scale invariance of the radiation spectraare analytically revealed and numerically validated. They are the laws for theradiation energy as the $6$th power of the motion constant exactly, also as theapproximate negative $6$th power with respect to the initial axial momentum andlaser intensity in a certain of conditions, respectively, and finally animportant self-similar law, i.e., when the circular laser frequency, theenvelope modulation frequency, and the modified cyclotron frequency aresimultaneously increased by a factor, the radiation energy will be increased bythe second power of that factor without changing the shape of the spectrum.With the application of these laws, especially the last one, a much higherradiation energy can be obtained and the harmonic at which the maximumradiation occurs can be precisely tuned without changing its amplitude. Thesefindings provide a possible way to advance radiation technology in many fieldssuch as medicine, communications, astrophysics, and security.

Thomson backscattering in combined uniform magnetic and envelope modulating circularly-polarized laser fields