Scaling law for the photon spectral density in the nonlinear Thomson-Compton scattering

We derive a new scaling law for the photon spectral density in nonlinear Thomson/Compton scattering, extending the findings of Heinzl, Seipt, and K\"ampfer [Phys. Rev. A 81, 022125 (2010)]. This allows one to easily include the effects of general scattering geometries, e.g., side injection, and of a finite-size detector on the photon spectrum. The scaling law is employed to study substructures emerging in the nonlinear Thomson/Compton spectra due to temporally shaped laser pulses scattering off relativistic electrons. We determine optimum scattering geometries for an experimental verification of these substructures.