Saturated pair-photon cascades on isotropic background photons

The problem of high-energy gamma-rays and electrons entering an isotropic background radiation field is considered. The gamma-rays produce electron-positron pairs by photon-photon interactions with the background photons, and the electrons Compton upscatter the background photons, giving rise to a pair-photon cascade. The Compton interactions are in the quantum Klein-Nishina regime. The spectra of the secondary produced gamma-rays are obtained in the case of large optical depths of the radiation field by solving a kinetic equation of the cascade. This process can be important for the early universe, gamma-ray bursts, jets, active galactic nuclei, and ultra-high-energy gamma-rays and electrons crossing the cosmic microwave background.