Photon vortex generation by synchrotron radiation experiments in relativistic quantum approach

We formulate a theoretical approach to describe photon vortex production in synchrotron/cyclotron radiation from a helical moving electron under a uniform magnetic field in the relativistic quantum framework. In quantum theory, electron orbitals in a magnetic field are under Landau states. The Landau level density increases with decreasing the magnetic field strength, and it is practically impossible to calculate exactly the synchrotron radiation in possible magnetic fields in the laboratory. We present a method to calculate it by using asymptotic formulations of the emission amplitudes under the condition of Li-Lf≪Li, where Li and Lf are the initial and final Landau numbers, respectively. The wave function of the generated photon is the eigenstate of the z component of the total angular momentum (zTAM) when the magnetic field is parallel to the z axis. The approximation is applicable for photon vortex production of several tens ℏ of zTAM. We also calculate numerically the energy spectra and spatial distribution of photon vortices in magnetic field strengths of 10 and 20 T using electrons with energies of 150 MeV and 8 GeV and Larmor radii of 10 and 100 μm for the helical motion. Although the Landau number becomes up to 108 in these conditions, the radiations could be calculated using the approximation. The results indicate that it is possible to produce predominantly photon vortices with a fixed zTAM when we control the energy and entrance trajectory of electrons and the structure of a magnetic field. The present formula contributes to the prediction and verification of photon vortex generation.