Usually, photons are described by plane waves with a definite 4-momentum. In addition to plane-wave photons, "twisted photons" have recently entered the field of modern laser optics; these are coherent superpositions of plane waves with a defined projection m of the orbital angular momentum onto the propagation axis, where m is integer. In this paper, we show that it is possible to produce high-energy twisted photons by Compton backscattering of twisted laser photons off ultra-relativistic electrons. Such photons may be of interest for experiments related to the excitation and disintegration of atoms and nuclei, and for studying the photo-effect and pair production off nuclei in previously unexplored experimental regimes.PACS numbers: 13.60. Fz, 42.65.Ky, 27.70.Jj, 12.20.Ds Introduction.-An interesting research direction in modern optics is related to experiments with so-called "twisted photons." These are states of the laser beam whose photons have a defined value m of the angular momentum projection on the beam propagation axis where m is a (large) integer [1]. An experimental realization [2] exists for states with projections as large as m = 200. Such photons can be created from usual laser beams by means of numerically computed holograms. The wavefront of such states rotates around the propagation axis, and their Poynting vector looks like a corkscrew (see Fig. 1 in Ref. [1]). It was demonstrated that micron-sized teflon and calcite "particles" start to rotate after absorbing twisted photons [3].In this Letter, we show that it is possible to convert twisted photons from an energy range of about 1 eV to a higher energies of up to a hundred GeV using Compton backscattering off ultra-relativistic electrons. In principle, Compton backscattering is an established method for the creation of high-energy photons and is used successfully in various application areas from the study of photonuclear reactions [4,5] to colliding photon beams of high energy [6]. However, the central question is how to treat Compton backscattering of twisted photons, whose field configuration is manifestly different from plane waves. Below, we use relativistic Gaussian units with c = 1, = 1, α ≈ 1/137. We denote the electron mass by m e and write the scalar product of 4-vectors k = (ω, k) andTwisted photon.-We wish to construct a twisted photon state with definite longitudinal momentum k z , absolute value of transverse momentum κ and projection m of the orbital angular momentum onto the z axis (propagation axis). We start from a plane-wave photon state with 4-momentum k = (ω, k) and helicity Λ = ±1,
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