Generating Cylindrical Vector {\gamma} Rays via Beam-Target Interactions: Towards Structured Light at High Energies

Structured {\gamma} rays, particularly cylindrical vector {\gamma} rays, offer promising tools for sub-nuclear imaging and polarization-sensitive probes in fundamental research and applications, but conventional optical methods face great challenges at such photon energy. Here, we put forward a novel method generating such {\gamma} rays through relativistic beam-target interactions. For instance, radially polarized {\gamma} rays can be generated by using a dense electron beam striking a multifoil target. We find that the radial polarization is transferred from the generated coherent transition radiation (CTR) fields to $\gamma$ photons through nonlinear Compton scattering, with the high polarization preserved by phase matching. Three-dimensional spin-resolved simulations demonstrate radial polarization degrees approaching 60\%. Furthermore, these {\gamma} rays can decay into azimuthally spin-polarized positrons via the nonlinear Breit-Wheeler process, with their spins aligning along the CTR magnetic field. Our work extends the concept of structured light into the {\gamma}-ray regime, offering new prospects for broad fields such as nuclear structure probing, fundamental symmetries tests, polarization-sensitive studies in extreme conditions, and laboratory astrophysical observations.