Computation and numerical simulation of focused undulator radiation for optical stochastic cooling

Optical stochastic cooling (OSC) is a promising technique for the cooling of dense particle beams. Its operation at optical frequencies enables obtaining a much larger bandwidth compared to the well-known microwave-based stochastic cooling. In the OSC undulator radiation generated by a particle in an upstream “pickup” undulator is amplified and focused at the location of a downstream “kicker” undulator. Inside the kicker, a particle interacts with its own radiation field from the pickup. The resulting interaction produces a longitudinal kick with its value depending on the particles momentum which, when correctly phased, yields longitudinal cooling. The horizontal cooling is achieved by introducing a coupling between longitudinal and horizontal degrees of freedom. Vertical cooling is achieved by coupling between horizontal and vertical motions in the ring. In this paper, we present formulas for computation of the corrective kick and validate them against numerical simulations performed with a wave-optics computer program.