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Optical stochastic cooling (OSC) is a promising technique for the cooling ofdense particle beams. Its operation at optical frequencies enables obtaining amuch larger bandwidth compared to the wellknown microwave-based stochasticcooling. 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 ownradiation field from the pickup. The resulting interaction produces alongitudinal kick with its value depending on the particles momentum which,when correctly phased, yields to longitudinal cooling. The horizontal coolingis achieved by introducing a coupling between longitudinal and horizontaldegrees of freedom. Vertical cooling is achieved by coupling between horizontaland vertical motions in the ring. In this paper, we present formulae forcomputation of the corrective kick and validate them against numericalsimulations performed using a wave-optics computer program.

Computation and Numerical Simulation of Focused Undulator Radiation for Optical Stochastic Cooling