Approximations for large angle synchrotron radiation

A large-angle beamstrahlung detector at CESR appears feasible[1] except for the unknown synchrotron radiation (SR) emitted at large (γθ >> 1) angle by the beam line magnets, most notably the quadrupoles adjacent to the CLEO IR. The purpose of this note is to describe a simple algorithm to extend the validity of the classical SR approach to angles far exceeding the typical angle θ = 1/γ. The beamstrahlung detector could monitor instantaneously the beambeam interaction. By extracting information on the beam-beam overlap, it could drive a feedback system that maintains optimal beam-beam overlap. Gains in integrated luminosity of 10-20% at CESR could conceivably be had, with larger gains obtainable if the device can allow detailed studies of the beam-beam limit and further machine optimization[1]. Larger gains can be had at two-ring machines, such as PEP II, where machine optimization is far more complex. The problems tackled by this note are the following two. First, large angle radiation involves a coherent Fourier transform of the magnetic field of the entire magnet. Our current algorithms work by propagating a beam in small steps through magnetic fields, and adding the radiation from each step incoherently. Thus a method must be developed to simulate coherent radiation as the sum of many incoherent steps. Second, a cutoff in phase space must be established, below which the classical approximation is valid, and above which new formulae will be used. The method presented here is valid for one electron only. The complications of predicting the exact beam shape and trajectory through various