At the interaction point of a storage ring collider each beam is subject to perturbations due to the electromagnetic field of the counter-rotating beam. For flat beams, a well known approximation models the beam by a current sheet which is uniform in the horizontal plane, restricting the particle motion to the vertical direction. In this classical model a water-bag beam distribution is used to find working points and beam-beam tune shift parameters which lead to a stable beam distribution. We try to find stability criteria for a more realistic Gaussian equilibrium distribution. In order to analyze the instabilities, a linearized Vlasov equation is solved computing radial and angular modes to first order in the displacement from the design trajectory.

We survey the range of techniques for numerical simulation of the beam-beam interaction in circular colliders in which synchrotron radiation is present. These include techniques used in weak-strong, quasi-strong-strong, and strong-strong simulations. Self-consistent strong-strong simulations usually use macroparticle-in-cell methods with a variety of refinements to increase the computational speed. Other approaches include macroparticle sampling methods and numerical solution of the Vlasov equation. We also describe and compare the existing beam-beam simulation codes for circular lepton colliders.

We report results of calculations and measurements relating RF phase noise to longitudinal motion of a stored beam. Treating the beam as a noise driven coupled oscillator system, we have made calculations to determine what coupled bunch synchrotron oscillation amplitudes result from RF phase noise. Measurements have also been carried out at CESR of phase noise in the RF system and coupled bunch synchrotron oscillation amplitudes. We also consider the impact of this noise on the dynamic range of a longitudinal feedback system.

Synchrotron radiation in electron-positron colliders and synchrotron light sources ejects photoelectrons from the beam chamber walls. The resulting cloud of slow photoelectrons interacts with the beam, and may generate a transverse coupled bunch instability. In some high current storage rings a photoemission effect causes the dominant transverse instability, with a sub-millisecond risetime. Two types of photoemission instabilities have been observed. In CESR, photoelectrons are trapped by the combination of a quadrupole electrostatic field from the distributed ion pumps and the field of the bending magnets. This "trapped photoelectron instability" has a very long range, is nonlinear in beam current, and is predominantly horizontal. In the Photon Factory and BEPC, photoelectrons moving freely across the chamber interact with the positron beam to produce a transverse instability. This "free photoelectron instability" occurs for bunch-to-bunch intervals smaller than the transit time (a few tens of nanoseconds) of photoelectrons across the chamber, occurs in the absence of external fields, is approximately linear in bunch current, and is predominantly vertical.

We have developed a time domain transverse feedback system with the high bandwidth needed to control transverse instabilities when the CESR ee collider is filled with trains of closely spaced bunches. This system is based on parallel digital processors and a stripline driver. It is capable of acting on arbitrary patterns of bunches having a minimum spacing of 14 ns. Several simplifying features have been introduced. A single shorted stripline kicker driven by one power amplifier is used to control both counter-rotating beams. The desired feedback phase is achieved by sampling the bunch position at a single location on two independently selectable beam revolutions. The system adapts to changes in the betatron tune, bunch pattern, or desired damping rate through the loading of new parameters into the digital processors via the CESR control system. The feedback system also functions as a fast gated bunch current monitor. Both vertical and horizontal loops are now used in CESR operation. The measured betatron damping rates with the transverse feedback system in operation are in agreement with the analytical prediction and a computer simulation developed in connection with this work.

We have developed a time domain transverse feedback system with the high bandwidth needed to control transverse instabilities when the CESR e/sup +/e/sup -/ collider is filled with trains of closely spaced bunches. This system is based on parallel digital processors and a stripline driver. It is capable of acting on arbitrary patterns of bunches having a minimum spacing of 14 ns. Several simplifying features have been introduced. A single shorted stripline kicker driven by one power amplifier is used to control both counter-rotating beams. The desired feedback phase is achieved by sampling the bunch position at a single location on two independently selectable beam revolutions. The system adapts to changes in the betatron tune, bunch pattern, or desired damping rate through the loading of new parameters into the digital processors via the CESR control system. The feedback system also functions as a fast gated bunch current monitor. Both vertical and horizontal loops are now used in CESR operation. The measured betatron damping rates with the transverse feedback system in operation are in agreement with the analytical prediction and a computer simulation developed in connection with this work.

An anomalous damping or growth of transverse coupled bunch modes has long been observed in CESR. The growth rates and tune shifts of these modes are a highly nonlinear function of current. The effect is associated with the operation of the distributed ion pumps, as it disappears when the pumps are not powered. We show that this effect can be explained by the presence of electrons trapped in the CESR chamber by the field of the dipole magnets and the electrostatic leakage field of the distributed ion pumps. Photoelectrons are introduced into the chamber by synchrotron radiation and can be ejected from the chamber by the passage of an e/sup +/ or e/sup -/ bunch. The transverse position of the beam thus modulates the trapped photoelectron charge density, which in turn deflects the beam, creating growth or damping and a tune shift for each coupled bunch mode. Predictions of the dependence of growth rate and tune shift on bunch current, bunch pattern, and mode frequency by a numerical model of this process are in approximate agreement with observations.