Theory of Microwave Instability and Coherent Synchrotron Radiation in Electron Storage Rings∗

Bursting of coherent synchrotron radiation has been observed and in fact used to generate THz radiation in many electron storage rings. In order to understand and control the bursting, we return to the study of the microwave instability. In this paper, we will report on the theoretical understanding, including recent developments, of the microwave instability in electron storage rings. The historical progress of the theories will be surveyed, starting from the dispersion relation of coasting beams, to the work of Sacherer on a bunched beam, and ending with the Oide and Yokoya method of discretization. This theoretical survey will be supplemented with key experimental results over the years. Finally, we will describe the recent theoretical development of utilizing the Laguerre polynomials in the presence of potential-well distortion. This self-consistent method will be applied to study the microwave instability driven the impedances due to the coherent synchrotron radiation.