Suppression of radiation excitation in focusing environment

Radiation damping and quantum excitation in an electron damping ring and a straight focusing channel are reviewed. They are found to be the two limiting cases in the study of a general bending and focusing combined system. In the intermediate regime where the radiation formation length is comparable to the betatron wavelength, quantum excitation can be exponentially suppressed by focusing field. This new regime may have interesting applications in the generation of ultra-low emittance beams