Achromatic dissipative focusing

The energy loss due to synchrotron radiation during beam-beam collisions can be exploited to obtain an achromatic focusing lens. The authors consider the successive collisions between a relativistic electron and two short bunches with the same number of particles: the first is a bunch of electrons and the second of positrons. Owing to the energy loss during the first collision, the two successive radial kicks do not compensate each other and, in thin lens approximation, there is a residual focusing effect which is dependent of the initial electron energy. A proper choice of the radial density profile of the two bunches leads to a linear achromatic focusing. This principle could be applied to reduce the final spot size in future electron-positron colliders with very large energy spread