Emittance compensation of elliptical beams

The technique known as emittance compensation was always considered in circular-symmetrical systems until now. It was owing to that (i) record brightness injectors were used for linacs only and could be made circular-symmetrical, that is without dipoles and quadrupoles, and (ii) the analysis of an elliptically-symmetrical system is much more complicated as it has two modes of charge oscillations that possess different wave numbers while the phases of both ones are to be appropriate to reach emittance compensation. Therewith, these record brightness injectors are necessary for energy recovery accelerators recently. The injection energy in these machines is not so high and an injection beamline always contains dipoles, so ellipticity should be taken into account while analyzing emittance compensation in these systems. An analytical description for the transverse dynamics of elliptically-symmetrical space-charge dominated beams is provided in the paper. A way to compensate both charge oscillation modes is pointed. A sample beamline containing an achromatic bend was optimized analytically and numerically.