Simple theory of emittance sharing and exchange due to linear betatron coupling

The influence of linear coupling due to skew quadrupoles on the transverse emittances is discussed using the simplified model of a smooth synchrotron with a constant-in-time skew gradient. New formulae for the coupled transverse emittances are derived that include the uncoupled emittances, the coupling strength (the tune difference on the coupling resonance) and the tune distance from the resonance. The classical formulae generally used only show a sharing of the emittances as coupling increases. For full coupling, i.e. on the resonance, a perfect sharing of the emittances is obtained. The new formulae reveal, in addition, the possibility to exchange the emittances as coupling decreases after the resonance crossing, as predicted and observed in the PSB by C. Carli.