Analytical Considerations for Reducing the Effective Emittance with Variable Dipole Field Strengths

The basic optics design scope in lepton rings is to match the sections in either side of the bending magnets in order to minimise the equilibrium emittance. A further important emittance reduction can be achieved by incorporating dipoles for which the deflecting field varies along the electron beam path in the magnet. The figure of merit for such lattices when used in a synchrotron light source is the minimisation of the so-called effective emittance. The effective emittance is computed in the middle of the undulator straight section as the product of the rms size and divergence and therefore includes contributions from the betatron emittance and from the electron energy spread. In this paper, analytical formulas are obtained for the minimum betatron and effective emittance in arbitrary dipole fields and the associated optics functions at the dipole entrance. Examples are given for specific dipole field functions and their properties with respect to the effective emittance minimisation. Finally, the effective emittance is parameterised with respect to standard cell optics properties.