Emittance minimisation with longitudinal dipole field variation

The reduction of horizontal emittance beyond the conventional limit is pursued by introducing a longitudinal variation of the field in a bending magnet. For a given dipole field, the search for the minimal emittance is formulated analytically under different boundary conditions, starting from the achromat condition, preferred for light source rings. With a dipole field rapidly decaying along the longitudinal position, it is shown that the emittance can be further reduced, essentially by rendering the H function to be out of phase with the third power of the local curvature. Under the minimal emittance condition analytically obtained, the optimal dipole field distribution is searched numerically with a polynomial function, analysing at the same time the mechanism of the emittance reduction. The minimal emittance is argued as a function of the required peak field and the field distribution. Adaptability of longitudinally varying dipole fields in a standard magnet lattice for light sources is also addressed.