Edge Focusing

Beam transport matrix elements describing the linearly falling fringe field of a combined function bending magnet are expanded in powers of the fringe field length by iteratively solving the integral form of Hill´s equation. The method is applicable to any linear optical element with variable focusing strength along the reference orbit. Results for the vertical and horizontal focal lengths agree with previous calculations for a zero gradient magnet and an added correction to the dispersion is found for this case. A correction to the fringe field gradient peculiar to a combined-function magnet with strong edge focusing is also found. The influence of edge focusing on the tunes and chromaticities of the NSLS rings is described. The improved chromaticity calculation for the booster was of particular interest since this ring has bending magnets with poletips shaped to achieve small positive chromaticities.