Fringe Effects in MAD PART I, Second Order Fringe in MAD-X for the Module PTC

The various versions of MAD contain a second order fringe field effect for the ideal bend. The expressions appeared in the famous SLAC-75 report where it is reported that they were derived by Hindmarsh and Brown in an unpublished work. We would like to rederive these expressions and check them. Moreover, we want to create maps for them which are not expansions around a “design orbit” but true operators valid around arbitrary orbits or at least a larger class of orbits. As in the original SLAC-75[1] report, our expression are expansions and are not meant to replace precise integration. We simply want a knob which is “PTC-compliant,” that is to say, exact in the Talman sense. Such PTC compliant maps have the virtue that they could be replaced by a thin exact “sandwich” made out of exact integrators. In fact the numerical checks of the SLAC-75 results, we precisely made out of such integrators.