Analytical expressions of transfer functions for a hard edge dipole magnet using a basic geometrical approach

In charged particle accelerators, higher order optics studies become important from various points of view, such as dynamic aperture, emittance dilution, beam loss, etc. For some new applications, nonlinear study has become important in single pass optics also. For studying the higher order optics, each magnetic element is represented by a higher order transfer function (map, i.e., a function that relates output coordinates of a trajectory with initial coordinates and momentum deviation). Here in this paper we have provided an alternate method to obtain the analytical formulation of the transfer function for a dipole magnet. This formulation is obtained on the basis of basic geometrical analysis and is exact up to all orders under hard edge approximation. Being an analytical expression, the estimation of higher order effects of dipole magnet can be studied quickly. For checking the correctness of this formulation, we separated terms up to third order, which can be used to obtain the standard transfer matrices up to same order. An example of emittance growth and bunch length modification for a C-chicane-type electron beam bunch compressor is provided using the analytical expression. DOI: 10.1103/PhysRevSTAB.16.014001