Analytic descriptions of soft-edge quadrupoles for beamline design in multi-particle simulations

All physical quadrupoles have a fringe field falloff, giving rise to intrinsic higher order aberrations. It is especially important to consider these aberrations when designing beamlines with strong, longitudinally cramped optics used to capture and transport large emittance beams. This paper presents relatively simple analytical formulae which describe the fields of quadrupoles including their higher order aberrations for implementation in beamline design. We review the analytic model for quadrupoles described by a $\mathrm{sech}^2$ strength function and subsequently expand it to quadrupoles whose strength function instead follows a tanh function, adding the effective length of the quadrupole as a free parameter. The implementation of these analytic field descriptions in envelope code TRANSOPTR and multi-particle codes GEANT4 and FLUKA is presented using the optics design of the DarkLight experiment at TRIUMF as example.