Intrinsic Third Order Aberrations in Electrostatic and Magnetic Quadrupoles

TL;DR

This paper rederives third order intrinsic aberration formulas for electrostatic and magnetic quadrupoles using Hamiltonian formalism, comparing their effects and discussing improvements in transport code handling of fringe fields.

Contribution

It introduces a Hamiltonian-based derivation of aberration formulas and compares electrostatic and magnetic quadrupoles, highlighting situations where each performs better.

Findings

Electrostatic quadrupoles can have lower aberrations in certain conditions.

Magnetic quadrupoles outperform electrostatic ones in other scenarios.

Current transport codes have pitfalls in handling quadrupole fringe fields.

Abstract

Intrinsic aberrations are those which occur due to the finite length of the desired field configuration. They are often loosely ascribed to the fringing field. This is misleading as it implies that the effects can be minimized by shaping the fields. In fact, there is an irreducible component related to the broken symmetry. It is present even in the hard-edge limit, and moreover, the other (soft-edge) effects can be simply ascribed to the intrinsic aberration spread over a finite length. We rederive the aberration formulas for quadrupoles using a Hamiltonian formalism. This allows for an easy comparison of electrostatic and magnetic quadrupoles. For different combinations of large and small emittances in the two transverse planes, it is found that in some situations electrostatic quadrupoles have lower aberrations, while in others, magnetic quadrupoles are better. As well, we discuss the ways in which existing transport codes handle quadrupole fringe fields. Pitfalls are pointed out and improvements proposed.