Quadrupoles, to Third Order

TL;DR

This paper derives the third order optics of quadrupoles, showing that third order aberrations are unaffected by fringe field shaping, enabling efficient nonlinear effect evaluation.

Contribution

It introduces a method to derive third order quadrupole optics that ignores fringe field derivatives, simplifying analysis and simulation.

Findings

Third order aberrations are insensitive to fringe field shaping.

Provides a method for quick evaluation of nonlinear effects.

Enables efficient third order tracking.

Abstract

I derive the third order optics of quadrupoles. I transform away the derivatives of the strength function, and thus demonstrate that third order aberrations are insensitive to fringe field shaping. The results can be used for efficient tracking to third order or for simple and quick evaluation of nonlinear effects.