Optics modules for circular accelerator design

This paper is intended as a comparison paper to “Circular Machine Design Techniques and Tools” presented at this conference by Roger Servranckx. The intent here is to provide a tutorial discussion on the basic optics of circular particle accelerators for the benefit of those readers who have a fundamental knowledge of charged particle optics but do not make it their profession to design particle accelerators. We begin the tutorial by presenting the solutions of the first-order differential equations of motion for a single particle in a closed circular machine introducing the concepts of phase shift, beta functions, and the Courant-Snyder invariant. From these solutions we derive the transfer matrix between two points in the machine as a function of the phase shift and the parameters contained in the Courant-Snyder invariant. We then introduce typical optical building blocks (modules) used in circular machine designs and relate them to their characteristic transfer matrix elements, the phase shift through them, and the Courant-Snyder-Twiss parameters, β, α, and γ. Next we discuss the systematics of some elementary phase ellipse matching problems between optical modules. The report ends with a discussion of second-order optical modules and how they are used to provide the momentum bandwidth needed for the design of a typical circular machine.