Introduction to Optics Design

TL;DR

This work presents a comprehensive introduction to linear optics design for accelerators through a symplectic, matrix-based framework. It formalizes particle motion with 6D canonical coordinates and a transfer matrix $M$, derives invariants such as the Courant-Snyder invariant $J_{CS}$ and the rms emittance $\epsilon_{rms}$, and connects matrix methods to Hill's equation $x''(s)+K(s)x(s)=0$. The text develops multiple factorizations of the One-Turn-Map $M_{OTM}$ (Diagonal, R, Twiss) to extract stability and optics functions, and extends the discussion to ensembles via the sigma matrix and matched-beam concepts. It also covers dispersion, chromaticity, closed-orbit computation, and the transition from single-particle dynamics to beam statistics, providing a solid foundation for linear optics codes and accelerator lattice design with practical computational tools. Significance lies in delivering a rigorous, implementable framework that underpins both theoretical understanding and practical optimization of accelerator lattices, enabling robust linear-optics and setting the stage for nonlinear extensions.

Abstract

This lecture provides an overview of the principles and methodologies involved in linear optics design. It aims to introduce key concepts such as the matrix formalism, the symplecticity, the quantities that are preserved in the single particle evolution, e.g., the Courant-Snyder invariant. It also covers the concept of beam emittance and matching conditions for an ensemble of particles. The goal is to equip readers with the foundational tools needed for both theoretical understanding and practical application in linear optics and accelerator design.

Figures (7)

• Figure 1: Reference systems and reference orbit, closed orbit and betatron oscillations.
• Figure 2: Examples of orthogonal and symplectic transformations.
• Figure 3: Similarity of two One-Turn-Map matrices referred to two different points $s_0$ and $s_1$.
• Figure 4: Betatron oscillations from $s_1$ to $s_2$ in the physical and normalized trace-spaces (referred as $X$ and $\bar{X}$, respectively). In this example we assumed a tune of $\mu_{OTM}/2\pi=1/8$ and a phase advance between $s_1$ and $s_2$ of $\Delta\mu=\pi/2$. To be noted that all the transformations between these 4 trace-spaces are symplectic, therefore the $4\times8$ positions represented in the figure have the same $J_{CS}$. The number of the markers indicates the turn number.
• Figure 5: The trace-space of an ensemble of particles.