Courant-Snyder formalism of longitudinal dynamics

X J Deng,A W Chao,C X Tang,W H Huang

doi:10.1103/physrevaccelbeams.24.094001

Abstract

Originating from the stochastic nature of the photon emission process, the diffusion of the electron longitudinal coordinate exists even if the global phase slippage of a storage ring is zero, as we cannot zero all the local phase slippages simultaneously. This quantum diffusion is viewed as the most fundamental limit of the lowest bunch length realizable in an electron storage ring from the single-particle dynamics perspective. Here, we present an analysis of this effect using the Courant-Snyder parametrization in the longitudinal dimension. Analytical formulas for the longitudinal emittance, energy spread, and bunch length are derived. The same formalism is used to discuss the application of multiple radio frequency systems for longitudinal strong focusing. The presented work is expected to be useful in the development of novel light source mechanisms like steady-state microbunching, where a short bunch length or small longitudinal emittance is needed.

Highlights

Short bunches and small longitudinal emittance are desired by synchrotron light source and lepton collider communities for coherent radiation generation and luminosity increase
From the single-particle dynamics perspective, there is a fundamental effect limiting the lowest bunch length realizable in an electron storage ring originating from the stochasticity of photon emission time, which was first originally analyzed by Shoji et al [2,3]
This stochasticity results in a diffusion of the electron longitudinal coordinate z even if the global phase slippage of the ring is zero, as we cannot make all the local or partial phase slippages zero simultaneously

Summary

INTRODUCTION

Short bunches and small longitudinal emittance are desired by synchrotron light source and lepton collider communities for coherent radiation generation and luminosity increase. From the single-particle dynamics perspective, there is a fundamental effect limiting the lowest bunch length realizable in an electron storage ring originating from the stochasticity of photon emission time, which was first originally analyzed by Shoji et al [2,3]. This stochasticity results in a diffusion of the electron longitudinal coordinate z even if the global phase slippage of the ring is zero, as we cannot make all the local or partial phase slippages zero simultaneously. The application of multiple radio frequency (rf) cavities, or laser modulators in an SSMB storage ring, for longitudinal strong focusing are discussed using the same formalism with important observations made

COURANT-SNYDER FORMALISM

Classical jηj scaling pffiffiffiffiffi

Discussions

Campbell’s theorem

Analysis

SUMMARY