Theory of Beam Echoes

TL;DR

This paper develops a comprehensive theoretical framework for beam echoes in circular accelerators, analyzing effects of nonlinear kicks, momentum spread, and multiple kicks, with extensions to 2D and longitudinal cases.

Contribution

It introduces detailed expressions for echo amplitude and pulse width under various conditions, including nonlinear effects and multiple kicks, extending existing theories.

Findings

Derived formulas for echo amplitude and pulse width.

Analyzed effects of nonlinear quadrupole and dipole kicks.

Explored enhancement of echo amplitude with multiple kicks.

Abstract

We develop the theory of beam echoes in circular accelerators under several different conditions. We derive detailed expressions for the echo amplitude and pulse width with nonlinear quadrupole and dipole kicks, first without and then with momentum spread. We use the theory with the linearized dipole and quadrupole kicks to solve the diffusion equation for different dependencies of the diffusion coefficient on the action. We then consider the use of multiple quadrupole kicks to increase the maximum echo amplitude. We have extended these calculations partially to the 2D case and we also have partial results for longitudinal echoes.