# Energy and RF Cavity Phase Symmetry Enforcement in Multi-turn ERL Models

## TL;DR

This paper discusses methods for enforcing phase symmetry in RF cavities within multi-turn energy recovery linacs, simplifying optimization and ensuring energy recovery and target beam energies.

## Contribution

It introduces symmetric modeling approaches for ERL cavities that reduce optimization complexity and demonstrate their application to the CBETA accelerator.

## Key findings

- Symmetric models satisfy energy and cavity loading targets.
- Symmetry reduces the number of parameters in optimization.
- Application to CBETA shows potential for symmetric operation.

## Abstract

In a multipass energy recovery linac (ERL), each cavity must regain all energy expended from beam acceleration during beam deceleration, and the beam should achieve specific energy targets during each loop that returns it to the linac. For full energy recovery, and for every returning beam to meet loop energy requirements, we must specify and maintain the phase and voltage of cavity fields in addition to selecting adequate flight times. These parameters are found with a full scale numerical optimization program. If we impose symmetry in time and energy during acceleration and deceleration, fewer parameters are needed, simplifying the optimization. As an example, we present symmetric models of the Cornell BNL ERL Test Accelerator (CBETA) with solutions that satisfy the optimization targets of loop energy and zero cavity loading. An identical cavity design and nearly uniform linac layout make CBETA a potential candidate for symmetric operation.