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

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.