Studies on a Triple-Turn Energy-Recovery Mode at the S-DALINAC

The electron accelerator S-DALINAC at TU Darmstadt was successfully operated in single- and double-turn energy-recovery mode. The latter was realized using a shared beam-transport where two beams are superimposed in the first recirculation beamline. Due to its current design, the S-DALINAC can be upgraded with reasonable effort to be operated in triple-turn energy-recovery mode with shared beam-transport. Here, two beams are superimposed in both the first and the second recirculation beamline. This mode is particularly challenging since it does not allow for on-the-fly beam tuning compared to a triple-turn energy-recovery mode with individual beam-transport. Therefore, the triple-turn energy-recovery mode requires precise determination of the accelerator setup obtained from beam-dynamics simulations prior to beam-tuning. First results of the necessary beam-dynamics simulations for this mode are presented.


Introduction
The economic performance of a linear accelerator (LINAC) can be significantly increased if the energy required to accelerate particles is recovered afterwards by efficient deceleration [1].The recovered energy is used to accelerate subsequent electrons, reducing the external radio frequency (RF) power required.For a given external RF power supplied, orders of magnitude higher beam currents, I, can be achieved using an energy-recovery LINAC (ERL) compared to a conventional LINAC.The superconducting Darmstadt linear electron accelerator (S-DALINAC, Fig. 1) [2] at Technische Universität Darmstadt is capable to be operated in several energyrecovery modes.By using superconducting cavities, the recovered energy can be efficiently stored in the electric field of the cavities, enabling low-loss reuse.The design of the S-DALINAC provides several operation modes; a single-turn as well as a double-turn ERL have been realized previously [3,4].By increasing the turn number, the very same LINAC can be used to increase the maximum beam energy, E. In this way, multi-turn ERLs enable significant increase of beam power, P , since P ∝ E × I. Here, costs can be saved if the costs of recirculation beamlines are less than costs for longer or more powerful LINACs to reach the same maximum beam energy with less turns.In these studies, a potential realization of the S-DALINAC as a triple-turn ERL is addressed to investigate special challenges of this mode.Multiple passing of the main LINAC is enabled due to the three recirculation beamlines.At maximum energy, the electron beam will be collided with the laser beam of the LCB source.Afterwards, a phase shift of ca.180 • is induced and the electron beam is decelerated to injection energy by again passing the main LINAC thrice.Picture taken from [4] and modified.

Current accelerator layout and required upgrades
Multi-turn acceleration as well as energy recovery requires returning the beam back to the accelerator, which is done by using recirculation beamlines.The S-DALINAC has three recirculation beamlines (see Fig. 1), which provides a conventional quadruple acceleration mode and -with a small upgrade as described below -triple-turn energy-recovery mode.
Multi-turn acceleration requires the time of flight, t rec,a , through a recirculation beamline connecting two acceleration steps, a, to be an integer multiple, N a , of the oscillation period of the alternating electric fields inside the cavities, t RF , plus a small offset time, t off,a .That is The offset time is used to take speed differences between electrons and speed of light into account as well as to influence the longitudinal phase space in a sophisticated way by causing off-crest acceleration.In order to decelerate the beam after the last acceleration step, a phase shift of ca.180 • relative to the alternating electric fields has to be induced.This is done by providing for the time of flight, t rec,b , through the recirculation beamline connecting the last acceleration with the first deceleration, b, with an associated offset time, t off,b .For further deceleration, the already induced phase shift has to be kept which is why the time of flight, t rec,d , through a recirculation beamline connecting two deceleration steps, d, must fulfill with an associated offset time, t off,d .Beamlines for recirculation after acceleration and deceleration can be either identical (shared beam-transport) or separated (individual beam-transport) [5].Due to its layout, the S-DALINAC can only be operated in the shared beam-transport model.
The third recirculation beamline (see Fig. 1) can currently be adjusted to fulfill the relation (1) to provide a conventional quadruple acceleration mode.By upgrading the path length adjustment system (PLAS) of this recirculation beamline, the relation (2) can also be fulfilled, which enables a different single-turn energy-recovery mode as well as a triple-turn energy-recovery mode.Both of these modes enable larger electron beam currents for the Laser-Compton Backscattering (LCB) source that is currently under construction in the third recirculation beamline [6].

Beam-dynamics simulations
Using the shared beam-transport model does not allow for on-the-fly tuning of the cavities and PLAS per recirculation beamline.Furthermore, sophisticated values for the longitudinal dispersion should be set per arc section in order to influence the longitudinal phase space properly.Therefore, beam-dynamics simulations prior to beam-tuning are necessary to find optimum values for these parameters [7,8].Similar to [4], the simulations have been performed using the tracking software elegant [9] with an extended version of the code that was already used to realize the double-turn energy-recovery mode at the S-DALINAC [10].
In the beam-dynamics simulations, the problem min f (p i (A 1 , ..., A 8 , ϕ 1 , ..., ϕ 8 , L F , L S , L T ), p 0,i , ϵ) subject to (A 1 , ..., A 8 , ϕ 1 , ..., ϕ 8 , L F , L S , L T ) ∈ M was solved first to find a solution ensuring that the centroid momenta, pi , after each main-LINAC pass, i, differ from the target momenta, p 0,i , by no more than ϵ (see Table 1 for values).Here, A 1 to A 8 are the peak on-axis longitudinal electric fields inside the main LINAC's cavities, ϕ 1 to ϕ 8 are the phases of these electric fields, and L F , L S and L T are the path length adjustments in the first, second and third recirculation beamline, here specified such that L = 0 corresponds to a circumference (the length of the main LINAC plus the length of a recirculation beamline) that is an integer multiple of 100 mm which is the wavelength of the alternating electric fields.These degrees of freedom can be freely selected within their individual technical limits, here indicated by the bounded set M (see Table 1).The function f is a penalty function defined by which provides an intrinsic stop criterion for the algorithm solving the problem once the objective function has a value of zero.
For the simulations, an initial beam with momentum spread σ δ = 6.5 × 10 −4 and bunch length σ t = 0.36 ps was used.These parameters are identical to those used in [4] which allows a comparison to the double-turn ERL.
Starting from a solution of the above mentioned problem, the first-order longitudinal dispersions of the injector arc, R 56,I , of the first recirculation beamline, R 56,F , of the second recirculation beamline, R 56,S , and of the third recirculation beamline, R 56,T , were varied according to [4,10] to find an optimum momentum spread.Technical limits for the R 56 values are listed in Table 1.
The results of the beam-dynamics simulations for the triple-turn ERL show that the centroid momenta reach the target momenta after each main-LINAC pass within a precision of ϵ = 630 eV/c (see Fig. 2).Furthermore, the momentum spread is small at the intended location for the interaction of the electron beam and the laser beam of the LCB source.Although a certain electron bunch length was not an objective in these simulations, it still has a suitable small value for the interaction of the electrons with the laser beam.Additionally, the electrons' momentum spread is sufficiently small along the entire accelerator so that the beam is kept within the momentum acceptance.The minimum ϵ is 630 eV/c in the case of the present results for the triple-turn ERL, compared to 1 eV/c for the double-turn ERL [4].For the tripleturn ERL, a higher value results due to the increased number of restrictive conditions ('reach p 0,5 and p 0,6 ') compared to the double-turn ERL; although there is one additional degree of freedom (L T ) the same precision as in [4] can not be achieved.In contrast, significantly lower values for the momentum spread at the intended interaction point (IP) can be achieved in the case of the triple-turn ERL compared to the double-turn ERL, since an additional degree of freedom to influence this quantity has been added (R 56,T ); due to a higher number of 'acting' dispersion terms and main-LINAC passes upstream of the intended IP, a more sophisticated influence of the longitudinal phase space can be exerted.
To set up the cavities of the main LINAC, j, the on-crest and off-crest momentum gains per cavity for the first main-LINAC pass, ∆p on-crest,j and ∆p off-crest,j , are needed [4].The resulting values are listed in Table 1.

Figure 1 .
Figure1.Picture of the S-DALINAC.After generating and boosting the electron beam in the injector section, it is accelerated to maximum energy by passing the main LINAC thrice.Multiple passing of the main LINAC is enabled due to the three recirculation beamlines.At maximum energy, the electron beam will be collided with the laser beam of the LCB source.Afterwards, a phase shift of ca.180 • is induced and the electron beam is decelerated to injection energy by again passing the main LINAC thrice.Picture taken from[4] and modified.

Figure 2 .
Figure 2. Simulated longitudinal quantities, starting after the injector LINAC.In these simulations, the effect of the longitudinal dispersion is implemented by elements of zero length at the end of the corresponding sections leading to abrupt changes of the bunch length.