Compensation of wake-field-driven energy spread in Energy Recovery Linacs

Energy Recovery Linacs provide high-energy beams, but decelerate those beams before dumping them, so that their energy is available for the acceleration of new particles. During this deceleration, any relative energy spread that is created at high energy is amplified by the ratio between high energy and dump energy. Therefore, Energy Recovery Linacs are sensitive to energy spread acquired at high energy, e.g. from wake fields. One can compensate the time-correlated energy spread due to wakes via energy-dependent time-of-flight terms in appropriate sections of an Energy Recovery Linac, and via high-frequency cavities. We show that nonlinear time-of-flight terms can only eliminate odd orders in the correlation between time and energy, if these terms are created by a beam transport within the linac that is common for accelerating and decelerating beams. If these two beams are separated, so that different beam transport sections can be used to produce time-of-flight terms suitable for each, also even-order terms in the energy spread can be eliminated. As an example, we investigate the potential of using this method for the Cornell x-ray Energy Recovery Linac. Via quadratic time-of-flight terms, the energy spread can be reduced by 66%. Alternatively, since the energy spread from the dominantly resistive wake fields of the analysed accelerator is approximately harmonic in time, a high-frequency cavity could diminish the energy spread by 81%. This approach would require bunch-lengthening and recompression in separate sections for accelerating and decelerating beams. Such sections have therefore been included in Cornell's x-ray Energy Recovery Linac design.


I. INTRODUCTION
Energy Recovery Linacs (ERLs) accelerate highcurrent particle beams to high energy in a linac. These are then used in x-ray [1,2], FEL [3] or nuclear physics [4,5] experiments. Subsequently, the beams are sent into the same linac at a decelerating phase to recover the particles' energy. This energy is then used to accelerate new bunches of particles [6]. Only with such energy-recycling does it become feasible to accelerate high-current beams to high energies in a linac. Today's high-current, highenergy beams are produced by storage rings, which reuse energetic electrons for millions of turns. The beam emittance is then limited by the equilibrium emittance that is established during these turns. ERLs use each bunch of electrons only once, and therefore have the potential of producing significantly smaller emittances.
One of the problems ERLs face is the wake-field-driven energy spread that builds up during a pass through the ERL. The energy spread at high velocity is relatively small, but relevant because it limits the bandwidth of the x-ray radiation. More importantly, it is multiplied during deceleration by the ratio between high energy and dump energy, which is approximately 500 in the case of the Cornell ERL. Decelerated bunches must be decoupled from accelerated bunches, via magnetic fields, in a demerger region before the dump. Here, particles with too large energy are not bent sufficiently to reach the dump; similarly, particles with too little energy hit the beam pipe before the dump, or are even decelerated to zero energy before leaving the superconducting linac, and are then lost in the cryogenic environment. Therefore, the energy spread that bunches have at the end of their deceleration has to be limited.
To illustrate the proposed techniques of wake-field compensation, we use the Cornell ERL, shown in Fig. 1. Bunches enter the ERL via the injector at (1). They are accelerated first by linac A at (2) and then by linac B at (4) after traversing the turn-around loop connecting the linacs at (3). They then traverse the Cornell Electron Storage Ring (CESR) at (6), producing x-rays in beamline sections (5) and (7). Returning to linac A, the bunches are decelerated, passed through the turn-around loop, and further decelerated by linac B. Finally, the demerger leads decelerated bunches into the beam dump at (8).
One can describe the propagation of a bunch's longitudinal phase-space distribution through the ERL via transformations for separate ERL components. In the Cornell ERL, shown schematically in Fig. 2, the bunch center is to be the design particle with t 0 = 0 and energy E c 0 . The bunch first traverses linac A, which has fre- quency ω, mapping the initial longitudinal phase-space position {t 0 , E 0 } for each particle to Here, the time coordinate after linac A is chosen so that the bunch center arrives at t 1 = 0; similarly, we choose t i = 0 at the bunch center for each other section. ∆E A is the energy the linac adds to the bunch center. To limit the peak accelerating voltage ∆EA e cos(φA) , φ A will be kept below 15 o . Next, the particles traverse the first turnaround loop with a time-of-flight mapping where the energy of the bunch center is E c 1 = E c 0 + ∆E A . Subsequently, linac B applies another accelerating field of frequency ω to the bunch, mapping {t 2 , E 2 } to Also φ B is limited to 15 o . Next, we add half the accumulated effect of all wake fields to E 3 : The particles then traverse CESR, with a time of flight mapping {t 4 , E 4 } to Next, the second half of the accumulated effect of wake fields is added: Now, the bunch returns to linac A, the second turn-around loop, and linac B, which apply Eqs. (1), (2) and (3) to yield where φ ′ A = φ A + ωδt CE , with φ ′ B = φ B + ω(δt CE + δt TA2 ). In this state, the bunch leaves the ERL for the dump. While energy recovery of ∆E A in linac A demands that the bunch center return to linac A after an odd multiple of half the RF period, a possible deviation δt CE is included above. Similarly, the decelerated beam might require a slightly different time than the accelerated beam to pass from linac A to linac B, which is described by δt TA2 . TABLE I: Sources and magnitudes of wake-driven energy spread [7]. Listed magnitudes are to be multiplied by the bunch charge of 77pC.

III. COMPENSATION METHODS FOR WAKE DRIVEN ENERGY SPREAD
A. Time of flight for wake correction A bunch's correlated energy spread can be reduced by decreasing its slope and curvature in time-energy phase space. This can be done by accelerating the bunch offcrest in the linacs, choosing an energy-dependent time of flight, and decelerating the bunch off-crest. We sketch this procedure in Fig. 5. There, arrows show the phasespace motion of a particle. Curve 1 shows the initial bunch profile; curve 2 shows the bunch after acceleration by the linacs; curve 3 shows the bunch after application of time-of-flight terms; curve 4 shows the decelerated bunch, where the bunch's initial curvature, i.e. secondorder correlation between time and energy, is eliminated. The arrows follow one particle of the bunch through this process.

Linear time of flight for linear-wake correction
We first eliminate the average slope ∆E9 of the phase-space bunch profile before the dump, where T is chosen to be six times as large as σ t , the rms temporal bunch length. Eliminating ∆E 9 may not necessarily decrease the energy spread, but serves to symmetrize the wake for higher-order compensation. We minimize |∆E 9 | using time-of-flight terms. Additionally, T 56 TA1 can be used to minimize the energy spread |∆E 4 | in CESR. This minimization has to be done numerically, and we start with the T 56 TA2 that eliminates the phasespace slope dE9 dt9 at the bunch center before the dump. We obtain dE9 dt9 from with dti dt0 and dEi dt0 from where we have chosen dE0 dt0 = 0 for phase-space distributions that enter the linac without linear time-energy correlation. Here, describes the transfer matrices in longitudinal phase space;

Nonlinear time of flight for nonlinear-wake correction
With the average phase-space slope of the bunch profile before the dump eliminated, the next step is to reduce the phase-space curvature d 2 E9 dt 2 9 at the bunch center before the dump. Unfortunately, minimizing the absolute curvature at the bunch center does not necessarily minimize the total energy spread. Therefore, we again use numerical minimization, which we start with analytically determined parameters that eliminate d 2 E9 The following iteration determines d 2 zi where H[f (x, y)] is the Hessian matrix of f (x, y); and the d zi dt0 are determined by Eq. (11). Equation (22) leads to being the curvature in longitudinal phase space with which the bunches enter the linac. We here assume that the bunch enters the ERL without time-energy correlation; i.e.
The Hessian matrices evaluated at the bunch center are zero except for

CESR time of flight
As explained in Fig. 5, a first-order energy correlation has to be admitted in CESR to influence the curvature of the longitudinal distribution. Consequently, the bunch entering CESR has a large energy spread, which undesirably broadens the bunch's x-ray spectrum, rendering it unfeasible to use time-of-flight terms in CESR for wake compensation.

Common turn-around loop for two ERL beams
In this section, we choose φ ′ A = φ A and φ ′ B = φ B for balanced acceleration and deceleration in each linac. Then, for the reference energy of the turn-around loop to be the same for both accelerating and decelerating beams, The slope, dE4 dt4 , at the bunch center in CESR is to be eliminated to have small energy spread for x-ray experiments. Consequently, dE4 dt4 = 0 and so Eq. (10) gives the slope at the bunch center in the second pass through the turn-around loop as dE7 dt7 = ω∆E A tan(φ A ) + W ′ (0) 2 . Eq. (10) also gives the slope at the bunch center in the first pass through the turn-around loop, dE1 dt1 = −ω∆E A tan(φ A ), and the beam therefore enters the turn-around loop with roughly opposite slopes on each pass; dE1 dt1 ≃ − dE7 dt7 . As a result, the net curvature change in the turnaround is zero, as explained in Fig. 6. First, linac A accelerates off-crest to produce a correlated energy spread here with positive slope. Second, the turn-around loop adds a second order time-of-flight shift, here to the right and thus effectively to the bottom of the bunch. In the second pass through the turn-around loop, the slope is negative, so that the second order time-of-flight terms shift effectively to the top of the bunch, compensating the curvature change in the first pass. The curvature from the first turn-around loop is counterbalanced by the curvature from the second turn-around loop. Evidently, this method cannot change even-order time-energy correlations. Only odd-order time-energy correlations at the dump can be eliminated when dE4 dt4 is to be eliminated.

Separate turn-around loops for each ERL beam.
In the previous two sections, we have seen how longitudinal phase-space curvature reduction can diminish the bunch energy spread, but can be feasibly carried out in neither CESR nor a single turn-around loop. However, passing the bunch through different turn-around loops before and after CESR enables such curvature reduction.
In this section, we choose φ ′ A = φ A and φ ′ B = φ B for balanced acceleration and deceleration in each linac. The variable T 56 TA1 is chosen to eliminate the slope, dE4 dt4 from Eq. (10), at the bunch center in CESR; and T 56 TA2 is chosen to eliminate dE9 dt9 at the dump; Similarly, T 566 TA1 will be used to eliminate the curvature, d 2 E4 dt 2 4 from Eq. (27), at the bunch center in CESR, and T 566 TA2 to eliminate that at the dump, leading to This choice compensates the curvature of the wake, as well as the curvature of the RF cosine wave. For the Cornell ERL, W ′ (0) ≃ 0, simplifying Eqs. (39) and (40). The part in Eqs. (39) and (40) that remains when setting W ′ (0) = W ′′ (0) = 0 corrects the curvature of the acceleration and deceleration cosine function [8]; the rest is responsible for correcting the curvature of the wake fields. The phases φ A and φ B can be chosen independently. Subsequently we choose φ B so that T 56 TA1 = T 56 TA2 = 0 when no wake field is present; i.e. ∆E A tan(φ A ) = ∆E B tan(φ B ).
We then numerically find the combination of T 56 TA1 and T 566 TA1 that minimizes the maximum energy difference between bunch particles at CESR, starting with Eqs. (37) and (39). Subsequently, we numerically find the combination of T 56 TA2 and T 566 TA2 that minimizes the maximum energy difference between bunch particles at the dump, starting with Eqs. (38) and (40). As an example we use ∆E A = ∆E B = 2495MeV and φ A = −φ B = 15 o . Figure 7 shows the best result, which has been obtained for T 56 TA1 = 4.4 × 10 −5 psMeV −1 , T 566 TA1 = FIG. 7: Bunch profiles after nonlinear-wake correction with separate turn-around loops for each ERL beam. Left ordinate gives energy at CESR while right ordinate gives energy at dump. Black-top: Cosine-like correlated longitudinal phase space from accelerating on crest with a σt = 2ps bunch length. Blue-middle: Longitudinal profile after suffering half the Cornell ERL's wake field. Red-bottom: Longitudinal profile at dump with residual energy spread due to higher-order correlations. 7.7 × 10 −4 psMeV −2 , T 56 TA2 = −5.6 × 10 −3 psMeV −1 , T 566 TA2 = 1.5 × 10 −3 psMeV −2 and T 56 CE = T 566 CE = 0, with the maximum energy difference between bunch particles decreasing by 66%, from 3.2 to 1.1Mev. The residual energy spread is due to higher-order terms in W (t). Hence, we acquire a large decrease in energy spread, without having had to send the bunch into CESR with an undesirable energy spread.
B. Harmonic-wake correction

High-frequency cavity
Instead of attempting to minimize phase-space curvature at the bunch center before the dump by nonlinear time-of-flight terms, one could add energy to the bunch to flatten its longitudinal phase-space distribution. We simulate this by inserting a cavity with period T approximately six times as large as σ t , the rms temporal bunch length, immediately after CESR. T is chosen to ensure the cavity is a multiple of the linac frequency, so subsequent bunches have the same phase at the cavity. The cavity maps the phase-space position after CESR, {t 6 , E 6 } from Eq. (6), to so t 6 and E 6 in Eq. (7) become t ′ 6 and E ′ 6 respectively. In Eq. (41), t * 1 and ∆E 1 are determined from Fourier analysing W (t): We again choose φ ′ A = φ A and φ ′ B = φ B for balanced acceleration and deceleration in each linac, and we choose φ B to satisfy Eq. (37), which eliminates the slope at the bunch center in CESR.
Using a single turn-around loop, we numerically find the combination of T 56 TA ≡ T 56 TA1 ≡ T 56 TA2 , t * 1 and ∆E 1 that minimizes the maximum energy difference between bunch particles at the dump, starting with Eq. (42). As an example we use ∆E A ≃ ∆E B ≃ 2495MeV and φ A = 10 o , which is non-zero because there has to be a slope in longitudinal phase space to eliminate the first-order correlation via T 56 TA . Figure 8 shows the best result, which has been obtained for t * 1 = −9.0 × 10 −2 ps, ∆E 1 = 1.5MeV, T 56 TA = −1.3 × 10 −3 psMeV −1 , φ B = −9.9 o , and T 566 TA1 = T 566 TA2 = T 56 CE = T 566 CE = 0, with the maximum energy difference between bunch particles decreasing by 81%, from 3.2 to 0.60MeV. The residual energy spread is due to higher harmonics in W (t). The decrease in energy spread is larger than that obtained using nonlinear-wake correction, indicating the effectiveness of harmonic correction.
Although the decrease in energy spread is substantial, the required cavity frequency 2π T ≃ 80GHz is too high to be economically feasible. The energy that has to be added to the bunch center is approximately 2MV, for a beam current of 0.1A. While power sources of up to 40GHz could be made available, a power of 200kW seems unfeasible. An X-band power sources at approximately 11.7GHz might be feasible, but the bunch length would have to be increased to do harmonic-wake correction with this lower frequency.

Bunch expansion and recompression
a. Common turn-around loop for two ERL beams. Reducing the frequency to 11.7GHz requires expanding the 6σ t of the bunch by 7 from 12ps to 84ps. Because a longer bunch acquires more energy spread in the cosinelike RF field, we should recompress the bunch immediately after the 11.7GHz cavity.
Resizing the bunch in CESR requires energy spread in CESR, and is therefore unfeasible. Using the turnaround loop allows for the compensation of energy spread due to φ A via a suitable φ B . Here we simulate a cavity in where t * 1 and ∆E 1 are determined from Eq. (42), with W (t) replaced by the difference between E 7 and E (0) 7 , which is E 7 obtained from setting W (t) = 0. Here, T is approximately six times the new rms temporal bunch length. T is chosen to ensure the cavity frequency is an odd multiple of the linac frequency, so the bunch has the opposite phase on both passes through the cavity. The second half of the turn-around loop maps where we choose T 56 ′ TA = −T 56 TA to correct the bunch length change from the first half of the turn-around loop. The numerical minimization will lead to a deviation of these two time-of-flight terms in order to find the smallest energy spread. For simplicity, we assume T 566 ′ TA = −T 566 TA = 0, and do not use time-of-flight terms in CESR. Eqs. (45) and (46) imply that t i and E i in Eqs. (3) and (9) become t ′′ i and E ′′ i respectively, for i = 2, 8.
We again choose φ ′ A = φ A and φ ′ B = φ B for balanced acceleration and deceleration in each linac, and φ B to eliminate the slope at the bunch center in CESR.
To determine the T 56 TA suitable for this method, we obtain the expansion factor dt8 dt7 = dt 8 dt 0 dt 7 dt 0 , using dt8 dt0 and dt7 dt0 from Eq. (11). Then, at the bunch center, where W (t) and ∆E 1 have been set to zero. To expand the decelerating beam by a factor of 7, we require T 56 TA = − 6 ω∆EB tan(φB) . During the first pass through the turn-around loop, we obtain with ∆E A tan(φ A ) = −∆E B tan(φ B ). The bunch is therefore expanded in the high-frequency cavity during both passes through the turn-around loop. We then numerically find the combination of T 56 TA ′ , t * 1 and ∆E 1 that minimizes the maximum energy difference between bunch particles at the dump, starting with Eq. (42). As an example we use ∆E A ≃ ∆E B ≃ 2495MeV, φ A = 15 o . Figure 9 shows the best result, which was obtained for t * 1 = 5.9 × 10 −2 ps, ∆E 1 = 0.14MeV, T 56 TA ′ = −1.1psMeV −1 and φ B = −15 o , with the maximum energy difference between bunch particles decreasing by 78%, from 3.2 to 0.72MeV. This result is decent, but the method is unfeasible because the bunch is over-compressed during the first pass, and therefore has to pass a point where it is fully compressed to very short length. At this point, Coherent Synchotron Radiation (CSR) damage to the beam can be very severe.
b. Separate turn-around loops for each ERL beam. The problem of over-compression and therefore very short bunches with strong CSR effect can be overcome by passing the bunches through different turn-around loops before and after CESR, such that only one of the turnaround loops carries out harmonic-wake correction. The harmonic correction is performed in the turn-around loop carrying the bunch after CESR, while the turn-around loop carrying the bunch before CESR has T 56 TA1 chosen according to Eq. (37) to eliminate the slope at the bunch center in CESR. The same expansion factor, dt8 dt7 = 7, and parameter choices, with the exceptions of t * 1 = −0.61ps, ∆E 1 = 0.21MeV, allow us to decrease the maximum energy difference between bunch particles by 78%, from 3.2 to 0.69Mev. The result is shown in Fig. 10.

IV. CONCLUSION
We have investigated the potential of using time-offlight terms in ERL loops to reduce wake-driven correlated energy spread. As an example, we have used the Cornell ERL. One could use the time of flight in the ERL return pass that contains the x-ray sources, but this method requires undesirable correlated energy spread in this pass. We found that time-of-flight terms in the linac that are common to the accelerating and the decelerating beams cannot compensate even orders in the time-energy correlation. However, when the accelerating and the decelerating beams are separated in the linac, time of flight for the two passes through the linac can be used to reduce the energy spread in the return loop, as well as in the beam dump. The correlated energy spread is reduced by 66% in our example.
Furthermore, we have looked into the potential of us- ing high-frequency cavities to reduce the energy spread. We find that placing an 80Ghz cavity after CESR could eliminate the first harmonic of the energy spread, and decreases the energy variation by 81% in our example. Unfortunately, a cavity with such a high frequency is unfeasible, necessitating bunch-lengthening to permit a lower cavity frequency. The method we have analysed involves expanding the bunch in the first half of the turn-around loop, correcting the first harmonic of the time-expanded energy spread in a cavity, then recompressing the bunch in the second half of the turn-around loop. Separate turnaround loops would be needed to avoid strong CSR forces during overcompression, yielding a 78% decrease in en-ergy variation in our example.
Since the wake-induced energy spread is better described by a sinusoid than a parabola, harmonic correction is somewhat more effective than nonlinear time-offlight terms at minimizing the energy spread at the dump. But bunch compression and decompression, as well as a section with high frequency structures, are hard to devise. In both cases, two turn-around loops are required, and have thus been included in the Cornell ERL design.
FIG. 11: Results. Black-top: Longitudinal profile at dump without wake-correction. Blue-middle: Dump profile with harmonic wake-correction. Red-middle: Dump profile with nonlinear time-of-flight wake-correction. Harmonic wakecorrection reduces energy spread more but is less feasible than nonlinear wake-correction.