1. JLEIC CCR Path Length and Gap Formation
Jiquan Guo, Haipeng Wang
Rev Apr 5, 2018

2. CCR Definitions E1 K2 K1 Points: Harmonic Number of CCR NhCCR
K1: center of injection harmonic kicker
Circumference of CCR divided by βERLλERL
K2: Center of extraction harmonic kicker
In the current design βERLλERL≡ m constant for 20-55MeV cooler with 952.6MHz ERL cavities
E1: entrance of the ERL linac
Path lengths
Sometimes (if noted) Nh refer to the circumference divided by bunch spacing
L1: path length from K1 to K2
Frequency range 0.133MHz, β range
L2: K2 to K1
L3: E1 to K1
Number of CCR turns Nt
L4: K2 to E1
How many times a electron bunch passes through the cooling section before extraction
L1+L2 = CCR circumference

3. Requirement between NhCCR and Nt (1)
K2 timing is L2/ ahead of K1. Assume K2 turns on at t=(0+n×Nt)×Tbucket, K1 turns on at t=(L2/+n×Nt)×Tbucket, n=0,1,2,3…
A bunch injected at K1 at time L2/×Tbucket .
The bunch arrives K2 at NhCCR×Tbucket after 1st turn, and n×NhCCR×Tbucket after n turns.
When n×NhCCR is a multiple of Nt, the bunch is ejected
To let the bunch circulate Nt turns, Nt must be co-prime with NhCCR
If each bunch circulates Nt turns, and Tinj/Tbucket=Nt, the bunches will fill the CCR
Extracted bunches K2
Nh,CCR=20
Tinj/Tb=Nt=3
Tinj
Injecting bunches K1 Tb Tk

4. Requirement between NhCCR and Nt (1)
If Nh,CCR and Tinj/Tb have a max common factor M, the bunch can only circulate for Tinj/Tb/M turns; only 1/M of buckets can be filled (should avoid)
Extracted bunches
Nh,CCR=18
Tinj/Tb=3
M=3
Nt=1
Tinj
Injecting bunches Tb Tk

5. Nh,CCR and Nt, reduced reprate operation
To have both CCR and source running at reduced reprate (only 1/N of the buckets are filled, N=Tinj/Tk=Tbunch/Tbucket)
Nt=Tinj/Tbunch=Tkicker/Tbucket still needs to be co-prime of Nh,CCR
Nh,CCR needs to be multiple of all possible ratio of N; otherwise, if Nc is the max common factor of Nh,CCR and N, 1/Nc of the CCR buckets will be filled (factor of N/Nc more than required), but only during part (Nc /N) of the time
Extracted bunches
Nh,CCR=20
Tinj/Tbunch=Nt=3
Tinj/Tk= Tbunch/Tbucket =2
Tinj
Tbunch
Tbucket Tk

6. Nh,CCR and Nt, reduced reprate operation
It will be nice to choose a prime number for Nt, so it poses less restriction on the choice of NhCCR and N
Nt=11 was suggest by A. Hutton, previously thought about 20, 10, 12
Nt=11 for 952.6MHz makes cavities easier to build (than 476MHz), and makes future upgrade to 952.6MHz bunch rep-rate in the CCR possible

7. NhCCR vs Nt, exact cancelation of kicker slope
Y. Huang, Phys. Rev. Accel. Beams 19,
A circulating bunch kicked by K2 with slope needs to be compensated by K1 in the following kick with same slope and (2n+1)π betatron phase advance, assuming K1/K2 kicking direction is the same (both upward or both downward). Failure to match the sequence of the two kickers’ slope will result in significant emittance growth of the cooling electron beam.
NhCCR=M×Nt+Δ. Is there any restriction on Δ?
Figure above shows that the cases of Δ=1 and 3 will have perfect cancelation
Actually as long as we arrange the K2 kick out pulse right ahead of K1 kick in pulse, we will have the perfect cancelation.
For the kicker slope cancelation issue, no further restriction on choice of Δ if Nh,CCR and Nt=Tinj/Tb are co-prime

8. Ion Trapping For CW bunched beam
Providing typical CCR parameters, ion trapping will be a problem for CCR
𝑁 𝑏 =2× , 𝐿 𝑠𝑒𝑝 =0.63 to 2.52m, 𝛽 𝑥,𝑦 ≈12m, 𝜖 𝑔𝑥,𝑦 =10nm, 𝜎 𝑥,𝑦 ≈0.35mm
𝐴 𝑡𝑟𝑎𝑝 =0.04 ~ 0.16≪1
It seems like a gap is mandatory
Rui Li, Mar 2018 JLEIC coll meeting

9. Forming gaps in CCR for Ion Clearing etc
For each missing bunch in ERL accelerating beam, the CCR will keep the bunch missing for Nt=11 turns
Trying to match CCR gap with the ion ring gap is impossible and unnecessary
It’s possible to have a gap of ~100ns for each turn in CCR
The other option is to ramp down the ERL current to 0 in a long gap (needs a few ms, which is a few times of the total stored beam energy of ~50J divided by the ERL klystron power of ~100kW), possible to work if ion trapping accumulation time will be more than 100ms (probably not the case for CCR).

10. Forming a gap per turn in CCR
ERL’s 3.2nC charge per bunch is huge, and mismatch of one pair acc-deacc bunches in the bunch train might cause non-negligible gradient droop
If a ~100ns gap in the deacc bunch followed by one acc bunch, it must ends with an acc bunch following a deacc bunch, and all the following gaps should do the same. A single bunch mismatch will cause at least a difference of ~0.13% in the energy of the two acc bunch trains (twice the 0.06% energy spread specs); correcting this transient by feed-forward requires multi-MW level pulsed RF power, which is not practical
We must have a supercycle of injection bunch train with length in the range of the ERL path length  1 bunch spacing, so the acc/deacc bunch train will overlap in the ERL, which basically eliminates the transient beam loading in the ERL linac
ERL path length = Nt×(L1+L2)+L3+L4-L2 ≈ (Nt+1) ×(L1+L2) [~12×CCR circumference]
We need to have one gap for each injection bunch train from ERL (bunch train length ≈CCR circumference). If this gap is generated by laser, it will produce serious transient beam loading in the gun and the booster cavities.
We should either generate the gap by kicking some bunches out with harmonic kickers, or compensate the transient with RF feed-forward.
ERL linac starts with gaps from each ERL injection bunch train aligned in the CCR
NhCCR=M×Nt+Δ, Δ needs to be co-prime to Nt
Δ=1 gives perfect gaps, if bunch train for each turn of CCR injection has a length of M×Nt, and ERL path length range within (Nt+1) × M×Nt Nt
The case of Δ=2 will form a step in the gap; larger Δ will form multi gaps.

11. Gap forming with NhCCR=Δ+M×Nt, Δ=1
NhCCR=Δ+M×Nt=320, Nt=11, M=29, Δ=1
# of train 1 2 3 4 5 6 7 8 9 10 11
Start bucket
320
319
318
317
316
315
314
313
312
310
309
2nd bucket
3rd bucket 22 21 20 19 18 17 16 15 14 13 12 …
End bucket
264
274
273
272
271
270
269
268
267
266
265
Gap
Injector will have 43.3MHz (952.6/22) bunch trains, with 25 bunches and 4 empty buckets in each train
Assumes the gaps in the injector is periodical, i.e., for case of Nhccr=320, the period is 319 buckets (11×29). The supercycle will be 12×319 buckets
A sharp gap will be formed in such case

12. Gaps and ERL path length, NhCCR=Δ+M×Nt, Δ=1
Assumes Nhccr=320, and the CCR has 11 turns (plus ~1 turn for ERL)
To minimize transient beam loading, the 1st bunch train needs to return to the ERL cavity for de-acceleration at about the same timing of 13th bunch train for acceleration, and the difference has to be odd multiple of ½ wavelength of 31cm (1/4 CCR bunch spacing of 63cm), and smaller than 11 RF buckets of 952MHz, considering future upgrade of 952.6MHz operation.
As a result, the ERL path length (=11×L1+10×L2+L3+L4 ≈12×(L1+L2)) should be in the range of 7656(=638*12)10.5 wavelengths of 952.6MHz.
Pattern repeats every MHz cycles
~1 turn in CCR

13. Injected bunch time structure
Assumes CCR operates at 476MHz and the injection is at 43.3MHz
Forming the source bunch pattern for NhCCR=320, bunch rep-rate 43.3MHz
25 bunches
Period of 29
gap=4
T (in unit of ns)

14. Gap forming with NhCCR=Δ+M×Nt, Δ=2
# of train 1 2 3 4 5 6 7 8 9 10 11
Start bucket
332
330
328
326
324
322
320
318
316
314
312
2nd bucket
331
329
327
325
323
3rd bucket 22 20 18 16 14 12
2nd last
264
262
260
258
256
254
252
250
248
246
244
End bucket
275
273
271
269
267
265
263
261
259
257
255
Full gap , half gap ,
In general, the rise/fall time of the beam current in the CCR gap will be ∆ −1 ×𝑁𝑡×bunch spacing.

15. Reduced bunch rep-rate operation
We need to consider the case of reduced bunch rep-rate operation for JLEIC and the CCR
There might be 3 ways to generate reduce rep-rate bunches in the CCR with gaps
1. change the CCR circumference when we need to change the rep-rate, maintains Δ=1. For example, when CCR bunch rep-rate is at 119MHz, we can choose harmonic number of 78 (C= m); at 238MHz, we choose Nh=155 (C= m); at 476MHz, Nh=309 (C=194.45m)
2. choose the harmonic number to optimize the lowest rep-rate case (Nh=78/156/312), and live with the long rise time of the gap for 476MHz case (if we choose to remove 4 out every 28 bunches, we won’t have a 0 current gap)
3. Choose optimal harmonic number for higher rep-rate (476MHz), i.e. Nh=320. In this case, the periodicity of the bunch train of 319 buckets can’t be preserved for low rep-rate, but we still need to preserve the periodicity of every 11 bunch trains (3520 buckets)
Might be more difficult for generating the gap in the source/booster (laser rise/fall time, booster transient beam loading, etc). In the cases with periodicity of 28 buckets (of 43.3MHz) in the injector bunch train, we can generate CW beam in the source and transfer through the booster, use kickers to kick out 4 of every 28 bunches.

16. NhCCR=320 with reduced bunch rep-rate
Assumes we operate at 119MHz bunch rep-rate in CCR.
Injects bunches in 476/44MHz, or MHz, every ns.
Each injected bunch train has 6 or 7 bunches, and 1 empty bunch between bunch trains. Totally 25 filled bunches for every 4 bunch trains (2.679μs), in a pattern of 7, 6, 6, 6. 7 6 6 6 7 6 6 6 7 6 6 6
1 ERL supercycle
=3 laser cycles = 8.047μs
=ERL path length 10.5
=12×(CCR circumference-Δ)
Period of 29
gap=4

17. Possible NhCCR choices
Nh (476MHz)
C (m)
Adjustable, optimized for MHz differently
11*7*4+1~4
11*8*4+1~4
Optimized for 476MHz,
Will work in 119MHz/238MHz
320
201.37
11×29+1, 23×14-2
340
213.96
11×31-1, 4×5×17
364
229.06
11×33+1, 4×7×13
384
241.6
11×35-1, 128×3
Optimized at 119MHz, gap rise/fall time will be long for 238/476MHz
312
196.34
356
224.03
After discussion with Shukui about the capability to generate the laser pattern shown in page 16, I believe the NhCCR requirements 4N AND 11M1 will be the best choice, leaving us a few options, 320, 340, 364 and 384.
Gradient droop in the booster could still be challenging. The feed-forward needs to take action every ~0.7μs (with unequal spacing). However, the bunch pattern is also possible to be generated with CW beam in the booster if we have a kicker repeats at 1.49MHz and flat top of ~92.375ns