1. TESLA Damping Ring: Injection/Extraction Schemes with RF Deflectors
D. Alesini, F. Marcellini

2. Summary CTF3-like Injection/Extraction scheme for TESLA DR
NO Bunch Length
2 (or 3) freq. in 4 (or 6) RF Defl. [2 (or 3) inj. + 2 (or 3) extr.]
2 (or 3) freq. in 2 RF Defl. [1 inj. + 1 extr.]
WITH Bunch Length
Effects of Errors in DR parameters
(phase advance between RF Deflectors, RF Amplitudes and Phase,...)

3. Injection scheme in the CR
CTF3 INJECTION SCHEME
The long bunch train with an intra-bunch distance of 20 cm is converted into a series of short bunch trains, with an intra-bunch distance of 2 cm. This is done in 2 steps: by a factor of 2 in the DL, then by a factor of 5 in the CR. DL
Beam axis
2nd Deflector
1st Deflector
Injection scheme in the CR

4. What we have done in Frascati
- BEAM DYNAMICS STUDY: BEAM LOADING IN THE RF DEFLECTORS
- RF DEFLECTORS DESIGN
What we have done in Frascati
1st turn - 1st bunch train from linac
Recomb.
in the CTF3
Prelim. phase
2nd turn
3rd turn
4th turn

5. CTF3-LIKE INJECTION/EXTRACTION SCHEME
(simple scheme)
LINAC TRAIN
Extraction
Injection
If the filling time (F) of the deflectors is less than TDR it is possible to inject or extract the bunches without any gap in the DR filling pattern.
 should be  * depending on the ring optics and septum position. Considering a single RF frequency 
 /MAX=1-cos(2/F)
Rec. factor

6. (first evaluations made by J.P. Delahaye)
IN THE TESLA CASE
(first evaluations made by J.P. Delahaye)
QUANTITY
SYMBOL
VALUE
LINAC
Number of bunches in the Linac NB
2820
Bunch time spacing in the Linac
TL
337 ns
Bunch spacing
LL= TL*c
101 m
Total length of the bunch train
LB= NB* LL
285 km DR
Recombination factor F 20
Total length of the DR
LDR = LB/F LL/F
14 km
Bunch spacing in the DR
LDR= LL/F
5 m
Bunch time spacing in the DR
TDR = TL/F
16.85 ns RF
DEFL.
Number of RF deflectors ND 2
(1 inj. + 1 extr.)
Freq. of the RF deflectors
fRF= n*1/ TL
1.3 GHz
(= 438*1/ TL)
Delta deflection between the extr./inj. bunches and the stored ones
*
0.6 mrad
Deflection of the extr./inj. Bunches
MAX
(= / (1-cos(2/F)))
12 mrad!!
/MAX  5% 1
F  ns !!
TW RF Deflectors
GAP
TESLA klystrons 2
inj./extr. with more RF frequencies near 1.3 GHz

7. 1
TW RF DEFLECTORS GAP DR
nB=1 1 bunch over 141  TG=337 ns

8. 2 = INJ./EXTR. WITH MORE RF FREQUENCIES NEAR 1.3 GHz (F=20)
 maximization of MAX
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 no bunch length
2 distant freq. case (every possible freq. in the considered range) =
2 close freq. case (frequencies differ for 1/ TL)
Extracted bunches
DEFLECTOR PARAMETERS
(Mode /2)
4 Deflectors (2 inj. + 2 extr.)
Cell dimensions
a = [mm]
b = [mm]
D = [mm]
t = [mm]
Defl 1  fRF1 = 432*1/ TL = [MHz]
Defl 2  fRF2 = 431*1/ TL = [MHz]
Total beam deflection = 1.36 [mrad]
Deflection defl.1 = 0.68 [mrad]
Deflection defl.2 = 0.68 [mrad]
MAX = 44 %
P = 9 [MW]
L = 1.51 [m]
F = 112 [nsec]
n. Cells/defl = 26
P = 5 [MW]
L = 2.03 [m]
F = 150 [nsec]
n. Cells/defl = 35

9. 2 Deflectors (1 inj. + 1 extr.)
1 RF Deflector powered with 2 (or 3 frequencies)
DEFLECTOR PARAMETERS
(Mode /2)
2 Deflectors (1 inj. + 1 extr.)
Defl 1  fRF1 = 432*1/TL = [MHz]
 fRF2 = 431*1/ TL = [MHz]
Total beam deflection = 1.36 [mrad]
Deflection fRF1 = 0.68 [mrad]
Deflection fRF2 = 0.68 [mrad]
P = + [MW]
1/*=95% (fRF1 = f*+1/2 TL)
2/* =95% (fRF2 = f*-1/2 TL)
L = 1.59 [m]
F = 118 [nsec]
n. Cells/defl = 28
P = + [MW]
1/*=90% (fRF1 = f*+1/2 TL)
2/* =90% (fRF2 = f*-1/2 TL)
L = 2.25 [m]
F = 167 [nsec]
n. Cells/defl = 39

10. DEFLECTOR PARAMETERS (/2) 6 Deflectors (3 inj. + 3 extr.)
3 Frequencies
maximization of MAX
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 no bunch length
3 distant freq. case 
3 close freq. case
MAX = 69 %
DEFLECTOR PARAMETERS (/2)
6 Deflectors (3 inj. + 3 extr.)
Defl 1  fRF1 = 433*1/ TL = [MHz]
Defl 2  fRF2 = 438*1/ TL = [MHz]
Defl 3  fRF3 = 443*1/ TL = [MHz]
Total beam deflection = 0.87 [mrad]
Deflection defl.1 = 0.29 [mrad]
Deflection defl.2 = 0.29 [mrad]
Deflection defl.3 = 0.29 [mrad]
P = 9 [MW]
L = 0.64 [m]
F = 48 [nsec]
n. Cells/defl = 11
P = 5.00 [MW]
L = 0.86 [m]
F = 64 [nsec]
n. Cells/defl = 15

11. DEFLECTOR PARAMETERS (/2) 2 Deflectors (1 inj. + 1 extr.)
maximization of MAX
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 no bunch length
3 distant freq. case 
3 close freq. case
IDEAL CASE
MAX = 67 %
DEFLECTOR PARAMETERS (/2)
2 Deflectors (1 inj. + 1 extr.)
Defl 1  fRF1 = 436*1/ TL = [MHz]
 fRF2 = 437*1/ TL = [MHz]
 fRF3 = 435*1/ TL = [MHz]
P = fRF1 + fRF2 + fRF3 [MW]
Total beam deflection = 0.91 [mrad]
fRF1= 0.31 [mrad]
fRF2= 0.3 [mrad]
Deflection @ fRF3= 0.3 [mrad]
MAX = 66 %
L = 0.69 [m]
F = 51 [nsec]
n. Cells/defl = 12
P = fRF1 + fRF2 + fRF3 [MW]
Total beam deflection = 0.92 [mrad]
fRF1 = 0.32 [mrad]
fRF2 = 0.3 [mrad]
fRF3 = 0.3 [mrad]
MAX = 65 %
L = 0.96 [m]
F = 71 [nsec]
n. Cells/defl = 17
REAL CASE:
w.p. with vphc 
 no perfect synchronism

12.  different recombination factors F
 2 or 3 freq. optimization in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 no bunch length
 different recombination factors F
2 distant freq.
2 close freq.
3 distant freq.
3 close freq.

13. FINITE BUNCH LENGTH
z=6 mm, the same 2 freq. optimized in the previous
case give:
Extracted bunch
1 = 9 %
New optimization procedure:
to increase 1
(if possible) to reduce the RF
slope over the bunch length
How to avoid the effect of the RF curvature on the extr. bunches

14. DEFLECTOR PARAMETERS (/2) 4 Deflectors (2 inj. + 2 extr.)
2 Frequencies
 maximization of 1
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 bunch length z=6 mm
2 distant freq. case 
2 close freq. case
DEFLECTOR PARAMETERS (/2)
4 Deflectors (2 inj. + 2 extr.)
Defl 1  fRF1 = 447*1/ TL = [MHz]
Defl 2  fRF2 = 432*1/ TL = [MHz]
Total beam deflection = 2.66 [mrad]
Deflection defl.1 = 1.33 [mrad]
Deflection defl.2 = 1.33 [mrad]
1 = 22.5 %
P = 9.00 [MW]
L = 2.9 [m]
F = 215 [nsec]
n. Cells/defl = 50
P = 5.00 [MW]
L = 3.97 [m]
F = 294 [nsec]
n. Cells/defl = 68

15. 2 Deflectors (1 inj. + 1 extr.)
 maximization of 1
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 bunch length z=6 mm
2 distant freq. case 
2 close freq. case
1 = 14 %
NO SOLUTION WITH
2 Deflectors (1 inj. + 1 extr.)

16. DEFLECTOR PARAMETERS (/2) 6 Deflectors (3 inj. + 3 extr.)
3 Frequencies
 maximization of 1
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 bunch length z=6 mm
3 distant freq. case 
3 close freq. case
DEFLECTOR PARAMETERS (/2)
6 Deflectors (3 inj. + 3 extr.)
Defl 1  fRF1 = 444*1/ TL = [MHz]
Defl 2  fRF2 = 437*1/ TL = [MHz]
Defl 3  fRF3 = 435*1/ TL = [MHz]
Total beam deflection = 1.05 [mrad]
Deflection defl.1 = 0.35 [mrad]
Deflection defl.2 = 0.35 [mrad]
Deflection defl.3 = 0.35 [mrad]
1 = 57 %
P = 9 [MW]
L = 0.78 [m]
F = 58 [nsec]
n. Cells/defl = 13
P = 5.00 [MW]
L = 1.04 [m]
F = 77 [nsec]
n. Cells/defl = 18

17. DEFLECTOR PARAMETERS (/2) 2 Deflectors (1 inj. + 1 extr.)
 maximization of 1
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 bunch length z=6 mm
3 distant freq. case 
3 close freq. case
IDEAL CASE
DEFLECTOR PARAMETERS (/2)
2 Deflectors (1 inj. + 1 extr.)
Defl 1  fRF1 = 435*1/ TL = [MHz]
 fRF2 = 436*1/ TL = [MHz]
 fRF3 = 434*1/ TL = [MHz]
1 = 40 %
P=9+9+9 [MW]
Total beam defl. = 1.57 [mrad]
fRF1= 0.57 [mrad]
fRF2= 0.5 [mrad]
Defl. @ fRF3= 0.5 [mrad]
MAX = 38 %
L = 1.26 [m]
F = 93 [nsec]
n. Cells/defl = 22
P=6+6+6 [MW]
Total beam defl. = 1.68 [mrad]
fRF1 = 0.67 [mrad]
fRF2 = 0.51 [mrad]
fRF3 = 0.51 [mrad]
MAX = 35 %
L = 1.83 [m]
F = 135 [nsec]
n. Cells/defl = 32
REAL CASE:
w.p. with vphc 
 no perfect synchronism

18. 6 Deflectors (3 inj. + 3 extr.)
 minimization of the bunch slope (with 1  30%)
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 bunch length z=6 mm
3 “distant” freq.
DEFLECTOR PARAMETERS
(Mode /2)
6 Deflectors (3 inj. + 3 extr.)
Defl 1  fRF1 = 442*1/ TL = [MHz]
Defl 2  fRF2 = 438*1/ TL = [MHz]
Defl 3  fRF3 = 435*1/ TL = [MHz]
Total beam deflection = 1.96 [mrad]
Deflection defl.1 = 0.65 [mrad]
Deflection defl.2 = 0.65 [mrad]
Deflection defl.3 = 0.65 [mrad]
1 = 31 %
P = 9 [MW]
L = 1.45 [m]
F = 108 [nsec]
n. Cells/defl = 25
P = 5.00 [MW]
L = 1.94 [m]
F = 144 [nsec]
n. Cells/defl = 34

19. HOW TO COMPENSATE THE DISTORTION DUE TO THE FINITE
BUNCH LENGTH IN THE EXTRACTION PROCESS

20. DEFLECTOR PARAMETERS (/2) 6 Deflectors (3 inj. + 3 extr.)
F=45  LDR6.3 Km
 maximization of 1
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 bunch length z=6 mm
3 distant freq.
1 = 30 %
DEFLECTOR PARAMETERS (/2)
6 Deflectors (3 inj. + 3 extr.)
Defl 1  fRF1 = 447*1/ TL = [MHz]
Defl 2  fRF2 = 440*1/ TL = [MHz]
Defl 3  fRF3 = 438*1/ TL = [MHz]
Total beam deflection = 2.02 [mrad]
Deflection defl.1 = 0.67 [mrad]
Deflection defl.2 = 0.67 [mrad]
Deflection defl.3 = 0.67 [mrad]
P = 9 [MW]
L = 1.5 [m]
F = 111 [nsec]
n. Cells/defl = 26
P = 5.00 [MW]
L = 2.00 [m]
F = 149 [nsec]
n. Cells/defl = 35

21. DEFLECTOR PARAMETERS (/2) 6 Deflectors (3 inj. + 3 extr.)
F=100  LDR2.85 Km
 maximization of 1
in the range [430*1/ TL 450*1/ TL] =1.276  GHz
 bunch length z=2 mm
3 distant freq.
1 = 28 %
DEFLECTOR PARAMETERS (/2)
6 Deflectors (3 inj. + 3 extr.)
Defl 1  fRF1 = 447*1/ TL = [MHz]
Defl 2  fRF2 = 440*1/ TL = [MHz]
Defl 3  fRF3 = 436*1/ TL = [MHz]
Total beam deflection = 2.16 [mrad]
Deflection defl.1 = 0.72 [mrad]
Deflection defl.2 = 0.72 [mrad]
Deflection defl.3 = 0.72 [mrad]
P = 9 [MW]
L = 1.6 [m]
F = 119 [nsec]
n. Cells/defl = 28
P = 5.00 [MW]
L = 2.15 [m]
F = 160 [nsec]
n. Cells/defl = 37

22. EFFECT OF ERRORS Possible Error sources in the
Injection/Extraction process
Errors in the Injection process can be damped
after some turns in the DR

23. First extracted bunches
EFFECT OF ERRORS IN THE EXTRACTION PROCESS
ERROR IN THE
PHASE ADVANCE 2-1
(nominal value 180 deg)
Example:
-2 distant freq. case
-F=20
-Ph. Adv. 1-2 = 150 and 200 deg;
-RF1=  RF2 =50 m;  RF1= RF2 =0
-Ph. Adv. 2-1 = 179 deg;
First extracted bunches
In each particular case it is possible to define this two quantities. The position (or angle) of the bunches at the extraction point are between xMAX and xMIN (or MAX and MIN)

24. CASE 1 -2 dist. freq. case compared with 3 dist. freq. case -F=20
-Ph. Adv. 1-2 = deg;
-RF1=  RF2 =50 m;  RF1= RF2 =0
-Ph. Adv. 2-1 = 179 deg;

25. CASE 2 -2 dist. freq. case compared with 3 dist. freq. case -F=20
-Ph. Adv. 1-2 = 200 deg;
-RF1=  RF2 =50 m;  RF1= RF2 =0
-Ph. Adv. 2-1 = deg;

26. RF AMPLITUDE (Deflector 1)
CASE 3
-2 dist. freq. case compared with 3 dist. freq. Case (F=20)
-Ph. Adv. 1-2 = 200 deg;
-RF1=  RF2 =50 m;  RF1= RF2 =0
-Amplitude variation VRF1
ERROR IN THE
RF AMPLITUDE (Deflector 1)
2 dist. freq.
3 dist. freq.

27. ERROR IN THE RF PHASE (Deflector 1)
CASE 4
-2 dist. freq. case compared with 3 dist. freq. Case (F=20)
-Ph. Adv. 1-2 = 200 deg;
-RF1=  RF2 =50 m;  RF1= RF2 =0
-Phase variation VRF1
ERROR IN THE
RF PHASE (Deflector 1)
2 dist. freq.
3 dist. freq.

28. 2 dist. freq. 3 dist. freq. CASE 5
-2 dist. freq. case compared with 3 dist. freq. Case (F=20)
-Ph. Adv. 1-2 = deg;
-RF1=  RF2 =50 m;  RF1= RF2 =0
-Amplitude VRF1=99% of the nominal value (each single freq.)
2 dist. freq.
3 dist. freq.

29. Conclusions Further analysis
The Injection/Extraction process in the DR with RF deflectors is feasible. In particular the cases of 2-3 RF frequencies and different recombination factors have been illustrated and discussed;
The problem of the finite bunch length has been discussed;
Best results have been obtained using 3 RF frequencies (eventually powering the same RF deflector);
TW RF Defl. 1.3 GHz reach the required performances in term of F and deflection;
The effects induced by errors in the DR parameters or RF Amplitude/Phase have been discussed. The study allows determining the tolerances in the power supply of the magnets or in the RF jitter in Amplitude and Phase.
Further analysis
-Effects of the RF Deflectors in the S.B./M.B. Beam dynamics of the DR;
(...)