Astra bunch 0.4 nC, at z=1.45 m
current / A
20
I A
bunch current
15
10
5
calculation with
107
particles
0
-6
-4
-2
0
2
4
6
-3
x 10
“slice 1”
(~1keV energy spread, I=17.6 A)
4
2
(slice energy - 4999749.7291) / eV
x 10
E
0
“slice 2”
(small energy spread, I=13.1 A)
s

 E0 eV
slice energy
-2
-4
-6
-6
-4
-2
0
2
4
6
-3
x 10
slice energy spread / eV
 E eV
2500
2000
slice energy
spread
1500
1000
500
0
-6
-4
-2
0
2
4
6
-3
x 10
“slice 1”
~1keV energy spread, I=17.6 A
0.3mm, about 110E6 particles
compression: 17.6A --> 110A --> 1200A
“slice 2”
~250eV energy spread, I=13.1 A
0.3mm, about 82E6 particles
compression: 13.1A --> 85.5A --> 944A
4
4
x 10
x 10
0
0
-5
-5
-10
-6
-4
-2
0
2
4
6
8
-10
-6
-4
-2
0
2
4
6
-3
x 10
4
3
3
2
2
1
1
0
-8
-6
-4
-2
0
2
4
6
8
-3
x 10
x 10
4
x 10
8
-3
x 10
0
-8
-6
-4
-2
0
2
4
6
8
-3
x 10
slicing and densification:
4
x 10
0
-5
-10
-6
-4
-2
0
2
4
6
8
criticism:
this introduces new
noise as it ignores
relaxation effects (due
to plasma oscillations)
-3
4
3
x 10
x 10
Astra distribution with 10E6 particles, but 0.4 nC /e = 2.5E9
2
we need 250 particles
per Astra-particles
 1D model starting
after gun (z=0.5m)
1
procedure: select
all-6particles
in the0 given
slice,
0
-8
-4
-2
2
4
6
8
extract correlations with z (to 3rd order) x 10
keep information about linear z-energy correlation
throw away z-information
multiply (replicate x 250) and mix the 5D-distribution
use random generator for z-coordinates
regenerate linear z-energy correlation
-3
full shot noise simulation with Xtrack starts after solenoid field (z=1.45 m)
Xtrack simulations
“slice 1” (~1keV energy spread, 110E6 particles) compression: 17.6A --> 110A --> 1200A
“slice 2” (~250eV energy spread, 82E6 particles) compression: 13.1A --> 85.5A --> 944A
“slice 1”long.(~1keV
energy spread, 110E6 particles) compression: 17.6A --> 110A --> 1200A
position and energy
energy /eV
5
1.5
x 10
5
25.90m E=1.450e+08 per=4.7613e-05
2
x 10
3
after BC2
1
5
49.30m E=2.975e+08 per=4.7373e-05
after ACC2
x 10
67.30m E=4.500e+08 per=4.7358e-05
before BC3
2
1
1
0.5
0
0
0
-1
-1
-0.5
-2
long. phase space (full period)
-1
-2
-1
0
1
bunch coordinate / m
4
5
x 10
-2
2
-2
-1
0
1
-5
-5
x 10
x 10
rms, total=5829.4 slice center=502.635
6
long. p.s. without z-correlation
-3
2
4
4
3
2
2
0
1
-2
4
x 10rms, total=12363.8742 slice center=8476.8337
-2
-1
0
1
2
E,tot  13.5 kEV E  9.6 kEVx 10
-5
5
4
x 10rms, total=13544.4661 slice center=9584.5669
0
0
-2
-1
0
1
-4
2
-2
-1
0
1
-5
x 10
5
2.28
x 10
-5
2
-2
-1
0
1
-5
5
2.26
x 10
-5
x 10
plasma oscillation
relative fluctuation=0.012741
2
x 10
5
relative fluctuation=0.0073061
2.26
x 10
relative fluctuation=0.0069003
long. density
2.26
2.24
2.24
2.22
2.22
2.22
2.2
2.2
2.2
2.18
2.18
2.24
2.18
2.16
2.14
-2
-1
0
1
2.16
2
-5
x 10
-2
-1
0
1
2.16
2
-5
x 10
-2
-1
0
1
2
-5
x 10
“slice 1” (~1keV energy spread, 110E6 particles) compression: 17.6A --> 110A --> 1200A
5
4
x 10
6
81.45m E=4.500e+08 per=5.0334e-06
1.5
after BC3
2
x 10
6
138.60m E=7.000e+08 per=5.019e-06
3
before F2kicker
190.70m E=7.000e+08 per=4.3331e-06
after F2seed chicane
2
1
0
x 10
1
0.5
-2
0
0
-4
-1
-0.5
-6
-8
-10
2
-2
-1
-2
-1
0
1
E  127 kEV
-1.5
2
-3
-2
-1
0
1
-6
-6
x 10
5
x 10rms, total=127436.338 slice center=4496.4511
-4
2
x 10
1
6
x rms,
10 total=415780.1531 slice center=380204.9082
-2
-1
Flash2
3
0
1
E  1.3 MeV
2
-6
x 10
6
xrms,
10 total=1269178.3162 slice center=1043434.5292
2
0
0.5
1
-2
0
-4
-0.5
-6
-1
0
-1
-2
-8
-2
-1
0
1
I  70 A
5
2.5
x 10
-1.5
2
-3
-2
-1
0
1
-6
-1
0
1
5
2.5
x 10
x 10
5
relative fluctuation=0.076546
5
2.4
2
-6
x 10
relative fluctuation=0.083756
2.3
-2
-6
x 10
2.4
-4
2
x 10
relative fluctuation=0.50467
I  600 A
4
2.3
3
2.2
2.2
2.1
2
2.1
2
1.8
1
2
1.9
-2
-1
0
1
1.9
2
-6
x 10
-2
-1
0
1
0
2
-6
x 10
-2
-1
0
1
2
-6
x 10
1d-models
effective impedance (long. impedance transversely averaged)
tracking from 0.4m (direct after gun) to 150m (before Flash1/Flash2)
1d-particles (non-linear, real noise)
longer sample (~ 1.1mm)
distribution from random generator
very high bandwidth
LGM (linear gain model)
frequency domain
integral equation
1d particles
random generator: uniform in z, gaussian in energy
“slice 1”
~1keV energy spread, I=17.6 A
400E6 particles, about 1.1 mm
compression: 17.6A --> 110A --> 1200A
“slice 2”
~250eV energy spread, I=13.1 A
398E6 particles, about 1.1 mm
compression: 13.1A --> 85.5A --> 944A
distribution at z =150 m
Current
Current
1400
1500
1200
1000
1000
3 m  200 m/Ctot
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
500
7 m  500 m/Ctot
0
0.5
1
x 10
x 10
6
5
6
Distribution at end
x 10
5
Distribution at end
x 10
0
0
-5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-5
0
0.5
1
x 10
x 10
6
6
Distribution at end without chirp
x 10
5
x 10
Distribution at end without chirp
0
0
-2
1.5
-5
-5
2
1.5
-5
-5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-5
bunch coordinate / m x 10
slice rms  120 keV
total rms  0.6 MeV
-5
0
0.5
1
1.5
bunch coordinate / m x 10-5
slice rms  250 keV
total rms  1.2 MeV
“slice 1”, compression: 17.6A --> 110A --> 1200A
5
Current Rms
150
8
BC2
BC3
Energy Rms
x 10
6
100
4
50
2
0
0
50
100
150
0
beamline coordinate / m
127 keV
0
50
log scale
4
10
100
150
100
150
beamline coordinate / m
log scale
2
10
4
0
10
10
-2
10
0
50
100
150
0
50
“slice 2”, compression: 13.1A --> 85.5A --> 944A
3D particle tracking
5
Current Rms
250
15
Energy Rms
x 10
200
10
150
100
5
50
0
253 keV
0
50
100
150
log scale
4
10
0
0
50
100
150
100
150
log scale
6
10
2
10
4
10
0
10
-2
10
0
50
100
150
0
50
spectra: comparison 1d particles, LGM
start
direct before BC2
Gain and (noise ratio)0.5 for  E=250eV, I0=13.1A
 = 30 m
Gain and (noise ratio)0.5 for  E=1000eV, I0=17.6A
3
10
3
10
2
2
10
10
1
1
10
10
0
0
10
10
-1
10
11
10
“slice 2”
 = 30 m
“slice 1”
-1
12
13
10
/mm  1.5
10
 2
14
10
10
11
10
/mm  0.15
effect of plasma oscillations
these oscillations are frozen in ACC1
yes: original white noise is reduced (in some spectral ranges)
12
10
13
10
 2
14
10
spectra: comparison 1d particles, LGM
direct after BC2
direct before BC3
Gain and (noise ratio)0.5 for  E=250eV, I0  85.5A
 = 30 m
Gain and (noise ratio)0.5 for  E=1000eV, I0  110A
3
10
3
10
2
?
2
10
10
1
1
10
10
0
0
10
10
-1
10
11
10
“slice 2”
 = 30 m
“slice 1”
-1
12
10
13
10
 2
14
10
10
11
10
12
10
13
10
 2
unexpected growth of energy spread
 needs further investigations
14
10
spectra: comparison 1d particles, LGM
direct after BC3
at 150m
Gain and (noise ratio)0.5 for  E=250eV, I0  944A
 = 30 m
Gain and (noise ratio)0.5 for  E=1000eV, I0  1200A
3
10
3
10
2
before bc2
after bc2
before bc3
after bc3
at 150m
2
10
10
1
1
10
10
0
0
10
10
-1
10
11
10
“slice 2”
 = 30 m
“slice 1”
-1
12
10
13
10
 2
14
10
10
11
10
12
10
huge disagreement due to strong non-linear effects
even for “slice 1”, but this is not very obvious from the plot
13
10
 2
14
10
energy spread versus initial energy spread
“slice 1”
“slice 2”
but with slightly different optic
but for 1.2 GeV
Erms vs initial E rms
5
10
x 10
8
8
6
6
4
4
2
2
0
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
log scale
6
10
“heating”
before BC2
after BC2
before BC3
after BC3
end
10
2000
3000
4000
5000
6000
7000
8000
9000 10000
log scale
before BC2
after BC2
before BC3
after BC3
end
5
10
4
3
10
2
2
10
1000
10
conservation
of phase space
3
0
6
C2
4
10
x 10
10
C3
5
10
Erms vs initial E rms
5
10
10
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
rms current fluctuations spread versus initial energy spread
“slice 1”
“slice 2”
but with slightly different optic
but for 1.2 GeV
Irms vs initial E rms
Irms vs initial E rms
250
250
200
200
150
150
100
100
50
50
0
0
1000
2000
3000
4000
5000
6000
7000
8000
0
9000 10000
0
1000
2000
3000
4000
6000
7000
8000
9000 10000
log scale
log scale
2
before BC2
after BC2
before BC3
after BC3
end
10
1
10
0
2
before BC2
after BC2
before BC3
after BC3
end
10
1
10
0
10
10
-1
-1
10
10
-2
10
5000
-2
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
10
0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
ACC1, slice E-spread
q = 1nC, Xtrack
cavity amplitudes =
[1.00,1.00,1.00,1.00, 1.00,1.00,1.00,1.00]
[0.75,0.75,0.75,0.75, 1.25,1.25,1.25,1.25]
case 100,100
case 75,125
optics with matching:
hor and vert beta functions, tracked and design
60
50
40
horizontal/vertical
solid: case 100,100
dashed: case 75,125
30
20
10
0
2
4
6
8
10
12
14
16
18
20
slice energy spread
after ACC1
at 19.8m (after ACC39), with matching
0.012
1:1
0.75 : 1.25
0.01
slice energy spread [MeV]
0.008
0.006
0.004
0.002
0
-6
-4
-2
0
s [m]
2
4
6
-3
x 10
“slice 2” (~250eV energy spread, 82E6 particles) compression: 13.1A --> 85.5A --> 944A
6
-1.55
x 10
6
25.90m E=1.450e+08 per=4.5949e-05
-2.2
after BC2
-1.6
x 10
6
49.30m E=2.975e+08 per=4.5709e-05
-2.9
after ACC2
-2.3
x 10
67.30m E=4.500e+08 per=4.5694e-05
before BC3
-3
-1.65
-3.1
-2.4
-1.7
-3.2
-2.5
-1.75
-3.3
-2.6
-1.8
-1.85
-2
-1
0
1
-2.7
2
-3.4
-2
-1
0
1
-5
-5
x 10
-1.685
6
x 10 rms, total=3825.0146 slice center=1194.869
-3.5
2
x 10
-2.35
6
x 10rms, total=21147.838 slice center=14990.0336
-2
-1
E,tot  23 kEV
-3.1
0
1
2
E  16.1 kEVx 10
-5
6
rms,
total=23076.5571 slice center=16108.9514
x 10
-1.69
-3.15
-2.4
-1.695
-3.2
-1.7
-2.45
-1.705
-3.25
-1.71
-2.5
-3.3
-1.715
-1.72
-2
-1
0
1
-2.55
2
-2
-1
0
1
-5
x 10
4
1.8
x 10
-3.35
2
-2
-1
0
1
-5
4
1.7
1.75
1.68
1.7
1.66
1.65
1.64
1.6
1.62
1.55
1.6
x 10
-5
x 10
plasma oscillation
relative fluctuation=0.028225
2
x 10
4
relative fluctuation=0.0095985
1.7
x 10
relative fluctuation=0.0090615
1.68
1.66
1.64
1.5
-2
-1
0
1
1.58
2
-5
x 10
1.62
-2
-1
0
1
1.6
2
-5
x 10
-2
-1
0
1
2
-5
x 10
“slice 2” (~250eV energy spread, 82E6 particles) compression: 13.1A --> 85.5A --> 944A
6
-1.5
x 10
6
81.45m E=4.500e+08 per=4.167e-06
-1.5
after BC3
-2
-2
-2.5
-2.5
-3
-3
-3.5
-3.5
-4
-4
x 10
6
138.60m E=7.000e+08 per=4.1529e-06
-1
before F2kicker
x 10
190.70m E=7.000e+08 per=3.4847e-06
after F2seed chicane
-2
-3
-4
-4.5
-2
-2
-1
0
E  253 kEV
1
2
-4.5
-2
-1
0
1
-6
-6
x 10
6
rms,
total=253051.2198 slice center=4329.1946
x 10
2
x 10
-1.5
6
rms,
total=272200.2074 slice center=70173.6747
x 10
-5
-1.5
-1
Flash2
-1
-0.5
0
0.5
1
E  407 keV
1.5
-6
x 10
6
x rms,
10 total=407245.0436 slice center=221224.5605
-2
-2.5
-2
-2.5
-3
-3
-3
-3.5
-3.5
-4
-4
-4.5
-4
-2
-1
0
1
I  15 A
4
1.75
x 10
2
-4.5
-2
-1
0
1
-6
2
-5
-1.5
-1
-0.5
0
0.5
1
-6
x 10
-6
x 10
4
relative fluctuation=0.016414
1.72
x 10
1.5
x 10
4
relative fluctuation=0.014115
2
relative fluctuation=0.14792
x 10
1.7
1.7
1.8
1.68
1.66
1.65
1.6
I  139 A
1.64
1.62
1.6
1.4
1.6
1.55
-2
-1
0
1
2
-6
x 10
1.58
-2
-1
0
1
2
-6
x 10
1.2
-1.5
-1
-0.5
0
0.5
1
1.5
-6
x 10
plasma oscillations
bunch coordinate
s μm
charge
density
plasma oscillation for typ=10m,   = 1E-6
 L/m
250
200
period of plasma oscillation / m
LINAC coordinate
Sp
17.6A, 10MeV
17.6A, 145MeV
110A, 145MeV
110A, 450MeV
1200A, 450MeV
1200A, 700MeV
150
100
50

0
-8
10
-7
10
-6
-5
10
10
/m
-4
10
-3
10