K. Floettmann 30th Nov. 2009
XFEL Injector matching
The present XFEL injector foresees operation of the first four cavities of the first module to operate at > 12MV/m
Can we also operate at a higher gradient?
Yes, but….
K. Floettmann 30th Nov. 2009
Classical Brillouin flow
Cancellation of the linear defoucusing space charge by a solenoid field requires:
2
2 4z
rmse rms
eB Px
m c x
P = generalized perveance
0 3 3
30 0
2
4
A
eA
IP
I
m cI
e
K. Floettmann 30th Nov. 2009
Classical Brillouin flow
The beam size stays constant if the contribution of the emittance can be ignored and if
eq erms
z
m cx P
eB
is fulfilled.
Small mismatches lead to an oscillatory solution.
K. Floettmann 30th Nov. 2009
Generalized Brillouin flow
2 3 20
02 A
rms
C Ix x x x
I x
Envelope equation for a beam in an acceleration section (ignoring emittance contribution):
for a cavity, on crest, linear energy gain:
2
8C
0 z
K. Floettmann 30th Nov. 2009
Generalized Brillouin flow
2 2 2
2 2 2 2 20 0 0,
30
4 2 8 2rms rms rms rmsArms
Ix x x x
I x
Cancellation of space charge and focusing term requires:
2 2 00,rms rmsx x
2
2
2
3
4
rms rms
rms rms
x x
x x
K. Floettmann 30th Nov. 2009
Generalized Brillouin flow
It follows:0, 0 0
2
3 Arms
I
x I
0, 0,02
rms rmsx x
What happens in case of small mismatches?
2 3 20
02 A
rms
C Ix x x x
I x
Damped oscillator?
K. Floettmann 30th Nov. 2009
Generalized Brillouin flow
Invariant envelope (solid) and envelope with mismatch of initial angle (± 50%).
K. Floettmann 30th Nov. 2009
Generalized Brillouin flow
The solution describes not a static equilibrium but a dynamic equilibrium. Direct comparison of space charge and focusing force reveal that the focusing force would need to be a factor sqrt(3) larger for a static compensation.
The space charge field works against the incoming beam divergence.
K. Floettmann 30th Nov. 2009
A simple Model for emittance compensation
Development of the beam size in a drift with space charge. Beam parameters: , ,Dashed line = analytic approximation
10 710n 6(10, 5,1, 0) 10P
2
30
4rms
rmsrms rms
Px
x x
K. Floettmann 30th Nov. 2009
A simple Model for emittance compensation
Slice size development of the high perveance slice (solid line) and the low perveance slice (broken line) after a focusing kick from a numerical integration of the envelope equation.
K. Floettmann 30th Nov. 2009
A simple Model for emittance compensation
Slice size development of the high perveance slice (solid line) and the low perveance slice (broken line) after a focusing kick from a numerical integration of the envelope equation.
K. Floettmann 30th Nov. 2009
A simple Model for emittance compensation
x
x’
xh, rmsxl, rms
K. Floettmann 30th Nov. 2009
A simple Model for emittance compensation
Development of the beam emittance as calculated from the high perveance and low perveance slices. The broken line ignores the slice emittance, i.e. the slices are treated as lines.
K. Floettmann 30th Nov. 2009
A simple Model for emittance compensation
Emittance (solid line) and slice size (broken lines) in an accelerating section with
41
Emittance (solid line) and slice size (broken lines) in an accelerating section with:
82
K. Floettmann 30th Nov. 2009
Cross over trajectories
x
px
K. Floettmann 30th Nov. 2009
Cross over trajectories
Slice size development for the high perveance slice (solid line) and a slice without space charge (broken line).
Development of the slopes of the phase space ellipses for the high perveance slice (solid line) and a slice without space charge (broken line).
K. Floettmann 30th Nov. 2009
Cross over trajectories
Development of the weighted emittance as calculated from the high perveance slice and the slice without space charge.
K. Floettmann 30th Nov. 2009
Cross over trajectories
x x
x’ x’
Phase space ellipses at the first (left) and second (right) coincidence of the correlation axis.
K. Floettmann 30th Nov. 2009
XFEL Injector with 20MV/m in the Module
K. Floettmann 30th Nov. 2009
XFEL Injector with 20MV/m in the Module
K. Floettmann 30th Nov. 2009
XFEL Injector with 20MV/m in the Module
K. Floettmann 30th Nov. 2009
XFEL Injector with 20MV/m in the Module
K. Floettmann 30th Nov. 2009
XFEL Injector with 20MV/m in the Module
K. Floettmann 30th Nov. 2009
XFEL Injector with 20MV/m in the Module
K. Floettmann 30th Nov. 2009
XFEL Injector with 20MV/m in the Module
1.0
0.75