CSR computation with
arbitrary cross-section of
the beam pipe
Work in progress
L.Wang, H.Ikeda, K.Ohmi, K. Oide D. Zhou
3.14.2012
KEK Accelerator Seminar
Outline

Parabolic equation and general formulation of
the problem
 Illustration of the method: numerical examples
 Excise of micro-wave instability simulation
with Vlasov solver
 Conclusions
2
Motivation

Parabolic equation has been solved in FEL, CSR, and
Impedance calculations, etc

Most present codes are limited for simple boundary, for
instance, rectangular cross-section for CSR and zero
boundary for FEL (need large domain)

CSR is important for Super-KEKB damping ring. To
estimate the threshold of micro-wave instability
Impedance calculation

Gennady Stupakov, New Journal of Physics 8 (2006)
280
Axis ymmetric geometry
2D parabolic solver for
Impedance calculation

L. Wang, L. Lee, G. Stupakov, fast 2D solver (IPAC10)
0.5
2.5
0.4
Echo2, sigl=0.1mm
1.5
1
r (cm)
r (mm)
2
0.3
0.2
0.5
0.1
0
0
10
20
30
40
50
60
0
0
z (mm)
2
4
6
8
10
z (cm)
0.14
0.07
0.05
dot-lines: FEM code
0.1
0.04
0.03
0.02
0.01
0.08
0.06
0.04
0.02
0
0
-0.01
0
Real, ECHO2
-Imaginary, ECHO2
0.12
ReZ, ImZ (k )
ReZ, ImZ (k )
0.06
Real, ECHO2
Imaginary, ECHO2
Real, FEM code
Imaginary, FEM code
200
400
600
f (GHz)
800
1000
-0.02
0
200
400
600
f (GHz)
800
1000
FEL
FEL (for example, Genesis by sven reiche)
Set the field ZERO out the domain of interest
CSR
For example, CSR in bend magnet
(Tomonori Agoh, Phys. Rev. ST Accel. Beams 7, 054403 (2004)
•Agoh, Yokoya, PRSTAB 054403
•Gennady, PRSTAB 104401
•Demin, PH.D thesis
•K. Oide, PAC09
Nonlinear Optics
BUNCH LENGTH MEASUREMENTS WITH LASER/SR
CROSS-CORRELATION*
8
GENERALITY
IF We neglect the 1st term
Why Finite Element Method (FEM)?
Advantages of FEM
Irregular grids
Arbitrary geometry
Easy to handle boundary
Small beam in a large domain (FEL in undulator)
CPU (strongly depends on the solver)
Accuracy(higher order element, adaptive mesh, symmetry, etc)
Disadvantage & Challenge:
Complexity in coding (irregular grid, arbitrary geometry, 3D…)
•Arbitrary geometry of beam pipe
•Any shape of beam
Mesh of chamber & beam
Impedance of
Grooved surface
Adaptive method
Speedup and improve accuracy

CSR computation
Straight beam pipe+ Bend magnet + straight beam pipe…
Straight beam pip
Straight beam pipe
12
Assumptions of our CSR
problem

We assume perfect conductivity of the walls
and relativistic particles with the Lorentz
factor =
 The characteristic transverse size of the
vacuum chamber a is much smaller than the
bending radius R (a<<R)
 Constant cross-section of beam chamber
13
Fourier transform

The Fourier transformed components of the
field and the current is defined as
Where k=/c and js is the projection of the beam current onto
~r
s. The transverse component of the electric field E is a
~r
~r ~r

(
E
two-dimensional E
vector
.x , E y ) , The longitudinal

~r
component is denoted by E
s
14
CSR Parabolic Equation
(Agoh, Yokoya, PRSTAB 054403)
  2 2k 2 x 
 ~
e
   

  2ki E      n0

R 
s 
0

 2 Eb 
E  Er  Eb
  2 2k 2 x 
 ~ r
2k 2 x b
   
E   

2
ki
E



R 
s 
R

Initial field at the beginning of the bend magnet
~r
 E (s  0)  0
2

After the bend magnet
 ~ r
 2
    2ki E  0
s 

1
0
  0
Required for Selfconsistent
computation
2  2x  2y
  (x, y)
2
(r   r )

e
r
Erb 
e 
4 0  2r (r   )
r
2


 r
15
FEM equation
3D problem (Treat s as one space dimension)

E  n  0
~r
ME  J
It’s general case, for instance, the cross-section can vary
for the geometry impedance computation.
Problem: converge slowly
2D problem (Treat s as time)
~r
~ r
ME  DE  J

E  n  0
It converges faster, the stability need to be treated carefully
Current status: Bend only

1
Z (k )    Es ( xc , yc , s)ds
I0
16
The ways to improve the
accuracy
High order element
The (longitudinal) impedance is
calculated from the transverse field,
which is nonlinear near the beam as
shown late,
r
i  E xr E y 
Es 

0


k  x
y 
Fine girds on the curved boundary
…..

E  n  0
17
Test of the code
Bend Length =0.2meter, radius =1meter
Rectangular Cross-section with dimension 60mmX20mm
Wang
Zhou
k=10e3m-1, (142.2, 119.1)
(140.6, 119.5) (@end of the bend)
18
Sample 1
SuperKEKB damping ring bend magnet
Bend
Length[m]
Bending angle
# of elements
B1
.74248
.27679
32
B2
.28654
.09687
38
B3
.39208
.12460
6
B4
.47935
.15218
2
Geometry:
(1) rectangular with width 34mm and height 24mm
(2) Ante-chamber
(3) Round beam pipe with radius 25mm
19
Rectangular(34mmX24mm)
k=1x104m-1 s=0.4m
20
k=1x104m-1 s=0.6m
21
k=1x104m-1 s=0.7m
22
k=2x103m-1 s=0.55m
23
Ante-chamber
Leakage of the field into antechamber, especially high frequency
field
(courtesy K. Shibata)
24
k=1x104m-1 s=0.55m
25
k=2x103m-1 s=0.55m
26
Round chamber, k=1x104m-1 s=0.65m
Round
chamber,
radius=25mm
Coarse mesh
used for the
plot
27
Simulation of Microwave
Instability using
Vlasov-Fokker-Planck code
Y. Cai & B. Warnock’s Code, Vlasov solver
(Phys. Rev. ST Accel. Beams 13, 104402 (2010)
Simulation parameters:
qmax=8, nn=300 (mesh), np=4000(num syn. Period),
ndt=1024 (steps/syn.); z=6.53mm, del=1.654*10-4
CSR wake is calculated using Demin’s code with rectangular
geometry (34mm width and 24mm height)
Geometry Wake is from Ikeda-san
28
Case I: Geometry wake only
29
Case II: CSR wake only
Single bend wake computation
30
CSR only (single-bend model)
Red: <z>, Blue: <delta_e>,
Green: sigmaz; pink: sigmae
In=0.03
In=0.04
In=0.05
Only CSR (1-cell model)
In=0.04
p4
In=0.03
In=0.05
Inter-bunch Communication through CSR in
Whispering Gallery Modes
Robert Warnock
33
Next steps
•Complete the field solver from end of the bend to infinite
•Accurate computation with fine mesh & high order element,
especially for ante-chamber geometry.
•Complete Micro-wave instability simulation with various CSR
impedance options (multi-cell csr wake)
34
Integration of the field from
end of bend to infinity
•Agoh, Yokoya, Oide
Gennady, PRSTAB 104401 (mode expansion)
Other ways??
35
Summary
A FEM code is developed to calculated the CSR with arbitrary
cross-section of the beam pipe
Preliminary results show good agreement. But need more detail
benchmark.
Fast solver to integrate from exit to infinite is the next step.
Simulation of the microwave instability with Vlasov code is under
the way.
36
Acknowledgements
Thanks Prof. Oide for the support of this work and discussion;
and Prof. Ohmi for his arrangement of this study and discussions
Thanks Ikeda-san and Shibata-san for the valuable information
Thanks Demin for detail introduction his works and great help
during my stay.
Thanks Agoh -san for the discussions.
37
Backup slides
38
CSR + Geometry wake
34mmX24mm rectangular
More complete model
(Courtesy, H. Ikeda@ KEKB 17th review)
39
3
x 10
4
2.5
Zr/Zi ( )
2
round,real
round,Imag
rectangular, real
rectangular, imag
1.5
1
0.5
0
-0.5
1000
2000
3000
4000
5000
6000
k (m-1)
7000
8000
9000
10000
40
Fast ion Instability in
SuperKEKB Electron ring
Wang, Fukuma and Suetsugu
Ion species in KEKB vacuum chamber (3.76nTorr=5E-7 Pa)
Ion
Species
H2
H2 O
CO
CO2
Mass
Number
2
18
28
44
Cross-section Percentage in
(Mbarn)
Vacuum
0.35
25%
1.64
10%
2.0
50%
2.92
15%
Beam current 2.6A
2500 bunches, 2RF bucket spacing
emiitacex, y 4.6nm/11.5pm
41
20 bunch trains
total P=1nTorr
Total P=3.76nTorr
Growth time 0.26ms and 0.04ms
in x and y
Growth time 1.0ms and 0.1ms
in x and y
10
x=1.0771 ms, y=0.1044ms
1
x=0.264ms, y=0.0438 ms
10
10
0
Agoh
10
-1
X,Y ()
X,Y ()
10
10
10
0
-1
-2
10
10
1
-2
-3
x
y
0.5
1
1.5
2
2.5
Time (ms)
3
3.5
4
10
x
y
-3
2
4
6
Time (ms)
8
10
43
x
HIGHER ORDER ELEMENTS

Tetrahedron elements
4=
20 nodes, cubic:
10 nodes, quadratic:
4 nodes, linear:
4
4
l
=constant
3= k
1=
i
Q
=1
16
11
2
7
7
10
1
2= j
=1
17
3
9
10
5
6
2
8
19
5 18 15
1
5
P
y
13
20
8
=0
z
14
12
9
6
3
k=1x104m-1 s=0.46m
45