HOMDYN(Higher Order Modes DYNamics)
Acknowledgements: J. B. Rosensweig, L. Serafini,
J.P. Carneiro, J. Clendenin, H. Edwards, V. Fusco, L. Giannessi, M. Migliorati, A. Mosnier, D.T. Palmer, L. Palumbo, Ph. Piot, M. Quattromini ,T. Ronsivalle, J. Sekutowicz, F. Tazzioli.
Q ui ckTi me™ and a TI FF (Uncompressed) decompressor are needed to see thi s pi cture.
Historical motivation:
Superconducting Photo-injectors
Requirements:
Relativistic and Non-Relativistic Beams==> Space Charge Effects
Long term multi-bunch beam cavity interaction (msec-1000 bunches)
==> Transients, Beam Loading, HOMexcitation
Space Charge
2 E 1
c2
E
t
J
t
Fb q E vB
bb Fp
),( LRfJ
Self Consistent Code and Reasonable CPU Time
The HOMDYN model
Normal Modes Expansion of Cavity fields
We describe the field evolution under the slowly varying envelope approximation for each resonant mode
Analytical propagation of field from bunch to bunch, including an external generator
Longitudinal and Transverse Beam Laminarity
We describe the bunch dynamics of a uniform charged cylinder under the Multi-Slice approximation
Analytically computation of slice Space Charge Fields
2c2kp
Rs s
G( s , Ar )
s3 1 s
2 G(s , Ar )
s
4nthc
s
21
Rs3 4n
Bc
s
21
Rs3
ss2 Ý s Ý R s Ks
sol Ksrf Rs
Ý Ý Z
Ý Ý R
e
mocs3
Ezsc s , t
Ezwake ss, t
e
mocs3
Ezacc zs , t
Ezsc s , t
Artistic view of the Envelope Equations
L(t)Rs(t)
t
t
Multi-Slice approximation and Envelope Equations:
Ý Ý R s ss2 Ý s Ý R s Ks
sol Ksrf Rs
2c2kp
Rss
G( s, Ar )
s3 1s
2 G(s ,Ar )
s
4nthc
s
21
Rs3 4n
Bc
s
21
Rs3
Ý z s cs
Ý s e
mocs3
Ezacc zs , t Ez
wake ss , t Ezsc s , t Ez
sc s, t
Ezsc s
Q
2o Rs2 1 s L 2 Ar ,s
2 s L 2 Ar, s
2 1 s L s L
s zs zt
MV
m
Ar,s Rs s L Aspect Ratio:
Radial Component (on envelope)
Longitudinal Component (on axis)
Ersc(r,z )
o
dEz (r 0,z )
dz
R
2
Space Charge Field:
Ersc s
Q
4o Rs L
1 s L
1 s L 2 Ar ,s2
s L
s L 2 Ar, s2
Q
4o Rs LG s, Ar,s
Ersc s
Q
4o Rs L
1 s L
1 s L 2 Ar ,s2
s L
s L 2 Ar, s2
Q
4o Rs LG s, Ar,s
L
RsRs
= 1 = 5 = 10
Ar,s Rs s L
Beam Generation and Cathode Image Charges:
0.0005 0.001 0.0015 0.002Z @mD
-25
-20
-15
-10
-5
5
Esc @MVêmD
0.0005 0.001 0.0015 0.002Z @mD
-10
-5
5
10
Esc @MVêmD
No Image Charges
With Images Charges
e-e+
Elliptical cross section bunch
Ex I
co
x
X X Y
Ey I
co
y
Y X Y Er I
2co R2 r
X
Y R*R*
X Y
2
Ex X, z Ey Y ,z Er R*,z
Kssol
eBz zs 2mos
2
Ksrf
e
2 smo
Ez
zs
c
Ez
t
kq ec
mB
kpoles cKkx , y
2
kp I s 2Io
Space Charge
RF Focusing SW+TW
Quadrupoles
Undulator
Solenoid
Beam Line
Ý Ý X s kqXs kpolesXs 2c2kp
Rs*s
G( s, Ar )
s3
4nthc
s
21
Xs3
4nBc
s
21
Xs3
Ý Ý Y s kqYs kpolesYs 2c2kp
Rs*s
G( s, Ar )
s3
4nthc
s
21
Ys3
4nBc
s
21
Ys3
Ý Ý R s ss2 Ý s Ý R s Ks
sol Ksrf Rs
2c2kp
Rss
G( s, Ar )
s3 1s
2 G(s ,Ar )
s
4nthc
s
21
Rs3 4n
Bc
s
21
Rs3
Cylindrical Symmetry
Elliptical Symmetry
Undulator (Compressor?)
Ý z c K2
2 1 cos 2kzz
Ý Ý X cKkx
2
X
Ý Ý Y cKky
2
Y kx2 ky
2 kz2
2u
2
Lc
K
R56 Lc2
K e
m
Brms
ckz
Brms ˆ B
2
TW structures
EzTW z, t ˜ z
SW z sin t o kd ˜ zSW z d sin t o
˜ zSW z z
SW z sin kd
0 0.0005 0.001 0.0015 0.002s
75
100
125
150
175
200
225
w
W// s Zoc
a a2 8.6s
Wake Fields
0 0.00001 0.00002 0.00003 0.00004dz
-0.25
-0.2
-0.15
-0.1
-0.05
0
MVam
Ez s 2QcZo
lb 8.61 1
8.6s
a2
Delta Wake Function
Convolution with a Uniform charge
distribution
Wake field along a 40 m bunch
V/pc/m
m m
MV/m
0
0.5
1
1.5
2
2.5
3
3.5
0 2 4 6 8 10Z_[m]
GunLinac
rms beam size [mm]
rms norm. emittance [um]
-0.04
-0.02
0
0.02
0.04
0 0.001 0.002 0.003 0.004 0.005 0.006
z=0.23891
Pr
R [m]
-0.05
0
0.05
0 0.0008 0.0016 0.0024 0.0032 0.004
z=1.5
Pr
R [m]
-0.04
-0.02
0
0.02
0.04
0 0.0008 0.0016 0.0024 0.0032 0.004
z=10
pr_[
rad]
R_[m]
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
-0.003 -0.002 -0.001 0 0.001 0.002 0.003
z=0.23891
Rs
[m]
Zs-Zb [m]
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
-0.003 -0.002 -0.001 0 0.001 0.002 0.003
Z=10
Rs
[m]
Zs-Zb [m]
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
-0.003 -0.002 -0.001 0 0.001 0.002 0.003
z=1.5
Rs
[m]
Zs-Zb [m]
0
0.5
1
1.5
2
2.5
3
3.5
0 100 200 300 400 500 600 700 800
Emittance evolution for different laser rise times
emit (ideal)emit (0.5 psec rise)emit (1 psec rise)
n
(m
m-m
rad)
z (cm)
Comparison with PARMELA/UCLA
==> good agreement
Comparison HOMDYN with ASTRA (Ph. Piot)
Experimental validation at the A0 photoinjector(J. P. Carneiro et al.)
1.6 cells gun, 1.3 GHz, 40 MV/m peak field, 1.3 KG solenoid
9 cells Tesla structure, 1.3 GHz, 15 MV/m accelerating field
9-cells TESLA structureGunPepperPo
t
Quads
Solenoid Scan @ 8nC
Emittance (mm-mrad) computed by HOMDYN
and PARMELA/Orsay
HomdynParmela
INPUTDECK
FIELD.SEK FIELD.POI
HOMDYN
HSUM.OUT
HBUNCH.OUT HTWISS.OUT
HFEL.OUT
HSCREEN.OUT
HSPACE.OUT
HSLICES.OUT
INPUTDECK - 1 (the beam line)
t0 TEND H1 NLOOP iprint IPR2-1.d-1 200.d-9 1.d-12 3000 1 10ZETA0 ZSTOP EMEV SPREAD0.00045d0 33.d0d0 1.d-6 0.d0MAXB reprate IPART CHARGE DLBUNCH IGA carflut LCOOP1 1000 1 1.d-9 -11.653d-12 55 0.d0 300.d-6ISAVE IREAD ZSAVE0 0 8.1d0iflat0XEMIT XBUNCH XPRIME ITWISS0.3d-6 1.d-3 0.d0 0YEMIT YBUNCH YPRIME ICAT BZCAT0.3d-6 1.d-3 0.d0 0 0IPOI ISOL ISHIELD IQUAD ICAV IFOUR ITWA ISWA IWIG iplas ifoil ZSCREEN3 0 0 13 1 0 2 0 1 0 0 33.d0ISCAN STEP11 -0.1d0
RF gun
Solenoids
TWstructures
Quadrupoles Undulator with Quadrupoles
INPUTDECK - 2 (the bunch)
t0 TEND H1 NLOOP iprint IPR2-1.d-1 200.d-9 1.d-12 3000 1 10ZETA0 ZSTOP EMEV SPREAD0.00045d0 33.d0d0 1.d-6 0.d0MAXB reprate IPART CHARGE DLBUNCH IGA carflut LCOOP1 1000 1 1.d-9 -11.653d-12 55 0.d0 300.d-6ISAVE IREAD ZSAVE0 0 8.1d0iflat0XEMIT XBUNCH XPRIME ITWISS0.3d-6 1.d-3 0.d0 0YEMIT YBUNCH YPRIME ICAT BZCAT0.3d-6 1.d-3 0.d0 0 0IPOI ISOL ISHIELD IQUAD ICAV IFOUR ITWA ISWA IWIG iplas ifoil ZSCREEN3 0 0 13 1 0 2 0 1 0 0 33.d0ISCAN STEP11 -0.1d0
HREAD.COR
INPUTDECK - 3 (integration step)
t0 TEND H1 NLOOP iprint IPR2-1.d-1 200.d-9 1.d-12 3000 1 10ZETA0 ZSTOP EMEV SPREAD0.00045d0 33.d0d0 1.d-6 0.d0MAXB reprate IPART CHARGE DLBUNCH IGA carflut LCOOP1 1000 1 1.d-9 -11.653d-12 55 0.d0 300.d-6ISAVE IREAD ZSAVE0 0 8.1d0iflat0XEMIT XBUNCH XPRIME ITWISS0.3d-6 1.d-3 0.d0 0YEMIT YBUNCH YPRIME ICAT BZCAT0.3d-6 1.d-3 0.d0 0 0IPOI ISOL ISHIELD IQUAD ICAV IFOUR ITWA ISWA IWIG iplas ifoil ZSCREEN3 0 0 13 1 0 2 0 1 0 0 33.d0ISCAN STEP11 -0.1d0
INPUTDECK - 4 (the output)
t0 TEND H1 NLOOP iprint IPR2-1.d-1 200.d-9 1.d-12 3000 1 10ZETA0 ZSTOP EMEV SPREAD0.00045d0 33.d0d0 1.d-6 0.d0MAXB reprate IPART CHARGE DLBUNCH IGA carflut LCOOP1 1000 1 1.d-9 -11.653d-12 55 0.d0 300.d-6ISAVE IREAD ZSAVE0 0 8.1d0iflat0XEMIT XBUNCH XPRIME ITWISS0.3d-6 1.d-3 0.d0 0YEMIT YBUNCH YPRIME ICAT BZCAT0.3d-6 1.d-3 0.d0 0 0IPOI ISOL ISHIELD IQUAD ICAV IFOUR ITWA ISWA IWIG iplas ifoil ZSCREEN3 0 0 13 1 0 2 0 1 0 0 33.d0ISCAN STEP11 -0.1d0
HSAVE.COR
saving output data in HBUNCH.OUT and HTWISS.OUT
every
dt1=TEND/NLOOP
saving output data in HSPACE.OUT every
dt2=dt1*IPR2
Zw
saving outputs every DZ=ZSCREEN in files HSCREEN.OUT and HFEL.OUT
INPUTDECK - 5 (parameters scan)
t0 TEND H1 NLOOP iprint IPR2-1.d-1 200.d-9 1.d-12 3000 1 10ZETA0 ZSTOP EMEV SPREAD0.00045d0 33.d0d0 1.d-6 0.d0MAXB reprate IPART CHARGE DLBUNCH IGA carflut LCOOP10 1000 1 1.d-9 -11.653d-12 55 0.d0 300.d-6ISAVE IREAD ZSAVE0 0 8.1d0iflat0XEMIT XBUNCH XPRIME ITWISS0.3d-6 1.d-3 0.d0 0YEMIT YBUNCH YPRIME ICAT BZCAT0.3d-6 1.d-3 0.d0 0 0IPOI ISOL ISHIELD IQUAD ICAV IFOUR ITWA ISWA IWIG iplas ifoil ZSCREEN3 0 0 13 1 0 2 0 1 0 0 33.d0ISCAN STEP11 -0.05d0
Range of the parameter scan MAXB*STEP
INPUTDECK - 6(elements)
SOLMAP ZSOLEN BMAX'sanel' 1.3223500d0 0.06d0'sanel' 4.900d0 0.0d0'gtfbz' 0.000d0 0.27d0ZQUAD QGRAD QLENGTH8.636d0 4.7828d0 0.132d08.918d0 -4.7828d0 0.132d012.60d0 -0.2281d0 0.132d012.882d0 0.2281d0 0.132d014.875d0 4.8562d0 0.132d0Z0CAV FI0 EPEAK IRISWF0.d0 27.5d0 120.d6 -11.6d-3ZK taug rfoff0.00045d0 -10.d-6 0.801d0nmod ik itrans1 0 0MODO WN QEX USTOR VSFH'new45' 2856.d6 23616.74 0.39478e-05 0.999892d0MODO Z0FR DNCEL ZINC WN PSI FI0 EP bl IRISWF'twnew' 1.5d0 86 0.05248467 2856.d6 120. -8. 33.33d6 1 11.6d-3Z0TWA FI0TW EPTWA5.d0 -13.d0 33.33d6Z0WIG ZPER NW BRMS KX NMODULES DRIFT18.1d0 3.d-2 71 0.504d0 0.d0 6 0.36d0
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HomDyn HomeWork for the SPARC project
RF gun
Solenoids
TWstructures
Quadrupoles Undulator with Quadrupoles