HOMDYN ( H igher O rder M odes DYN amics) Massimo.Ferrario@LNF.INFN.IT Acknowledgements:

HOMDYN(Higher Order Modes DYNamics)

Massimo.Ferrario@LNF.INFN.IT

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