Stanislav BaturinEuclid TechLabs

10/26/2016

• Introduction to the wakefield concepts

• Simple calculation of the wakefields

• Wakefield based devices

Outline

Wakefield. Friend or foe?

• Beam energy spread

• Parasitic transverse momentum induction

• Negative effect on beam dynamics

Source of the wakefield

Disk loaded structureCollimators

and pipe transitionsStructures with

the “retarding” layer

Definition of the wakefield

Transverse wakefield potential

Longitudinal wakefield potential

Definition of the wakefield potential

1

1 ,z zz sW E z t dz

Q c

∞

−∞

+ = − = ∫

1

1 , z sW F z t dzQ c

∞

⊥−∞

+ = = ∫

1 2 ( )zE Q Q W s∆ = −Energy change of the test particle

Problem: How to calculate wakefield?

Finite differencemethods

Semi-analyticalmethods Pen and paper

What are the steps?

Pen and paper

The Idea

The Method

The Result

zW Ws⊥

⊥

∂= ∇

∂

(0 ) 2 (0)z zG G+ =

Panofsky-Wenzel theorem

Beam loadingtheorem

Important theorems

A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators (Wiley

and Sons, New York, 1993).

One more theorem

Relativistic Gauss Theorem

(0 ) 0D

Dz

S

E dS+ =∫∫

The Idea

Relativistic Gauss Theorem

z

S

DHdl qnV dSt

⊥

∂ = + ∂ ∫ ∫∫

0

0

(0 )V

Vz

S

qE dSε

+ = −∫∫1 ( )z

S

DHdl dS qc

δ ζζ

⊥

∂= +

∂∫ ∫∫

S.S. Baturin, A.D. Kanareykin. PRL113, 214801, 2014

0 0( ) (y ) (z )n x x y Vtδ δ δ= − − −

Vt zζ = −

(0 ) 0V

Vz

S

dH S+ =∫∫

The Method

Relativistic Gauss Theorem S.S. Baturin, A.D. Kanareykin. PR-AB19, 051001 (2016)

0zE⊥∆ =0zH⊥∆ =

Inside vacuum cannel

Relativistic Gauss TheoremThe Method

x iyω = +

x ye E iE= +

)( z zZg E iHχ ζ

∂−

∂=

∂ ∂

· 4 z zE Hi iπρζ ζ⊥ ⊥ ⊥ ⊥

∂ ∂∇ + ∇ × = + −

∂ ∂E E

· 4 z zE Hi i iπρζ ζ⊥ ⊥ ⊥ ⊥

∂ ∂∇ + ∇ × = + − ∂ ∂

H H

Maxwell equations in vacuum channel

x yh H iH= +

ct zζ = −

2 2 ( ) ( ) Zge Q x yδ δω ω

∂∂ ′ ′= +∂ ∂

Relativistic Gauss TheoremThe Method

*

2 2 ( ) ( ) Zge dQ x ydχ δ δ

χ ω χ∂∂ ′ ′= + ∂ ∂

0zH =zE const=

0Ze g− = 0Ze g− =

0

*

2(0 ) (0 ) 4z z

Q d dE iHa d d

ω ω

χ χω ω

+ +

=

− = −

0

*2

2 2( ) 4( )x yeQ d dF iF sa d

sd

sω ω

χ χω ω

=

=

+

Relativistic Gauss Theorem

Longitudinal electromagnetic field

Transverse Lorentz force

The Result

0s +→

The Result

Conformal mapping

2

20

(0 )16

Vz

c

qEa

ππε

+ = −2

0z

c

qEa

χ

πε= −

General formula

( ) tanh4

ia

π ωωχ = −

2dJdχω

=

(0 ) (0 )z zE J E χω + +=

0

*

2 20

2R) e(0zc

Q Q d dEa a d d

ω ω

χ χπε ω ω

+

=

= − −

Relativistic Gauss Theorem S.S. Baturin, A.D. Kanareykin. PR-AB19, 051001 (2016)

Conformal mapping

02

0

( ) cc

raa r

χ ωωω−

=−

40

320

0

2

1

( )r

c

c

rF eQa r

a

ss πε

= −

0

*2

2 20

( )

c

rF eQ d da d

ss d

ω ω

χ χπε ω ω

=

=

General formula

The Result

Relativistic Gauss Theorem

A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators (Wiley and Sons, New York, 1993).

S.S. Baturin, A.D. Kanareykin. PRL113, 214801, 2014

Cylinder Cylinderdisplaced

Planar Rectangular

The Result

Relativistic Gauss Theorem

1f =( ) 22

0

1(1 / )

dpc

fr a

=+

2

16plf π= 0.86sqf =

S.S. Baturin, A.D. Kanareykin. PRL113, 214801, 2014

0

*

2 20

2R) e(0zc

Q Q d dEa a d d

ω ω

χ χπε ω ω

+

=

= − −

Example of Dielectric Wakefield Accelerator:▫ Dielectrics: ε = 4.4 (cordierite-10 (alumina) – low-loss ceramic,

ε = 5.7 – diamond, ε = 3.7 - quartz▫ Electron beam parameter: σz = 20-30 um, Q = 1-3 nC, frequency 0.3-1.2 THz, ▫ Wakefield maximum 0.1-1.0 GV/m

Wakefield accelerator

W RW+ −∆ =

Transformer ratio

max

maxz

z

ER

E

+

−=

Now we define limits on wakefield maximum and transformer ratio.

Wakefield accelerator

The Idea

Limiting Gradient

( )z zE E E R+ + −→ →

( )z zE E R+ +→

max

maxz

z

ER

E

+

−=

0 0 00

( ) cos[ ( )] ( )s

zE s A k s s s dsρ− = −∫ 2

0

( ) ( )( )s

z zE s E ss k dsA A

ρ− −

= + ∫

The Method

Limiting Gradient

0 0 00

( ) cos[ ( )] ( )l

zE s A k s s s dsρ+ = −∫ ( )z zE E E+ + −→

STEP 1

STEP 2

Estimate the integrals in ( )zE E+ −

Limiting Gradient S.S. Baturin, A. Zholentsto be published

2 20

2max( )1z

Qf REa Rπε

+ ≤+

The Result

2 21R k l≤ +

The Result

Argonne studies of Wakefield Accelerator*

Driver charge

Aperture Driver energy

Desired energy gain

Transformer ratio

8nC 2 mm 400 MeV 1.6 Gev 5

max( ) 109zMVEm

=

*A. Zholents et al, Nuclear Instruments and Methods in Physics Research Section A Accelerators Spectrometers Detectors and Associated Equipment 829 February 2016

Now we will discuss wakefield application to the bunch

diagnostics.

30

Bunch diagnostics in principle

Slice Emittance Measurement

31

Retractable structure

Induced transversemomentum

Transverse wake

Quadruple

slice beam size

on Screen

Slice Emittance Measurement

S. Bettoni, P. Craievich, A. A. Lutman,2 and M. Pedrozzi1 PR-AB 19, 021304 (2016) Experimental results

Slice Emittance Measurement

[ ]8n A

p

Ismrad Iε

γ≈ 'n r rε γσ σ=

17.045AI kA=

Energy 140 MeV 1.4 GeV 14 GeV

bunch length 1 mm 100um 30 um

current 300 A 1 kA 3.5 kA

resolution 75 um 10 um 1.7 um

1n mm mradε = ×

There are more applications that may take advantage of controllable

wakefields.

1) S. Antipov, S. Baturin et al. PRL 112, 114801 (2014)Energy Spread Compensation2) P. Emma, M. Venturini, K. L. F. Bane, G. Stupakov et al. PRL 112, 034801

S. Antipov et al. PRL 108, 144801 (2012)

Subpicosecond Bunch Train Production

U=50 MVPh. -10 deg.

U=70 MVPh. 15 deg.

U=250 MVPh. 30.8 deg.

R56= 0 mm

Drive bunch In

R56= 20 mm

Drive bunch Out

Time reversible tracking

Includes:• Wakefields• Long. space charge• T566

CSR is next

Large energy chirp

~ 4 kA~ 200 A

Bunch Shaping

• Wakefields are very important part of accelerator science and technology

• Many novel ideas on how we can benefit from wakefieldsappeared and proved to be working during past years

• New tools for calculation of the wakefields and beam dynamics have been developed and more to come

• Analytical tools for the plasma wakefields

Conclusion