1 BROOKHAVEN SCIENCE ASSOCIATES Recent Developments in Wakefield and Impedance Computation Boris...

1BROOKHAVEN SCIENCE ASSOCIATES

Recent Developments in Wakefield Recent Developments in Wakefield and Impedance Computationand Impedance Computation

Recent Developments in Wakefield Recent Developments in Wakefield and Impedance Computationand Impedance Computation

Boris Podobedov [email protected]

January 16, 2014

Boris Podobedov, Jan. 16, 2014

• It’s a pretty well-studied subject

• Maxwell’s equations are 150 years old

• Many textbooks and hundreds of papers exit

• Codes are getting ever more powerful

Motivation: Could There Be Anything New in Motivation: Could There Be Anything New in Wakefields & Impedances?Wakefields & Impedances?

Motivation: Could There Be Anything New in Motivation: Could There Be Anything New in Wakefields & Impedances?Wakefields & Impedances?

CST Studio SuiteCST Studio Suite


More Powerful Codes are AvailableMore Powerful Codes are AvailableMore Powerful Codes are AvailableMore Powerful Codes are Available

26-27 June 2013 CERN

Electromagnetic field simulations for accelerator optimization

http://indico.cern.ch/contributionDisplay.py?contribId=11&confId=243336

An Excellent Review of Codes was An Excellent Review of Codes was Recently Presented by Recently Presented by Alexej Grudiev Alexej Grudiev at at

CERNCERN

An Excellent Review of Codes was An Excellent Review of Codes was Recently Presented by Recently Presented by Alexej Grudiev Alexej Grudiev at at

CERNCERN


MotivationMotivationMotivationMotivation

• Knowledge of wakefields, incl. geometric ones, is critically important for accelerator beam dynamics.

• Detailed wakefield calculations for realistic vacuum chambers are done with time domain EM solvers, which calculate the fields due to finite length bunches.

• Extremely fine meshes are needed to compute wakes at small distances, so calc’s are slow and lots memory is req’d.

• This is especially true if the structures are very big (compared to bunch length) and smoothly varying.

Electromagnetic

Code forHandlingOfHarmfulCollective

Effects

EM solvers do not calculate point-charge wakefields


• For cavity-like structures we use the diffraction model:

• Wake-potentials for all cavity shapes (tapered or not, deep or shallow, etc.) converge to this model for short enough bunches and distances.

• Model is easily expandable to 3D geometries.

Asymptotic Model for Short-Bunch Asymptotic Model for Short-Bunch Wakefields of Cavity-like StructuresWakefields of Cavity-like StructuresAsymptotic Model for Short-Bunch Asymptotic Model for Short-Bunch Wakefields of Cavity-like StructuresWakefields of Cavity-like Structures

1/2( ) , 0d dW z k z z

2 2 2/4 1 18 4 4 4 4( ) ( I( , ) sign( ) I( , ) )s s sf s e s s

02

/ 2

d

Z ck g

a

( )

dd

kW z f z

g2 a

I(…) are Bessel functions

wake-function

wake-potential


Diffraction ModelDiffraction ModelDiffraction ModelDiffraction Model

• Fresnel diffraction (from A. Chao's book).• Wakefield is found from energy losses of an ultra-relativistic beam to

diffracted waves. • Plane waves diffracted by a straight-edge plane screen is an adequate

approximation for beam in cylindrical geometry.


• For collimator-like structures use the optical model:

• Turns out this model describes all collimator-like structures, including 3D; A recipe to calculate geometry-dependent kopt exists [see Stupakov, Bane, Zagorodnov, PRST-AB 10, 054401 (2007) ]

( ) ( )opt optW z k z 2

21/2 1 2( ) (2 )z

opt optW z k e

0 ( / ) /optk Z c Ln a b

2b

2 awake-function

wake-potential

Asymptotic Model for Short-Bunch Asymptotic Model for Short-Bunch Wakefields of Collimator-like StructuresWakefields of Collimator-like Structures

Asymptotic Model for Short-Bunch Asymptotic Model for Short-Bunch Wakefields of Collimator-like StructuresWakefields of Collimator-like Structures

0

( 0)

( )

opt

opt

W z

W z


Asymptotic Short-Bunch Wakes are Known, Asymptotic Short-Bunch Wakes are Known, so What’s Missing?so What’s Missing?

Asymptotic Short-Bunch Wakes are Known, Asymptotic Short-Bunch Wakes are Known, so What’s Missing?so What’s Missing?

• Asymptotic wakefields are approximate expressions (strongest singularity) at z=0.

• They fall down very sharply with z, and one cannot say where they get overtaken by other (non-singular ) parts of the wake.

• They don’t give us complete picture, i.e. W(z) for any z.


Can We Approach Point-Charge Wakes Can We Approach Point-Charge Wakes through Geometry Scaling?through Geometry Scaling?

Can We Approach Point-Charge Wakes Can We Approach Point-Charge Wakes through Geometry Scaling?through Geometry Scaling?


Scaling LawsScaling LawsScaling LawsScaling Laws

• Stupakov, Bane, and Zagorodnov show that the following longitudinal impedance and wakefield scaling holds true for 1) structures of general shape at high frequencies (i.e. >> pipe cutoff)2) small angle transitions at all frequencies

Here R and u are functions of one variable, andis the longitudinal scale factor between the geometries.

• In terms of the wake-potential due to a finite length bunch, Eqs.(1-2) are equivalent to where W and W stand, respectively, for the wake-potentials due to the original and the

scaled structures, and relates the longitudinal dimensions by L=L.

• Similar scaling laws are derived for transverse.

( ; ) ( ) (1)

( ; ) ( ). (2)

Z R

W z u z

1 /( ) ( / ). (3)W z W z


Wake for a 3D Tapered TransitionWake for a 3D Tapered TransitionWake for a 3D Tapered TransitionWake for a 3D Tapered Transition

G. Stupakov, K. L. F. Bane, I. Zagorodnov, PRST-AB 14, 014402 (2011)


Scaling Breakdown Example: Small Angle Scaling Breakdown Example: Small Angle Tapered Cavity Tapered Cavity

Scaling Breakdown Example: Small Angle Scaling Breakdown Example: Small Angle Tapered Cavity Tapered Cavity

• ECHO results for the original and =1/5 scaled geometries are shown• Clearly there is a disagreement in the long range wake• In -domain this disagreement extends way above the cutoff• In general, scaling in a very large range<<1, doesn’t seem accurate• More on breakdown is in A. Blednykh, S. Krinsky, PRSTAB 15, 054405 (2012)

Geometrybefore Scaling

rmin=1 cm, rmax=1.05 cm, g=20 cmL=2.5 cm

gL


Recent Method to Find Point-Charge WakesRecent Method to Find Point-Charge WakesRecent Method to Find Point-Charge WakesRecent Method to Find Point-Charge Wakes

• In WEODB1 NAPAC’13 paper we extended this method to transverse wakes


-0.5 0 0.5 1 1.5

-15

-10

-5

0

z (cm)

W (

V/p

C )

-0.5 0 0.5 1 1.5-0.2

0

0.2

0.4

0.6

0.8

z (cm)

D , D

( V

/pC

)

=5 mm =2 mm =1 mm

=5 mm =2 mm =1 mm=0

Introducing the Method: Wake of a Step-Out Introducing the Method: Wake of a Step-Out Introducing the Method: Wake of a Step-Out Introducing the Method: Wake of a Step-Out

• Wake-potentials are singular at →0• Subtracting singular part (optical

model) we obtain a well-defined limit (black line) at →0

• This function is approximated by

• Coefficients and can be found by fitting (next VG).

• Thus we reconstruct point-charge wakefield (at short z-range)

2

20 21/2 3/2

(3)( ) ( )

2

z

s opt

Z c LnW z W z e

( ) ( ) ( )sD z W z W z

0( ) lim ( )D z D z

( ) ( )D z z H z

2 cm6 cm

step-functionfrom causality


Wake of a Step-Out Con’t: Wake of a Step-Out Con’t: fitting for fitting for and and Wake of a Step-Out Con’t: Wake of a Step-Out Con’t: fitting for fitting for and and

• Point-charge and Gaussian bunch functions are related:

• and can be found by fitting D(z) from EM solver for i.e. |z/|<.• Take =2 mm and apply the fitting. Then use and obtained to reconstruct wakes for

other values of

• Reconstructed wakes agree well with direct ECHO calculation

( ) ( )D z z H z 2

222 2 2

1 erf( ) e .( )z

z zD z

-0.5 0 0.5 1 1.5

-15

-10

-5

0

z (cm)

W || (

V/p

C )

=5 mm =2 mm =1 mm

-0.5 0 0.5 1 1.5-0.2

00.20.40.6

z (cm)

D || (

V/p

C )

=5 mm =2 mm =1 mm =0

rmax=3 cm

rmin=1 cm

dots: ECHO lines: rec'd from 2mm

ECHOW


• Why did the =2 mm fit work well? Becauseg.• Parameter g>0 is the first location of the wake singularity (or singularity

of its derivatives) closest to z=0. • cannot be extended beyond z=g since the

wake derivative is singular (“kink”).

How to Pick How to Pick in EM solverin EM solverHow to Pick How to Pick in EM solverin EM solver

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

z (cm)

D (

V/p

C )

=5 mm=2 mm= 1 mm=0.1 mm

g= 2 cm

( ) ( )D z z H z

2 cm6 cm

min2 2cmg r •Run EM solver with g , typically g=0.1-0.15 is O.K.

•Running with shorter bunch gives no new information about the wake!

•g can be found by simple geometry analysis.


g

g

gg Parameter for a CavityParameter for a Cavitygg Parameter for a CavityParameter for a Cavity

rmin

2 2min(2 )g r g g

• Red ray (spherical wave front) eventually catches up with ALL particles in the bunch, thus affecting the wakefield for all values of z.

• Green ray travels g/c behind and it will never catch up with the front of the bunch, so g emerges in the front portion of the wake.

• For other ratios between rmin, rmax, and g, other combinations may define g , i.e. g =2g for a short cavity or, for a shallow one,

rmax

2 2max min4( )g r r g g

1/2max min min/ 2, / 2r r g r

Similarly g one can find for arb. geometry (see PRST-AB paper)


-50 0 50 100

-250

-200

-150

-100

-50

0

z, m

W ,

V/p

C

=16 m=8 m=4 m=2 m=1 m

21/2

Shallow Cavity ExampleShallow Cavity ExampleShallow Cavity ExampleShallow Cavity Example

• Our method is NOT shortening the bunch until the EM solver wakes converge (this can fail! )

• For this cavity wakes are ~Gaussian; they double as shortens each factor of 2, suggesting

• This is totally incorrect!• It is diffraction model instead

Important to Know g before Running the EM Solver

1/2( ) ~W z z

( ) ~ ( )W z z

rmin=10 mmrmax=11 mmg=10 cm

-0.1 0 0.1 0.2-25

-20

-15

-10

-5

0

5

z, cm

W ,

V/p

C

=400 m=200 m=100 m=50 m

2

2max min2( ) / 20 μmg r r g


g

Why Why g g is Easy to Find for Arbitrary is Easy to Find for Arbitrary GeometryGeometry

Why Why g g is Easy to Find for Arbitrary is Easy to Find for Arbitrary GeometryGeometry

rmin

1

2

1 2min( , )g

• Green ray does not affect short-range wake for • Brown ray does not affect short-range wake for

This is longer for deep cavities with rmax>2rmin.• By causality, any cavity with radial boundary, r(s), that coincides with the

figure for r(s)<2rmin, but otherwise is arbitrarily complex, must have the same short-range wake for z<g .

• =>g is defined by the geometry near rmin

rmin

2 2min(2 )z r g g

2 2max min4( )z r r g g

rmax


How It All Works TogetherHow It All Works TogetherHow It All Works TogetherHow It All Works Together

1. Determine analytical singular wake model:2. Determine g

3. Calculate the wake-potential with your favourite EM solver for g :

4. Subtract the singular wake:5. Fit the remainder, , with the

function: (fit range |z/|< works well)

6. Short-bunch wake (for arb. is then:

7. For point-charge:

2

20 0

0

22 2 2

1 erf( ) ez

z z

0 0 0( ) ( ) ( )ECHO sD z W z W z

( ) & ( )s sW z W z

0 ( )ECHOW z

0 ( )D z

2

222 20 2

1 erf( )( 3 ) ( )e s

zz zW z W z

00( 3 ) ( )ECHOW z W z

0( 3 ) ( ) ( )sW z z H z W z 00( 3 ) ( )ECHOW z W z

Transverse is very similar


The Method Was Applied to Many The Method Was Applied to Many GeometriesGeometries

The Method Was Applied to Many The Method Was Applied to Many GeometriesGeometries

It worked well for all of them


Simple Cavity ExampleSimple Cavity Example

• Diffraction-model behaviour near z=0• Pickmg

• Short-bunch wake reconstructed well.2 2

min(2 ) 1.24 cmg r g g

-1 0 1 2 3 4 5-6

-5

-4

-3

-2

-1

0

1

2

z (cm)

W (

V/p

C )

g = 1

.24

cm

=2 mm =1 mm =0.5 mm =0.2 mm

-0.05 0 0.05 0.1 0.15 0.2

0

0.2

0.4

0.6

0.8

z (cm)

D , D

( V

/pC

)

=200 m =100 m =50 m =0

dots: ECHO lines: rec'd from 200μm

ECHOW

0 0 0( ) ( ) ( )ECHO dD z W z W z

rmin=1 cmrmax=5 cmg=1 cm


Transverse Wake for the Same CavityTransverse Wake for the Same CavityTransverse Wake for the Same CavityTransverse Wake for the Same Cavity

• The same algorithm works well in the transverse (except the transverse diffraction model is non-singular).

• Reconstructed wakes from =2 mm agree perfectly with direct ECHO calculations

dots: direct ECHO lines: rec'd from 2mm

ECHOW


g

• Wake length-scales can be understood from causality.• Beam fields scattered at corner #1 (shown in green) will never catch up

with the head, as they must clear corner #2 getting (at least) g/c delay.• Thus g emerges in the front portion of the wake-field. W(z<g) depends

on rmin and onlywhile at larger z it depends on L and rmax as well.• If rmax>9rmin, a shorter length scale, g=4rmin, shows up, because fields

that “cut across” (shown in red) acquire a delay of 4rmin< L/2

• W(z<g) is very simple (see BP IPAC2012 for analytical results).

L

gg Parameter for a Small-Angle Linear Taper Parameter for a Small-Angle Linear Taper gg Parameter for a Small-Angle Linear Taper Parameter for a Small-Angle Linear Taper

2 2 2max min( ) / 2, ( 1)g r r L L L

rmax

#1#2

rmin

4rmin


Example of a Structure with a Linear TaperExample of a Structure with a Linear TaperExample of a Structure with a Linear TaperExample of a Structure with a Linear Taper

• Since g is known, tapered structures can be easily handled with our method.

0 0.05 0.1 0.15

0

2

4

6

8

10

12

14

16

18

20

22

z (cm)

D ,

D (

V/p

C )

gcavi

ty =

439

m

D200mECHO

D50mECHO

D20mECHO

D20m from D

50mECHO

D from D

50mECHO

21min2min( ,4 )taper

g L r

1 /

tapergcavity

g L g

-6 -4 -2 0 2 4 60

1

2

3

4

5

=0.1)

s (cm)

r (c

m)

L =4.8 cm

g =10.6 cm


Smooth Non-Linear TransitionsSmooth Non-Linear TransitionsSmooth Non-Linear TransitionsSmooth Non-Linear Transitions

• Smooth non-linear transitions are common (i.e. SC cavities)• Esp. important is the case with the slope, r’(s), matched to zero at rmin

• This case results in additional wake singularity.• Simplest case is a quadratically varying boundary near rmin .

• Wake-potential can be expressed through hypergeometric functions. • For simple cases can be found analytically, see BP IPAC2012.

1/3(0 )smoothsmooth gW z z

122

0

1 '( )L

smoothg r s ds L

L

rmax

rmin


Diversion to Creeping WavesDiversion to Creeping WavesDiversion to Creeping WavesDiversion to Creeping Waves

From Wikipedia

• Diffraction on a smoothed out edge is related to creeping waves• A creeping wave is diffracted around the shadowed surface of a smooth body.


Example of a Smooth Parabolic TransitionExample of a Smooth Parabolic TransitionExample of a Smooth Parabolic TransitionExample of a Smooth Parabolic Transition

• Since singular wakefields and g are known, such structures can be easily handled with our method (for complex structures coefficient could be found by fitting).

1/3 10 min6( '') (2 )Z c r r

-5 -2.5 0 2.5 50

0.5

1.0

1.5

2.0

s (cm)

r (m

m)

parabolic transition, L=1 cm

g=5 cm

0 10 20 30 400

200

400

600

800

1000

1200

z (m)

W -W

opt (

V/p

C )

g = 1

6.6

m

=2 m =1 m =0.5 m

dots – direct calculations with ECHOlines – reconstructed from ECHO calc’s for 2m bunch


NSLS-II Landau CavityNSLS-II Landau Cavity

• 1.5 GHz dual cell cavity, rside_pipe= 6 cm

• Final results for the short-range wakes:

To find 10m bunch wake

Brute force: ~480 hours of Intel(R) Xeon(R) [email protected] GHz CPU to zmax=1 cm.

Our method: uses only 50m

calc’s, saves a factor of 53 on CPU time and 52 on memory. Gives a model of the point-charge wake as a bonus.

-40 -20 0 20 40 60-1

0

1

2

3

4

5

6

7

z (m)

W -W

opt (

V/p

C )

=5 m =10 m =20 m

dots: direct ECHO lines: rec'd from 50μm

ECHOW

Jim Rose


0 0.5 1 1.5 20

50

100

150

200

z, cm

W y, V

/pC

/m

=500 m =250 m

3D Example3D Example3D Example3D Example

2b=2 cm

6 cm

g=6.5 cm

Wakes by I. Zagorodnov, 3D ECHO + CST mesher

Constant (5 cm full-) width collimator

2 2(2 ) 3 mmg b g g

g

• Observe g , where expected• Expected behavior near the origin; can easily fit point-charge wake & • Same for longitudinal (+ optical model), and for quadrupolar wakes

~(1+erf(z//21/2)) behavior


SummarySummarySummarySummary

• Wakefield calculation is important task for modern accelerators. For large and smooth accelerator structures and short bunches, direct EM solver calc’s can be extremely time-consuming.

• We reviewed several recent developments in geometric wakefield calculations, including new (or more general) short-bunch asymptotic wakefield models, use of scaling laws for small angle transitions, and a new method to obtain point-charge wakefields from an EM solver.

• All the works reviewed provide valuable physics intuition, while some of them also directly result in great savings in computing time required to calculate wake-potentials due to very short bunches.

• The works described are applicable to 2D or 3D geometries.

Thank you

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