EUROFEL-Report-2007-DS2-084 a EUROPEAN FEL Design Study€¦ · in the low-frequency (inductive) regime [5] which is not the case for FERMI. A 3D numerical code would allow one to

EUROFEL-Report-2007-DS2-084 a

EUROPEAN FEL Design Study

Deliverable N°: D 2.11

Deliverable Title: Wakefield effects studied by simulations and experiments - Numerical Code Evaluation for the Impedance Budget of FERMI

Task: DS-2

Authors: C. Bontoiu, P. Craievich

Contract N°: 011935

Project funded by the European Community under the “Structuring the European Research Area” Specific Programme

Research Infrastructures action

1. Introduction

This is a report on the results obtained after a few months of using threespecialized wakefield calculation codes, namely: ABCI [1] which is a 2D code,CST Particle Studio (CST PS) [2] and GdfidL [3] which are 3D codes. Themain goals of this trial period were to determine how suitable is each code inmodelling various components likely to be found on the final FERMI layout [4],what are the reliability limits in terms of bunch length and mesh lines densityand to what extent do the results change as the mesh lines density is modified.Varying the bunch length, reliability tests of the codes have been carried in termsof longitudinal and transverse geometric wake potentials of a pillbox cavity forwhich the theory is well established. At the end dispersion of the longitudinalwake potential maxima is analyzed with respect to the mesh quality. The reportends with a recommendation on the choice of a 3D numerical code between CSTPS and GdfidL, based on the results of the tests carried hereby.

2. Codes usefulness for the FERMI Project

The importance of wakefield minimization is well documented and appreciatedin the design of the modern FELs. However, although a large number oftheoretical studies are available for the design of various accelerator components,like beam collimators, bellows, vacuum pumping slots, kickers (protrusions)etc, the availability of a 3D modelling code is essential. Short-range wakefieldscannot be measured and mitigated once the components have been installed andtherefore, the free electron lasing itself relies on the numerical and analyticalpossibilities to keep them under control, a priori. An example regarding thelimitations of the theory and consequently the applicability of a numerical codeis the vacuum chamber of the undulator intra-sections, which will be excitedby very short bunches (i.e. 300 µm). The undulator intra-section must allowroom for beam focusing quadrupoles, slots for the vacuum pumps connectionand the necessary of beam instrumentation. Thus, to allow the installation ofthe quadrupoles the vacuum chamber cross-section has to be reshaped fromrectangular (with rounded ends) to circular and thus due to the limited gapbetween the undulators, the resulting step-out transitions (although tapered)are steep enough to induce noticeable wakefields. In order to minimize them,having constrains on the bunch current, one can try to numerically model thetapers themselves. In addition, the pumping slots have to be optimized sothat their impedance is reduced to minimum possible, keeping simultaneouslythe vacuum conductance at optimum value. Thus, the vacuum chamber profilewould intuitively look as shown in Fig.1. Although analytical formulations of theimpedance exist for particular hole shapes, neither they take into account thewall thickness nor the wall curvature; the only reference to the wall thicknessproblem appears in paper inspired from Bethe’s diffraction theory applicablein the low-frequency (inductive) regime [5] which is not the case for FERMI.A 3D numerical code would allow one to get an insight into the wakefield

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resulting through the coupling modes between the vacuum chamber and theshort cylinder shaped into the curved wall. It is further challenging to investigatethe possibility of placing the vacuum ports transversally, on the step-in/out wallsand this can be done numerically only.

z

y

x

beam axis

Figure 1: Artist’s view of the vacuum chamber for the undulator intra-sections.

3. Case study: a pillbox cavity

The interaction between an ultra-relativistic charged bunch and a simple pillboxcavity as the one drawn in Fig.2 falls into three frequency regimes with respectto the cut-off frequency of the pipe ωc on which the cavity is installed. If ω ≈

g = 40 mm

a = 15 mmb = 30 mm

z

beam path

Figure 2: Simple pillbox cavity whose interaction with short Gaussian bunchesis analyzed numerically and field lines obtained numerically with ABCI as thebunch progresses from left to right.

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ωc the bunch feeds the whole cavity and there will be trapped resonant modesfrom which the bunch itself can reabsorb energy while in the limit ω << ωc,the cavity impedance is purely inductive; these two cases are treated withinthe so-called broad-band resonator model1 [6]. As the bunch becomes shorter,that is ω >> ωc, the broad-band resonator model does not give an accuratedescription of the impedance at high frequencies because it predicts a purelycapacitive impedance. A new model inspired from the Fresnel diffraction at thestraight edge has been developed for this frequency range: the diffraction model

[7]. Thus, for positive z values the pillbox longitudinal and transverse wakefunctions are defined respectively as:

w||(z) =Z0 c

√2 π2 a

√

g

z(1)

w⊥(z) =23/2 Z0 c

π2 a3

√g z (2)

For the pillbox cavity shown in Fig.2 the longitudinal and transverse wakefunctions are shown in Fig.3. Particular wake potentials can be further obtained

Figure 3: The longitudinal (left) and transverse (right) wake functions of apillbox cavity with pipe radius a = 15 mm, the cavity radius b = 30 mm andgap g = 40 mm.

convolving (1, 2) with the longitudinal bunch profile under consideration. Havinga concise formulation and being well tested, these theoretical results are takenas a gauge for the numerical codes as long as the bunch Fourier components aremostly above ωc.

Numerically obtained wake potentials for the pillbox cavity defined in Fig.2(left)are shown versus the equivalent theoretical values within the diffraction regimefor Gaussian bunches of standard deviation σ = 250, 500, 600 and 750 µm

in Figs.4, 5, 6 and 7 respectively. The absolute error is indicated as ∆W|| =

Wtheory|| − Wnumerical

|| and ∆W⊥ = Wtheory⊥ − Wnumerical

⊥ for the longitudinal

and transverse wake potential respectively.

1The broad-band resonator model ignores the possible existence of cavity modes. These

cavity modes occur at frequencies below the cutoff frequency ωc.

3

250Bunch

CST PS

ABCI

GdfidLDiffraction model (theory)

Relative error Relative error

Longitudinal wake potential Transverse wake potential

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Figure 4: Numerically obtained longitudinal and transverse wake potentials for aGaussian bunch with σ = 250 µm plotted against the corresponding theoreticalvalue.

4

500Bunch

CST PS

ABCI




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It is remarkable that above σ = 500 µm all three codes yield the longitudinalwake potential with comparable error (around 7% or less) at the theoreticalpeak location (see Tables 1, 2 and 3 below). The differences might originatein several reasons like: insufficient meshing, material filling errors etc. For asshort bunches as σ = 250 µm there is a visible difference between the outputsof the codes in the sense that GdfidL due to its wake windowing algorithm canmanage finer meshing and therefore produce more reliable results; about 3%error at the theoretical peak location compared with about 7% in the case ofCST PS. On the other hand, it is proven that ABCI is the most reliable and shallbe used whenever the structure under investigation has azimuthal symmetry.With the peak absolute error ∆W|| defined as the difference between the valueof the theoretical peak and the value of the one computed numerically, Tables1, 2, 3 show the relative numerical errors obtained at the computation of thelongitudinal wake potential. They are listed as ∆W||/W

theory|| for the peak value

and ∆z0/z0theory = (z0

numerical−z0theory)/z0

theory for the peak location on thez-axis.

Table 1: ABCI numerical error of the peak longitudinal wake potential.

z0 [µm] W|| [V/pC]∣

∣

∣

∆z0

z0theory

∣

∣

∣

∣

∣

∣

∆W||

W theory

∣

∣

∣

Theory ABCI Theory ABCI [%] [%]

σ = 250 µm 201 180 -6.78 -6.60 10.44 2.61

σ = 500 µm 397 350 -4.83 -4.58 11.83 5.14

σ = 600 µm 474 425 -4.42 -4.15 10.33 6.00

σ = 750 µm 591 520 -3.96 -3.67 12.01 7.26

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Table 2: CST PS numerical error of the peak longitudinal wake potential.

z0 [µm] W|| [V/pC]∣

∣

∣

∆z0

z0theory

∣

∣

∣

∣

∣

∣

∆W||

W theory

∣

∣

∣

Theory CST PS Theory CST PS [%] [%]

σ = 250 µm 201 355 -6.78 -6.29 76.61 7.22

σ = 500 µm 397 485 -4.83 -4.61 22.16 4.60

σ = 600 µm 474 511 -4.42 -4.18 7.80 5.26

σ = 750 µm 591 572 -3.96 -3.69 3.21 6.74

Table 3: GdfidL numerical error of the peak longitudinal wake potential.

z0 [µm] W|| [V/pC]∣

∣

∣

∆z0

z0theory

∣

∣

∣

∣

∣

∣

∆W||

W theory

∣

∣

∣

Theory GdfidL Theory GdfidL [%] [%]

σ = 250 µm 201 251 -6.78 -6.56 24.87 3.17

σ = 500 µm 397 438 -4.83 -4.57 10.32 5.40

σ = 600 µm 474 515 -4.42 -4.14 8.64 6.11

σ = 750 µm 591 649 -3.96 -3.69 9.81 6.69

In what concerns the transverse wake potentials there is not a standard wayto compare the results since W⊥(z) does not show a peak within the bunchrange. If one considers the bunch centre (z=0) as a reference point the relativeerror for σ ≥ 500µm is around 20-22% for GdfidL and around 4-8% for CST PS.However, these error ranges are somewhat less accurate since the computationshave been run without symmetry planes being applied to the pillbox cavity,which consequently involved a coarser meshing in order to stay within thememory limits of the computer used. On the other hand one notices differentbehaviour of the numerically computed transverse wakefields. Namely, whilethe GdfidL and ABCI output doesn’t intersect the theoretical prediction (therelative error doesn’t change the slope), CST PS does (the relative error changesthe slope). This behaviour provides an apparent convergence of the error linesat the tail of the bunch.

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Running simulations for a number of different Gaussian bunches at the samecomputational resources and meshing facilities, the degree to which numericalresults match the theory can vary substantially. It is hence important toparameterize the convergence of the wake potential output in terms of itsmaximum absolute value and peak position for each code. Taking the parameteras the bunch length (i.e. the standard deviation σ), a reliability chart, has beenobtained as shown in Fig.8.

Longitudinal wake potential peaks Loss factors

Figure 8: Absolute dispersion of the longitudinal peak wake potential as σ

decreases. The dots represent theoretically or numerically computed valueswhile the solid lines joining them represent a spline fit in each case. Numericalerrors are visible as well from the disperssion of the loss factors (bottom right).

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Thus, with respect to the plots at the bottom of Fig.8, having more memoryresources available, the codes would yield results closer to the theory black lines,respectively for the peak wake potentials (left) and loss factors (right) and wouldpreserve a better local parallelism to them as one goes from long bunches (theupper-right corners) to short bunches (the lower-left corners).

As an extension of the error tables shown above, Fig.9 assigns the relative errorin calculating the peak wake potential value and its location for a wider bunchrange. It can be noticed that at very short bunches, although the numericaloutput doesn’t match the theory reasonably, the computed peak value mightapproach the theoretical equivalent with high precision as for example less than4% for σ ≤ 300 µm. Its location instead is very erroneously computed and onlyat σ ≥ 500 µm falls within a margin of about 10% error.

s mm

200 400 600 800 1000

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s mm

200 400 600 800 1000

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50

60

70

%th

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<

<

%th

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z0

z0

ABCI

CST PS

GdfidL

Figure 9: Relative variation of the longitudinal peak wake potential (left) andof the longitudinal wake potential peak location (right) as σ decreases.

It is also important to investigate the accuracy of the numerical codes at veryshort bunch lengths, because the wake potentials of such bunches can be regarded,within a certain range of error, as the wake functions for a given structure. Atvery short bunch length (σ = 50 µm) a test has been carried on the same cavity,modifying the mesh steps as shown in Fig.10 only for the GdfidL code which,due to its windowing algorithm, could resolve such a high frequency content.Table 4 labels the relative error of the peak potential value W|| and that of its

location z0. Thus, although W|| is obtained reasonably accurate, z0 suffers fromlarge error.

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-150 -100 -50 0 50 100 150

0

2

4

6

8

Longitudinal wake potential

D

Absolute error Relative error

mesh 1:

mesh 2:

mesh 3:

mesh 4:

trans. long. mesh max. active mesh memory computationsteps steps cells cells step consumption time

962 66 61.08 mil. 39.32 mil. 20 m 2.088 GB

30 m 0.631 GB 58 min.

482 35 8.13 mil. 5.19 mil. 40 m 0.276 GB 21 min.

382 28 4.09 mil. 2.60 mil. 50 m 0.138 GB 10 min.

m

m

m

m

4h 16 min.

642 45 18.55 mil. 11.89 mil.

%th

eory

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-100 -50 0 50 100

0

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30

40

50

60

70

Figure 10: Mesh accuracy influence in evaluating the longitudinal wake potentialwith GdfidL for a standard deviation σ = 50 µm. The dots represent numericalcomputed values in each case while the solid lines are their polynomial fits.

5. Conclusions

First, the main conclusion is that for the case study exposed above with bunchesof σ ≥ 500 µm all three codes have produced reasonably good results; theycan be further improved carrying simulations with larger memory available.However, the accuracy depends greatly on the ratio between the bunch lengthand the size of the object under analysis and Figs.4, 5, 6and7 illustrate well thisaspect. At the same amount of CPU memory, for shorter bunches than thoseused above, a better agreement with the theory would imply the reduction ofthe pillbox cavity dimensions so that the higher frequency content is betterresolved by the mesh. Secondly all these codes have only Gaussian bunchesembedded and thus the computed wake potentials would not describe entirelythe problems of wakefields for FERMI. An expedient solution is to use very

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Table 4: Overview of the peak longitudinal wake potential sensitivity on themesh accuracy.

z0 W||(z0)∣

∣

∣

∆z0

z0theory

∣

∣

∣

∣

∣

∣

∆W||

W theory

∣

∣

∣

[µm] [V/pC] [%] [%]

Theory 42 -14.675 - -

mesh 1 70 -14.656 66.66 0.12

mesh 2 89 -14.040 111.90 4.32

mesh 3 109 -13.336 259.52 9.12

mesh 4 131 -12.609 311.90 14.07

short Gaussian bunches in order to obtain pseudo-wake functions which canbe further convoluted with any particular charge distribution. But this implieshuge memory resources and as it has been proven, GdfidL only can cope withthis challenge. Table 5 gives a synoptic view of the FERMI components likelyto induce strong geometric wakefields.

Table 5: The most important wakefield generating objects at FERMI.Object Geometry Bunch RangeOutput/input Laser 2D/3D 2.4mm-3mm

Spectrometer vacuum chamber 3D 2.4mm-3mm

Stripline BPMs 3D 300µm - 3mm

Tapers 2D/3D 300µm - 3mm

Bellows 2D 300µm - 3mm

Linearizer 2D/3D 2.4mm-3mm

Low energy RF deflectror 2D/3D 300µm - 3mm

High energy RF deflector 2D/3D 300µm

Cavity BPMs 3D 300µm

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Bibliography

[1] ABCI website: http://abci.kek.jp/abci.htm

[2] Computer Simulation Technology website:http://www.cst.com/Content/Products/PS/Overview.aspx

[3] GdfidL website: http://www.GdfidL.de/

[4] FERMI Conceptual Design Report, January 2007;

[5] N. McDonald, Electric and magnetic coupling through small aperturesin shield walls of any thickness, IEEE Trans. on Microwave Theory andTechniques, Vol. MTT-20, No. 10, Oct. 1972, p.689;

[6] Al. Wu Chao, Physics of Collective Beam Instabilities in High EnergyAccelerators, John Wiley and Sons, Inc., 1993;

[7] K. Bane and M. Sands, Wakefields of very short bunches in an acceleratingcavity, SLAC-PUB-4441, Nov. 1987;

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EUROFEL-Report-2007-DS2-084 a EUROPEAN FEL Design Study€¦ · in the low-frequency (inductive) regime [5] which is not the case for FERMI. A 3D numerical code would allow one to

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