In this note we study the geometric wakefields generated by a driving electron bunch in the SPARC LAB COMB chamber. Due to the change in iris radius of the beam pipe leading into the chamber, the electron beam will induce wakefields, which can have an effect on the bunches in the train used for the SL COMB experiment. Here, we present wakefields and impedances simulations and determine the effects that they may have on the beam properties

Brynes, Alex

Castorina, Giovanni

Frasciello, O.

Marcelli, Augusto

Spataro, Bruno

English

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               Laboratori Nazionali di Frascati   ISTITUTO NAZIONALE DI FISICA NUCLEARE                  INFN-16-10/LNF           30th June 2016     Studies of geometric wakefields and  impedances due to collimators  Alex Brynes1, Giovanni Castorina1, O. Frasciello1, Augusto Marcelli1;2 and  Bruno Spataro1 1)INFN-Laboratori Nazionali di Frascati Via E. Fermi 40, Frascati, Italy 2) RICMASS - Rome International Center for Materials Science Superstripes, Via dei Sabelli 119A, 00185 Roma, Italy     Abstract   In this note we study the geometric wakefields generated by a driving electron bunch in the SPARC LAB COMB chamber. Due to the change in iris radius of the beam pipe leading into the chamber, the electron beam will induce wakefields, which can have an effect on the bunches in the train used for the SL COMB experiment. Here, we present wakefields and impedances simulations and determine the effects that they may have on the beam properties.         Pubblicato da SIDS–Pubblicazioni      Laboratori Nazionali di Frascati 1 IntroductionA new technique to produce a short train of electron bunches, which can be used in beam-driven plasma wakefield acceleration experiments has been proposed at SPARC LAB [1].If a significant acceleration of the witness bunch is to be achieved the requirements on theproperties of the driving electron bunches are of great importance. Due to the dimensionsof the plasma cavity, the beam pipe radius will change, effectively acting as a collimator.In this note, we present simulations of the geometric wakefields due to various designs ofthe collimator, their associated impedances and loss factors. We will then determine theeffects that these wakefields will have on the electron bunches.2 Wake potentials and loss factorsHere we will give a brief overview of how wakes, impedances and loss factors are de-fined for both the longitudinal and transverse cases. A more thorough treatment is givenin Refs [2] and [3]. A charged particle beam travelling in a vacuum chamber can induceelectromagnetic fields in the structure, which can then in turn have an effect on beamdynamics and bunch properties, particularly when the vacuum chamber presents irregu-larities or discontinuities. These wake fields appear behind the driving bunch (as longas β = 1, which is the case for bunches at SPARC LAB upon emission from the RFgun). The longitudinal wake potential (in Volts) is defined as the change in voltage of aunit charge trailing behind a leading charge. If either charge has an offset from the beampipe axis, the trailing charge can experience a transverse momentum kick, also knownas the transverse wake potential (in Volts). The wake functions are defined as the wakepotentials per unit charge. If we consider an electron bunch, with a charge Q, travellingthrough a cylindrically symmetric cavity along the z-axis it is followed by a test charge,with a charge q, at a distance s from some point in the bunch, for example the centre ofthe bunch. The energy change of the test charge can be calculated using the longitudinalLorentz force, F∥(s, z) produced by the bunch:U∥ = −∫ ∞−∞dzF∥(s, z) (1)In addition, if the bunch has a transverse offset, x, from the beam pipe axis, thenit can induce a transverse electromagnetic force, F⊥(s, z), which can affect the beamdynamics of the bunch and of any bunches trailing behind it. This transverse wake hasa dependence on x in addition to s. The momentum kick experienced by a bunch with agiven transverse offset is given by:2U⊥ =∫ ∞−∞dzF⊥(s, z) (2)The longitudinal wake function, W∥(s), can have an effect also on the energy spreadof the trailing charge measured as V/C. The transverse wake potential, W⊥(s), can affectthe transverse emittance of the bunch and any bunches trailing behind it and it is measuredas V/cm:W∥ =U∥Qq(3)W⊥ =U⊥Qqx(4)If we consider a bunch with a longitudinal charge density λ(s), and defining theimpulse wake functions (i.e., for a point-like bunch) as W∥0,⊥0(s), we may also define thewake functions by using a convolution integral:W∥(s) =∫ ∞0ds′λ(s− s′)W∥0(s′) (5)W⊥(s) =∫ ∞0ds′λ(s− s′)W⊥0(s′) (6)In the ultra-relativistic limit, due to causality the impulse wake functions will benegligible. We can also determine the total energy lost by the test charge after passingthrough the collimator, defined as the longitudinal loss factor k∥. The transverse lossfactor, k⊥, describes the amplitude of the transverse momentum kick experienced by thetrailing charge:k∥ =U∥q2(7)k⊥(x) =U⊥q2x(8)The Fourier transform of the impulse wake functions then gives the longitudinal andtransverse impedances, Z∥(ω) and Z⊥(ω)Z∥(ω) =1c∫ ∞0dsW∥0(s) exp(iωs/c) (9)Z⊥(ω) = −ic∫ ∞0dsW⊥0(s) exp(iωs/c) (10)We may decompose these impedances into the real and the imaginary part:3Z∥(ω) = R∥(ω) + iX∥(ω) (11)Z⊥(ω) = R⊥(ω) + iX⊥(ω) (12)measured as Ω and Ω/m, respectively, and behave like a regular circuit impedancein the ultra-relativistic approximation.3 Simulation parametersIn Fig. 1 are shown some of the designs of the various collimators we considered forthe SL Comb chamber. Fig. 1(a) shows a simple collimator, where the beam pipe radiusdecreases sharply. Subsequent designs include tapering collimators of decreasing angles,in order to observe the effect of a smoother transition between radii. Ref [4] gives expres-sions for the impedances due to collimators with smooth tapering angles. The beam piperadius for the SL Comb chamber, b1, is 20 mm, b2, the final radius of the collimator, is2.5 mm, and the collimator has a length, L, of 10 cm. Fig. 1(e) shows the structure withthe smallest tapering angle of 0.41241 radians.4 Wake Potentials and loss factors4.1 Wake potentialsThe ABCI [6] code allows the user to calculate the wake potential, impedance and lossfactor, as a function of s (the distance behind the head of the bunch), of a Gaussian bunchpassing through an axisymmetric structure, in both the longitudinal and transverse planes.A variable radial mesh can be implemented for cases where the radius of the beam pipechanges. The sizes of the mesh in both r and z directions allows for fast, accurate compu-tation. By using Napoly’s integration method [7], the calculation can be performed withbeam pipes which are short at both ends.We also determined the transverse wake potential as a function of the tapering an-gle. Plots corresponding to W∥ and W⊥ for the cavities shown in Fig.1(a e) are given inFigs.2 and 3. We see that as the tapering angle decreases, the peak of the transverse wakepotential decreases, as this allows for a smoother transition down to the smaller beam piperadius.4(a) (b)(c) (d)(e)Figure 1: Collimator designs used for ABCI simulations. All collimators are axisymmet-ric4.2 Loss factorsIn terms of the effects of the longitudinal wakes on the properties of the bunch, the in-tegrated longitudinal wake potential is probably the most significant parameter, as this5Figure 2: The Wake potential (in V/pC) as a function of the tapering angleFigure 3: The transverse wake potential as a function of the tapering anglegives the total energy loss of the test charge. We allowed the beam pipe length to varyand calculated the longitudinal loss factors for the cavities in Fig. 1(a) and 1(e) as weexpected to see the largest variations between these two cavities (see Fig.4a and 4b).As the beam pipe length is extended, the levelling off of the loss factor is expected.As the length becomes sufficiently long, the wake fields behind the bunch will becomenegligible. In order to observe the trend of how both k∥ and k⊥ depend on the taperingangle, we listed these values in Table 1. Data clearly show that both loss factors decreaseas the tapering angle decreases.6(a) (b)Figure 4: Longitudinal loss factor (in V/pC) as a function of beam pipe length for thecollimators in Figs. 1(a) and 1(e)Table 1: k∥ and k⊥ as a function of for a beam pipe 50 cm longTapering Angle k∥ (V/pC) k⊥ (V/pC/mm)α = π/2 19.11 2.493α = 1.05165 17.32 2.241α = 0.71883 15.25 2.042α = 0.528074 13.19 1.842α = 0.41241 11.23 1.7035 Effects on the beam5.1 Kick factorsAs previously mentioned, the longitudinal wakefields can increase the energy spread ofthe electron bunch. Various schemes have been suggested [5] which, depending on thegeometry under consideration, can generate analytical formulae which are useful for ourpurposes. The so-called diffraction regime is appropriate when:kb2L& j201 (13)Where k = ωcand j201 is the first root of the Bessel function J0. For ω ≥ 50GHz,this condition holds. It has been shown that in the diffraction regime, the longitudinalimpedance of a round step collimator at high frequency can be given as:Z∥,HF =Z0πln(b1b2) (14)Approximate values for the longitudinal loss factor and the wakefield-induced en-ergy spread for a Gaussian bunch in the high frequency limit can be estimated [8]:7k∥,HF =c2Z∥,HF√πσ(15)krms∥,HF =√∫ +∞−∞dsλ(s)[W∥(s)− k∥,HF ]2 ≈ 0.4k∥,HF (16)Considering the values of our geometry we estimated k( ∥, HF ) ≈ 21.11 V/pC,and thus the energy spread krms∥,HF ≈ 8.4 V/pC. The ABCI simulations give a value ofk∥ = 19.11 V/pC for the cavity shown in Fig.1(a). For the SL COMB experiment, thebunch charge should not exceed 50 pC, giving a wakefield-induced energy spread of 400V. For a beam energy of 120 MeV, the energy spread is negligible. Moreover, as we arein the diffractive regime, we can also assume that the beam pipe is relatively long. In thisway, if it has an offset from the beam pipe axis we may obtain the transverse momentumkick that the beam may experience. Assuming that our collimator is long (L ∼ a2/σ) wecan calculate the transverse impedance:Z⊥,HF =Z0cωπ(1b22− 1b21)(17)From the value of Z⊥, we can calculate the transverse kick factor and the inducedmomentum spread of a Gaussian bunch:k⊥,HF =ω2Z⊥ (18)krms⊥,HF =k⊥,HF√3(19)Giving an estimate for the high-frequency transverse kick factor of k⊥,HF ≈ 5.66V/pC/mm, the induced momentum spread due to this kick is krms,HF ≈ 3.29 V/pC/mm.We can also compare the above numerical results with analytical results regarding theimaginary part of the transverse impedance at zero frequency [9]. The transverse lossfactor, k⊥, is related to the imaginary part of the transverse impedance at zero frequencyin the following way:X⊥(ω = 0) = −2√πσck⊥ (20)As an example, the numerical value calculated for the cavity 1(e) is X⊥(ω =0) = 19.583 kΩ/m. The transverse loss factor is 1.703 kV/pCm, and thus we obtainIm[Z⊥(ω = 0)] = 20.137 kΩ/m. Since the code is within 3% of the analytical value wecan consider the numerical results accurate enough for our purposes. We also find that thetransverse loss factor is approximately 25% larger for the step collimator.85.2 Transverse emittance growthIn order to determine the effects that the collimator wakefields may have on the transversephase space of a real beam, we can calculate the mean centroid kick given to a bunch withan offset from the beam axis x. This kick is then given by [10]:θ =xQkavg⊥E(21)Where Q is the bunch charge, kavg⊥ is the transverse kick factor averaged over thelength of the beam, and E is the energy of the beam. For the SL COMB experiment, thedriving bunches have a charge of 50 pC, and the witness bunch has a charge of 25 pC, allwith E = 120MeV . We can then determine the transverse emittance growth due to thiskick:ϵx =√(ϵ2x;0(1 +βxϵx;0θ2))(22)where ϵx,0 is the x-emittance at the entrance of the collimator, ϵx is the x-emittanceafter the kick, and βx is the betatron function at the entrance of the collimator. ABCIgives a value of kavg⊥ = 2.453 V/pC/mm. If we estimate ϵx,0 = 1 mmmrad, βx = 10m and an offset in x of 1 mm, a negligible emittance growth for all collimator designs isobtained.6 Collimator with a capillary6.1 Perfectly conducting capillaryWe have also considered a layout with a short capillary within the collimator with theaim of studying the effects of the beam passing through a dielectric material. Our initialABCI studies were limited to characterize the behaviour of geometric wakefields of aperfectly conducting structure. Below are presented some of our results. The geometriesof cavities 1(a) and 1(e) were modified to include capillaries of different radii: 500 m,750 m and 1000 m, in the centre of the collimators, with a length of 2 cm (see Fig.5).We calculated that the longitudinal loss factors are largely unchanged when a capillary isincluded in the geometry, but Z∥ and Z⊥ are around an order of magnitude larger than inthe initial designs. If we assume that the Equation (15) is still valid (which may not be thecase, due to the more complex geometry), we first determine if our simulations are givingreasonable values. A comparison between the numerical values of X⊥(ω = 0) and thatobtained using the Equation (15) is given in Table 2.9Figure 5: Schematic of the step collimator in Fig.1(a) with a short capillary included inthe geometry.We can consider these numerical values sufficiently accurate for our purposes. Fig.6gives plots of Z∥ and Z⊥for the three capillary radii given above. The impedances for thegeometries with capillary radii of 750 µm and 1000 µm appear to level off or decrease inthe high-frequency limit, but the geometry with a capillary radius of 500 µm continues toincrease. In order to check the validity of these simulations, we did a cross-check with the3-D EM code GdfidL [11]. Simulations run for the same geometry with a capillary radiusof 500 µm, both with a perfectly conducting capillary and with an aluminium capillary.Results for impedances and wake potentials for this geometry as generated by both codesare given in Fig.7. We can see that both codes match up quite well. Discrepancies betweenthe two sets of simulations may be due to a difference in the mesh sizes. It was alsodetermined that the resistive wakes due to the aluminium capillary have a small effect incomparison with the geometric wakes.10Table 2: Comparison between analytical and numerical calculations of X⊥(ω = 0),[kΩ/m]Capillary Radius 500 µm 750 µm 1000 µmABCI 320.92 197.06 140.12Analytical 308.38 184.35 152.86% difference 4 6 8(a) (b)(c) (d)Figure 6: Z∥ and Z⊥ for various capillary radii, as computed by the ABCI code6.2 Dielectric capillaryA modified version of the ABCI code allows the user to include dielectric materials intheir simulations [12]. The inclusion of a dielectric material can induce trapped modes,which give rise to resonances in the impedance of the structure. We have simulated theimpedances in the structure shown in Fig. 5 for a range of dielectric constants for thecapillary.11(a) (b)(c) (d)(e) (f)Figure 7: Z∥, Z∥, W∥ and W⊥ as computed by the GdfidL and ABCI codes, for a perfectlyconducting capillary7 ConclusionsWe have simulated the longitudinal and transverse wakefields and impedances due to dif-ferent designs of collimators for the SL COMB chamber. By analysing various taperingangles, we have determined that both the longitudinal and transverse loss factors becomesmaller as the tapering angle decreases. Our numerical results calculated using the ABCI12code, correspond reasonably well with analytical formulae. It allows to determine theeffect that these wakefields will have on the bunches used in the SL COMB experiment.It is expected that, unless the beam has a large offset from the axis of the beam pipe, thesewakefields will significantly affect the beam.We have also considered a modified geometry in which a thin capillary is insertedinside the beam pipe. Initially, our simulations were run using both ABCI and GdfidLfor a perfectly conducting capillary. The results of both codes for the real and imaginaryparts of the longitudinal and transverse impedances in this structure were found to matchup reasonably well for both codes, as did the wake potentials. In GdfidL an aluminiumcapillary was also considered, and we determined that the resistive wakes did not have alarge impact on the total wakes generated. By using a modified version of the ABCI code,which allows the user to include dielectric structures in the geometry, we were also able tosimulate the resonances in the impedance of capillaries with various dielectric constants.As a final comment the CST software [13] gave a similar transverse kick estimation.References[1] A. Mostacci et al, Proc. of IPAC 11, THYB01 (2011)[2] K. L. F. Bane, P. B. Wilson, T. Weiland, Wake Fields and Wake Field Acceleration,SLAC-PUB-3528 (1984)[3] L. Palumbo, V. G. Vaccaro, M. Zobov, Wake Fields and Impedance, LNF-94/041(1994)[4] K. Yokoya, Impedance of Slowly Tapered Structures, CERN SL-90-88 (1990)[5] G. V. Stupakov, Impedance of Small-Angle Collimators in High-Frequency Limit,SLAC-PUB-8857 (2001)[6] ABCI Code, http://abci.kek.jp/[7] O. Napoly, Y. H. Chin, B. Zotter, A Generalized Method for Calculating Wake Po-tentials”, Nucl. Instrum. Methods, A334, 255 (1993).[8] M. Dohlus, I. Zagorodnov, O. Zagorodnova, Impedances of Collimators in EuropeanXFEL, TESLA-FEL 2010-04 (2010)[9] G. V. Stupakov, Geometrical Wake of a Smooth Taper, SLAC-PUB-7086 (1995)13[10] J. Resta-Lpez, Single-bunch Transverse Emittance Growth due to Collimator Wake-field Effects, http://arxiv.org/pdf/1309.0480.pdf (2013)[11] GdfidL Code, http://www.gdfidl.de/[12] Y. Shobuda, Y. H. Chin, K. Takata, Impedance of a Ceramic Break and its ResonanceStructures, PR-STAB 17, 091001 (2014)[13] CST Code, https://www.cst.com/14

Studies of geometric wakefields and impedances due to collimators

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Studies of geometric wakefields and impedances due to collimators

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