Bane, Karl Leopold Freitag

Zagorodnov, I

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Wakefield Calculations for 3D Collimators ∗Igor Zagorodnov, DESY, Hamburg, GermanyKarl L.F. Bane, SLAC, Menlo Park, CA 94025, USAEUROTeV-Report-2006-074AbstractUsing a recently developed time domain numerical ap-proach we calculate the short-range geometric wakefieldsof 3D collimators and compare with analytical models. Wefind, in the diffractive regime, that the transverse mode kickfactor can be approximated from the change in field energybetween the beam pipe and the collimator if the collimatoris long, or using a “field clipping” estimate if it is short. Forcollimators of past and present measurements at SLAC, nu-merical, analytical, and measurement results are compared.INTRODUCTIONDesigning collimators is an important task for acceler-ator and FEL projects. In this report we study the short-range geometric wake of 3D (rectangular) collimators us-ing a time domain numerical method [1]. This method,together with an indirect 3D integration algorithm, allowsone to obtain accurate results for the difficult combinationof short bunches and long, gently tapered collimators [2].In this report, we consider symmetric collimators of thetype sketched in Fig. 1. Depending on bunch length, thewake can be thought of as being inductive, diffractive, orof an intermediate type [3]. If the collimator is round andρ1 ≡ αb2σ−1 ¿ 1, it is inductive, with kick factor [4]k⊥ =Z0c α2pi3/2σ(1b2− 1b1), (1)where σ is the rms bunch length (we consider only Gaus-sian bunches), Z0 = 377 Ω, and c is the speed of light. Theinductive regime for round collimators was studied numer-ically in [5], and the results were in agreement with Eq. 1.If the collimator has a rectangular cross-section, it is in-ductive when ρ2 ≡ αh2σ−1b−12 ¿ 1, with h the collimatorwidth; in this case [3]k⊥ =Z0c αh4pi1/2σ(1b22− 1b21). (2)If ρ1 ¿ 1 but ρ2 & pi2, then a rectangular collimator is inthe intermediate regime andk⊥ = 0.215AZ0c√ασb32, (3)with A a factor to be defined in the next section.When ρ1 À 1 both round and rectangular collimatorsare diffractive, a regime we study in the next section. Fi-nally, in the last section we demonstrate that, in fact, Eq. 1agrees well with numerical results for rectangular collima-tors with arbitrary ρ2 (only requiring ρ1 ¿ 1).∗Work supported by the Department of Energy, contract DE-AC02-76SF00515, the EUROFEL and the EUROTeV projects.Figure 1: Top half of a symmetric collimator.DIFFRACTIVE REGIMEWhen a beam encounters a collimator it takes time forthe primary fields of the beam to adjust themselves to anew steady-state configuration. Hence, we expect wake-field effects for short and long collimators to differ.In the longitudinal case, the impedance of a collimatorat high frequencies (the diffractive regime) [6, 7]Z‖ ≈ 2Ze , (4)with Ze the impedance associated with the beam’s (static)fields. We suggest that depending on the collimator lengthd, Ze should be found in two different ways. For a longcollimator, Ze is equal to the change in the field energybetween the beam pipe and the collimator; it has the formZe =1Q2Z0(∫Ω1(∇ϕ1)2ds−∫Ω2(∇ϕ2)2ds), (5)where ϕi is the solution to the Poisson equation∆ϕi(~x) = Z0Qδ(~x− ~x0) , ~x ∈ Ωiϕi(~x) = 0 , ~x ∈ ∂Ωi , i = 1, 2 ; (5′)Q is bunch charge, Ω1, Ω2, are the beam pipe and collima-tor cross-sections. For a short collimator, Ze is obtainedfrom the field energy clipped away by the collimatorZe =1Q2Z0∫Ω1−Ω2(∇ϕ1)2ds . (6)The loss and kick factors of a Gaussian bunch are ap-proximately given byk‖ =c√piσZe(0) , (7)k⊥ =c∆2[Ze(∆)− Ze(0)] , (8)where ∆ is beam offset from the axis.For a monopole mode the steady-state field pattern doesnot depend on pipe radius and both Eqs. 5 and 6 giveZe =Z02piln(b1b2), (9)and hencek‖ =Z0c2pi3/2σln(b1b2). (10)The dipole steady-state field pattern, however, depends onthe pipe radius and Eqs. 5 and 6 give different results. Fora long collimator (d→∞) we expectk⊥ =Z0c2pi(1b22− 1b21), (11)for a short collimator (d→ 0)k⊥ =Z0c4pi(1b22− b22b41). (12)In order to test Eqs. 11, 12, and to obtain a feeling about“long” and “short” we have calculated the loss and kickfactors for example collimators as functions of collimatorlength d. For the first example α = pi/2, b1 = 19 mm, andb2 = 1.9 mm; bunch lengths σ = 0.5, 0.3, 0.1 mm. Wefind that the numerically obtained loss factors are not sen-sitive to collimator length d, and Eq. 9 approximates themwell. The calculated kick factors are shown in Fig. 2 (leftframe). The straight gray lines give the asymptotic (long,short) approximations, Eqs. 11, 12. The numerical resultsagree well with the analytical estimates. Also, the directsums of the kicks for “in”- and “out”-transitions (calculatedseparately) are shown. We see that they agree with the kickfor a sufficiently long collimator, as expected.Figure 2: Kick factor vs. collimator length. A roundcollimator (left), a square or rectangular collimator (σ =0.3 mm, right).For 3D collimators, the loss factor for short and long col-limators is, in general, no longer nearly the same. In thiscase the energy impedance Ze can be found numerically bysolving the 2D problem (5’) (see, e.g. [7]). We have per-formed both 3D time-dependent and 2D electrostatic cal-culations for square and rectangular collimators with aper-ture half-height b2 = 1.9 mm, aperture width h = 3.8 mm(square) or 38 mm (rectangular), beam pipe half-height andhalf-width b1 = 19 mm. The results are presented in Fig. 2(the right frame) and Table 1. The gray lines in the figureshow the approximate kicks for the square collimator. Wesee from the table that the rectangular collimator gives akick very similar to the round one. Of the three collima-tors, the smallest kick is for the square one. We see thatEqs. 11, 12, can be useful for all three types of collimators.Table 1: Loss and kick factors as estimated by 2D electro-static calculation. The bunch length σ = 0.3 mm. “Short”means using Eq. 6, “long” Eq. 5.k‖ [V/pC] k⊥ [V/pC/mm]Typeshort long short longround 78 78 2.50 5.01rect. 56 72 2.43 6.11square 74 78 1.99 4.25From the above results we see that the kicks for long andshort step collimators are related bykshort⊥ ≈12klong⊥ . (13)Hence, in Eq. 3 we suggest one use A = 1 for a long col-limator (d → ∞) and A = 12 for a short one (d → 0). Fora collimator in the inductive regime, however, the resultshould be independent of d.The good agreement we have found between direct time-domain calculation [1] and the approximations (5, 6), sug-gests that the latter method can be used to approximateshort-bunch wakes for a large class of 3D collimators.RECTANGULAR COLLIMATORSIn our simulations for 3D collimators we used a time-domain numerical scheme [1] combined with an indirectintegration algorithm [2]. We consider now two sets of col-limators that were measured in experiment at SLAC. Thefirst set includes four collimators measured in 2001 [8].The parameters are given in Table 2. All rectangular col-limators have width h = 2b1 = 38 mm and no flat region(d = 0).Table 2: Geometry of SLAC collimators of 2001.Coll. # 1 2 3 4Type rect. square rect. rect.b2 [mm] 1.9 1.9 1.9 3.8α [mrad] 168 335 335 298In order to check the accuracy of the 3D discretizationwe have calculated wakefields for round collimators withb1 = 19 mm, b2 = 1.9 mm, α = 335 mrad using the 2.5Dand 3D codes. The two results are nearly indistinguishable(see Fig. 3 on the left). The wakes for square collimator#2 and rectangular collimator #3 are also plotted, for twobunch lengths, in Fig. 3.Figure 3: Transverse wake of Gaussian bunch, with σ =0.65 mm (left) and σ = 0.3 mm (right).In Fig. 4 a comparison of simulation (open symbols) andexperiment (closed symbols) for the SLAC collimators ismade. We estimate the accuracy in calculated k⊥ to beabout 1%. We note good agreement for rectangular colli-mators #1 and #4. On the other hand the short bunch resultsfor collimators #2 and #3 agree very well with the calcu-lations of the previous section, but they disagree with theexperimental data (by 20%). Tables 3, 4, compare simu-lated results, measurements, and the analytical estimates.Entries in bold type are the appropriate estimates for thesimulated results. Note that Eq. 1 results agree surprisinglywell with simulation for rectangular collimators with arbi-trary ρ2 (only requiring ρ1 ¿ 1).Figure 4: Kick factor vs. σ for various collimators.Finally, we have calculated loss and kick factors for theset of collimators used in the recent, 2006 SLAC experi-ment. The collimator description and measured data canbe found in [9]. (During the process of preparing this paperthe data had not yet been analyzed.) Perhaps the most inter-esting result in Table 5 is a noticeable difference in k⊥ forcollimators #2 and #3. The collimators have identical taperangles and apertures. However, for collimator #2 d = 0,for collimator #3 d = 1 m. In agreement with the analyticmodels discussed in this paper, a long collimator length re-sults in a kick factor increase by a factor ∼ 2. In orderto check the results of Table 5 we have done 2.5D simu-lations for round collimators with the same profiles. Theresults show the same qualitative behavior as those for therectangular collimators of Table 5.Table 3: Kick factor [V/pC/mm]; σ = 0.65 mm. Measure-ment errors are given in parentheses.Coll. # 1 2 3 4ρ1/ρ2 0.5/50. 1.0 1./98. 1.7/43.simul. 1.28 1.75 1.72 0.50meas. 1.2(0.1) 1.4(0.1) 1.4(0.1) 0.54(0.05)Eq. 1 1.24 2.5 2.5 1.0Eq. 3 2.4 3.3 3.3 1.1Eq. 12 2.5 2.5 2.5 0.62Table 4: Kick factor [V/pC/mm]; σ = 1.2 mm. Measure-ment errors are given in parentheses.Coll. # 1 2 3 4ρ1/ρ2 0.3/27. 0.5 0.5/53. 0.9/24.simul. 0.90 1.35 1.30 0.41meas. 0.78(0.13) 1.2(0.1) 1.08(0.1) 0.49(0.15)Eq. 1 0.7 1.3 1.3 0.5Eq. 3 1.7 2.4 2.4 0.8Eq. 12 2.5 2.5 2.5 0.6Table 5: Loss and kick factors for new set of collimators.k‖ [V/pC] k⊥ [V/pC/mm]Coll. # σ = 0.3 σ = 0.5 σ = 0.3 σ = 0.5mm mm mm mm1 50 28 1.9 1.72 60 33 3.6 3.13 63 33 6.1 5.14 40 24 0.7 0.85 81 47 7.1 6.86 51 24 2.9 2.37 60 34 3.1 2.78 56 28 3.0 2.4AcknowledgementsWe thank M. Dohlus, G. Stupakov and T. Weiland forhelpful discussions on wakes and impedances, and CSTGmbH for letting us use CST MICROWAVE STUDIO forthe meshing.REFERENCES[1] I. Zagorodnov, T. Weiland, Phys Rev STAB 8, 042001(2005).[2] I. Zagorodnov, DESY 06-081, May 2006.[3] G.V. Stupakov, SLAC-PUB-8857, June 2001.[4] K. Yokoya, CERN-SL/90-88 (AP), July 1990.[5] I. Zagorodnov et al., PAC’03, p. 3252 (2003).[6] S. Heifets and S. Kheifets, Rev Mod Phys 63, 631 (1991).[7] K.L.F. Bane, I. Zagorodnov, PAC’06, THPCH072.[8] P. Tenenbaum et al., PAC’01, p. 418 (2001).[9] N.K. Watson et al., PAC’06, MOPLS066.

Wakefield Calculation for 3D Collimators

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