Napoly, O

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Particle Accelerators, 1991, Vol. 36, pp.15-23Reprints available directly from the publisherPhotocopying permitted by license only@1991 Gordon & Breach Science Publishers, S.A.Printed in the United Kingdom.(1)THE WAKE POTENTIALS FROM THE FIELDS ONTHE CAVITY BOUNDARYO.NAPOLYDPhN-STAS, CE Saclay, Orme des Merisiers, F-91191 Gif/Yvette(Received 14 June 1991; in final form 18 July 1991)The wake potentials of axi-symmetrical structures are expressed, to their leading multipole order, in termsof the e.m.fields on the metallic boundary, with no contribution from the beam pipe. For the longitudinalpotential, the expression applies to any type of structure including re-entrant ones or with unequal in and outbeam-pipe radii. For the transverse case, however, the beam-pipe radii must be equal. These expressionsgeneralize the usual method of calculating the wake potentials over the cavity gap and extends its applicabilityto more general structures like collimators, tapers or re-entrant cavities. It should in particular be useful forcomputer calculations of the wake fields in the time domain.1. INTRODUCTIONThe beam dynamics of high-current circular and linear accelerators is dominated by theeffect of the impedances or wake potentials 1 of the metallic structures surrounding thebeam trajectory. The longitudinal and transverse wake potentials are defined bydef 1 j+ooWz(r, e, s) = - Q -00 dz EAr, e, z, t(z, s))andd f 1 j+ooW..l(r, e, s) ~ Q -00 dz (E..l + V x B)(r, e, z, t(z, s)) (2)where s is the distance behind a given origin zo == vt in the exciting bunch, of charge Q,and t(z, s) == (z + s)/v. Throughout this paper, we will concentrate exclusively on theultra-relativistic case where v == c and ~ == 00 and on structures with circular symmetryaround the z-axis. Then, given ro == (ro,Oo == 0) the radial position of the excitingbunch, the potentials obey the following multipole expansion 200andd~(r, 0, s) == '" rmr[f cos(mO) -w(m)(s)L..t dsm=O(3)00W--L(r, 0, s) == L m rm-1r[f(cos(mO) f - sin(mO) 0) w(m)(s) (4)m=l1516 O.NAPOLYKeeping only the leading tenns of this expansion, i.e. for particles close to the axis, andwith a convenient slight change of notations, the potentials reduce to~(r, 0, s) == wo(s)roW-L(r,O,s) == - Wl(S)ro(m == 0)(m == 1)(5)(6)The main consequence of Eqs.(5,6) is that, at the leading order in the multipole expansion,the longitudinal and transverse potentials experienced by a test particle behind the bunchdoes not depend on its transverse coordinates (r, B). This property is used in computercodes solving Maxwell's equations in the time domain, like TBCI 3 or ABCI 4, tocompute the wake potentials of cavities with side tubes of radius a from the longitudinaland azimuthal potentials evaluated at the tube radius r == a. Since the fields E z , Eeand B r are zero oli the surface of the tube, supposed infinitely conducting, the integralsgiving W z and We in Eqs.(1,2) reduce to finite range integrals over the gap of the cavity.This method improves the accuracy of the calculation for two reasons:• it avoids evaluating the wake fields on or near the bunch axis where the fields Erand Be are singular and where, therefore, the calculation of the regular componentsof the fields suffers the most from numerical noise.• as shown very clearl~ by the so-called 'field matching' techniques for solvingMaxwell's equations in simple geometries, the cavity is the cause of multiple-scattering of the electromagnetic wave accompanying the charged particle bunch.At any distance from the cavity, there are always scattered waves propagatinginside the beam tube and contributing to the wake potentials through the integralsof Eqs.(1,2). Cutting these integrals to a finite range over the beam tubes, whichis mandatory in a computer calculation, is therefore an approximation unless oneintegrates at r == a.However there are cases for which this method cannot be employed, e.g. structureswith unequal side-tube radii, like tapers, or structures for which there is no clear lineof sight lying on the side tubes, like collimators or re-entrant cavities. In general, allstructures for which one cannot draw, within the vacuum or dielectric region, a straightline at constant radius lying on both the external beam-tubes are not suitable for thismethod.In this paper, we derive expressions for the wake potentials which enables one toextend this method to structures of arbitrary shape. The wake potentials are given asintegrals of e.m.fields over the boundary of the metallic structure such that side tubesgive no contribution· to the integrals. In fact, one can also chose, when convenient, tointegrate over the more general set of "contours" lying on the side tubes and followingpartly the boundary partly a line at constant radius in the vacuum region. Such contoursare illustrated in Fig.2. The only restriction to applying these results is, in the case ofthe transverse potential, that the side tubes have equal radii.After introducing some definitions, the longitudinal potential is calculated in Section2 and the transverse one in Section 3. In both cases, we begin by re-deriving the r-independence of the wake potentials described by Eqs.(5,6) as an intennediate result.WAKE POTENTIAL FROM BOUNDARY FIELD 17~-:Zl(r~--+---~---~IIr IIIv-e__E:: _FIGURE 1 A simple cavity: notations.SaulzFIGURE 2 A multicell cavity with re-entrant shape. The contour ac(r) is composed of the three contoursaC l (r), aC2 (r) and aC3 (r) indicated in the figure.2. CALCULATION OF THE LONGITUDINAL POTENTIAL (m == 0)We consider an axi-symmetrical cavity of arbitrary shape with side tubes of radius ainand aout as shown in Fig.l. We denote by if, z the coordinates of a point on the boundary,and by ifmin and ifmax the minimum and maximum radius of the cavity, including thetubes. We assume that the bunch which excites the wake fields in the cavity has velocityv == C, total charge Q and is characterized by a longitudinal charge distribution A(s)normalized to one, i.e.]+00-00 >.(s) ds = 1 (7)Concentrating on the longitudinal wake-fields in the mode m 0 for which theazimuthal dependence drops out, it is convenient to decompose the electromagnetic fields18as followsO.NAPOLYE == E(O) + E(r)B == B(O) + B(r) (8)where (E(O), B(O)) are the e.m. fields which would be generated by the charge distributionin a smooth pipe, given byB(O)(r,s) == _Q- A(S)e 21rCEorE~O) (r, s) == c B~O) (r, s)(9)(10)(11 )the other components being zero. The wake potentials W z and W..l are entirely given bythe additional fields (E(r), B(r)), sometimes called 'radiated fields', which are solutionsof homogeneous Maxwell's equations and vanish at infinite distance z == ±oo in thetubes.In analogy with the definition of the longitudinal potential given by Eq.( l), we intro-duce the two following quantitiesWz(r,Si zl,Z2) ~f - ~ 1:2 dz E~r)(r,z,t(z,s))where Zl and Z2 are longitudinal coordinates such that the line element [(r, Zl), (r, Z2)]lies inside of the cavity (see Fig.l), and(12)where Z1 (r) and Z2 (r) denote the longitudinal coordinates of the boundary at the radiusr and where the sum indicates that one eventually has to add up the contributions of N rdisconnected line elements as shown in Fig.2 for N r == 2. Notice that for small enoughr, zl(r) or z2(r) can be infinite. In fact, for r < rmin, both zl(r) and z2(r) are infiniteand it is a matter of definition to write(13)We now investigate the radial dependence of the generalized functions Wz and W z.Using the fact that, for m == 0, the radiated fields reduce to (E~r), E~r), B~r)) and that, asalready mentionned, they satisfy homogeneous Maxwell's equations, it is easy to derivethe following equalities== 8z Er (r, Z, t(z, s)) + 8t Be(r, z, t(z, s))leading to. ) _ 1 rZ2 d ((r) (r))( ())8rW z (r, S, Zl, Z2 - - Q lZl dz dz Er + cBo r, z, t Z,S (15)WAKE POTENTIAL FROM BOUNDARY FIELD 19(22)up to terms in 1/ ~2. Therefore, denoting by ~Zl ,Z2 the difference operator between Z2and Zl, one can write in the limit v ~ c8rWz (r, s; Zl, Z2) = - ~~Zl'Z2(E~r) + cB~r))(r, Z, t(z, s)) (16)For r < rmin, this equation shows, by taking Zl == -00 and Z2 == +00 for which theradiated fields vanish, that the longitudinal potential Wz (r, s) does not depend on r, asexpressed by Eq.(5).Turning to W z, one getsd - _ " [ ._ _ 1 ((r) _ d:Z2drWzCr,s) - ~ 8rWzCr,s,zl(r),z2(r)) - Q Ez (r,z2(r),t) d,;(r)-Eir)(r, Zl (r), t) dd; (r)) ] (17)Using Eq.(16), this can be conveniently written as1 2NrdWzCr,s) = Q LEi [(E~r) +cB~r))dr+Eir)dZ] (r,zi, t(Zi, s)) (18)i=lwhere i denotes the index of the boundary points at radius r, starting from Z == -00,and Ei == (_I)i-l. Integrating Eq.(18) from r == 0, where Wz(r,s) == wo(s), tor 2: sup(ain, aout) givesWz(r,s) -wo(s) == Ql ( [(E~r) +cB~r))dr+Eir)dZ] (19)Jac(r)where the contour of integration ac(r ), as illustrated in Fig.2, is in general composedof several pieces following the boundary of the cavity. By relating, through Eq.(8), theradiated fields to the fields (E, B) which satisfy the metallic boundary conditionE r dr + E z d:Z == 0 (20)and the fields (E(O), B(O») given by Eqs.(9,10), this integral simplifies towo(s) == Wz(r, s) - QC ( Be(r, z, t(z, s)) dr + _I_In (aout ) '\(s) (21)Jac(r) 1rEo ainThe important property of the above expression is that the parts of the contour such thatdr == 0, and in particular the side tubes, do not contribute to the integral. Eq.(21) canalso be written in terms of the radiated field B~r):WO(s) == Wz(r,s) - QC ( B~r)(r,z,t(z,s)) dr+ _I_In (aout ) '\(s)JaC(r) 21rEo ainMoving r to the maximum radius rmax of the cavity allows to express the wakepotential in terms of the fields Be on cavity boundary as followsWO(s)==-QC ( Be(r,z,t(z,S))dr+_l-In(aout) '\(s) (23)Jac 1rEo ainwhere now ac denotes .the full contour of the cavity.20 O.NAPOLY3. CALCULATION OF THE TRANSVERSE POTENTIAL (m == 1)The calculation of the transverse potential is parallel to the preceding one, althoughslightly more complicated. First, some care has to be taken in the definition of thefields (E(O), B(O)) since, unlike in the m == 0 case, the smooth-pipe solution depends onthe pipe radius. We assume that (E(O), B(O)) refers to the fields in the incoming tubewith radius ain in such a way that the radiated fields (E(r),B(r)) vanish for z ~ -00.In the other limit z ~ +00, they tend to the difference between the outgoing andincoming smooth-pipe solutions. Therefore, 'in both limits, their longitudinal componentsE~r), B~r), as well as their tranverse combination (EY) + v x B(r)) vanish for v ~ c.We introduce the generalized transverse wake functionsand1 j Z2. ~ (r) (r)W-t(r,O,s,zl,Z2) - Q Zl dz (E-t +v x B )(r,O,z,t(z,s)) (24)(25)with the same notations as for Eqs.(11,12). Using the homogeneous Maxwell's equations,one can derive, as was done in Eqs.(14,15) for the longitudinal case, the followingidentities(26)(27)(~8r(rWe) - ~8eWr) (r,O,S;ZI, Z2) = - ~ b.z1 ,z2 Bir)(r,O,z,t(z,s))neglecting tenns of order 1/12 as in Eq.(14). Expressing the azimuthal dependence ofthe e.m. fields projected in the mode m == 1, as followsone can writeE r == er cos () ,B r == br sin () ,Ee == ee sin () ,Be == be cos () ,E z == ez cos ()B z == bz sin () (28)Wr == W r cos ()We == -We sin ()The differential equations (25,26) then lead to(1 1). _ C (r);:8r(rwe) - ;:wr (r, s, ZI, Z2) - Q b.Z1 ,z2 bz (r, z, t(z, s))Introducing the sum and the difference of Wr and Wea == Wr + We8 == Wr - We(29)(30)(31)(32)WAKE POTENTIAL FROM BOUNDARY FIELD 21the differential system translates toBra-(r, S; Zl, Z2) = - ~ ~Zl,Z2 (e~r) - cb~r)) (r, z, t(z, s)) (33)r12 Br (r2t5)(r, S; Zl, Z2) = - ~ ~Zl,Z2 (e~r) + cb~r)) (r, z, t(z, s)) (34)For r < rmin, with ZI == -00 and Z2 == +00, these are equations for the sum anddifference of the actual radial and azimuthal wake potentials. In view of the discussionbeginning the section, their right-hand side vanish and the equations are easily solved toyieldW r == we == WI independent of r (35)(40)reproducing Eq.(6).Turning to the functions a == wr + we and "8 == wr - we derived from W -l definedby Eq.(25), one can show, in the same way as in the longitudinal case, thatd;(j = ~ [(e~r) - cb~r))dr - (e~r) - cb~r) - e~r) - cb~r))dZ] (36)d(r28) = ~r2 [(e~r) + cb~r))dr - (e~r) - cb~r) + e~r) + cb~r))dZ] (37)Integrating these equations from r == 0 where a(r, s) == 2WI (s) and "8 == 0 to a radius rlarger than the tube radii, yieldsa(r, s) - 2WI (s) == Q1 r [(e~r) - cb~r))df - (e~r) - cb~r) - e~r) - cb~r) )dZ] (38)JacCr)r2"8(r, s) == Q1 r 1'2 [(e~r) + cb~r))df - (e~r) - cb~r) + e~r) + cb~r))dZ] (39)JacCr)which simplifies again when Eq.(8) is used to translate it into the fields (E, B) satisfyingthe boundary conditionsEe == 0 and B r dE - B z dr == 0and into the smooth-tube solutions (ECO), BCO)) such thatEiO) == BiO) == E~O) - cB~O) == E~O) + cB~O) == 0 (r == (0) (41)everywhere. One therefore ends up with2WI(S) == a(r, s) - Q1 r [ezdf - (er - cbo)dZ] (42)JacCr)r2"8(r, s) == Q1 r 1'2 [ezdf - (er - cbo)dZ] (43)JacCr)Unlike what happens in the longitudinal case, the integration over the beam tubes doesnot drop out from Eq.(42) due to the presence of the term in dE. It can however beeliminated between Eq.(42) and Eq.(43) but only when the tube have equal radii(44)22leading toO. NAPOLY(a2 - r2 )wr (r, s) + (a2 + r2 )We(r, s)+Ql r (r2 - a2 ) [ezcLr - (er - cbo)dZ]Jac(r)(45)(46)Moving r to r max , one gets the transverse potential as an integral over the contour BCof the cavity with no contribution from th~ beam tubes(47)If the beam tubes have unequal radii, it is straightforward to eliminate the contributionof either of the tubes leading to the same expressions as in Eqs.(45,46) with a replacedby either ain or aout. For calculations where s == Iz - ct I is not too large, one shouldpresumably eliminate the contribution from the outgoing tube since the incoming one isaffected by the back-scattering of the incoming wave only over distances of the order ofs.4. CONCLUSIONWe have derived expressions of the longitudinal (m == 0) and transverse (m == 1)wake potentials, for axi-symmetrical metallic structures, in terms of integrals of theelectromagnetic fields, at constant distance Iz - ctI behind the exciting bunch, which donot involve the value of the fields in the incoming and exiting beam tubes. A familyof finite contours of integration can be used, namely the ones involving line elementsat constant but not necessarily equal radii, and elements of the metallic boundary ofthe structure. In particular one can chose to integrate over the contour following theboundary of the structure. In the case of the longitudinal potential, the integral over theboundary involves only the magnetic field Be and is such that all boundary elementsparallel to the axis, including the beam tubes, drop out from the integral. This resultapplies to structures of arbitrary shape.These expressions should be especially useful when implemented in time-domainwake-field computer programs since they generalize and extend the applicability of theusual method which consists in integrating the wake fields, when possible, over the gap ofthe cavity. In the case of re-entrant cavity shape, for instance, the use of the WAKCORpost-processor of the TBCI program can be avoided by integrating directly the wakefields along the cavity boundary.ACKNOWLEDGEMENTI thank A. Mosnier for useful discussions and G. Dome for careful reading of themanuscript.REFERENCES1. A.W. Chao, AlP Conf. Proc 105 (AlP, New York, 1983)WAKE POTENTIAL FROM BOUNDARY FIELDK. Bane, T. Weiland and P.B. Wilson, AlP Conf. Proc 127 (AlP, New York, 1985)P.B. Wilson, SLAC-PUB-4547 (1989)S.A. Heifets and S.A. Kheifets, SLAC-PUB-5297 (1990)2. T. Weyland, NIM 216 (1983), 313. T. Weyland, DESY 82-015 (1982)4. C. Chin, CERN/LEP-TH/88-3 (1988)23

The Wake Potentials From The Fields On The Cavity Boundary

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