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Wake fields effects due to surface roughness in a circular pipe
Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
Wake fields effects due to surface roughness in a circular pipe M. Angelicia, F. Frezzab, A. Mostaccic, L. Palumboa,* b
a Dip. Energetica, Universita ‘‘La Sapienza’’, Via Scarpa 14, 00161 Rome, Italy Dip. Ing. Elettronica, Universita ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Rome, Italy c CERN, CH-1211 Geneve 23, Switzerland
Received 21 May 2001; received in revised form 26 February 2002; accepted 28 February 2002
Abstract The problem of the wake field generated by a relativistic particle travelling in a long beam pipe with rough surface has been studied by means of a standard theory based on the hybrid modes excited in a periodically corrugated waveguide with circular cross-section. Slow waves synchronous with the particle can be excited in the structure, producing wake fields whose frequency and amplitude depend on the depth of the corrugation. An analytical expression of the wake field is given for very small corrugations. r 2002 Elsevier Science B.V. All rights reserved. PACS: 29.17; 29.27 Keywords: Wakefield; Coupling impedance; Circular pipe; LCLS
1. Introduction The effect of surface roughness is a subject arisen in the design of machines with extremely short bunches of the order of tens of microns. In this case, in fact, the surface roughness may be a source of wake fields which might significantly increase the beam emittance and the energy spread. Even the surface roughness due to residual defects in workmanship, may be responsible of the longitudinal emittance growth due to wakefields, which is particularly harmful in the design of SASE Free Electron Lasers [1,2] because of the roughness in the beam tube. After any machining process the metal wall of the pipe shows a random behaviour with valleys *Corresponding author. E-mail address: [email protected] (L. Palumbo).
and crests with typical r.m.s. depth much smaller than the crests distance. Several models have been studied considering both periodic and random roughness [3–7,12,13]. The periodic systems, due to the multiple coherent reflections of waves from the perturbations, can sustain slow harmonics which are in synchronism with the beam and can be described in terms of wake fields. In the case of random surfaces, the multiple reflections are not coherent and the field acting on the beam is not stationary producing, therefore, a weaker effect. Although the random case is more realistic, the periodic geometry has been often considered in order to give a conservative estimate of the roughness effects. A synchronous wake due a slow wave propagating in a corrugated pipe has been first derived in Ref. [4] and applied to the LCLS case in Ref. [6]. It predicts that the amplitude of the resonant wake does not depend on the
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 0 7 9 5 - 7
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
corrugation height when this last is comparable to the period. However, it has been predicted in Refs. [9–11] that when the height of the periodic wall corrugations becomes smaller than the period, the loss factor for the modes rapidly decreases. We believe that the latter models, predicting a wakefield amplitude that vanishes for vanishing depth of the corrugations, are more appropriate to describe the effect of beam pipes with corrugations. In general, the different models predict an effect that depends upon the radius and the length of the pipe, and on the details of the surface roughness. In this paper we study the problem of the wake fields produced by an ultra-relativistic charge travelling inside a beam tube with a periodic corrugation making use of a standard theory based on the hybrid modes propagating in the waveguide [8,9]. Aim of the paper is to find an analytical expression of the wake fields, expliciting the dependence upon the corrugation height for the particular case of very small corrugation. The paper is structured as follows: in Section 2 we describe the method used, in Section 3 we present the results: first the dispersion relation for the fields and the frequency where the wave is synchronous with the charge can be excited; then, the amplitude of the field excited by the charge, the wake function and the coupling impedance.
2. The method Let us consider a periodically corrugated infinite waveguide with circular cross-section, with inner radius a and outer radius b: We model the wall roughness as a series of periodic (with period L) obstacles of height h ðh ¼ b aÞ and thickness t (see Figs. 1 and 2). The charge travels along the zaxis; we assume t5L; L5l and the ohmic losses in the material negligible. The periodicity of the geometry along the z-axis allows us the use of Floquet’s theorem which implies a field solution independent of the period L (obtained from a single cell). The steps are the following: at first we solve the homogeneous problem, finding the modes propagating in the waveguide and their features (the dispersion equation, the cut-off
11
z h
h Fig. 1. Relevant geometry.
Fig. 2. Schematic view of the waveguide and notation adopted.
frequency and the frequency where the synchronous wave is excited). The dispersion relation is found by applying the continuity conditions for the field components over the boundary between the slot (the space inside the corrugation) and the internal region of the waveguide. The field inside the waveguide is considered as generated by the magnetic and the electric Hertz potentials along the z-axis. Then we apply the Lorentz reciprocity principle, including the charge as an impulsive source, finding the coefficients used to express the electric field along the z-axis.
3. Results 3.1. The homogeneous problem The electromagnetic fields inside the corrugation are considered to be those due to propagating
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
12
radial modes; higher-order evanescent modes are considered negligible, this assumption is justified under the hypothesis that the wavelength is much greater than the distance between two corrugations ðlbLÞ: In a cylindrical coordinate system ðr; f; zÞ; the components of the electromagnetic field inside the corrugation field are ErC ¼ 0;
ð1Þ
EfC ¼ 0; X EzC ¼ ½Cn Jn ðk0 rÞ þ Dn Yn ðk0 rÞ cosðnfÞ;
ð2Þ ð3Þ
n
X
HrC ¼
n
n ½Cn Jn ðk0 rÞ þ Dn Yn ðk0 rÞ sinðnfÞ; jomr ð4Þ
HfC ¼ HzC
k0 X ½Cn Jn0 ðk0 rÞ þ Dn Yn0 ðk0 rÞ cosðnfÞ ð5Þ jom n
¼0
ð6Þ
jðotb0n zÞ
where e is assumed and suppressed and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b0n ¼ k02 kt2 ð7Þ Where b0n is p the propagation conffiffiffiffiffiffiffiffihybrid-mode ffi stant, k0 (o= m0 e0 ) is the free-space propagation constant and kt is the transverse wave number. The field inside the waveguide is considered as generated by the Hertz potentials along the z-axis: X 0 Pez ¼ An Jn ðkt rÞ cosðnfÞejðotbn zÞ ; ð8Þ n
Pmz ¼
X
0
Bn Jn ðkt rÞ sinðnfÞejðotbn zÞ
ð9Þ
n
which are related to the field by the equations: E ¼ jomr Pm þ ðk2 þ rrÞPe ;
ð10Þ
H ¼ joer Pe þ ðk2 þ rrÞPm
ð11Þ
where Pm ¼ Pmz z0 ;
Pe ¼ Pez z0 :
ð12Þ
The potential Pez generates TMz modes and the potential Pmz generates TEz waves. The superposition of both kind of modes gives origin to hybrid modes. So the expressions of the components of the hybrid modes in the internal region of
the waveguide are given by Eqs. (10) and (11) using relations (8) and (9): h i n ern ¼ jBn om Jn ðkt rÞ jAn b0n kt Jn0 ðkt rÞ r 0 cosðnfÞejðotbn zÞ ; ð13Þ h i 0 n efn ¼ jBn omkt Jn0 ðkt rÞ þ jAn b0n Jn ðkt rÞ ejðotbn zÞ ; r ð14Þ 0
ð15Þ ezn ¼ An kt2 Jn ðkt rÞ cosðnfÞejðotbn zÞ ; h i 0 n hrn ¼ jBn b0n kt Jn0 ðkt rÞ jAn oe Jn ðkt rÞ ejðotbn zÞ ; r ð16Þ h i n hfn ¼ jBn b0n Jn ðkt rÞ jAn oekt Jn0 ðkt rÞ r 0 cosðnfÞejðotbn zÞ ; ð17Þ 0
hzn ¼ Bn kt2 Jn ðkt rÞ sinðnfÞejðotbn zÞ :
ð18Þ
Applying the boundary condition at r ¼ b (the tangential electric field in the perfectly conducting wall vanishes) from Eq. (3) we find Cn Jn ðk0 bÞ EzC ðr ¼ bÞ ¼ 0 ) Dn ¼ ð19Þ Yn ðk0 bÞ and imposing the continuity of the tangential components of the field at r ¼ a: EzC ¼ Ez ;
HfC ¼ Hf
for r ¼ a
ð20Þ
that is, using Eq. (19) Jn ðk0 bÞ Cn Yn ðk0 aÞ ¼ An kt2 Jn ðkt aÞ ð21Þ Cn Jn ðk0 aÞ Yn ðk0 bÞ k0 Jn ðk0 bÞ Cn Jn0 ðk0 aÞ Cn Yn0 ðk0 aÞ Yn ðk0 bÞ jom 0 n ¼ jBn bn Jn ðkt aÞ jAn oekt Jn0 ðkt aÞ: ð22Þ a From Eqs. (21) and (22) we find the dispersion relation for the hybrid modes: Jn0 ðk0 aÞYn ðk0 bÞ Jn ðk0 bÞYn0 ðk0 aÞ Jn ðk0 aÞYn ðk0 bÞ Jn ðk0 bÞYn ðk0 aÞ ¼
k0 Jn0 ðkt aÞ b02 n2 Jn ðkt aÞ j 2n 3 : kt Jn ðkt aÞ a kt k0 Jn0 ðkt aÞ
ð23Þ
The modes of interest are the TM0m : the TEz and the TMnm (with na0) modes do not give any
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
In the hypothesis of small corrugations ða-bÞ the cut-off frequency is found to be (see Appendix A): c x01 fco ¼ ð25Þ 2p a þ h where x01 is the first zero of the Bessel function of first kind and order zero. It has to be noted that, for h-0; (25) tends to the cut-off frequency of the TM01 mode in the smooth waveguide of radius a; as expected. In the same hypothesis of small corrugations we found the frequency where the synchronous wave is excited (crossing frequency): c x01 pffiffiffiffiffi: f%cr ¼ ð26Þ 2p ah It shows pffiffiffithe typical behaviour of proportionality to 1= h: Figs. 3 and 4 report the Brillouin diagrams when h=a ¼ 0:1 and 0.8. The dashed line is the dispersion curve for the TM01 mode in the corrugated waveguide and the continuous one is the straight line b00 ¼ k0 : For b00 > k0 the wave is slow and can be synchronous with the charge. This is just the fundamental mode
200
150
'0
contribution in the application of the reciprocity principle. The dispersion relation (23) for n ¼ 0 becomes J00 ðk0 aÞY0 ðk0 bÞ J0 ðk0 bÞY00 ðk0 aÞ k0 J00 ðkt aÞ ¼ : ð24Þ J0 ðk0 aÞY0 ðk0 bÞ J0 ðk0 bÞY0 ðk0 aÞ kt J0 ðkt aÞ
13
100
50
0 10
20
30
40 k0
50
60
70
Fig. 4. Brillouin diagram for a circular cross-section waveguide of radius a with periodic corrugations of depth h and h=a ¼ 0:8:
which becomes synchronous at a given frequency. It is apparent that other modes can propagate in the beam tube, and that the coupling effects can in principle subtract energy to the synchronous wave. However, one can see after some calculations that the coupling factors for shallow corrugations are proportional to h2 : 3.2. Including the sources Once derived the modes of structures, having solved the homogeneous problem, the field generated by a point charge can be found by means of the Lorentz reciprocity principle [14]: Z Z 7 7 ðEn H E Hn Þ n dS ¼ J E7 n dV : ð27Þ
100 80
S
60
V
'0
The sign þ is for the wave travelling along þz and the sign is for the wave travelling along z: The current density of a point charge travelling onaxis, J; used in the reciprocity principle, is modelled as an impulsive source
40 20 0 40
50
60
70
80
90
k0
Fig. 3. Brillouin diagram for a circular cross-section waveguide of radius a with periodic corrugations of depth h and h=a ¼ 0:1:
Jðr; f; z; oÞ ¼ q
dðrÞ dðfÞejðz=vp Þo z0 r
ð28Þ
where z0 is the unit vector along z-axis, q is the charge and vp is the charged particle velocity. The electromagnetic field can be expressed using
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
14
the modal expansion [14]: X E¼ ðan Eþ n þ bn En Þ;
domain is Ez ðz; tÞ ¼
ð30Þ
where c is the light velocity in free-space and Z0 is the free-space characteristic impedance. From Eq. (38) it can be noted that Ez has a p phase difference with the charge (the sign), meaning that it is a decelerating field and that the height of the corrugation h fixes not only the crossing frequency, but also the field amplitude through the factor h=a3 : Another feature of the electric field component along the z-axis that can be seen from Eq. (38) is the independence from the radial coordinate r:
n
H¼
X
ðan Hþ n þ bn H n Þ
n
where jbzn z Eþ ; n ¼ ðetn þ ezn z0 Þe þjbzn z E : n ¼ ðetn ezn z0 Þe
ð31Þ
jbzn z Hþ ; n ¼ ðhtn þ hzn z0 Þe þjbzn z H : n ¼ ðhtn þ hzn z0 Þe
ð32Þ
The coefficients an and bn are found by applying the Lorentz principle (27). For n ¼ 0; the present case, they are given by (see Appendix A): a00 ¼ b00 ¼
qdððk0 =bÞ b00 Þ ; 2poeb00 a2 F ðkt aÞ
qdððk0 =bÞ þ b00 Þ 2poeb00 a2 F ðkt aÞ
ð33Þ
ð34Þ
where b is the relativistic factor and F ðkt aÞ ¼
J12 ðkt aÞ
J0 ðkt aÞJ2 ðkt aÞ
ð35Þ
and J0 ðkt aÞ; J1 ðkt aÞ and J2 ðkt aÞ are the Bessel functions of first kind and order 0, 1 and 2, respectively. Therefore, it becomes possible to find the expression of the electric field in the frequency domain along the z-axis using Eqs. (29) and (31) for n ¼ 0 Ez ¼ a0 eþ z0 b0 ez0
0 8qZ0 ch cosðo % cr tÞejb0 z 2 3 ðx01 Þ pa
ð29Þ
ð36Þ
which using the coefficients (33) and (34) becomes q Ez ðr; f; z; oÞ ¼ 0 2 o % cr eb0 a F ðkt aÞ 0 k0 k0 b00 þ d þ b00 ejb0 z kt J0 ðkt rÞ d b b ð37Þ where o % cr is the crossing angular frequency (expressed by 2p times the Eq. (26)). In the hypothesis of small corrugation (h-0), for ultrarelativistic particles (b-1) and close to the crossing frequency (kt -0) the electric field in the time
ð38Þ
3.3. Longitudinal coupling impedance and wake function Following the standard definition of the longitudinal coupling impedance per unit length Zz ðoÞ [15]: @Zz ðoÞ 1 ¼ Ez ðx ¼ 0; y ¼ 0; z; oÞejoz=c ; ð39Þ @z q from Eq. (37) we get @Zz ðoÞ 4Z0 ch ¼ ½dðo o % cr Þ þ dðo þ o % cr Þ: ð40Þ @z ðx01 Þ2 pa3 Again from the definition [15], it is easy to get the longitudinal wake function per unit length wz ðtÞ @wz ðtÞ Ez ðz; tÞ joz=c ¼ e : ð41Þ @z q From Eq. (38): @wz ðz; tÞ 8Z0 c h ¼ cosðo % cr tÞ: @z ðx01 Þ2 p a3
ð42Þ
Both the longitudinal coupling impedance (40) and the longitudinal wake function (42) are proportional to the depth height. This result agrees with Ref. [6] for what concerns the resonant frequency but, in contrast, it predicts an amplitude which scales with h=a: The disappearance of the wake amplitude when h-0 is somehow expected from the physical point of view and agrees with the general behaviour of the wake predicted other authors [11]. The availability of a simple analytical expression of the wake, although within the limit of validity of the model, represents a certain
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
advantage for the estimate of the wake effects on the beam. Just as an example we estimate the effect of a periodic roughness considering the LCLS parameters [2]. We assume a rectangular bunch, of temporal dimension 2T; where T is given by T ¼ pffiffiffi ð 3=cÞsl ; being sl (¼ 15 mm) the longitudinal dimension of the bunch. We find that the crossing frequency is given by 1 f%cr ¼ 2:29 1011 pffiffiffi ðHzÞ h
ð43Þ
and the amplitude of the wake function per unit length is given by w00 ¼ 3:2017 1018 h ðV=CmÞ:
15
4. Conclusions We have derived the longitudinal wake due to a periodic corrugation in a circular beam pipe in the case of very small corrugation, and for a wavelength of the wake larger than the period. A simple analytical expression of the wake shows that the amplitude of the sinusoidal wake function is proportional to the corrugation height h and the slow wave frequency, p synchronous with the beam, ffiffiffi is proportional to 1= h: This result agrees with the behaviour of a wake decreasing with the corrugation height that has been recently predicted by other authors.
ð44Þ Appendix A
The normalised energy spread, is given by Ref. [6] DE rms w00 DQ 1 sinð4o % cr TÞ ¼ 1 2E0 2ðo 4o E0 % cr T % cr TÞ2 4 )1=2 sinðo % cr TÞ ð45Þ o % cr T where E0 ¼ 14:35 GeV is the total energy of the electron beam, D ¼ 112 m is the total length of the path followed by the beam and Q ¼ 0:1 nC is the bunch charge. In Fig. 5 is reported the energy spread vs. h (m). The design parameters require the energy spread to be o5 104 ; which is verified for value of h of the order of tens of microns.
A.1. Solution of the dispersion relation In the hypothesis of small corrugations and from the definition of Wronskian, it is possible to use the following approximations [16] 2 J00 ðk0 aÞY0 ðk0 bÞ J0 ðk0 bÞY00 ðk0 aÞ ¼ ðA:1Þ pk0 a J0 ðk0 aÞY0 ðk0 bÞ J0 ðk0 bÞY0 ðk0 aÞ ¼ so the dispersion relation becomes kt J0 ðkt aÞ ¼ k0 h: k0 J00 ðkt aÞ
2h pa
ðA:2Þ
ðA:3Þ
Considering that k02 ¼ kt2 þ kz2 ; and that at cut-off kz ¼ 0; k0 ¼ kt ; we get J0 ðk0 aÞ ¼ k0 h ðA:4Þ J00 ðk0 aÞ
5 10-5
4 10-5
∆Erms/E0
and solving this equation for k0 3 10-5
k0 ¼ 2 10-5
1 10
-5
0 0
1 10-5
2 10-5
3 10-5
4 10-5
5 10-5
6 10-5
7 10-5
h
Fig. 5. Energy spread for the circular cross-section waveguide vs. h (m).
k0S a aþh
ðA:5Þ
where k0S is the propagation constant for the smooth waveguide. It is immediate to find the cutoff frequency from the propagation constant expressed by Eq. (A.5). For the smooth waveguide the dispersion relation is given by J0 ðkt aÞ ¼ 0
)
kt a ¼ x0m
ðA:6Þ
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
16
where x0m is the mth zero of the Bessel function of the first kind of order 0. Being x the distance from the zero x0m and making x-0; it is possible to write x J00 ðkt aÞ ¼ J00 ðx0m xÞ- J1 ðx0m Þ J1 ðx0m Þ; x0m ðA:7Þ
A00 ¼ 2oeb00 kt2
J0 ðkt aÞ ¼ J0 ðx0m xÞ-J1 ðx0m Þx:
Solving the integral we find
ðA:8Þ
From Eqs. (A.7), (A.8), (A.1), (A.2) it is possible to get another approximated expression of the dispersion relation 1 x k0 1 ðA:9Þ 1þ ¼ : x x0m kt k0 h Solving this second-order equation in x x¼
k02 ha x0m
Ann ¼ 2
Z
ðetn htn Þ z0 dS
ðA:16Þ
S
that for n ¼ 0 is Z
a 0
½J00 ðkt rÞr dr
A00 ¼ 2poeb00 kt2 a2 F ðkt aÞ:
Z
2p
df:
ðA:17Þ
0
ðA:18Þ
From Eqs. (A.15) and (A.18) it is easy to find expression (33). Likewise, we find the expression for coefficient (34) from Eq. (A.13).
References ðA:10Þ
we get kt a ¼ x0m
ðk0 aÞ2 h x0m a
ðA:11Þ
Imposing the crossing condition kt ¼ 0 we get the propagation constant at the crossing frequency x 2:40 o % ffiffiffiffiffi ¼ pffiffiffiffiffi ¼ : ðA:12Þ k%0 ¼ p0m c ah ah It is immediate to get the crossing frequency from this equation. A.2. Calculation of the coefficients for the modal expansion The Lorentz reciprocity principle assumes the expression Z X X bm Anm am Bnm ¼ J Eþ ðA:13Þ n dV m
X
and
V
m
bm Cnm
m
X m
am Dnm ¼
Z V
J E n dV
ðA:14Þ
where the coefficients Anm ; Bnm ; Cnm and Dnm are related to the transverse of the nth mode. In the case under examination n ¼ m; Eq. (A.14) becomes Z J E ðA:15Þ an Ann ¼ n dV V
[1] A VUV FEL at TTF Linac, Conceptual Design Report TESLA-FEL 95-03, DESY Hamburg, 1995. [2] LCLS Design Study Group, SLAC Report No. SLAC-R521, 1998. [3] K.L.F. Bane, C.K. Ng, A.W. Chao, Estimate of the impedance due to wall surface roughness, Report SLACPUB-7514, SLAC, 1997. [4] A. Novokhatski, A. Monier, Proceedings of 1997 Particle Accelerator Conference, IEEE Press, New Jersey, 1997. [5] G.V. Stupakov, Physical Review Special Topics, AB, Vol. 1, 1998. [6] K.L.F. Bane, A. Novokhatski, The resonator impedance model of surface roughness applied to the LCLS parameters, Report SLAC-AP-117, SLAC, 1999. [7] K.L.F. Bane, G.V. Stupakov, Impedance of a beam tube with small corrugations, Report SLAC-PUB-8599, SLAC, 2000. [8] G.H. Bryant, Proc. IEE 116 (1969) 203. [9] A. Mostacci, L. Palumbo, F. Ruggiero, S. Ugoli, Wake fields effects due to surface roughness, Eighth International Workshop on Linear Colliders, Frascati, 1999. [10] G. Stupakov, R.E. Thomson, D. Walz, R. Carr, Effects of beam-tube roughness on X-ray free electron laser performance, Physical Review Special Topics, AB, Vol. 2, 1999. [11] G. Stupakov, Surface roughness impedance, Proceedings of the 19th Advanced ICFA BD Workshop, Arcidosso 2000, Vol. 581, AIP, New York, 2001. [12] A.V. Agonov, A.N. Lebedev, Effects of channel roughness on beam energy spread, Proceedings of the 19th Advanced ICFA BD Workshop, Arcidosso 2000, Vol. 581, AIP, New York, 2001. [13] E. Di Liberto, F. Frezza, L. Palumbo, Effects of statistical roughness on the propagation of electromagnetic fields in a circular waveguide, Proceedings of the 19th Advanced ICFA BD Workshop, Arcidosso 2000, Vol. 581, AIP, New York, 2001.
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17 [14] R.E. Collin, Field Theory of Guided Waves, 2nd Edition, IEEE Press, New York, 1991. [15] L. Palumbo, V. G. Vaccaro, M. Zobov, in: S. Turner (Ed.), Proceedings of the CERN Accelerator School: Advanced Accelerator Physics Course, Rhodes, 1993, No. 95-06 in
17
CERN Yellow Report, European Laboratory for Particle Physics, Geneva, Switzerland, 1995, pp. 331–390. [16] C.A. Balanis, Advanced Engineering Electromagnetics, Wiley, New York, 1989.
Wake fields effects due to surface roughness in a circular pipe M. Angelicia, F. Frezzab, A. Mostaccic, L. Palumboa,* b
a Dip. Energetica, Universita ‘‘La Sapienza’’, Via Scarpa 14, 00161 Rome, Italy Dip. Ing. Elettronica, Universita ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Rome, Italy c CERN, CH-1211 Geneve 23, Switzerland
Received 21 May 2001; received in revised form 26 February 2002; accepted 28 February 2002
Abstract The problem of the wake field generated by a relativistic particle travelling in a long beam pipe with rough surface has been studied by means of a standard theory based on the hybrid modes excited in a periodically corrugated waveguide with circular cross-section. Slow waves synchronous with the particle can be excited in the structure, producing wake fields whose frequency and amplitude depend on the depth of the corrugation. An analytical expression of the wake field is given for very small corrugations. r 2002 Elsevier Science B.V. All rights reserved. PACS: 29.17; 29.27 Keywords: Wakefield; Coupling impedance; Circular pipe; LCLS
1. Introduction The effect of surface roughness is a subject arisen in the design of machines with extremely short bunches of the order of tens of microns. In this case, in fact, the surface roughness may be a source of wake fields which might significantly increase the beam emittance and the energy spread. Even the surface roughness due to residual defects in workmanship, may be responsible of the longitudinal emittance growth due to wakefields, which is particularly harmful in the design of SASE Free Electron Lasers [1,2] because of the roughness in the beam tube. After any machining process the metal wall of the pipe shows a random behaviour with valleys *Corresponding author. E-mail address: [email protected] (L. Palumbo).
and crests with typical r.m.s. depth much smaller than the crests distance. Several models have been studied considering both periodic and random roughness [3–7,12,13]. The periodic systems, due to the multiple coherent reflections of waves from the perturbations, can sustain slow harmonics which are in synchronism with the beam and can be described in terms of wake fields. In the case of random surfaces, the multiple reflections are not coherent and the field acting on the beam is not stationary producing, therefore, a weaker effect. Although the random case is more realistic, the periodic geometry has been often considered in order to give a conservative estimate of the roughness effects. A synchronous wake due a slow wave propagating in a corrugated pipe has been first derived in Ref. [4] and applied to the LCLS case in Ref. [6]. It predicts that the amplitude of the resonant wake does not depend on the
0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 0 7 9 5 - 7
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
corrugation height when this last is comparable to the period. However, it has been predicted in Refs. [9–11] that when the height of the periodic wall corrugations becomes smaller than the period, the loss factor for the modes rapidly decreases. We believe that the latter models, predicting a wakefield amplitude that vanishes for vanishing depth of the corrugations, are more appropriate to describe the effect of beam pipes with corrugations. In general, the different models predict an effect that depends upon the radius and the length of the pipe, and on the details of the surface roughness. In this paper we study the problem of the wake fields produced by an ultra-relativistic charge travelling inside a beam tube with a periodic corrugation making use of a standard theory based on the hybrid modes propagating in the waveguide [8,9]. Aim of the paper is to find an analytical expression of the wake fields, expliciting the dependence upon the corrugation height for the particular case of very small corrugation. The paper is structured as follows: in Section 2 we describe the method used, in Section 3 we present the results: first the dispersion relation for the fields and the frequency where the wave is synchronous with the charge can be excited; then, the amplitude of the field excited by the charge, the wake function and the coupling impedance.
2. The method Let us consider a periodically corrugated infinite waveguide with circular cross-section, with inner radius a and outer radius b: We model the wall roughness as a series of periodic (with period L) obstacles of height h ðh ¼ b aÞ and thickness t (see Figs. 1 and 2). The charge travels along the zaxis; we assume t5L; L5l and the ohmic losses in the material negligible. The periodicity of the geometry along the z-axis allows us the use of Floquet’s theorem which implies a field solution independent of the period L (obtained from a single cell). The steps are the following: at first we solve the homogeneous problem, finding the modes propagating in the waveguide and their features (the dispersion equation, the cut-off
11
z h
h Fig. 1. Relevant geometry.
Fig. 2. Schematic view of the waveguide and notation adopted.
frequency and the frequency where the synchronous wave is excited). The dispersion relation is found by applying the continuity conditions for the field components over the boundary between the slot (the space inside the corrugation) and the internal region of the waveguide. The field inside the waveguide is considered as generated by the magnetic and the electric Hertz potentials along the z-axis. Then we apply the Lorentz reciprocity principle, including the charge as an impulsive source, finding the coefficients used to express the electric field along the z-axis.
3. Results 3.1. The homogeneous problem The electromagnetic fields inside the corrugation are considered to be those due to propagating
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
12
radial modes; higher-order evanescent modes are considered negligible, this assumption is justified under the hypothesis that the wavelength is much greater than the distance between two corrugations ðlbLÞ: In a cylindrical coordinate system ðr; f; zÞ; the components of the electromagnetic field inside the corrugation field are ErC ¼ 0;
ð1Þ
EfC ¼ 0; X EzC ¼ ½Cn Jn ðk0 rÞ þ Dn Yn ðk0 rÞ cosðnfÞ;
ð2Þ ð3Þ
n
X
HrC ¼
n
n ½Cn Jn ðk0 rÞ þ Dn Yn ðk0 rÞ sinðnfÞ; jomr ð4Þ
HfC ¼ HzC
k0 X ½Cn Jn0 ðk0 rÞ þ Dn Yn0 ðk0 rÞ cosðnfÞ ð5Þ jom n
¼0
ð6Þ
jðotb0n zÞ
where e is assumed and suppressed and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b0n ¼ k02 kt2 ð7Þ Where b0n is p the propagation conffiffiffiffiffiffiffiffihybrid-mode ffi stant, k0 (o= m0 e0 ) is the free-space propagation constant and kt is the transverse wave number. The field inside the waveguide is considered as generated by the Hertz potentials along the z-axis: X 0 Pez ¼ An Jn ðkt rÞ cosðnfÞejðotbn zÞ ; ð8Þ n
Pmz ¼
X
0
Bn Jn ðkt rÞ sinðnfÞejðotbn zÞ
ð9Þ
n
which are related to the field by the equations: E ¼ jomr Pm þ ðk2 þ rrÞPe ;
ð10Þ
H ¼ joer Pe þ ðk2 þ rrÞPm
ð11Þ
where Pm ¼ Pmz z0 ;
Pe ¼ Pez z0 :
ð12Þ
The potential Pez generates TMz modes and the potential Pmz generates TEz waves. The superposition of both kind of modes gives origin to hybrid modes. So the expressions of the components of the hybrid modes in the internal region of
the waveguide are given by Eqs. (10) and (11) using relations (8) and (9): h i n ern ¼ jBn om Jn ðkt rÞ jAn b0n kt Jn0 ðkt rÞ r 0 cosðnfÞejðotbn zÞ ; ð13Þ h i 0 n efn ¼ jBn omkt Jn0 ðkt rÞ þ jAn b0n Jn ðkt rÞ ejðotbn zÞ ; r ð14Þ 0
ð15Þ ezn ¼ An kt2 Jn ðkt rÞ cosðnfÞejðotbn zÞ ; h i 0 n hrn ¼ jBn b0n kt Jn0 ðkt rÞ jAn oe Jn ðkt rÞ ejðotbn zÞ ; r ð16Þ h i n hfn ¼ jBn b0n Jn ðkt rÞ jAn oekt Jn0 ðkt rÞ r 0 cosðnfÞejðotbn zÞ ; ð17Þ 0
hzn ¼ Bn kt2 Jn ðkt rÞ sinðnfÞejðotbn zÞ :
ð18Þ
Applying the boundary condition at r ¼ b (the tangential electric field in the perfectly conducting wall vanishes) from Eq. (3) we find Cn Jn ðk0 bÞ EzC ðr ¼ bÞ ¼ 0 ) Dn ¼ ð19Þ Yn ðk0 bÞ and imposing the continuity of the tangential components of the field at r ¼ a: EzC ¼ Ez ;
HfC ¼ Hf
for r ¼ a
ð20Þ
that is, using Eq. (19) Jn ðk0 bÞ Cn Yn ðk0 aÞ ¼ An kt2 Jn ðkt aÞ ð21Þ Cn Jn ðk0 aÞ Yn ðk0 bÞ k0 Jn ðk0 bÞ Cn Jn0 ðk0 aÞ Cn Yn0 ðk0 aÞ Yn ðk0 bÞ jom 0 n ¼ jBn bn Jn ðkt aÞ jAn oekt Jn0 ðkt aÞ: ð22Þ a From Eqs. (21) and (22) we find the dispersion relation for the hybrid modes: Jn0 ðk0 aÞYn ðk0 bÞ Jn ðk0 bÞYn0 ðk0 aÞ Jn ðk0 aÞYn ðk0 bÞ Jn ðk0 bÞYn ðk0 aÞ ¼
k0 Jn0 ðkt aÞ b02 n2 Jn ðkt aÞ j 2n 3 : kt Jn ðkt aÞ a kt k0 Jn0 ðkt aÞ
ð23Þ
The modes of interest are the TM0m : the TEz and the TMnm (with na0) modes do not give any
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
In the hypothesis of small corrugations ða-bÞ the cut-off frequency is found to be (see Appendix A): c x01 fco ¼ ð25Þ 2p a þ h where x01 is the first zero of the Bessel function of first kind and order zero. It has to be noted that, for h-0; (25) tends to the cut-off frequency of the TM01 mode in the smooth waveguide of radius a; as expected. In the same hypothesis of small corrugations we found the frequency where the synchronous wave is excited (crossing frequency): c x01 pffiffiffiffiffi: f%cr ¼ ð26Þ 2p ah It shows pffiffiffithe typical behaviour of proportionality to 1= h: Figs. 3 and 4 report the Brillouin diagrams when h=a ¼ 0:1 and 0.8. The dashed line is the dispersion curve for the TM01 mode in the corrugated waveguide and the continuous one is the straight line b00 ¼ k0 : For b00 > k0 the wave is slow and can be synchronous with the charge. This is just the fundamental mode
200
150
'0
contribution in the application of the reciprocity principle. The dispersion relation (23) for n ¼ 0 becomes J00 ðk0 aÞY0 ðk0 bÞ J0 ðk0 bÞY00 ðk0 aÞ k0 J00 ðkt aÞ ¼ : ð24Þ J0 ðk0 aÞY0 ðk0 bÞ J0 ðk0 bÞY0 ðk0 aÞ kt J0 ðkt aÞ
13
100
50
0 10
20
30
40 k0
50
60
70
Fig. 4. Brillouin diagram for a circular cross-section waveguide of radius a with periodic corrugations of depth h and h=a ¼ 0:8:
which becomes synchronous at a given frequency. It is apparent that other modes can propagate in the beam tube, and that the coupling effects can in principle subtract energy to the synchronous wave. However, one can see after some calculations that the coupling factors for shallow corrugations are proportional to h2 : 3.2. Including the sources Once derived the modes of structures, having solved the homogeneous problem, the field generated by a point charge can be found by means of the Lorentz reciprocity principle [14]: Z Z 7 7 ðEn H E Hn Þ n dS ¼ J E7 n dV : ð27Þ
100 80
S
60
V
'0
The sign þ is for the wave travelling along þz and the sign is for the wave travelling along z: The current density of a point charge travelling onaxis, J; used in the reciprocity principle, is modelled as an impulsive source
40 20 0 40
50
60
70
80
90
k0
Fig. 3. Brillouin diagram for a circular cross-section waveguide of radius a with periodic corrugations of depth h and h=a ¼ 0:1:
Jðr; f; z; oÞ ¼ q
dðrÞ dðfÞejðz=vp Þo z0 r
ð28Þ
where z0 is the unit vector along z-axis, q is the charge and vp is the charged particle velocity. The electromagnetic field can be expressed using
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
14
the modal expansion [14]: X E¼ ðan Eþ n þ bn En Þ;
domain is Ez ðz; tÞ ¼
ð30Þ
where c is the light velocity in free-space and Z0 is the free-space characteristic impedance. From Eq. (38) it can be noted that Ez has a p phase difference with the charge (the sign), meaning that it is a decelerating field and that the height of the corrugation h fixes not only the crossing frequency, but also the field amplitude through the factor h=a3 : Another feature of the electric field component along the z-axis that can be seen from Eq. (38) is the independence from the radial coordinate r:
n
H¼
X
ðan Hþ n þ bn H n Þ
n
where jbzn z Eþ ; n ¼ ðetn þ ezn z0 Þe þjbzn z E : n ¼ ðetn ezn z0 Þe
ð31Þ
jbzn z Hþ ; n ¼ ðhtn þ hzn z0 Þe þjbzn z H : n ¼ ðhtn þ hzn z0 Þe
ð32Þ
The coefficients an and bn are found by applying the Lorentz principle (27). For n ¼ 0; the present case, they are given by (see Appendix A): a00 ¼ b00 ¼
qdððk0 =bÞ b00 Þ ; 2poeb00 a2 F ðkt aÞ
qdððk0 =bÞ þ b00 Þ 2poeb00 a2 F ðkt aÞ
ð33Þ
ð34Þ
where b is the relativistic factor and F ðkt aÞ ¼
J12 ðkt aÞ
J0 ðkt aÞJ2 ðkt aÞ
ð35Þ
and J0 ðkt aÞ; J1 ðkt aÞ and J2 ðkt aÞ are the Bessel functions of first kind and order 0, 1 and 2, respectively. Therefore, it becomes possible to find the expression of the electric field in the frequency domain along the z-axis using Eqs. (29) and (31) for n ¼ 0 Ez ¼ a0 eþ z0 b0 ez0
0 8qZ0 ch cosðo % cr tÞejb0 z 2 3 ðx01 Þ pa
ð29Þ
ð36Þ
which using the coefficients (33) and (34) becomes q Ez ðr; f; z; oÞ ¼ 0 2 o % cr eb0 a F ðkt aÞ 0 k0 k0 b00 þ d þ b00 ejb0 z kt J0 ðkt rÞ d b b ð37Þ where o % cr is the crossing angular frequency (expressed by 2p times the Eq. (26)). In the hypothesis of small corrugation (h-0), for ultrarelativistic particles (b-1) and close to the crossing frequency (kt -0) the electric field in the time
ð38Þ
3.3. Longitudinal coupling impedance and wake function Following the standard definition of the longitudinal coupling impedance per unit length Zz ðoÞ [15]: @Zz ðoÞ 1 ¼ Ez ðx ¼ 0; y ¼ 0; z; oÞejoz=c ; ð39Þ @z q from Eq. (37) we get @Zz ðoÞ 4Z0 ch ¼ ½dðo o % cr Þ þ dðo þ o % cr Þ: ð40Þ @z ðx01 Þ2 pa3 Again from the definition [15], it is easy to get the longitudinal wake function per unit length wz ðtÞ @wz ðtÞ Ez ðz; tÞ joz=c ¼ e : ð41Þ @z q From Eq. (38): @wz ðz; tÞ 8Z0 c h ¼ cosðo % cr tÞ: @z ðx01 Þ2 p a3
ð42Þ
Both the longitudinal coupling impedance (40) and the longitudinal wake function (42) are proportional to the depth height. This result agrees with Ref. [6] for what concerns the resonant frequency but, in contrast, it predicts an amplitude which scales with h=a: The disappearance of the wake amplitude when h-0 is somehow expected from the physical point of view and agrees with the general behaviour of the wake predicted other authors [11]. The availability of a simple analytical expression of the wake, although within the limit of validity of the model, represents a certain
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
advantage for the estimate of the wake effects on the beam. Just as an example we estimate the effect of a periodic roughness considering the LCLS parameters [2]. We assume a rectangular bunch, of temporal dimension 2T; where T is given by T ¼ pffiffiffi ð 3=cÞsl ; being sl (¼ 15 mm) the longitudinal dimension of the bunch. We find that the crossing frequency is given by 1 f%cr ¼ 2:29 1011 pffiffiffi ðHzÞ h
ð43Þ
and the amplitude of the wake function per unit length is given by w00 ¼ 3:2017 1018 h ðV=CmÞ:
15
4. Conclusions We have derived the longitudinal wake due to a periodic corrugation in a circular beam pipe in the case of very small corrugation, and for a wavelength of the wake larger than the period. A simple analytical expression of the wake shows that the amplitude of the sinusoidal wake function is proportional to the corrugation height h and the slow wave frequency, p synchronous with the beam, ffiffiffi is proportional to 1= h: This result agrees with the behaviour of a wake decreasing with the corrugation height that has been recently predicted by other authors.
ð44Þ Appendix A
The normalised energy spread, is given by Ref. [6] DE rms w00 DQ 1 sinð4o % cr TÞ ¼ 1 2E0 2ðo 4o E0 % cr T % cr TÞ2 4 )1=2 sinðo % cr TÞ ð45Þ o % cr T where E0 ¼ 14:35 GeV is the total energy of the electron beam, D ¼ 112 m is the total length of the path followed by the beam and Q ¼ 0:1 nC is the bunch charge. In Fig. 5 is reported the energy spread vs. h (m). The design parameters require the energy spread to be o5 104 ; which is verified for value of h of the order of tens of microns.
A.1. Solution of the dispersion relation In the hypothesis of small corrugations and from the definition of Wronskian, it is possible to use the following approximations [16] 2 J00 ðk0 aÞY0 ðk0 bÞ J0 ðk0 bÞY00 ðk0 aÞ ¼ ðA:1Þ pk0 a J0 ðk0 aÞY0 ðk0 bÞ J0 ðk0 bÞY0 ðk0 aÞ ¼ so the dispersion relation becomes kt J0 ðkt aÞ ¼ k0 h: k0 J00 ðkt aÞ
2h pa
ðA:2Þ
ðA:3Þ
Considering that k02 ¼ kt2 þ kz2 ; and that at cut-off kz ¼ 0; k0 ¼ kt ; we get J0 ðk0 aÞ ¼ k0 h ðA:4Þ J00 ðk0 aÞ
5 10-5
4 10-5
∆Erms/E0
and solving this equation for k0 3 10-5
k0 ¼ 2 10-5
1 10
-5
0 0
1 10-5
2 10-5
3 10-5
4 10-5
5 10-5
6 10-5
7 10-5
h
Fig. 5. Energy spread for the circular cross-section waveguide vs. h (m).
k0S a aþh
ðA:5Þ
where k0S is the propagation constant for the smooth waveguide. It is immediate to find the cutoff frequency from the propagation constant expressed by Eq. (A.5). For the smooth waveguide the dispersion relation is given by J0 ðkt aÞ ¼ 0
)
kt a ¼ x0m
ðA:6Þ
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17
16
where x0m is the mth zero of the Bessel function of the first kind of order 0. Being x the distance from the zero x0m and making x-0; it is possible to write x J00 ðkt aÞ ¼ J00 ðx0m xÞ- J1 ðx0m Þ J1 ðx0m Þ; x0m ðA:7Þ
A00 ¼ 2oeb00 kt2
J0 ðkt aÞ ¼ J0 ðx0m xÞ-J1 ðx0m Þx:
Solving the integral we find
ðA:8Þ
From Eqs. (A.7), (A.8), (A.1), (A.2) it is possible to get another approximated expression of the dispersion relation 1 x k0 1 ðA:9Þ 1þ ¼ : x x0m kt k0 h Solving this second-order equation in x x¼
k02 ha x0m
Ann ¼ 2
Z
ðetn htn Þ z0 dS
ðA:16Þ
S
that for n ¼ 0 is Z
a 0
½J00 ðkt rÞr dr
A00 ¼ 2poeb00 kt2 a2 F ðkt aÞ:
Z
2p
df:
ðA:17Þ
0
ðA:18Þ
From Eqs. (A.15) and (A.18) it is easy to find expression (33). Likewise, we find the expression for coefficient (34) from Eq. (A.13).
References ðA:10Þ
we get kt a ¼ x0m
ðk0 aÞ2 h x0m a
ðA:11Þ
Imposing the crossing condition kt ¼ 0 we get the propagation constant at the crossing frequency x 2:40 o % ffiffiffiffiffi ¼ pffiffiffiffiffi ¼ : ðA:12Þ k%0 ¼ p0m c ah ah It is immediate to get the crossing frequency from this equation. A.2. Calculation of the coefficients for the modal expansion The Lorentz reciprocity principle assumes the expression Z X X bm Anm am Bnm ¼ J Eþ ðA:13Þ n dV m
X
and
V
m
bm Cnm
m
X m
am Dnm ¼
Z V
J E n dV
ðA:14Þ
where the coefficients Anm ; Bnm ; Cnm and Dnm are related to the transverse of the nth mode. In the case under examination n ¼ m; Eq. (A.14) becomes Z J E ðA:15Þ an Ann ¼ n dV V
[1] A VUV FEL at TTF Linac, Conceptual Design Report TESLA-FEL 95-03, DESY Hamburg, 1995. [2] LCLS Design Study Group, SLAC Report No. SLAC-R521, 1998. [3] K.L.F. Bane, C.K. Ng, A.W. Chao, Estimate of the impedance due to wall surface roughness, Report SLACPUB-7514, SLAC, 1997. [4] A. Novokhatski, A. Monier, Proceedings of 1997 Particle Accelerator Conference, IEEE Press, New Jersey, 1997. [5] G.V. Stupakov, Physical Review Special Topics, AB, Vol. 1, 1998. [6] K.L.F. Bane, A. Novokhatski, The resonator impedance model of surface roughness applied to the LCLS parameters, Report SLAC-AP-117, SLAC, 1999. [7] K.L.F. Bane, G.V. Stupakov, Impedance of a beam tube with small corrugations, Report SLAC-PUB-8599, SLAC, 2000. [8] G.H. Bryant, Proc. IEE 116 (1969) 203. [9] A. Mostacci, L. Palumbo, F. Ruggiero, S. Ugoli, Wake fields effects due to surface roughness, Eighth International Workshop on Linear Colliders, Frascati, 1999. [10] G. Stupakov, R.E. Thomson, D. Walz, R. Carr, Effects of beam-tube roughness on X-ray free electron laser performance, Physical Review Special Topics, AB, Vol. 2, 1999. [11] G. Stupakov, Surface roughness impedance, Proceedings of the 19th Advanced ICFA BD Workshop, Arcidosso 2000, Vol. 581, AIP, New York, 2001. [12] A.V. Agonov, A.N. Lebedev, Effects of channel roughness on beam energy spread, Proceedings of the 19th Advanced ICFA BD Workshop, Arcidosso 2000, Vol. 581, AIP, New York, 2001. [13] E. Di Liberto, F. Frezza, L. Palumbo, Effects of statistical roughness on the propagation of electromagnetic fields in a circular waveguide, Proceedings of the 19th Advanced ICFA BD Workshop, Arcidosso 2000, Vol. 581, AIP, New York, 2001.
M. Angelici et al. / Nuclear Instruments and Methods in Physics Research A 489 (2002) 10–17 [14] R.E. Collin, Field Theory of Guided Waves, 2nd Edition, IEEE Press, New York, 1991. [15] L. Palumbo, V. G. Vaccaro, M. Zobov, in: S. Turner (Ed.), Proceedings of the CERN Accelerator School: Advanced Accelerator Physics Course, Rhodes, 1993, No. 95-06 in
17
CERN Yellow Report, European Laboratory for Particle Physics, Geneva, Switzerland, 1995, pp. 331–390. [16] C.A. Balanis, Advanced Engineering Electromagnetics, Wiley, New York, 1989.