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Klystron instability of a relativistic electron beam in a bunch compressor
Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
Klystron instability of a relativistic electron beam in a bunch compressor E.L. Saldina, E.A. Schneidmillera,*, M.V. Yurkovb a
Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany b Joint Institute for Nuclear Research, Dubna, 141980 Moscow Region, Russia Received 18 February 2002; received in revised form 5 April 2002; accepted 5 April 2002
Abstract In this paper, we consider a klystron-like mechanism of amplification of parasitic density modulations in an electron bunch passing a magnetic bunch compressor. Analytical expressions are derived for the small-signal gain. The effect of wakefields in front of the bunch compressor is analyzed by using a model of linear compression which assumes linear correlated energy chirp and linear dependence of a path length on energy deviation. Analysis of the density modulation growth due to coherent synchrotron radiation inside bends of the magnetic bunch compressor is done for the simplified case of no correlated energy chirp (no compression). Analytical results of this paper can be used for benchmarking numerical simulation codes. r 2002 Elsevier Science B.V. All rights reserved. PACS: 41.75.Ht; 29.27.Bd; 41.60.Ap Keywords: Bunch compressor; Beam dynamics; Wakefields; Coherent synchrotron radiation; Amplification
1. Introduction Magnetic bunch compressors are designed to obtain short electron bunches with a high peak current for linac-based short-wavelength FELs [1– 5] and future linear colliders [5–7]. The basic principle of compression is very simple. A relativistic electron bunch accumulates energy chirp while passing RF accelerating structures off-crest and then gets longitudinally compressed due to an energy-dependent path length in the magnetic compressor (for instance, in a chicane). Since, *Corresponding author. Tel.: +49-40-8998-2676; fax: +4940-8998-4475. E-mail address: [email protected] (E.A. Schneidmiller).
however, electron bunches are very short and intensive, collective effects like coherent synchrotron radiation (CSR) [8] can seriously influence beam dynamics in compressors [9]. In the recent experiments with bunch compressors [10–12] and in numerical simulations [13], the strong high-frequency perturbations of longitudinal phase space have been observed. The selfconsistent simulations [14] of beam dynamics in the TESLA Test Facility (TTF) bunch compressor chicane (BCC), taking into account CSR effects, have also shown phase space fragmentation. It has been explained by strong enhancement of CSR effects due to the locally peaked (non-Gaussian) density distribution created during compression process because of RF nonlinearity (see also Ref. [15]).
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 9 0 5 - 1
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E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
It has been pointed out in Refs. [14,16] that another mechanism can be considered which is also relevant for the ideal linear RF modulation (or, even without modulation). Namely, high-frequency components of the beam current spectrum (higher than typical inverse pulse duration) cause energy modulations at the same frequencies due to wakefields. The energy modulation is converted into an induced density modulation while the beam is passing the bunch compressor. If the wakefields are strong enough, the induced modulation can be much larger than the initial one. In other words, the system can be treated as a high-gain klystron-like amplifier. The general tendency is that higher frequencies (to some extent) are going to get amplified stronger so that they may become much better pronounced in comparison with the case of undisturbed compression. Thus, the electron beam is potentially unstable and its longitudinal phase space can be essentially modified. In this paper we study such a mechanism analytically in linear approximation. Since it is difficult to measure (simulate) small high-frequency perturbations in the initial state of the beam, one cannot exactly predict its final state. Thus, our goal is to calculate (estimate) the gain as a function of frequency. If the gain is large, then one may expect significant modifications of longitudinal phase space in the bunch compressor, and vice versa. In Section 2, we study the dynamical aspect of the problem assuming linear energy chirp along the beam and the given amplitude of parasitic energy modulation at some frequency. In Section 3, we consider the case when these energy perturbations are created due to wakefields upstream of bunch compressor. In Section 4, we study CSR-induced growth of the density modulation in BCC. We consider there a simplified case of no compression (no correlated energy chirp). In Section 5, we estimate effective density modulations due to the shot noise in electron beam. A numerical example is presented in Section 6.
distribution of the beam with DC current, linear energy chirp along the beam and Gaussian energy spread can be described with the following function: " # I0 ðdg hg0 zÞ2 f ðz; dgÞ ¼ pffiffiffiffiffiffi exp ð1Þ 2s2g 2psg
2. Compression of the beam with linear energy chirp and superimposed sinusoidal modulation
C¼
In this paper we consider 1-D model of the electron beam. An undisturbed phase space
where z is the coordinate along the beam (particles with positive values of z are placed behind the particle with z ¼ 0Þ; g0 ¼ E0 =ðmc2 Þ is the nominal energy in units of the rest energy, m is electron’s mass, g0 b1; c is the velocity of light, dg ¼ ðE E0 Þ=ðmc2 Þ is the energy deviation from the nominal value, sg ¼ sE =ðmc2 Þ is the rms local energy spread, h ¼ dðdgÞ=ðg0 dzÞ describes linear energy chirp along the beam, I0 is the beam current. Normalization is chosen in such a way that after integration over dg we get the current. We assume g0 to be large and consider small energy deviations dg5g0 although formally we let dg extend from N to N: The model of DC current allows us to exclude edge effects from consideration and to deal with small sinusoidal modulations. To describe phase space transformation in the bunch compressor, we assume a linear dependence of path length in the compressor on dg=g0 described by the element of a linear transfer matrix: R56 ¼
qz : qðdg=g0 Þ
Then a particle position in the beam before and after compression, zi and zf ; are connected by dg zf ¼ zi þ R56 : g0 Therefore, to describe the final state of the beam we should substitute z in (1) by z R56 dg=g0 : Then the new distribution will have the form of (1) where I; sg ; and h are substituted by CI; Csg ; and Ch; respectively. Here C is the compression factor: 1 : 1 þ hR56
For compression one should provide hR56 o0: For instance, R56 o0 for the chicane so that h has to be positive in this case. In addition, in this paper we
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
restrict our consideration by the condition 1 þ hR56 > 0; i.e. the beam is undercompressed. Now let us consider an energy modulation at some frequency o on top of the linear chirp. In front of the bunch compressor the phase space distribution has the form ( ) I0 ½dg hg0 z þ Dg sinðkzÞ2 f ðz; dgÞ ¼ pffiffiffiffiffiffi exp 2s2g 2psg
3
For kCðCR56 sg =g0 Þ1 this happens when DgXsg : It is worth mentioning that sg always stands for the initial energy spread (before compression). In this paper we will use linear approximation assuming that CkjR56 jDg=g0 51: In other words, we will look for a small-signal gain. This leaves us with only the first harmonic of the beam current ðJ1 ðX ÞCX =2Þ:
ð3Þ IðzÞCCI0 ½1 þ rind sgnðR56 Þ cosðCkzÞ where k ¼ o=c and Dg is the amplitude of energy modulation. As it was done above, we substitute z where sgnðR56 Þ is the sign of R56 and rind is the by z R56 dg=g0 to describe the change of distribuamplitude of the first harmonic in the final state of tion function in the bunch compressor. Then we the beam: integrate over dg in order to get current as a function of z: ( ) Z N I0 ½dgð1 þ hR56 Þ hg0 z þ Dg sinðkz kR56 dg=g0 Þ2 IðzÞ ¼ pffiffiffiffiffiffi ddg exp : 2s2g 2psg N After change of variables x ¼ ½dgð1 þ hR56 Þ hg0 z the integral takes the following form: Z N CI0 IðzÞ ¼ pffiffiffiffiffiffi dx exp 2psg N ( ) ½x þ Dg sinðCkz CkR56 x=g0 Þ2 : 2s2g The integral of such a form is known to describe the process of density bunching starting from initial sinusoidal energy modulation but without linear energy chirp (see, for instance, Ref. [17]). Making integration and Fourier expansion, one gets " N X Dg IðzÞ ¼ CI0 1 þ 2 Jn nCkR56 g0 n¼1 ! # 1 2 2 2 2 s2g exp n C k R56 2 cosðnCkzÞ : ð2Þ 2 g0 Here Jn is the Bessel function of nth order. Without compression ðh ¼ 0; C ¼ 1Þ Expression (2) is reduced to the well-known one [17]. Analyzing Eq. (2), we see that the frequency range (of initial modulation), in which the beam can be effectively bunched, is limited by kpðCR56 sg =g0 Þ1 : Within this range the condition CkjR56 jDg=g0 X1 means that the beam is completely bunched and the phase space is fragmented.
rind
! Dg 1 2 2 2 s2g ¼ CkjR56 j exp C k R56 2 : g0 2 g0
ð4Þ
We have considered here the model of infinitely long beam. The results of this paper can be used for a bunch with finite length s as soon as the following condition is satisfied: ksb1:
ð5Þ
3. Wakefields upstream of a bunch compressor Let us assume that upstream of the bunch compressor there is a small density perturbation ri at some frequency: IðzÞ ¼ I0 ½1 þ ri cosðkzÞ:
ð6Þ
Due to some wakefields upstream of compressor, the beam gets modulated in energy at the same frequency with the amplitude Dg: Describing the action of wakefields by longitudinal impedance ZðkÞ we can connect the amplitudes of energy and density modulations as follows: Dg ¼
jZðkÞj I0 r Z 0 IA i
ð7Þ
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
4
where Z0 ¼ 377 O is the free-space impedance and IA ¼ 17 kA is the Alfven current. Then, using Eq. (4), we calculate the amplitude of the induced density modulation at the end of bunch compressor. In general case, to find final density modulation rf one should sum up induced modulation and (transformed to the end of compressor) initial one, taking care of phase relations. But in this paper we use the approximation
where Sc is the length of a path through compressor and b is the beta-function. This condition is always met in practice. The formulas of this section can be used when the effect of wakefields in front of compressor is much stronger than any collective effects inside the bunch compressor.
ri 5rind 51:
4. CSR in a bunch compressor chicane
In other words, rf Crind and the gain in density modulation r r G ¼ f C ind ri ri
Wakefields can also exist inside bunch compressors. We consider here coherent synchrotron radiation which is an intrinsic feature of magnetic compressors. CSR effects can be minimized there but not avoided. A CSR-induced beam instability in storage rings has been investigated in Ref. [18]. That instability develops continuously, in small increments, like most instabilities of relativistic electron beams. We analyze here quite different klystron-like mechanism of the instability in a bunch compressor.
is assumed to be high, Gb1 (otherwise the effect, considered in this paper, is not of great importance). Under this approximation the gain depends neither on phase of ZðkÞ nor on sign of R56 and is equal to ! I0 jZðkÞj 1 2 2 2 s2g G ¼ CkjR56 j exp C k R56 2 : ð8Þ 2 g0 I A Z 0 g0 For broadband nonresonant wakefields the product kjZðkÞj is usually a growing function of k: For such cases the maximal gain is achieved at sg 1 C jR56 jC: ð9Þ kopt g0 The optimal final frequency (when the beam is compressed) is almost independent of compression factor C: A crude estimate for the maximal gain is Gmax C
I0 jZðkopt Þj : sg IA Z0
ð10Þ
The term I0 =ðsg IA Þ is proportional to a longitudinal brightness (particles density in longitudinal phase space). In practice, the phase space distribution can be of complex shape. We note that the local energy spread should be taken for estimations of amplification effect. In the considered case the influence of beam emittance e on longitudinal dynamics is a secondorder effect and is negligible when keSc =b51
ð11Þ
4.1. The model In this section, we consider a simplified case when the beam is not compressed (no energy chirp: h ¼ 0; C ¼ 1Þ: While the formulae of the previous section are rather general and do not depend on a type of the bunch compressor, in this section we have to choose a specific model. We consider a symmetric three-dipole chicane1 where the first and the last dipoles have the length Ld ; and the middle one is as long as 2Ld : The bending angle in the first dipole y is small, y ¼ Ld =R51 ðR is the bending radius), and the distance DL between two subsequent dipoles is much larger than the dipole length: DLbLd :
ð12Þ
The R56 can then be expressed in a simple form: R56 C 2DLy2 :
ð13Þ
To describe CSR we use the steady-state model neglecting transient effects. The domain of validity 1 Under limitations, accepted in this section, all the results are valid for a four-dipole chicane, too.
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
of this model can be estimated as 2
Ld bð24R =kÞ
1=3
:
ð14Þ
It is assumed that inequality kR=g30 51 always holds. We neglect the influence on CSR of transverse beam size and of the screening effect of the vacuum chamber requiring that [9,20] ðR=b3 Þ1=2 5k5ðR=s3> Þ1=2
ð15Þ
where b is the transverse size of vacuum chamber and s> is that of electron beam. Since we analyze the dynamics inside bunch compressor, we cannot use condition (11) for neglecting emittance effect on longitudinal dynamics. The relevant condition will be presented below. We also neglect here collective transverse forces [19]. Under conditions (14) and (15) the module of CSR impedance in the first and the last dipoles can be expressed as [20] jZðkÞj jZl ðkÞjLd 2Gð2=3ÞLd k1=3 ¼ ¼ Z0 Z0 31=3 R2=3
ð16Þ
and in the middle dipole it is two times larger. Here Zl ðkÞ is the impedance per unit length and Gð?Þ is the complete gamma-function, Gð2=3ÞC1:354: One can use impedance (16) to calculate energy modulation only if the relative change of density modulation is small on the scale of formation length ð24R2 =kÞ1=3 (see Ref. [21] for more details). In our model the density changes on the scale of Ld and this condition is equivalent to Eq. (14).
5
where s1 is a coordinate along the reference trajectory, s1 ¼ 0 at the entrance to the first dipole. The amplitude of the induced density modulation in the second dipole can then be calculated using simple generalization of formula (4) for ‘‘cold’’ beam: Z Ld dDg r2 ðs2 Þ ¼ k ds1 R56 ðs1 -s2 Þ ð18Þ g 0 0 ds1 where s2 ¼ 0 at the entrance to the second dipole and R56 ðs1 -s2 Þ connects energy kick at s1 and a change of position z at s2 : Under condition (12), it is well approximated by DL R56 ðs1 -s2 ÞC 2 ðLd s1 Þs2 ð19Þ R Then we get r2 ðs2 Þ ¼
1jZl ðkÞj I0 kDLL2d ri s2 : 2 Z0 g0 IA R2
ð20Þ
Since we assumed that the induced modulation r2 is much larger than the initial one (gain is large), the change of CSR-induced energy modulation in the second dipole can be approximated by dDg jZl ðkÞj I0 ¼ r ðs2 Þ ð21Þ g0 ds2 Z 0 g0 I A 2 with negligibly small initial value. Then we do similar calculations for the third dipole ðs3 ¼ 0 at its entrance): Z 2Ld dDg r3 ðs3 Þ ¼ k ds2 R56 ðs2 -s3 Þ ð22Þ g 0 0 ds2 where DL ð2Ld s2 Þs3 : ð23Þ R2 Finally, using Eqs. (20)–(23) we obtain the total gain as a ratio between final and initial amplitudes of density modulation: r3 ðs3 ¼ Ld Þ 2 jZl ðkÞj2 I0 2 k2 DL2 L6d G¼ ¼ : ri 3 Z02 g0 IA R4
R56 ðs2 -s3 ÞC 4.2. ‘‘Cold’’ electron beam Let us first consider the case when the energy spread can be neglected. In the framework of the accepted model we consider the two-stage amplification in the bunch compressor. The gain is assumed to be large in each stage. An initial density perturbation (6) causes energy modulation in the first dipole (see Eqs. (7),(16)). This modulation increases linearly inside the dipole:
With the help of Eqs. (13) and (16) we rewrite the expression for the gain in a different form 2G2 ð2=3Þ I0 2 k8=3 jR56 j2 L2d : ð24Þ G¼ g0 I A 35=3 R4=3
dDg jZl ðkÞj I0 ¼ r g0 ds1 Z0 g0 IA i
When deriving this expression we neglected density bunching inside a dipole caused by the energy
ð17Þ
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
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modulation induced in the same dipole. In other words, we ignored a self-consistent process inside a dipole which may eventually lead to an exponential growth of the modulation. Thorough analysis shows that expression (24) is accurate as soon as
and the function f is # ¼ 3k#2=3 expðk#2 =2Þ f ðkÞ " # pffiffiffi p k#2 2 2 # expðk# =4Þ erfðk=2Þ ð27Þ 1þ 2 k#
Ld pLg
where
where g IA 1=4 1=3 2=3 Lg ¼ 0 k R : I0
erfðxÞ ¼ 2p1=2
Z
x
dt expðt2 Þ
0
In the case when Ld bLg one could expect an exponential growth inside dipoles with the growth rate about L1 g : In practice, however, for any reasonable set of electron beam parameters, this regime cannot be achieved because the frequency range will be limited by the energy spread and/or emittance. The exponential growth (with a different scaling for the growth rate) inside dipoles of a bunch compressor has been predicted in Ref. [22] but the results of that paper are incorrect.2
# is the error function. For k51; when the influence # k#8=3 of energy spread is negligible, we get f ðkÞC and Eq. (26) is reduced to Eq. (24). The maximal value of f is achieved at k#opt ¼ 2:15 and is equal to 1:98: Thus, the maximal gain is Gmax ¼ 1:16 g20
ð28Þ
in agreement with a simple estimate. The accurate result (28) differs by o10% from the result of Ref. [24], although an incorrect assumption of constant density modulation inside dipoles has been used in Ref. [24]. 4.4. Estimation of emittance effect
4.3. Gaussian energy spread Let us first make a simple estimation of the gain. If we assume that energy modulation happens only in the first dipole, we can use (with C ¼ 1Þ relations (9), (10) and (16) to estimate the gain at Gmax Cg0 ; where I0 g0 1=3 Ld g0 ¼ : ð25Þ 2 s g IA s g ðR jR56 jÞ1=3 Actually, we have two-stage amplification and should expect the dependence Gmax Cg20 : The accurate calculations [23] confirm this estimate. Leaving out the details of calculations we present here the final result for the gain G¼
2G2 ð2=3Þ 2 # g0 f ðkÞ: 35=3
ð26Þ
Here, sg k# ¼ jR56 jk g0 2
An incorrect equation of longitudinal motion has been used.
One of the limitations due to emittance (15) was already presented. More stringent limitation, however, comes from the longitudinal motion inside dipoles.3 Indeed, a particle with an offset x from the reference orbit moves along the beam: dz x ¼ : ds R pffiffiffiffiffi For a typical offset xC eb; the maximal pffiffiffiffiffichange of coordinate z can be estimated at ebLd =R: Therefore, the above-presented results for the gain can be used if pffiffiffiffiffi k ebLd 51: ð29Þ R The energy spread cuts off the gain at 1 kCs1 g g0 jR56 j : Using Eq. (13) we estimate that formula (26) is valid in the entire wavelength range when pffiffiffiffiffi sg Ld eb5 DL : ð30Þ g0 R 3 The net effect through the whole compressor is of the second order, see (11).
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
In other words, transverse size, defined by emittance, should be smaller than dispersioninduced transverse size in the middle dipole. In opposite case, thepcut-off will be defined by the ffiffiffiffiffi condition kCR=ð ebLd Þ: To roughly estimate maximal gain one can substitute this expression for k into formula (24).
5. Estimation of shot noise effect In this paper, we considered a bunch compressor as a high-gain linear amplifier of klystron type. It amplifies initial density modulations within some frequency band. For broadband wakefields (like CSR) we have a broadband amplifier with DoCo0 ; where o0 is some optimal frequency defined by the energy spread (eventually by emittance). Even without macroscopic density modulation, the initial signal for such an amplifier always exists because of the shot noise in electron beam. Let us consider the beam consisting of randomly emitted electrons with an average current I0 : Its spectrum is a ‘‘white’’ noise which will be filtered by the amplifier. The time domain fluctuations of the current at the amplifier entrance within a frequency band Do are defined by Schottky formula: /i2 S ¼
eI0 Do : p
ð31Þ
Since DoCo0 in the considered case, we can write down the following expression for relative initial fluctuations: /r2i Ssh ¼
/i2 S eo0 1 C C pI0 Nl I02
ð32Þ
where Nl is a number of particles per wavelength l ¼ 2pc=o0 : Thus, effective initial density modulation due to the shot noise is 1 ðri Þsh Cpffiffiffiffiffiffi: Nl
ð33Þ
The coherence length for such a broadband system is about lc C2pc=o0 ¼ l: Thus, the effective shot noise bunching is about inverse square root of number of particles per coherence length.
7
Relative amplitude of AC current behind the bunch compressor is then of the order of Gmax rf Cpffiffiffiffiffiffi: Nl
ð34Þ
Due to the lack of coherence this current will constitute irregular, spiky structure on a time scale of o1 0 : For typical parameters of electron beams one can estimate ðri Þsh at 104 : Thus, for a sufficiently large gain (especially in a chain of bunch compressors) the shot noise bunching can become important. Finally, we note that when simulating bunch compressors with the help of macroparticles approach, one should eliminate artificial noise connected with a relatively small number of macroparticles. If, for instance, one uses Nm macroparticles in a bunch and they are distributed in phase space randomly, thenpthe noise effect will ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be overestimated by a factor Nb =Nm ; where Nb is actual number of particles in the bunch.
6. Numerical example Let us consider an example of amplification of high-frequency modulations due to CSR for the case when the beam goes through the chicane without compression. Although such a case is not very interesting for practice, analytical results of this paper can be used for benchmarking numerical simulation codes. The parameters of the four-dipole chicane and of the electron beam are as follows [25]: Ld ¼ 0:5 m; DL ¼ 5 m; R ¼ 10:35 m; R56 ¼ 0:025 m; electron energy is 5 GeV; uncorrelated rms energy spread is 100 keV; rms bunch length is 20 mm; peak current is 6 kA: In Fig. 1, we present the gain curves for two cases: energy spread is included (curve 1) and ‘‘cold’’ beam (curve 2). Triangles (for the beam with energy spread) and circles (for ‘‘cold’’ beam) are the results of numerical simulations with 1-D code [26]. One can notice rather good agreement between analytical and numerical results.
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E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
Fig. 1. The small-signal gain versus modulation wavelength. Curve (1) was calculated with the help of formula (26) taking into account the energy spread while formula (24) was used to compute curve (2) for the ‘‘cold’’ beam case. Triangles (for beam with energy spread) and dots (for ‘‘cold’’ beam) are the results of numerical simulations [26].
Acknowledgements We thank R. Brinkmann, M. Dohlus, K. . Flottmann, T. Limberg, Ph. Piot, J. Rossbach and D. Trines for useful discussions.
References [1] [2] [3] [4]
J. Andruszkow, et al., Phys. Rev. Lett. 85 (2000) 3825. S.V. Milton, et al., Science 292 (2001) 2037. J. Rossbach, Nucl. Instr. and Meth. A 375 (1996) 269. The LCLS Design Study Group, Linac Coherent Light Source (LCLS) Design Report, SLAC-R-521, 1998. [5] R. Brinkmann, et al. (Eds.), TESLA Technical Design Report, DESY 2001-011.
[6] T.O. Raubenheimer (Ed.), Zeroth Order Design Report for the Next Linear Collider, SLAC Report 474, 1996. [7] N. Akasaka, et al., JLC Design Study, KEK Report 97-1, 1997. [8] L.V. Iogansen, M.S. Rabinovich, Sov. Phys. JETP 37 (10) (1960) 83. [9] Ya.S. Derbenev, et al., DESY print TESLA-FEL 95-05, 1995. [10] M. Huning, . Ph. Piot, H. Schlarb, DESY print TESLAFEL-2000-05, 2000; M. Huning, . Ph. Piot, H. Schlarb, Nucl. Instr. and Meth. A 475 (2001) 348. [11] H.H. Braun, et al., Phys. Rev. Special Topics—Accelerators and Beams 3 (2000) 124 402. [12] Ph. Piot, et al., Proceedings of EPAC 2000, Vienna, Austria, 2000, p. 1546. [13] M. Borland, Nucl. Instr. and Meth. A 483 (2002) 268. [14] T. Limberg, Ph. Piot, E.A. Schneidmiller, DESY print TESLA-FEL-2000-05, 2000; T. Limberg, Ph. Piot, E.A. Schneidmiller, Nucl. Instr. and Meth. A 475 (2001) 353. [15] R. Li, Proceedings of EPAC 2000, Vienna, Austria, 2000, p. 1312. [16] E.L. Saldin, E.A. Schneidmiller M.V. Yurkov, Nucl. Instr. and Meth. A 483 (2002) 516. [17] L.H. Yu, Phys. Rev. A 44 (1991) 5178. [18] S. Heifets, G. Stupakov, SLAC-PUB-8761, 2001. [19] Ya.S. Derbrnev, V.D, Shiltsev, FNAL-TM-1974, 1996. [20] J.B. Murphy, S. Krinsky, R.L. Gluckstern, Part. Accel. 57 (1997) 9. [21] S. Heifets, SLAC-PUB-9054, 2001. [22] S. Heifets, G. Stupakov, SLAC-PUB-8988, 2001. [23] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, CSRinduced klystron instability in bunch compressors, in preparation. [24] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, DESY print DESY-01-129, 2001. [25] http://www.desy.de/csr, Benchmark Example. [26] M. Dohlus, Field calculations for bunch compressors, Presented at the CSR workshop, Zeuthen, Germany, January 2002, http://www.desy.de/csr.
Klystron instability of a relativistic electron beam in a bunch compressor E.L. Saldina, E.A. Schneidmillera,*, M.V. Yurkovb a
Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany b Joint Institute for Nuclear Research, Dubna, 141980 Moscow Region, Russia Received 18 February 2002; received in revised form 5 April 2002; accepted 5 April 2002
Abstract In this paper, we consider a klystron-like mechanism of amplification of parasitic density modulations in an electron bunch passing a magnetic bunch compressor. Analytical expressions are derived for the small-signal gain. The effect of wakefields in front of the bunch compressor is analyzed by using a model of linear compression which assumes linear correlated energy chirp and linear dependence of a path length on energy deviation. Analysis of the density modulation growth due to coherent synchrotron radiation inside bends of the magnetic bunch compressor is done for the simplified case of no correlated energy chirp (no compression). Analytical results of this paper can be used for benchmarking numerical simulation codes. r 2002 Elsevier Science B.V. All rights reserved. PACS: 41.75.Ht; 29.27.Bd; 41.60.Ap Keywords: Bunch compressor; Beam dynamics; Wakefields; Coherent synchrotron radiation; Amplification
1. Introduction Magnetic bunch compressors are designed to obtain short electron bunches with a high peak current for linac-based short-wavelength FELs [1– 5] and future linear colliders [5–7]. The basic principle of compression is very simple. A relativistic electron bunch accumulates energy chirp while passing RF accelerating structures off-crest and then gets longitudinally compressed due to an energy-dependent path length in the magnetic compressor (for instance, in a chicane). Since, *Corresponding author. Tel.: +49-40-8998-2676; fax: +4940-8998-4475. E-mail address: [email protected] (E.A. Schneidmiller).
however, electron bunches are very short and intensive, collective effects like coherent synchrotron radiation (CSR) [8] can seriously influence beam dynamics in compressors [9]. In the recent experiments with bunch compressors [10–12] and in numerical simulations [13], the strong high-frequency perturbations of longitudinal phase space have been observed. The selfconsistent simulations [14] of beam dynamics in the TESLA Test Facility (TTF) bunch compressor chicane (BCC), taking into account CSR effects, have also shown phase space fragmentation. It has been explained by strong enhancement of CSR effects due to the locally peaked (non-Gaussian) density distribution created during compression process because of RF nonlinearity (see also Ref. [15]).
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 9 0 5 - 1
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It has been pointed out in Refs. [14,16] that another mechanism can be considered which is also relevant for the ideal linear RF modulation (or, even without modulation). Namely, high-frequency components of the beam current spectrum (higher than typical inverse pulse duration) cause energy modulations at the same frequencies due to wakefields. The energy modulation is converted into an induced density modulation while the beam is passing the bunch compressor. If the wakefields are strong enough, the induced modulation can be much larger than the initial one. In other words, the system can be treated as a high-gain klystron-like amplifier. The general tendency is that higher frequencies (to some extent) are going to get amplified stronger so that they may become much better pronounced in comparison with the case of undisturbed compression. Thus, the electron beam is potentially unstable and its longitudinal phase space can be essentially modified. In this paper we study such a mechanism analytically in linear approximation. Since it is difficult to measure (simulate) small high-frequency perturbations in the initial state of the beam, one cannot exactly predict its final state. Thus, our goal is to calculate (estimate) the gain as a function of frequency. If the gain is large, then one may expect significant modifications of longitudinal phase space in the bunch compressor, and vice versa. In Section 2, we study the dynamical aspect of the problem assuming linear energy chirp along the beam and the given amplitude of parasitic energy modulation at some frequency. In Section 3, we consider the case when these energy perturbations are created due to wakefields upstream of bunch compressor. In Section 4, we study CSR-induced growth of the density modulation in BCC. We consider there a simplified case of no compression (no correlated energy chirp). In Section 5, we estimate effective density modulations due to the shot noise in electron beam. A numerical example is presented in Section 6.
distribution of the beam with DC current, linear energy chirp along the beam and Gaussian energy spread can be described with the following function: " # I0 ðdg hg0 zÞ2 f ðz; dgÞ ¼ pffiffiffiffiffiffi exp ð1Þ 2s2g 2psg
2. Compression of the beam with linear energy chirp and superimposed sinusoidal modulation
C¼
In this paper we consider 1-D model of the electron beam. An undisturbed phase space
where z is the coordinate along the beam (particles with positive values of z are placed behind the particle with z ¼ 0Þ; g0 ¼ E0 =ðmc2 Þ is the nominal energy in units of the rest energy, m is electron’s mass, g0 b1; c is the velocity of light, dg ¼ ðE E0 Þ=ðmc2 Þ is the energy deviation from the nominal value, sg ¼ sE =ðmc2 Þ is the rms local energy spread, h ¼ dðdgÞ=ðg0 dzÞ describes linear energy chirp along the beam, I0 is the beam current. Normalization is chosen in such a way that after integration over dg we get the current. We assume g0 to be large and consider small energy deviations dg5g0 although formally we let dg extend from N to N: The model of DC current allows us to exclude edge effects from consideration and to deal with small sinusoidal modulations. To describe phase space transformation in the bunch compressor, we assume a linear dependence of path length in the compressor on dg=g0 described by the element of a linear transfer matrix: R56 ¼
qz : qðdg=g0 Þ
Then a particle position in the beam before and after compression, zi and zf ; are connected by dg zf ¼ zi þ R56 : g0 Therefore, to describe the final state of the beam we should substitute z in (1) by z R56 dg=g0 : Then the new distribution will have the form of (1) where I; sg ; and h are substituted by CI; Csg ; and Ch; respectively. Here C is the compression factor: 1 : 1 þ hR56
For compression one should provide hR56 o0: For instance, R56 o0 for the chicane so that h has to be positive in this case. In addition, in this paper we
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
restrict our consideration by the condition 1 þ hR56 > 0; i.e. the beam is undercompressed. Now let us consider an energy modulation at some frequency o on top of the linear chirp. In front of the bunch compressor the phase space distribution has the form ( ) I0 ½dg hg0 z þ Dg sinðkzÞ2 f ðz; dgÞ ¼ pffiffiffiffiffiffi exp 2s2g 2psg
3
For kCðCR56 sg =g0 Þ1 this happens when DgXsg : It is worth mentioning that sg always stands for the initial energy spread (before compression). In this paper we will use linear approximation assuming that CkjR56 jDg=g0 51: In other words, we will look for a small-signal gain. This leaves us with only the first harmonic of the beam current ðJ1 ðX ÞCX =2Þ:
ð3Þ IðzÞCCI0 ½1 þ rind sgnðR56 Þ cosðCkzÞ where k ¼ o=c and Dg is the amplitude of energy modulation. As it was done above, we substitute z where sgnðR56 Þ is the sign of R56 and rind is the by z R56 dg=g0 to describe the change of distribuamplitude of the first harmonic in the final state of tion function in the bunch compressor. Then we the beam: integrate over dg in order to get current as a function of z: ( ) Z N I0 ½dgð1 þ hR56 Þ hg0 z þ Dg sinðkz kR56 dg=g0 Þ2 IðzÞ ¼ pffiffiffiffiffiffi ddg exp : 2s2g 2psg N After change of variables x ¼ ½dgð1 þ hR56 Þ hg0 z the integral takes the following form: Z N CI0 IðzÞ ¼ pffiffiffiffiffiffi dx exp 2psg N ( ) ½x þ Dg sinðCkz CkR56 x=g0 Þ2 : 2s2g The integral of such a form is known to describe the process of density bunching starting from initial sinusoidal energy modulation but without linear energy chirp (see, for instance, Ref. [17]). Making integration and Fourier expansion, one gets " N X Dg IðzÞ ¼ CI0 1 þ 2 Jn nCkR56 g0 n¼1 ! # 1 2 2 2 2 s2g exp n C k R56 2 cosðnCkzÞ : ð2Þ 2 g0 Here Jn is the Bessel function of nth order. Without compression ðh ¼ 0; C ¼ 1Þ Expression (2) is reduced to the well-known one [17]. Analyzing Eq. (2), we see that the frequency range (of initial modulation), in which the beam can be effectively bunched, is limited by kpðCR56 sg =g0 Þ1 : Within this range the condition CkjR56 jDg=g0 X1 means that the beam is completely bunched and the phase space is fragmented.
rind
! Dg 1 2 2 2 s2g ¼ CkjR56 j exp C k R56 2 : g0 2 g0
ð4Þ
We have considered here the model of infinitely long beam. The results of this paper can be used for a bunch with finite length s as soon as the following condition is satisfied: ksb1:
ð5Þ
3. Wakefields upstream of a bunch compressor Let us assume that upstream of the bunch compressor there is a small density perturbation ri at some frequency: IðzÞ ¼ I0 ½1 þ ri cosðkzÞ:
ð6Þ
Due to some wakefields upstream of compressor, the beam gets modulated in energy at the same frequency with the amplitude Dg: Describing the action of wakefields by longitudinal impedance ZðkÞ we can connect the amplitudes of energy and density modulations as follows: Dg ¼
jZðkÞj I0 r Z 0 IA i
ð7Þ
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
4
where Z0 ¼ 377 O is the free-space impedance and IA ¼ 17 kA is the Alfven current. Then, using Eq. (4), we calculate the amplitude of the induced density modulation at the end of bunch compressor. In general case, to find final density modulation rf one should sum up induced modulation and (transformed to the end of compressor) initial one, taking care of phase relations. But in this paper we use the approximation
where Sc is the length of a path through compressor and b is the beta-function. This condition is always met in practice. The formulas of this section can be used when the effect of wakefields in front of compressor is much stronger than any collective effects inside the bunch compressor.
ri 5rind 51:
4. CSR in a bunch compressor chicane
In other words, rf Crind and the gain in density modulation r r G ¼ f C ind ri ri
Wakefields can also exist inside bunch compressors. We consider here coherent synchrotron radiation which is an intrinsic feature of magnetic compressors. CSR effects can be minimized there but not avoided. A CSR-induced beam instability in storage rings has been investigated in Ref. [18]. That instability develops continuously, in small increments, like most instabilities of relativistic electron beams. We analyze here quite different klystron-like mechanism of the instability in a bunch compressor.
is assumed to be high, Gb1 (otherwise the effect, considered in this paper, is not of great importance). Under this approximation the gain depends neither on phase of ZðkÞ nor on sign of R56 and is equal to ! I0 jZðkÞj 1 2 2 2 s2g G ¼ CkjR56 j exp C k R56 2 : ð8Þ 2 g0 I A Z 0 g0 For broadband nonresonant wakefields the product kjZðkÞj is usually a growing function of k: For such cases the maximal gain is achieved at sg 1 C jR56 jC: ð9Þ kopt g0 The optimal final frequency (when the beam is compressed) is almost independent of compression factor C: A crude estimate for the maximal gain is Gmax C
I0 jZðkopt Þj : sg IA Z0
ð10Þ
The term I0 =ðsg IA Þ is proportional to a longitudinal brightness (particles density in longitudinal phase space). In practice, the phase space distribution can be of complex shape. We note that the local energy spread should be taken for estimations of amplification effect. In the considered case the influence of beam emittance e on longitudinal dynamics is a secondorder effect and is negligible when keSc =b51
ð11Þ
4.1. The model In this section, we consider a simplified case when the beam is not compressed (no energy chirp: h ¼ 0; C ¼ 1Þ: While the formulae of the previous section are rather general and do not depend on a type of the bunch compressor, in this section we have to choose a specific model. We consider a symmetric three-dipole chicane1 where the first and the last dipoles have the length Ld ; and the middle one is as long as 2Ld : The bending angle in the first dipole y is small, y ¼ Ld =R51 ðR is the bending radius), and the distance DL between two subsequent dipoles is much larger than the dipole length: DLbLd :
ð12Þ
The R56 can then be expressed in a simple form: R56 C 2DLy2 :
ð13Þ
To describe CSR we use the steady-state model neglecting transient effects. The domain of validity 1 Under limitations, accepted in this section, all the results are valid for a four-dipole chicane, too.
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
of this model can be estimated as 2
Ld bð24R =kÞ
1=3
:
ð14Þ
It is assumed that inequality kR=g30 51 always holds. We neglect the influence on CSR of transverse beam size and of the screening effect of the vacuum chamber requiring that [9,20] ðR=b3 Þ1=2 5k5ðR=s3> Þ1=2
ð15Þ
where b is the transverse size of vacuum chamber and s> is that of electron beam. Since we analyze the dynamics inside bunch compressor, we cannot use condition (11) for neglecting emittance effect on longitudinal dynamics. The relevant condition will be presented below. We also neglect here collective transverse forces [19]. Under conditions (14) and (15) the module of CSR impedance in the first and the last dipoles can be expressed as [20] jZðkÞj jZl ðkÞjLd 2Gð2=3ÞLd k1=3 ¼ ¼ Z0 Z0 31=3 R2=3
ð16Þ
and in the middle dipole it is two times larger. Here Zl ðkÞ is the impedance per unit length and Gð?Þ is the complete gamma-function, Gð2=3ÞC1:354: One can use impedance (16) to calculate energy modulation only if the relative change of density modulation is small on the scale of formation length ð24R2 =kÞ1=3 (see Ref. [21] for more details). In our model the density changes on the scale of Ld and this condition is equivalent to Eq. (14).
5
where s1 is a coordinate along the reference trajectory, s1 ¼ 0 at the entrance to the first dipole. The amplitude of the induced density modulation in the second dipole can then be calculated using simple generalization of formula (4) for ‘‘cold’’ beam: Z Ld dDg r2 ðs2 Þ ¼ k ds1 R56 ðs1 -s2 Þ ð18Þ g 0 0 ds1 where s2 ¼ 0 at the entrance to the second dipole and R56 ðs1 -s2 Þ connects energy kick at s1 and a change of position z at s2 : Under condition (12), it is well approximated by DL R56 ðs1 -s2 ÞC 2 ðLd s1 Þs2 ð19Þ R Then we get r2 ðs2 Þ ¼
1jZl ðkÞj I0 kDLL2d ri s2 : 2 Z0 g0 IA R2
ð20Þ
Since we assumed that the induced modulation r2 is much larger than the initial one (gain is large), the change of CSR-induced energy modulation in the second dipole can be approximated by dDg jZl ðkÞj I0 ¼ r ðs2 Þ ð21Þ g0 ds2 Z 0 g0 I A 2 with negligibly small initial value. Then we do similar calculations for the third dipole ðs3 ¼ 0 at its entrance): Z 2Ld dDg r3 ðs3 Þ ¼ k ds2 R56 ðs2 -s3 Þ ð22Þ g 0 0 ds2 where DL ð2Ld s2 Þs3 : ð23Þ R2 Finally, using Eqs. (20)–(23) we obtain the total gain as a ratio between final and initial amplitudes of density modulation: r3 ðs3 ¼ Ld Þ 2 jZl ðkÞj2 I0 2 k2 DL2 L6d G¼ ¼ : ri 3 Z02 g0 IA R4
R56 ðs2 -s3 ÞC 4.2. ‘‘Cold’’ electron beam Let us first consider the case when the energy spread can be neglected. In the framework of the accepted model we consider the two-stage amplification in the bunch compressor. The gain is assumed to be large in each stage. An initial density perturbation (6) causes energy modulation in the first dipole (see Eqs. (7),(16)). This modulation increases linearly inside the dipole:
With the help of Eqs. (13) and (16) we rewrite the expression for the gain in a different form 2G2 ð2=3Þ I0 2 k8=3 jR56 j2 L2d : ð24Þ G¼ g0 I A 35=3 R4=3
dDg jZl ðkÞj I0 ¼ r g0 ds1 Z0 g0 IA i
When deriving this expression we neglected density bunching inside a dipole caused by the energy
ð17Þ
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
6
modulation induced in the same dipole. In other words, we ignored a self-consistent process inside a dipole which may eventually lead to an exponential growth of the modulation. Thorough analysis shows that expression (24) is accurate as soon as
and the function f is # ¼ 3k#2=3 expðk#2 =2Þ f ðkÞ " # pffiffiffi p k#2 2 2 # expðk# =4Þ erfðk=2Þ ð27Þ 1þ 2 k#
Ld pLg
where
where g IA 1=4 1=3 2=3 Lg ¼ 0 k R : I0
erfðxÞ ¼ 2p1=2
Z
x
dt expðt2 Þ
0
In the case when Ld bLg one could expect an exponential growth inside dipoles with the growth rate about L1 g : In practice, however, for any reasonable set of electron beam parameters, this regime cannot be achieved because the frequency range will be limited by the energy spread and/or emittance. The exponential growth (with a different scaling for the growth rate) inside dipoles of a bunch compressor has been predicted in Ref. [22] but the results of that paper are incorrect.2
# is the error function. For k51; when the influence # k#8=3 of energy spread is negligible, we get f ðkÞC and Eq. (26) is reduced to Eq. (24). The maximal value of f is achieved at k#opt ¼ 2:15 and is equal to 1:98: Thus, the maximal gain is Gmax ¼ 1:16 g20
ð28Þ
in agreement with a simple estimate. The accurate result (28) differs by o10% from the result of Ref. [24], although an incorrect assumption of constant density modulation inside dipoles has been used in Ref. [24]. 4.4. Estimation of emittance effect
4.3. Gaussian energy spread Let us first make a simple estimation of the gain. If we assume that energy modulation happens only in the first dipole, we can use (with C ¼ 1Þ relations (9), (10) and (16) to estimate the gain at Gmax Cg0 ; where I0 g0 1=3 Ld g0 ¼ : ð25Þ 2 s g IA s g ðR jR56 jÞ1=3 Actually, we have two-stage amplification and should expect the dependence Gmax Cg20 : The accurate calculations [23] confirm this estimate. Leaving out the details of calculations we present here the final result for the gain G¼
2G2 ð2=3Þ 2 # g0 f ðkÞ: 35=3
ð26Þ
Here, sg k# ¼ jR56 jk g0 2
An incorrect equation of longitudinal motion has been used.
One of the limitations due to emittance (15) was already presented. More stringent limitation, however, comes from the longitudinal motion inside dipoles.3 Indeed, a particle with an offset x from the reference orbit moves along the beam: dz x ¼ : ds R pffiffiffiffiffi For a typical offset xC eb; the maximal pffiffiffiffiffichange of coordinate z can be estimated at ebLd =R: Therefore, the above-presented results for the gain can be used if pffiffiffiffiffi k ebLd 51: ð29Þ R The energy spread cuts off the gain at 1 kCs1 g g0 jR56 j : Using Eq. (13) we estimate that formula (26) is valid in the entire wavelength range when pffiffiffiffiffi sg Ld eb5 DL : ð30Þ g0 R 3 The net effect through the whole compressor is of the second order, see (11).
E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
In other words, transverse size, defined by emittance, should be smaller than dispersioninduced transverse size in the middle dipole. In opposite case, thepcut-off will be defined by the ffiffiffiffiffi condition kCR=ð ebLd Þ: To roughly estimate maximal gain one can substitute this expression for k into formula (24).
5. Estimation of shot noise effect In this paper, we considered a bunch compressor as a high-gain linear amplifier of klystron type. It amplifies initial density modulations within some frequency band. For broadband wakefields (like CSR) we have a broadband amplifier with DoCo0 ; where o0 is some optimal frequency defined by the energy spread (eventually by emittance). Even without macroscopic density modulation, the initial signal for such an amplifier always exists because of the shot noise in electron beam. Let us consider the beam consisting of randomly emitted electrons with an average current I0 : Its spectrum is a ‘‘white’’ noise which will be filtered by the amplifier. The time domain fluctuations of the current at the amplifier entrance within a frequency band Do are defined by Schottky formula: /i2 S ¼
eI0 Do : p
ð31Þ
Since DoCo0 in the considered case, we can write down the following expression for relative initial fluctuations: /r2i Ssh ¼
/i2 S eo0 1 C C pI0 Nl I02
ð32Þ
where Nl is a number of particles per wavelength l ¼ 2pc=o0 : Thus, effective initial density modulation due to the shot noise is 1 ðri Þsh Cpffiffiffiffiffiffi: Nl
ð33Þ
The coherence length for such a broadband system is about lc C2pc=o0 ¼ l: Thus, the effective shot noise bunching is about inverse square root of number of particles per coherence length.
7
Relative amplitude of AC current behind the bunch compressor is then of the order of Gmax rf Cpffiffiffiffiffiffi: Nl
ð34Þ
Due to the lack of coherence this current will constitute irregular, spiky structure on a time scale of o1 0 : For typical parameters of electron beams one can estimate ðri Þsh at 104 : Thus, for a sufficiently large gain (especially in a chain of bunch compressors) the shot noise bunching can become important. Finally, we note that when simulating bunch compressors with the help of macroparticles approach, one should eliminate artificial noise connected with a relatively small number of macroparticles. If, for instance, one uses Nm macroparticles in a bunch and they are distributed in phase space randomly, thenpthe noise effect will ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be overestimated by a factor Nb =Nm ; where Nb is actual number of particles in the bunch.
6. Numerical example Let us consider an example of amplification of high-frequency modulations due to CSR for the case when the beam goes through the chicane without compression. Although such a case is not very interesting for practice, analytical results of this paper can be used for benchmarking numerical simulation codes. The parameters of the four-dipole chicane and of the electron beam are as follows [25]: Ld ¼ 0:5 m; DL ¼ 5 m; R ¼ 10:35 m; R56 ¼ 0:025 m; electron energy is 5 GeV; uncorrelated rms energy spread is 100 keV; rms bunch length is 20 mm; peak current is 6 kA: In Fig. 1, we present the gain curves for two cases: energy spread is included (curve 1) and ‘‘cold’’ beam (curve 2). Triangles (for the beam with energy spread) and circles (for ‘‘cold’’ beam) are the results of numerical simulations with 1-D code [26]. One can notice rather good agreement between analytical and numerical results.
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E.L. Saldin et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 1–8
Fig. 1. The small-signal gain versus modulation wavelength. Curve (1) was calculated with the help of formula (26) taking into account the energy spread while formula (24) was used to compute curve (2) for the ‘‘cold’’ beam case. Triangles (for beam with energy spread) and dots (for ‘‘cold’’ beam) are the results of numerical simulations [26].
Acknowledgements We thank R. Brinkmann, M. Dohlus, K. . Flottmann, T. Limberg, Ph. Piot, J. Rossbach and D. Trines for useful discussions.
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[6] T.O. Raubenheimer (Ed.), Zeroth Order Design Report for the Next Linear Collider, SLAC Report 474, 1996. [7] N. Akasaka, et al., JLC Design Study, KEK Report 97-1, 1997. [8] L.V. Iogansen, M.S. Rabinovich, Sov. Phys. JETP 37 (10) (1960) 83. [9] Ya.S. Derbenev, et al., DESY print TESLA-FEL 95-05, 1995. [10] M. Huning, . Ph. Piot, H. Schlarb, DESY print TESLAFEL-2000-05, 2000; M. Huning, . Ph. Piot, H. Schlarb, Nucl. Instr. and Meth. A 475 (2001) 348. [11] H.H. Braun, et al., Phys. Rev. Special Topics—Accelerators and Beams 3 (2000) 124 402. [12] Ph. Piot, et al., Proceedings of EPAC 2000, Vienna, Austria, 2000, p. 1546. [13] M. Borland, Nucl. Instr. and Meth. A 483 (2002) 268. [14] T. Limberg, Ph. Piot, E.A. Schneidmiller, DESY print TESLA-FEL-2000-05, 2000; T. Limberg, Ph. Piot, E.A. Schneidmiller, Nucl. Instr. and Meth. A 475 (2001) 353. [15] R. Li, Proceedings of EPAC 2000, Vienna, Austria, 2000, p. 1312. [16] E.L. Saldin, E.A. Schneidmiller M.V. Yurkov, Nucl. Instr. and Meth. A 483 (2002) 516. [17] L.H. Yu, Phys. Rev. A 44 (1991) 5178. [18] S. Heifets, G. Stupakov, SLAC-PUB-8761, 2001. [19] Ya.S. Derbrnev, V.D, Shiltsev, FNAL-TM-1974, 1996. [20] J.B. Murphy, S. Krinsky, R.L. Gluckstern, Part. Accel. 57 (1997) 9. [21] S. Heifets, SLAC-PUB-9054, 2001. [22] S. Heifets, G. Stupakov, SLAC-PUB-8988, 2001. [23] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, CSRinduced klystron instability in bunch compressors, in preparation. [24] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, DESY print DESY-01-129, 2001. [25] http://www.desy.de/csr, Benchmark Example. [26] M. Dohlus, Field calculations for bunch compressors, Presented at the CSR workshop, Zeuthen, Germany, January 2002, http://www.desy.de/csr.