Effects of Potential Energy Spread on Particle Dynamics in Magnetic

Proceedings of FEL2014, Basel, Switzerland THP032 EFFECTS OF POTENTIAL ENERGY SPREAD ON PARTICLE

DYNAMICS IN MAGNETIC BENDING SYSTEMS ∗ R. Li , Jeﬀerson Lab, Newport News, VA 23606, USA

Abstract transverse dynamics have been analyzed earlier [5–7], and Understanding CSR effects for the generation and trans- it is found that their harmful effects related to the potent port of high brightness electron beams is crucial for designs feature of strong transverse dependence are cancelled. After of modern FELs. Most studies of CSR effects focus on the the cancellation, the transverse dynamics of particles is per- impacts of the longitudinal CSR wakefield. In this study, turbed by the remaining driving factors such as the effective we investigate the impact of the initial retarded potential longitudinal CSR force and the centrifugal force related to energy of particles, due to bunch collective interaction, on particles’ initial potential energy. the transverse dynamics of particles on a curved orbit. It is In this study, we present the role of potential energy in shown that as part of the remnants of the CSR cancellation the particle transverse dynamics after the cancellation effect effect when both the longitudinal and transverse CSR forces is taken into account. We will show that the initial slice are taken into account, this initial potential energy at the potential energy spread of a bunch, which we call pseudo entrance of a bending system acts as a pseudo kinetic en- slice energy spread, is indistinguishable from the usual slice ergy, or pseudo energy in short, because its effect on particle kinetic energy spread in its perturbation to the transverse par- optics through dispersion and momentum compaction is in- ticle optics via both dispersion and momentum compaction. distinguishable from effect of the usual kinetic energy offset This effect is measurable only when the peak current of from the design energy. Our estimation indicates that the the bunch is high enough such that the pseudo slice energy resulting effect of pseudo energy spread can be measurable spread is appreciable compared to the slice kinetic energy only when the peak current of the bunch is high enough such spread. The implication of this study on simulations and that the slice pseudo energy spread is appreciable compared experiments of CSR effects will be discussed. to the slice kinetic energy spread. The implication of this . study on simulations and experiments of CSR effects will be discussed. ROLE OF POTENTIAL ENERGY IN BUNCH TRANSVERSE DYNAMICS INTRODUCTION In this section the CSR cancellation effect is briefly re- When a high brightness electron beam is transported viewed. We show how a centrifugal force term, which is through a curved orbit in a bending system, the particle related to the initial potential energy of particles, emerges dynamics is perturbed by the coherent synchrotron radiation as one of the remnant of the cancellation. We also discuss (CSR) forces, or the collective Lorentz force as a result of the role of initial potential energy in transverse particle dy- the Lienard-Wiechert fields generated by particles in the namics. bunch. The longitudinal CSR interaction takes place when Consider an ultrarelativistic electron bunch moving on the fields generated by source particles at bunch tail over- a circular orbit with design radius R and design energy take the motion of the test particles [1] at bunch head and 2 E0 = γ0mc . Let x = r R be the radial offset of par- cause changes of kinetic energy for the head particles. For ticles from the design orbit.− The single particle optics is parameters currently used in most machine designs and op- determined by the configuration of the external magnetic erations, the approximation of the longitudinal CSR force fields, while for a bunch with high peak current, this design by that calculated using 1D rigid-line bunch model [2] often optics will be perturbed by the Lorentz force Fcol result- gives good description of the observed CSR effects [3]. ing from the collective electromagnetic interaction amongst In addition to the longitudinal CSR force, the transverse particles within the bunch. For transverse dynamics, such CSR force [4] can directly perturb transverse particle dy- perturbation is expressed in terms of the first order equation namics. This force features energy independence and, due to divergent contribution from nearby-particle interaction, has d2 x x ∆E Fcol strong nonlinear dependence on the transverse (and longitu- + = + x , (1) c2dt2 R2 RE E dinal) positions of particles inside the bunch. Meanwhile, 0 0 the potential energy change, as a result of both longitudinal col with Fx being the radial component of the collective and radial CSR or Coulomb forces, can cause change of Fcol = cole cole ∆ = Lorentz force Fs s + Fx r , and E E E0 kinetic energy of the particles and impact transverse particle being the deviation of the kinetic energy from the design− dynamics via dispersion. The joint effects of both the trans- energy. The existence and effect of transverse CSR force verse CSR force and the kinetic energy change on bunch col Fx were first pointed out by Talman [4] when he analyzed ∗ col Work supported by Jefferson Science Associates, LLC under U.S. DOE Fx for space charge interaction of a bunch on a circular Contract No. DE-AC05-06OR23177. orbit using Lienard-Wiechert fields. His study shows that ISBN 978-3-95450-133-5

eff as a result of logarithmic divergence of nearby-particle in- with the longitudinal effective force term Fv defined as teraction, this force is dominated by a term with the potent ∂Φcol ∂Acol undesirable feature of strong nonlinear dependence over par- Feff = e β . (7) ticles’ transverse position within the bunch. Besides its role v c∂t − · c∂t ! in the second driving term on the RHS of Eq. (1), the CSR (r v ) force also causes change of particle’s kinetic energy For a bunch with phase space density distribution f , ,t , the retarded potentials (Φcol,Acol) in the above expression t are given by ∆E(t) = ∆E (t = 0) + Fcol vdt′, (2) Z0 · f (r′,v′,t′) Φcol(r,t) = e dr′dv′, which impacts the beam optics via dispersion as depicted Z r r′ ′| −′ |′ ′ by the first driving term in Eq. (1). As an example, the case v (r v ) Acol r = f , ,t r′ v′ col = col = ( ,t) e ′ d d (8) of zero bunch charge corresponds to Fx Fs 0 and Z r r ∆E = E(0) E , when Eqs. (1) and (2) are reduced to equa- | − | − 0 ′ ′ tions for single particle dynamics. For the case of 1D CSR for t = t r r /c, and with Ne electrons in the bunch, − | − | col = model as used in ELEGANT simulation, one has (1) Fx 0 col r v r v = and (2) Fs in Eq. (2) is obtained for a 1D bunch moving f ( , ,t)d d Ne . (9) along the design orbit with its line-charge density distribu- Z tion obtained by projecting the actual 3D bunch distribution col The advantage of Eqs. (3)-(7) is that the potent term in Fx (at the time of force calculation) onto the design orbit, with of Eqs. (1) and (3), contributed from the divergent local the assumption that this 1D projected distribution has been interactions, is now cleanly represented by the centrifugal frozen as it is for all retarded times. In general, however, the space charge force FCSCF in Eq. (4), leaving the remaining kinetic energy can be changed by the usual longitudinal CSR eff effective radial force Fx free from the energy-independent force acting on the bunch as well as by the potential energy local divergence of the order of FCSCF. Similarly, all the change caused by various ways of particle-bunch interac- local-interaction contributions to ∆E(t) are summarized by tion. Examples of such particle-bunch interaction include e[Φcol(t) Φcol(0)] in Eq. (6), leaving the remaining con- (1) betatron motion in the potential well set by the radial − − eff tributions from the effective longitudinal force Fv also free CSR force for a coasting beam [8], (2) noninertial space from the energy-independent local divergence. Substitut- charge force related to the radiative part of the longitudinal ing Eqs. (3) and (6) into Eq. (1), one obtains the transverse Lienard-Wiechert electrical field experienced by off-axis dynamical equation particles interacting with a line bunch on a circular orbit [9], 2 (3) longitudinal space-charge interaction for a converging d x x δE (0) + = + Gˆ cor (10) bunch on a straight section right before the bunch entering c2dt2 R2 R into the last dipole of a bunch compression chicane [10]. In total all these examples, the collective-interaction-induced poten- with δE (0) the relative energy deviation from design tial energy eΦcol shares the same feature of strong nonlinear energy dependence of particles’ transverse position as that shown (0) E0 in the Talman’s force. δE (0) = E − , for (0) = E(0) + Eφ (0) (11) E E The relation between the two driving terms in Eq. (1) 0 col ∆ becomes clear when both Fx and E for the test particle with col are written in terms of retarded potentials [5]. The Taman’s Eφ (0) = eΦ (0), (12) force is written as and Gˆ cor contains all the terms related to the interaction col = eff CSCF Lagrangian [7] that has negligible energy-independent local- Fx Fx + F , (3) interaction contributions CSCF with the centrifugal space charge force F term and the t eff ˆ cor = 1 1 eff ′ ′ eff effective tranverse force Fx term defined respectively as G Fv (t )dt + Fx + Gres (13) E0 R Z0 ! col e βs A FCSCF = s , (4) with the residual of cancellation r Φcol Acol dAcol Φcol( ) col( ) eff = ∂ ∂ x t βs As t Fx e β e . (5) Gres = e − − ∂x − · ∂x ! − cdt − RE0 2 γ−2 + θ2/2 Meanwhile the change of kinetic energy for the test particle e 0 r′ v′ ′ r′ v′ ′ f ( , ,t )d d , (14) is ≃ − R Z r r | − | t for θ = [s(t) s′(t′)]/R being the angular distance be- ∆ ( ) = ∆ ( ) Φcol( ) Φcol( ) eff( ′) ′ E t E 0 e[ t 0 ] + Fv t cdt , tween the test particle− at the observation time and the source − − Z0 (6) particle at retarded time. As we can see, the terms with ISBN 978-3-95450-133-5

col ∆ ∆ ′ ∆ ∆ potent local-divergence contributions in both Fr /E0 and in Let ( xc , xc , zc , δc ) be the phase space perturbation cor ∆E/(RE0) of Eq. (1) are collected in Gres in Eq. (14). More- generated by Gˆ in Eq. (10). Then after combining Eq. (10) over, since θ 0 when r′ r, the local divergence of (for s ct ) with dz/ds = x/R(s), one finds that the initial the two terms→ cancels, leaving→ the residual with no energy- kinetic≃ energy offset and− potential energy of a particle at independent contributions from local interaction for any the entrance of a bending system always work together, as bunch distribution, as indicated by Eq. (14) [7,11, 12]. the joint entity δE0, in causing transverse and longitudinal Unlike the usual longitudinal CSR force, which is the re- chromatic effects for single particle optics, namely, sult of EM field generated by tail of the bunch overtaking the x R R 0 R x ∆x head of the bunch [1], analyses of the radial CSR force based    11 12 16   0   c  x′ R R 0 R x′ ∆x′ on Lienard-Wiechert fields show [11,13] that Fcol is domi- = 21 22 26 0 + c , x  z   R R 1 R   z   ∆z  nated by contributions from head-tail interaction that has a    51 52 56   0   c   δ   0 0 0 1   δ   ∆δ  sudden turn-on behavior as the bunch moves from straight  H     H0   c  (15) section onto a circular orbit. It can be shown the dominant col CSCF for head-tail contributions in Fx are all included in F of δE = δ + δ , (16) Eq. (4), and it turns on suddenly when the bunch moves from 0 k0 φ0 = straight section r to r R. Likewise the head-tail con- with δk0 = (E(0) E0)/E0 the usual relative kinetic energy → ∞ Φcol ∆ − tributions are equally important for e (t) term in E of offset, and δφ0 = Eφ0/E0. The significance of the impact ∆− Eq. (6). Since the perturbation of E on optics in Eq. (1) is of δφ0 on beam dynamics depends on the comparison of via the driving term ∆E/(E0 R) in Eq. (1), its effect on the the initial potential energy spread with the initial kinetic en- transverse dynamics also has a sudden turn-on behavior as ergy spread. The discussion of Lorentz gauge as the natural the bunch enters from straight section (R ) to a circular choice of gauge for exhibiting the CSR cancellation effect → ∞ arc (R finite). Consequently the two terms involved in the can be found in Ref. [7]. cancellation in Eq. (14) share common behavior and the residual of the cancellation is orders of magnitude smaller THE SLICE TOTAL ENERGY SPREAD than each of the two terms [12]. FOR A GAUSSIAN BUNCH Φcol As the two potent terms e (t)/(RE0) and As a simplest example, we consider the impact of E on col − φ e βs As (t)/(RE0)—contained in the first and second the slice spread of total energy for an electron bunch with driving terms of Eq. (1) respectively—are cancelled, an idealistic cylindrically symmetric 3D Gaussian density Φcol a centrifugal force term e (0)/RE0 emerges as the distribution remnant of the cancellation. The effect of this term on particle dynamics is the focus of this paper. One important 1 r2 z2 P(r, z) = exp . (17) observation [11, 13] is that similar to features of the 3/2 2 2 2 (2π) σr σz − 2σr − 2σz ! centrifugal space charge force, eΦcol(0) in Eq. (11) also contains contribution from the head-tail particle interaction The bunch arrives at the entrance of a bending system at t = 0 with local divergence, and its effect on the transverse (we drop t dependence in the following discussion) after dynamics has the sudden turn-on behavior which is not moving ultrarelativistically on a straight path with vs = βc cancelled away. On the other hand, it was noted [6, 12] and vx = vy = 0. Here we have σx = σy = σr and 2 2 2 that since δH (0) in Eq. (10) acts as the initial relative r = x + y . The probability distribution of particles over col col energy offset, so unlike the eΦ (t) term, here eΦ (0) in the total energy offset ∆ = ∆E + Eφ, with ∆E, Eφ and ∆ E E Eq. (11) does not cause emittance growth when the bunch is being random variables, is transported through an achromatic bending system such as a magnetic bunch compression chicane. PE (∆ ) = PE (∆ Eφ )PE (Eφ )dEφ, (18) E Z E − φ Another residual term left from the cancellation is Gˆ cor in Eq. (10). It is dominated by effects from the longitudi- where we assume a Gaussian distribution for the kinetic nal tail-head CSR interaction, for which 1D CSR model is energy usually a good approximation. Exceptional situations occur 1  (∆E)2  PE (∆E) = exp . (19) when the Derbenev’s criteria [5] for the validity of 1D CSR √ − 2σ2 2πσE  E  model is violated. Such situations include CSR interaction during roll-over compression [14] and cases of CSR-induced In this section we summarize how the distribution of total microbunching instability when either the modulation wave- energy PE (∆ ), for the central slice (z = 0) of the bunch, E length is very short or the bunch transverse size is very large. deviates from the usual kinetic energy distribution PE (∆E)

The correct evaluation of retarded potentials/ﬁelds in such as a result of the potential energy distribution PEφ (Eφ ) of situations requires extra care in identifying the source parti- the particles. We note that PE (∆ ) = PE (E) when the E = cles by ﬁnding the intersection of the past light cone of test potential energy vanishes, i.e., PEφ (Eφ ) δ(Eφ ). particles with the 2D/3D bunch distribution in the history The potential energy distribution PEφ (Eφ ) can be of bunch motion [14]. uniquely determined from the probability distribution of ISBN 978-3-95450-133-5

particles in the bunch P(r,t) and the dependence of potential energy on the particle spatial distribution Eφ (r,t). For our example of cylindrical Gaussian distribution, with 2 (r˜, z˜) = (r/σr , z/σz ) and w = r˜ , the probability for particles to lie between w and w + dw for the central z = 0 slice is deduced from Eq. (17) ∞ −w/2 Pw (w) = e /2, with Pw (w)dw = 1. (20) Z0 The potential energy of a test particle at (x,t) x x ρ(xr ,t r /c) 3 Eφ (x,t) = e − | − | d xr (21) Z x xr | − | can be obtained [15] by applying Lorentz transformation on the scalar potential from the bunch comoving frame to 2 the lab frame. For our example, with α = (σr /γσz ) , this yields

2 Eφ (r, z) = Eφ0 f (r˜, z˜), with Eφ0 = mc Ip/IA (22)

for the peak current Ip = Ne ec/(√2πσz ) and Alfven current IA = e/(re c) = 17 kA, and

∞ dτ r˜2 z˜2 f (r˜, z˜) = exp . Z0 (1 + τ)√1 + ατ − 2(1 + τ) − 2(1 + ατ) ! (23) The dependence of the form function f (x˜, y˜, z˜) is shown in Fig. 1. For the central slice at z = 0, we have from Eq. (22) Figure 1: Behavior of f (x˜, y˜, z˜) in Eq. (23) for a cylindrical = Eφ (r,0) Eφ0 U(w) (24) bunch over (a) the x˜ = 0 plane and (b) the z˜ = 0 plane for α = 5.7 10−7 . for the normalized potential energy ×

∞ dτ w U(w) = exp . (25) 0.51. An estimation of the rms spread of the total energy Z0 (1 + τ)√1 + ατ − 2(1 + τ) ! resulted from Eq. (18) is given by

Combining U(w) with Pw (w) in Eq. (20), one gets the prob- 2 2 σE σE + (Eφ0σU ) . (29) ability for the value of potential energy of a particle to reside ≃ q between φ and φ + d φ E E E This relation shows that the rms of the total energy can be appreciably larger than that of the kinetic slice energy spread PE ( φ ) d φ = PU (U)dU = Pw (w)dw (26) φ E E when Eφ0σU or ξ 1 (30) Pw (w) ≡ σE ≥ PU (w) PEφ ( φ ) φ0 = . (27) ≡ E E dU(w)/dw implying simutaneously both high peak current Ip and small | | kinetic energy spread σ . The semi-analytical results of probability distribution P E U To quantitatively compare the slice potential energy on U is then obtained from the parametric dependence of spread with the usual slice kinetic energy spread, and in (U(w), P (w)) on w [15], as shown in Fig. 2, which au- U particular, to compute the slice spread of the total energy, tomatically satisﬁes P (U)dU = 1. The probability of U here we use the following parameters for the bunch [16, 17] distribution for the totalR energy of particles in Eq. (18) then becomes = = n = = E0 135MeV, σz 750µm, ǫ x,y 1µm, βx,y 6 m, (31) ∆ = ∆ − PE ( ) PE ( Eφ0U)PU (U)dU. (28) which corresponds to α = 5.7 10 7. We further choose E Z E − × Ip = 120 A and thus Eφ0 = 3.6 keV. Substituting the semi- For the example in Fig. 1, the average and rms of particle analytical results PU (U) as shown in Fig. 2 into Eq. (28), one distribution over U are respectively < U >= 15.0 and σU = gets the distribution of total energy spread PE (∆ ) for cases E ISBN 978-3-95450-133-5

PU HUL HaL ΣE k=3 keV 1.0 0.14 0.8 0.12 0.10 0.6 0.08 E U P

P 0.06 0.4 0.04 0.2 0.02

< U > 0.00 0.0 -15 -10 -5 0 5 10 15 12 13 14 15 16 E HkeVL U HbL ΣE k=1 keV Figure 2: Probability distribution PU (U) for the central slice 0.4 of a cylindrical Gaussian bunch with α = 5.7 10−7. × 0.3 when the slice kinetic energy spread takes (1) the typical E 0.2 value σE = 3 keV [18] when ξ = 0.6 and (2) σE = 1 P keV when ξ = 1.8. The final results are shown in Fig. 3. 0.1 It confirms our expectation that when ξ 1 the potential energy has negligible effect on the slice total≤ energy spread 0.0 σE , as displayed by Fig. 3a, yet when ξ 1 the potential -15 -10 -5 0 5 10 15 energy spread can cause appreciable widening≥ of the total E keV energy spread, as indicated by Fig. 3b in which the red H L curve for PE (∆ ) is much wider than the green curve for Figure 3: PE (∆ ) for z = 0 slice of a cylindrical Gaus- E E = = −7 PE (E). Numerical evaluation using PE (∆ ) of Eq. (28) sian bunch with Eφ0 3.6 keV, α 5.7 10 for (a) E × yields σE = 2.09 keV for Fig. 3b, which agrees well with the σE = 3keV and (b) σE = 1 keV. Green line: Gaussian estimation by Eq. (29) for σE = 1 keV. The semi-analytical distribution for kinetic energy PE (E); Brown line: semi- results of PE (∆ ), shown by the red curves in Fig. 3, are analytical results for distribution of total energy PE (∆ ); E also in good agreementE with results from the Monte Carlo Solid black line: Gaussian distribution using the estimated approach presented by the red dots in Fig. 3. Here for the rms in Eq. (29); Red dots: Monte Carlo results of PE (∆ ). E Monte Carlo approach we populate N = 104 particles with random Gaussian distributions in both the 2D configuration space (in the z = 0 plane) and in the kinetic energy offset ∆ ∆ i = ∆ i i E. One then evaluates the total energy E + Eφ approximately a constant when the bunch gets compressed = i E by a magnetic chicane with compression factor C, since both of the i-th (i 1, N) particle by using Eφ in Eq. (22), and further obtain the histogram of particle distribution in ∆ . Eφ0 in Eq. (22) and the slice kinetic energy spread σE are E increased by C. The potential energy spread for a general 3D DISCUSSIONS bunch distribution requires careful numerical calculations. In this paper, after a brief review of the cancellation effect For beam and machine parameters used in present de- in the CSR-induced perturbation on bunch transverse dynam- signs, the 1D CSR model adopted by ELEGANT simula- ics for an ultrarelativistic electron bunch moving through tion often gives results in good agreement with measured a magnetic bending system, it is shown how, after cancel- CSR effects [3]. From the point of view of cancellation lation, the effective CSR forces and the potential-energy effect in CSR, the explanation of such success resides in the related centrifugal force emerge as the net driving factors for fact that in the 1D model both parts of the cancellation— col particle transverse dynamics. The behavior of longitudinal the radial CSR force Fr and the potential energy change effective CSR force for an energy-chirped Gaussian bunch e[Φcol(t) Φcol(0)]/R—-are set to be zero, and only the has been studied earlier [14], and in this presentation we impact− resulted− from the dominant longitudinal CSR force eff summarized that role and behavior of the initial potential Fv is being kept. The 2D/3D CSR effects are expected energy term. The main conclusion is that the initial poten- to show up in experiments for the unusual cases when the tial energy always works together with the initial kinetic bunch distribution in x-z plane does not satisfy the Derbenev 2 1/3 eff energy in perturbing the particle transverse dynamics, and criteria σx/(σz R) 1, thence the behavior of Fv will for the example of a Gaussian bunch, we find that the poten- deviate [14] from that for≪ a 1D rigid-line bunch, and when the tial energy effect is important only when the peak current bunch parameters are such that Eq. (30) is satisfied, thence of the bunch is high and the slice kinetic energy spread is the effect ot potential energy will appear as enlargement of small, i. e., ξ 1. Note that ξ in Eq. (30) should remain the measured slice energy spread or lengthening of bunch ≥ ISBN 978-3-95450-133-5

length for a bunch at maximum compression as compared [4] R. Talman, Phys. Rev. Lett. 56, 1429 (1986). to results from the 1D CSR model. [5] Ya. S. Derbenev and V. D. Shiltsev, SLAC-Pub-7181 (1996). Correct modeling of the cancellation effect poses signifi- [6] R. Li, Proceedings of the 2nd ICFA Advanced Accelerator cant challenge for 2D/3D CSR simulations, since it requires Workshop, p.369 (1999). the two parts involved in the cancellation, e[Φcol(t) Φcol col − − [7] R. Li and Ya. Derbenev, JLAB-TN-02-054 (2002). (0)]/R and the radial CSR force Fr , be calculated with the same accuracy. Because e[Φcol(t) Φcol(0)] is an inte- [8] E. Lee, Part. Accel. 25, 241 (1990). − grated effect of longitudinal and transverse CSR force on the [9] B. E. Carlsten, Phy. Rev. E 54, p.838 (1996). circular orbit or Coulomb forces on a straight path, the can- [10] K. L. F. Bane and A. W. Chao, Phys. Rev. ST Accel. Beams cellation effect can be taken care automatically only when 5, 104401, (2002). both the CSR and space charge interaction are fully taken into et al. account and the dynamics are advanced self-consistently. In [11] G. Geloni , DESY 02-48, (2002). Φcol Φcol col [12] R. Li and Ya. Derbenev, Proceedings of 2005 Particle Accel- addition, all the potential terms (t), (0) and As (t) have sensitive dependence on the 3D bunch density distribu- erator Conference, Knoxville (2005). tion, as shown in Fig. 1. Without adequate care, incomplete [13] G. Geloni et al., DESY 03-44 (2002) modeling could result in partial or none cancellation and [14] R. Li, Phys. Rev. ST Accel. Beams 11, 024401 (2008). cause artificial errors. More detailed discussions can be [15] R. Li, arXiv:1401.2868 (2014) found in Ref. [15]. [16] R. Akre et al., Phys. Rev. ST Accel. Beams 11, 030703 (2008). REFERENCES [17] Z. Huang et al., Phys. Rev. ST Accel. Beams 13, 020703 [1] Ya. S. Derbenev et al., DESY-TESLA-FEL-95-05 (1995). (2010). [2] E. Saldin, E. Schneidmiller and M. Yurkov, Nucl. Instrm. [18] D. Ratner, A. Chao, and Z. Huang, Proceedings of FEL08, Methods Phys. Res. Sect. A 398, 373 (1997). Gyeongju, Korea, (2008). [3] K. L. F. Bane et al., Phys. Rev. ST Accel. Beams12, 030704 (2009)

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