Kudryavtsev, A M

Lotov, K V

Skrinsky, A N

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Particle Accelerators, 1997, Vol. 57, pp. 1-8Reprints available directly from the publisherPhotocopying permitted by license only© 1997 OPA (Overseas Publishers Association)Amsterdam B.V. Published in The Netherlands underlicense by Gordon and Breach Science PublishersPrinted in IndiaON ENERGY LOSS OF NARROW ANDDENSE ULTRA-RELATIVISTIC BUNCH INPLASMAA.M. KUDRYAVTSEV*, K.V. LOTOV and A.N. SKRINSKYBudker Institute of Nuclear Physics, 630090, Novosibirsk, Russia(Received 10 October 1996; In final form 8 January 1997)Energy balance of a particle bunch propagating through the plasma is considered forthe bunches of a high density (compared to plasma) and a small radius (compared toanomalous skin depth). Energy loss to scattering of near-axis plasma electrons is esti-mated. This component of energy balance is shown to be important but not excessivelyhigh for the parameters interesting for high energy physics. In particular, the energyloss to near-axis electrons is proved to remain finite as the bunch radius tends to zero.Keywords: Plasma wakefield acceleration; Ionisation lossesEnergy balance of a particle bunch propagating through the plasmais one of the key questions for plasma-based advanced acceleratortechniques 1,2. Earlier 3, this problem was studied for the bunches of alow density (as compared to that of the plasma) with the most atten-tion paid to beam loading issues. In this paper, we consider the ener-gy balance of a dense (compared to plasma) and narrow (comparedto the anomalous skin depth c/wp ) axisymmetric bunch. In parti-cular, we show that the energy transferred from the bunch to near-axis plasma electrons remains finite as the bunch radius tends to zero(while the bunch length remains finite).The latter issue is especially important for potential applications ofplasma to linear collider for the TeV energy Tange. In this case, theaccelerated beams have to be of extremely small emittances and radii*Corresponding author.2 A.M. KUDRYAVTSEV et al.to provide acceptable luminosity. The bunch density is much higherthan that of the plasma (no), though the bunch population shouldremain much smaller than the number of plasma electrons per(ejwp )3 (otherwise the distortion of the plasma wave would result intoo large energy spread of the accelerated bunch).The total energy balance of such a bunch is composed of twoparts: the energy loss due to the acceleration of near-axis plasmaelectrons and the energy extracted from (or transferred to) remoteplasma layers (at radii rv ej wp ). The latter does not contain anypeculiar to narrow bunch features. It can be easily found in particle-in-cell 3 or finite-difference 4 simulations via calculation of the longi-tudinal electric field. In the non-linear plasma wave (with the electricfield amplitude of the order of J47rnome2), this long-range interac-tion can provide the energy gain up to me wp per particle per unitlength, where m is the electron mass.The energy lost to near-axis electrons cannot be calculated sostraightforwardly. Fluid codes fail to solve this problem because ofessentially non-hydrodynamical motion of plasma electrons near thedense bunch. The correct value of energy losses can, in principle, beobtained with existing particle-in-cell codes, but introduction of theadditional length scale (associated with the bunch radius) greatlyincreases the number of particles required and, thus, prohibits thestudy of extremely narrow bunches. The aim of this paper is to give acorrect estimate of this non-hydrodynamic energy loss for very nar-row bunches of very high density. Note that for this estimate it is notnecessary to specify the profile of the plasma wave accelerating thebunch. Since the energy acquired by near-axis plasma electrons ismuch greater than their initial energy (in the plasma wave), the waveprofile affects the energy loss only through the change of plasmaelectron density.To establish the upper limit of energy losses, we consider an infi-nitely narrow bunch (composed of Nb particles) propagating throughthe unperturbed plasma of density no. The particles are assumed to beuniformly distributed over the bunch length I. Even for zero radiusbunch, the energy loss per unit length (W) turns out to be finite:me 2W == Nb -I-F(L). (1)HereENERGY LOSS OF DENSE BUNCH 3Nb lwp1rno(cjwp)3 . -c-' (2)Wp == J 41rnoe2 / m is the plasma electron frequency and other notationis conventional. The dimensionless functions F(L) are shown inFigure 1 by solid lines. Similar functions for the bunch of a finite width(of the radius c/ wp , with the particles being uniformly distributed overthe bunch cross-section) are shown by dots. The difference betweenthese graphs represents the energy loss to the near-axis plasma layer. Itis seen that the energy loss of the narrow bunch can be several timeshigher than that of a "wide" bunch. The near-axis losses scale as L 2,while the energy transferred to remote plasma layers is proportional toL. Thus, the fraction of near-axis losses increases with the increase ofNb , I, or no.Expression (1) and the function F(L) are obtained as follows. Theradial electric field Er and the azimuthal magnetic field H<p of the in-finitely narrow ultrarelativistic bunch may be approximated by theformula2eNb (rwp )Er(r) == H<p(r) == --;:texp -~ (3)F(L)L0.080.060.040.02. . . . . . . . . . .. . .e, P. . .. .o 0.04 0.08 0.12 0.16 LFIGURE 1 The energy loss for electron (e) and positron (p) bunches.4 A.M. KUDRYAVTSEV et al.(we use cylindrical coordinates with the z-axis being the direction pfbunch propagation). This approximation is good for r « e/wp , whilefor r~ e/ wp the screening of the bunch field by the plasma is takeninto account only qualitatively through the exponential factor. Inparticular, we neglect temporal changes of the fields in far regioncaused by the bunch-driven displacement of plasma electrons. Actu-ally, the law of field screening is somewhat different 5. However, fullyself-consistent treatment of remote plasma layers would not givemore accurate value of near-axis losses, but unnecessarily compli-cates the calculations. It is self-evident that the bunch can be treatedas "rigid" and unaffected by the fields appearing in plasma.Within the considered model, the energy of plasma electrons andthe z-component of their generalized momentum are conserved:"(me2 - e~ == const, e,,(mvz - - ~ == const,e(4)where ~(r) is the electrostatic potential and the z-component of vectorpotential at the same time:(5)"( is the relativistic factor of plasma electrons, and e is the elementarycharge (e > 0). The total energy loss is(6)where ro corresponds to the initial position of a plasma electron(before the bunch disturbs the plasma), and rf is its final radius (afterthe bunch passage).For the electron bunch, we can find rf for any specified ro by merelyequating the electron travel time and the duration of field action:l rf dr I 11 rf Vz- == -+- -drro Vr e e ro Vr (7)(the integral on the right-hand side of (7) represents the distanceENERGY LOSS OF DENSE BUNCHtravelled by the electron in z-direction). Since, from (4),5J2M ~Vr == e 1 + ~ ' Vz == e 1+ ~'we can rewrite (7) aswhere Ll<l> = e((r) - 2<1> (ro) ) ,me(8)l rf drI = ro V2Ll<l> ,whence the dependence rr(ro) can be easily found numerically. Thennumerical integration of (6) yields the formula (1) and the functionF(L).For the positively charged bunch, the energy loss can be found ina similar way with the little complication arising from the oscillatorymotion of near-axis plasma electrons (Figure 2). Energy loss for widebunches is calculated analogously.The finiteness of the energy loss can be proved analytically as well.For the electron bunch this is evident, since the energy acquired byaplasma electron may be majorized by the function e('<1>(oo) - <1>(ro)),which has the integrable logarithmic singularity at ro == o.0.20.150.10.05o 0.02 0.04 0.06 0.08 ri W Ieo pFIGURE 2 The final radius of plasma electrons as the function of their initial radiusfor electron (e) and positron (p) bunches with L = 0.1.6 A.M. KUDRYAVTSEV et al.For the positively charged bunch the situation is more compli-cated, since the infinite energy is acquired by electrons for whichfro dr IJoJ2~-2k+l (k=O,1,2, ... )or(here we take into account the logarithmic behaviour of <P near theaxis). The vicinity of each logarithmical singularity (9) makes a finitecontribution to W, but there is an infinity of them. To majorize thetotal energy loss, we consider the contribution ~Wk of the ringo Lc Lcrk = 2y0r(k+ l)wp < ro < rk = y0r(2k+ l)wp · (10)The final radius rr is determined from the requirementy0r(2k + 1)wp i rf dr· wp / cL == ro + ,c 0 Jln(ro/r)whencedrf vi-d == -v:ff(2k + 1)ln(ro/rr).roThen(11 )ENERGY LOSS OF DENSE BUNCH 7It is easy to show that the contribution of the ring rk+l < ro < r2 is lessthan ~Wk. Since the seriesconverges, we conclude that the energy loss of the positively chargedbunch is finite.It is interesting to note why the calculation of energy loss in termsof "instant push" gives the erroneous result:elErll 1prv--CX-,C ro J p2 JdroW <X 21fro . 2m dro <X ~ (diverges). (13)This is because near-axis plasma electrons rapidly pass to the regionof either smaller field (electron bunch) or opposite-sign field (posi-tron bunch). As a result, the estimate (13) turns out to be unac-ceptably overstated.The general conclusion of the paper is that the non-hydrodynamicenergy losses of a dense bunch travelling in plasma are important butnot excessively high for the parameters interesting for high energyphysics. To illustrate this conclusion let us estimate the energy lossfor the bunch with Nb == 109 particles. For this bunch to be effec-tively accelerated in plasma, the condition(14)should be met, which gives no « 2 . 1016 cm-3 in our case. Also, thebunch length has to be small compared to c/wp (to provide smallenergy spread). Choosing 1 == 0.05mm and no == 1015 cm-3 (c/wp ~0.17 mm), we satisfy both conditions. From (2) we find L ~ 0.15,which corresponds to near-axis losses of about 75 MeV/m per parti-cle. Note that this is the average value; actual loss per particledepends on particle position along the bunch, so the particles at theback front of the bunch lose several times greater energy. The plasmaof the given density can provide the accelerating gradient up to3 GeV1m. Thus, non-hydrodynamical energy losses are relatively8 A.M. KUDRYAVTSEV et al.small, but they should be taken into account in in-depth studies ofplasma-based accelerator techniques.References[1] J.S. Wurtele (1993). The role of plasma in advanced accelerators. Phys. Fluids B,5(7), 2363-2370.[2] E. Esarey, P. Sprangle, J. Krall, and A. Ting (1996). Overview of plasma-basedaccelerator concepts. IEEE Trans. Plasma Sci., 24(2), 252-288.[3] S. Wilks, T. Katsouleas, J.M. Dawson, P. Chen, and J.J. Su (1987). Beam loadingin plasma waves. IEEE Trans. Plasma Sci., PS-15(2), 210-217.[4] B.N. Breizman, T. Tajima, D.L. Fisher, and P.Z. Chebotaev (1992). Excitation ofNonlinear Wake Field in a Plasma for Particle Acceleration. In: Research Trends inPhysics: Coherent Radiation and Particle Acceleration, edited by A. Prokhorov, pp.263-287 (AlP Press, New York).[5] R. Keinigs and M.E. Jones (1987). Two-dimensional dynamics of the plasma wake-field accelerator. Phys. Fluids, 30(1), 252-263.

On energy loss of narrow and dense ultrarelativistic bunch in plasma

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