no. I? C. Joshi KOTiGE., r.. _, fr PORTIONS Of THIS REPORT flre ILLEBBBIE. Electrical Engineering Department University of California, Los Angeles

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1 no. I? THE SURFATRON LASER-PLASMA ACCELERATOR: PROSPECTS AND LIMITATIONS CO:;p-P n p o. Q n 5 A o 7 C. Joshi KOTiGE., r.. _, fr PORTIONS Of THIS REPORT flre ILLEBBBIE. Electrical Engineering Department University of California, Los Angeles (t has ba^a reproduced from the best California, U.S.A available copy to permit the broadest possible availability. The surfatron laser-plesma accelerator ' is an extension of :he 3 4 plasma beat wave accelerator scheme '. The basic idea behind this scheme, as in all the laser accelerator schemes, is to utilize the 9 10 very intense electric fields, V/cm, associated with focussed laser beams to accelerate particles. Unfortunately, apart from having very large electric fields, focussed lase' -cams have few other characteristics that are desirable for accelerating particles. First, being an electromagnetic (e.m.) wave, a laser beam hase_ perpendicular to _k; whereas for particle acceleration, we require a component of E^ in the direction we wish to accelerate particles. Second, the field propagates at the speed of light c in a vacuum so that particles cannot be made to stay in phase with the wave: the group velocity of the e.m. wave must be made less than but very close to c to ensure that the particles interact with the field for a very long time. Third, even if both problems mentioned above are overcome, there is the problem of laser wavelength. Today's intense lasers have wavelengths in the 10 \im pm range: too short to accelerate particles to high energies. Lastly, in vacuum the laser beams cannot be made to stay focussed greater than a Rayleigh length because of diffraction. If the laser beam is propagated in a high density, but underdense plasma, the situation can be quite favorable for particle acceleration By a high density underdense plasma we mean 10 < n e < 10 cm such that i>y- = 4ire n /m < ID /4. Here w p e o o is the laser frequency and <JO_. is P the plasma frequency. Under certain conditions an intense transverse electric field of the laser can be very efficiently transformed into a longitudinal electric field of a plasma density wave. In the laser accelerator scheme known ss the plasma beat wave accelerator, such a & plasma wave can be driven by beating two colinear laser beams u Q, k o ind ui, 1 ki J in a plasma such that ui = u - J), and k = k - k,. epw o 1 epw o 1 Physically,, a plasma wave is nothing more th in regions of space-charge,1 _,_ // 3c i - v / j / - V_ / i~- "3 Uiinr.iijiiiitJN ui- IHIi D0CIMNI

2 01, 0 L PLAMA WAVI (a) (b) Fig. 1. In the "Plasma Beat Wave Accelerator", particles (e, H ) are accelerated by an electric field generated by a plasma wave traveling close to the speed of light. The plasma wave in turn is excited by two colinear laser pulses beating in a plasma at the plasma frequency (a). In the "SURFATRON" accelerator scheme a perpendicular B field is applied to the plasma wave such that an electron trapped by a potential trough moving at v. sees and electric field, B/c, which accelerates it across the wave f ro.i; (b).

3 separation which arise because the spatial gradient of the electric field squared of the beat wave envelope (Fig. la), which is in the direction of propagation, exerts a periodic force on the plasma electrons at wavenumber k. The plasma wave is thus an electrosta- epw tic wave with E parallel to k, w «The dispersion relation for an e.m. wave inside the plasma is w ui + c k and that for a plasma wave is u - OJ + 3k v with op o epw p epw e v 2 m kt e /m. In the very underdense plasma u, u, >>to and assuming that 3k 2 v 2 << u) n it is easy to show that to a high degree of epw e V. approximation the phase velocity of the plasma wave is equal to the 2 2 L group velocity of the beat wave, (v.) * ( V B^BW = c^ ~ u /%). Since all three waves are locked into synchronism and the group 2 velocity of the plasma wave 3v /v. << c, the plasma wave can grow to a large amplitude. As the plasma wave is an electrostatic wave, we can use Poisson's equation to estimate the maximum potential that we might expect if we assume that the perturbed electron density is equal to the initial density. Since the plasma wave is propagating at relativistic speed, Lorentz transformation gives the energy in the laboratory frame as 2(u o /u ) me or (n c /n e ) MeV where n c is the critical density of the plasma where w Q = nip. To the first order, this is the maximum amount of energy that a particle can gain from the wave. It can be seen that for a plasma wave excited by a CO2 laser in a (n c /n e ) = 10 or n e = 10^3 Cm~3 plasma the maximum eneigy gain can be a TeV. So what is the problem? First, the above example assumes that we can obtain and utilize the critical or "wavebreaking" fields to accelerate particles. Computer simulations show that as the electric field of the plasma wave approaches it's maximum value, (E e pw^crit " 4Tren e /k e p W, it begins to exibit significant harmonic content. It also begins to pick-up and accelerate a significant number of plasma electrons. This is shown in Fig. 2. Both of these effects are undesirable if we wish to accelerate a bunch of injected particles. we might expect to be able to use -0.l(E epw ) crit. In reality If we assume that we require a minimum field of 1 GeV/m, this puts a severe restriction on the laser frequency to plasma frequency ratio and hence on the maximum energy we can expect to gain at a given plasma density for a certain laser wavelength, The second problem is that of the acceleration

4 nrv th ee Fig. 2. ( 0 1 Results of a self consistent 1 2/2-D particle simulation show that when the electric field of a longitudinal electron plasma wave becomes very large it quickly develops harmonics (a). A sizeable number of plasma electrons are trapped and accelerated to high energies by the plasma wave electric field (b ). (b)

5 length being much greater than the distance over laser beams can be focussed down. For tu 0,uij_ >> (u p, the wavelength of the plasma wave is approximately a factor (w o /u) ) 2 larger than the laser wavelength in the wave frame. It is important to remember that particles see an accelerating field and therefore gain energy over only half a wavelength of the plasma wave. This acceleration distance is 2(to 0 /u>p)2 c/wp in the laboratory frame. Thus ever, for a modest energy gain of 1 TeV in the above example, even after assuming E crit, the acceleration length is 3.35 km. This length is much greater than the Rayleigh length over which the lasers can be focussed at the required intensity. In any case the electric field is -300 MeV/m, a factor 10 improvement over present day linacs^ but still below the 1 GeV/m limit. The parameter regime for the plasma beat wave accelerator scheme is depicted in Fig. 3. It can be seen that 1 GeV/m field requirement restricts the C02 laser for particle energies below 60 MeV. A 1 urn laser is promising for accelerating particles up to 16 GeV, whereas, a 0.26 yin laser (the forth harmonic of the 1.06-um Nd: glass laser ) may be useful for accelerating particles to 100 GeV. From Fig. 3b it is evident that the distance to reach a certain energy is proportional to the laser wavelength but so is the Rayleigh length - ±2F 6/6-.,. Here F is the f number of the focussing optics, X is the laser wavelength, 6 is the beam divergence containing 85% of the laser energy and 6_ T is the diffraction limited divergence of the beam. Even after manipulating the plasma and laser spatial profiles the prospects of keeping a laser beam focussed over a distance greater than a few meters are very remote. In this scheme therefore one has to be prepared for multiple staging. In a single stage the best we can hope to achieve is to accelerate electrons to a few GeV using a 0.26 ym laser. But the penalty one hat to pay in terms of laser energy needed is very high (Table 1) because the oscillatory velocity of the electron squared is proportional to IX. In table 1 we list the parameters for laser accelerators based on three laser wavelengths assuming that the objective is to obtain 10 electrons with an energy spread Ay/y = 10 It can be seen that the laser energy requirements stretch the current laser technology by an order of mangitude. The problem with the plasma beat wave accelerator now becomes immediately obvious. At low values of u) O /u) p the electric fields can

6 Figure 3 (a). The maximum energy gained by a particle in the "Plasma Beat Wave Accelerator" as a function of the laser frequency to plasma frequency ratio assuming that the maximum electric field of the plasma wave is 10% of the critical or wavebreaking field. The minimum electric field requirement of 1 GeV.'m rules out the CO2 laser for high energy accelerator requirements. The shaded areas show the usefulness of 1.06 pm and 0.26 pm lasers.

7 S z I 10" ENERGY (GeV) Figure 3 (b). Acceleration length vs. energy for three laser wavelengths in the "Plasma Beat Wave Accelerator".

8 be extremely high but the field-particle interaction, /E-dL, limits the maximum energy to low values (Fig. 3a). For example, at 0.26 um and (ti> /(Dp)» 10 or n e = 1.6 * 10*" cm" 3 we might conservatively expect to gain 100 GeV/m. In spite of this the maximum energy gain 1B only about 10 MeV because the particles eventually outrun the wave. In the 1 2 "SURFATRON" laser accelerator scheme ' a way is found to phase-lock the particles in the electric field so that particles do not out-run the wave. The basic idea behind this scheme is very simple (Fig. 1b). A high phase velocity plasma wave is excited in the presence of a perpendicular magnetic field by beating two laser beams in a plasma such that their frequency difference is now the upper hybrid frequency. Thus 2 2 h Ad) = u - ID, = w, = (w + u) ) where m r is the nonrelativistic eleco 1 h P c «- tron cyclotron frequency eb/mc. Consider a longitudinal plane wave electric field of the plasma wave propagatit.g at v_^ and the applied magnetic field of the form _E = E o sin (kx - a>t) a^ (1) B = B a z (2) A trapped particle in such a wave now experiences a -v *B^ force which accelerates it parallel to the wavefront of the accelerating wave, while at the same time keeping it in phase with it. The condition for a particle to be trapped in the potential well of the wave is simply that there has to be net force due to the electric field on the particle in the direction of the wave. Thus in the wave frame the x componet of the force is F = e (E sin kx. + y v 1 E/c) (3) x o 1 pn y 2 2 -h where y, = (1 - v /c ), v, = ID /k, x. = x - v, t, and ph ph ' t>h epw epw' 1 ph ' v' is the y velocity in the wave frame. If y B < E, a trapped particle can never become out of phase with the electric field and thus can be accelerated indefinitely. By solving the equation'; of motion of a trapped particle of charge e and rest mass m in the laboratory frame ee.d (yv ) = o (sin kx - ut) + u> v (A) X c y dt m

9 TABLE 1: Parameter Regime for the Beat Wave Accelerator Scheme Laser Wavelength (urn) Plasma Density (cm-3) Laser Pulsewidth KWHM (ns) Bunch Length (cm) Laser Energy ~(kj) Maximum Energy (GeV) 10.6 io io 19 - io *10~ xl xl * *10 A Energy width of accelerated particles Number of particles accelerated Particles per bunch Initial energy of injected particles Bunch cross section Laser cross section Bunch density Laser intensity Maximum plasma wave electric field ioio 2* cm x10-2 cm ne v o /c = (E ).. epw crit.

10 jl (y V ) o -0) V (5) dt y c x where Y B (1 ~ v^/c^ x - v*"/c^) y, Katsouleas K t l and d D have shown that the particle gains energy at a rate [h ^ = 30 GeV/cm (6) Ay S =0.1GeV/c m U ^ - ( n ^ (7) che direction of the wave, theoretically to arbitrary energies. Here B,,_ is the magnetic field in kilogauss, n., is the plasma density., lo in the units of 10 cm ^ and X is the laser wavelength in microns. The quantity in the parenthesis B 1.,,/n. ),X 1 is the fraction of the critical or wavebreaking electric field which as before we take to be equal to 0.1. Several examples are presented in Table 2 for accelerating particles to 100 GeV. We will discuss the example based on using a 1 urn laser to show how in practice we can hope to achieve such a device. It would at first seem an impossible task to excite a plasna that is 3.16 meters long and 0.33 meters wide on laser energy considerations alone (even if such a plasma could be created). therefore has to be tackeled the following way. The problem First, we have to keep the laser beams as small in diameter as possible to achieve the necessary intensities (v Q /c ~ 0(1)) commensurate with achieving a reasonable Rayleigh ltngth. Second, we realize that the electron beam makes an angle with respect to the plasma wave in the laboratory frame. In this case, this angle is 6. Now if instead of using colinear geometry for optical mixing we utilize some finite angle optical mixing then (because k device width (Fig. 4)- << k, k,) we can considerably reduce the To calculate the geometry for optical mixing we have to realize only two constraints. First, the only region of the plasma we need to excite is the region traversed by the electron beam. Second, in this region k makes an angle of 6 with respect to the high fre- 2 quency beam k. In this example (w /to ) = 100 and assuming that o op

11 TABLE 2: Sample Parameters To Reach 100 GeV Based on the SURFATRON Concept Laser Wavelength (urn) _3 Plasma Density (cm ) io B KG /n 16\ Magnetic Field (kg) Device width W (m) *10-2 Device length L (m) Angle Between Particles and kave (degrees) 18.3

12 k v «u, ID = to and energy conservation gives u> «10to epw e p epw P P and ui - 9 u. The dispersion relations give k ~»^99 w /c and k. ~ *^80 ID /c. Using the cosine rule we obtain k " 1.02 u) p /c and the angle between k 0 and k-^ as 0.8. Over the device length of 3.16 meters the two laser beams diverge by only 4.4 cm. the width of the low frequency beam. This we take to be The width of the high frequency beam can be much smaller than this as long as (v 0 /c) Uo (v o /c) ui > 0.1. Thus for a 100 GeV accelerator we are taking about a plasma this is around 10 cm in diameter and a few meters long. By reducing the width (r) of the high frequency beam we also help reduce the beam emittance 2 2 which is roughly on the order 4(r /R )TT steradian fo: the highest energy particles, R = W^ + L. The energy spread A-y/y of the particles should also be excellent since the energy gain of the particles is much larger than y,, their initial energy. Now what about the laser power requirement? Suppose that by using F/10 optics v e can focus a 1 pi laser beam to a 1 mm spot size over the required length (~3 meters). laser pulse we require 78 kj laser energy. To obtain (v o /c) - 1 in a 10 ps (FWHM) The low frequency beam is 100 times less intense but it is spread over a 44 mm area, so we need ~34 kj in this beam. These are only very rough energy estimates. To keep the energy requirement at a reasonable level one has to go to very short (10 ps) laser pulses. Unfortunately, laser systems developed for laser-fusioi; research have typical pulsewidths in the ns range. Thus considerable research and development effort is necessary in the area of high brightness, short-pulse lasers and focussing, transporting and manipulating such beams, specifically related to high energy accelerator needs. An advantage of using a short pulse is that the plasma should be stable against many undesirable instabilities that 7 8 occur on ion acoustic timescale, e.g.- ponderomotive, thermal, and 9 "0 resonant-self-focussing and Brillouin scattering" 1 ". Another advantage when a short laser pulse is used is that the main body of the plasma should be relatively unperturbed. Simulations show that propagation of an inr.ense, modest length (100 ps) laser pulse through an underdense plasma can heat the plasma up to MeV temperatures due to non-linear wave-wave and wave-particles interactions. Fully self-consistent, relativistic simulations, using a 1 2/2-D 2 electromagnetic particle code, have been carried out which confirm

13 ACCELERATED " Fig. h. In the "SURFATRON" concept the injected particles are accelerated at an angle w.r.t. the plasma wave. Optical mixing at a small angle must be employed such that the high frequency laser beam and the election beam are colinear. In the wave frame the electrons move parallel to the plasma wave fronts.

14 that the particles do gain energy at a rate given by equations (6) and (7). We are at present carrying out two-dim^nsiond simulations in order to investigate transverse stability, self-consistent B field generation and sidescattering problems. The mcst severe of these problems appears to be the self-consistent or self-induced B field. the plasma wave amplitude become so large that the wave can trap significant numbers of plasma electrons then this can give rise to an azimuthal, teegagsuss B field which will completely dominate the applied d.c. B field. Experimentally of course the major problem is going to be one of producing a plasma in the presence of a perpendicular magnetic field. Before we can contemplate building a 100 GeV laser electron accelerator based on the surfatron concept considerable effort is necessary on the development of appropriate plasma sources and very short pulse systems. ACKNOWLEDGEMENTS I would like to thank Prof. J. M. Dawson and F. F. Chen; and W. Mori and T. Katsouleas for many stimulating discussions on the topic of laser acceleration of particles. This work was supported by the Lawrence Livermore Laboratory, University Research Program and NSF grant ECS REFERENCES 1. T. Katsouleas and J. M. Dawson, Paper presented at the 1983 Particle Acceleration Conference, Santa Fe, N.M., to be published i^ Aug. 1983, IEEE J. Nucl, Soc. 2. W. Mori, C. Jcshi, and J. M. Dawson, ibid. 3. C. Joshi, T. Tajima, and J. M. Dawscts, Phys. Rev. Lett. 7, 1285 ( T. Tajima and J. M. Dawson, Phys. Rev. Lett. _43, 267 (1979). 5. Laser Acceleration of Particles, AIP Conference Proceedings No. 91, Ed. by P. Channel (1982). 6. W, Mori - Private communication. 7. C. E. Max, Phys. Fluids _19.» 1733 (197o). 8. V. K. Tripathi and L. A. Pitale, J. Appl. Phys. 48_, 3288 (1977). 9. C. Joshi, C. E. Clayton, and ~. F. Chen, Phys. Rev. Lett 48^ 874 (1982). 10. C. E. Clayton, C. Joshi, A. Yasuda, and F. F. Chen, Phys,, Fluids. 11. J. M. Kindel, Private communication. If

15 DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.