t. tajima- laser accelerator by plasma waves for ultra-high energies

8/3/2019 T. Tajima- Laser Accelerator by Plasma Waves for Ultra-High Energies

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LASER ACCELERATOR BY PLASMA WAVES FOR ULTRA-HIGH ENERGIEST. Tajima

Department of Physics and Ins t i tu te for Fusion StudiesUniversity of Texas, Austin, Texas 78712

ABSTRACTParalle l intense laser beams uJo,ko and bJ1,k1 shone on a plasma withfrequency separation equal to the plasma frequency Wp is capable ofcreating a coherent large elect ros ta t ic f ie ld and accelerating part ic lesto high energies in large flux. The photon beat exci tes through the

forward Raman scattering large amplitude plasmons whose phase velocityis equal to (wo-W1)/(ko-k1), close to c in an underdense plasma. Theplasmon t raps electrons with elect ros ta t ic f ie ld EL = yY2 mCI.J.I/e, of theorder of a few GeV/cm for plasma density 1018cm-3 . Because of the phasevelocity of the f ie ld close to c th is f ield carries trapped electrons tohigh energies: W = 2mc 2 (ul(:)!Wp)2. The (multiple) forward Ramanins tab i l i ty saturates only when a sizable electron population i s trappedand most of the electromagnetic energy is cascaded down to the frequencyclose to the cut-off ( ~ ) . Preaccelerated part icles coherent with theplasmon f ie lds can also be accelerated. In order to acceleratepart ic les to ultra-high energies, a series of focused l aser l igh t beamsare phase-matched with accelerating par t ic les . There seems a favorableregime sat isfying both conditions for the self- t rapping ins tabi l i ty an dfor avoiding the f i lamentation ins tab i l i ty . The former ins tab i l i tyconfines the l igh t beam from otherwise spreading over Rayleigh's length,and th e l a t t e r ins tab i l i ty would make the beam cross-section nonuniform.Because of the ultra-high energy of par t ic les , the longitudinaldefocussing between the plasma wave and part icles happens. To overcometh is di f f icu l ty , the r e la t iv i s t i c forward Bril louin scattering processis proposed and discussed.

I . INTRODUCTIONI t is becoming clearer that extrapolat ion of the present highenergy accelerator i s not enough to meet today's challenge 1 ofultra-high energies. While a possibi l i ty exists that unt i l the par t ic le

energy reaches the grand unified mass of 1013CeV there i s a vast deser tdomain of no new physics (1 02 - 1014 GeV) , there exists a possibi l i ty ofhaving a jungle of various part ic les (such as Higgs part ic les) an d newphysics in the much lower energy range. Supposing that we try toachieve energies of 100-1000TeV within a reasonable physical size( i . e . for example, within a s ta te) , we real ize that the necessary

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electr ic field to accelerate part icles is so large that under thepresent technology no metallic surface would withstand such a hugeelectr ic f ield. For example, an electr ic f ield of O.lGeV/cm ( r f or dc)corresponds to 1eVI'X and thus severely modifies the electronwavefunction within the atom, leading to sparking, breakdown etc.Besides this dif f icul ty , the surface heating contr ibutes to anothercause for breakdown with high f ield. Another point immediately becomesevident. Since the part icles accelerated are in such a high energy thatthey t ravel with a velocity very close to th e speed of l ight c, thephase velocity of th e accelerating wave (or structure) must be againvery close to the speed of l igh t c.Besides these fundamental physics questions, we have to consideravailable technologies to achieve high energies. When we look overvarious electromagnetic power generation techniques, the most intensef ie lds are delivered by the laser technology at th e present time,although this does not preclude other technologies in the future. Forexample, millimeter wave lengths may be provided by the free electronlaser or gyrotron-type laser . The tendency toward a much shorterwavelength of accelerating st ructure than the present microwavetechnology i s not only due to th e available high field values in theshorter electromagnetic wave generation techniques, but also due to theconsiderat ion of luminosity of the part icle beams. From Heisenberg'suncertain principle , AE6t < 0, the cross-section 0 of part ic le reactionat 6E + is proportional to E c ~ , where is the energy in theD Ecmcenter-of-the-mass. Because of th is , the cross-section becomes smallerand smaller in the higher energy. This means that the luminosity has toincrease as E ~ in order to keep the number of events in unit time to beconstant . 2 This puts a very stringent condition to the acceleratorconcept in high energies. As we discuss in more detai l la ter , theluminosity is inversely proport ional to the length of the beam bunch.Therefore, in order to relax the severe condit ion on the luminosity, i tis advisable to go for a short beam bunch, presumably obtainable againby a short wavelength wave. This again points toward the lasertechnology. In the following we review the part ic le acceleration in theelectromagnetic waves.A part ic le in an electromagnetic f ield of small amplitude isaccelerated perpendicular to the direct ion of this electromagneti wavek and executes osci l lat ions along the electr ic f ie ld direction E. Nonet acceleration, therefore, i s achieved. This holds true even I f theelectromagnetic field amplitude is large and there is the magneticacceleration (see Fig. 1). The magnetic acceleration with the electr icacceleration makes th e part ic le execute an Figure 8 orbi t with no netacceleration. A spatia l ly or temporally localized packet ofelectromagnetic waves cannot cancel a l l the osci l latory motion and thusleaves a net acceleration. The amount of acceleration, however, remainssmall.

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E

~ - - + - - - - r - - - - - - i - k

e e-B small largeamplitude amplitude\ IVno accelerationFig. 1: Part icle motion in a propagating plane electromagnetic wave.The wavenumber k is perpendicular to electr ic field E and magnetic fieldB. No net acceleration along the ~ - d i r e c t i o n is achieved.

In order to gain net acceleration by electromagnetic waves, therehave been many attempts, which may be categorized into two: the vir tualphoton approach and the real photon approach. Most of the conventionalaccelerators including (proposed) collective accelerators are in thef i r s t category. Consider Fig. 2(a). In order to obtain an electr icfield component paral lel to the wave propagation kn , some (metal)ref lector is placed. The wave has En component, but unfortunately thephase velocity of the wave is larger than th e speed of l ightw/kn > w/k = c. Thus no coupling. One may confine the electromagneticfield by two conductors in a wave-guide [Fig. 2(b)] instead of asemi-infinite case in Fig. 2(a). The character is t ics of wave phasevelocity of a wave-guide is well-known [See Fig .2 (c) ] . The phasevelocity is always larger than c: v h = c/ vll-(w /w)2 where wc is theccut-off frequency for the wave-guige. An accelerator such as SLAC'salleviates this problem by implementing a periodic structure in thewave-guide ( i r ises) . A periodically rippled wave-gUide introduces theso-called Brillouin effect into the phase velocity characterist ics. TheBrillouin diagram for the frequency vs. wavenumber in th e ripplewave-guide is depicted in Fig. 2(d). Here sections of phase velocityless than c are realized. There are other ways to make phase velocityless than c, such as the dielectr ic coating. All these techniques maybe collectively called a technique for slow-wave st ructure. Part ic lescan now surf on such field crests and obtain net acceleration. Theintensity of th e fields is l imited by materials considerations such aselectr ic breakdowns. Almost invariably, the localized high electr icfield results a t and near the slow-wave structure, making the breakdowneasier there. I t may be possible to create a slow wave structure in th eform of a plasma wave guide (or plasma "optical f iber") . A narrow wave

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(0 )

- ! t i l

(b l

o

Fig. 2: Virtual photons and photons in a plasma. (a) A planeelectromagnetic wave reflects on the metallic surface. There is a fieldcomponent paral lel to k II ,E sin0. In the metal th e EM fieldexponentially decays. (b ) If we put two metallic plates together, weget a waveguide. Again a field component paral lel to exists . (c)llThe dispersion relat ion of the EM wave in the waveguide. k is th eparal lel wavenumber. (d) The dispersion relation of the EM waves in theuniform and ripple waveguides. (e) The dispersion relation of the EMwave in a plasma.

Fig. 3. The surrounding plasma wave-guide structure (plasma f iberoptics) plays a role of a slow-wave structure. If one matches vph - c,th e wave f ield component paral lel to the laser propagation may beemployed to accelerate part ic les.

guide with rippled surface accentuated either by a plasma or by plasmaand magnetic fields (see Fig. 3).

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When we uti l ize the real photon in a plasma, there is no physicallimitation due to th e materials considerations. The characterist ics ofthe phase velocity of th e real photon in a plasma is similar to th e onein a wave-guide. See Fig. 2(e). The phase velocity of theelectromagnetic wave in a plasma is vph = c/ V l-(wp/w )2, always largerthan c in an underdense plasma, indicating again th e diff icul ty todirectly couple the wave to accelerate electrons. Here wp is the plasmafrequency. I t is possible, however, to nonlinearly couple to th e plasmai f the amplitude of the electromagnetic waves are suff iciently large.In Refs. 3-5 we discussed a laser electron accelerator scheme byexciting a large amplitude Langmuir wave created ei ther by a strongphoton wavepacket with a very short spatial pulse length as a photonwake or by two beating photons. In Ref. 5 we concluded that using twophoton beams is much more effective in acceleration than using a veryshort photon pulse. We primarily focus on the two beam case althoughthe physics involved in the case of short wavepacket is quite commonwith the present case.The basic mechanism of part icle acceleration is as follows. Thetwo injected laser beam induce plasmons (or a Langmuir wave) through theforward Raman scattering process. This may be regarded as opticalmixing. The resultant large amplitude plasma wave with phase velocityvery close to the speed of l ight grows and is sustained by th e laserl ights . It grows unti l the amplitude becomes re la t iv is t ic , i . e . thequivering velocity of th e electrostat ic f ield becomes c, so that thewave begins trapping electrons in the t a i l of distribution andaccelerating them. The trapped electrons can be accelerated to highenergies since th e electrostat ic wave is propagating with a phasevelocity very close to c. The trapped electrons form a bunch in density.Since the energy dependence of the accelerated electron velocity isnonlinear, th e detrapping time of electrons is very long. This is avery significant advantage of the l inear acceleration of this type incomparison with a circular machine. In a circular machine such as abuncher th e energy dependence of the electron angular velocity isapproximately l inear,6 thus the detrapping time is much shorter.In terms of the available technology, the pulsed intense electronor io n beam technology] delivers an electr ic f ield of ~ 1 0 ] V/cm and apower density of 10 13 W/cm2 On th e other hand, the laser technology iscapable of delivering a power density of 10 18 W/cm2 for a glass laserand 1016 W/cm2 for a C02 laser . As we shall see la ter , th eelectrostat ic f ield can reach 109 V/cm for the glass laser case and asimilar but somewhat smaller value for th e C02 laser case.In the present paper, we present a possible scheme to accelerateions (as well as electrons) to high energies, exploiting the extremelyhigh value of the longitudinal electr ic f ield associated with the beatplasmon. Although i t is perhaps inevitable to have multi-stageaccelerating modules to reach a very high energy, i t is crucial that th elaser l ight is focused along the propagation direction as long aspossible. I t is also important that th e laser l ight does not break upin pieces in th e radial direction, tha t may lead to incoherentaccelerating and side scat ter ing.

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In Sec. I I we present the basic concept of the laser beataccelerator. In Sec. I I I we examine the role of forward Ramanscattering and discuss simulation and experimental results . Sec. IVdiscusses th e ul t rare la t iv is t ic waves. The io n acceleration scheme ispresented and self-focusing and fi1amentation ins tabi l i t ies arediscussed in order to determine a possible operation regime for io nacceleration in Sec. V.Also, in th e following two papers Dr. C. Joshi and Dr. D. Sullivandiscuss this problem through experiments and simulation, respectively.

I I . BEAT WAVE ACCELERATORTwo large amplitude travelling electromagnetic waves, (wO,kO) and(wl,kI), injected in an underdense plasma induce a plasma wave(wp,kO-kI) through th e beating of two electromagnetic waves if thefrequency separation of two electromagnetic waves is equal to th e plasmafrequency:

oscil la t ions. 3 ,4

Wo - wI = wp (1)kO kl = kp (2)where kp is th e wavenumberelectromagnetic waves gives of ther ise to plasma wave.a nonlinear The beat of twoponderomotive forcewhich sets off th e plasma This process may also beregarded as a nonlinear optical mixing6 as well as a forward Ramanscattering. 4 ,9,lO It may be possible to achieve the objective throughthe forward Raman ins tabi l i ty , 9 i . e . the second electromagnetic wave(wl,kl) grows from a thermal noise. Rosenbluth et ale have discussedplasma heating through beat of electromagnetic wave. I l , l2 I t isimportant that the plasma is sufficiently underdense so that wo is muchlarger than wp (see Fig. 4). This will ensure that th e phase velocityof the plasma wave vp is very close to the speed of l ight:

W

Wo t ~ ---------~ , ' ~ " : ",WI - - - - - - - - - - - ... :

IIIIIII: kp : - , - - _ - - - , 1 . . . . : . 4 _ " ' ; ~ : . . . , j 1 " -a k kxI ko

Fig. 4: Dispersion relation for the EM waves. Two laser beams beat.

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--(3)

In the l imit of oop/ooo 1, Eq. (3 ) yields

velocity of the plasma is much less than The resultant plasma

p = = = c( 1 o o ~ ) 1/ 2006 (4)th e relation we used in Refs. 3 and 4.Suppose that 000 is not very much larger than oo p then the phasewave c.wave can quickly trap electrons and saturates. In th e course ofinteraction the nonlinear effects may change the phase velocity of theplasma wave. In the case of oop/ooo not small, the interaction of l ightwaves and plasma is strong and th e l ight waves suffer strong feed-backfrom th e plasma. (The l ight waves may be called "plast ic" or "soft" inthis case.) In our case of oop/ooo 1, the interaction of l ight wavesand plasma is less intense and the characteristics of l ight waves arelargely preserved. As we shal l see l a ter , the ratio of the energydensity of the electrostat ic plasma wave to the electromagnetic wave is(oo p/ooO)2, Le . the l ight wave dominated. Therefore, the plasma waveremains reinforced or "regulated" by th e beating two laser beams

koclJ l - w ~ / o o 6 '" kOc= k 1cl J I - W ~ / o o r '" k 1c

(The l ight waves may be called "hard" in our case.) Since the phasevelocity vp is very close to c for oop/ooo 1, the electron trappingand, therefore, saturation, occur only when th e electrostat ic wave growsup to an amplitude so large that i t becomes re la t iv is t ic . The otherimportant consequence of oop/ooo 1 is that part icles wil l be in phasewi th the plasma wave for a long time and achieve a large amount ofacceleration, because, again, th e phase velocity vp c and theparticles would no t exceed vp easi ly .Let us consider the energy gain of an electron trapped in theelectrostat ic wave with phase velocity vp = ooe/kp. We go to th e restframe of th e photon-induced longitudinal wave (plasma wave). Since thewave has the approximate phase velocity Eq. (4), 6 = vp/c and Y = ooo/oo pNote that this frame is also th e rest frame for tlie photons in theplasma: in this frame th e photons have no momentum and th e photonwavenumber is zero. The Lorentz transformations of th e momentumfour-vectors for the photons and the plasmons (the plasma wave) are

y i6Y) (kO ) (5)( -i6Y Y iOOO/c

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( 6)

where the right-hand side refers to the rest frame quantities with(kwaverespect to th e plasma wave = k /y), kO is the photon wavenumberin th e laboratory frame and the well-kqown dispersion relation for th ephoton in a plasma 00 0 = ( o o ~ + kbc2)1/2 was used. Equation (5) i sreminiscent 13 of the relation between meson and the massless (vacuum)photon: Eq. (5) indicates that th e photon in th e plasma (dressed photon)

has th e rest mass oop/c, because the electromagnetic interaction shieldedby plasma electrons can reach only the collisionless skin depth c/oo p inthe plasma. This is just as the nuclear force reaches the inverse ofthe meson mass and Yukawa predicted th e meson energy as00 = Vc2/a2+k2c2,where a is the nuclear radius. 13 Compare:

Meson ++ Vacuum Photon(interaction length a) (interaction length ~

00 00 = kc

Photon in Plasmas ++ Vacuum Photon(interaction length c/oo p)

00 00 2 + k 2c 2p 00 = kcAt the same time, the Lorentz transformation gives the longitudinalelectr ic field associated with the plasmon as invariant ( E ~ a v e = EL) .The electrostat ic wave amplitude can be evaluated by a fewdifferent (independent) ways yielding the same resul t . An argumentresorting to th e wave breaking l imit was employed in Ref. 4. Here le tus discuss in terms of the available electron density. When mos t ofelectrons are bunched as a result of the beat electromagnetic waves, wemay estimate the electrostat ic f ield by assuming most of the electronsgive r ise to this f ield:

'iJ E = -4'JTen or (7)where n is the electron density. The maximum electrostat ic field is ,therefore,

We may derive th e maximum electrostat ic f ield Eq. (8 ) by another method.

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As we shal l discuss in Sec. I I I , the electrostat ic wave saturates onlywnen the trapping width of the wave becomes wide enough to begintrapping,the ta i l of electrons. This condition may be written aseEL 1/2vp - ve v t r = (row vp) (9)p

where ve is the electron thermal velocity. I f we neglect ve incomparison with vp and approximate vp by c, Eq. (9) now yields Eq. (8).The condition (8) is valid even i f the trapped electrons are highlyaccelerated as long as the bulk of electrons remain nonrelat ivist ic.However, when the bulk of electrons obtain kinetic energy, say th eperpendicular energy, then th e formula needs corrections. 14 According toRef. 14 , the obtainable electrostat ic field increased to a valueErel y ~ / 2 mcwp/e, where Yp is the effective re la t iv is t ic factor. Whenstrong heating of electrons occurs, mismatching of conditions (1) and(2 ) arises and we need more detailed study on th e attainable electr icfield in this case.The electr ic potential due to the plasma wave evaluated in th elaboratory frame is

A IlI.\lpC ce4l = ef ELdx = e -- (-) = (10)wp eGoing to the wave frame, we obtain the potential in th e wave frame

(11)This energy in th e wave frame corresponds to the laboratory energy bythe Lorentz transformation

(:BY ~ i B Y ) (:::2) ( : ~ ~ 1 ~ 2 ) ' (12)where the right-hand side refers to the laboratory frame quant i t ies .Thus we obtain the maximum energy electrons can achieve by th e plasmawave trapping as

wmax = ymaXmc2 = 2 Y ~ C 2 = 2(w O) 2mc2 ( 13)wpThe time to reach energies of Eq. (13) may be given by

wo 22(-) /w p (14)wpand the length of acceleration to reach the Eq. (13) energy as

(15)

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For th e ~ l a s s laser of 1 wavelength shone on a plasma of density1018(1019)cm- i t would require under the present mechanism a power of10 18 (1018 ) W/cm2 to accelerate electrons to energies wmax of 109(108)eVover the distance of 1(0.03)cm with the longitudinal field EL of109(3 x 109)V/cm. For th e C02 laser of 1 wavelength these numbersscale accordingly.To demonstrate the present mechanism for e l e ~ t r o n acceleration wehave carried out computer simulations employing 1 - - D (one spat ial andthree velocity and f ield dimensions) fully self-c6nsistent re la t iv is t icelectromagnetic code. 15 Two parallel electromagnetic waves (wOkO) and(wlkl) are imposed on an in i t ia l ly uniform thermal electron plasma.The direction of the photon propagation as well as the allowed spatialvariation is taken as the x-direction. The system length is Lx = 10246th e speed of l ight c = lOwn6 th e photon wavenumber kO = 2w x 68/10246the number of electrons 10240 and th e particle size 16 with a Gaussianshape and the ions are fixed and uniform where 6 is the grid spacing.The thermal velocity ve = lwn6. The photon frequencies are taken aswQ = 4.29w p and wI = 3.t9wp while the amplitudes arevi = eEi/mwi = c (i = Q or 1).Figure 5 shows the phase space of electrons accelerated by the beatplasma wave kp wR/c. High energy electrons are seen in every ridge ofeach length of the resonantly excited electron plasma wave. Thehorizontally stretched arms in Fig. 5(a) are separated by lengthA = 2w/kp The maximum electron energy was 85mc2 in th is case higherthan the value given by Eq. (13). One reason for this discrepancy maybe that we now have two intense electromagnetic waves so that magneticacceleration associated with Vo x Bl and VI x BO also begins to p layarole . The distribution function f(PII) or f(Yn) is shown in Fig. 5(b)>>exhibiting strong main body heating as well as a high energy t a i l .Figure 5 shows the electrostat ic f ield profile in space at t = 30wp1 (an early t ime). The field amplitude already reachedEx EL = mwpc/e. One can also see i ts coherent field pattern. Theobserved wavelength is 2w/kp = 2w/(kO-kl).

The second case is that of injection of a wavepacket of a singlephoton (wO,ko )>> whose packet length Lt = A/2 = wc/w ' as discussed inRefs. 3 and 4. Using the same code with parameters pLx = 5126, c = 5ve ,the photon wavenumber kO = 2w/156, th e number of electrons 5120eEO/mwO = eBO/mwO = c, Lt = wc/w p, PO = eEO/wO and wo = ( w + kQc 2) 1/2,we s tar t th e system with electromagnetic pulse in th e plasma within i t i a l conditions

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Fig. 5: Photon beat acceleration by two laser beams (wO,kO) and (wl,k1) .(a) The electron phase space (x,ps) at t = 2 4 ( } , J ~ ! The maximum Yn forelectrons is 85 in this case. (b) The logarithm of the electrondistribution function at t = 1 3 5 w ~ ! (c) The electron dis t r ibut ionfunction at t = 1 3 5 w ~ !

Ey = EO s in kO(x-xO)Bz BO sin kO(x-xO)Py = Pthermal + Po cos kO(x-xO)

for the period of x = [506, 81.46] and xO = 506. With th e assignment,the wavepacket 1ffs a spectrum in k with a peak around k = kO andw = ( w + kfic2) , and propagates in the forward x-directionapproximately retaining the original polarization.Figure 7 shows an early stage of the system development. Thephase-space plot [Py vs. x in Fig. 7(b)] indicates a strong modulationin th e Py distribution within the photon packet location. The kinkstructure extends beyond the packet ending. Figure 7(a) shows Px vs. x.The intense longitudinal momentum osci l lat ions are clearly appearing,beginning at the photon packet and extending to i t s in i t ia l start ingpoint. This is th e wake plasma wave set off by the photon packet seenin Fig. 7(b). As in Fig. 6, the long stretching arm-like phase-spacepattern [Fig. 7(a)] appears with i ts momentum keeping increasing. Thewake plasmon structure is also apparent in the longitudinal fields[F ig .7 (c ) ] . In th is case th e electrostat ic f ield reaches values aroundEL 0.6 mowp/e. In Fig. 8 we plot th e maximum electron energy observedin our simulations as a function of (wO/w p)2. The prediction Eq. (13)is written in a solid l ine to compare with the simulation values.Agreements are reasonable.


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U03EO"-xW V ~ h II

~ ~ 1o 512 1024

x -Fig. 6: The electrostat ic f ie ld Ex vs. x a t t = 30wp The amplitude ofth e field EL already reached EL - IDWpc/e.

Fig. 7: Wake-plasmon excitation by a short laser wavepacket and trappingof electrons. The head of the photon packet has proceeded forward tox = 310 at t = 24wp1 oooloop = 4.3. (a) The longitudinal momentumj{px = PII) vs. position of electrons. (b) p - x phase space. (c) Thelongitudinal f ie ld EL = Ell vs. position. (d) Particle acceleration intime (solid line) and electr ic field intensity in time (dashed l ine) .I I I . FORWARD RAMAN INSTABILITY AND EXPERIMENTS

We study th e spectral distribution of photons in time. From th esimulation ru n with two photons (ooo,ko) and (oo ltk1) we observe in Fig. 9a clear cut energy cascade via multiple Raman forward scat ter ing. Theoriginal waves with equal amplitude Ei = mc 00 i l e ( i = 0 or 1) cascadetoward smaller k as seen in Fig. 9(a) and 9(b). A small amount ofenergy is up-converted. The spectrum i s sharply peaked at a part iculardiscrete wavenumber kn = ko - nkp where n i s an integer. The spectralintensity S{k,oo) for the electrostat ic component shows overwhelming peakat k = kp and no significant energy in any frequency at the backscatterwavenumbers. This strongly suggests that a l l possible backscattering


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70

60

50N

40>-30

20

10

00 10 20 30 40(",I",plZ

Fig. 8: Maximum electron energy vs. (wO/w p)2 in the short wavepacketcase. The dots are from simulations and the solid l ine is fromEq. (13) .

processes are suppressed or saturated at a very low level in our presentproblem. The electrostat ic spectral density S(k,w) shows peaks atk = kp as well as k = nkp All these observations confirm that thedownward photon cascade is due to the multiple forward Raman scattering.A similar downward photon cascade is observed in the case with aphoton packet of one plasma wave (wo,ko ) . Figure 10 shows thewavenumber spectrum of the electromagnetic pulse at successive times.The original smooth-shaped spectrum evolves into a multipeak structurewith a roughly equal, but sl ightly increasing, separation in wavenumber

(0)

S(k)

o 102

Fig. 9: The electromagnetic energy distribution (spectrum) as a functionof mode numbers. Two laser beam pumps kO and k1 are i n d i c ~ t e d byarrows. (a) t = 142.5wp} (b ) t = 24Owp!


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as k approaches kp ' This again indicates that the photon (wo,ko) decaysinto (00 1,k1)' (ooZ,kZ)'" by successive or multiple forward Ramanins tabi l i ty .The reason why the backscattering is suppressed but the forwardscattering is very prominent is the following: When the backscatteringplasma wave is excited, enhanced Landau damping or electron trapping bythis plasma wave saturates i t at a low level, thus l imiting thebackscattering to a small value. The wavenumber kb of the plasma waveproduced by th e backscattering process in the case of oop/ooo 1 iskb = 2ko The phase velocity of the backscatter ing plasma wave is

oo p c oo pv = - -= - - (16)P 2k 2 00o 0Trapping of electrons by this wave begins happening when thetrapping width Vtr becomes wide enough to reach the t a i l of the thermalelectron dist r ibut ion. An approximate trapping width may be written as

b 1/2b l/ Z eEL (17)v = ( !L) = ( - v)t r m mw ppwhere the superscripts b refer to the backscattering electrostat icwave16 The formula is nonrelat ivis t ic , but is suff icient for thepresent purpose. Also recal l the discussion given af ter Eq. (9). Thecondition that a large number of electrons are trapped is given17 by

1/2v - = 2(Te /m) (18)p vt r < 2veThe maximum electrostat icby sett ing ve = 0:

eE bL =1II.II p

wave amplitude is obtained for a cold plasma

(19)

-3I -2I -II .tI

Fig. 10: Electromagnetic spectral intensity in wavenumber. The arrowwith 0 indicates the rough position of the original laser beamwavenumber peak; n indicates k ' = k nk 'p

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In both cases of Fig. 9 and Fig. 10 we did not detect a large peak ofelectrostat ic spectrum S(k,w) at k = 2ko Thus the forward Ramaninstabil i ty or process appears to be the las t parametric process tosaturate in a hot underdense plasma. In fact i t can be argued that i twill saturate only when the original electromagnetic wave has completelycascaded by multiple forward Raman process to waves near w wp Inthis case most of the electromagnetic energy may be extracted from thelaser l ights to electrostat ic wave energy and eventually to kineticenergy. The idealized efficiency, therefore, may be given byn = 1 - (wp/wO)2.An experimental observation of th e forward Raman ins tabi l i ty andassociated electron acceleration and heating has recently been done9 inconjunction with th e present concept and physical discussion. A C02laser is shone on an underdense plasma producing electrons of energy upto 1.4 MeV. The laser power density is such that eEo/mwoc 0.3 and thefrequencies are wp/wo 0.46. The plasma was created by the laser l ightshone on 1 3 0 ~ thick carbon foi l producing th e in i t i a l plasma temperatureof ~ 2 0 k e V . In the experiment the laser emits only one beam so that thebeat has to grow from the noise. I t i s , therefore, in general, possibleto have other competing processes such as side sca t te r , backscatter, twoplasmon decay simultaneously taking place. In spi te of these competingprocesses, lower quivering velocity of the laser , and lower wO/w p, th eexperiment shows 9 high energy electrons in the forward direct ion.Simulations are carried out in order to see the wave spectrum andto compare th e dist r ibut ion function of electrons with the experiment.Using a similar set-up as before, we set the plasma parameters same asthe experiment: 9 Te 20keV, and uniform plasma, wp/wo 0.46, thepropagating electromagnetic wave has eEO/mwoc 0.3. The electronparal le l distribution function f(p") along with the electrostat ic wavespectra are displayed in Fig. 11. The temperature and the maximumelectron energy observed in the simulation dist r ibut ions are similar toth e experimentally measured values. For example, simulations show theforward electron maximum energy of 1.3MeV and temperature of 100keV incomparison with the experimental values of 1.4MeV and 90 100keV,respectively. In the backward direction, simulations show the electronmaximum energy of 0.9MeV and temperature of 60keV compared to theexperimental values of 0.8MeV and 40 50keV. The electrostat ic wavespectrum, Fig. II(b), shows that th e backscattering mode kb (which growsini t ial ly) is swamped by other modes with a smaller wavenumber, the mostintense of which is the plasma wave associated with forward scatteringkp In addition, there are some wavenumbers which are less than kp Thus the heated electron distributions obtained by the experiment9 andthe simulations agree well with most of the electron heating due to theforward Raman ins tabi l i ty , but is no t so much due to the backwardprocess. In th e present case th e phase velocity of th e backscatteringplasma wave w Ikb 1.6ve Thus this wave is heavily Landau damped tobegin with an I as i t grows in amplitude, more and more electrons will betrapped by i t and and the damping will grow. In view of Eq. (18),


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Simulations have been performed to study this parametricdependence. 18 Again the same type of set-ups and code as in the shortwavepacket case is used. Parameters we use are Lx = 1 0 2 4 ~ , ve = 1 w p e ~ ' the in i t i a l photon wavenumber kO = 2'11" / l 0 ~ , c = 5 w p e ~ ' 1 0 2 4 ~ electronsand ions each, and eE0/mwOc i s varied from 1 to 20 . The packet lengthis chosen to be Lt 'I1"c/wp The obtained energy scaling is displayed inFig. 12, showing ymax ex: \)2 = (eEo/mwoc)2. The coefficient of theproportionality i s also reasonably f i t with Eq. (21). More detai ls areshown in Refs. 18 and 19. In a similar study Sullivan and Godfrey alsoinvestigated 14 th e accelerated electron energy dependence on the laserfield strength.

V. HIGH ENERGY PARTICLE ACCELERATORWe have introduced the concept of laser accelerator using twoparal le l intense laser beams and a resultant beat plasma wave. Wediscussed various character is t ics of this scheme and demonstrated i t viacomputer simulations and to a certain degree via experiments. Althoughthese investigations are preliminary, they are certainly encouraging.In the present paper we propose and demonstrate the way to accelerate

particles to high energies without losing regular structure of thef ie ld. This is important since in a very high energy accelerator i t isl ikely that any material cannot withstand the very strong acceleratingf ield , the system has to regulate i t se l f ; in th e present case the plasmaand the laser beams have to regulate themselves. The regularity of theaccelerating fields is kept by the reinforcing two intense laser beams.Since the plasma is underdense, the laser beams are able to impresstheir periodic structure on the plasma and the p l a ~ m a oscil la t ion.Their f r e q l 1 e D S i e ~ and wavenumbers are wo = kacl .,J1-wp /wd .... kac andw1 = k1 c / J l - w ~ / w .... kl c and ka, kl reinforcing the plasma frequencyand wavenumber. The plasma particle-wave interaction cannot destroy thestructure easily either unt i l the forward Raman instabi l i ty saturates ,2,10;:

Imax10'

102 2,102 3,10' 4,102 (A..)2Z; mw cFig. 12: The maximum electron energy vs. laser electr ic f ie ld intensi ty(eEO/mwOc)2 in the case of ul t rare la t iv is t ic laser beam. The straight

l ine is y max = ~ (w 0) 2 \) 22 v wg p

DC"


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which does not happen at a low level . This is because th e excitedplasma wave has a phase velocity very close to c. Once th e particle isaccelerated and becomes re la t iv is t ic , i t will stay with the acceleratingfield for a long time in a coherent fashion. Since the density ofelectrons is high, the created f ield can be quite respectable of theorder of l09V/ cm This scheme may be useful no t only for acceleratingelectrons to high energies in large flux, but also for bunchingelectrons with a regular structure.A large regular electr ic field of 1Q9V/cm range propagating withphase velocity very close to c is certainly attractive for acceleratingions, as well . One possible way to do this is to preaccelerate ions toa moderately to highly re la t iv is t ic energies and then to inject ontothis f ield. A schematic diagram is depicted in Fig. 13 . In order toaccelerate to multi-TeV energies i t is l ikely to need many modulars ofsuch an accelerator, since the laser focus and other problems may arise .In this case i t may be a technical challenge to make ion clumps in phasewith the positive electric field in a l l those modulars. Even if weassume that we can accelerate ions almost a l l the time, with the fieldof l09V/ cm , i t would take l05cm to reach an energy of lOOTeV.

We address here a couple of crucial questions associated withacceleration of part icles to high energies with the present concept.The f i rs t problem is th e longitudinal deterioration of acceleration: th edephasing between the accelerating electrostat ic plasma wave and th epart icles being accelerated. The second is the transversedeterioration: the defocusing of the laser l ight due to the laser opticsas well as due to the plasma nonlinear effects on the laser l ight .

Pph-- t - - - - - - - - - - - i ; , ; i FIons-!- -8- 4- - e - - ~ - - e - - ~ - -8--1-e-

:\ } .:\ !, \;1 Tail electrons.... ; .... : .." i

plasmaFig. 13: Schematic phase space diagram with bulk electrons, acceleratedt a i l electrons, and injected preaccelerated ions at the right phase.Pph is the phase momentum of the plasma wave excited by the beating twolaser beams. Pb is the injected ion beam momentum. The ion clumpseparation is c/w p

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After ions reach or exceed the phase velocity of the plasma wave,they overtake the phase of th e wave and thus get ou t of phase. Whenions are in very high energies, they are quickly overtaking the wave.The ion velocity relat ive to th e phase velocity of the wave is

_ _ 1 w 2l1v:; c vph ="2 (w) c. (22)oThe dephasing can take place over a quarter wavelength. Therefore, thedephasing time is obtained as

'td A/4'" - tw = WO2 1

'IT (-1 - ,w J w (23)p pwhere is the wavelength of th is plasma wave. This time (23) i s verymuch similar to Eq. (14). On the other hand, time for ions toaccelerate by energy M c 2 ~ y is given as

(24)

where M is the ion mass. Since the electron perpendicular temperaturecannot be extremely high, yl/2 is of order unity (perhaps 2 3). Theenergy gain within the dephasing time, therefore, ist, y < IT ( ~ (. WO )2y 1/ 2- M ' (0 1 (25)P

If the ions are protons and (wO/wp)2 103, t,y is of th e order of unity.This indicates that i f one wants to accelerate up to 100TeV, one wouldneed 105 times of re-phasing operations between ions and the wave. Ifone wants to accelerate electrons beyond th e energy (13), one also needsre-phasing. Again, i f one wants to accelerate electrons up to 100TeV,one would need 105 times of re-phasing.

One way to avoid these many rephasing processes is to increase theelectron mass from m to y1m by increasing the perpendicular electrontemperature or adding a helical perturbation. In th is case thefrequency separation should be reduced by a factor 1/y1/2 from wp totl1p/yi!2. The possible electrostat ic f ield achieved by t-be plasma waveis now

(26)and the laboratory potential drop is

2em y 1 mc (27)Consequently, th e maximum energy corresponding to Eq. (13) is

(28 )


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Extending th i s idea fur ther t we arr ive a t the concept of ther e l a t iv i s t i c forward Bri l lou in sca t t e r ing process ins tead of the forwardRaman process. In th i s we propose to use two l a se r l igh ts with (wOtkO)and (h)ltk1) obeying wo - wI = (llpi an d kO - k1 = Wpi/Ct where (,lpi i s theion plasma frequency: (,'pi = (4nne 2/M) l /Z . The in tense two l aser beamswil l create e l ec t ros t a t i c osci l l a t ions a t the ion plasma frequencYtwhich is forced osci l l a t ions (quasi-mode) in the l imi t of the soundveloci ty put to be c with the wavelength C/u.lpi. The most s igni f i cantadvantage of th i s scheme i s tha t the frequency separat ion of two l aser sand the resonant ion plasma frequency are so much smaller than the l aserfrequency tha t the phase veloci ty i s much c lose r to c . This can be doneby the plasmon resonance (Raman) process by reducing (llp/ (llOt but th isonly leads to reduced accelera t ing f ie ld s t rength Eq. ( I8) . From as imi la r argument above [Eq. (26)] t it might be tha t EL "" M !.l'pi cleoHowever t th i s turns out not to be the case from our s imulat ion s tudy.We f ind tha t

(29)an d

(30)The ampl i f ica t ion of energy by the Lorentz t ransformation y ' 2 i s nowgiven by y' = WO/Wpi t thus yield ing

max (WO)2 rM)1/2 2W < 2 --. - mc. (31)- 'wpi ' mIn pract ice the energy of Eq. (31) i s hard to obtain . There are

severa l complications: modes d i f f e r en t from kO - k1 are exci ted and thee l ec t ros t a t i c f ie ld prof i le i s not as coherent as in the Raman process .In addi t ion , s ince the frequency difference and wavenumber dif ferenceare much smaller t the previous ly unimportant backscat ter ing process nowcomes into play a f te r some i n i t i a l times when the desired forwardBr i l louin sca t t e r ing takes p lace. Some of our preliminary s tudies bycomputer s imulat ion are shown in Fig. 14. Parameters are now:v e = 1 W p ~ t kn = 2n x 7 7 / 1 0 2 4 ~ t k1 = 2n x 7 3 / 1 0 2 4 ~ t M/m = 25 tv i = eEi/m(,)i = /5 c fo r i = 0 and I t c = 9 U1p l'"t Lx = 1024t t and numbersof elect rons an d ions are 10240 each, and !.l10 - (,)1 i s kept to be w i .Certainly much higher energies are achieved. Clear ly much research hasto be done here in the fu ture .

The second problem of defocusing comes in because we need3- 5 ,14 avery high l aser f ie ld E such tha t eE/mwoc "" 1. Since the present lyavai lable l aser cannot de l ive r such a high intensi ty of l i gh t withoutfocusing it by opt ics . the l aser l igh t has a focal length associatedwith i t , the Rayleigh length . This length zR :: 1'W0/-"'.p, with AR an d wothe l aser wavelength and the wais t would have to be of the l aseraccelera t ion length:


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" ..:....":': ::'(.:

, ...:. . '.... . ..' . ". : . ~ : ~ .... . . :........:...;

r.: ," .t to .':/':' . : ' . ~ : ::. ..,'.. .. ': ..

oIo

Fig. 14 : Rela t iv i s t i c Bri l lou in backscat ter ing process . Elec t ron p - xxphase space i s shown. Two beat ing l aser s w ' wI with wO-w1 = w i a rein jec ted in to a plasma from l e f t . Extremelyohigli energy e l e c f r o n ~ a reseen in four bunches whose separat ion i s c / w . much wider than theRaman bunch c / w . p1p

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(26)Outside of this regime, the l aser power would be to o low. Fortunately,the self- t rapping of the l igh t beam can take place beyond a certainthreshold la se r power20 ; when the lase r power is a t the threshold, thebeam propagates without defocusing, overcoming the natural tendency ofspreading over the Rayleigh length. When the laser power exceeds, thela se r beam propagates with i t s envelope resembling a sausage, s t i l lsat isfactory to ou r purpose. Felber20 gives the self- t rapping conditionas

(32)

where f,.D i s th e Debye length, T i s the electron temperature, and a isthe l igh t beam cross-section radius. The condition (32) i s not terr iblydi f f icu l t to fu l f i l l . The filamentation ins tab i l i ty i s anotherins tabi l i ty that develops perpendicular to th e beam propagationdirect ion. This ins tab i l i ty makes th e orginally uniform beam prof i leinto spiky nonuniform one, leading to possible di f f icu l ty to accelerateuniformly. The mechanism and condition to avoid th is ins tab i l i ty wasstudied. 21 We may be able to have the above mentioned self- trapping andto avoid the f i lamentation ins tabi l i ty i f we choose a certa in domain inthe lower frame of Fig. 7 of Ref. 21 , L e . the r ight upper port ion ofthe graph above the broken l ine but below th e hatched area. Thiscorresponds to th e regime with enough beam intensity and enough beamradius.The issue of luminosity becomes unavoidable as the energy goes up.As th e uncertain principle dic ta tes , the in teract ion time f:..t reducesf:..t ~ / ~ E . The cross-section a t asymptotic energy i s

(33)

where ~ the center-of-mass energy. Because the number of eventscm 'is nco times luminosity, the luminosity L has to be given asE2L 0:: cm (34)

On the other hand, since the luminosity i s inversely proport ional to thebeam bunch length, the shor t wavelength may be favorable on th is issue.There are other two or three dimensional phenomena we have notaddressed. Among these, the self-magnetic f ie ld effects generated bythe accelerated electrons and thei r currents, need to be invest igated.The s ide-scat ter ing Raman ins tabi l i ty tends to increase emittance of theelectron beam. The emittance may also be determined by the plasma

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electron temperature, density, laser power, etc. These questions as wellas some unencountered problems have to be tackled and answered. Some ofthe problems will be addressed by th e following speakers, Drs. C. Joshiand D. Sullivan.

ACKNOWLEDGMENTS

The author would l ike to thank Drs. J . M. Dawson, C. Joshi,D. Sullivan, A. Sessler, W. Willis, F. Felber, R. O. Hunter,M. N. Rosenbluth, A. Salam, and N. Bloembergen for stimulatingdiscussions, encouragements and interes ts . The work was supported byth e U. S. Department of Energy, Contract No. DE-FGOS-80-ETS3088 andNational Science Foundation Grants ATM81-10S39. Much of this work wascarried ou t in collaboration with Dr. J . M. Dawson. Dr. F. Felberpointed out the possibi l i ty of self-trapping of laser beams.


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REFERENCES1. A. Salam, in this Conference.2. W. Willis, in this Conference.3. T. Tajima and J . M. Dawson, IEEE Trans. Nucl. Science NS-26,4188(1979).4. T. Tajima and J . M. Dawson, Phys. Rev. Lett. 43, 267(1979).5. T. Tajima and J . M. Dawson, IEEE Trans. Nucl. Science NS-28,3416(1981).6. T. Tajima, J . Vomvoridis, F. Felber, B. Spivey, R. O. Hunter,to be published.7. B. Bernstein and I . Smith, IEEE Trans. Nucl. Science 1,294(1973).8. N. Kroll, A. Ron, and N. Rostoker, Phys. Rev. Lett. 11,83(1964).9. C. Joshi, T. Tajima, J . M. Dawson, H. A. Baldis, and N. A.Ebrahim, Phys. Rev. Lett. 47, 1285(1981).10. Y. R. Shen and N. B l o e m b e r g e ~ Phys. Rev. 137, A1787(1965).11. M. N. Rosenbluth and C. S. Liu, Phys. Rev. Lett. ~ 701(1972).12. B. I . Cohen, A. N. Kaufman, and K. M. Watson, Phys. Rev.Lett. 29, 581(1972).13. H. Yukawa, Proc. Phys. Math. Soc. Jpn. 17, 48(1935).14. D. J. Sullivan and B. B. Godfrey, IEEE Trans-.- Nucl. ScienceNS-28, (1981).15. A. T. Lin, J . M. Dawson, and H. Okuda, Phys. Fluids! l ,

1995(1974).16. Sometimes Vtr is defined as 12 times the value of Eq. (17).17. J. M. Dawson and R. Shanny, Phys. Fluids 11, 1506(1968).18. M. Ashour-Abdalla, J . N. Leboeuf, T. T a j i m ~ J . M. Dawson, andC. F. Kennel, Phys. Rev. A23, 1906(1981).19. J . N. Leboeuf, M. Ashour-Abdalla, T. Tajima, C. F. Kennel, F.V. Coroniti, and J . M. Dawson, Phys. Rev. A25, 1023(1982).20. F. S. Felber, Phys. Fluids 23, 1410(1980). -21. F. S. Felber and D. P. Chernin, J . Appl. Phys. ~ 7052(1981).


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DISCUSSIONWinterberg. I should like to mention.another related idea, which is alogical extension and perhaps no t so crazy as i t seems. This is to usesolid densities, where wavelengths are much shorter, and an X-ray laser .Such a laser has been established with the help of a fission bomb; i fwe can someday make micro-explosions leading to high densities there isl i t t le doubt that we can again make a soft X-ray lasers. This wil l allowhigher density of operations and even higher electr ic f ie lds.Nation. Can you comment on the ini t ia l trapping. Do you need highdensity high energy electron beams to s tar t with?Tajima. For the particles to be trapped in the electrostatic wave theymust have relat ivist ic energies.Nation. How many particles can be trapped and accelerated?Tajima. In my simulation 1% of the particles are accelerated. Thiswould be about 1016 per cm3Participant. Have you considered side scatter?Tajima. We have thought about i t , but cannot do conclusivecalculations with our 2D code. My hunch is that i t is no t terr iblyimportant.Bingham. You will get longitudinal de tuning of the wave from the trappedpart ic les.Tajima. I t does no t happen as strongly as you might think. The reasonis that a l l the trapped particles are propagating at th e speed of l ight ,and only 1% of th e particles are trapped.Reiser. What about gas scattering?Tajima. This might be important at densit ies of order 1021 cm-3


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t. tajima- laser accelerator by plasma waves for ultra-high energies

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