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Laser accelerator developments for future high-energy accelerators
Nuclear Instruments and Methods in Physics Research A 410 (1998) 514—519
Laser accelerator developments for future high-energy accelerators Kazuhisa Nakajima High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305, Japan
Abstract Laser-driven accelerators will be realized as the next generation particle accelerators in the near future. Their development has been accelerated by success of high-energy gain electron acceleration by means of a laser wakefield excited via interaction of intense ultrashort laser pulses with underdense plasmas. On the basis of achievements of laser wakefield acceleration (LWFA) experiments, a design of LWFA for the second-generation study is considered from the point of view of application to high-energy accelerators, such as future linear colliders. ( 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Laser—plasma accelerators have been conceived to be the next-generation particle accelerators, promising ultrahigh field particle acceleration and a compact size compared to conventional accelerators [1]. Among a number of laser—plasma accelerator concepts, the laser wakefield accelerator [2] has been revived because of a simple mechanism that plasma waves excited by intense short laser pulses generate accelerating fields and the availability of high peak power, ultrashort pulse lasers and high-gradient acceleration as well. Therefore this method is suitable to build high-energy accelerators as long as it is possible to propagate intense laser pulses in underdense plasmas over a long distance. In the meantime some laser acceleration methods in vacuum, such as a vacuum beat wave accelerator [3], may be useful for the injector or the buncher producing microbunches because of no field limitation contrary to plasma-based
accelerators. The first-generation studies on laser—plasma accelerators have succeeded in demonstrating the proof-of-principle experiments to prove ultrahigh field generation in a plasma and electron acceleration by plasma waves. It is of importance for practical applications to generate a high-energy gain with a good beam quality as well as high gradient acceleration. The world-wide scientific community has started the study with the aim of realizing the second-generation high energy laser—plasma accelerators [4,5]. Their efforts are concentrated on overcoming two difficulties. The one is the development of optical guiding methods in plasmas to extend the acceleration length for laser-based schemes. The other is the development of spatial and temporal matching technologies of electron beams to be accelerated by plasma waves. The former issue is a topical subject on laser—plasma interaction physics. The latter issue includes a very low emittance, short-bunch electron source, such as a photocathode RF gun [6] and
0168-9002/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 1 5 8 - 2
K. Nakajima/Nucl. Instr. and Meth. in Phys. Res. A 410 (1998) 514—519
a plasma cathode driven by intense ultrashort laser pulses [7], and a femtosecond synchronization technology. This paper describes research for a practical accelerator design based on laser wakefield acceleration which was mainly for the purpose of achieving high-energy gain to be able to apply to future linear colliders. Characteristics of laser—plasma accelerators including laser acceleration method in vacuum are pointed out in Section 2. A design concept is presented from the point of view of high-energy gain optimization in Section 3. In conclusion, it is discussed which technology developments are necessary to achieve high-energy laser acceleration.
2. Characteristics of laser accelerators The most prominent feature of laser acceleration methods is the use of ultraintense laser fields, which is defined by strength parameter a , where a , 0 0 eA /m c2 is the normalized peak amplitude of the 0 % laser vetor potential A , given by 0 a "(2e2j2I/pm2c5)1@2+0.85]10~9j I1@2, 0 0 % 0
(1)
for the peak intensity I in units of W/cm2, the laser wavelength j "2pc/u in units of lm, and laser 0 0 frequency u . The laser transverse electric field is 0 obtained from E [TV/m]"m c2ka +3.2a /j K2.7]10~9I1@2. L % 0 0 0 (2) In principle, there is no limitation of the maximum laser field in vacuum. The vacuum laser acceleration is free from any field breaking in a classical sense. Meanwhile in the plasma-based accelerators, the accelerating field gradients are limited to the wave breaking field given by E [V/cm]"m cu /eK0.96n1@2, 0 % 1
(3)
for a nonrelativistic plasma wave [8], where u "(4pn e2/m )1@2 is the electron plasma fre1 % % quency and n is the ambient electron plasma den% sity in units of cm~3. It means that the plasma density of n "1018 cm~3 can sustain the acceler% ation gradient of 100 GeV/m. For a relativistic
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plasma wave in a 1-D cold plasma, the wave-breaking field is given by E "E J2(c !1)1@2, where WB 0 1 c "(1!v2/c2)~1@2 is the relativistic factor asso1 1 ciated with the phase velocity of the plasma wave, v "c(1!u2/u2)1@2 in an underdense plasma [9]. 1 1 0 In 2-D analysis, however, the wave breaking occurs at much lower wave amplitudes than the 1-D wave-breaking field [10]. Physically, the wave breaking appears when the plasma electron velocity exceeds the phase velocity of the plasma wave. It is implied that the wave breaking causes acceleration of plasma electrons trapped by plasma waves so that a coherent plasma oscillation breaks due to a strong wave energy dissipation. One of the unfavourable features for laser acceleration is the acceleration length being limited to an extremely short distance. In fact, since a laser pulse undergoes diffraction in vacuum or uniform plasmas, the laser spot size increases as r "r (1#z2/Z2)1@2 for 4 0 R a Gaussian laser beam, where r is the minimum 0 spot size at the focal point z"0 and Z "pr2/j is R 0 0 the vacuum Rayleigh length. Therefore, the acceleration distance is limited to the diffraction length, ¸ "pZ . In case optical guiding makes laser— $*& R plasma interaction distance longer than many Rayleigh lengths in plasmas, acceleration length will be limited to the dephasing length or the pump depletion length. As the electrons accelerated by a plasma wave with constant phase velocity of v (c increase their velocity toward the speed of 1 light c, the electrons eventually outrun a correct acceleration phase of the plasma wave into a deceleration phase. The dephasing length is defined as ¸ "j /2(1!v /c)Kj c2, assuming c A1 [2]. 1 $ 1 1 1 1 The pump depletion length in which the laser pulse loses a half of its total energy to excite plasma waves is estimated by equating the laser pulse energy to the energy left behind in the wakefield, E2¸ "(1/2)E2¸, where E is the laser field [11]. L L ; 1$ In the LWFA driven by a Gaussian laser pulse, the pump depletion length is given by ¸ K 1$ 2.65j c2a~2. 1 1 0 As far as the accelerator technology is concerned, the laser accelerator structure is built in a compact size compared to the conventional accelerator as a result of ultrahigh acceleration gradients and a short accelerating wavelength which is 1—10 lm for vacuum-based accelerators and 10—100 lm for
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plasma-based accelerators. On the contrary, the transverse and longitudinal phase space regions capable of acceleration are 103—105 times smaller than the conventional accelerators. It implies that it is necessary to make not only the transverse and longitudinal beam matching with accelerating waves but also an ultrafast timing of the order of femtosecond with &¹Hz bandwidth.
3. Design of laser electron accelerator In general, a practical electron accelerator is composed of the injector to produce an electron beam pre-acceleration, the buncher to make microbunches, and the main accelerating structure to cause high-energy acceleration of the injected electron beam. As an illustration of a design of the electron laser accelerator, the system, composed of the photocathode RF gun as the electron injector, the vacuum beat wave accelerator as the microbunching injector, and the laser wakefield accelerator as the main high—energy accelerator, is considered.
Fig. 1. Photoinjector setup.
3.2. Laser injector 3.1. Photoinjector In order to obtain a low emittance, a short bunched electron beam, the best injector available at present is the RF photoinjector consisting of the photocathode RF gun and a solenoidal magnet for transverse emittance compensation [6]. As an illustration, the photoinjector developed by KEK/ BNL/SHI collaboration are shown in Fig. 1. [12]. The photoelectron beam of &1 nC emitted by frequency quadrupled UV (263 nm) illuminating the copper cathode is accelerated to 4.5 MeV by the field of 100 MV/m in the 1.6 cell S-ban RF gun powered by the klystron RF pulse at 6 MW and the pulse width of 4 ls. The normalized emittance of &1p mm mrad can be obtained by adjusting the solenoidal magnet. The diode pumped modelocked YLF oscillator generates 79.3 MHz, 18 ps pulses synchronized with 2856 MHz RF frequency. The output electron pulse with&7 ps width will be compressed down to the order of 100 fs with a bunch compressor consisting of the chicane magnets [13] or the a-magnet [14].
The vacuum beat wave acceleration based on the nonlinear ponderomotive force serves as the injector to make microbunching of the injected electron beam to the order of laser wavelength due to acceleration mechanism similar to the inverse free electron laser [15]. Two co-propagating laser beams of different frequencies generate the axial component of the ponderomotive force to act on the injected electrons, given by F "DeD(]B/c) z z Ka a *k(m c2/c)sin(*kz!*u#*/), (4) 01 02 % where u !u "*u"c*k'0, */ is the phase 2 1 constant, c"(1!b2)~1@2 is the relativistic factor of the electron beam, and b"v/c. Assuming the acceleration length to be twice the Rayleigh length and P "P "P[¹¼]"2.15]10~2a2(r /j )2, 0 0 0 1 2 the maximum energy gain yields *¼[MeV]K22(j /j !1)1@2P1@2[TW]. 1 2
(5)
K. Nakajima/Nucl. Instr. and Meth. in Phys. Res. A 410 (1998) 514—519
The phase velocity of the accelerating field near the focus of the two laser beams is v K 1 c[1!(1!Z /Z )/(*kZ )], which is less than c R1 R2 R1 for Z 'Z and can be controlled by choosing R2 R1 two laser spot sizes. Since the accelerating gradient is inversely proportional to the electron energy, the acceleration up to &100 MeV will be effectively obtained by this method [16]. As an example, with j "2j "1 lm and P"20 TW, the energy gain 1 2 is 100 MeV. The wavelength of the beat wave is *j"j j /(j !j )"1 lm. The length of each 1 2 1 2 microbunch will be less than 1 lm during acceleration. By adjusting overlapping duration between two ultrashort pulses to the order of &10 fs, only a single accelerated bunch will be injected into the plasma wave. A direct injection of electrons into accelerating bucket of plasma waves is proposed as a plasma cathode [7]. In LWFA an injection laser pulse intersecting the wave transversely or colinearly can accelerate a fraction of plasma electrons to be trapped in the wakefield. In underdense plasmas the phase velocity is further less than c. If two beating laser pulses with shorter pulse width of the order of&10 fs than a half of the plasma wavelength copropagate in the wakefield excited by the driving laser pulse, the beat wave acceleration force pumps plasma electrons to be trapped by the acceleration phase of the wakefield. Another method is proposed where two counter-propagating injection pulses are used [17]. These schemes of injecting electrons into the plasma wave is referred to as the plasma cathode. 3.3. High-energy gain design of ¸¼FA Consider the single-stage energy gain limit of the laser wakefield accelerator due to laser diffraction, electron detuning, and pump depletion. When the laser wakefield is excited by a laser pulse with spatial and temporal Gaussian profiles of pulse duration q fs, spot size r lm and peak power P TW, the 0 maximum amplitude of the axial wakefield yields (6) eE [GeV/m]K8.6]104Pj2/(qr2c ), 0 0 0 z.!9 where c "(1#a2/2)1@2 takes account of nonlinear 0 0 relativistic effects, and a "6.8j P1@2/r . This max0 0 0 imum amplitude occurs at the plasma wavelength,
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j [lm]K0.57q in a plasma with the resonant elec1 tron density, n [cm~3]K3.5]1021/q2. The energy % gain limited by vacuum diffraction is given by *¼ epZ E which, in the limit a2@1, is written as 7 R z 0 *¼ [MeV]K850Pj /q. If diffraction is overcome 7 0 by some method of optical guiding, the energy gain limited by dephasing is given by *¼ "(2/p)eE ¸ , $ z $ which can be written as *¼ [GeV]K0.01Pq2/r2. $ 0 In case the acceleration length is limited by the pump depletion, the energy gain given by *¼ " 1 eE ¸ can be written as z 1$ (7) *¼ [GeV]"2mc2c2K3.3]10~4q2/j2 , 0 1 1$ where c "u /u . It implies that the energy gain 1 0 1 does not depend on the peak power, but on the pulse duration as long as it is limited by the pump depletion. In order to maximize the energy gain in the single-stage, the acceleration length should be set to the pump depletion length less than the dephasing length, ¸ (¸ , which requires the laser strength 1$ $ parameter a '1.6. The wakefield amplitude must 0 be suppressed to less than the wave-breaking field, E /E ( J2(c !1)1@2K(2*c )1@4, (8) z 0 1 1$ where c "*¼ /m c2. For the wakefield driven by 1$ 1$ % a Gaussian laser pulse, E /E "Jpe~1a2/2, the maxz 0 0 imum laser strength parameter must be a (1.8*c1@8. 0 1$ In order to obtain a large acceleration phase space, the longitudinal component of the wakefield should be larger than transverse component. For the Gaussian laser pulse with width p "cq/(2J2), the z peak amplitudes of both components are DeE D "Jpe~1mc2a2/p 'DeE D 0 z r .!9 z .!9 "Jpe~3@2mc2a2/r , (9) 0 0 which gives the condition, DE /E DK0.6(p /r )41. r z z 0 This condition is written as r [lm]50.11q. 0 If the pump laser wavelength is chosen to be j lm, the pulse duration requirement to obtain the 0 given energy gain *c"*¼/m c2 is % q[fs]"1.25j Dc1@2. (10) 0 The acceleration length necessary to attain the given energy gain is ¸ [cm]"0.94]10~4j *c3@2a~2. !# 0 0
(11)
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The peak power requirement to attain the given energy gain is
Table 1 Design parameters of the single-stage laser wakefield accelerator
P[TW]"2.15]10~2a2(r /j )254]10~4a2*c. 0 0 0 0 (12)
Energy gain *¼ (GeV) Laser wavelength j (lm) 0 Pulse duratuon q (fs) Laser strength a 0 Peak power P (TW) Pulse energy (J) Spot radius r (lm) 0 c "u /u 1 0 1 Plasma density n (cm~3) % Plasma wavelength j (lm) 1 Acceleration length ¸ (cm) !# Injection energy c * Bunch length p (lm) ;" Beam emittance e (lm) n N .*/ N (109) .!9
The maximum number of electrons capable of accelerating with 100% energy spread is estimated [18] to be N &1.4]1010Pj2/q'4.5]106j a2*c1@2. (13) 0 0 0 .!9 In a vacuum the electron beam envelope spreads outward due to the space-charge force and the transverse thermal motion (the emittance) without the external focusing field. In a plasma the beam electrons undergo self-focusing forces due to selfinduced wakefields inside the beam. The beam envelope equation on the r.m.s. beam radius, p , can rb be expressed as
A
B
d2p rN b2 e2 rb# % ! n "0, (14) dz2 b2c2p3 J2pp cprb rb zb where e is the normalized emittance of the beam, n p is the r.m.s bunch length, r "2.818]10~13 cm zb % is the classical electron radius and N is the total number of electrons in the bunch [19]. The equilibrium beam radius can be obtained from equating the focusing force and the thermal space-charge spreading as p "(J2pp /r N)1@2e /(b2Jc). The rb zb % n minimum number of electrons contained in the bunch is given by the condition, p (r /2. rb 0 p e2 4J2pp e2 zb n K3.56]109 zb n , N 5 (15) .*/ b4c r2 r b4c r2 * *0 % *0 where c "(1!b2)1@2 is the relativistic factor of the * * injection electron beam, and p , e , and r are meazb n 0 sured in lm. As an illustration, the design parameters of the single-stage laser wakefield accelerator with the energy gain more than 1 GeV are shown in Table 1. 3.4. Plasma waveguide It is essential for the laser wakefield accelerator to achieve a long laser—plasma interaction ranging from a few cm to 1 m in order to increase the energy gain from tens of MeV to the order of GeV. In
1 0.8 44 1.6—2.9 2—6.5 0.1—0.3 '4.8 31 1.8]1018 25 2.5—0.75 10 1 1 1.5]107 0.4—1.3
5 0.8 98 1.6—3.2 10—40 1—3.9 '11 70 3.6]1017 56 28—7 10 1 1 2.9]106 0.9—3.7
10 0.8 139 1.6—3.4 20—90 2.8—12.5 '15 99 1.8]1017 79 80—18 10 1 1 1.6]106 1.3—5.8
order to extend the acceleration length over the diffraction length, optical guiding has been proposed as a promising way of propagating a highpower laser pulse over many Rayleigh lengths in a plasma [20]. A laser beam may be guided through a plasma of which the refractive index along the optical axis is sufficiently so high as to compensate diffraction. The relativistic self-guiding in a homogeneous plasma has been predicted to occur above the critical power, given by P "17(u2/w2) GW [21,22]. For example, the criti# 0 1 cal power of a 800 nm laser pulse is P "30 TW in # a plasma n "1018 cm~3. A problem in this mecha% nism is to make a stable long distance channeling. It is known that optical guiding of a Gaussian laser pulse with a focal spot radius of r can be 0 made through the plasma density channel with a parabolic electron-density profile given by n(r)"n(0)#*nr2/r2. If the channel density depth 0 satisfies *n"1/(pr r2), where r is the classical elec%0 % tron radius, propagation of a laser pulse occurs with a constant spot size r . In the channel-guided 0 LWFA, the plasma density channel must be preformed by a prepulse before exciting wakefields. A possible method to generate refractive index structure is the use of shock waves produced by a laser-induced gas breakdown [23]. We have observed self-channeling of a 2 TW laser pulse with 90 fs duration occurring over a few
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cm even under the relativistic critical power [24]. It is implied that a mechanism of this self-channeling may be attributed to leaking mode self-effect due to field-induced saturable ionization recently presented as a new mechanism of self-channeling of an ultrashort laser pulse in underdense plasmas [25]. When a plasma filament with a sharply bounded cross-section is created via tunneling ionization owing to strong dependence of the field ionization on the field intensity, a laser pulse is guided in the form of a leaking mode with small losses over many Rayleigh lengths by a strong reflection of an electromagnetic wave on a sharp plasma boundary. Assuming a uniform electron density to be saturated at a level of n inside the plasma wavegu% ide with the radius of r , the ratio of the leakage 1 distance Z to the vacuum Rayleigh length Z is %&& R given by Z /Z "k r3/r2, where r is the spot size %&& R 11 0 0 of a laser pulse focused on a gas. For k r 51, 11 a long distance self-channeling, Z AZ ,takes %&& R place in the plasma waveguide with the radius, r Ar . For example, the self-channeling length 1 0 of a laser pulse at j "0.8 lm will be Z "74 cm 0 %&& in a plasma of n "1018 cm~3 for r "10 lm and % 0 r "100 lm. 1 4. Conclusions What the kind of developments that are necessary to accomplish high-energy laser-plasma accelerators, such as laser linear colliders are summarized. There will be need to develop the ultralow emittance beam injector with the normalized emittance less than (1p mm mrad, the microbeam buncher producing the bunch length of 1 lm, the laser wakefield accelerator with the single-stage energy gain more than 1 GeV, and the beam synchronization of inter-stage with the order of femtosecond. As an example of the low emittance, microbunched beam injector, a scheme consisting of the photocathode RF gun and the bunch compressor or the vacuum beat wave accelerator
519
are considered. The plasma-cathode will be a possible technique to make a synchronized beam injection into the plasma wave. The laser requirements of the GeV-range laser wakefield accelerator are a peak power of &100 TW, a pulse duration of &100 fs, and a pulse energy of &10 J with the repetition rate of &100 Hz. The high-energy laser-plasma accelerators require a long distance stable optical guiding of a &1 m length.
References [1] T. Tajima, J.M. Dawson, Phys. Rev. Lett. 43 (1979) 267. [2] P. Sprangle et al., Appl. Phys. Lett. 53 (1988) 2146. [3] E. Esarey, P. Sprangle, J. Krall, Phys. Rev. E 52 (1995) 5443. [4] E. Esarey et al., IEEE Trans. Plasma Sci. 24 (1996) 252. [5] W.P. Leemans et al., IEEE Trans. Plasma Sci. 24 (1996) 331. [6] X.J. Wang et al., Nucl. Intr. and Meth. A 356 (1995) 159. [7] D. Umstadter, J.K. Kim, E. Dodd, 76 (1996) 2073. [8] J.M. Dawson, Phys. Rev. 133 (1959) 383. [9] A.I. Akhiezer, R.V. Polovin, Zh. Eksp. Teor. Fiz. 30 (1956) 915. [10] S.V. Bulanov et al., Phys. Rev. Lett. 78 (1997) 4205. [11] W. Horton, T. Tajima, Phys. Rev. A 34 (1986) 4110. [12] F. Sakai et al., Proc. 11th Symp. on Accelerator Science and Technology, Harima Science Garden City, 1997, pp. 473—475. [13] B.E. Carlsten et al., Phys. Rev. E 53 (1996) R2072. [14] P. Kung et al., Phys. Rev. Lett. 73 (1994) 967. [15] P. Sprangle et al., Opt. Commun. 124 (1996) 69. [16] A. Ting et al., AIP Conf. Proc. 7th Advanced Accelerator Concepts Lake Tahoe, 1996, vol. 398, pp. 443. [17] E. Esarey et al., Phys. Rev. Lett. 79 (1997) 2682. [18] S. Wilks et al., IEEE Trans. Plasma Sci. PS-15 (1987) 210. [19] K. Nakajima et al., Nucl. Intr. and Meth. A 375 (1996) 593. [20] E. Esarey et al., Phys. Fluids B 5 (1993) 2690. [21] G.Z. Sun, E. Ott, Y.C. Lee, P. Guzdar, Phys. Fluids 30 (1987) 526. [22] P. Sprangle, C.M. Tang, E. Esarey, IEEE Trans. Plasma Sci. PS-15 (1987) 145. [23] C.G. Durfee III, H.M. Milchberg, Phys. Rev. Lett. 71 (1993) 2409. [24] M. Kando et al., AIP Conf. Proc. of the 7th Advanced Accelerator Concepts Lake Tahoe, 1996 vol. 398, pp. 390. [25] A.M. Sergeev, M. Lontano, A.V. Kim, OSA 1997 Technical Digest Series 7 (1997) 118.
VI. ACCELERATOR DESIGN
Laser accelerator developments for future high-energy accelerators Kazuhisa Nakajima High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305, Japan
Abstract Laser-driven accelerators will be realized as the next generation particle accelerators in the near future. Their development has been accelerated by success of high-energy gain electron acceleration by means of a laser wakefield excited via interaction of intense ultrashort laser pulses with underdense plasmas. On the basis of achievements of laser wakefield acceleration (LWFA) experiments, a design of LWFA for the second-generation study is considered from the point of view of application to high-energy accelerators, such as future linear colliders. ( 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Laser—plasma accelerators have been conceived to be the next-generation particle accelerators, promising ultrahigh field particle acceleration and a compact size compared to conventional accelerators [1]. Among a number of laser—plasma accelerator concepts, the laser wakefield accelerator [2] has been revived because of a simple mechanism that plasma waves excited by intense short laser pulses generate accelerating fields and the availability of high peak power, ultrashort pulse lasers and high-gradient acceleration as well. Therefore this method is suitable to build high-energy accelerators as long as it is possible to propagate intense laser pulses in underdense plasmas over a long distance. In the meantime some laser acceleration methods in vacuum, such as a vacuum beat wave accelerator [3], may be useful for the injector or the buncher producing microbunches because of no field limitation contrary to plasma-based
accelerators. The first-generation studies on laser—plasma accelerators have succeeded in demonstrating the proof-of-principle experiments to prove ultrahigh field generation in a plasma and electron acceleration by plasma waves. It is of importance for practical applications to generate a high-energy gain with a good beam quality as well as high gradient acceleration. The world-wide scientific community has started the study with the aim of realizing the second-generation high energy laser—plasma accelerators [4,5]. Their efforts are concentrated on overcoming two difficulties. The one is the development of optical guiding methods in plasmas to extend the acceleration length for laser-based schemes. The other is the development of spatial and temporal matching technologies of electron beams to be accelerated by plasma waves. The former issue is a topical subject on laser—plasma interaction physics. The latter issue includes a very low emittance, short-bunch electron source, such as a photocathode RF gun [6] and
0168-9002/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 1 5 8 - 2
K. Nakajima/Nucl. Instr. and Meth. in Phys. Res. A 410 (1998) 514—519
a plasma cathode driven by intense ultrashort laser pulses [7], and a femtosecond synchronization technology. This paper describes research for a practical accelerator design based on laser wakefield acceleration which was mainly for the purpose of achieving high-energy gain to be able to apply to future linear colliders. Characteristics of laser—plasma accelerators including laser acceleration method in vacuum are pointed out in Section 2. A design concept is presented from the point of view of high-energy gain optimization in Section 3. In conclusion, it is discussed which technology developments are necessary to achieve high-energy laser acceleration.
2. Characteristics of laser accelerators The most prominent feature of laser acceleration methods is the use of ultraintense laser fields, which is defined by strength parameter a , where a , 0 0 eA /m c2 is the normalized peak amplitude of the 0 % laser vetor potential A , given by 0 a "(2e2j2I/pm2c5)1@2+0.85]10~9j I1@2, 0 0 % 0
(1)
for the peak intensity I in units of W/cm2, the laser wavelength j "2pc/u in units of lm, and laser 0 0 frequency u . The laser transverse electric field is 0 obtained from E [TV/m]"m c2ka +3.2a /j K2.7]10~9I1@2. L % 0 0 0 (2) In principle, there is no limitation of the maximum laser field in vacuum. The vacuum laser acceleration is free from any field breaking in a classical sense. Meanwhile in the plasma-based accelerators, the accelerating field gradients are limited to the wave breaking field given by E [V/cm]"m cu /eK0.96n1@2, 0 % 1
(3)
for a nonrelativistic plasma wave [8], where u "(4pn e2/m )1@2 is the electron plasma fre1 % % quency and n is the ambient electron plasma den% sity in units of cm~3. It means that the plasma density of n "1018 cm~3 can sustain the acceler% ation gradient of 100 GeV/m. For a relativistic
515
plasma wave in a 1-D cold plasma, the wave-breaking field is given by E "E J2(c !1)1@2, where WB 0 1 c "(1!v2/c2)~1@2 is the relativistic factor asso1 1 ciated with the phase velocity of the plasma wave, v "c(1!u2/u2)1@2 in an underdense plasma [9]. 1 1 0 In 2-D analysis, however, the wave breaking occurs at much lower wave amplitudes than the 1-D wave-breaking field [10]. Physically, the wave breaking appears when the plasma electron velocity exceeds the phase velocity of the plasma wave. It is implied that the wave breaking causes acceleration of plasma electrons trapped by plasma waves so that a coherent plasma oscillation breaks due to a strong wave energy dissipation. One of the unfavourable features for laser acceleration is the acceleration length being limited to an extremely short distance. In fact, since a laser pulse undergoes diffraction in vacuum or uniform plasmas, the laser spot size increases as r "r (1#z2/Z2)1@2 for 4 0 R a Gaussian laser beam, where r is the minimum 0 spot size at the focal point z"0 and Z "pr2/j is R 0 0 the vacuum Rayleigh length. Therefore, the acceleration distance is limited to the diffraction length, ¸ "pZ . In case optical guiding makes laser— $*& R plasma interaction distance longer than many Rayleigh lengths in plasmas, acceleration length will be limited to the dephasing length or the pump depletion length. As the electrons accelerated by a plasma wave with constant phase velocity of v (c increase their velocity toward the speed of 1 light c, the electrons eventually outrun a correct acceleration phase of the plasma wave into a deceleration phase. The dephasing length is defined as ¸ "j /2(1!v /c)Kj c2, assuming c A1 [2]. 1 $ 1 1 1 1 The pump depletion length in which the laser pulse loses a half of its total energy to excite plasma waves is estimated by equating the laser pulse energy to the energy left behind in the wakefield, E2¸ "(1/2)E2¸, where E is the laser field [11]. L L ; 1$ In the LWFA driven by a Gaussian laser pulse, the pump depletion length is given by ¸ K 1$ 2.65j c2a~2. 1 1 0 As far as the accelerator technology is concerned, the laser accelerator structure is built in a compact size compared to the conventional accelerator as a result of ultrahigh acceleration gradients and a short accelerating wavelength which is 1—10 lm for vacuum-based accelerators and 10—100 lm for
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plasma-based accelerators. On the contrary, the transverse and longitudinal phase space regions capable of acceleration are 103—105 times smaller than the conventional accelerators. It implies that it is necessary to make not only the transverse and longitudinal beam matching with accelerating waves but also an ultrafast timing of the order of femtosecond with &¹Hz bandwidth.
3. Design of laser electron accelerator In general, a practical electron accelerator is composed of the injector to produce an electron beam pre-acceleration, the buncher to make microbunches, and the main accelerating structure to cause high-energy acceleration of the injected electron beam. As an illustration of a design of the electron laser accelerator, the system, composed of the photocathode RF gun as the electron injector, the vacuum beat wave accelerator as the microbunching injector, and the laser wakefield accelerator as the main high—energy accelerator, is considered.
Fig. 1. Photoinjector setup.
3.2. Laser injector 3.1. Photoinjector In order to obtain a low emittance, a short bunched electron beam, the best injector available at present is the RF photoinjector consisting of the photocathode RF gun and a solenoidal magnet for transverse emittance compensation [6]. As an illustration, the photoinjector developed by KEK/ BNL/SHI collaboration are shown in Fig. 1. [12]. The photoelectron beam of &1 nC emitted by frequency quadrupled UV (263 nm) illuminating the copper cathode is accelerated to 4.5 MeV by the field of 100 MV/m in the 1.6 cell S-ban RF gun powered by the klystron RF pulse at 6 MW and the pulse width of 4 ls. The normalized emittance of &1p mm mrad can be obtained by adjusting the solenoidal magnet. The diode pumped modelocked YLF oscillator generates 79.3 MHz, 18 ps pulses synchronized with 2856 MHz RF frequency. The output electron pulse with&7 ps width will be compressed down to the order of 100 fs with a bunch compressor consisting of the chicane magnets [13] or the a-magnet [14].
The vacuum beat wave acceleration based on the nonlinear ponderomotive force serves as the injector to make microbunching of the injected electron beam to the order of laser wavelength due to acceleration mechanism similar to the inverse free electron laser [15]. Two co-propagating laser beams of different frequencies generate the axial component of the ponderomotive force to act on the injected electrons, given by F "DeD(]B/c) z z Ka a *k(m c2/c)sin(*kz!*u#*/), (4) 01 02 % where u !u "*u"c*k'0, */ is the phase 2 1 constant, c"(1!b2)~1@2 is the relativistic factor of the electron beam, and b"v/c. Assuming the acceleration length to be twice the Rayleigh length and P "P "P[¹¼]"2.15]10~2a2(r /j )2, 0 0 0 1 2 the maximum energy gain yields *¼[MeV]K22(j /j !1)1@2P1@2[TW]. 1 2
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The phase velocity of the accelerating field near the focus of the two laser beams is v K 1 c[1!(1!Z /Z )/(*kZ )], which is less than c R1 R2 R1 for Z 'Z and can be controlled by choosing R2 R1 two laser spot sizes. Since the accelerating gradient is inversely proportional to the electron energy, the acceleration up to &100 MeV will be effectively obtained by this method [16]. As an example, with j "2j "1 lm and P"20 TW, the energy gain 1 2 is 100 MeV. The wavelength of the beat wave is *j"j j /(j !j )"1 lm. The length of each 1 2 1 2 microbunch will be less than 1 lm during acceleration. By adjusting overlapping duration between two ultrashort pulses to the order of &10 fs, only a single accelerated bunch will be injected into the plasma wave. A direct injection of electrons into accelerating bucket of plasma waves is proposed as a plasma cathode [7]. In LWFA an injection laser pulse intersecting the wave transversely or colinearly can accelerate a fraction of plasma electrons to be trapped in the wakefield. In underdense plasmas the phase velocity is further less than c. If two beating laser pulses with shorter pulse width of the order of&10 fs than a half of the plasma wavelength copropagate in the wakefield excited by the driving laser pulse, the beat wave acceleration force pumps plasma electrons to be trapped by the acceleration phase of the wakefield. Another method is proposed where two counter-propagating injection pulses are used [17]. These schemes of injecting electrons into the plasma wave is referred to as the plasma cathode. 3.3. High-energy gain design of ¸¼FA Consider the single-stage energy gain limit of the laser wakefield accelerator due to laser diffraction, electron detuning, and pump depletion. When the laser wakefield is excited by a laser pulse with spatial and temporal Gaussian profiles of pulse duration q fs, spot size r lm and peak power P TW, the 0 maximum amplitude of the axial wakefield yields (6) eE [GeV/m]K8.6]104Pj2/(qr2c ), 0 0 0 z.!9 where c "(1#a2/2)1@2 takes account of nonlinear 0 0 relativistic effects, and a "6.8j P1@2/r . This max0 0 0 imum amplitude occurs at the plasma wavelength,
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j [lm]K0.57q in a plasma with the resonant elec1 tron density, n [cm~3]K3.5]1021/q2. The energy % gain limited by vacuum diffraction is given by *¼ epZ E which, in the limit a2@1, is written as 7 R z 0 *¼ [MeV]K850Pj /q. If diffraction is overcome 7 0 by some method of optical guiding, the energy gain limited by dephasing is given by *¼ "(2/p)eE ¸ , $ z $ which can be written as *¼ [GeV]K0.01Pq2/r2. $ 0 In case the acceleration length is limited by the pump depletion, the energy gain given by *¼ " 1 eE ¸ can be written as z 1$ (7) *¼ [GeV]"2mc2c2K3.3]10~4q2/j2 , 0 1 1$ where c "u /u . It implies that the energy gain 1 0 1 does not depend on the peak power, but on the pulse duration as long as it is limited by the pump depletion. In order to maximize the energy gain in the single-stage, the acceleration length should be set to the pump depletion length less than the dephasing length, ¸ (¸ , which requires the laser strength 1$ $ parameter a '1.6. The wakefield amplitude must 0 be suppressed to less than the wave-breaking field, E /E ( J2(c !1)1@2K(2*c )1@4, (8) z 0 1 1$ where c "*¼ /m c2. For the wakefield driven by 1$ 1$ % a Gaussian laser pulse, E /E "Jpe~1a2/2, the maxz 0 0 imum laser strength parameter must be a (1.8*c1@8. 0 1$ In order to obtain a large acceleration phase space, the longitudinal component of the wakefield should be larger than transverse component. For the Gaussian laser pulse with width p "cq/(2J2), the z peak amplitudes of both components are DeE D "Jpe~1mc2a2/p 'DeE D 0 z r .!9 z .!9 "Jpe~3@2mc2a2/r , (9) 0 0 which gives the condition, DE /E DK0.6(p /r )41. r z z 0 This condition is written as r [lm]50.11q. 0 If the pump laser wavelength is chosen to be j lm, the pulse duration requirement to obtain the 0 given energy gain *c"*¼/m c2 is % q[fs]"1.25j Dc1@2. (10) 0 The acceleration length necessary to attain the given energy gain is ¸ [cm]"0.94]10~4j *c3@2a~2. !# 0 0
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The peak power requirement to attain the given energy gain is
Table 1 Design parameters of the single-stage laser wakefield accelerator
P[TW]"2.15]10~2a2(r /j )254]10~4a2*c. 0 0 0 0 (12)
Energy gain *¼ (GeV) Laser wavelength j (lm) 0 Pulse duratuon q (fs) Laser strength a 0 Peak power P (TW) Pulse energy (J) Spot radius r (lm) 0 c "u /u 1 0 1 Plasma density n (cm~3) % Plasma wavelength j (lm) 1 Acceleration length ¸ (cm) !# Injection energy c * Bunch length p (lm) ;" Beam emittance e (lm) n N .*/ N (109) .!9
The maximum number of electrons capable of accelerating with 100% energy spread is estimated [18] to be N &1.4]1010Pj2/q'4.5]106j a2*c1@2. (13) 0 0 0 .!9 In a vacuum the electron beam envelope spreads outward due to the space-charge force and the transverse thermal motion (the emittance) without the external focusing field. In a plasma the beam electrons undergo self-focusing forces due to selfinduced wakefields inside the beam. The beam envelope equation on the r.m.s. beam radius, p , can rb be expressed as
A
B
d2p rN b2 e2 rb# % ! n "0, (14) dz2 b2c2p3 J2pp cprb rb zb where e is the normalized emittance of the beam, n p is the r.m.s bunch length, r "2.818]10~13 cm zb % is the classical electron radius and N is the total number of electrons in the bunch [19]. The equilibrium beam radius can be obtained from equating the focusing force and the thermal space-charge spreading as p "(J2pp /r N)1@2e /(b2Jc). The rb zb % n minimum number of electrons contained in the bunch is given by the condition, p (r /2. rb 0 p e2 4J2pp e2 zb n K3.56]109 zb n , N 5 (15) .*/ b4c r2 r b4c r2 * *0 % *0 where c "(1!b2)1@2 is the relativistic factor of the * * injection electron beam, and p , e , and r are meazb n 0 sured in lm. As an illustration, the design parameters of the single-stage laser wakefield accelerator with the energy gain more than 1 GeV are shown in Table 1. 3.4. Plasma waveguide It is essential for the laser wakefield accelerator to achieve a long laser—plasma interaction ranging from a few cm to 1 m in order to increase the energy gain from tens of MeV to the order of GeV. In
1 0.8 44 1.6—2.9 2—6.5 0.1—0.3 '4.8 31 1.8]1018 25 2.5—0.75 10 1 1 1.5]107 0.4—1.3
5 0.8 98 1.6—3.2 10—40 1—3.9 '11 70 3.6]1017 56 28—7 10 1 1 2.9]106 0.9—3.7
10 0.8 139 1.6—3.4 20—90 2.8—12.5 '15 99 1.8]1017 79 80—18 10 1 1 1.6]106 1.3—5.8
order to extend the acceleration length over the diffraction length, optical guiding has been proposed as a promising way of propagating a highpower laser pulse over many Rayleigh lengths in a plasma [20]. A laser beam may be guided through a plasma of which the refractive index along the optical axis is sufficiently so high as to compensate diffraction. The relativistic self-guiding in a homogeneous plasma has been predicted to occur above the critical power, given by P "17(u2/w2) GW [21,22]. For example, the criti# 0 1 cal power of a 800 nm laser pulse is P "30 TW in # a plasma n "1018 cm~3. A problem in this mecha% nism is to make a stable long distance channeling. It is known that optical guiding of a Gaussian laser pulse with a focal spot radius of r can be 0 made through the plasma density channel with a parabolic electron-density profile given by n(r)"n(0)#*nr2/r2. If the channel density depth 0 satisfies *n"1/(pr r2), where r is the classical elec%0 % tron radius, propagation of a laser pulse occurs with a constant spot size r . In the channel-guided 0 LWFA, the plasma density channel must be preformed by a prepulse before exciting wakefields. A possible method to generate refractive index structure is the use of shock waves produced by a laser-induced gas breakdown [23]. We have observed self-channeling of a 2 TW laser pulse with 90 fs duration occurring over a few
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cm even under the relativistic critical power [24]. It is implied that a mechanism of this self-channeling may be attributed to leaking mode self-effect due to field-induced saturable ionization recently presented as a new mechanism of self-channeling of an ultrashort laser pulse in underdense plasmas [25]. When a plasma filament with a sharply bounded cross-section is created via tunneling ionization owing to strong dependence of the field ionization on the field intensity, a laser pulse is guided in the form of a leaking mode with small losses over many Rayleigh lengths by a strong reflection of an electromagnetic wave on a sharp plasma boundary. Assuming a uniform electron density to be saturated at a level of n inside the plasma wavegu% ide with the radius of r , the ratio of the leakage 1 distance Z to the vacuum Rayleigh length Z is %&& R given by Z /Z "k r3/r2, where r is the spot size %&& R 11 0 0 of a laser pulse focused on a gas. For k r 51, 11 a long distance self-channeling, Z AZ ,takes %&& R place in the plasma waveguide with the radius, r Ar . For example, the self-channeling length 1 0 of a laser pulse at j "0.8 lm will be Z "74 cm 0 %&& in a plasma of n "1018 cm~3 for r "10 lm and % 0 r "100 lm. 1 4. Conclusions What the kind of developments that are necessary to accomplish high-energy laser-plasma accelerators, such as laser linear colliders are summarized. There will be need to develop the ultralow emittance beam injector with the normalized emittance less than (1p mm mrad, the microbeam buncher producing the bunch length of 1 lm, the laser wakefield accelerator with the single-stage energy gain more than 1 GeV, and the beam synchronization of inter-stage with the order of femtosecond. As an example of the low emittance, microbunched beam injector, a scheme consisting of the photocathode RF gun and the bunch compressor or the vacuum beat wave accelerator
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are considered. The plasma-cathode will be a possible technique to make a synchronized beam injection into the plasma wave. The laser requirements of the GeV-range laser wakefield accelerator are a peak power of &100 TW, a pulse duration of &100 fs, and a pulse energy of &10 J with the repetition rate of &100 Hz. The high-energy laser-plasma accelerators require a long distance stable optical guiding of a &1 m length.
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