- Email: [email protected]
“Unlimited” particle acceleration by an intense plasma wave with a varying phase velocity
Volume 150, number 5,6,7
PHYSICS LETTERSA
12 November 1990
"Unlimited" particle acceleration by an intense plasma wave with a varying phase velocity B. M e e r s o n
Centerfor Plasma Physics, Racah Institute of Physics, Hebrew Universityof Jerusalem, Jerusalem, 91904Israel Received 2 July 1990; accepted for publication 6 September 1990 Communicatedby R.C. Davidson
An "unlimited" electron and ion accelerationis shown to occur in the field of an intense "tapered" plasma wave. This acceleration schemedoes not require high injection energiesand pre-bunchingof the acceleratedparticlesand could operate at moderate plasma wave amplitudes. A possibleway of the phase velocity"tapering" is suggested.
Mechanisms of high-gradient electron acceleration by relativistic plasma waves received much attention after the work by Tajima and Dawson [ 1 ], where the beat-wave acceleration concept was proposed (for a review of this and related schemes see ref. [2 ] ). Since the phase velocity of a laser-driven plasma wave can be made to be close to the speed of light c, the electrons, resonantly interacting with the wave, can be accelerated significantly [ 1 ]. The first simultaneous observations of a beat-wave excited plasma wave and accelerated electrons have been recently reported [ 3 ]. For the resonance interaction between a charged particle and a relativistic plasma wave to be efficient, either a sufficiently large injection energy of the particle, or a sufficiently large plasma wave amplitude is necessary. The required injection energies become beyond practical application for ions, therefore for the ion acceleration the beat-wave acceleration scheme has to be properly modified. For electrons, it would be desirable to overcome the waveparticle dephasing [ 1,2 ] and thus ensure an "unlimited" acceleration in a one-dimensional acceleration scheme, because the creation of a coherent two-dimensional plasma wave structure in the Surfatron concept [4,5 ] presents very difficult problems. Such a one-dimensional "unlimited" acceleration of both electrons and ions can be achieved ifone employs a plasma wave with variable phase velocity. The idea is to start with a wave, which is able to trap the 290
accelerated particles (it must be a relatively slow wave in the case of ions), and then, by gradually increasing the phase velocity, to keep the particles trapped (phase-locked) and accelerated. In the following we shall find conditions under which (a) such a scheme could work for a large fraction of pre-injeered electrons or ions, (b) the initial particle bunching is unnecessary and (c) the precise form of the phase velocity spatial (or temporal) profde is unessential. Finally, we shall briefly discuss a possible way of the phase velocity "tapering". The essence of the proposed acceleration mechanism can be demonstrated by a simplified model of a sinusoidal plasma wave with constant frequency OJo, propagating in the x-direction in a weakly inhomogeneous plasma (generalization to the slowly varying frequency case is straightforward). In the geometric optics approximation, the plasma wave accelerating field has the form X
E(x, t ) = E o ( x ) s i n t ~ k ( x ' ) d x ' - c O o t + ~ o ) .
(1)
The relativistic equation of motion of a charged particle in the wave field can then be written as
dp _ ZeE(x,t) (l+m~c2/p2)~/2 , dx c
(2)
where mo is the mass of the particle, Ze is its charge
0375-9601/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-HoUand)
Volume 150,number 5,6,7
PHYSICSLETTERSA
and p is its momentum. Introduce a phase variable
~= i k(x') dx'-coot+~o,
(3)
corresponding to the reference frame traveling with the wave. Then eq. (2) can be rewritten as a set of two first-order equations, d~ = N ( z ) - (1 + p - 2 ) 1/2 dz
(4)
dP
(5)
dz = - F ( z ) (1 +p-2) 1/2 sin ~,
where z=cooX/C and P=p/moc are the dimensionless coordinate and momentum of the particle, N(z) =N(x(z))=ck(x(z))/coo is the refractive index of the plasma, and
F(z) = - ZeEo (x(z))/moCcoo is the dimensionless electric field amplitude. The Lorentz factor Y of the particle is simply Y= ( 1 +p2)~/2. Prior to consideration of the acceleration process, as described by eqs. (4), (5), let us briefly analyze the case of N = N o = c o n s t and F = F o = c o n s t . In this case the autonomous set of equations (4), (5) can be easily integrated, and its integral has the following form, No( 1 + p 2 ) l / 2 _ p _ F ° cos ~=const.
(6)
The phase diagram of the system consists of trapped (phase-locked) and untrapped particles, and the momentum (as well as the energy) of each particle undergoes oscillations with a "nonlinear frequency" v. In the vicinity of the stable equilibrium points ~ = 2 ~ n , P . = (N 2 - 1 )-1/2 (n is an integer) the oscillations are harmonic, and their frequency Vo= F1/2 ~ mrO2 ~ 1 ) 3/4 The maximum momentum gain by 0 ~,~* a trapped particle, which at the point ~= 0 has a momentum Po > 0, can be found from (6): 2No(1 +Po2) i/2_ (1 +N2)po
Pm~=
N2_I
(7)
When Po << 1, this value becomes independent of Po:
I'm = 2 N o / ( N ~ - l ). As No is close to l, P~ is large, making the acceleration scheme of Tajima and Dawson very advantageous. Now, the trapping condition for a particle can be written as
12 November 1990
2Fo >No( 1 +Po2) l/2--Po -- (N~ - 1 )1/2.
(8)
For the electron acceleration by the Langmuir wave, the corresponding values of the electric field amplitude are always lower than the wave-breaking limit (see re£ [6] and references therein), so hopefully they can be achieved. However, as far as lower-energy electrons and moderate-amplitude plasma waves are concerned, the particle trapping requires No > 1. For weakly relativistic ions inequality (8) can be fulfilled only if No> 1, regardless of the (achievable) Langmuir wave amplitude. Before returning to eqs. (4), (5) with a variable N(z), let us consider the "universal nonlinear resonance" [7 ] limit of the equations with N = No = const and F = Fo = const. This limit describes the trapped particles with small relative changes of P: I P - P . I <
d2¢/dz2+Fo(N~- I )3/2 sin ¢ = 0 ,
(9)
solutions of which can be written in terms of the elliptic functions [ 7 ]. In this case, the maximum size of the trapped particle region of the phase diagram has the following form, APmax =4F~/2No (N 2 - 1 ) -3/4.
(10)
The inequality IP - P . I <
( 11 )
If the wave is very intense, i.e., Fo> 0.5 for the electrons, condition ( 11 ) cannot be satisfied regardless of the value of No. On the contrary, it can be easily fulfilled either for electrons in weaker wave fields, or for ions. Let us return now to eqs. (4), (5) with variable N ( z ) (still keeping F = Fo = const for simplicity ) and describe the "unlimited" electron acceleration. Suppose that an ensemble of electrons is injected at z = 0 into the plasma with the uniform distribution of initial phases ~o and some distribution of the initial momenta Pin>0. The acceleration process must start with an initial value of the phase velocity co/k(O ) < c (N(0) > 1 ), such that a large fraction of the particles find themselves trapped by the wave of the (dimensionless) amplitude Fo. The required N(0) can be determined from condition (8), which in this case 291
Volume 150,number 5,6,7
PHYSICSLETTERSA
12 November 1990
locity (i.e. a decrease in N(z) ) will make the average electron velocity follow the wave phase velocity, which means acceleration on the average (such a regime has been called "the dynamic autoresonance" [8] ). As the phase velocity approaches c (i.e. N ( z ) goes to 1 ), the stable equilibrium point of fig. 1 tends to infinity, the electrons experiencing a monotonous "unlimited" acceleration. This asymptotic (ultra-relativistic) stage of the acceleration is described by the limit of P--*~ of eqs. (4), (5), which immediately follows,
4.0
3.0
g
dPdz- - F 0 s i n ( f N ( z ' ) d z ' - z + ~ ) ,
(13,
0 1.0
0
eo° o I
0,0
-3.0
-2.0
.II.0
!
0.0
i
1,0
i
2.0
3.0
Fig. 1. An exampleof a closed trajectorywith a very small positive Po for Fo=0.25 and No= 1.3.
where ~ is a constant for each particle, depending on the acceleration "prehistory" of the particle. It is clear from eq. (13) that P grows proportionally to z as long as N ( z ) --, 1 and the phase ~ is appropriate. Fig. 2 illustrates the electron acceleration. While numerically solving eqs. (4), (5), we employed a conservative value of Fo = 0.25 and chose the following profile of N ( z ) , N ( z ) = (No +otz 2) ( 1 -I" OdZ 2 )
is rather soft. For initially non,relativistic particles (with dimensionless momenta Po<< 1 at ~ = 0 ) , the minimum value of N ( 0 ) must be N ( O ) = F o + ( 4 F o ) -~ , a t F o < 0 . 5 , =1,
at F o > 0 . 5 .
(12)
It is seen from (12) that a constraint on the initial refractive index is imposed only at F0 <0.5. For given Fo, all the electrons, whose initial coordinates in the ~-P plane are inside the closed trajectory of the system (4), (5) with N o = N ( 0 ) from (12) and a very small Po (an example is presented in fig. 1 ), will be initially trapped by the wave. If we now start to vary the phase velocity (and hence N ( z ) ) slowly (adiabatically) on the scale of the spatial period of the above-mentioned nonlinear oscillations of the trapped electrons, the majority of those electrons will remain phase-locked #~. An increase in the phase ve~t The adiabatieitycondition cannot be satisfied in a narrow region near the ~paratrix, dividingthe trappodand passing partides, so the corresponding (small) fraction of them may be lost, regardlesshow slowN(z) is. 292
-1
,
(14)
The chosen values of parameters demonstrate the unlimited acceleration of the electrons with small injection energies (whose unlimited acceleration by a plasma wave with Fo<0.5 and a constant phase velocity would be impossible even in the case of No= 1 ). In these numerical examples, the ultra-relativistic stage of the acceleration proceeds in a very good agreement with eq. (13). Similar results are also obtained for different initial momenta and phases of the trapped electrons and for other slow dependences of N(z). At smaller a the initial stage of the acceleration has an oscillatory character. For large injection energies, the net increase in the phase velocity of the wave is, o f course, very small. Let us proceed to the ion acceleration. At a given plasma wave amplitude, the absolute value of the dimensionless parameter Fo becomes now much smaller (by a factor of Zme/mi) than in the case of electrons (me and mi are the electron and ion rest masses, Z is the ion charge number). This means a large increase in the initial refractive index N ( 0 ) required for the particle trapping, for reasonable injection energies. For instance, let us consider a plasma
Volume 150,number 5,6,7
PHYSICSLETTERSA
10
8
6
4
2
i
| 40
DISTANCE
0.0
(b) -0.5
-1.0
-1.5
-2.0
-2.50
'
110
'
20
'
| 30
40
DISTANCE
Fig. 2. A numerical example of an unlimited electron acceleration, as describedby eqs. (4), ( 5 ) and the N(z ) profilefrom eq. (14) for Fo=0.25, No= 1.3 and at=0.02: (a) Pand (b) ~ versus the distance. wave close to the wave-breaking limit. In this case the parameter IFol (which would be equal to, say, 1 for the electrons) would become IFol = 5 . 4 × 10 -4 and require No=463, if we want to accelerate protons with very small energies, so that Po << IFo I. Such a regime is similar to the electron acceleration regime considered above. A more practical limit is characterized 'by a larger injection momentum, eo >> IFo I, and a small momentum spread: 8P<< 4F~/2No(N 2 - 1 )-3/4
(15)
12 November 1990
In this case, if we choose No= (1 +P0-2) 1/2, a large fraction of the ions will get trapped by the wave. Since Fo has now the opposite sign, the stable equilibrium points become ?_.=n(2n+ 1 ), Po (with an obvious change of the trapping phase interval). This regime presents a modification of the above-mentioned "universal nonlinear resonance" limit [ 7 ], with an adiabatically slow coordinate dependence added. If N(z) decreases slowly on the scale of the distance I;-1 and finally approaches 1, the ions will be "unlimitedly" accelerated. In this regime, the initial stage of the acceleration always has the character of a relatively slow (upward) momentum drift imposed on the fast nonlinear momentum oscillations. When sufficiently large momenta are achieved, the oscillations vanish (it is seen from the ultra-relativistic equation (13) ). The presence of a small parameter in the problem (i.e. relatively small particle oscillation amplitudes, proportional to F~/2 ) makes it possible to describe this acceleration regime analytically, similarly to a recent work on the "Rydberg accelerator" [ 8 ]. The oscillatory component of the acceleration can be efficiently described in terms of the adiabatic invariant, corresponding to the slowly evolving nonlinear oscillations of P and ~. In the present work we restrict ourselves to a numerical example (see fig. 3). Calculations for different initial phases and momenta of the trapped particles and other slow dependences N(z) give similar results. Asymptotically, the momentum grows according to eq. (13). It should be noted that the idea of the resonance ion acceleration by a space-charge wave with a varying phase velocity was discussed earlier [9 ]. Ref. [9 ] was devoted to the ion acceleration by a relativistic electron beam-supported space-charge wave, where a proper change in the phase velocity is achieved in a converging wave guide. The scheme proposed in the present work can be applied for electrons, as well as for ions. Also, we have performed calculations of the particle orbits and discussed criteria for the phaselocking. The feasibility of the proposed acceleration schemes depends on whether it will be found practical to create the appropriate phase velocity profiles. Generally, such profiles can be achieved by changing the plasma wave frequency a n d / o r the wavenumber with distance a n d / o r in time. In order 293
Volume 150, number 5,6,7
PHYSICSLETTERSA
4.0
3.0
'!
m 1
o
2.0
o) == D.
1.0
0.0 1 0 '0 0
2 0 0' 0
3 o'o 0
'
40 o 0
5000
DISTANCE
Fig. 3. A numerical example of an unlimited ion acceleration, as described by eqs. (4), (5) and the N(z) profile from eq. (14) for For -5.4)< 10-4, Nor 10.05 and a = 5× 10-4: (a) Pand (b) versus the distance. Initial conditions are Po= 0.1, ~ = 3.5. that the speed of light be a stiff limit on the wave phase velocity, one can employ a plasma-filled wave guide. As an example, we consider here the Trivelpiece-Gould space-charge mode [ 10] in a strongly magnetized plasma wave guide. At sufficiently large longitudinal wavelengths, the phase velocity of this mode, cah,/~,.,2~.tuprwop'a'"'2"ll/2,is always less than c but approaches it arbitrarily close as tOp/tOo increases. Since the " v a c u u m " wave guide frequency tOo is inversely proportional to the wave guide radius [ 10 ], while the plasma frequency is determined by the electron concentration ne, the parameter (Of/to O c a n be gradually increased in a diverging wave guide, possibly with an increasing electron concentration. The controlled increase of nc can be achieved via an increase of the ionizing laser beam intensity in the process of irradiation of a multiply charged ion plasma [ 11 ]. The ion acceleration will necessarily, require a small initial toe/tOo which means a rather small initial wave guide radius. For example, when 294
12 November 1990
ne= 10 I5 cm -3, the initial radius must be as small as 0.1 mm, implying a (diverging) plasma fiber. Using the dispersion curves of the two Trivelpiece-Gould modes [ 10 ] and checking the resonance conditions for the frequencies and wave numbers, it is easy to see that the beat-wave excitation technique in such a fiber can employ not only laser beam (co- or counter-propagating), but also submiUimeter radiation beams close to the fiber cutoff. The latter regime may be preferable as far as the radial mode control in the fiber is concerned, however, the power of existing sources in this range is smaller. The energy gain in this scheme is limited by the coherence time of the space-charge wave. In any case, this time cannot exceed the background electron-ion collision time. For electron acceleration this limitation is not essential. However, the ion acceleration will require multi-staging. Of course, many serious issues in the diverging plasma fiber scheme, or related schemes (optimization of the fiber mode excitation, radial focusing of the accelerated particles, radial and longitudinal mode control in the diverging fiber, the Raman cascade of the fiber modes, etc.) have to be addressed and require considerable analysis. Some of these problems are typical for the beatwave accelerator scheme [2,5 ], while others introduce a new element (a diverging plasma fiber). I would like to thank L. Friedland and Y. Carmel for very helpful discussions. This work was completed during my stay at the Laboratory for Plasma Research, University of Maryland at College Park. I am very grateful to the Laboratory staff, and especially to B. Levush, who made my stay there enjoyable and fruitful.
References [ 1] T. Tajima and J.M. Dawson, Phys. Rev. Lett. 43 (1979) 267. [2] J.L. Bobin, in: New developments in particle acceleration techniques, Vol. 1, ed. S. Turner (CERN, Geneva, 1987) p. 58. [ 3 ] Y. Kitagawa,T. Matsumoto, T. Minamihata, K. Sawai, K. Mima, IC Nishihara, H. Azeehi, ICA. Tanaka, H. Takabe and S. Nakai, preprint 9001, Osaka University, Osaka (1990). [4] T. Katsouleasand J.M. Dawson,Phys. Rev. Lett. 51 (1983) 392.
Volume 150, number 5,6,7
PHYSICS LETTERS A
[5] C. Joshi, W.B. Mori, T. Katsouleas, J.M. Dawson, J.M. Kindel and D.W. Forslund, Nature 311 (1984) 525. [6] T. Katsouleas and W.B. Mori, Phys. Rev. Lett. 61 (1988) 90. [7] B.V. Chirikov, Phys. Rep. 52 (1979) 263. [ 8 ] B. Meerson and L. Friedland, Phys. Rev. A 41 (1990) 5233.
12 November 1990
[9] P. Sprangle, A.T. Drobot and W.M. Manheimer, Phys. Rev. Lett. 36 (1976) 1180. [ 10] A.W. Trivelpiece and R.W. Gould, J. Appl. Phys. 30 (1959) 1784. [ 11 ] A. Zigler, private communication.
295
PHYSICS LETTERSA
12 November 1990
"Unlimited" particle acceleration by an intense plasma wave with a varying phase velocity B. M e e r s o n
Centerfor Plasma Physics, Racah Institute of Physics, Hebrew Universityof Jerusalem, Jerusalem, 91904Israel Received 2 July 1990; accepted for publication 6 September 1990 Communicatedby R.C. Davidson
An "unlimited" electron and ion accelerationis shown to occur in the field of an intense "tapered" plasma wave. This acceleration schemedoes not require high injection energiesand pre-bunchingof the acceleratedparticlesand could operate at moderate plasma wave amplitudes. A possibleway of the phase velocity"tapering" is suggested.
Mechanisms of high-gradient electron acceleration by relativistic plasma waves received much attention after the work by Tajima and Dawson [ 1 ], where the beat-wave acceleration concept was proposed (for a review of this and related schemes see ref. [2 ] ). Since the phase velocity of a laser-driven plasma wave can be made to be close to the speed of light c, the electrons, resonantly interacting with the wave, can be accelerated significantly [ 1 ]. The first simultaneous observations of a beat-wave excited plasma wave and accelerated electrons have been recently reported [ 3 ]. For the resonance interaction between a charged particle and a relativistic plasma wave to be efficient, either a sufficiently large injection energy of the particle, or a sufficiently large plasma wave amplitude is necessary. The required injection energies become beyond practical application for ions, therefore for the ion acceleration the beat-wave acceleration scheme has to be properly modified. For electrons, it would be desirable to overcome the waveparticle dephasing [ 1,2 ] and thus ensure an "unlimited" acceleration in a one-dimensional acceleration scheme, because the creation of a coherent two-dimensional plasma wave structure in the Surfatron concept [4,5 ] presents very difficult problems. Such a one-dimensional "unlimited" acceleration of both electrons and ions can be achieved ifone employs a plasma wave with variable phase velocity. The idea is to start with a wave, which is able to trap the 290
accelerated particles (it must be a relatively slow wave in the case of ions), and then, by gradually increasing the phase velocity, to keep the particles trapped (phase-locked) and accelerated. In the following we shall find conditions under which (a) such a scheme could work for a large fraction of pre-injeered electrons or ions, (b) the initial particle bunching is unnecessary and (c) the precise form of the phase velocity spatial (or temporal) profde is unessential. Finally, we shall briefly discuss a possible way of the phase velocity "tapering". The essence of the proposed acceleration mechanism can be demonstrated by a simplified model of a sinusoidal plasma wave with constant frequency OJo, propagating in the x-direction in a weakly inhomogeneous plasma (generalization to the slowly varying frequency case is straightforward). In the geometric optics approximation, the plasma wave accelerating field has the form X
E(x, t ) = E o ( x ) s i n t ~ k ( x ' ) d x ' - c O o t + ~ o ) .
(1)
The relativistic equation of motion of a charged particle in the wave field can then be written as
dp _ ZeE(x,t) (l+m~c2/p2)~/2 , dx c
(2)
where mo is the mass of the particle, Ze is its charge
0375-9601/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-HoUand)
Volume 150,number 5,6,7
PHYSICSLETTERSA
and p is its momentum. Introduce a phase variable
~= i k(x') dx'-coot+~o,
(3)
corresponding to the reference frame traveling with the wave. Then eq. (2) can be rewritten as a set of two first-order equations, d~ = N ( z ) - (1 + p - 2 ) 1/2 dz
(4)
dP
(5)
dz = - F ( z ) (1 +p-2) 1/2 sin ~,
where z=cooX/C and P=p/moc are the dimensionless coordinate and momentum of the particle, N(z) =N(x(z))=ck(x(z))/coo is the refractive index of the plasma, and
F(z) = - ZeEo (x(z))/moCcoo is the dimensionless electric field amplitude. The Lorentz factor Y of the particle is simply Y= ( 1 +p2)~/2. Prior to consideration of the acceleration process, as described by eqs. (4), (5), let us briefly analyze the case of N = N o = c o n s t and F = F o = c o n s t . In this case the autonomous set of equations (4), (5) can be easily integrated, and its integral has the following form, No( 1 + p 2 ) l / 2 _ p _ F ° cos ~=const.
(6)
The phase diagram of the system consists of trapped (phase-locked) and untrapped particles, and the momentum (as well as the energy) of each particle undergoes oscillations with a "nonlinear frequency" v. In the vicinity of the stable equilibrium points ~ = 2 ~ n , P . = (N 2 - 1 )-1/2 (n is an integer) the oscillations are harmonic, and their frequency Vo= F1/2 ~ mrO2 ~ 1 ) 3/4 The maximum momentum gain by 0 ~,~* a trapped particle, which at the point ~= 0 has a momentum Po > 0, can be found from (6): 2No(1 +Po2) i/2_ (1 +N2)po
Pm~=
N2_I
(7)
When Po << 1, this value becomes independent of Po:
I'm = 2 N o / ( N ~ - l ). As No is close to l, P~ is large, making the acceleration scheme of Tajima and Dawson very advantageous. Now, the trapping condition for a particle can be written as
12 November 1990
2Fo >No( 1 +Po2) l/2--Po -- (N~ - 1 )1/2.
(8)
For the electron acceleration by the Langmuir wave, the corresponding values of the electric field amplitude are always lower than the wave-breaking limit (see re£ [6] and references therein), so hopefully they can be achieved. However, as far as lower-energy electrons and moderate-amplitude plasma waves are concerned, the particle trapping requires No > 1. For weakly relativistic ions inequality (8) can be fulfilled only if No> 1, regardless of the (achievable) Langmuir wave amplitude. Before returning to eqs. (4), (5) with a variable N(z), let us consider the "universal nonlinear resonance" [7 ] limit of the equations with N = No = const and F = Fo = const. This limit describes the trapped particles with small relative changes of P: I P - P . I <
d2¢/dz2+Fo(N~- I )3/2 sin ¢ = 0 ,
(9)
solutions of which can be written in terms of the elliptic functions [ 7 ]. In this case, the maximum size of the trapped particle region of the phase diagram has the following form, APmax =4F~/2No (N 2 - 1 ) -3/4.
(10)
The inequality IP - P . I <
( 11 )
If the wave is very intense, i.e., Fo> 0.5 for the electrons, condition ( 11 ) cannot be satisfied regardless of the value of No. On the contrary, it can be easily fulfilled either for electrons in weaker wave fields, or for ions. Let us return now to eqs. (4), (5) with variable N ( z ) (still keeping F = Fo = const for simplicity ) and describe the "unlimited" electron acceleration. Suppose that an ensemble of electrons is injected at z = 0 into the plasma with the uniform distribution of initial phases ~o and some distribution of the initial momenta Pin>0. The acceleration process must start with an initial value of the phase velocity co/k(O ) < c (N(0) > 1 ), such that a large fraction of the particles find themselves trapped by the wave of the (dimensionless) amplitude Fo. The required N(0) can be determined from condition (8), which in this case 291
Volume 150,number 5,6,7
PHYSICSLETTERSA
12 November 1990
locity (i.e. a decrease in N(z) ) will make the average electron velocity follow the wave phase velocity, which means acceleration on the average (such a regime has been called "the dynamic autoresonance" [8] ). As the phase velocity approaches c (i.e. N ( z ) goes to 1 ), the stable equilibrium point of fig. 1 tends to infinity, the electrons experiencing a monotonous "unlimited" acceleration. This asymptotic (ultra-relativistic) stage of the acceleration is described by the limit of P--*~ of eqs. (4), (5), which immediately follows,
4.0
3.0
g
dPdz- - F 0 s i n ( f N ( z ' ) d z ' - z + ~ ) ,
(13,
0 1.0
0
eo° o I
0,0
-3.0
-2.0
.II.0
!
0.0
i
1,0
i
2.0
3.0
Fig. 1. An exampleof a closed trajectorywith a very small positive Po for Fo=0.25 and No= 1.3.
where ~ is a constant for each particle, depending on the acceleration "prehistory" of the particle. It is clear from eq. (13) that P grows proportionally to z as long as N ( z ) --, 1 and the phase ~ is appropriate. Fig. 2 illustrates the electron acceleration. While numerically solving eqs. (4), (5), we employed a conservative value of Fo = 0.25 and chose the following profile of N ( z ) , N ( z ) = (No +otz 2) ( 1 -I" OdZ 2 )
is rather soft. For initially non,relativistic particles (with dimensionless momenta Po<< 1 at ~ = 0 ) , the minimum value of N ( 0 ) must be N ( O ) = F o + ( 4 F o ) -~ , a t F o < 0 . 5 , =1,
at F o > 0 . 5 .
(12)
It is seen from (12) that a constraint on the initial refractive index is imposed only at F0 <0.5. For given Fo, all the electrons, whose initial coordinates in the ~-P plane are inside the closed trajectory of the system (4), (5) with N o = N ( 0 ) from (12) and a very small Po (an example is presented in fig. 1 ), will be initially trapped by the wave. If we now start to vary the phase velocity (and hence N ( z ) ) slowly (adiabatically) on the scale of the spatial period of the above-mentioned nonlinear oscillations of the trapped electrons, the majority of those electrons will remain phase-locked #~. An increase in the phase ve~t The adiabatieitycondition cannot be satisfied in a narrow region near the ~paratrix, dividingthe trappodand passing partides, so the corresponding (small) fraction of them may be lost, regardlesshow slowN(z) is. 292
-1
,
(14)
The chosen values of parameters demonstrate the unlimited acceleration of the electrons with small injection energies (whose unlimited acceleration by a plasma wave with Fo<0.5 and a constant phase velocity would be impossible even in the case of No= 1 ). In these numerical examples, the ultra-relativistic stage of the acceleration proceeds in a very good agreement with eq. (13). Similar results are also obtained for different initial momenta and phases of the trapped electrons and for other slow dependences of N(z). At smaller a the initial stage of the acceleration has an oscillatory character. For large injection energies, the net increase in the phase velocity of the wave is, o f course, very small. Let us proceed to the ion acceleration. At a given plasma wave amplitude, the absolute value of the dimensionless parameter Fo becomes now much smaller (by a factor of Zme/mi) than in the case of electrons (me and mi are the electron and ion rest masses, Z is the ion charge number). This means a large increase in the initial refractive index N ( 0 ) required for the particle trapping, for reasonable injection energies. For instance, let us consider a plasma
Volume 150,number 5,6,7
PHYSICSLETTERSA
10
8
6
4
2
i
| 40
DISTANCE
0.0
(b) -0.5
-1.0
-1.5
-2.0
-2.50
'
110
'
20
'
| 30
40
DISTANCE
Fig. 2. A numerical example of an unlimited electron acceleration, as describedby eqs. (4), ( 5 ) and the N(z ) profilefrom eq. (14) for Fo=0.25, No= 1.3 and at=0.02: (a) Pand (b) ~ versus the distance. wave close to the wave-breaking limit. In this case the parameter IFol (which would be equal to, say, 1 for the electrons) would become IFol = 5 . 4 × 10 -4 and require No=463, if we want to accelerate protons with very small energies, so that Po << IFo I. Such a regime is similar to the electron acceleration regime considered above. A more practical limit is characterized 'by a larger injection momentum, eo >> IFo I, and a small momentum spread: 8P<< 4F~/2No(N 2 - 1 )-3/4
(15)
12 November 1990
In this case, if we choose No= (1 +P0-2) 1/2, a large fraction of the ions will get trapped by the wave. Since Fo has now the opposite sign, the stable equilibrium points become ?_.=n(2n+ 1 ), Po (with an obvious change of the trapping phase interval). This regime presents a modification of the above-mentioned "universal nonlinear resonance" limit [ 7 ], with an adiabatically slow coordinate dependence added. If N(z) decreases slowly on the scale of the distance I;-1 and finally approaches 1, the ions will be "unlimitedly" accelerated. In this regime, the initial stage of the acceleration always has the character of a relatively slow (upward) momentum drift imposed on the fast nonlinear momentum oscillations. When sufficiently large momenta are achieved, the oscillations vanish (it is seen from the ultra-relativistic equation (13) ). The presence of a small parameter in the problem (i.e. relatively small particle oscillation amplitudes, proportional to F~/2 ) makes it possible to describe this acceleration regime analytically, similarly to a recent work on the "Rydberg accelerator" [ 8 ]. The oscillatory component of the acceleration can be efficiently described in terms of the adiabatic invariant, corresponding to the slowly evolving nonlinear oscillations of P and ~. In the present work we restrict ourselves to a numerical example (see fig. 3). Calculations for different initial phases and momenta of the trapped particles and other slow dependences N(z) give similar results. Asymptotically, the momentum grows according to eq. (13). It should be noted that the idea of the resonance ion acceleration by a space-charge wave with a varying phase velocity was discussed earlier [9 ]. Ref. [9 ] was devoted to the ion acceleration by a relativistic electron beam-supported space-charge wave, where a proper change in the phase velocity is achieved in a converging wave guide. The scheme proposed in the present work can be applied for electrons, as well as for ions. Also, we have performed calculations of the particle orbits and discussed criteria for the phaselocking. The feasibility of the proposed acceleration schemes depends on whether it will be found practical to create the appropriate phase velocity profiles. Generally, such profiles can be achieved by changing the plasma wave frequency a n d / o r the wavenumber with distance a n d / o r in time. In order 293
Volume 150, number 5,6,7
PHYSICSLETTERSA
4.0
3.0
'!
m 1
o
2.0
o) == D.
1.0
0.0 1 0 '0 0
2 0 0' 0
3 o'o 0
'
40 o 0
5000
DISTANCE
Fig. 3. A numerical example of an unlimited ion acceleration, as described by eqs. (4), (5) and the N(z) profile from eq. (14) for For -5.4)< 10-4, Nor 10.05 and a = 5× 10-4: (a) Pand (b) versus the distance. Initial conditions are Po= 0.1, ~ = 3.5. that the speed of light be a stiff limit on the wave phase velocity, one can employ a plasma-filled wave guide. As an example, we consider here the Trivelpiece-Gould space-charge mode [ 10] in a strongly magnetized plasma wave guide. At sufficiently large longitudinal wavelengths, the phase velocity of this mode, cah,/~,.,2~.tuprwop'a'"'2"ll/2,is always less than c but approaches it arbitrarily close as tOp/tOo increases. Since the " v a c u u m " wave guide frequency tOo is inversely proportional to the wave guide radius [ 10 ], while the plasma frequency is determined by the electron concentration ne, the parameter (Of/to O c a n be gradually increased in a diverging wave guide, possibly with an increasing electron concentration. The controlled increase of nc can be achieved via an increase of the ionizing laser beam intensity in the process of irradiation of a multiply charged ion plasma [ 11 ]. The ion acceleration will necessarily, require a small initial toe/tOo which means a rather small initial wave guide radius. For example, when 294
12 November 1990
ne= 10 I5 cm -3, the initial radius must be as small as 0.1 mm, implying a (diverging) plasma fiber. Using the dispersion curves of the two Trivelpiece-Gould modes [ 10 ] and checking the resonance conditions for the frequencies and wave numbers, it is easy to see that the beat-wave excitation technique in such a fiber can employ not only laser beam (co- or counter-propagating), but also submiUimeter radiation beams close to the fiber cutoff. The latter regime may be preferable as far as the radial mode control in the fiber is concerned, however, the power of existing sources in this range is smaller. The energy gain in this scheme is limited by the coherence time of the space-charge wave. In any case, this time cannot exceed the background electron-ion collision time. For electron acceleration this limitation is not essential. However, the ion acceleration will require multi-staging. Of course, many serious issues in the diverging plasma fiber scheme, or related schemes (optimization of the fiber mode excitation, radial focusing of the accelerated particles, radial and longitudinal mode control in the diverging fiber, the Raman cascade of the fiber modes, etc.) have to be addressed and require considerable analysis. Some of these problems are typical for the beatwave accelerator scheme [2,5 ], while others introduce a new element (a diverging plasma fiber). I would like to thank L. Friedland and Y. Carmel for very helpful discussions. This work was completed during my stay at the Laboratory for Plasma Research, University of Maryland at College Park. I am very grateful to the Laboratory staff, and especially to B. Levush, who made my stay there enjoyable and fruitful.
References [ 1] T. Tajima and J.M. Dawson, Phys. Rev. Lett. 43 (1979) 267. [2] J.L. Bobin, in: New developments in particle acceleration techniques, Vol. 1, ed. S. Turner (CERN, Geneva, 1987) p. 58. [ 3 ] Y. Kitagawa,T. Matsumoto, T. Minamihata, K. Sawai, K. Mima, IC Nishihara, H. Azeehi, ICA. Tanaka, H. Takabe and S. Nakai, preprint 9001, Osaka University, Osaka (1990). [4] T. Katsouleasand J.M. Dawson,Phys. Rev. Lett. 51 (1983) 392.
Volume 150, number 5,6,7
PHYSICS LETTERS A
[5] C. Joshi, W.B. Mori, T. Katsouleas, J.M. Dawson, J.M. Kindel and D.W. Forslund, Nature 311 (1984) 525. [6] T. Katsouleas and W.B. Mori, Phys. Rev. Lett. 61 (1988) 90. [7] B.V. Chirikov, Phys. Rep. 52 (1979) 263. [ 8 ] B. Meerson and L. Friedland, Phys. Rev. A 41 (1990) 5233.
12 November 1990
[9] P. Sprangle, A.T. Drobot and W.M. Manheimer, Phys. Rev. Lett. 36 (1976) 1180. [ 10] A.W. Trivelpiece and R.W. Gould, J. Appl. Phys. 30 (1959) 1784. [ 11 ] A. Zigler, private communication.
295