Short, relativistically strong laser pulse in a narrow channel

21 November 1994 PHYSICS LETTERS A

ELSEVIER

Physics Letters A 195 (1994) 84-89

Short, relativistically strong laser pulse in a narrow channel S.V. Bulanov a, F.F. Kamenets b, F. Pegoraro c, A.M. Pukhov b a General Physics Institute of the Russian Academy of Sciences, I/avilov Street 38, 117942 Moscow, Russian Federation b MOSCOWInstitute for Physics and Technology, DolgoprudnyL Moscow Region, Russian Federation c Department of Theoretical Physics, University of Turin, via P. Giuria 1, 10125 Turin, Italy

Received 20 July 1994; revised manuscript received 3 August 1994; accepted for publication 7 September 1994 Communicated by M. Porkolab

Abs~a~

We present the results of an analytical study and of a two-dimensional particle-in-cell simulation of a relativistically strong laser pulse propagating in a narrow channel which eliminates the pulse spreading due to diffraction. In an empty channel with sharp boundaries, the main absorption mechanism is "vacuum heating" of the electrons expelled from the walls. These electrons fill the channel and form a charged cloud which moves at a relativistic velocity behind the pulse and produces a longitudinal electric field that can be used to accelerate charged particles. This cloud can also act as a mirror and, interacting with electromagnetic radiation, upshift its frequency. The interaction with the channel wall depends on the pulse intensity and polarization. TM-polarized pulses undergo greater losses than TE-pulses, and cause the formation of a charged cloud, of high harmonics and of a quasi-static magnetic field. In the case of a channel filled by an underdense plasma, an ultrashort pulse excites a strong wake wave with a longitudinal electric wake field which can accelerate charged particles.

1. Introduction

Super-intense, ultrashort laser pulses with pulse length o f the order o f 100 fs have been discussed recently with renewed interest and major progress has been achieved in producing them [ 1 ]. The quiver energy o f electrons in the electric field o f such pulses, which have a typical intensity o f the order o f 1018 W / cm 2, approaches relativistic conditions and can even be greater than their rest mass. In the electric field o f these pulses, which is more intense than the field inside atoms, matter behaves as if instantaneously ionized, but the plasma that is created does not have sufficient time to ablate during the pulse time. Therefore the dynamics o f the electrons is most important. Among the many applications made possible by such laser pulses, we recall charged particle acceleration [2], frequency upshifting o f electromagnetic radia-

tion [ 3 ], and the development of X-ray sources [ 4 ]. However, in order to achieve the conditions necessary for efficient acceleration o f electrons and photons, the problems that arise from the diffractive spread o f the pulse and from the optimization of the processes that can excite a strong electric field in the plasma must be solved. The propagation o f a laser pulse in a narrow channel, which eliminates the diffraction spreading o f the pulse, was examined theoretically in Refs. [ 5-8 ] and experimentally in Ref. [ 9 ]. The channel can also be used to create a nonneutral plasma behind the pulse [ 10 ]. The longitudinal electric field produced by the leading front of this plasma cloud moves at relativistic velocities and can accelerate charged particles. In this Letter we discuss the propagation in a narrow cylindrical channel of a short pulse, its interaction with the wall, the creation o f an electron cloud

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S. V. Bulanov et al. / Physics Letters A 195 (1994) 84-89

and the acceleration of charged panicles. In addition we present a numerical simulation of the strong wake field excited by a short pulse in a narrow channel filled by an underdense plasma. As indicated by the numbers in front, for these numerical simulations we have used a 2 ( 3 / 3 ) - P I C code. This particle-in-ceU code solves the two-dimensional evolution equations of a relativistic collisionless plasma in an electromagnetic field: all quantities depend on x, y, t, while the fields and velocities have all three components. The number of grid-points is 128 × 256, the number of panicles per cell is ~ 10 and the total panicle number is ~ 2 X 10 5.

2. Pulse propagation

85

the description of the nonlinear dynamics of the electrons in the super-intense electromagnetic radiation, we follow the approach formulated in Ref. [ 14 ], and used in Refs. [ 12,13 ] for the ease of a plane geometry, and perform a Lorentz transformation from the laboratory frame K to the frame K' moving along the x-axis with group velocity vs. In the K' frame, E'r and k~ vanish while the amplitudes of E~ and H i are equal to E0. In the following we assume the fields in the electromagnetic wave to be given and introduce the Lagrangian coordinates r0 and T. These are related to the Eulerian variables r' and t' by r' = ro + ~(ro, z), z=t', where ~(ro, z) is the displacement of an electron from its initial position ro and v" = O~/Oz. The action of the wave on the electrons results in a radial electric field given by ~(ro,O

When considering the propagation of a laser pulse along the axis (denoted as the x-axis) of a narrow cylindrical channel of radius R, the TM-polarization is the most interesting one for the creation of a charged plasma, while the TE-polarization is of interest for channeling the electromagnetic radiation. In a plane geometry TM- and TE-modes correspond to the p-polarization and to the s-polarization, respectively [ 11 ], which have been investigated recently in detail with PIC numerical simulations [ 12,13 ]. For azimuthally symmetric propagation, TM-modes have a longitudinal component of the electric field Ex (we denote its amplitude by Eo), a radial component Er and an azimuthal component H , of the magnetic field, only. The pulse frequency co depends on the longitudinal wave-number kx as to 2 = k2xc2 + r2c 2, where K is determined by requiring that Ex vanishes at the wall. This implies Jo (r-R) = 0, with Jo a Bessel function, and gives r ~ 2.2/R. The group velocity of the radiation propagating along the channel is v~= kxc2/o~= k~c/ (k2+~:2) ~/2. We assume that the laser light is reflected by the walls. This requires to2-k2c2 1.2/R2r¢ [7] where r,=e2/m,c 2 is the classical electron radius. Further, we assume x/k~<< 1, i.e., kxR ~ 1 and kx/p >> 1, with lp the pulse length. The radial component E~ of the TM-mode does not vanish at the channel wall and accelerates electrons into the cha=nel leading to the formation of an electron cloud behind the laser pulse. In order to simplify

47re ~ [n~(ro+s)](ro+s)ds. E'~(ro, T)= ro+-----~

(1)

o

Here n~ (r') is the density distribution of the ions in the K' frame. If the quiver radius rz of the electrons is larger than the plasma characteristic nonuniformity length L, i.e., if rE = epEoT/meo~ 2 > L with r = v~/c and 7 = ( 1 - p2) - 1/2, electrons with initial coordinate close to the boundary are thrown out of the wall into the channel within one oscillation period of the pulse. This results in electron "vacuum heating" [ 15 ], i.e., in the breaking of the flow of the electrons, self intersection of their trajectories and stochastization of their motion. In order to describe these processes, we assume that the ion density distribution vanishes inside the channel, n~ = 0 for r ' = (ro+~) R . Inserting this expression into Eq. ( 1 ) we obtain for the radial electric field inside the channel

R2_r 2 E'~(r', t')=2~en'o (ro +~-----~"

(2)

The hydrodynamic equations that describe the relativistic electron motion in Lagrangian variables have the energy integral I-D 2

/.2

mec2 + 2 7 r e 2 n ' o ~ - ~ - + r2oIn ro [ 1 _f12_ (O~/cOz)2] 1/2 R + ( R 2 - r 2 ) In ~ ] = c o n s t .

(3)

S. K Bulanov et al. /Physics Letters A 195 (1994) 84-89

86

The electrons are decelerated in their motion towards the channel axis and stop at the radius r'~n, r~in=Rexp

(

U~)2~2 z 2 o)ve(ro - R 2)

1 r2 +--In 2 r2-R 2

R) '

(4) with v~ = v~ (ro) their initial velocity. In the nonrelativistic limit, which is obviously valid near the point where the particle stops, Eq. (3) can be written in the form

O~ (o)2e(r2_R2) ~ , / 2 Or--\ ~ In r m i n ]

(5)

1/2

1 In (ro,+ ~)) ~ ( ~ , 3, rmi--"----~/ (6)

Here ~ ( a , b; x) is a confluent hypergeometric function [ 16 ] and "~'min)the time it takes the electrons to cross the channel and reach train, is given by 27rmin Tmin--~" o ) p e ( r o2 _ R 2) 1/2 _

-l/z

"1

~'[-,3,1n..,R-ff--) \2 rmin]

(7) "

Using the asymptotic expansion of the function J / ( a , b; x) [ 16 ] and calculating the Jacobian, IOro/orpl,2-,2 we find the electron density near the channel axis for "t",~, train) n~2v~272 n'~= o)2 r,2[ln(R/r, ) ]2.

AS~eEill~¢¢~eEillJ(1-fl)~eEill)l(kxR)

2.

(9)

The acceleration length l ~ and the energy A8 exceed the nonuniformity length Iii and the electrostatic energy in the cloud by the factor (kxR)2 >> I.

4. Numerical simulations

_ o)~(r 2 _ R 2) 1/2 m (T-- Train) . 2Train

×(ln..,R-~--] \ rmi,]

The typical values of the electrostatic potential in the channel and of the electric field at the leading edge of the electron cloud can be estimated as ~ ~ rnecEtl 2/ 2e and Eli ,~ ¢/lll, with lit ~ min{R, ZminVg} the scale of the cloud nonuniformity. The energy gain of a relativistic particle is

,

and has the following solution (ln~)

3. Particle acceleration

(8)

This value tends to infinity for r'~O, but the total number of electrons near the axis remains finite. The mean electron density in the frame K is ( n e ) 2nov2/o)2 R 2 which corresponds to the equipartition between the mean electron potential and kinetic energies in the channel. From this expression we obtain the pulse depletion length due to electron vacuum heating: lvh~ lp(Rkx/a) 2 with a=eEo?/mjoc the pulse dimensionless amplitude.

In order to overcome the restrictions on the parameters of the pulse and of the channel that are needed in the above analytical treatment, it is necessary to solve the fully relativistic, kinetic, nonlinear problem numerically. The results of the simulation of the pulse propagation in an initially empty channel are presented in Figs. 1-3 in a 2-D x, y plane geometry. The interaction of a TM-polarized laser pulse, which has a nonvanishing Efcomponent of the electric field, is shown in Fig. 1 for o)t/2n= 15, 20. The coordinates x and y are normalized to 2=2nc/o). At t = 0 the pulse has a plane front along y with amplitude a = 10, length lp= 182 along x and is incident on a narrow channel which begins at x = 5 and has a width 22. The plasma density corresponds to tope= 3.3o). In frames (a) the iso-energetic pulse contours are shown, Frames (b) give the x-dependence, at y = 0 , of the pulse energy density and show the strong pulse decay due to the expulsion from the wall and heating of the electrons inside the channel. The electron distribution in the channel is shown in frames (c) and in frames (d) their y-averaged phase space (Px, x) is given. It is seen that electrons are heated up to energies of the order of a2/2 ~ 50. No regular wake field is excited in this regime. The interaction ofa relativistically strong pulse with a sharp-boundary, overdense plasma is accompanied by the efficient generation both of high harmonics [ 13 ] and low frequency radiation [ 13,17 ], and of a quasi-static magnetic field [ 18 ]. High harmonics at the channel axis are shown in Fig. 2a: the energy in

S. E Bulanov et al. ~Physics Letters A 195 (1994) 84-89

87

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X × Fig. 1. Differentstages of the interactionwith the channelofa TM-polarizedpulse. Parts (I), (II) correspondto cot/2x= 15, 20, respectively;x, y are in units of2: (a) contoursof equal e.m. energydensity; (b) e.m. energydensitydistribution alongx for y= 0; (c) electron distribution inside the channel; (d) y-averagedphase space (Px, x). the harmonics decreases together with the pulse amplitude. The quasi-static (zero frequency) magnetic field is shown at different points along the channel in Figs. 2b-2d. The pulse polarization changes radically the interaction of an ultrastrong pulse with a sharp boundary overdense plasma [ 13 ]. This is apparent in Fig. 3 for a TE-polarized pulse ( E z # 0 ) propagating in a narrow channel for the same parameters of Fig. 1. Frames (a) and (b) show a very small decay of the pulse energy. Interference between waves with different values o f x created at the channel entry leads to the modulation of the pulse amplitude which is shown in frame (b). Electrons fill the channel much less efficiently, as is seen by comparing Figs. 3c and lc. Electron heating is also less efficient, see Fig. 3d where their y-averaged phase space (Px, x) is given, and, for these parameters, no high harmonic generation is observed. The weak decay of a short TE-polarized pulse makes its use convenient for generating a strong wake electric field i n a channel filled by an underdense plasma. This process was described in Ref. [ 7 ] using the envelope approximation for the pulse propaga-

tion and the quasi-static approach for the pulse interaction with the plasma waves. The following requirements must be satisfied in order to make this scheme of wake field particle acceleration advantageous in comparison with the excitation of plasma waves behind a wide short pulse propagating in a uniform plasma. First, for a given total pulse power P=cR21rE2/47r, the Rayleigh length IR=~rR2/A must be longer than the depletion length ld~,= min{lp (Rkx/ a) 2, (c/~2pa)(o~/~2p)2}, with Qp the plasma frequency inside the channel. Second, the depletion length must exceed the acceleration length la~ = ~ / ( l - r ) where /~l=rain{R, 23/2a/1%} is the electric field nonuniformity scale in the channel filled by the underdense plasma and lq, = c/£2p. Here we have considered a pulse with a sharp leading front ( < 1/l%a), a >> 1 and Ip < 23/2a/I%. A detailed analysis of the depletion process can be found in Refs. [ 8,19 ]. For the nonlinear plasma wave not to be spoiled by the finite transverse size of the channel, and therefore of the pulse, we must have R > 2 2/3a/k~,: in this case the difference between the pulse group velocity, which coincides with the phase velocity of the wake, and the speed of light in vacuum is still determined by (~2p/ co)2: vg~c (1-£2~p/2to2). However, the depletion

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Fig. 2. High harmonic and quasi-static magnetic field generation: (a) excitation of high harmonics as a function o f x along the channel; (b), (c), (d) magnetic field spectrum at x = 5 , 10, 15 respectively. 10 Y

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X Fig. 4. Wake field excitation behind a TE-polarized short pulse, /0=52 and a=6, in a channel with R=3,~ for tot/2:t=40: (a) contours of equal e.m. energy density; (b) e.m. energy density distribution along x for y = 0; (c) y-averaged phase space ( P,, x ); (d) longitudinal electric field along the axis.

S. V.. Bulanov et al. ~Physics Letters A 195 (1994) 84-89 length/d~= (c/£2pa) (to/~2p) 2 turns out to be shorter than the acceleration length lace= (ca/~2p) (to/~2p) 2. The condition that the Rayleigh length lR be longer than ld~ requires a > (co/g2 v) i/3. The wake field excitation is shown in Fig. 4 for a TE-polarized pulse with initial amplitude a = 6 in a narrow channel, R = 32, filled by plasma: f2v = 0.15to. Frame (a) shows the iso-energetic pulse contours at tot/27t=40. Frame (b) gives the x-dependence, at y = 0 , of the pulse energy density. No heavy losses are seen. In frame (c) the y-averaged phase space (Px, x) is given: a group of electrons is accelerated by the field behind the pulse up to energies of order 30mcc 2. In frame (d) the wake electric field is shown as a function of x: the first period of the wake field has a regular structure while the remaining part is rather irregular due to the effect of the transverse inhomogeneity caused by the channel.

5. Conclusions When an ultrashort laser pulse propagates in a narrow channel its diffraction spread is eliminated. The pulse decay depends on its polarization: the energy losses arise from the expulsion of the electrons from the walls into the channel and from the transformation of the pulse energy into quiver energy of the electrons. Losses are large for a TM-polarized pulse leading to fast absorption, plasma heating and high harmonic generation. On the contrary, a short TEpolarized ultrastrong pulse suffers relatively weak losses and can be used to excite a strong wake electric field and to accelerate particles, However, the transverse inhomogeneity of the plasma tends to spoil the regular structure of the wake and a more detailed analysis is needed in order to determine the real capabilities of this method of particle acceleration.

89

Acknowledgement Two of the authors (S.B. and A.P. ) would like to acknowledge the hospitality of the University of Turin. Financial support from the Italian Ministry for Research (MURST), from the Associazione per lo Sviluppo Scientifico e Tecnologico del Piemonte and from the Russian Foundation for the Promotion of Basic Research (grant N. 94-02-16922) is gratefully acknowledged.

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