1E.EE TRANSACTJONS ON PLASMA SCIENCE, VOL. 24, NO. 2, APRIL 1996
363
Linear Theory of Resonance Self-mModulation of an Intense Laser Pulse in Homogeneous Plasma and Plasma Channels N. E. Andreev, V. I. Kirsanov, L. M. Gorbunov, and A. S. Sakharov
Abstract-The analytical solutions describing the linear stage of the intense-laser-pulse self-modulation, which results in a strong plasma wakefield excitation, are studied in terms of the paraxial approximation. The attention is focused on phase relations that were ignored in the previous studies. It is shown that the value of the phase velocity of the plasma wake wave differs from the pulse group velocity so that under some specific conditions, the relativistic factor corresponding to the phase velocity can be siubstantially less than that for the group velocity. This may be important for the particle acceleration in the self-modulatedlaser wakefield accelerator.
I. INTRODUCTION
E
XCITATION of electron-density waves by short laser pulses in an underdense plasma (where the electron plasma frequency wP is much less than the carrier frequency of the laser pulse W O ) is widely discussed in relation to the advanced laser accelerator concept [ 11, [2]. From the standpoint of charged-particle acceleration, the properties of these waves, including their amplitude and phase velocities and also their relation to the pulse evolution, are very important. Even in the case of a fairly short pulse (7 U;'), the pulse evolution can result in not only a variation of the amplitude of the excited wave, but also can cause a decrease (compared to the group velocity of the laser radiation) of the phase velocity of this wave [31, 141. For longer pulses, when the self-modulation of the laser pulses can occur [5]-[7], the study of the dynamics of tlhe plasma-wave excitation (including the amplitude and plhase time dependencies) is of special interest for the selfmodulated-laser wakefield accelerator (SMLWFA) [5], [6]. This configuration of laser accelerator essentially involves the laser pulse dynamics, and, in this very case, one may expect the effect of variation of the wakefield properties to be the most strongly pronounced. N
Manuscript received January 1, 1996; revised February 15, 1996. This work vvas performed as part of the research programs of the Russian Ministry of Science and was supported in part by the Russian Foundation for Basic Research under Grants 94-02-03843 and 94-02-03847 and in part by the International Scientific Foundation under Grant M83000. N. E. Andreev and V. I. Kirsanov are with the High Energy Density Research Center, Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow 127412 Russia (e-mail:
[email protected]). L. M. Gorbunov is with the P. N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow 117924 Russia. A. S. Sakharov is with the Institute of General Physics, Russian Academy of Sciences, Moscow 117942 Russia. Publisher Item Identifier S 0093-3813(96)04481-5.
Although the amplitude characteristics of a plasma wave excited by a laser pulse during the pulse self-modulation have been the subject of several studies (see, for example, [8]-[10]), no attention as yet has been paid to the study of the phase velocity of the excited plasma wave. The self-modulation of laser pulses in an underdense plasma can be regarded as a pulse instability against an amplitude modulation with a characteristic wavenumber of about w p / c . Because of the resonance character of the electron-plasmawave excitation involved, here we will refer to this instability as the resonance self-modulation instability (RSMI). Such an instability was first discovered in numerical simulations of laser-plasma interactions [5]-[7] and was used to obtain a generation of a strong plasma-wave in a recent experiment on laser acceleration [ 113. Analytical study of resonance self-modulation based on the analysis of the dispersion relation for electromagnetic radiation propagating in an underdense plasma was introduced in [7]-[9], where such a modulation was associated with the forward and near-forward stimulated Raman scattering instability. The study of the RSMI, in the case when it is connected with a modulation of the pulse spot size, was first conducted in [lo], where the asymptotic expressions describing the growth of perturbations were obtained for several instability regimes. However, because the asymptotic analysis fails to provide a detailed picture of the onset of instability, a more precise analytical study of the RSMI allowing a description of the phase characteristics of the excilted plasma waves is of interest. In this paper, the analytical solutions describing the linear stage of the RSMI related to the pulse spot-size modulation are examined in terms of the paraxial approximation for a flattop pulse with a sharp leading edge. The attention is focused on phase relations that were ignored in the previous studies. It is shown that the phase velocity of the plasma wake wave can vary during the onset of iinstability so that under some specific conditions the relativistic factor corresponding to the phase-velocity can be substantially less (even in the order of magnitude) than for the pulse group velocity. This seems to be of crucial importance for the SMLWFA [5], [12], [13]. 11. BASIC IZQUATIONS
We assume the field of a laser pulse E in the form
0093-3813/96$05.00 0 1996 IEEE
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 24, NO. 2, APRIL 1996
364
where a is the complex dimensionless envelope of the pulse field which varies slowly in time and space, ko = +(WO” w,”)1/2 is the wavenumber of the laser field, e is a unit vector of ( 4 ~ e ~ n o / m , )is’ the / ~ electron plasma the polarization, w, frequency, and no is the unperturbed plasma density. The self-consistent evolution of the axisymmetric laser pulse can be described by a system of equations for a and for the normalized plasma-electron-density perturbation N = &,/no which, in an underdense plasma (w, k;’ when the transverse part of the Laplace operator in (3) can be neglected. Then, substituting (4)-(6) into (2) and
In the absence of a plasma, we have 7 = 1; whereas in the case Q = 0, when a transportation of the pulse (without a change in the pulse spot size) over a large distance is possible (either due to the relativistic optical guiding or in a preformed plasma channel), this time is infinitively large, 7 = CO. Further, we will consider the instability related to short-wavelength
ANDREEV et al.: LINEAR THEORY OF RESONANCE SELF-MODULATION
365
perturbations during the time intervals that are small compared to the characteristic time 7 and assume N 1. In this case, for the quantities corresponding to shortwavelength perturbations, assuming f > l),the difference between the phase and group velocities Sv tends to a constant negative value which depends on k~ and R only Sv
'Uph - 'Ug
= -cn(kpz,)-'
The corresponding value of
7,h
-2c(koR)-'.
R
(39)
equals -112
?Iph= 7s
kpzR
(I+%) kOR
.
(40)
Fig. 1 shows (a) the dependences [obtained from (32)] of the amplification factor in the wake K , and (b) the difference between the group and phase velocities Sv normalized to Svo = c / ( l c , x ~ )as functions of time for akPL = 1. It can be seen that the early stage of the instability ( K , N 1, i.e., for chosen parameters, it corresponds to T < 20) is characterized by strong pulsations of K , and Sv. Note that the similar pulsations of the wakefield were observed in numerical simulations [13], [14]. At the late stage ( K , >> I), Sv approaches the value determined by (39). In the strong-coupling case ( a k p L >> R2), for T < ak,L/R3 [see (36) and (37)], we obtain
1 is a weakly varying function of the Here, F(K,) amplification factor which reaches its maximum equal to 1.94 at K , P 1.23 and then falls very slowly with an increase in K , (as 33/4(1nK,)-1/2; see Fig. 2). N
z (b) Fig. 3. The same dependences as in Fig. 1 in the case of strong coupling for ak,L = 100.
Fig. 3 shows the time dependences [obtained from (35)] of c ) a k p L = 100. In contrast to (a) K , and (b) ( , k p . z ~ ) ( S v /for the weak-coupling (channel-guiding) case (see Fig. l), there is a smooth transition (without oscillations of K , and Sv) from the early to the late stage of the instability. This transition is accompanied by a steady growth of the phase velocity of the wake wave. Our results were obtained for the initial and boundary conditions (27) corresponding to a pulse with a sharp leading edge. We note that a variation in the initial and boundary conditions (for example, when the instability is triggered by a second weak-intensity frequency-shifted laser pulse [131) mainly affects the results for the early stage of the instability; whereas in the late stage, the behavior of the solutions weakly changes.
368
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 24, NO. 2, APRIL 1996
IV. CONCLUSIONS Our resultsrelated to the linear stage of the RSMI, which are obtained in the paraxial approximation, can be summarized as follows. _. There are two typical regimes of RSMI that differ in the characteristic time of the instability onset; this time can be either less (if ak,L 5 1)or greater (when ak,L >> 1) than Z R / C (weak and strong coupling, respectively). During the onset of the instability, the phase velocity of the excited wake wave vph changes, and the maximum variations in ?+h correspond to the early stage of the instability, i.e., when the amplification factor is small: K, 1. At the late stage of the instability ( K , >> I), ‘vph is always less than the pulse group velocity vs for both instability regimes. For 7,” > SZR, at the late stage of the instability, Y p h is substantiallv less than Y,. During the transition to the late stage of the instability, in the strong coupling case, v p h grows steadily. For the case of weak coupling, we carried out numerical simulations of the linear stage of RSMI for a laser pulse propagating in a plasma channel [using (2) and (3)]. The parameters were chosen close to those in [9], [13], and [14]: ys = 50, k p z 2 ~ 5oR = 400,a = 0.1, and k,L < 40. The deviation of the phase velocity from ug was AV c/2@@,i.e., Y p h N yg/5. This is in good agreement with the analytical prediction given by (39) and (40). In conditions of a numerical experiment [9], [5] ( a ‘v 2.5 and k P L N 20, strong coupling), for yg = 22.5, k , z ~N 4500, assuming that F 1.0-0.7 (which corresponds to the late 20 t lo4), we can find that at the stage of the RSMI, K , linear stage of instability yph is approximately 20-30% lower than ys.This agrees with the simulation results presented in [9, Fig. 6(b)]. The reduction in the phase velocity of the wake wave (compared to vg) can be important for particle acceleration. Our results indicate that because of the decrease in the acceleration length I , = 27rkp1y&, the actual acceleration may occur only over the distance not exceeding ZR [see (40) and (41)]. However, because our analysis is limited to the linear stage of the instability, the verification of this conclusion needs a study of the nonlinear stage of the RSMI. The other fact, which is interesting for particle acceleration, relates to the transitional stage of the RSMI in the strong coupling case when V p h increases [see Figs. 2 and 3 together with (41)], and an improvement of the conditions for the particle trapping in the plasma wave is possible. It is interesting to compare the instability considered (which is related to a transverse redistribution of the pulse energy in each pulse cross section) with the similar instability that exists in the one dimensional (1-D) approximation (which also leads to a longitudinal modulation of the pulse and strong wakefield excitation, but is related to an axial transport of energy) and can be associated with direct-forward Raman scattering [16], [17]. For description of this 1-D instability, the term with a cross-derivative 2c(d2/dz’dt’)a should be incorporated in (2). N
‘Y
N
N
In our analysis, this term is assumed to be less than the term c 2 A i , and this instability is ignored. The direct comparison of analytical solutions describing the RSMI of the pulse with a sharp leading edge in the 1-D [17] and in the paraxial (35) approximation shows that the 1-D mechanism for the RSMI is unimportant for the sufficiently narrow laser pulse
~;L:,A; < 16
7,”
5, L + A i 2 .
REFERENCES
[I] T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett., vol. 43, pp. 267-270, 1979. [2] J. S. Wurtele, “The role of plasma in advanced accelerators,” Phys. Fluids B, vol. 5, pp. 2363-2370, July 1993. [3] H. H. Kuehl, C. Y . Zhang, and T. Katsouleas, “Interaction of weakly nonlinear laser pulse with a plasma,” Phys. Rev. E, vol. 47, pp. 1249-126 1, 1993. [4] S. V. Bulanov, V. I. Kirsanov, and A. S. Sakharov, “Limiting electric field of the wakefield plasma wave,” Sov. JETP Lett., vol. 53, no. 11, pp. 565-569, 1991. [5] N. E. Andreev, L. M. Gorbunov, V. I. Kirsanov, A. A. Pogosova, and R. R. Ramazashvili, “Resonant excitation of wake fields by a laser pulse in a plasma,” Sov. JETP Lett., vol. 55, no. 10, pp. 571-576, 1992. [6] P. Sprangle, E. Esarey, J. Krall, and G. Joyce, “Propagation and guiding of intense laser pulses in plasmas,” Phys. Rev. Lett., vol. 69, pp. 2200-2203, 1992. [7] T. M. Antonsen, Jr. and P. Mora, “Self-focusing and Raman scattering of laser pulses in tenuous plasmas,’’ Phys. Rev. Lett., vol. 69, pp. 2204-2207, 1992. [8] __ , “Self-focusing and Raman scattering of laser pulses in tenuous plasma,” Phys. Fluids B , vol. 5, pp. 1440-1452, 1993. [9] N. E. Andreev, L. M. Gorbunov, V. I. Kirsanov, A. A. Pogosova, and R. R. Ramazashvili, “The theory of laser self-resonant wakefield excitation,” Physica Scripta, vol. 49, pp. 101-109, 1994. [lo] E. Esarey, J. Krall, and P. Sprangle, “Envelope analysis of intense laser pulse self-modulation in plasmas,” Phys. Rev. Lett., vol. 72, pp. 2887-2890, 1994. [ 1 I ] K. Nakajima et al., “Observation of ultrahigh gradient electron acceleration by a self-modulated intense short laser pulse,” KEK Preprint 94-179: KEK, Japan, 1995; to appear in Phys. Rev. Lett. [I21 J. Krall, A. Ting, E. Esarey, and P. Sprangle, “Enhanced acceleration in a self-modulated-laser wakefield accelerator,” Phys. Rev. E, vol. 48, pp. 2157-2161, 1993. [13] N. E. Andreev, V. I. Kirsanov, A. A. Pogosova, and L. M. Gorbunov, “Laser wakefield accelerator in a plasma pipe with self-modulation of the laser pulse,” Sov. JETP Lett., vol. 60, no. 10, pp. 713-717, 1994. [14] N. E. Andreev, L. M. Gorbunov, and V. I. Kirsanov, “Stimulated processes and self-modulation of a short intense laser pulse in the laser wakefield accelerator,” Phys. Plasmas, vol. 2, no. 6, pp. 2573-2582, 1995. [15] V. V. Goloviznin, P. W. van Amersfoort, N. E. Andreev, and V. I. Kirsanov, “Self-resonant plasma wakefield excitation by laser pulse with steep leading edge for particle acceleration,” Phys. Rev. E., vol. 52, no. 5 , pp. 5327-5332, 1995. [16] W. B. Mori, C. D. Decker, D. E. Hinkel, and T. Katsouleas, “Raman forward scattering of short-pulse high-intensity lasers,” Phys. Rev. Lett., vol. 72, pp. 1482-1484, 1994. [17] A. S. Sakharov and V. I. Kirsanov, “Theory of Raman scattering for a short ultrastrong laser pulse in a rarefied plasma,” Phys. Rev. E, vol. 49, pp. 3274-3282, 1994.
N. E. Andreev, photograph and biography not available at the time of publication.
V. I. Kirsanov, photograph and biography not available at the time of publication.
ANDREEV et al.: LINEAR THEORY OF RESONANCE SELF-MODULATION
L,. M. Gorbunov, photograph and biography not available at the time of publication.
369
A. S. Sakharov, photograph and biography not available at the time of publication.
