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Role of stochastic heating in wakefield acceleration monitored by optical injection
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AID:22048 /SCO Doctopic: Plasma and fluid physics
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Role of stochastic heating in wakefield acceleration monitored by optical injection
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A. Bourdier ∗ , S. Rassou, M. Drouin
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CEA, DAM, DIF, 91297 Arpajon, France
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Article history: Received 12 April 2013 Accepted 21 June 2013 Available online xxxx Communicated by F. Porcelli
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Keywords: Wake field acceleration Stochastic acceleration
It is shown that stochastic heating can play an important role in Laser Wake Field Acceleration. When considering low density plasma interacting with a high intensity wave perturbed by a low intensity counterpropagating wave, stochastic heating can provide electrons with the right momentum for trapping in the wake field. The influence of stochastic acceleration on the trapping of electrons is compared to the one of cold injection by considering several polarizations of the colliding pulses. When the plasma density exceeds some value, stochastic heating becomes important and is necessary in some circumstances to get electrons trapped into the wakefield. © 2013 Published by Elsevier B.V.
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1. Introduction
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In this Letter, the role of stochastic acceleration on electron trapping in laser-wakefield acceleration (LWFA) [1–3] when a counterpropagating pulse is taken into account is explored. We rather select moderate laser intensity I 2 × 1019 W cm−2 and plasma density ne 8 × 1018 cm−3 close to values proposed by Lu et al. [4–8] to achieve controlled and stable blowout of the electrons. In this study, dedicated to the analysis of the mechanisms at stake when the two counterpropagating electromagnetic waves collide, we tend to avoid self-injection into the wake by lowering the pump pulse intensity. The relative influence of stochastic heating and beat wave force on the injection mechanism is discussed. Many different combinations of polarizations can be chosen for both waves, each of these possibilities results in particular forces acting on the plasma electrons, when the two waves collide [9]. The goal of this Letter is to discuss the influence of the different forces in the electron trapping by the accelerating cavity versus the different physical parameters. Two electromagnetic counterpropagating waves with the same frequency are considered, a high intensity one and a perturbing mode. 2Dx3Dv PIC simulations are achieved with code CALDER [10] in which underdense mm-long plasma interact with two counterpropagating laser pulses with the same duration. We focus on the force best suited for injection of electrons into a wakefield, dependence to plasma density is explored. For a given length of interaction of the two waves, at rather high plasma densities, it is found that the case when the two waves are linearly polarized in the same plane is more efficient to trap particles than the case
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Corresponding author. Tel.: +33169266137. E-mail address: [email protected] (A. Bourdier).
0375-9601/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physleta.2013.06.033
when they are circularly polarized and rotating in the same direction. When one is close or below the cold injection threshold [11] and above threshold for stochastic acceleration, the trapped charge when the two waves are linearly polarized the trapped charge is far more important than in the other case. Moreover, when the intensity of the high intensity wave is higher, the trapped charge can be higher when they are linearly polarized. This is no longer true at lower densities. A single particle approach is presented first. Then, preliminary PIC code simulations results are given, some of these results are confirmed by a theoretical model next. Finally, it is shown in a discussion that this problem depends on two main parameters.
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2. Influence of stochastic heating compared to the one of the stationary force on electron trapping in the wakefield
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2.1. Single particle approach
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When the two counterpropagating waves are linearly polarized in the same plane (P-linear polarizations), the normalized vector potentials of the two waves can be assumed to be given by: aˆ = a cos(tˆ − zˆ )ˆex and aˆ 1 = −a1 sin(tˆ + zˆ )ˆex ( zˆ = k0 z, tˆ = ω0 t). When the generalized momentum is such that Pˆ x = Pˆ y = 0
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( Pˆ x, y ,z = P x, y ,z /mc). The normalized force acting on the charged
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particle in the direction of propagation of the waves is given by ˆ /∂ zˆ , where Hˆ is the normalized Hamiltonian of an d Pˆ z /dtˆ = −∂ H electron in the two waves. One has
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sin 2(tˆ + zˆ ).
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2 ˆ ˆf z = d P z = − aa1 cos 2 zˆ − a sin 2(tˆ − zˆ ) γ 2γ dtˆ
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(1)
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It contains a beatwave term: ˆf zB W = −(aa1 /γ ) cos 2 zˆ and other terms which render the problem chaotic, electrons are accelerated to much higher velocities in the z-direction due to the stochastic process (Fig. 1). This situation is, a priori, the most promising to trap electrons in the wake field. The case when the perturbing mode is polarized perpendicularly to the polarization plane of the high intensity wave is also considered (S-linear polarizations). The normalized vector potentials of the two waves are assumed to be: aˆ = a cos(tˆ − zˆ )ˆex , and aˆ 1 = a1 sin(tˆ + zˆ )ˆe y . In this case, when Pˆ x = Pˆ y = 0, the normalized transverse mechanical momentum contains no first-order term as aˆ .ˆa1 = 0. These polarizations rule out the electron acceleration in the z-direction due to the beatwave force. Still, some stochastic heating due to the other terms can be found when considering a single particle. The situation when the two waves are circularly polarized is also considered. Two cases can be considered: the two wave vectors can rotate in the same direction (positive circular polarizations: C+ polarizations) or in two opposite directions (negative circular polarizations: C− polarizations). In both cases the problem is integrable (Appendix A). When the two vectors rotate in two opposite directions, there is no force accelerating particles in the direction of propagation of the high intensity wave (Appendix A) (Fig. 1). When the two vectors rotate in the same direction, the beatwave force ˆf zB W = a1 a/γ sin 2 zˆ (Appendix A) is the only one to accelerate the electrons in the direction of propagation of the wave. This term can be quite efficient to pre-accelerate plasma background electrons as its spatial scale is λ0 /2, where λ0 is the laser wave length [9–12] (Fig. 1).
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2.2. Plasma PIC code simulations
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Fig. 1. Coordinates of the mechanical momentum. a = 3, a1 = 0.3. (a) C− polarizations, (b) C+ polarizations, (c) P-linear polarizations, (d) S-linear polarizations.
2.2.1. Preliminary results Two 30 fs laser pulses with wavelength λ = 0.8 μm are considered now. The waves interact with mm-size plasma. The pump pulse which creates the accelerating wakefield is focused to a 18 μm full wiδ τ h at half maximum (FWHM). The low intensity pulse is counterpropagating and is focused to a 31 μm focal spot. In the first place, it is assumed that the pump pulse interacts with plasma with a density: ne = 4.3 × 10−3 nc , where nc is the plasma critical density. The role of stochastic acceleration is studied by comparing the effect of the collision of two linearly polarized waves (P- and S-linear polarizations) to the one of two circularly polarized lasers considering the cases of positive and negative circular polarizations. To do so, 2D PIC code simulations were performed. The electron energy distribution function is calculated long after the collision of the two waves. It was assumed first that a1 is above the threshold for stochastic heating [16–20]. In the case when the waves are circularly polarized, the problem
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Fig. 2. Electron energy distribution obtained above the stochastic threshold value for a1 . ne = 4.3 × 10−3 nc . (a) a = 1.5, a1 = 0.4, (b) a = 2, a1 = 0.4.
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of a single electron interacting with them is integrable, as a consequence, no stochastic acceleration is expected when the two waves propagate in low density plasma. In C− polarizations cases and in the S-linear polarizations case, simulation results show that almost no electron is trapped and accelerated in the wakefield (Fig. 2). In these situations there is no beatwave and the stochastic heating in the S polarizations case is very weak. On the contrary, a significant charge is injected and consequently accelerated in the P-linear polarizations case (Fig. 2(a)). In good agreement with previously published theoretical work, the C+ polarized waves do not trap any charge when a 2 (value below the cold electron trapping: a = 2) [11] (Fig. 2). At this plasma density a significant charge can be trapped in the P polarizations case when a1 is above the threshold. Then, the P polarizations case plays a major role. In this case, stochastic heating provides electrons with enough momentum for trapping in the wake field. Most particles are trapped after passing through the front of the accelerating cavity. In the case when a = 2 and a1 = 0.4, the phase plot which has been perform just at the end of the collision of the two laser pulses do show that one has more particles with a high momentum in the direction of propagation of the wave in the P polarizations case (Fig. 3). When a1 is below the threshold, no charge is trapped in the P polarizations case and in the C+ polarizations one. When a = 1.5 (value below the cold electron trapping) and a1 = 0.05 (value below the threshold), no charge is trapped in the P or C+ polarizations cases. When a = 2 and a1 = 0.05, just about the same small charge is trapped in both polarizations case. When a > 2, the P polarizations case prevails when a1 is above the threshold for stochastic heating (Fig. 4). With the P polarization the trapped charge is QP = 25 pC/μm while it is QC+ = 17 pC/mm with the C+ polarizations.
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Fig. 3. Mechanical momentum density. ne = 4.3 × 10−3 nc . a = 2, a1 = 0.4. (a) P polarizations case, (b) C+ polarizations case.
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Fig. 5. Electron energy distribution. a = 3, ne = 2.5 × 10−4 nc . (a) a1 = 0.3 (value above the threshold value), (b) a1 = 0.05 (value below the threshold value).
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A lower density is considered next, it is assumed that ne = 2.5 × 10−4 nc . The effects of two polarizations are compared in Fig. 5. The C+ polarizations case can even be more efficient than the P-linear polarizations case. In this situation, cold injection [11,21] due to the beating force only can diphase electrons and trigger their injection into the wake. This is true above and below the threshold value in a1 for stochastic heating. At this low density, simulations have shown that when a = 1.5 no particle is trapped in the case of C+ polarized waves (this value of a is below the cold injection threshold). When a1 is above the stochastic threshold a small charge is trapped when the waves are linearly polarized. We shall only consider a pump pulse intensity a = 3 above the threshold for cold electron trapping (a 2) further. The transverse momentum distribution (Fig. 6) shows that when stochastic heating is triggered, one has more electrons with a high transverse momentum.
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Fig. 4. Electron energy distribution. a = 3, ne = 4.3 × 10−3 nc . (a) a1 = 0.05, (b) a1 = 0.3.
2.2.2. Test particles approach A statistical survey of electron trapping was made in this case. Considering 260 test particles with very close initial conditions it was shown that 16.8% are trapped after interacting with two circularly polarized waves while only 7.25% are trapped in the P polarizations case. When one has a lot of stochastic acceleration, particles which interact with the two waves can take a strong transverse momentum. If the density of the bubble is too low, many electrons do not make their way along the rim of the bubble
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Fig. 8. Trajectories of trapped test particles in the accelerating cavity. The red particles interact with two P polarized waves while the green particles interact with two C+ polarized waves. The slow-evolving electron density map is plotted as background. a = 3, a1 = 0.3, ne = 2.5 × 10−4 nc . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
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Fig. 6. Transverse momentum distribution. a = 3, a1 = 0.3, ne = 2.5 × 10−4 nc . (a) Plinear polarizations, (b) C+ polarizations.
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Fig. 7. Motion of two test particles in the vicinity of the bubble with the same initial conditions. The red particle interacts with two P polarized waves while the green one interacts with two C+ polarized waves. The slow-evolving electron density map is plotted as background. a = 3, a1 = 0.3, ne = 2.5 × 10−4 nc . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
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as they are not dragged back well enough by the space charge field created by the electron free cavity. Fig. 7 shows a test particle which is trapped in the bubble when it interacts with the two circularly polarized waves while it is not in the other situation. Fig. 8 presents trajectories of test particles in the wake frame passing through the front of the bubble before being trapped. The test particles are initially at rest, they are arranged in a volume which is smaller than the one of interaction of the two waves. Fig. 8 brings to light that more particles are trapped after passing through the front of the bubble when interacting with two
Fig. 9. Electron trajectory. R = 27.9 μm, d = 0.5. (a) P polarizations, (b) C+ polarizations.
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circularly polarized waves. The longitudinal electric field maximum value is higher on the propagation axis. Off-axis injected electrons with large transverse momenta will take more time to be brought back by the transverse electric field. They will not get a high enough longitudinal momentum to be trapped into the wake. This mechanism explains the results highlighted by Figs. 5 and 8.
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2.2.3. Theoretical model Kostyukov’s model has also been used to confirm these results [22,23] (Fig. 8). Two particles with two different initial transverse momentum were chosen in the momentum distributions (Fig. 6). The two values were chosen close to the maximum values for the P and C+ polarizations cases respectively (Fig. 9). This model says, once more, that the charged particle is trapped only after interacting with two C+ polarized waves.
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2.2.4. Discussion We are going to describe the transition between a P and a C+ dominated regime. Two parameters take part in this transition: the plasma density ne and ρ = L/R where L is the interaction length of the two laser pulses and R the radius of the accelerating cavity. The pulse duration is denoted by t. When ne = 4.3 × 10−3 nc (ρ = 1.34), in the P polarizations case, the trapped charge increases with a1 (Fig. 10(a)) because ρ is high which means that the length of interaction is close to the radius of the bubble. Then, many particles undergo stochastic acceleration inside the bubble (Fig. 11(a)) facilitating their trapping. For high densities, in the C+ polarizations case, the beatwave force is the only one responsible for the entry of electrons through the front of the bubble (Appendix A). Wake inhibition increases with a1 and the density, and this phenomenon is stronger in the C+ case [11,24]. Then, the electrostatic field inside the cavity is not strong enough for trapping electrons. This explains why the trapped charge stagnates or decreases when a1 is increased. In the P polarizations case, a part of the electrons are not trapped by the stationary wave and undergo stochastic heating (Fig. 11(a)). These electrons are easily trapped in the accelerating cavity. Fig. 10(b) describes a situation (a = 3, ne = 2.5 × 10−3 nc , ρ ≈ 1) which is close to the transition between the two regimes. At low density ne < 2.5 × 10−3 nc when t = 30 fs then ρ 1, therefore the situation becomes more favorable to cold injection. In fact, the number of particles trapped and accelerated is higher in the C+ polarizations case (Fig. 10(c), (d)). In the P polarizations case, some electrons at the front of the bubble are lost due to the longitudinal decelerating electric field. Moreover many off-axis electrons undergoing stochastic acceleration gain a high transverse momentum (Fig. 6) which may prevent their trapping (Fig. 8(a)). This transverse effect is detrimental for the P polarizations case. The C+ polarizations case becomes more efficient as more particles get dephased in the stationary wave and less charge is lost due to the transverse dynamics (Fig. 8(b)). In this range of densities there is almost no wakefield inhibition (Fig. 12(b)). The importance of parameter ρ is pointed out in Fig. 10(e). When the two pulses have a longer duration ( t = 60 fs), then parameter ρ increases and the P polarizations case becomes more efficient just like in the high density case with a shorter duration. Fig. 11 displays the momentum density distribution in the P polarizations case just after the collision of the two waves. At low density, it shows that the heated electrons stand at the front of the bubble (Fig. 11(b)). In this region electrons are decelerated and may be lost for trapping. This situation contrasts with high density, then electrons are spread in all the bubble (Fig. 11(a)). It has also been observed that when a high charge is trapped, the beam modifies the transverse electric field inside the wake, then the bubble shape can be altered which might result in a weaker accelerating field for the trapped beam [25–27].
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3. Conclusion
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The problem of injection of electrons in the wake field created by a moderately high intensity wave perturbed by a counterpropagating wave has been studied in this Letter. The role of stochastic acceleration has been discussed. The effect on the trapped charge in LWFA of stochastic acceleration has been compared to the one of the beatwave force. Several polarizations were considered in order to study the influence of the different forces applied to electrons in the direction of propagation of the wave. A transitional regime has been observed through PIC code simulations. For a given length of the two laser pulses, at rather high
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plasma densities, two linearly polarized waves in the same plane lead to more particles trapped than in the case when the two waves are circularly polarized (and rotating in the same direction). The difference of trapped charge with the other case is spectacular when one is close or below the cold injection threshold and above the threshold for stochastic acceleration. When the intensity of the high intensity wave is higher, the trapped charge can still be higher when the waves are linearly polarized. For lower density cold injection leads to higher charges injected in the accelerating cavity.
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Acknowledgement
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We acknowledge stimulating discussions with Dr. X. Davoine.
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Appendix A
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Let us consider a single electron interacting with two circularly polarized waves propagating in two opposite directions. Two situations interesting can be considered. One can consider first two plane waves with their wave vectors rotating in the same direction
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aˆ 1 = √ cos(tˆ + zˆ )ˆex + √ sin(tˆ + zˆ )ˆe y . 2 2
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This is called the positive circular polarizations. The normalized Hamiltonian of the electron reads
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This is a three-dimensional problem with two obvious constants of ˆ is a motion, Pˆ x and Pˆ y . It can be shown easily that yˆ Pˆ x − xˆ Pˆ y + H third constant of motion. As these three constants are independent and in involution the problem is integrable, no chaotic trajectory can be found and no stochastic acceleration can take place. When Pˆ x = Pˆ y = 0, one has
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ˆ. This is a one-dimensional problem with one constant of motion H The force which can pre-accelerate the electron to trap it into the wake field is given by
∂ hˆ a1 a =− = sin 2 zˆ . ∂ zˆ γ dtˆ
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(A.3)
This beatwave force which accelerates the electrons in the z-direction, can be enough, in some circumstances, to trap particles in the wake field. The situation when the wave vectors are rotating in two opposite directions can also be considered (negative circular polarization), then, the wave vector √ can be chosen in the √ following way: for the main wave: aˆ = (a/ 2 ) cos(tˆ − zˆ )ˆex + (a/ 2 ) sin(tˆ − zˆ )ˆe y ;
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Fig. 10. Electron charge trapped in the cavity versus a1 . a = 3. (a) t = 30 fs, ne = 4.3 × 10−3 nc , ρ = 1.34. (b) t = 30 fs, ne = 2.5 × 10−3 nc , ne = 1 × 10−3 nc , ρ = 0.65. (d) t = 30 fs, ne = 2.5 × 10−4 nc , ρ = 0.32. (e) t = 60 fs, ne = 1 × 10−3 nc , ρ = 1.29.
ρ = 1.02. (c) t = 30 fs,
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This three-dimensional problem has three constants of motion, Pˆ x , Pˆ y and yˆ Pˆ x − xˆ Pˆ y + Pˆ z which are independent and in involution. As a consequence, the problem is integrable, and no stochastic acceleration can take place. When Pˆ x = Pˆ y = 0, one has
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Fig. 11. Longitudinal momentum distribution. P polarizations. a = 3, a1 = 0.5. (a) ne = 4.3 × 10−3 nc , ρ = 1.34. (b) ne = 2.5 × 10−4 nc , ρ = 0.32.
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Fig. 12. Longitudinal electrostatic field on the axis before the collision (tˆ = 2400) and during the collision (tˆ = 2640). C+ polarizations case. a = 3, a1 = 0.5. (a) ne = 4.3 × 10−3 nc . (b) ne = 2.5 × 10−4 nc .
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√
and for the√low intensity perturbing wave: aˆ 1 = −(a1 / 2 ) cos(tˆ + zˆ )ˆex + (a1 / 2 ) sin(tˆ + zˆ )ˆe y . In this case the Hamiltonian reads
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References
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[13] [14] [15]
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Uncited references
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This is a time-dependent one-dimensional problem with one constant of motion Pˆ z . As a consequence this problem is integrable. There is no force to pre-accelerate the electrons in the z-direction and no stochastic acceleration takes place.
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hˆ =
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[1] T. Tajima, J.M. Dawson, Phys. Rev. Lett. 43 (1979) 267. [2] J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J.-P. Rousseau, F. Burgy, V. Malka, Nature 431 (2004) 541. [3] S.P.D. Mangles, C.D. Murphy, Z. Najmudin, A.G.R. Thomas, J.L. Collier, A.E. Dangor, E.J. Divall, P.S. Foster, J.G. Gallacher, C.J. Hooker, D.A. Jaroszynski, A.J. Langley, W.B. Mori, P.A. Norreys, F.S. Tsung, R. Viskup, B.R. Walton, K. Krushelnick, Nature 431 (2004) 535. [4] S.F. Martins, R.A. Fonseca, W. Lu, W.B. Mori, L.O. Silva, Nature 6 (2010) 311. [5] W. Lu, C. Huang, M. Zhou, W.B. Mori, T. Katsouleas, Phys. Rev. Lett. 96 (2006) 165002. [6] W. Lu, M. Tzoufras, C. Joshi, F.S. Tsung, W.B. Mori, J. Vieira, R.A. Fonseca, L.O. Silva, Phys. Rev. ST Accel. Beams 10 (2007) 061301. [7] W. Lu, C. Huang, M. Zhou, M. Tzoufras, F.S. Tsung, W.B. Mori, T. Katsouleas, Phys. Plasmas 13 (2006) 056709. [8] V. Malka, J. Faure, Y.A. Gauduel, E. Lefebvre, A. Rousse, K.T. Phuoc, Nature 4 (2008) 447. [9] X. Davoine, E. Lefebvre, J. Faure, C. Rechatin, A. Lifschitz, V. Malka, Phys. Plasmas 15 (2008) 113102. [10] E. Lefebvre, N. Cochet, S. Fritzler, V. Malka, M.-M. Aléonard, J.-F. Chemin, S. Darbon, L. Disdier, J. Faure, A. Fedotoff, O. Landoas, G. Malka, V. Méot, P. Morel, M. Rabec Le Gloahec, A. Rouyer, Ch. Rubbelynck, V. Tikhonchuk, R. Wrobel, P. Audebert, C. Rousseaux, Nucl. Fusion 43 (2003) 629. [11] X. Davoine, A. Beck, A. Lifschitz, V. Malka, E. Lefebvre, New J. Phys. 12 (2010) 095010. [12] G. Fubiani, E. Esarey, C.B. Schroeder, W.P. Leemans, Phys. Rev. E 70 (2004) 016402. [13] H. Kotaki, S. Masuda, M. Kando, J.K. Koga, Phys. Plasmas 11 (2004) 3296. [14] J. Faure, C. Rechatin, A. Norlin, A. Lifschitz, Y. Glinec, V. Malka, Nature Lett. 444 (2006) 737. [15] X. Davoine, E. Lefebvre, J. Faure, C. Rechatin, A. Lifschitz, V. Malka, Phys. Plasmas 15 (2008) 113102. [16] J.T. Mendoça, Phys. Rev. A 28 (1983) 3592. [17] J.M. Rax, Phys. Fluids B 4 (12) (1992) 3962. [18] A. Bourdier, D. Patin, E. Lefebvre, Phys. D 206 (2005) 1. [19] A. Bourdier, M. Drouin, Laser and Particle Beams 27 (2009) 545. [20] D. Patin, Le chauffage stochastique dans l’interaction laser-plasma à très haut flux, Thèse, Université Paris XI, 2006. [21] X. Davoine, E. Lefebvre, C. Rechatin, J. Faure, V. Malka, Phys. Rev. Lett. 102 (2009) 065001. [22] I. Kostyukov, A. Pukhov, S. Kiselev, Phys. Plasmas 11 (2004) 5256. [23] Alain Bourdier, Sébastien Rassou, Guillaume Girard, Mathieu Drouin, J. Modern Phys. 3 (2012) 1018. [24] C. Rechatin, J. Faure, A. Lifschitz, V. Malka, Phys. Plasmas 14 (2007) 060702. [25] M. Tzoufras, W. Lu, F.S. Tsung, C. Huang, W.B. Mori, T. Katsouleas, J. Vieira, R.A. Fonseca, L.O. Silva, Phys. Rev. Lett. 101 (2008) 145002. [26] C. Rechatin, X. Davoine, A. Lifschitz, A. Ben Ismail, J. Lim, E. Lefebvre, J. Faure, V. Malka, Phys. Rev. Lett. 103 (2009) 194804. [27] C. Rechatin, J. Faure, X. Davoine, O. Lundh, J. Lim, A. Ben-Ismaïl, F. Burgy, A. Tafzi, A. Lifschitz, E. Lefebvre, V. Malka, New J. Phys. 12 (2010) 045023.
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Role of stochastic heating in wakefield acceleration monitored by optical injection
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Article history: Received 12 April 2013 Accepted 21 June 2013 Available online xxxx Communicated by F. Porcelli
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Keywords: Wake field acceleration Stochastic acceleration
It is shown that stochastic heating can play an important role in Laser Wake Field Acceleration. When considering low density plasma interacting with a high intensity wave perturbed by a low intensity counterpropagating wave, stochastic heating can provide electrons with the right momentum for trapping in the wake field. The influence of stochastic acceleration on the trapping of electrons is compared to the one of cold injection by considering several polarizations of the colliding pulses. When the plasma density exceeds some value, stochastic heating becomes important and is necessary in some circumstances to get electrons trapped into the wakefield. © 2013 Published by Elsevier B.V.
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1. Introduction
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In this Letter, the role of stochastic acceleration on electron trapping in laser-wakefield acceleration (LWFA) [1–3] when a counterpropagating pulse is taken into account is explored. We rather select moderate laser intensity I 2 × 1019 W cm−2 and plasma density ne 8 × 1018 cm−3 close to values proposed by Lu et al. [4–8] to achieve controlled and stable blowout of the electrons. In this study, dedicated to the analysis of the mechanisms at stake when the two counterpropagating electromagnetic waves collide, we tend to avoid self-injection into the wake by lowering the pump pulse intensity. The relative influence of stochastic heating and beat wave force on the injection mechanism is discussed. Many different combinations of polarizations can be chosen for both waves, each of these possibilities results in particular forces acting on the plasma electrons, when the two waves collide [9]. The goal of this Letter is to discuss the influence of the different forces in the electron trapping by the accelerating cavity versus the different physical parameters. Two electromagnetic counterpropagating waves with the same frequency are considered, a high intensity one and a perturbing mode. 2Dx3Dv PIC simulations are achieved with code CALDER [10] in which underdense mm-long plasma interact with two counterpropagating laser pulses with the same duration. We focus on the force best suited for injection of electrons into a wakefield, dependence to plasma density is explored. For a given length of interaction of the two waves, at rather high plasma densities, it is found that the case when the two waves are linearly polarized in the same plane is more efficient to trap particles than the case
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*
Corresponding author. Tel.: +33169266137. E-mail address: [email protected] (A. Bourdier).
0375-9601/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physleta.2013.06.033
when they are circularly polarized and rotating in the same direction. When one is close or below the cold injection threshold [11] and above threshold for stochastic acceleration, the trapped charge when the two waves are linearly polarized the trapped charge is far more important than in the other case. Moreover, when the intensity of the high intensity wave is higher, the trapped charge can be higher when they are linearly polarized. This is no longer true at lower densities. A single particle approach is presented first. Then, preliminary PIC code simulations results are given, some of these results are confirmed by a theoretical model next. Finally, it is shown in a discussion that this problem depends on two main parameters.
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2. Influence of stochastic heating compared to the one of the stationary force on electron trapping in the wakefield
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2.1. Single particle approach
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When the two counterpropagating waves are linearly polarized in the same plane (P-linear polarizations), the normalized vector potentials of the two waves can be assumed to be given by: aˆ = a cos(tˆ − zˆ )ˆex and aˆ 1 = −a1 sin(tˆ + zˆ )ˆex ( zˆ = k0 z, tˆ = ω0 t). When the generalized momentum is such that Pˆ x = Pˆ y = 0
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( Pˆ x, y ,z = P x, y ,z /mc). The normalized force acting on the charged
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It contains a beatwave term: ˆf zB W = −(aa1 /γ ) cos 2 zˆ and other terms which render the problem chaotic, electrons are accelerated to much higher velocities in the z-direction due to the stochastic process (Fig. 1). This situation is, a priori, the most promising to trap electrons in the wake field. The case when the perturbing mode is polarized perpendicularly to the polarization plane of the high intensity wave is also considered (S-linear polarizations). The normalized vector potentials of the two waves are assumed to be: aˆ = a cos(tˆ − zˆ )ˆex , and aˆ 1 = a1 sin(tˆ + zˆ )ˆe y . In this case, when Pˆ x = Pˆ y = 0, the normalized transverse mechanical momentum contains no first-order term as aˆ .ˆa1 = 0. These polarizations rule out the electron acceleration in the z-direction due to the beatwave force. Still, some stochastic heating due to the other terms can be found when considering a single particle. The situation when the two waves are circularly polarized is also considered. Two cases can be considered: the two wave vectors can rotate in the same direction (positive circular polarizations: C+ polarizations) or in two opposite directions (negative circular polarizations: C− polarizations). In both cases the problem is integrable (Appendix A). When the two vectors rotate in two opposite directions, there is no force accelerating particles in the direction of propagation of the high intensity wave (Appendix A) (Fig. 1). When the two vectors rotate in the same direction, the beatwave force ˆf zB W = a1 a/γ sin 2 zˆ (Appendix A) is the only one to accelerate the electrons in the direction of propagation of the wave. This term can be quite efficient to pre-accelerate plasma background electrons as its spatial scale is λ0 /2, where λ0 is the laser wave length [9–12] (Fig. 1).
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2.2. Plasma PIC code simulations
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Fig. 1. Coordinates of the mechanical momentum. a = 3, a1 = 0.3. (a) C− polarizations, (b) C+ polarizations, (c) P-linear polarizations, (d) S-linear polarizations.
2.2.1. Preliminary results Two 30 fs laser pulses with wavelength λ = 0.8 μm are considered now. The waves interact with mm-size plasma. The pump pulse which creates the accelerating wakefield is focused to a 18 μm full wiδ τ h at half maximum (FWHM). The low intensity pulse is counterpropagating and is focused to a 31 μm focal spot. In the first place, it is assumed that the pump pulse interacts with plasma with a density: ne = 4.3 × 10−3 nc , where nc is the plasma critical density. The role of stochastic acceleration is studied by comparing the effect of the collision of two linearly polarized waves (P- and S-linear polarizations) to the one of two circularly polarized lasers considering the cases of positive and negative circular polarizations. To do so, 2D PIC code simulations were performed. The electron energy distribution function is calculated long after the collision of the two waves. It was assumed first that a1 is above the threshold for stochastic heating [16–20]. In the case when the waves are circularly polarized, the problem
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Fig. 2. Electron energy distribution obtained above the stochastic threshold value for a1 . ne = 4.3 × 10−3 nc . (a) a = 1.5, a1 = 0.4, (b) a = 2, a1 = 0.4.
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of a single electron interacting with them is integrable, as a consequence, no stochastic acceleration is expected when the two waves propagate in low density plasma. In C− polarizations cases and in the S-linear polarizations case, simulation results show that almost no electron is trapped and accelerated in the wakefield (Fig. 2). In these situations there is no beatwave and the stochastic heating in the S polarizations case is very weak. On the contrary, a significant charge is injected and consequently accelerated in the P-linear polarizations case (Fig. 2(a)). In good agreement with previously published theoretical work, the C+ polarized waves do not trap any charge when a 2 (value below the cold electron trapping: a = 2) [11] (Fig. 2). At this plasma density a significant charge can be trapped in the P polarizations case when a1 is above the threshold. Then, the P polarizations case plays a major role. In this case, stochastic heating provides electrons with enough momentum for trapping in the wake field. Most particles are trapped after passing through the front of the accelerating cavity. In the case when a = 2 and a1 = 0.4, the phase plot which has been perform just at the end of the collision of the two laser pulses do show that one has more particles with a high momentum in the direction of propagation of the wave in the P polarizations case (Fig. 3). When a1 is below the threshold, no charge is trapped in the P polarizations case and in the C+ polarizations one. When a = 1.5 (value below the cold electron trapping) and a1 = 0.05 (value below the threshold), no charge is trapped in the P or C+ polarizations cases. When a = 2 and a1 = 0.05, just about the same small charge is trapped in both polarizations case. When a > 2, the P polarizations case prevails when a1 is above the threshold for stochastic heating (Fig. 4). With the P polarization the trapped charge is QP = 25 pC/μm while it is QC+ = 17 pC/mm with the C+ polarizations.
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Fig. 3. Mechanical momentum density. ne = 4.3 × 10−3 nc . a = 2, a1 = 0.4. (a) P polarizations case, (b) C+ polarizations case.
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Fig. 5. Electron energy distribution. a = 3, ne = 2.5 × 10−4 nc . (a) a1 = 0.3 (value above the threshold value), (b) a1 = 0.05 (value below the threshold value).
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A lower density is considered next, it is assumed that ne = 2.5 × 10−4 nc . The effects of two polarizations are compared in Fig. 5. The C+ polarizations case can even be more efficient than the P-linear polarizations case. In this situation, cold injection [11,21] due to the beating force only can diphase electrons and trigger their injection into the wake. This is true above and below the threshold value in a1 for stochastic heating. At this low density, simulations have shown that when a = 1.5 no particle is trapped in the case of C+ polarized waves (this value of a is below the cold injection threshold). When a1 is above the stochastic threshold a small charge is trapped when the waves are linearly polarized. We shall only consider a pump pulse intensity a = 3 above the threshold for cold electron trapping (a 2) further. The transverse momentum distribution (Fig. 6) shows that when stochastic heating is triggered, one has more electrons with a high transverse momentum.
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Fig. 4. Electron energy distribution. a = 3, ne = 4.3 × 10−3 nc . (a) a1 = 0.05, (b) a1 = 0.3.
2.2.2. Test particles approach A statistical survey of electron trapping was made in this case. Considering 260 test particles with very close initial conditions it was shown that 16.8% are trapped after interacting with two circularly polarized waves while only 7.25% are trapped in the P polarizations case. When one has a lot of stochastic acceleration, particles which interact with the two waves can take a strong transverse momentum. If the density of the bubble is too low, many electrons do not make their way along the rim of the bubble
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Fig. 8. Trajectories of trapped test particles in the accelerating cavity. The red particles interact with two P polarized waves while the green particles interact with two C+ polarized waves. The slow-evolving electron density map is plotted as background. a = 3, a1 = 0.3, ne = 2.5 × 10−4 nc . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
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Fig. 6. Transverse momentum distribution. a = 3, a1 = 0.3, ne = 2.5 × 10−4 nc . (a) Plinear polarizations, (b) C+ polarizations.
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Fig. 7. Motion of two test particles in the vicinity of the bubble with the same initial conditions. The red particle interacts with two P polarized waves while the green one interacts with two C+ polarized waves. The slow-evolving electron density map is plotted as background. a = 3, a1 = 0.3, ne = 2.5 × 10−4 nc . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
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as they are not dragged back well enough by the space charge field created by the electron free cavity. Fig. 7 shows a test particle which is trapped in the bubble when it interacts with the two circularly polarized waves while it is not in the other situation. Fig. 8 presents trajectories of test particles in the wake frame passing through the front of the bubble before being trapped. The test particles are initially at rest, they are arranged in a volume which is smaller than the one of interaction of the two waves. Fig. 8 brings to light that more particles are trapped after passing through the front of the bubble when interacting with two
Fig. 9. Electron trajectory. R = 27.9 μm, d = 0.5. (a) P polarizations, (b) C+ polarizations.
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circularly polarized waves. The longitudinal electric field maximum value is higher on the propagation axis. Off-axis injected electrons with large transverse momenta will take more time to be brought back by the transverse electric field. They will not get a high enough longitudinal momentum to be trapped into the wake. This mechanism explains the results highlighted by Figs. 5 and 8.
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2.2.3. Theoretical model Kostyukov’s model has also been used to confirm these results [22,23] (Fig. 8). Two particles with two different initial transverse momentum were chosen in the momentum distributions (Fig. 6). The two values were chosen close to the maximum values for the P and C+ polarizations cases respectively (Fig. 9). This model says, once more, that the charged particle is trapped only after interacting with two C+ polarized waves.
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2.2.4. Discussion We are going to describe the transition between a P and a C+ dominated regime. Two parameters take part in this transition: the plasma density ne and ρ = L/R where L is the interaction length of the two laser pulses and R the radius of the accelerating cavity. The pulse duration is denoted by t. When ne = 4.3 × 10−3 nc (ρ = 1.34), in the P polarizations case, the trapped charge increases with a1 (Fig. 10(a)) because ρ is high which means that the length of interaction is close to the radius of the bubble. Then, many particles undergo stochastic acceleration inside the bubble (Fig. 11(a)) facilitating their trapping. For high densities, in the C+ polarizations case, the beatwave force is the only one responsible for the entry of electrons through the front of the bubble (Appendix A). Wake inhibition increases with a1 and the density, and this phenomenon is stronger in the C+ case [11,24]. Then, the electrostatic field inside the cavity is not strong enough for trapping electrons. This explains why the trapped charge stagnates or decreases when a1 is increased. In the P polarizations case, a part of the electrons are not trapped by the stationary wave and undergo stochastic heating (Fig. 11(a)). These electrons are easily trapped in the accelerating cavity. Fig. 10(b) describes a situation (a = 3, ne = 2.5 × 10−3 nc , ρ ≈ 1) which is close to the transition between the two regimes. At low density ne < 2.5 × 10−3 nc when t = 30 fs then ρ 1, therefore the situation becomes more favorable to cold injection. In fact, the number of particles trapped and accelerated is higher in the C+ polarizations case (Fig. 10(c), (d)). In the P polarizations case, some electrons at the front of the bubble are lost due to the longitudinal decelerating electric field. Moreover many off-axis electrons undergoing stochastic acceleration gain a high transverse momentum (Fig. 6) which may prevent their trapping (Fig. 8(a)). This transverse effect is detrimental for the P polarizations case. The C+ polarizations case becomes more efficient as more particles get dephased in the stationary wave and less charge is lost due to the transverse dynamics (Fig. 8(b)). In this range of densities there is almost no wakefield inhibition (Fig. 12(b)). The importance of parameter ρ is pointed out in Fig. 10(e). When the two pulses have a longer duration ( t = 60 fs), then parameter ρ increases and the P polarizations case becomes more efficient just like in the high density case with a shorter duration. Fig. 11 displays the momentum density distribution in the P polarizations case just after the collision of the two waves. At low density, it shows that the heated electrons stand at the front of the bubble (Fig. 11(b)). In this region electrons are decelerated and may be lost for trapping. This situation contrasts with high density, then electrons are spread in all the bubble (Fig. 11(a)). It has also been observed that when a high charge is trapped, the beam modifies the transverse electric field inside the wake, then the bubble shape can be altered which might result in a weaker accelerating field for the trapped beam [25–27].
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The problem of injection of electrons in the wake field created by a moderately high intensity wave perturbed by a counterpropagating wave has been studied in this Letter. The role of stochastic acceleration has been discussed. The effect on the trapped charge in LWFA of stochastic acceleration has been compared to the one of the beatwave force. Several polarizations were considered in order to study the influence of the different forces applied to electrons in the direction of propagation of the wave. A transitional regime has been observed through PIC code simulations. For a given length of the two laser pulses, at rather high
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plasma densities, two linearly polarized waves in the same plane lead to more particles trapped than in the case when the two waves are circularly polarized (and rotating in the same direction). The difference of trapped charge with the other case is spectacular when one is close or below the cold injection threshold and above the threshold for stochastic acceleration. When the intensity of the high intensity wave is higher, the trapped charge can still be higher when the waves are linearly polarized. For lower density cold injection leads to higher charges injected in the accelerating cavity.
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Acknowledgement
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We acknowledge stimulating discussions with Dr. X. Davoine.
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Appendix A
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Let us consider a single electron interacting with two circularly polarized waves propagating in two opposite directions. Two situations interesting can be considered. One can consider first two plane waves with their wave vectors rotating in the same direction
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This is called the positive circular polarizations. The normalized Hamiltonian of the electron reads
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This is a three-dimensional problem with two obvious constants of ˆ is a motion, Pˆ x and Pˆ y . It can be shown easily that yˆ Pˆ x − xˆ Pˆ y + H third constant of motion. As these three constants are independent and in involution the problem is integrable, no chaotic trajectory can be found and no stochastic acceleration can take place. When Pˆ x = Pˆ y = 0, one has
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This beatwave force which accelerates the electrons in the z-direction, can be enough, in some circumstances, to trap particles in the wake field. The situation when the wave vectors are rotating in two opposite directions can also be considered (negative circular polarization), then, the wave vector √ can be chosen in the √ following way: for the main wave: aˆ = (a/ 2 ) cos(tˆ − zˆ )ˆex + (a/ 2 ) sin(tˆ − zˆ )ˆe y ;
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Fig. 10. Electron charge trapped in the cavity versus a1 . a = 3. (a) t = 30 fs, ne = 4.3 × 10−3 nc , ρ = 1.34. (b) t = 30 fs, ne = 2.5 × 10−3 nc , ne = 1 × 10−3 nc , ρ = 0.65. (d) t = 30 fs, ne = 2.5 × 10−4 nc , ρ = 0.32. (e) t = 60 fs, ne = 1 × 10−3 nc , ρ = 1.29.
ρ = 1.02. (c) t = 30 fs,
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This three-dimensional problem has three constants of motion, Pˆ x , Pˆ y and yˆ Pˆ x − xˆ Pˆ y + Pˆ z which are independent and in involution. As a consequence, the problem is integrable, and no stochastic acceleration can take place. When Pˆ x = Pˆ y = 0, one has
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Fig. 11. Longitudinal momentum distribution. P polarizations. a = 3, a1 = 0.5. (a) ne = 4.3 × 10−3 nc , ρ = 1.34. (b) ne = 2.5 × 10−4 nc , ρ = 0.32.
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Fig. 12. Longitudinal electrostatic field on the axis before the collision (tˆ = 2400) and during the collision (tˆ = 2640). C+ polarizations case. a = 3, a1 = 0.5. (a) ne = 4.3 × 10−3 nc . (b) ne = 2.5 × 10−4 nc .
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√
and for the√low intensity perturbing wave: aˆ 1 = −(a1 / 2 ) cos(tˆ + zˆ )ˆex + (a1 / 2 ) sin(tˆ + zˆ )ˆe y . In this case the Hamiltonian reads
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References
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[13] [14] [15]
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Uncited references
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This is a time-dependent one-dimensional problem with one constant of motion Pˆ z . As a consequence this problem is integrable. There is no force to pre-accelerate the electrons in the z-direction and no stochastic acceleration takes place.
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a21
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hˆ =
2 a1 ˆP x + √a cos(tˆ − zˆ ) − √ ˆ cos(t + zˆ ) 2
2
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