Raman-seeded laser wakefield acceleration

Transcription

1 Raman-seeded laser wakefield acceleration Mykhailo Fomytskyi, Charles Chiu and Michael Downer Department of Physics, University of Texas at Austin Austin, Texas 7871 February 1, Introduction Because electrostatic fields in a relativistic plasma wave can exceed those in rf linacs by 3 orders of magnitude, plasma- based accelerators can provide a compact source of high energy electron bunches [Tajima79]. Of the various methods of driving large-amplitude plasma waves [Esarey96], the self-modulated laser wakefield accelerator (SM-LWFA) [Sprangle9, Antonsen9, Andreev9] has so far yielded electron bunches of the highest energy (tails to > 00 MeV [Fritzler0]), charge (> 1 nc/bunch [Wagner97, Leemans01]) and collimation (transverse emittance ɛ < 0.01pmm mrad [Chen99]). In the SM-LWFA the plasma density n e is chosen such that the duration τ p of the driving pulse (typically 0.1 < τ p < 1 ps) significantly exceeds the plasma period ωp 1 = m/(4πn e e? (typically < n e < 10 0 cm 3 ). If, in addition, the power of the driving pulse exceeds the critical power P c = 17(ω0 /ω p)gw for relativistic selffocusing [Max74], a forward Raman scattering (FRS) instability - i.e. conversion of the EM wave (ω 0, k 0 ) into a plasma wave (ω p, k p ) and Stokes (ω 0 ω p, k 0 k p ) (?0 -?p, k0 - kp) and anti- Stokes(ω 0 + ω p, k 0 + k p ) EM sidebands - can grow [Sprangle9, Antonsen9, Andreev9]. Under these conditions, a collimated beam of multi- MeV electrons, resulting from capture and acceleration of electrons from the surrounding plasma by the plasma wave, has been observed in several laboratories [Modena95, Ting97, Wagner97, Malka97, Leemans01]. The beam properties achieved in recent work are favorable enough that near-term applications of the SM-LWFA - including table-top nuclear activation of rare isotopes [Leemans01], injectors for conventional high-energy physics (HEP) accelerators [Pitthan01], and radiation oncology [Tajima 97] - are now emerging. A disadvantage of the SM-LWFA is the high peak power of the driving laser pulses that is required to achieve the beam energy, collimation and peak current required for these applications. This restricts laser repetition rate to a few Hertz, and average current to nano-amp levels. Most demonstrations of SM-LWFA have, in fact, been essentially single-shot (<< 1 Hz) [Modena95, 1

2 Ting97, Wagner97, Malka97]. Only recently has high quality SM-LWFA has been achieved at repetition rates as high as 10 Hz [Leemans01]. However, kilohertz repetition rates (micro-amp average currents) are desirable for most applications. Recently laser systems capable of delivering pulses approaching 1TW at 1 khz repetition rate have indeed been developed [Bagnoud00]. Currently, however, these pulses fall short of the threshold required for efficient SM-LWFA. Therefore it is timely to investigate schemes for seeding the growth of FRS instability that have the potential to reduce the threshold energy for SM-LWFA to levels achievable by khz laser systems, while retaining the favorable beam characteristics (high peak current, collimation, energy). In this paper we investigate through 1D and D PIC simulations a Ramanseeding mechanism, originally proposed by Fisher and Tajima [Fisher96], in which a driving pulse with field E(ω) is seeded by a weak co-propagating, superposed pulse E (ω seed of frequency ω seed ω ω p. Fisher and Tajima showed that a seed as small as 0.01 E(ω) could greatly enhance FRS compared to the unseeded case. This analysis, however, was confined to 1D simulations assuming exact Raman resonance (ω seed ω ω p ), and did not consider the intensity-dependence of wakefield enhancement nor the yield and spectrum of accelerated electrons. In view of recent demonstrations of high quality SM-LWFA beams, khz TW laser systems and potential applications that could benefit from reduced threshold, it is imperative to examine the quantitative 1D and D features of Raman-seeded laser-wakefield acceleration (RS-LWFA) explicitly. Recent demonstration of experimental schemes for generating Raman-shifted seed pulses by chirped-pulse stimulated Raman scattering (SRS) in Raman-active crystals [Zhavoronkov01] additionally motivate the present study. Raman forward scattering instability in presence of Raman Seed pulse The effect of Raman seed can be understood qualitative based on a spatialtemporal theory of Raman forward scattering of Decker et al.[decker96]. In this section we look the part of their work which is relevant to the present Raman seeding problem. Consider a plane electromagnetic pump waves traversing in a cold plasma medium. The laser-plasma interaction of concern is the Raman scattering in the forward or near forward direction. Taking into account the incident, the Stokes and Anti-Stokes side bands, the normalized complex vector potential of the electromagnetic waves takes on the approximate form a = 1 [ a0 e iθ0 + a e iθ + a + e iθ+ + c.c. ] (1) and the modified electron density fluctuation function is written as χ δn a = χ 0 + χ s eiθe + c.c. ()

3 Here the phase angle of the waves of type-i (where i = 0, +, and e) are given by θ i = p iµ x µ, with the four momenta of e.m. waves and plasma waves being p 0µ = (k 0, 0, ω 0 ), p ±µ = (k 0 ± +k z, ±k, ω 0 ± ω), p eµ = (k z, k ; ω). (3) For plasma waves, the dominant contribution is near ω = ω p and k z = k p. The space-time 4-vector x µ = (x, x ; ct). In terms of the speed-of-light Galilean frame space-time 4 vectors: x µ c = (ψ, x ; cτ), the phase angle of the type-i waves is θ i = p iµ x µ c. From Maxwell s equation, the linearized continuity equation, linearized Lorentz force equation within the plasma fluid, the assumption that the amplitudes of the waves a i s, χ s are small and they are smooth functions of ψ and τ, and working to leading order in 1 γ p = ωp ω 0, one finds that temporal derivatives of the Stokes and anti-stokes waves take on the following form: iω a τ Here the mismatched parameter with the dispersion functions: = χ s a ω 0 p, and iω a + + τ = χ s a 0 ω p a +. (4) = D + D ω4 p ω 0 (ck ), (5) D ± = (ω ± (ck z ) ) ± (ωω 0 c k z k 0 ) (ck ). (6) The Stokes waves in a plane wave approximation, i.e. ignoring the ψ and τ dependence in a, at ω = ω p, satisfy the dispersion relation withd = 0, and the anti-stakes, in a same fashion, the dispersion relation D + = 0. As the mismatch parameter is increasing from 0, the instability evolves from 4-waves, to 4-waves non-resonance (where the anti-stokes waves participate in a nonresonance way) and to the 3waves (where anti-stokes waves no longer participate processes). The present consideration together with the assumption that χ s is slowly varying, i.e. ( / τ + c / ψ)χ s << ω p χ s, within their approximation, Decker et al obtained following closed form for the growth of the density fluctuation instability: χ s = χ s0 n=0 ( ) n ψ/c exp z I n(z) χ s0, (7) τ ψ/c πz where z = γ jw (τ ψ/c)ψ/c, with the subscript jw indicating the 4-waves or the 3-waves cases: γ 4w = a 0 8 ω 4 p ω 0, γ 3w = a 0 16 ω 3 p ω 0 = γ p γ 4w > γ 4w. (8) 3

4 The last step in eq.(7) indicates the asymptotic behavior in z. For a fixed psi and large τ, z γ jw ψτ/c. Although the coefficient χ s0 may be associated with various noise-sources in the laser-plasma system, Decker et al observed that a short laser pulse may lead to an important contribution to χ s0, compared to e.g. the thermal noise. To estimate the effect due to the shortness of the pulse, one would proceed to solve the differential equation ( ψ + k p ) χ s = a. (9) Using the standard Green s function method, this solution takes on the general form ψ ( ) a χ s (ψ) = dξ 0 (ξ) G(ψ ξ) (10) For eq.(9), G(x) = 1 k p sin k p x, for x 0, and 0 elsewhere. To obtain an analytic dependence of χ s0 on laser pulse parameters such as the intensity and the pulse-length, Decker et al used the following pulse-envelope a(ξ) = a 0 sin πξ, for 0 < ξ < L. (11) L and a(ξ) = 0 elsewhere. This leads to ( χ s = χ s0 cos k 0 ψ k ) pl, where χ s0 = C s0 sin k pl, (1) with C s0 and the rms-average of χ s0 are respectively ( ) 4 a C s0 = 0 (L/λ p ), χ avg s0 C s0. (13) 1 4 For Raman seeding case, we parameterize the incident e.m. waves by ( ) πξ a = a 0 + a seed = a 0 sin [sin k 0 ξ + η sin ((k 0 k p )ξ)]. (14) L for 0 < ξ < L, where for the seed amplitude a seed = ηa 0. In the integrand of eq.(10), replacing the function a 0 (ξ) by a 0 (ξ)a seed (ξ) and γ4w = a 0 a seed ωp 4/(4ω ), we obtain ( L χ seed = χ s0 sin k p ψ, withχ s0 = π λ p ) ( a0seed a 0 ) ( ) a 0. (15) 4 Numerical example: For numerical ilustration, we choose to work with laser wavelength λ 0 = 1µ, plasma density such that γ p = 5.8, and a 0seed = 0.1a 0. For 4

5 the pulselength L = 150f s, which corresponds to a laser pulse with longitudinal gaussian-width 100f s. (Misha: please check) In terms of the abscissa variable, the tail end of the pulse is at ξ = Lω p /c 50. General comments: Plots here simply illustrate the approach. Misha: after careful considerations please redo the plots.) Fig 1a illustrates the instability growth plot: χ s vs τω p for a 0 = 0.35, for the exact solution (solid curve), the asymptotic solution (dashed-dot curve) and the rms-average of the nonseeded case (dashed curve). The same plot for a 0 = 0.75 is shown in Fig 1b. Note that for a 0 = 0.35 the Raman instability for the seeded case begin to be noticieable, i.e. x s = 0.1 at about τ = 100. Up to the time considered in our 1D-PIC simulaltion (see Sec. 3), say τω p 500, the contribution of the nonseeded case is relatively unimportant. On the other hand at a 0 = 0.7, the seeded and unseeded contribution becomes comparable at τω p 00. In Fig 1c, we show the instability growth as a function of a 0 especially at the same location i.e. the tail end of the pulse, and a typical intermediate value τω p = 50. One sees that, at the time chosen, it is at around a 0 0.6, the instability contribution due to the seed is comparable to the average value of the nonseeded case. To sum up either from the comparison of the analytic expressions of the nonseeded case eq(1) or the seeded case eq(15), and also from the plots, one sees that in the region i.e. a 0 < 0.5, i.e. the domain we have explored in our 1D and D PIC simulations, the seed-pulse plays a substantive role in enhancing the wakefield waves. 3 Raman-seeded self-modulated LWFA Figure shows the experimental method that we are using to generate the required seed pulse. The main (pump) pulse at 800nm is produced by a conventional 4TW Ti:sapphire chirped pulse amplification system. The seed pulse is formed from 10% of the amplified, chirped 800nm pulse that reflects specularly from (i.e. into the zero-order of) the first compressor grating. This normally wasted light, still chirped, passes through a barium nitrate crystal that shifts 10% of its energy to a Stokes band at centered at 873nm by stimulated Raman scattering (SRS) (Fig., right). SRS must be done with the chirped pulse to avoid continuum generation, self-focusing and other competing third-order nonlinear optical processes[zhavoronkov01]. The shifted light is then filtered and compressed to form a seed pulse that is perfectly synchronized with the main pulse. We now simulate situations in which the main pulse is focused to intensity 0.5 < a 0 < 0.5, i.e. spot radius 15 < w 0 < 0µm for the pulses described above. For near-resonant (ω p ω omega seed plasma density, k p w 0 0, so the nominal criterion k p w 0 >> 1 for 1D simulation is satisfied at 5

6 t = 0. At t > 0, 1D effects will continue to dominate at early times; D effects become important later. To estimate when the 1D D transition occurs, consider the 1D effects of longitudinal bunching and photo acceleration, and the D effect of transverse focusing. The variations of the pulse envelope at a fixed value of the coordinate ψ = ct x for the two cases are respectively given by [Mori97]: 1 < a > < a = iω p δn > τ γp n, and 1 < a > < a > τ = 1 ( c ) δn γp r n. (16) At fixed ψ, if δn/n is approximately constant for 0 < t < τ, then < a > 1D grows as τ, while < a > D grows as τ. The transition from 1D to D occurs at: τ transition = 4L R. (17) cγ p For example, with Rayleigh length L R 1mm and γ p = 5.8, τ transition = 760ωp 1. Here we present 1D PIC simulations of wakefield acceleration up to τ transition. The main and seed pulses propagate in the x-direction, and are linearly polarized in the y- direction. Their electric fields are respectively ] E y = E 0 f(x x c, σ x, k), where f( x, σ, k) = exp [ x σ sin kx, (18) E y = E 0 f(x x c, σ x, k ), with ω = ω ω p, and k k k p. (19) For the simulation results below, the main pulse has λ = 1µm?? pulse length σ x = 100fs, the seed pulse σ x = σ x and x c = x c + 0.5σ x. The cold (T 0) plasma has density n e = cm 3, which gives λ p = 5.8µ, and γ p = λ p /λ = ω/ω p = 5.8µm. Fig. 3 shows the time evolution of (a) the normalized wakefield potential φ = ee x /(mω p c) and (b) the maximum electron momentum p max x for a 0 = 0.35 and seed amplitude a0 = 0 or a 0 = 0.1a 0. For the unseeded case, φ never exceeds 0.1, whereas for the seeded case φ reaches a maximum of 0.6 at t = 300ωp 1 < τ transition, sufficient to trap electrons[esarey95]. p max x shown in Fig. 3b, is reached when a trapped electron accelerates from the top of the potential barrier through a half plasma wavelength, where it dephases from the wake potential. In the lab frame, in the ideal case where φ is constant over the course of acceleration, the maximum dephasing time is t deph = πγp /ω p and maximum kinetic energy KE max = 4γpφ 0 mc [Esarey96]. Figure 4 shows how (a) φ max?and (b) KE max scale with a 0 for seeded SM- LWFA, while maintaining seed amplitude a 0? = 0.1a 0. The threshold for electron trapping and acceleration to MeV energies is near a KE max rises steadily up to a , then saturates, despite continuing increase of φ?, indicating deviation from the ideal case described above. Those electrons that reach 6

7 KE max 4MeV are accelerated before t deph 11 < τ transition, so the 1D approximation remains valid. The yield of MeV electrons can be derived for experimental parameters of interest. For example, if a 0 = 0.35 is realized experimentally with a main pulse of energy 170mJ, duration 100fs, focused to w o = 18µm our simulations predict that 1.3 nc per pulse of electrons with KE > 1 MeV are produced. The corresponding electron spectrum at t = 600(x 554µm) is shown at the left side of Figure. The Raman-seeded SM-LWFA superficially resembles the plasma beat-wave accelerator (PBWA) [Esarey96]. However, the two approaches differ markedly in the ratio b = I seed /I main, which is 0.01 for the simulations above, and 1 for the PBWA. This, in turn, leads to two important distinctions between the two methods. First, the smaller ratio b 0.01 is much easier to realize experimentally, since only a small fraction of the main pulse need be converted to the Stokes wavelength. Secondly, the growth of the Raman- seeded wakefield is much less sensitive than PBWA to detuning ω = ω seed (ω ω p ) of the seed pulse frequency from its optimum value ω seed = (ω ω p ), as illustrated in Fig. 4c. This plot compares KE max vs ω/ω p for the Raman-seeded LWFA and PBWA. For this comparison, we assumed a common primary pulse with intensity I 0 was split into a main pulse with I main = (1 α)/i 0 and a seed pulse with I seed = αηi 0, where η is the conversion efficiency into the Stokes frequency. Thus b = ηα/(1 α)i 0 and a 0 = for the Raman-seeded LWFA example (using α = 0.03, η = 0.3), which yields the electron spectrum shown in Fig. (left) and 1.3 nc/pulse as discussed above, while b = 1 and a 0,main = a 0,seed = for the PBWA example (using α = 0.769, η = 0.3). KE max is consistently higher for the seeded case than for the PBWA. More importantly, the seeded LWFA is more stable against deviation of the seed pulse frequency from ω ω p. Evidently, the detuned seed pulse self- corrects more quickly as self-modulation evolves when b is small. Thus the Raman-seeded LWFA is much more compatible than PBWA with plasmas of nonuniform density, such as those present in a gas jet. As b decreases from unity, the maximum KE plot evolves continuously from the narrow plot shown for the PBWA to the broader seeded LWFA curve. Of course, b cannot decrease indefinitely below 0.01, because the production rate of energetic electrons then falls eventually to the level of the unseeded SM-LWFA. 4 D PIC simulation results References [Andreev9] N. E. Andreeev, L. M. Gorbunov, V. I. Kirsanov, A. A. Pogosova, and R. R. Ramazashviki, JETP Lett. 55, 571 (199). [Antonsen9] T. M. Antonsen, Jr., and P. Mora, Phys. Rev. Lett. 69, (199). 7

8 [Bagnoud00] V. Bagnoud and F. Salin, Amplifying laser pulses to the terawatt level at a 1-kilohertz repetition rate, Appl. Phys. B70, S165-S170 (000). [Chen99] S.-Y. Chen, M. Krishnan, A. Maksimchuk, R. Wagner and D. Umstadter, Detailed dynamics of electron beams self-trapped and accelerated in a self-modulated wakefield, Phys. Plasmas 6, (1999). [Decker96] Phys. Plasma 3, 1360 (1996). [Esarey95] Esarey, E. and Pilloff, M., Phys. Plasmas, (1995). [Esarey96] E. Esarey, P. Sprangle, J. Krall, and A. Ting, Overview of plasmabased accelerator concepts, IEEE Trans. Plasma Science 4, 5-88 (1996). [Fisher96] D. L. Fisher and T. Tajima, Enhanced forward Raman scattering, Phys. Rev. E 53, (1996). [Fritzler0] S. Fritzler et al., 00 MeV Energy Gain in Self-Modulated Laser Wakefield Accelerator, Proceedings AAC 0 (00). [Leemans01] W. P. Leemans, D. Rodgers, P. E. Catravas, C. G. R. Geddes, G. Fubiani, E. Esarey, B. A. Shadwick, R. Donahue, and A. Smith, Gammaneutron activation experiments using laser wakefield accelerators, Phys. Plasmas 8, (001). [Malka97] G. Malka, E. Lefebvre, and J. L. Miquel, Phys. Rev. Lett. 78, 3314 (1997). [Max74] C. E. Max, J. Arons, and A. B. Langdon, Phys. Rev. Lett. 33, 09 (1974). [Modena95] A. Modena, Z. Najmudin, A. E. Dangor, C. E. Clayton, K. A. Marsh, C. Joshi, V. Malka, C. B. Carrow, C. Danson, D. Neely, and F. N. Walsh, Nature (London) 377, 606 (1995). [Mori97] Mori, W. B., IEEE J. Quant. Elec. 33, (1997). [Pitthan01] Pitthan, R., private communication, 00. [Sprangle9] P. Sprangle, E. Esarey, J. Krall, and G. Joyce, Phys. Rev. Lett. 69, 00 (199). [Tajima79] T. Tajima and J. M. Dawson, Laser electron accelerator, Phys. Rev. Lett. 43, (1979). [Tajima 97] T. Tajima, J. Jpn. Soc. Therap. Rad. Oncol. 9, -5 (1997). 8

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10 Table I. High Density Experimental Results Table 1: High Density Experiments. For names of experimental groups see Sec..1. ( ) 10 gp P (TW) a 0 n 19 ω L e cc τ (fs) ω p L (mm) L R N e E max (MeV) τ τ p > > Table II. Operation Parameters and Simulation Results Table : Operation and Simulation Results Laser power 1.7 TW Repetition Rate 10 Hz Pulse Length 100 fs Wavelength 1µ beam-diameter at mirror 5mm focal length of the mirror 17 cm spot-radius: r 18 µm Rayleigh length (L R = πr /λ) 1mm Electron density /cc Plasma waves λ p = 5.8µm, ω p = (3.1fs) 1 Wakefield: seeded/unseeded at 0.6L R 0.6mm 6 fold enhancement Number of electrons (KE > 1 MeV, at 0.6mm) , or 1.3 nc per pulse Maximum electron energy (at 0.6mm) 40 MeV Dose rate: 35 MeV with an 8MeV-bandwidth D 4.5 Gy / min 10