Relativistic self-focusing in underdense plasma

Physica D 152 153 2001) 705 713 Relativistic self-focusing in underdense plasma M.D. Feit, A.M. Komashko, A.M. Rubenchik Lawrence Livermore National Laboratory, PO Box 808, Mail Stop L-399, Livermore, CA 94550, USA Abstract We present the description of powerful beam self-focusing in underdense plasma. The importance of electron cavitation, i.e. total electron evacuation under the effect of ponderomotive forces, is emphasized. Cavitation results in suppression of filamentation and the possibility to channel power well above the nominal critical power of self-focusing for a distance of many Rayleigh lengths. 2001 Elsevier Science B.V. All rights reserved. PACS: 42.65.Jy; 42.68.Ay; 52.58.Ns; 52.60.+h Keywords: Underdense plasma; Cavitation; Self-focusing 1. Introduction Intense laser radiation propagated in a nonlinear medium with refractive index n = n 0 + n 2 E 2 can undergo self-focusing. To overcome the diffractive divergence the beam power P must exceed the critical value P cr = 0.148λ 2 /n 0 n 2. A beam with P P cr breaks into many filaments each with power about P cr. The filaments then form an intensity singularity in a finite distance. In transparent dielectrics, the intensity growth is typically stopped by material breakdown or other nonlinear interaction. Self-focusing in underdense plasma has an interesting nature. The plasma can sustain extremely intense electric fields. For short laser pulses ion motion can be disregarded. In this case the nonlinearity of light propagation is determined by the relativistic motion of electrons in the laser field. It was shown many years ago [1 3] that in the weakly relativistic regime, steady state self-focusing is similar to self-focusing in a Kerr medium: the focusing is described by a non- Corresponding author. linear Schrodinger equation NSE). The critical power P cr is equal to 16 n c /n GW, where n is the electron density and n c is the critical density value. Recent advances in laser technology [4], based on the chirped pulse amplification technique, make intensities greater than 10 18 W/cm 2 available for experiments. In these experiments the laser beam power can be 1000 times larger than the critical value. Having in mind NSE based self-focusing model one can expect multiple filamentation, focusing up to the point where the paraxial approximation breaks down in every filament and radiation scatters. But experiments [5] demonstrate stable radiation channeling limited only by the plasma thickness. This indicates that when the electron motion in the laser field becomes essentially relativistic, the self-focusing is qualitatively different. We will start with a discussion of the basic equations describing the self-focusing of very intense laser beams. Recent studies [6 11] demonstrate that self-focusing changes qualitatively for very intense beams. In this case, the laser field becomes so strong that the ponderomotive force evacuates electrons from 0167-2789/01/$ see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S0167-278901)00203-2

706 M.D. Feit et al. / Physica D 152 153 2001) 705 713 macroscopic regions electron cavitation). Stable channeling with confined power P P cr can take place inside the resultant empty cavity since further focusing cannot take place. We show below that the previously used description of cavitation leads to non-conservation of net plasma charge. Modification of the cavitation description and a comparison of our approach with previous results will be done in the second part of the paper. Finally, we will demonstrate that our model is able to reproduce the results of the experiment [5]. 2. Basic equations Propagation of ultra-intense radiation in plasma involves a number of highly nonlinear phenomena characterized by different spatial and temporal scales e.g. the scale of charge separation, the laser wavelength, the transverse beam size, and the longitudinal plasma scale which must be resolved in calculations. Also, the self-focusing of light in Kerr media is very different in two and three dimensions, thus reliable modeling should be three-dimensional. On the other hand, the disparate sizes of parameters allow a simplified description [12,13]. Equations for slowly varying envelopes can be derived from the fully coupled system of equations of relativistic hydrodynamics for the electron motion and from Maxwell s equations for the laser beam: A = 0 gauge choice), 2.1) 2 1 ) c 2 2 t A = 4π c en ev + 1 c tφ, 2.2) Φ = 4πeN 0 N e ), 2.3) t P + v )P = e Φ + e c ta e v A), c 2.4) t N e + N e v) = 0, 2.5) where A and Φ are respectively, the vector and scalar potentials, P is the electron momentum, N e is the electron density and N 0 the ion density. The force equation 2.4) can be cast into the equivalent form t P e ) [ c A v P e )] c A = e Φ m 0 c 2 γ 2.6) with the immediate consequence that the vorticity P e ) c A is conserved. The initial conditions for our problem, quiescent plasma and no field guarantee the initial absence of vorticity; therefore the vorticity vanishes at all later times. This simplifies 2.6) to t P e c A ) = e Φ m 0 c 2 γ, 2.7) These equations can be averaged over the rapid optical oscillations. The next temporal scale is determined by the plasma frequency ω p and the averaged equations are derived as an expansion in ω p /ω. Hence, we can use the derived equations only for self-focusing description in underdense plasma. The details of this derivation can be found in [12,13]. We consider the propagation of ultra-short pulses only. During the pulse, ions have no time to move and their motion is disregarded. As a result we do not consider the ponderomotive and thermal light self-focusing. Assuming that the process is stationary in a system of reference moving with the laser pulse quasistatic approximation) and that the fields are axi-symmetric, the averaged description can be reduced to the equations for the dimensionless envelope of the laser field vector potential a = ea/mc 2 and the electrostatic potential φ = eφ/mc 2 : 2i a t + a = ηa, η = 1 + φ 1 + φ, 2 φ ξ 2 = γη 1 γ + ξ 1 η φ ξ, γ = 1 + 1/η) φ/ ξ)) 2 + a 2 /2 + 1 + φ) 2. 21 + φ) 2.8) Here γ is the relativistic factor, and η the electron density in the pulse system of reference. Transverse coordinates are normalized to the plasma scale

M.D. Feit et al. / Physica D 152 153 2001) 705 713 707 L = c/ω p, propagation coordinate ξ is normalized to the Rayleigh length kl 2. The complicated structure of these equations corresponds to sophisticated physics associated with intense pulse propagation. Ponderomotive forces accelerate electrons on the front of the pulse and the wake field after the pulse. Charge separation creates a strong electrostatic field which affects electron motion. Movement of charge particles generates currents and powerful magnetic fields. Currents and magnetic fields can be calculated from electrostatic and laser field vector potentials with the help of simple relations. There are a variety of soliton-like localized solutions of 2.8): 1D and 2D solitons of laser field [6 10,14,15] and structures induced by the intensive laser field but existing long after the pulse [16]. For the weak field case 2.8) is reduced to the NSE. We consider below only the problem of laser beam filamentation for pulse long enough that one can disregard the charge separation along the propagation direction and wake field excitation. With this simple example we will indicate some additional complexity of system 2.8) related to the cavitation phenomenon. 3. Cavitation and relativistic channeling When the pulse duration is much longer than 1/ω p, a further simplification is to disregard charge separation in the direction of propagation. While such a model does not include wake field generation important e.g. for laser acceleration studies), it is adequate for present purposes. For propagation in a plasma with density near critical, this approximation is good even for pulses with duration as small as a few tens of femtoseconds. We show below that in the process of self-focusing, small-scale modulations of the pulse can arise. For pulse durations greater than 0.5 ps, pulse splitting can be important. In this approximation, steady state relativistic self-focusing in the co-moving system of reference 2.8) is reduced to the paraxial wave equation [5 10] 2ia z + a + 1 n ) a = 0. 3.1) γ Here a is the envelope of the vector potential normalized by the electron rest energy a = ea/mc 2, n the electron density in units of the critical density n = n e /n o, γ = 1+ 2 1 a 2 ) 1/2 is the usual relativistic factor, the transverse coordinates are normalized by c/ω p, with ω p being the plasma frequency, and z is normalized by the Fresnel length corresponding to the plasma wavelength ωc/ωp 2 c/ω. The electron density n in 3.1) is related to the light intensity by n = 1 + γ. 3.2) Eqs. 3.1) and 3.2) can be derived by simplification of 2.8) taking into account the assumptions discussed above. A problem with 3.2) is that at high intensity, the ponderomotive force dominates the electric field produced by charge separation in a macroscopic region and the density given by Eq. 3.2) can take non-physical, negative values. It was suggested in [5] that negative densities be avoided by setting the electron density to zero inside the cavitation zone, i.e. n = 0 if1+ γ<0 3.3) This type of cavitation model has been used in several published investigations [5 10]. Unfortunately, this model has a difficulty which makes quantitative results unreliable. To demonstrate the problem, consider a powerful beam propagating in plasma, with ponderomotive forces evacuating electrons up to a radius R. Atr = R, we must have 1 + γ = 0. The solution of Eq. 3.1) is completely determined by this specification, the requirement that the field should decay at large radius, and the fixed beam power P [8,9]. Now charge conservation requires that n 1) dv = 0. 3.4) The equation is thus over-determined and solutions with cavitation obtained in [5 9] have, generally speaking, non-zero net charge. Numerical calculations confirm the net charge appearance after cavitation takes place [17]. The above inconsistency could be removed by the introduction of surface charge on the cavitation

708 M.D. Feit et al. / Physica D 152 153 2001) 705 713 boundary which would make Eq. 3.1) consistent with both the boundary conditions and the additional condition 3.4). A more natural way to solve this problem, however, is to schematically take into account the finite plasma temperature T [17]. In this case, instead of Eq. 3.2) one can derive the equation n = 1 + γ + α ln n), α = T 1. 3.5) mc2 Eq. 3.5) has a simple physical meaning. The displacement of electron density produces an electrostatic potential Φ, according to the relation Φ = 1 n. Hence, Eq. 3.5) describes a Boltzmann electron distribution in joint electrostatic and ponderomotive potentials. It is clear that in this case the electron density in the region of laser field localization can be extremely small, but nevertheless, non-zero. There will be no singularities in the solution. The temperature correction is important only in a very thin boundary layer. The additional term regularizes and selects the physically correct solution of Eq. 3.1). Also, our calculations demonstrate that the solution is not sensitive to the exact temperature value chosen. To demonstrate the difference in the two descriptions of cavitation, we calculated the steady state solution of 3.1) a expiλ 2 t): 2λ 2 a + a + 1 n ) a = 0. γ In Fig. 1, the steady state solution for λ = 0.32 for both old and regular cavitation descriptions is shown. The solutions are similar, but correspond to noticeably different amounts of power. The decrease of the maximal intensity is a trend observed in all our simulations; this difference in amplitude becomes quite noticeable for self-focusing of very intense beams. For example, for P = 118.5 P cr, the maximum amplitude drops to about 50%. The maximum amplitude is important for interpretation of experimental data since it determines the energy of the so-called fast electrons. It is important to note that the steady state solution can carry arbitrarily large power. The cavitation problem is essential the for more general system 2.8) also. One can see that density can become negative under the effect of ponderomotive Fig. 1. Steady state amplitudes and densities profiles for λ = 0.32. Solid lines: old cavitation model, P = 1.52 P cr. Dashed lines: the regular description, P = 1.35 P cr, α = 0.003. forces. The proposed regularization will work for this situation and poses the regular studies of 2.8) solutions at high pulse intensities. The initial system of Maxwell s equations and relativistic hydrodynamics equations has the Hamiltonian structure [18]. The averaged equations can be derived by the direct averaging of the Hamiltonian over the rapid oscillations [7]. Hence the averaged equations 2.8) must have a Hamiltonian structure also. For the case of the long pulse self-focusing Eqs. 3.1) and 3.5) can be written in the Hamiltonian form 2ia z = δh δa 3.6) with the Hamiltonian being H = [ a 2 a 2 + 4γ n 1) 2n 1) 1 n 1) +4αn ln n n + 1)]dV. 3.7)

M.D. Feit et al. / Physica D 152 153 2001) 705 713 709 Together with the additional constraint δh/δn = 0, Eq. 3.6) is equivalent to Eq. 3.5). The Hamiltonian structure implies that Eqs. 3.1) and 3.5) conserve the Hamiltonian H in addition to the power P = a 2 dv. In the usual cavitation model, the equations have Hamiltonian structure outside the cavitation zone. H is given by Eq. 3.7) with n = 0. The equations are also Hamiltonian inside the cavity, but due to the moving surface effect, the overall Hamiltonian is not conserved within the cavitation model. The evolution of the Hamiltonian in the cavitation model is presented in Fig. 4. Jumps in the value of H are directly correlated with the appearance of cavitation. In our regular model, the Hamiltonian H of Eq. 3.7) was conserved with high accuracy during numerical evaluation of 3.1) and 3.5). Now consider steady state solutions of Eq. 3.1). Similar to steady state solutions for the NSE, these solutions realize extrema of H for fixed values of power P see e.g. the review [19]). It was shown in [20] that for T = 0, H is bounded from below for a fixed value of P. The bracket n lnn) n + 1) is always positive and hence the thermal corrections do not affect this result. The boundedness of H means that the solution corresponding to minimum H for a fixed P is stable, according to the Lyapunov theorem [18]. Hence, it is natural to expect stable channeling of intense radiation in underdense plasma. This description was used to model a straightforward relativistic self-channeling experiment [5]. In this experiment, 1 m wavelength radiation was focused on a supersonic He jet. The jet had a sharp gradient and a long uniform interaction region 1 mm in length. The plasma electron density was n = 4 10 19 cm 3 and the laser pulse duration was 400 fs. The critical power was about 430 GW. Our model is applicable to these conditions. Laser power up to 2.5 TW was tightly focused on the gas jet with a beam waist a = 6.7 m. Side imaging via Thompson scattering and nearfield imaging of the channel exit provided diagnostic information about the spatial extent of the laser channel. Due to the small focal spot, the Rayleigh length πa 2 /λ is much smaller than the length of the plasma and focused radiation that must disperse away. Experimentally, one finds a stepwise increase of laser channel length above a power of 1.5 TW. At 2 TW the length of the channel is limited by the extent of the plasma jet. A nearfield image at 1 TW shows the laser beam distributed over a large spot 0.1 mm) with local hot spots in the center. With power increased to 2 TW, the beam profile is observed to consist of a round spot, small in size plus a larger dim spot accounting for the unguided light. About 45% of the incident energy was guided in the experiment. Unfortunately, the experimental spatial resolution was not sufficient to resolve the transverse size of the channel. We modeled the experiment by propagating a focused Gaussian beam through a plasma slab. The focusing point was taken as 1 Rayleigh length inside the plasma. For power of 2 TW and beam size of 6.7 m, the peak intensity was 1.15 10 18 W/cm 2 or 0.9 in the dimensionless units introduced above. The plasma length was about 200 in these units and the spot size was 6.6. Results of simulations with power 2 TW are presented in Fig. 2. The beam intensity grows initially due to the focusing, then cavitation takes place and the beam is trapped in the regime of self-channeling, stable propagation. The empty cavitated core has a size of 2.4 m, well below experimental resolution. The intensity of the radiation on the axis is about 1.25 10 19 W/cm 2. The cavitated core contains about 60% of the initial beam energy in agreement with experimental results. When the power drops up to 1 TW our model still indicates self-channeling. When P = 0.7TW, we observe beam scattering after the focusing point. Fig. 3 presents the amplitude and density evolution for this case. We see that cavitation takes place at the focal point and the field intensity spike is high, but the beam expands rapidly immediately after the transition to self-channel propagation is sharp as power is increased in agreement with the experiment. Also we observed the oscillations of the field intensity and width for power over self-trapping threshold, similar to that observed in experiment. The difference between experimental and theoretical self-channeling thresholds can be attributed to

710 M.D. Feit et al. / Physica D 152 153 2001) 705 713 Fig. 2. Evolution of the amplitude ar, z) of the vector potential and averaged electron density for 2 TW pulse propagation. Initial distribution ar,z = 0) = 0.87 exp[ 1 + i)r 2 /9.47) 2 ]; a) field a r, z), b) density nr, z). energy losses, plasma ionization and ionization defocusing at the front of the beam which are not included in our model. Also the beam quality can be different from that of the ideal Gaussian used in the calculations. Now consider self-focusing of a beam carrying power many times P cr. Such situations occur in experiments involving the interaction of Petawatt class lasers with solid targets. Any prepulse from such a laser is powerful enough to ablate material leading to a plasma profile at the time the main pulse arrives. When electron density in this plasma is close to critical, the main pulse power can exceed thousands of P cr.

M.D. Feit et al. / Physica D 152 153 2001) 705 713 711 Fig. 3. Evolution of the amplitude ar, z) of the vector potential and averaged electron density for 0.7 TW pulse propagation. Initial distribution ar,z = 0) = 0.5 exp[ 1 + i)r 2 /9.47) 2 ]; a) field a r, z), b) density nr, z). Fig. 4 presents our results for high power beam self-focusing at P = 515 P cr. We see that, initially, the beam breaks into filaments with power well above that sufficient to produce cavitation. Then, the initial filaments coalesce as a result of nonlinear interaction finally forming one channel carrying most of the initial beam power. Qualitatively, such behavior can be explained as follows. With well-developed cavitation, the main determinant of H is the electrostatic energy of separated charges. Coalescence of empty channels decreases the total electrostatic energy and makes the formation of one channel to transport most of the beam energetically preferable. The length over which channel formation occurs at electron density close to critical is a few tens of microns. Of course, plasma inhomogeneity is important and we do not know the actual plasma profile. Nevertheless, the plasma tendency to focus and channel ultra-intense radiation must occur experimentally. Probably such focusing of radiation is

712 M.D. Feit et al. / Physica D 152 153 2001) 705 713 Fig. 4. Evolution of the amplitude ar, z) of the vector potential for a flat top pulse with a = 3, R = 20 and P = 515 P cr. responsible for the high conversion efficiency of laser beam energy into the energy of fast electrons observed in experiments. taking into account the real plasma profile are capable to model whether self-channeling will occur in a specific experimental setting. 4. Conclusion In conclusion, we demonstrated that a laser beam can be self-channeled by an underdense plasma if the laser intensity is high enough to produce electron cavitation. The more consistent description of electron cavitation used here does not qualitatively change the conclusions of previous studies. Our simulation results are in agreement with experiment, and we were able to estimate channel parameters unavailable experimentally. We found above that for pulses with power much greater than P cr, stable channeling of radiation through the underdense plasma can be realized. But beam power is not the only parameter needed to characterize such propagation. If the beam is not focused tightly, or is defocused during propagation through an inhomogeneous, underdense plasma, beam filamentation and scattering can take place even at high power. Thus, only detailed numerical calculations Acknowledgements This work was performed under the auspices of the US Department of Energy by the Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48. References [1] A.G. Litvak, Sov. Phys. JETP 30 1969) 344. [2] A.L. Berhoer, V.E. Zakharov, Sov. Phys. JETP 31 1970) 486. [3] C. Max, J. Arons, A.B. Langdon, Phys. Rev. Lett. 33 1974) 209. [4] M.D. Perry, G. Mourou, Science 64 1994) 1917. [5] S. Chen, G. Sarkisov, A. Maksimchuk, R. Wagner, D. Umstadter, Phys. Rev. Lett. 80 1998) 2610. [6] G.-Z. Sun, E. Ott, Y.C. Lee, et al., Phys. Fluids 30 1987) 526. [7] X.L. Chen, R.N. Sudan, Phys. Fluids B 5 1993) 1336. [8] A.B. Borisov, A.V. Borovskiy, O.B. Shiryaev, et al., Phys. Rev. A 45 1992) 5830. [9] A.B. Borisov, O.B. Shiryaev, A. McPherson, et al., Plasma Phys. Contr. F 37 1995) 569.

M.D. Feit et al. / Physica D 152 153 2001) 705 713 713 [10] A.M. Komashko, S.L. Musher, S.U. Turitsyn, et al., JETP Lett. 62 1995) 860. [11] P. Sprangle, E. Esarey, J. Krall, et al., Phys. Rev. Lett. 69 1992) 2200. [12] E. Esarey, P. Sprangle, et al., IEEE Trans. Plasma Sci. 24 1996) 252. [13] M.D. Feit, J.C. Garrison, A.M. Rubenchik, Phys. Rev. E 53 1996) 1068. [14] R. Sudan, Ya. Dimant, O. Shirayev, Phys. Plasmas 4 1997) 1489. [15] V. Kozlov, A. Litvak, V. Suvorov, Sov. Phys. JETP 76 1979) 148. [16] S. Bulanov, T. Esirkepov, N. Naumova, F. Pegoraro, V. Vshivkov, Phys. Rev. Lett. 82 1999) 3440. [17] M.D. Feit, A.M. Komashko, S.L. Musher, A.M. Rubenchik, S.K. Turitsyn, Phys. Rev. E 57 1998) 7122. [18] V.E. Zakharov, Sov. Phys. JETP 33 1971) 927. [19] E.A. Kuznetsov, A.M. Rubenchik, V.E. Zakharov, Phys. Rep. 142 1986) 105. [20] X.L. Chen, R.N. Sudan, Phys. Rev. Lett. 70 1993) 2082.