Commun. Theor. Phys. (Beijing, China) 49 (2008) pp. 753 758 c Chinese Physical Society Vol. 49, No. 3, March 15, 2008 Stable Propagating Waves and Wake Fields in Relativistic Electromagnetic Plasma XIE Bai-Song and YIN Xin-Tao The Key Laboratory of Beam Technology and Materials Modification of Ministry of Education, Beijing Normal University, Beijing 100875, China Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, China (Received May 14, 2007) Abstract Stable propagating waves and wake fields in relativistic electromagnetic plasma are investigated. The incident electromagnetic field has a finite initial constant amplitude meanwhile the longitudinal momentum of electrons is taken into account in the problem. It is found that in the moving frame with transverse wave group velocity the stable propagating transverse electromagnetic waves and longitudinal plasma wake fields can exist in the appropriate regime of plasma. PACS numbers: 52.35.Mw, 52.38.-r Key words: nonlinear wave propagation, laser-plasma interactions 1 Introduction Nowadays advanced laser techniques make the laser intensity so high over 10 18 W/cm 2 that results in relativistic laser-plasma physics. This high-field science ignites many new explosion such as laser-plasma accelerator for electrons and protons, dense electron-positron plasmas, which are of astrophysical interest and extreme matter state and so on. A review article of the recent advances in relativistic laser-plasma interactions is in Ref. [1]. In past decades stable propagating relativistic electromagnetic waves as well interesting solitary waves have attracted great attention because of their roles in nonlinear behaviors of physics of ultra-intense laser-plasma interaction. [2,3] Recently some efforts have been attempted to understand the relativistic electromagnetic waves propagation and its effect on the plasma wave due to ultra-intense laser plasma interaction in highly nonlinear processes of strong electromagnetic waves coupling with the plasmas wake waves. [4 10] Relativistic electromagnetic solitary waves of some analytical theories have been developed [4,11,12] and extensive numerical results have been also performed by particlein-cell simulations [5] and other schemes for solving the coupled equations directly. [8,9] It is indeed important to study the solitary waves for understanding many aspects of laser-plasma interaction such as laser wake field acceleration and laser overdense penetration and so on. For example Bulanov et al. [6] found that it is possible for existence of relativistic solitary waves in unmagnetized or/and magnetized plasmas. And they have indicated the importance of parameter of frequency of vector potential. In our previous studies [10] we have obtained an optimum wake field under different asymmetric prevailing laser field and it is found that the laser shape is important to the wake field strength. We also studied the steady bifurcation and, [8] single or/and multi peaks solitons in relativistic laser plasmas. [8,9] For the coupled equations of circularly polarized electromagnetic wave and plasma wake wave we have made a detailed bifurcation analysis based on the coexistence of three different equilibriums and have found the solitons numerically in both of underdense as well overdense plasma. In most practical situation, however, the existence of solitary wave is very delicate, which relies on the system parameters sensitively while there are other abundant types of waves that can propagate more easily than solitary waves in plasma. In the presence of relativistic electromagnetic field interacting with plasma the longitudinal momentum of electrons of plasma would become very important and they cannot be ignored. On the other hand, it is possible that the relativistic electromagnetic field has an initial amplitude before entering the plasma. Therefore in this paper we shall study involving problem by including the longitudinal momentum of electrons of plasma as well the initial non-zero incident laser field. It is found that the stable propagating transverse electromagnetic waves and longitudinal plasma wake fields can exist in the appropriate regime of plasma. The treatment of the problem, which is described in Sec. 2 by deriving the coupled equations of transverse electromagnetic wave and longitudinal electrostatic wave, is more effective in the moving frame of the laser propagation in comparison with that in the rest laboratory frame. Some typical numerical results are given in Sec. 3 in both of underdense and overdense plasmas. Finally some implications of our results to charged particles acceleration are discussed briefly. The project supported by National Natural Science Foundation of China under Grant No. 10475009 and partly by the New Century Excellent Talents in University of China
754 XIE Bai-Song and YIN Xin-Tao Vol. 49 2 Formulation of Problem We start from Maxwell s equations for the electromagnetic wave vector potential A and for the scalar potential ϕ, and the hydrodynamic equation for the cold electrons with the kinetic momentum p = γmv, where γ = (1 v 2 /c 2 ) 1/2 is the relativistic factor and v the fluid velocity of electrons. By introducing the following dimensionless quantities: t = ω p t, r = k p r, q = q/mc, v = v/c(= q/γ), n = n/n 0, a = ea/mc 2, and φ = eϕ/mc 2, where ω p = 4πn 0 e 2 /m is the electron plasma frequency, n 0 the unperturbed electron density, and m the electron rest mass, then the normalized equations for considered problem are 2 a 2 a t 2 = ϕ t + n γ q, (1) a = 0, (2) 2 ϕ = n 1, (3) (q a) t = (ϕ γ) + v (q a), (4) where γ = 1 + q 2 and the symbol for all normalized quantities have been dropped. We consider that the electromagnetic field propagates along z and the problem is in 1-D, that means x = y = 0. It is well known that the solution can be seen clearly in the moving frame, ξ = z v g t, with speed of group velocity v g = 1 ε 2, usually 0 ε 1. Assuming that a x + ia y = a(ξ) exp(iωτ), (5) where τ = t v g z since the phase velocity is 1/v g. From Eqs. (1) (4) and by introducing the infinite boundary conditions and the rest of electrons: ϕ = q = 0 but with initial a = a 0 we obtain the following coupling equations for wake wave and the circularly polarized transverse electromagnetic wave as where d 2 a dξ 2 + f(a, φ; ε)a ω2 a = ε 2, (6) d 2 φ f(a, φ; ε)( = dξ2 1 + a 2 0 + φ) 1 ε 2, (7) ε 2 = 1 vg 2, 1 ε 2 f(a, φ; ε) = (. (8) 1 + a 2 0 + φ)2 ε 2 (1 + a 2 ) 3 Numerical Results Equations (6) and (7) constitute a set of coupling equations for the problem. It is usually difficult for us to find the solitary wave solutions of Eqs. (6) and (7) since the fixed points and bifurcation analysis indicate that there are not pure saddle points but only saddle-focus type. So we have to resort numerical method to try to get some other physical reasonable solutions. In underdense plasmas by choosing carefully the system parameters we obtain some of the stable wave solutions under appropriate conditions that correspond to the closed single or multilooped phase orbits in phase space. As an example numerical results for one of these kinds of waves are depicted in Figs. 1 and 2 for relativistic electromagnetic wave and in Figs. 3 5 for electrostatic wake field. We have chosen the frequencies of electromagnetic vector potential ω 9.994 while the index of plasma ε = 0.1 is fixed. Clearly we see that these solutions can propagate very long distance in the underdense plasma since they are stable under perturbation. Fig. 1 Stable modulation laser field a with a 0 = 0.5025, ɛ = 0.1, and ω 9.994. Fig. 2 Phase portrait of laser field a vs. da/dξ. The system parameters are the same as in Fig. 1. In the situation of overdense plasma the problem would be more complex. It is worthy to make some simple but reasonable theoretical analysis before giving the involving numerical results. At first it reminds us that in overdense plasma the frequency of electromagnetic wave is less than that of plasma so that ω < 1. The EM wave dispersion relation should be taken into account for the relativistic correction to electron rest mass such that
No. 3 Stable Propagating Waves and Wake Fields in Relativistic Electromagnetic Plasma 755 ( ε = 1/ ω 4 1 + a 2 0 ). Therefore for initial constant EM wave a 0 with boundary conditions a = 0 and φ = φ = 0 at ξ = 0, equations (6) and (7) become very simple form as d 2 a/dξ 2 0 and d 2 φ/dξ 2 0. This means the plasma density is not changed and the EM wave is propagating without any modification by plasma. Obviously in most practical cases there are always some perturbations for plasma even if these disturbances are in noise level. Therefore after an enough propagation distance of EM wave the disturbance effect would be developed and induce the strong modulation for EM waves and plasma wake waves with each other. Indeed as the initial field strength a 0 increases the modulation wave behaviors occur more and more early when EM waves enter plasma. On the other hand as the wave travels enough long distance in overdense plasma and due to ponderomotive force by high EM field the electrons would be pushed and evenly depleted, i.e. n 0, such that d 2 a/dξ 2 + ω 2 a 0 and d 2 φ/dξ 2 1. The former equation indicates that the final stable periodic spatially EM wave would be formed in moving frame. And the later equation exhibits a very surprising fact that the wake electrostatic field strength, E z = φ = ξ, which is directly proportional to the distance of wave propagation in plasma. Certainly because that the limitation of wave propagation distance it is not possible to make the wake field strength to reach arbitrary high value. Meanwhile the possible large value of wake field is also inhibited by many other factors such as wave breaking effect, laser field filamentations, various instabilities if the interaction distance between laser and plasma becomes a long distance. Figs. 6 9 the three stages of wave development and evolution. In the case of a 0 = 0.81 and ω = 0.9795 the first phase of constant wave travels a more distance, the second phase of modulated wave have occurred. In the case of a 0 = 0.82 and ω = 0.9771 the first phase of constant wave travels a less distance, the second phase of modulated wave occurs more and more but the final phase has not occurred in the physically reasonable interaction distance, at which various instabilities and wave breaking have not occurred. However when a 0 = 0.825 and ω 0.975 865 098 the first phase of constant wave travels very less distance, the second phase of modulated wave occurs in less regime but the dominant behavior exhibits the final phase where electromagnetic wave is periodic and wake field is proportional to the distance of wave propagation. For the latest situation if we choose the frequency parameter a very little deviation from the requirement of exact dispersion relation as ω = 0.975 9 it is found that the waves would be explosive which is shown in Fig. 8. This indicates the very sensitive behavior of nonlinear waves on system parameters in the regime of the overdense plasma near the elliptic-center and saddle-focus bifurcation points. Fig. 4 Phase portrait of electrostatic potential φ vs. dφ/dξ. The system parameters are the same as in Fig. 1. Fig. 3 Stable modulation electrostatic potential φ. The system parameters are the same as in Fig. 1. Our numerical results are consistent with the theoretical analysis mentioned above. We choose a moderate system parameters for illustration of wave behaviors in overdense plasma. For fixed plasma index ε = 0.9 we show in Fig. 5 Stable modulation wake field dφ/dξ. The system parameters are the same as in Fig. 1.
756 XIE Bai-Song and YIN Xin-Tao Vol. 49 Fig. 6 Phase portraits [(a) and (b)], and waves [(c) and (d)] of vector and scalar fields with a 0 = 0.81, ɛ = 0.9 and ω 0.9795. Fig. 7 Phase portraits [(a) and (b)], and waves [(c) and (d)] of vector and scalar fields with a 0 = 0.82, ɛ = 0.9, and ω 0.9771.
No. 3 Stable Propagating Waves and Wake Fields in Relativistic Electromagnetic Plasma 757 Fig. 8 Phase portraits [(a) and (b)], and waves [(c) and (d)] of vector and scalar fields with a 0 = 0.825, ɛ = 0.9, and ω 0.975 865 098. Fig. 9 Phase portraits [(a) and (b)], and waves [(c) and (d)] of vector and scalar fields with a 0 = 0.825, ɛ = 0.9, and ω = 0.9759.
758 XIE Bai-Song and YIN Xin-Tao Vol. 49 4 Discussions and Conclusions In underdense plasma, quasi-periodic relativistic electromagnetic wave propagates with multi-loop phase portrait. In overdense plasma, however, in contrast to the case of the underdense plasma, the relativistic electromagnetic waves under very small disturbance propagate with different patterns of phase portrait in different travelling regimes. In general electromagnetic wave experiences three stages of phases: first it has almost the initial constant amplitude with fixed point in phase space followed by the second phase with modulated wave amplitude and finally it develops a stable spatial periodic motion with a single limit cycle in phase space. However the wave propagation is very sensitive to the matching or dismatching of frequency. Our numerical results entirely confirmed the theoretical analysis mentioned above. We choose a moderate system parameters for illustration of wave behaviors in overdense plasma. In summary, we have investigated modulated stable wave solutions with each other for the wake wave and transverse electromagnetic waves by including the longitudinal momentum of electrons and initial laser wave. It is found that in appropriate regime of frequency and initial amplitude of vector potential the stable waves exist. It is also found that the laser direct acceleration is more effective than the wake field acceleration. References [1] D. Umstadter, J. Phys. D: Appl. Phys. 36 (2003) R151. [2] K. White, C. Gahn, J. Meyer-ter-Vehn, G. Pretzler, A. Pukhov, and G. Tsakiris, Plasma Phys. Control. Fusion 41 (1999) B221. [3] R. Bingham, J.T. Mendonca, and P.K. Shukla, Plasma Phys. Control. Fusion 46 (2004) R1. [4] T. Zh. Esirkepov, F.F. Kamenets, S.V. Bulanov, and N.M. Naumova, JETP Lett. 68 (1998) 36. [5] S.V. Bulanov, Phys. Rev. Lett. 82 (1999) 3440. [6] D. Farina, M. Lontano, and S. Bulanov, Phys. Rev. E 62 (2000) 4146. [7] D. Farina and S.V. Bulanov, Phys. Rev. Lett. 86 (2001) 5289. [8] C. Hua, B.S. Xie, and K. He, Chaos, Solitons and Fractals 25 (2005) 1161. [9] B.S. Xie and S.C. Du, Phys. Plasmas 13 (2006) 074504. [10] B.S. Xie and N.C. Wang, Physica Scripta 65 (2002) 444. [11] P.K. Kaw, A. Sen, and T. Katsouleas, Phys. Rev. Lett. 68 (1992) 3172. [12] R.N. Sudan, Y.S. Dimant, and O.B. Shiryaev, Phys. Plasmas 4 (1997) 1489.