Electron Kinetic Effects on Raman Backscatter in Plasmas M. S. Hur, 1 R. R. Lindberg, 2, * A. E. Charman, 2 J. S. Wurtele, 2,3 and H. Suk 1,† 1 Center for Advanced Accelerators, KERI, Changwon, Kyongnam 641-120, Korea 2 Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA 3 Center for Beam Physics, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 17 March 2005; published 8 September 2005) We augment the usual three-wave cold-fluid equations governing Raman backscatter (RBS) with a new kinetic thermal correction, proportional to an average of particle kinetic energy weighted by the ponderomotive phase. From closed-form analysis within a homogeneous kinetic three-wave model and ponderomotively averaged kinetic simulations in a more realistic pulsed case, the magnitude of these new contributions is shown to be a measure of the dynamical detuning between the pump laser, seed laser, and Langmuir wave. Saturation of RBS is analyzed, and the role of trapped particles illuminated. Simple estimates show that a small fraction of trapped particles (6%) can significantly suppress backscatter. We discuss the best operating regime of the Raman plasma amplifier to reduce these deleterious kinetic effects. DOI: 10.1103/PhysRevLett.95.115003 PACS numbers: 52.35.Mw, 42.65.Yj, 52.38.Bv Electron kinetic effects on stimulated Raman scattering in plasmas have been explored intensively in various con- texts, especially in connection with the role of Raman backscatter (RBS) in the ignition phase of inertial confine- ment fusion [1,2]. These investigations were motivated by discrepancies between the results observed in fluid-based simulations and those based on kinetic models. In fully kinetic simulations [1], Vu et al. observed the saturation of Raman reflectivity followed by quasiperiodic bursting. These behaviors have been attributed to a nonlinear phase shift between the three waves associated with trapped particles [1], or to a breakup of the plasma wave by the trapped-particle instability [2]. However, the particle trapping effect described in [1] is more or less phenomenological, in the sense that certain physical terms responsible for the secular phase shift are omitted, and its ultimate dynamical origin remains some- what unclear. In this Letter, we derive a new kinetic thermal correction from averaging the dynamical equa- tions used in free-electron-laser and averaged-particle-in- cell (aPIC) models [3]. Its inclusion in the three-wave model provides a clear mathematical encapsulation of electron trapping and other nonlinear effects on RBS satu- ration and bursting. Analysis indicates that in certain re- gimes RBS may be saturated predominantly by small levels of electron trapping, rather than by the previously proposed mechanism invoking the breakup of the plasma wave [2]. Our kinetic three-wave model is also useful for the analysis of the Raman backward laser amplifier [3–8], which provided the basis for our study. We begin with the coupling between pump and seed lasers and the plasma electrons, corresponding to the wave equations implemented in the aPIC computer model [3], used for the kinetic simulations discussed below. Dynamics are derived under a number of simplifying as- sumptions: viz., field and particle data vary appreciably only in the longitudinal (z) direction; the plasma is under- dense; electron motion remains nonrelativistic; ions re- main immobile on the relevant time scales; the seed and pump lasers can be represented as eikonal fields with slowly varying envelopes modulating their respective car- rier oscillations, which are at most only weakly detuned from the Raman resonance. With these approximations, the counterpropagating seed and pump laser fields evolve ac- cording to @ @t a s  c @ @z a s  i! 2 p 2! 1 a p he i j i; (1a) @ @t a p  c @ @z a p  i! 2 p 2! 2 a s he i j i; (1b) where a s;p are the eikonal envelopes associated with nor- malized vector potential of the seed and pump lasers, respectively, in the Coulomb gauge; ! 1;2 are the carrier frequencies of the seed and pump, respectively; ! p is the plasma frequency of the unperturbed plasma of density n 0 . The averaged terms in the right-hand side (RHS) of Eq. (1) are the scaled components of transverse electron current density driving (and driven by) the seed and pump, respec- tively, normalized by en 0 c. Canonical momentum conser- vation was used to derive the averaged current density; the details can be found in [3]. The exponent  j k b z j  !t is the phase of the jth particle in the beat wave between pump and seed; !  ! 2  ! 1 is the beat fre- quency; and k b  k 1  k 2 is the beat wave number. The angular brackets denote a ponderomotive spatial-averaging operation, namely, hQ ‘ i P j:k b jz j z ‘ j Qz j ; j ; t=N 0 for any observable Qz;; t, where z j t is the position of the jth particle,  j  1 c d dt z j is its scaled velocity, and the sum extends over all particles within a ponderomotive bucket centered at z ‘ ; and N 0  2 k b n 1=3 0 is the initial number of particles in a ponderomotive wavelength. PRL 95, 115003 (2005) PHYSICAL REVIEW LETTERS week ending 9 SEPTEMBER 2005 0031-9007= 05=95(11)=115003(4)$23.00 115003-1  2005 The American Physical Society