Forward and Backward Stimulated Brillouin Scattering of Crossed Laser Beams

Transcription

1 Forward and Backward Stimulated Brillouin Scattering o Crossed Laser Beams Stimulated Brillouin scattering (SBS) in a plasma is the decay o an incident (pump) light wave into a requency-downshited (Stokes) light wave and an ion-acoustic (sound) wave. 1 It is important in direct and indirect 3 inertial coninement usion (ICF) experiments ecause it scatters the laser eams away rom the target, therey reducing the energy availale to drive the compressive heating o the nuclear uel. The SBS o an isolated eam has een studied in detail. Backward SBS was studied in numerous early papers, and near-orward, sideward, and near-ackward SBS were studied in some recent papers. 4 8 Because eams overlap in the coronal plasma surrounding the nuclear uel, it is also important to analyze SBS (and other parametric instailities) driven y two (or more) crossed eams. For some scattering angles the SBS geometries allow the pump waves to share daughter waves Because the growth o these daughter waves is driven y two pump waves (rather than one), the growth rates associated with these scattering angles are higher than the growth rates associated with other scattering angles. Such is the case or orward and ackward SBS, in which the Stokes wave vectors isect the angle etween the pump wave vectors. The outline o this article is as ollows: (1) We derive the equations governing orward and ackward SBS. () We solve the linearized equations governing the transient phase o the instaility. These equations dier rom the linearized equations governing the SBS o an isolated eam 7 ecause the orward and ackward SBS o crossed eams each involve one Stokes wave and two sound waves (rather than one). (3) We solve the nonlinear equations governing the steady state o the instaility. These equations descrie the nonlinear competition etween orward and ackward SBS. (4) We discuss the entire evolution o orward and ackward SBS. Finally, (5) we summarize the main results o the article. In the Appendix we show that, in steady state, the equations governing the simultaneous near-orward and near-ackward SBS o an isolated eam are equivalent to the equations governing the simultaneous orward and ackward SBS o crossed eams; thus, many results o this article also apply to the SBS o an isolated eam. Governing Equations The SBS o crossed eams is governed y the Maxwell wave equation 1 ( + ω c ) A = ω n A (1) tt e h e l h or the electromagnetic potential, together with the soundwave equation 1 ( tt + c s 1 ) nl = c s A h. () = 1 The electromagnetic potential Ah vh cs me mi is the quiver speed o electrons in the high-requency electric ield divided y a characteristic speed that is o the order o the electron thermal speed; n l is the low-requency electron density luctuation associated with the sound wave divided y the ackground electron density; and the signiy that only the low-requency response to the ponderomotive orce was retained. The geometry associated with orward SBS is shown in Fig (a). The orward SBS o eam 1 is suject to matching conditions o the orm ω = ω + ω, k = k + k, (3) 1 s1 1 s1 where (ω 1,k 1 ) and (ω, k ) satisy the light-wave dispersion equation ω = ω + e c k, and (ω s1, k s1 ) satisies the soundwave dispersion equation ω = cs k. Similar matching conditions apply to the orward SBS o eam. Because the sound requencies depend on the magnitudes o the sound-wave vectors, ut not on their directions, ω s = ω s1 = ω s. LLE Review, Volume

2 (a) k k 1 k k s k s1 tively. In Eqs. (7) and (8) ν s1 N* 1 and ν s N* are phenomenological terms that model the Landau damping o the sound waves, 1 and ν s1 n* and ν s n* are phenomenological terms that maintain the density luctuations associated with the sound waves at their common noise level n* in the asence o instaility. Because the Landau-damping rates depend on the magnitudes o the sound-wave vectors, ut not on their direction, ν s = ν s1 = ν s. () k k s k Equations (6) (8) descrie the initial (transient) evolution o SBS. In steady state, z µ 1, (9) da = A + A A P1863 k s1 k 1 where µ ω ω 4ω ω ν v. (1) = e s s s Figure Geometry associated with the SBS o crossed laser eams: (a) orward SBS; () ackward SBS. and By sustituting the Ansätze A = A exp( ik x iω t)+ A exp ik x iω t h [ ] + + A exp ik x iω t cc.. (4) 1 1 nl = N1exp iks1 x iωst + N exp ( ik x iω t)+ cc.. (5) s in Eqs. (1) and (), one can show that A = i( ω ω )( A N + A N ) z e s v * *, (6) 1 1 ( t + νs ) N* = ( i ω 1 1 s ωs) A* 1A + νs1n *, (7) ( + ν ) N* = ( i ω ω ) A* A + ν n *. (8) t s s s s Because ω s << ω, we made the approximation in Eq. (6) that the requency and group speed o the scattered light wave equal the requency and group speed v o the pump waves, respec- Apart rom a actor o A 1 or A, µ is the spatial growth rate o orward SBS in the strongly damped regime. 7 The orward-scattered intensity F = A satisies the equation df z = µ P1 + P F, (11) where P1 = A1 and P = A are the pump intensities. The geometry associated with ackward SBS is shown in Fig (). The ackward SBS o eam 1 is suject to matching conditions o the orm ω = ω + ω, k = k + k, (1) 1 s1 1 s1 where (ω 1,k 1 ) and (ω, k ) satisy the light-wave dispersion equation, and (ω s1, k s1 ) satisies the sound-wave dispersion equation. Similar matching conditions apply to the ackward SBS o eam : as in orward SBS, ω s = ω s1 = ω s. By adding to Ansatz (4) the term and to Ansatz (5) the terms A exp ( ik x iω t)+ cc.. (13) N1exp( iks1 x iωst)+ N exp ( iks x iω st)+ cc.. (14) associated with ackward SBS, one can show that 19 LLE Review, Volume 75

3 A = i( ω ω )( A N + A N ) z e v * *, (15) 1 1 ( t + νs) N = γ A + νsn, (3) ( t + νs ) N* = ( i ω 1 1 s ωs) A* 1A + νs1n *, (16) where ( t + νs ) N* = ( i ω s ωs) A* A + νsn *. (17) As in orward SBS, ν s = ν s1 = ν s. In its transient (linear) phase, ackward SBS is independent o orward SBS. In steady state, the ackward-scattered intensity B= A satisies the equation db z = µ P1 + P B, (18) where µ is given y Eq. (1) and the values o ω s and ν s associated with ackward SBS. Apart rom a actor o A 1 or A, µ is the spatial growth rate o ackward SBS in the strongly damped regime. 7 In the high-gain regime, the intensities o the scattered light waves as they exit the plasma are comparale to the intensities o the pump waves as they enter the plasma, and one must account or the depletion o the pump waves within the plasma. In steady state, the pump intensities satisy the equations dzp1 = µ FP1 µ BP1, (19) d P = µ FP µ BP, () z where we made the approximation that the evolution o the pump waves is one dimensional. One can veriy Eqs. (19) and () y applying the principle o power conservation to Eqs. (11) and (18). Linear Analysis o the Transient Phase The orward SBS o crossed eams consists o two mirrorimage processes that share the same Stokes wave and, hence, are governed y the coupled equations (6) (8). By making the 1 sustitutions ω A A, iω e N1 * 1 ω s N 1, i ω e N * 1 ω s N, iω e n* 1 ω s n, and z v z, one can rewrite these equations as za = γ1n1 + γn, (1) ( t + νs) N1 = γ1 A + νsn, () 1 1 e s 1 s γ ω ω A ω ω =, (4) 1 e s s γ ω ω A ω ω =. (5) A is proportional to the action amplitude o the Stokes wave, and N 1 and N are proportional to the action amplitudes o the sound waves. In the asence o damping, γ 1 and γ are the temporal growth rates o the orward SBS o eams 1 and, respectively, in an ininite plasma. By using the comined amplitudes where γ = γ + γ N+ = γ 1N1 + γ N γ, (6) N = γ N1 γ 1 N γ, (7) 1 1, one can rewrite Eqs. (1) (3) as za = γ N+, (8) ( t + νs) N+ = γ A + ν sn+, (9) ( t + νs) N = ν sn, (3) where n+ = n γ 1 + γ γ and n = nγ 1γ 1 1γ. Equations (8) and (9) are equivalent to the equations governing the orward SBS o an isolated eam, 7 and Eq. (3) is simple. Consequently, the solutions o Eqs. (8) (3) can e written in the orm A ( z, t)= ν n G z z, t t dz dt, (31) t z s + t z + s + + N ( z, t)= ν n G z z, t t dz dt, (3) t z s N ( z, t)= ν n G z z, t t dz dt, (33) LLE Review, Volume

4 where the Green unctions 1 G ( z, t)= γ I γ ( zt) exp νst, (34) 1 1 G+( z, t)= γ ( t z) I 1 γ ( zt) exp( νst)+ δ() z exp ( νst), = () ( s ) (35) G z, t δ z exp ν t. (36) In Eqs. (34) and (35), I m is the modiied Bessel unction o the irst kind, o order m. The original amplitudes N 1 and N are determined y Eqs. (3) and (33) and the inversion equations = N γ γ N γ γ N, (37) = + 1 N γ γ N γ γ N. (38) Solutions (31) (33) descrie the growth and dissipative saturation o orward SBS. By analyzing the time dependence o the Green unctions, one can show that the saturation time t ~ γ z ν. (39) s s The steady-state limits o solutions (31) (33) are = A z, n+νs γ exp γ z νs 1, (4) = N+ z, n+ exp γ z νs, (41) N ( z, )= n. (4) Notice that γ ν sv = µ ( A1 + A ), in agreement with Eq. (9). I the interaction length exceeds a ew gain lengths, one can model Stokes generation as Stokes ampliication with an incident amplitude A n+νs γ. = The ackward SBS o crossed eams also consists o two mirror-image processes that share a Stokes wave and are governed y Eqs. (1) (5), with replaced y and z replaced y l z; thus, Eqs. (6) (4), and the conclusions drawn rom them, also apply to ackward SBS. Equations (1) (3) apply to other parametric instailities driven y crossed pump waves, provided that one type o daughter wave is strongly damped. Nonlinear Analysis o the Steady State The simultaneous orward and ackward SBS o crossed eams is governed y Eqs. (11) and (18) (). By making the sustitution P 1 + P P, one can rewrite these equations as df z =µ PF, (43) db z = µ PF, (44) dp z = µ F+ µ BP. (45) Equations (43) (45) apply to other simultaneous parametric instailities driven y crossed pump waves, provided that one type o daughter wave is strongly damped. For SBS, µ = µ = µ, 7 and one can use the sustitution µz z to rewrite Eqs. (43) (45) in the simple orm df z = PF, (46) db z = PB, (47) dp z = ( F+ BP ). (48) The sustitutions F/P() F, B/P() B, P/P() P, and P()z z nondimensionalize Eqs. (46) (48) ut leave them unchanged in orm. Because the solutions o Eqs. (46) (48) are complicated, it is instructive to review the limiting solutions that apply to orward and ackward SBS separately. 1. Forward SBS In the asence o ackward SBS, Eqs. (46) (48) reduce to It ollows rom these equations that df z = PF, (49) dp z = FP. (5) P+ F = 1 + N, (51) where N = F() is incident (noise) intensity o the orwardscattered wave. Since P, it ollows rom Eq. (51) that 19 LLE Review, Volume 75

5 S 1 + N, (5) where S = F(l) is the output (signal) intensity o the orwardscattered wave and l is the gain length o orward SBS. Equation (5) relects the act that the signal intensity cannot exceed the total input intensity. By sustituting Eq. (51) in Eq. (49), one can show that ( 1 + ) = F N z log ( 1 +. (53) N N F ) Equation (53) determines the interaction distance z required to produce the orward-scattered intensity F. By inverting this equation, one inds that N 1+ N F( ζ)=, (54) N + exp ζ where ζ = 1+ N z. Solution (54) is consistent with Eq. (5). The normalized intensities o the pump and Stokes waves in a semi-ininite plasma are plotted as unctions o the gain distance z in Fig. 75.3, or the case in which N = 1 6. As the Stokes intensity increases, the pump intensity decreases, in accordance with Eq. (51). For uture reerence, notice that the initial growth o the Stokes wave rom noise is driven y an undepleted pump wave. Intensity P Distance Figure 75.3 Normalized intensities plotted as unctions o the gain distance or orward SBS in a semi-ininite plasma. The solid line represents the pump wave; the dashed line represents the Stokes wave. For orward SBS the output intensities rom a inite plasma depend on the plasma length in the same way that the intensities within a semi-ininite plasma depend on the distance rom the plasma oundary.. Backward SBS In the asence o orward SBS, Eqs. (46) (48) reduce to It ollows rom these equations that db z = PB, (55) dp z = BP. (56) P B= 1 S, (57) where S = B() is the output (signal) intensity o the ackwardscattered wave. Since P, it ollows rom Eq. (57) that S 1 + N, (58) where N = B(l) is the incident (noise) intensity o the ackward-scattered wave and l is the gain length o ackward SBS. Equation (58) relects the act that the signal intensity cannot exceed the total input intensity. By sustituting Eq. (57) in Eq. (55), one can show that ( 1 S ) z = log S ( 1 S + B) B. (59) The signal intensity is determined y Eq. (59) and the condition B(l) = N. By inverting Eq. (59), with S known, one inds that S 1 S B( ζ)= exp( ζ) S, (6) where ζ = 1 S z. Solution (6), which was irst otained y Tang, 13 is consistent with Eq. (58). The normalized output intensity o the Stokes wave is plotted as a unction o the gain length l in Fig (a), or the case in which N = 1 6. The normalized intensities o the pump and Stokes waves within the plasma are plotted as unctions o the gain distance z in Fig (), or the case in which N = 1 6 and l = 3. Because the pump and Stokes waves propagate in opposite directions, the initial growth o the Stokes wave rom noise is driven y a depleted pump wave [Fig ()]. Consequently, when pump depletion is important (l > 1), the rate at which the Stokes output intensity increases with gain length is slower or ackward SBS [Fig (a)] than or orward SBS (Fig. 75.3). Backward SBS scatters the pump power less eiciently than orward SBS. LLE Review, Volume

6 3. Simultaneous Forward and Backward SBS When orward and ackward SBS occur simultaneously, it ollows rom Eqs. (46) (48) that and Output intensity Intensity P Gain length P+ F B= 1 + N S (61) FB = N S. (6) Equation (61) is a generalization o equations that apply to the orward and ackward instailities separately, whereas Eq. (6) is peculiar to the comined instaility. Since P, it ollows rom Eq. (61) that S + S 1 + N + N. (63) Equation (63) relects the act that the total signal intensity cannot exceed the total input intensity. It ollows rom Eqs. (6) and (63) that 3 (a) () Distance Figure (a) Normalized output intensity o the Stokes wave plotted as a unction o the gain length l or ackward SBS. () Normalized intensities within the plasma plotted as unctions o the gain distance or l = 3. The solid line represents the pump wave; the dashed line represents the Stokes wave. S N N + N + N, (64) N S N +. (65) N + N By sustituting Eqs. (61) and (6) in Eq. (46), one can show that where ( + ) dz F = R+ F R F, (66) [ + ] 1 ± R± = 1+ N S ± 1+ N S 4NS. (67) It ollows rom Eq. (66) that ( + ) R+ N R F ( R+ + R ) z = log. (68) ( R+ F) ( R + N ) S is determined y Eq. (68) and the condition B(l) = N, which is equivalent to the condition Fl ()=( N N) S. By inverting Eq. (68), with S known, one inds that ( ) +( ) R+ R + N exp ζ R R+ N F( ζ)= R + N exp ζ R+ N, (69) where ζ = R+ + R z. Solution (69) is consistent with Eq. (64). For the common case in which 1 S >> N, one can use the approximate roots to rewrite Eqs. (68) and (69) as and R+ 1 S + N 1 S, (7) R NS 1 S, (71) + ( ) 1 S 1 NS S F ( 1 S) z log (7) ( 1 S F) N N 1 S exp ζ S F( ζ), (73) N exp( ζ)+ ( 1 S) respectively, where ζ ( 1 S ) z. 194 LLE Review, Volume 75

7 The normalized (total) output intensity o the (orward and ackward) Stokes waves is plotted as a unction o the gain length l in Fig (a), or the case in which N = N = 1 6. When pump depletion is unimportant (l < 1), the Stokes output intensity o the comined instaility is the sum o the Stokes output intensities o the orward and ackward instailities. The normalized intensities o the pump and Stokes waves within the plasma are plotted as unctions o the gain distance z in Fig () or the case in which N = N = 1 6 and l = 3. The initial growth o oth Stokes waves rom noise is driven y a depleted pump wave. Consequently, when pump depletion is important (l > 1), the rate at which the Stokes output intensity increases with gain length is slower or the comined instaility than or the orward instaility [Fig (a)]. In Figs , the noise intensities or orward and ackward SBS were equal. This choice made possile a air comparison o the intrinsic scattering eiciencies o the two instailities. The noise intensity or orward SBS is larger, however, than the noise intensity or ackward SBS ecause the action sources that generate the light waves [Eq. (4) or orward SBS and its analog or ackward SBS] are inversely proportional to the sound requencies. 14 To illustrate how this imalance aects the comined instaility, the normalized output intensity o the Stokes waves is plotted as a unction o the gain length in Fig (a) or the case in which N = 1 16 and N = 1 7. The normalized intensities o the pump and Stokes waves within the plasma are plotted as unctions o the gain distance in Fig () or the case in which N = 1 6, N = 1 7, and l = 3. It is clear rom the igures that orward SBS overwhelms ackward SBS in steady state. Output intensity Intensity P (a) () Gain length 1 15 Distance Output intensity Intensity P (a) () Gain length 1 15 Distance Figure (a) Normalized output intensities plotted as unctions o SBS gain length l. The orward and ackward noise intensities are equal. The solid line represents the total output intensity or the comined instaility. For comparison, the dot dashed and dashed lines represent the output intensities when orward and ackward SBS occur separately [Figs and 75.33(a), respectively]. () Normalized intensities within the plasma plotted as unctions o the gain distance or l = 3. The solid line represents the pump wave, the dot dashed line represents the orward Stokes wave, and the dashed line represents the ackward Stokes wave. Figure (a) Normalized output intensities plotted as unctions o SBS gain length l. The orward noise intensity exceeds the ackward noise intensity y a actor o 1. The solid line represents the total output intensity or the comined instaility. For comparison, the dot dashed and dashed lines represent the output intensities when orward and ackward SBS occur separately. () Normalized intensities within the plasma plotted as unctions o the gain distance or l = 3. The solid line represents the pump wave, the dot dashed line represents the orward Stokes wave, and the dashed line represents the ackward Stokes wave. LLE Review, Volume

8 Discussion Initially, pump depletion is unimportant, and orward and ackward SBS grow independently. This (linear) spatiotemporal growth is descried y Eqs. (8) (3). Since the growth rate γ ( sinφ) 1 [Eqs. (4) and (5)], the soundwave damping rate ν s sinφ, and the saturation time t s 1 ( sinφ ) [Eq. (39)], where φ is the scattering angle, ackward SBS grows and saturates more quickly than orward SBS. 7 The steady-state (nonlinear) spatial evolution o ackward SBS is descried y Eqs. (57) and (6). In the high-gain regime, ackward SBS depletes the pump wave signiicantly [Fig ()]; thus, the spatiotemporal growth o orward SBS is driven y a pump wave whose intensity varies with distance, and Eqs. (31) (36) do not apply as written. By making the sustitutions N± γ N±, n± γ n±, and z [ γ ( z )] dz z in Eqs. (8) (3), however, one can show that z A = N +, (74) ( + ν ) N = A + ν n, (75) t s + s + ( t + νs) N = ν sn. (76) Since Eqs. (74) (76) contain no variale coeicients, their solution can e inerred rom Eqs. (31) (36). It ollows that the (linear) saturation time o orward SBS is given y Eq. (39), with γ z z replaced y [ γ ( z )] dz. Since the saturation time is proportional to the (integrated) gain distance, the reduction o the gain distance y pump depletion shortens the saturation time o orward SBS. Since the steady-state (nonlinear) Eqs. (46) (48) have a unique solution, the spatial evolution o the comined instaility is given y Eqs. (61), (6), and (69), even though orward and ackward SBS grow at dierent rates and saturate at dierent times. It is clear rom Figs and that the output intensity o the ackward Stokes wave is lower in the presence o the orward Stokes wave than in its asence; thus, the comined instaility is characterized y a urst o ackward SBS ollowed y the ascendance o orward SBS. The major theme o the Nonlinear Analysis o the Steady State section and the preceding discussion is that orward and ackward SBS coexist and compete or the pump energy. One should rememer that several other processes also coexist and modiy this competition. These processes include doule SBS, 9 which is made possile y a sound wave whose wave vector is the sum o the pump-wave vectors, and the transer o energy etween the pump waves 15 1 and the Bragg scattering o the pump waves, 16 oth o which are made possile y a sound wave whose wave vector is the dierence o the pump-wave vectors; thus, the interaction physics is even richer than the physics discussed herein. Summary In this article we studied in detail the simultaneous orward and ackward SBS o crossed laser eams. We otained new analytical solutions or the linearized equations governing the transient phase o the instaility [Eqs. (1) (3)] and the nonlinear equations governing the steady state [Eqs. (46) (48)]. In their transient phases, orward and ackward SBS grow independently. Initially, ackward SBS grows more quickly than orward SBS. As the ackward Stokes wave grows, it depletes the pump wave and modiies the growth o the orward Stokes wave. In steady state, orward SBS dominates the comined instaility ecause the orward Stokes wave has a larger noise intensity rom which to grow and orward SBS scatters the pump power more eiciently. In the Appendix we show that the equations governing the simultaneous near-orward and near-ackward SBS o an isolated eam are equivalent to the equations governing the simultaneous orward and ackward SBS o crossed eams; thus, the results o this article also apply to the SBS o an isolated eam. ACKNOWLEDGMENT This work was supported y the National Science Foundation under contract No. PHY , the U.S. Department o Energy Oice o Inertial Coninement Fusion under Cooperative Agreement No. DE-FC3-9SF1946, the University o Rochester, and the New York State Energy Research and Development Authority. The support o DOE does not constitute an endorsement y DOE o the views expressed in this article. Appendix A: Forward and Backward SBS o an Isolated Laser Beam In this appendix we show that the equations governing the simultaneous orward and ackward SBS o an isolated eam are equivalent to the equations governing the simultaneous orward and ackward SBS o crossed eams. The geometry associated with the orward SBS o an isolated eam is shown in Fig (a). Each orward-scattering process is suject to matching conditions o the orm ω = ω + ω, k = k + k, (A1) s s where (ω, k ) and (ω, k ) satisy the light-wave dispersion equation ω = ω + e c k, and (ω s, k s ) satisies the sound- 196 LLE Review, Volume 75

9 (a) k k s ( + ν ) N * = i( ω ω ) A * A + ν n *. (A5) t s s s s () k 1 k k s1 In Eq. (A5), ν s N* is a phenomenological term that models the Landau damping o the sound wave, and ν s n* is a phenomenological term that maintains the density luctuations associated with the sound wave at their noise level n* in the asence o instaility. Because the Landau-damping rates depend on the magnitudes o the sound-wave vectors, ut not on their directions, ν s = ν s1 = ν s. By making the sustitutions 1 ω A A, iωen* 1 ωs N, iωen* 1 ωs n, and z v z, one can rewrite Eqs. (A4) and (A5) as k k k s za = γ N, (A6) k 1 k s1 where ( + ν ) N = γ A + ν n, (A7) t s s P1888 Figure Geometry associated with the SBS o an isolated laser eam: (a) near-orward SBS; () near-ackward SBS. 1 e s s γ ω ω A ω ω. (A8) = Equations (A6) and (A7) are equivalent to Eqs. (8) and (9), the solution o which was descried in the text. wave dispersion equation ω = cs k. Because the requencies o the daughter waves depend on the magnitude o their wave vectors, ut not on their directions, ω = ω 1 = ω and ω s = ω s1 = ω s. Equations (A4) and (A5) descrie the transient evolution o orward SBS. In steady state, da z = µ A A, (A9) By sustituting the Ansätze [ A = A exp( ik x iω t)+ A exp ik x iω t h 1 1 and ] + + A exp ik x iω t cc.. nl = N1exp iks1 x iωst + N exp ( iks x iω st)+ cc.. (A) (A3) where µ ω ω 4ω ω ν v. (A1) = e s s s Notice that µ = γ νsv is in agreement with Eqs. (A6) (A8). It ollows rom Eq. (A9) that the orwardscattered intensities F 1 = A 1 and F = A satisy the equations A df z = µ PF, (A11) 1 1 into Eqs. (1) and (), and making the slowly varying envelope approximation, one can show that each orward-scattering process is governed y equations o the orm = ( ω ω ) za i e AN v *, (A4) where P df z = µ PF, (A1) = A is the pump intensity. The geometry associated with the ackward SBS o an isolated eam is shown in Fig (). Each ackward- LLE Review, Volume

10 scattering process is suject to matching conditions o the orm ω = ω + ω, k = k + k, (A13) s s where (ω, k ) and (ω, k ) satisy the light-wave dispersion equation, and (ω s, k s ) satisies the sound-wave dispersion equation. As in orward SBS, ω = ω 1 = ω and ω s = ω s1 = ω s. By adding to Ansatz (A) the terms state, the pump intensity satisies the equation dp z = µ ( F1 + F) µ ( B1 + B). (A) By making the sustitutions F = F 1 + F and B = B 1 + B in Eqs. (A11), (A1), (A18), (A19), and (A), one can show that the simultaneous orward and ackward SBS o an isolated eam is governed y the equations df=µ PF, (A1) z A exp ik x iω t A exp ( ik x iω t)+ cc.. (A14) db z = µ PB, (A) dp z = µ F+ µ BP. (A3) and to Ansatz (A3) the terms N1exp iks1 x iωst + N exp ( ik x iω t)+ cc.. s s (A15) associated with ackward SBS, one can show that each ackward-scattering process is governed y equations o the orm = ( ω ω ) za i e AN v *, (A16) ( t + νs) N * = ( i ω s ωs) A * A + νsn *. (A17) As in orward SBS, ν s = ν s1 = ν s. It ollows rom Eqs. (A16) and (A17) that the transient evolution o ackward SBS is governed y Eqs. (A6) (A8), with replaced y and z replaced y l z. Equations (A6) and (A7) apply to other parametric instailities driven y an isolated pump wave provided that one type o daughter wave is strongly damped. In steady state, the ackward-scattered intensities B 1 = A 1 and B = A satisy the equations db z 1 = µ PB1, (A18) db z = µ PB, (A19) where µ is given y Eq. (A1), with replaced y. In the high-gain regime, the intensities o the scattered waves as they exit the plasma are comparale to the intensity o the pump wave as it enters the plasma, and one must account or the depletion o the pump wave within the plasma. In steady Equations (A1) (A3) are equivalent to Eqs. (43) (45), the solution o which was descried in the text. It is clear rom the derivation o Eqs. (A1) (A3) that one can interpret F as the intensity scattered orward over the entire range o angles or which propagation in the z direction is a reasonale approximation, and one can interpret B as the intensity scattered ackward over the entire range o angles or which propagation in the z direction is a reasonale approximation. Equations (A1) (A3) apply to other parametric instailities driven y an isolated pump wave, provided that one type o daughter wave is strongly damped. For SBS, µ = µ = µ, 7 and one can use the sustitution µz z to rewrite Eqs. (A1) (A3) in the orm o Eqs. (46) (48). REFERENCES 1. W. L. Kruer, The Physics o Laser Plasma Interactions, Frontiers in Physics, Vol. 73, edited y D. Pines (Addison-Wesley, Redwood City, CA, 1988).. R. L. McCrory and J. M. Soures, in Laser-Induced Plasmas and Applications, edited y L. J. Radziemski and D. A. Cremers (Dekker, New York, 1989), p J. D. Lindl, Phys. Plasmas, 3933 (1995). 4. M. R. Amin et al., Phys. Rev. Lett. 71, 81 (1993). 5. M. R. Amin et al., Phys. Fluids B 5, 3748 (1993). 6. C. J. McKinstrie, R. Betti, R. E. Giacone, T. Koler, and J. S. Li, Phys. Rev. E 5, 18 (1994). 7. R. E. Giacone, C. J. McKinstrie, and R. Betti, Phys. Plasmas, 4596 (1995); 5, 118 (1998). 8. C. J. McKinstrie, J. S. Li, and A. V. Kanaev, Phys. Plasmas 4, 47 (1997). 198 LLE Review, Volume 75

11 9. A. A. Zozulya, V. P. Silin, and V. T. Tikhonchuk, Sov. Phys. JETP 65, 443 (1987). 1. D. F. DuBois, B. Bezzerides, and H. A. Rose, Phys. Fluids B 4, 41 (199). 11. C. J. McKinstrie and M. V. Goldman, J. Opt. Soc. Am. B 9, 1778 (199). 1. Laoratory or Laser Energetics LLE Review 74, 113, NTIS document No. DOE/SF/ (1998). Copies may e otained rom the National Technical Inormation Service, Springield, VA C. L. Tang, J. Appl. Phys. 37, 945 (1966). 14. R. L. Berger, E. A. Williams, and A. Simon, Phys. Fluids B 1, 414 (1989). 16. V. V. Eliseev et al., Phys. Plasmas 3, 15 (1996). 17. C. J. McKinstrie, J. S. Li, R. E. Giacone, and H. X. Vu, Phys. Plasmas 3, 686 (1996). 18. C. J. McKinstrie, V. A. Smalyuk, R. E. Giacone, and H. X. Vu, Phys. Rev. E 55, 44 (1997). 19. C. J. McKinstrie, A. V. Kanaev, V. T. Tikhonchuk, R. E. Giacone, and H. X Vu, Phys. Plasmas 5, 114 (1998).. R. K. Kirkwood et al., Phys. Rev. Lett. 76, 65 (1996). 1. A. K. Lal, K. A. Marsh, C. E. Clayton, C. Joshi, C. J. McKinstrie, J. S. Li, and T. W. Johnston, Phys. Rev. Lett. 78, 67 (1997). 15. W. L. Kruer et al., Phys. Plasmas 3, 38 (1996). LLE Review, Volume