Two-Dimensional Analysis of the Power Transfer between Crossed Laser Beams

Transcription

1 Two-Dimensional Analsis of the Power Transfer between Crossed Laser Beams The indirect-drive approach to inertial confinement fusion involves laser beams that cross as the enter the hohlraum. Ionacoustic waves in the plasma at the overlap region can transfer power between the beams. Since this could adversel affect the implosion smmetr it is important to understand the mechanisms that make such a transfer possible. In this contet two studies have been made of the interaction of crossed laser beams mediated b an ion-acoustic wave. Kruer et al. performed a one-dimensional analsis of the stead-state power transfer emphasizing the effects of different beam frequencies and the inhomogeneit of the plasma. Eliseev et al. 3 performed two-dimensional simulations of the interaction of equal-frequenc beams in a homogeneous plasma. In addition to observing a time-dependent power transfer between the beams the observed several secondar processes and supplemented their numerical simulations with one-dimensional analses of certain processes. Here we present a two-dimensional analsis of the power transfer between beams of unequal frequenc in a homogeneous plasma for both the transient and steadstate regimes. wave divided b the background electron densit and the signif that onl the low-frequenc response to the ponderomotive force was retained. P55 Beam θ Beam ξ Figure 66.3 Geometr of the interaction of crossed laser beams. The beam widths are equal and denoted b w and the beam intersection angle is denoted b θ. The characteristic coordinates ξ and η measure distance in the propagation directions of beams and respectivel. η w Governing Equations Laser beams that cross interact via ion-acoustic waves in the irradiated plasma. The interaction geometr is shown in Fig and is governed b Mawell s wave equation 4 ( tt + ωe c ) Ah = ωe nlah () and B substituting the Ansaetze = ( ) Ah ξη t A ξη t ep i kξ ωt + A( ξη t) ep [ i( kη ωt) ]+ cc.. (3) for the electromagnetic potential together with the ion-acoustic (sound) wave equation 4 ( + v c ) n = c A. () tt s t s l s h nl( ξη t)= n( ξη t) ep [ i( kξ k η) ]+ cc.. (4) into Eqs. () and () and making the slowl varing envelope approimation one can show that = The electromagnetic potential Ah vh cs me mi is the quiver velocit of electrons oscillating in the high-frequenc electric field divided b a characteristic speed that is of the order of the electron thermal speed n l is the low-frequenc electron-densit fluctuation associated with the ion-acoustic A i ω ω v na ep iωt ξ A i ω = ( ω v ) n* A ep iωt η = e e (5) LLE Review Volume 66 73

2 and ( tt s t s ) = s + v + ω n ωa A * ep iωt (6) where v is the group velocit of the higher-frequenc beam; v is the group velocit of the lower-frequenc beam; ω = ω ω is the difference between the beam frequencies; k s = k k is the ion-acoustic wave vector; and ω s = c s k s is the ionacoustic frequenc. The characteristic variables ξ and η measure distance in the propagation directions of beams and respectivel. The time derivatives were omitted from Eqs. (5) because the time taken for the laser beams to cross the interaction region is much shorter than the time taken for the ion-acoustic wave to respond to the ponderomotive force. Stead-State Analsis In stead state the beams interact according to ξ A = iα β A A ηa = iα + β A A where the nonlinear coefficients ωeωs ωs ω α = ω ( ω ω) + ω v s 4v s ω ω ω β = e svs ω ( ω ω). + ω v s 4v s Since ω << ω the differences between ω and ω and v and v were neglected in the first of Eqs. (7). 5 In the Lorentzian approimation (7) (8) ω ω δ ω ω α e s β e sv s (9) 4ω v δ + v 4ω v δ + v ( s ) ( s ) where the frequenc-detuning parameter δ = ω ω s. The coefficients α and β characterize the real and imaginar parts respectivel of the ion-acoustic response to the ponderomotive force [see Eq. (6)]. The are plotted as functions of δ in Fig for the case in which v s ω s =.. Both coefficients are normalized to ωω e s 4ωωsv vs which apart from a factor of A is the spatial growth rate of stimulated Brillouin scattering (SBS) in the strong-damping limit. Although the Lorentzian approimation for α becomes less accurate as the magnitude of the frequenc-detuning parameter increases the Lorentzian approimation for β is accurate for arbitrar frequenc detuning. For values of v s ω s larger than. there are significant discrepancies between the approimate and eact epressions for both coefficients. Real response α Imaginar response β P Frequenc detuning δ Figure 66.3 Nonlinear coefficients α and β [Eqs. (7)] normalized to the resonant gain coefficient plotted as functions of the frequenc detuning parameter δω s for the case in which v s ω s =.. The coefficients α and β characterize the real and imaginar parts respectivel of the ion-acoustic response to the ponderomotive force. The solid lines represent the eact coefficients [Eqs. (8)] and the dashed lines represent the approimate coefficients [Eqs. (9)]. Equations (7) are solved subject to the boundar conditions = = () A η A A ξ ρa where ρ is the ratio of the amplitudes of the incident beams. B changing variables according to 74 LLE Review Volume 66

3 one can reduce Eqs. (7) to B = A ξ ep iα A d ξ η B = A ep iα A dη ξ η () B = β B B B = β B B. () It is convenient to define the normalized intensities I = B A and I = B A the normalized distances = β A ξ and = β A η is the power in the cross section of beam that is and P l a distance from the entrance to the interaction region. B combining Eqs. (3) one can show that from which it follows that P r ep P (6) = { } (7) P( )= log ep( r) ep ( ). It then follows from Eq. (7) and the relations I = P and I = rep ( P) that ep I( )= ep( r) + ep and the normalized beam width l = β A w sin θ r ep r I( )=. ep( r) + ep (8) where w is the phsical beam width and θ is the beam intersection angle (see Fig. 66.3). In terms of these dimensionless variables Eqs. () become I = I I I = I I (3) and the boundar conditions [Eqs. ()] become = = (4) I I r where r = ρ is the ratio of the beams intensities. Despite the fact that Eqs. (3) are nonlinear and describe beam propagation in two directions there is a wa to solve them analticall. 67 It is convenient to define P( )= I d P ( )= I d. (5) is the power in the cross section of beam Phsicall P l that is a distance from the entrance to the interaction region B combining Eqs. (3) one can also show that { } (9) = + [ ] P log ep ep r. Equation (9) and the relations I = P and I = ep( P) are consistent with solutions (8). The beam-intensit profiles are displaed in Fig for the case in which l = 3 and r =.. Notice that the intensit of beam is nearl constant and the intensit of beam is nearl independent of. When rl << as it is for Fig Eqs. (8) reduce to I and I r ep() in agreement with the linearized versions of Eqs. (3). The beam-intensit profiles are also displaed in Fig for the case in which l = 3 and r =.. Notice that the intensit profiles are highl two-dimensional. Beam is depleted as it propagates in the direction and beam is amplified as it propagates in the direction. Consequentl the depletion of beam along the characteristic = l is more rapid than its depletion along the characteristic = and the of beam along the characteristic = is more rapid than its along the characteristic = l. LLE Review Volume 66 75

4 It follows from Eqs. (7) and (9) that = () P P P P ; the power gained b beam must equal the power lost b beam. The power transfer P( l l) P( l ) is denoted b T(l) and is given b { } ()= ( )+ () ( ) T l log ep rl ep l ep rl. () Since the normalized incident power is l the fractional power transfer is Tl () l. This fractional power transfer is plotted as a function of l in Fig It is not difficult to show that Tl () l r[ ep() l ] for rl << and l ~ and Tl () l for l >>. Despite the compleit of the beam evolution which is twodimensional and nonlinear the power transfer is characterized b two dimensionless parameters. The first r is simpl the ratio of the incident beam intensities. The second l depends on several dimensional parameters that characterize the beams (a) (b) I P I Figure 66.3 Beam-intensit profiles [Eqs. (8)] for the case in which the normalized beamwidth l = 3 and the ratio of the incident beam intensities r =.. Notice that the intensit of beam is nearl constant and the intensit of beam is nearl independent of as linear theor predicts. (a) (b). I.5. P I Figure Beam-intensit profiles [Eqs. (8)] for the case in which the normalized beamwidth l = 3 and the ratio of the incident beam intensities r =.. Notice that the intensit profiles are highl two-dimensional. 76 LLE Review Volume 66

5 Power transfer P Beamwidth Figure Fractional power transfer from beam to beam [Eq. ()] plotted as a function of the normalized beamwidth l for two values of r the ratio of the incident beam intensities. The dashed line corresponds to r =. and the solid line corresponds to r =.. and the plasma. As a numerical eample suppose that the electron densit n e = cm 3 the electron temperature T e = kev the ion temperature T i =.5 kev the laser wavelength λ =.35 µm the laser intensit I = 5 Wcm and the beamwidth w = mm. For these parameters l.7. One can infer the value of l for other parameters b using the fact that with the electron-to-ion temperature ratio fied l is proportional to n e λ I w and is inversel proportional to T e. Since Eq. () is valid for < θ < π the angular dependence of l is also of interest. When δ = β A is the spatial growth rate of SBS. For an ion-acoustic wave subject to Landau damping this growth rate is independent of θ. 89 In this case l is inversel proportional to sin θ: the power transfer is larger for beams that are nearl parallel or antiparallel because the overlap for a longer distance. The importance of δ is measured relative to ω s and v s both of which are proportional to sin( θ ). Thus the power transfer is more sensitive to detuning when the beams are nearl parallel and less sensitive when the beams are nearl antiparallel. When δ the interaction of beams and causes their phases to be shifted b φ and φ respectivel. It follows from Eqs. () and (5) that φ ( )= αp ( l ) β I φ ( )= αp ( l) β I. () and P ( l ) are plotted as The normalized phase shifts P l functions of position in Fig An observer traveling with Phase shift of beam Phase shift of beam P Distance () Distance () Figure Normalized phase shifts [Eqs. ()] plotted as functions of position for the case in which the normalized beamwidth l = 3 and for two values of r the ratio of the incident beam intensities. The dashed lines correspond to r =. and the solid lines correspond to r =.. The spatial inhomogeneit of these phase shifts causes the beams to be deflected. either beam would measure a larger normalized phase shift on the left side of the beam. For beams of moderate width (l ~ ) the variation of phase with distance is approimatel linear and the beam deflection angles θ and θ are easil estimated. It follows from the laws of geometrical optics that [ ()] θ φ () l φ k w θ φ φ l k w. (3) B combining Eqs. () and (3) and neglecting the difference between k and k one can show that [ () ] θ θ α k sin θ T l l. (4) LLE Review Volume 66 77

6 Both beams are deflected in the same angular direction: anticlockwise when δ < and clockwise when δ >. Because the fractional power transfer depends on l and hence on β the beam deflection angle [Eq. (4)] depends on both α and β. It is evident from Fig that the magnitude of the beam deflection angle is largest when δ ~ v s. When δ = or δ = ω s the beams are not deflected. For wide beams (l >> ) the phase of beam still varies approimatel linearl with distance and the first of Eqs. (4) is still valid. Unfortunatel the variation with distance of the phase of beam is highl nonlinear and it is difficult to estimate the beam deflection angle and focusing distance. In this case however the power transfer from beam to beam is complete: the irradiation smmetr is destroed and the issues of beam deflection and focusing are irrelevant. = (5) N ξη t n ξη t ep iωt which satisfies the simplified equation = + v iδ N i ω A A *. (6) t s s Subject to the normalized boundar and initial conditions = = (7) A η t A ξ η which differ from the phsical conditions b a factor of ρa and the initial condition N( ξη )= (8) the solution of Eq. (6) and the second of Eqs. (5) is The beam deflection angle is larger for beams that are nearl parallel or antiparallel because the nonlinear phase shifts that deflect the beams are proportional to the power transfer. For the same reason the beam deflection angle is more sensitive to detuning when the beams are nearl parallel and less sensitive to detuning when the beams are nearl antiparallel. Transient Analsis Equations (8) and () describe completel the steadstate power transfer between beams and. However it is important to know how long the beam interaction takes to reach stead state. If this saturation time is comparable to the duration of the interaction the transient power transfer must also be determined. where t A( η t)= + γ ( η v t I t ) γ( η v) [ ] ep vs iδ t dt t N( η t)= ( i ωsa ) I t γ( η v) =( e s s) [ ] ep vs iδ t dt (9) γ ω ω A 4 ω ω (3) The case in which δ = ω s has been studied theoreticall and eperimentall. In stead state beam is unamplified. However the response of the ion-acoustic wave to a stead ponderomotive force includes a resonant transient that is required to satisf the initial conditions. This resonant response produces a frequenc-downshifted component of beam. In turn the frequenc-downshifted component of beam gives rise to a component of the ponderomotive force that drives the ion-acoustic wave resonantl. Because of this feedback mechanism the transient (SBS) grows considerabl and lasts for a time that is long compared to the damping time of the ion-acoustic wave. We consider here the complimentar case in which δ << ω s and the linearized equations can be simplified and solved eactl. It is advantageous to work in terms of the ionacoustic amplitude is the temporal growth rate of SBS in an infinite plasma and I m denotes a modified Bessel function of the first kind of order m (rather than a beam intensit). It is evident from this solution that the linear evolution of beam is one-dimensional. As t A t ( η ) ep γ η v vs iδ ( ) N t i A v i ( η ) ωs s δ ep γ η v vs iδ in agreement with the linearized versions of Eqs. (3). (3) The normalized intensit of beam is plotted as a function of v s t in Fig for the case in which the spatial growth parameter γ η vv s = 3 and for three values of δ. The oscil- 78 LLE Review Volume 66

7 Intensit Intensit (a) δ =.3 v s (b) δ =. v s For a fied value of η the saturation time is inversel proportional to sin ( θ ). 89 However in the present contet the maimal value of η is w sinθ and the saturation time is inversel proportional to sinθsin ( θ ). Thus the saturation time is longer for beams that are nearl parallel or antiparallel and the increase in saturation time is larger for beams that are nearl parallel. It is evident from Figs and that two-dimensional evolution signifies the convective depletion of beam which is a nonlinear effect. When the beams are onl moderatel wide or the ratio of the incident beam intensities is small the stead-state interaction of the beams is approimatel onedimensional and is consistent with the linearized Eqs. (3). For these cases we epect the transient evolution of the beams to be well described b Eqs. (9). However when the beams are wide or the incident intensities are comparable the depletion of beam is significant and the nonlinear Eqs. (5) and (6) must be used to determine the transient evolution of the beams. Intensit P (c) δ = 3. v s 5 5 Time (ν s t) Figure Intensit of beam [which follows from the first of Eqs. (9)] plotted as a function of v s t for the case in which the spatial growth parameter γ η vv s = 3. (a) δ =.3 v s ; (b) δ =. v s ; (c) δ = 3. v s. lations in beam intensit are due to the beating of the driven response and the resonant transient. As the magnitude of δ increases the maimal transient intensit decreases less than the stead-state intensit so the transient becomes more important. For the case in which δ = the linear saturation time is 89 ts γ η v vs. (3) When δ some oscillations persist for a longer time but the beam intensit is of the order of the stead-state intensit at this time: the saturation time does not depend sensitivel on δ and is well approimated b Eq. (3). Numerical Simulations Equations (5) and (6) were solved numericall and the total power passing through the eit boundar of each beam was determined as a function of time. The power of beam is plotted as a function of v s t in Fig for the case in which γ η vv s = 3 and for three values of δ. The dashed line corresponds to r =. for which the maimal is and the solid line corresponds to r =. for which the maimal is. Although plotting the power rather than the absolute power transfer disguises the fact that Tl () l it facilitates a comparison of the analtical and numerical results. In particular the deviation of the two numerical curves from one another signals the onset of nonlinearit. B comparing Figs and one notes that when r =. the predictions of linear theor are quantitativel correct for δ = 3. v s and qualitativel correct for δ =. v s and δ =.3 v s. When r =. the predictions of linear theor are qualitativel correct for δ = 3. v s and incorrect for δ =. v s and δ =.3 v s. The numerical results show that the onset of nonlinearit is more rapid and its effect on the transient and stead-state power is more dramatic when the incident intensit of beam is high or the normalized beam width is large [see the second of Eqs. (8) and (9) and Fig. 66.3]. When nonlinearit is important the interaction saturates more quickl than linear theor predicts. The etent to which LLE Review Volume 66 79

8 Power Power Power P (c) δ = 3. v s 4 (a) δ =.3 v s (b) δ =. v s 5 5 Time (ν s t) Figure Power of beam plotted as a function of v s t for the case in which the spatial growth parameter γ η vv s = 3. The power was determined numericall for two values of r the ratio of the incident beam powers. The dashed line corresponds to r =. for which the maimal is and the solid line corresponds to r =. for which the maimal is. (a) δ =.3 v s ; (b) δ =. v s ; (c) δ = 3. v s. beam is depleted changes in the time taken for the beams to cross the interaction region. In the model Eqs. (5) and (6) this transit time is instantaneous. Thus the depletion of beam allows both beam intensities to adjust to their stead-state profiles on a time scale that is short compared to the damping time scale inherent in linear theor. the Nonlinearit becomes important when γ ηt v argument of the amplifing terms in Eqs. (9) reaches a critical value that is inversel dependent on r. It follows from this condition that the nonlinear saturation time is inversel proportional to γ η. For fied η the nonlinear saturation time is inversel proportional to sin( θ ) as is the linear saturation time [Eq. (3)]. However in the present contet the maimal value of η is w sinθ and the nonlinear saturation time is proportional to cos( θ ). Summar The power transfer between crossed laser beams made possible b an ion-acoustic wave was studied in detail. Despite the compleit of the beam evolution which is two-dimensional and nonlinear a simple formula was derived for the stead-state power transfer. This power transfer depends on two dimensionless parameters: the ratio of the incident beam intensities and the normalized beamwidth. The normalized beamwidth is proportional to the phsical beamwidth and the intensit of the higher-frequenc beam and is inversel dependent on the detuning of the laser difference frequenc from the ion-acoustic frequenc. Numerical simulations showed that the transient power transfer is larger than the stead-state power transfer and usuall oscillates in time. The convective depletion of the higher-frequenc beam saturates the power transfer more quickl than the damping of the ion-acoustic wave. The deflection of each beam b the other was also studied briefl. The analsis of this article is based on the standard model in which the beams are assumed to be monochromatic and their interaction is assumed to be in stead state. This simplified model allows one to understand the basic phsics of the power transfer from the higher-frequenc beam to the lower-frequenc beam. A more realistic model would allow the beams to have man frequenc components. The analsis of such a model in both the transient and stead-state regimes will be the subject of future work. ACKNOWLEDGMENT This work was supported b the National Science Foundation under Contract No. PHY the U.S. Department of Energ (DOE) under Contract No. W-745-ENG-36 the U.S. Department of Energ Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC3-9SF946 the Universit of Rochester and the New York State Energ Research and Development Authorit. The support of DOE does not constitute an endorsement b DOE of the views epressed in this article. REFERENCES. J. D. Lindl Phs. Plasmas 3933 (995).. W. L. Kruer et al. Phs. Plasmas 3 38 (996). 8 LLE Review Volume 66

9 3. V. V. Eliseev W. Rozmus V. T. Tikhonchuk and C. E. Capjack Interaction of Crossed Laser Beams with Plasmas to be published in Phsics of Plasmas. 4. W. L. Kruer The Phsics of Laser Plasma Interactions Frontiers in Phsics Vol. 73 (Addison-Wesle Redwood Cit CA 988) Chaps. 7 and 8. A 5. The normalized action flu amplitudes A and ω v ω v satisf Eqs. (7) eactl. 6. M. Maier W. Kaiser and J. A. Giordmaine Phs. Rev (969). 7. T. Speziale Phs. Fluids (984). 8. C. J. McKinstrie R. Betti R. E. Giacone T. Kolber and J. S. Li Phs. Rev. E 5 8 (994). 9. R. E. Giacone C. J. McKinstrie and R. Betti Phs. Plasmas 4596 (995).. J. S. Li C. J. McKinstrie C. Joshi and K. Marsh Bull. Am. Phs. Soc (994).. K. Marsh C. Joshi and C. McKinstrie Bull. Am. Phs. Soc (994). LLE Review Volume 66 8