Beyond the random phase approximation: Stimulated Brillouin backscatter for finite laser coherence times
We develop a statistical theory of stimulated Brillouin backscatter (BSBS) of a spatially and temporally partially incoherent laser beam for laser fusion relevant plasma. We find a new collective regime of BSBS (CBSBS) with intensity threshold controlled by diffraction, an insensitive function of the laser coherence time, , once light travel time during exceeds a laser speckle length. The BSBS spatial gain rate is approximately the sum of that due to CBSBS, and a part which is independent of diffraction and varies linearly with . We find that the bandwidth of KrF-laser-based fusion systems would be large enough to allow additional suppression of BSBS.
Inertial confinement fusion (ICF) experiments require propagation of intense laser light through underdense plasma subject to laser-plasma instabilities which can be deleterious for achievement of thermonuclear target ignition because they can cause the loss of target symmetry, energy and hot electron production Lindl2004 (). Among laser-plasma instabilities, backward stimulated Brillouin scatter (BSBS) has long been considered a serious danger because the damping threshold of BSBS of coherent laser beams is typically several order of magnitude less then the required laser intensity for ICF. BSBS may result in laser energy retracing its path to the laser optical system, possibly damaging laser components Lindl2004 (); MeezanEtAlPhysPlasm2010 (). Recent experiments for a first time achieved conditions of fusion plasma and indeed demonstrated that large levels of BSBS (up to tens percent of reflectivity) are possible FroulaPRL2007 ().
Theory of laser-plasma interaction instabilities is well developed for coherent laser beam Kruer1990 (). However, ICF laser beams are not coherent because temporal and spatial beam smoothing techniques are currently used to produce laser beams with short enough correlation time, and lengths to suppress self-focusing Kruer1990 (); Lindl2004 (); MeezanEtAlPhysPlasm2010 (). The laser intensity forms a speckle field - a random in space distribution of intensity with transverse correlation length and longitudinal correlation length (speckle length) , where is the optic -number and is the wavelength (see e.g. RosePhysPlasm1995 (); GarnierPhysPlasm1999 ()). There is a long history of study of amplification in random media (see e.g vedenov1964 (); PesmeBerger1994 () and references there in). For small laser beam correlation time , the spatial instability increment is given by a Random Phase Approximation (RPA). Beam smoothing for ICF typically has much above the regime of RPA applicability. There are few examples in which implications of laser beam spatial and temporal incoherence have been analyzed for such larger . One exception is forward stimulated Brillouin scattering (FSBS). We have obtained in Refs. LushnikovRosePRL2004 (); LushnikovRosePlasmPhysContrFusion2006 () the FSBS dispersion relation for laser beam which has the correlation time too large for RPA relevance, but still small enough to suppress single laser speckle instabilities RoseDuBois1994 (). We verified our theory of this “collective” FSBS instability regime with 3D simulations. Similar simulation results had been previously observed SchmittAfeyan1998 ().
This naturally leads one to consider the possibility of a collective regime for BSBS (CBSBS). We present 2D and 3D simulation results as evidence for such a regime, and find agreement with a simple theory that above CBSBS threshold, the spatial increment for backscatter amplitude , is well approximated by the sum of two contributions. The first contribution is RPA-like without intensity threshold (we neglect light wave damping). The second contribution has a threshold in laser intensity. That threshold is in parameter range of ICF hohlraum plasmas such as at the National Ignition Facility (NIF) Lindl2004 () and the Omega laser facility (OMEGA) NiemannPRL2008 () experiments. The existence of threshold was first predicted in Ref. LushnikovRoseArxiv2007 () in the limit where is the speed of flight RemarkReplete (). The second contribution is collective-like because it neglects speckle contributions and is only weakly dependent on . CBSBS threshold is applicable for strong and weak acoustic damping coefficient . The theory also demonstrates a good quantitative prediction of the instability increment for small which is relevant for gold plasma near the wall of hohlraum in NIF and OMEGA experiments Lindl2004 (); NiemannPRL2008 ().
The paper is organized as follows. In Section II we introduce the basic equations of BSBS for laser-plasma interaction and the stochastic boundary conditions which correspond to the partial incoherence of laser beam. In Section III we analyze the linearized BSBS equations and find the dispersion relations. In Section IV the convective versus absolute instabilities are analyzed from the dispersion relations. Section V describes the details of the performed stochastic simulations of the full linearized equations. In section VI the conditions of applicability of the dispersion relation are discussed as well as the estimates for typical ICF experimental conditions are given. In Section VII the main results of the paper are discussed.
Ii Basic equations
Assume that laser beam propagates in plasma with frequency along . The electric field is given by
where is the envelope of laser beam and is the envelope of backscattered wave, , and c.c. means complex conjugated terms. Frequency shift is determined by coupling of and through ion-acoustic wave of phase speed and wavevector with plasma density fluctuation given by where is the slow envelope (slow provided ) and is the average electron density, assumed small compared to the critical electron density . We consider a slab model of plasma (plasma parameters are uniform). The coupling of and to plasma density fluctuations gives
, and is described by the acoustic wave equation coupled to the pondermotive force which results in the envelope equation
Here we neglected terms in the right-hand side (r.h.s.) which are responsible for self-focusing effects, is the Landau damping of ion-acoustic wave and is the scaled acoustic Landau damping coefficient. and are in thermal units (see e.g. LushnikovRosePRL2004 ()) defined so that if we add self-focusing term in r.h.s. of Eq. (II) then in equilibrium, with uniform , the standard is recovered.
We use a simple model of induced spacial incoherence beam smoothing LehmbergObenschain1983 () which defines stochastic boundary conditions at for spatial Fourier transform (over ) components , of laser beam amplitude LushnikovRosePRL2004 ():
is chosen as the idealized “top hat” model of NIF optics polarization (). Here means the averaging over the ensemble of stochastic realizations of boundary conditions, is the top hat width and the average intensity, determines the constant.
Iii Linearized equations and dispersion relations
In linear approximation, assuming so that only the laser beam is BSBS unstable, we neglect right hand side (r.h.s.) of Eq. (2). The resulting linear equation with boundary condition (II) has the exact solution as decomposition of into Fourier series,
Figures 1 show the increment of the spatial growth of backscattered light intensity as a function of the rescaled correlation time obtained from the numerical solution of the stochastic linear equations (3)-(III) (details of numerical simulations are provided in Section V), the scaled damping rate and the scaled laser intensity . These scaling quantities are defined as
Here has the meaning of the correlation time in units of the acoustic transit time along speckle. (Note that definition of is different by a factor from the definition used for FSBS LushnikovRosePRL2004 (); LushnikovRosePlasmPhysContrFusion2006 (), where units of the transverse acoustic transit time through speckle were used.) We use dimensionless units with as the unit in direction, and is the time unit. means averaging over the statistics of laser beam fluctuations (II). is the damping rate in units of the inverse acoustic propagation time along a speckle. (See also Figure 4 below for illustration of intensity normalization in comparison with physical units.)
We relate to the instability increments for and (we designate them and , respectively). In general, growth rates of mean amplitudes give a lower bound to . However, according to Figure 2, is almost coherent on a time scale justifying the use of mean values of amplitudes.
First we look for the expression for . Eq. (3) is linear in and implying that can be decomposed into with
Approximating r.h.s. of (II) as gives
which means that we neglect off-diagonal terms Since speckles of laser field arise from interference of different Fourier modes, we associate the off-diagonal terms with speckle contribution to BSBS RoseDuBois1993 (); RosePhysPlasm1995 (); RoseMounaixPhysPlasm2011 (). Neglecting off-diagonal terms requires that during time light travels much further than a speckle length, and that , where is the characteristic time scale at which BSBS convective gain saturates at each speckle MounaixPRL2000 ().
with the Fourier transformed given by
where is the Heaviside step function.
We assume that is slow in comparison with (consistent with Figure 2) which allows to approximate fluctuating terms in r.h.s. of (11) as which has the same form as the Bourret approximation PesmeBerger1994 () and provides the closed expression for as follows
where the kernel of the response function is the inverse Fourier transform of (12) and the laser beam correlation function is given by
We look for solution of (III) in exponential form , then the exponential time dependence of (III) allows to carry all integrations in (12) and (III) explicitly to arrive at the following dispersion relation in dimensionless units
In the continuous limit , sum in (III) is replaced by integral, giving for the most unstable mode :
which supports the convective instability with the increment only for , where is the convective CBSBS threshold given by
In the limit , the increment is independent of which suggests that we refer to it as the collective instability branch. For finite but small and there is sharp transition of as a function of from 0 for to -independent value of . That value can be obtained analytically from (16) for just above the threshold as follows: absolutecomment ().
The increment is obtained in a similar way by statistical averaging of equation (3) for which gives
with the Fourier transformed response function
Then the Bourret approximation (18) results in the following closed expression for :
We look for solution of (III) in exponential form , then the exponential time dependence of (III) allows to carry all integrations in (19) and (III) explicitly to arrive at the following dispersion relation in dimensionless units
Here we neglected the contribution to from diffraction and used the condition Equation (III) does not have a convective threshold (provided we neglect here light wave damping) while has near-linear dependence on for which is typical for RPA results. It suggests that we refer as the RPA-like branch of instability.
Solving the equations (16) and (III) numerically for allows to find and , respectively, for given . We choose in (16) and in (III) to maximize and , respectively. Figures 1a and 1b show that the analytical expression is a reasonably good approximation for numerical value of above the convective threshold (17) for which is the main result of this paper. Below this threshold analytical and numerical results are only in qualitative agreement and we replace by because in that case.
The qualitative explanation why is a surprisingly good approximation to is based on the following argument. First imagine that propagates linearly and not coupled to the fluctuations of , so its source is in r.h.s of (3). If grows slowly with (i.e. if changes a little over the speckle length and time ), then so will at the rate . But if the total linear response ( is the renormalization of bare response due to the coupling in r.h.s of (3)) is unstable then its growth rate gets added to in the determination of since in all theories which allow factorization of 4-point correlation function into product of 2-point correlation functions, . Here and etc. mean a set of all spatial and temporal arguments.
Iv Convective instability versus absolute instability
In this Section we show that the dispersion relations (16) and (III) predict absolute instability for large intensities. We first consider the dispersion relation (16) which has branch cut in the complex -plane connecting two branch points and .
Absolute instability occurs if the contour in the complex -plane cannot be moved down to real axis because of pinching of two solutions of (16) in the complex -plane Briggs1964 (),PitaevskiiLifshitzPhysicalKineticsBook1981 (). To describe instability one of these solutions must cross the real axis in -plane as the contour is moving down. The pinch occurs provided
The pinch condition (22) together with the requirement of crossing the real axis in -plane result in
for the absolute instability threshold intensity . Assuming we obtain from (IV) the explicit expression for the CBSBS absolute instability threshold
The absolute instability threshold for the second RPA-like branch (III) is obtained similarly with the pinch condition It gives the absolute instability threshold for RPA-like branch of instability
For the absolute threshold (25) reduces to the coherent absolute BSBS instability threshold
V Numerical simulations
We performed two types of simulations. First type is simulations (three spatial coordinates , and ) of Eqs. (3), (II) and (III) with the boundary and initial conditions (II),(6) in the limit (i.e., setting in (3) and (II)). It implies that the phases in (III) become only -dependent, . That formal limit , is consistent provided . Then in the linear instability regime, the laser field, , at any time may be obtained by propagation from while the scattered light field, is obtained by backward propagation from . Time scales are now set by the minimum of and the acoustic time scale for the density . We performed numerical simulation of and via a split-step (operator splitting) method. advances only due to diffraction and is determined exactly by (III). For given , is first advanced due to diffraction in transverse Fourier space, and then the source term (r.h.s. of (3) which is ) is added for all . The density is evolved in the strong damping approximation in which the term is omitted from equation (II). In the regimes of interest, in particular near the collective threshold (17) regime, the dimensionless damping coefficient in (II) increases with acoustic Landau damping coefficient, and even for its physically smallest value of , the scaled damping is approximately while is either or (an inverse speckle length). So given and , may be advanced in time at each , for each transverse Fourier mode, or since the transverse Laplacian term is estimated as unity in magnitude (base on the speckle width estimate of ), may be approximately advanced at each spatial lattice point.
chosen to mimic the extra factor in the integral over transverse direction of the full problem. That modified top hat choice ensures that the linearized equations of that problem give exactly the same analytical solutions (16) and (III) as for the full problem. We used again the split step method by integrating along the characteristics of and and solving for the diffraction by Fourier transform in the transverse coordinate .
We run simulations in the box . For both types of simulations the boundary conditions for were set at . We take these boundary conditions as the Fourier modes of in with random time-independent phases. These modes correspond to the random seed from the thermal fluctuations. The boundary conditions for were set to be zero. As the time progress from the beginning of each simulation, both and grow until reaching the statistical steady state if the is below the threshold of absolute instability (25). Figure 3 shows a typical time dependence of , where means averaging over the transverse coordinate . Because we solve linear equations (3) and (II), the maximum value of grows if we increase as well as the boundary condition is defined up to the multiplication by the arbitrary constant. -dependence of in the statistical steady state follows the exponential law well inside the interval . Near the boundaries and there are short transition layers before solution settles at law inside that interval. The particular form of the boundary conditions for and affect only these transition layers while law is insensitive to them.
To recover with high precision we performed simulations for long time after reaching statistical steady state and average over that time at each (i.e. we assumed ergodicity). E.g. for (the time the laser light travels along laser speckles) we use 256 transverse Fourier modes and discrete steps in dimensionless units with the typical length of the system ( speckle lengths) and a time step . For this particular set of parameters it implies . For simulation we typically wait time steps to achieve a robust statistical steady state and then average over another time steps (together with averaging over the transverse coordinates) to find with high precision. Figures 1a and 1b show extracted from and simulations, respectively.
For the practical purposes it is also interesting to estimate the time at which the initial thermal fluctuations of are amplified by to reach the comparable intensity with the laser pump. We obtained from simulations that for two laser speckles (relevant for gold plasma in ICF experiments and corresponds to in dimensionless units), and . In dimensional units for NIF conditions ps which is well below hydrodynamic time (several hundreds of ps).
Vi Applicability of the dispersion relation and estimates for experiment
and as well as . Here is the temporal growth rate of the spatially homogeneous solution given by Also is the bandwidth for and is the effective bandwidth for . is dominated by the diffraction in (3) giving in the dimensionless units . Then (29) reduces to and . Together with the condition used in the derivation of (16) and assuming that , it gives a double inequality which can be well satisfied for , i.e. for as in gold ICF plasma but not for as in low ionization number ICF plasma. Also implies that because otherwise, below that threshold, which would contradict . All these conditions are satisfied for for the parameters of Figure 1 with or (solid lines in Figure 1) but not for (dashed lines in Figure 1). Additionally, an estimate for from the linear part of the theory of Ref. MounaixPRL2000 () results in the condition which is less restrictive than above conditions. These estimates are consistent with the observed agreement between and from simulations (filled circles in Figures 1) for above the threshold (17). We conclude from Figures 1 that the applicability condition for the Bourret approximation is close to the domain of values for which .
For nominal NIF parameters Lindl2004 (); LushnikovRosePlasmPhysContrFusion2006 (), , and electron plasma temperature ( was recently updated from the old standard value RosenInvitedAPRPlasma2011 ()), we obtain from (17) that for gold plasma with which is in the range of NIF single polarization intensities. Fig. 4 shows in the limit from simulations, analytical result ( in that limit) and the instability increment of the coherent laser beam (see e.g. Kruer1990 ()). It is seen that the coherent increment significantly overestimates numerical increment especially around The convective increment has a significant dependence on if we include the effect of finite and finite as in Fig. 1b. Current NIF 3Å beam smoothing design has ps implying . In that case Fig. 1b shows that there is a significant (about 5 fold) change in between and . Similar estimate for KrF lasers (ps) gives which results in a significant () reduction of for compare with above NIF estimate.
The BSBS threshold may be reduced by self-induced temporal incoherence (see e.g. SchmittAfeyan1998 ()), which in its linear regime, includes collective FSBS (CFSBS) which reduces and laser correlation lengths. For low plasma, the CBSBS and CFSBS thresholds are close while the latter may be lowered by adding higher dopant.
In conclusion, we identified the collective threshold (17) of BSBS instability of partially incoherent laser beam for ICF relevant plasma. Above that threshold the CBSBS increment is well approximated by the sum of the collective-like increment and RPA-like increment . That result is in agreement with the direct stochastic simulations of BSBS equations. Values of and are comparable above threshold while in a small neighborhood of threshold the value of changes quickly with changing either correlation time or laser intensity to pass through collective threshold. With further increase of laser intensity the absolute instability also develops above the threshold (25).
Acknowledgements.We acknowledge helpful discussions with R. Berger and N. Meezan. P. L. and H. R. were supported by the New Mexico Consortium and Department of Energy Award No. DE-SCOO02238 as well as by the National Science Foundation under Grants No. PHY 1004118, and No. PHY 1004110. A. K. was partially supported by the Program “Fundamental problems of nonlinear dynamics” from the RAS Presidium and “Leading Scientific Schools of Russia” grant NSh-6170.2012.2.
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