Collective stimulated Brillouin backscatter
We develop the statistical theory of the stimulated Brillouin backscatter (BSBS) instability of a spatially and temporally partially incoherent laser beam for laser fusion relevant plasma. We find a new regime of BSBS which has a much larger threshold than the classical threshold of a coherent beam in long-scale-length laser fusion plasma. Instability is collective because it does not depend on the dynamics of isolated speckles of laser intensity, but rather depends on averaged beam intensity. We identify convective and absolute instability regimes. Well above the incoherent threshold the coherent instability growth rate is recovered. The threshold of convective instability is inside the typical parameter region of National Ignition Facility (NIF) designs although current NIF bandwidth is not large enough to insure dominance of collective instability and suggests lower instability threshold due to speckle contribution. In contrast, we estimate that the bandwidth of KrF-laser-based fusion systems would be large enough.
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 and hot electron production Lindl2004 (). Among laser-plasma instabilities, the backward stimulated Brillouin backscatter (BSBS) has been considered for a long time as serious danger because the damping threshold of BSBS of coherent laser beams is typically several order of magnitude lower compared with the required laser intensity for ICF. 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 possibleFroulaPRL2007 ().
Theory of laser-plasma interaction (LPI) 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 correlation time, and lengths to suppress laser-plasma interactions. 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 and is the wavelength (see e.g. RosePhysPlasm1995 (); GarnierPhysPlasm1999 ()). Beam smoothing is a part of most constructed and suggested ICF facilities. However, instability theory of smoothed laser beam interaction with plasma is not well developed. There are intense experimental and simulation ongoing efforts FroulaPRL2007 () to determine BSBS threshold for smoothed beams which appears to be in some cases quite low so that it is now under discussion that laser intensity at the National Ignition Facility (NIF) should lowered by a factor of few compared with original NIF designsLindl2004 () with intensities .
Here we develop a theory of collective BSBS instability (CBSBS), which is a new BSBS regime, for propagation of laser beam with finite in homogeneous plasma. CBSBS has threshold comparable with NIF intensities. CBSBS requires small enough to suppress contribution from speckles. If we additionally assume that then CBSBS threshold does not depend on . Such is accessible to KrF lasers Weaver2007 (), , but not for NIF glass lasers with beam smoothing up to 3Å at , implying at . This is consistent with the numerical simulations which show that BSBS threshold in NIF emulation experiments is dominated by speckles FroulaPRL2007 (); BergerPrivate2007 (). We show below that speckle-dominated threshold is lower by a factor 7 than CBSBS threshold. Since plasma inhomogeneity can only increase instability threshold Kruer1990 (), The CBSBS threshold is a lower bound. Fig. 1 depicts CBSBS between large speckle regime RoseDuBois1994 () and random phase approximation (RPA) vedenov1964 (); DuBoisBezzeridesRose1992 (); PesmeBerger1994 () regime.
Assume that laser beam propagates in plasma with frequency along with the electric field 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 with phase speed and wavevector with plasma density fluctuation given by where is the slow envelope and is the average electron density, assumed to be small compared to critical density, . The coupling of and to plasma density fluctuations gives, ignoring light wave damping,
, and is described by the acoustic wave equation coupled to the pondermotive force which results in the envelope equation
where we neglected terms in r.h.s. which are responsible for self-focusing effects, is the Landau damping of ion-acoustic wave and is the scaled acoustic damping coefficient. and are in thermal units (see e.g. LushnikovRosePRL2004 ()).
Assume that laser beam was made partially incoherent through induced spacial incoherence beam smoothing LehmbergObenschain1983 () which defines stochastic boundary conditions at for the spacial Fourier transform (over ) components , of laser beam amplitude LushnikovRosePRL2004 ():
chosen to the idealized ”top hat” model of NIF optics polarization (). Here and the average intensity, determines the constant.
In linear approximation, assuming so that only laser beam is BSBS unstable, we can neglect right hand side (r.h.s.) of Eq. (2). The resulting linear equation with top hat boundary condition (Collective stimulated Brillouin backscatter) has the exact solution as decomposition of into Fourier series, with Eq. (3) is linear in and which implies that can be also decomposed into We approximate r.h.s. of (4) as so that
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 (independent hot spot model RoseDuBois1993 (); RosePhysPlasm1995 ()). Speckle contribution can can be neglected if MounaixPRL2000 ()
where is the characteristic time scale at which BSBS convective gain saturates.
We use the linear part of the theory of Ref. MounaixPRL2000 () to estimate for speckle contribution to backscatter as , where and we choose the typical intensity of light in speckle , where is the spatial average of laser intensity RoseDuBois1993 (); GarnierPhysPlasm1999 (); MounaixPRL2000 (). In such a case , where here and below designates the scaled dimensionless laser intensity defined as . For typical NIF parameters Lindl2004 (); LushnikovRosePlasmPhysContrFusion2006 (), and we obtain from (7) the estimate which is not satisfied for low plasma ionization number plasma in NIF which typically has . However, CBSBS can still be relevant for NIF in gold plasma near hohlraum Lindl2004 () with . Similar estimate for KrF lasers () gives which is easier to satisfy because of smaller and suggests that KrF lasers are better suited for applicability of CBSBS.
where is the transverse unit of length, is the unit in direction, is the time unit and .
The dispersion relation (Collective stimulated Brillouin backscatter) is correct provided the temporal growth rate is small compare to inverse time of light propagation along speckle, , and if during time light travels much further than a speckle length, . That second condition ensures that term could be neglected in Eq. (3) allowing the time dependence of in Eqs. (3) and (Collective stimulated Brillouin backscatter) to be ignored and in such case density fluctuation evolves without fluctuations. E.g. for typical NIF parameters, we obtain that which is well satisfied for NIF optics Lindl2004 ().
In the continuous limit , sum in (Collective stimulated Brillouin backscatter) is replaced by integral which gives for most unstable mode :
Eq. (9) has branch cut in complex plane determined by points and . Standard analysis of convective vs. absolute instabilities (see e.g. Briggs1964 ()) should be modified to include that branch cut. In discrete case with instead of branch cut the discrete dispersion relation (Collective stimulated Brillouin backscatter) has solutions located near the line . These solutions are highly localized around some so they cannot be approximated by (9) but they are stable for . Generally there are two solutions of (9), however for one solution is absorbed into branch cut. Second solution is stable. Above the convective CBSBS threshold,
the first solution crosses real axis from below as so it describes instability of backscattered wave with
However, above the absolute CBSBS threshold, which can be approximated from solution of Eq. (9) as
the contour cannot be moved down to real axis because of pinching of two solutions of (9) which defines growth rate of absolute instability. We conclude that classical analysis of instabilities still holds for incoherent beam if we additionally allow the absorption of one solution branch into branch cut. This effect results from incoherence of pump beam which has infinitely many transverse Fourier modes in approximation of Eq. (9) and there is no counterpart of that effect for coherent beam.
For the absolute threshold (11) reduces to the coherent absolute BSBS instability threshold
For NIF parameters, with moderate acoustic damping, , we obtain in dimensional units and For high plasma (e.g. gold plasma near the wall of NIF hohlraum Lindl2004 (), ) we obtain and Typical intensity of NIF laser shots is between and so we conclude that in different parts of NIF plasma both convective and absolute instabilities are possible. Fig. 2 compares instability gain rate of coherent and incoherent beams for .
In contrast with Eq. (10), the convective instability threshold in coherent case is 0 because we neglect damping of in Eq. (3). Retaining collisional light damping gives finite threshold , where Kruer1990 () is the collisional damping of backscattered wave and is the electron-ion collision frequency. That threshold is several orders of magnitude smaller compared with (10) and is neglected here. Qualitatively incoherence of laser beam can be considered as effective damping of with effective damping rate .
Depending on laser incoherence we have a hierarchy of thresholds: (a)Spatially incoherent laser beam with large has threshold, which is dominated by intense speckles RoseDuBois1994 (). (b)Spatial and temporary incoherent beam with satisfying (7) is given by (10) which factor times higher compared with speckle threshold and does not depends on It indicates practical limit of how threshold of BSBS instability can be increased by decreasing (c)For much smaller , such that it is smaller than both inverse acoustic damping and inverse temporal growth rate , the classical RPA regime is recovered which has ignorable diffraction (vedenov1964 (); DuBoisBezzeridesRose1992 (); PesmeBerger1994 ()). This limit (e.g. for and it requires ) is not practical for ICF as is too small. Cases (a)-(c) are shown in Fig. 1.
Current NIF 3Å beam smoothing design is between regimes (a) and (b). KrF laser with would be in regime (b). Thus generally we expect that KrF-laser-based ICF allows access to CBSBS regime although CBSBS threshold for NIF can be possibly initiated by self-induced temporal incoherence (see e.g. SchmittAfeyan1998 ()). Another possibility for self-induced temporal incoherence is through collective forward stimulated Brillouin scatter (CFSBS) instability LushnikovRosePRL2004 (); LushnikovRosePlasmPhysContrFusion2006 (). Above CFSBS threshold correlation length decreases with beam propagation length and may decrease . For low plasma threshold for CFSBS is close to (10) LushnikovRosePRL2004 (). As increases (which can be achieved by adding high dopant), CFSBS threshold decreases below (10) and might result in decrease of
To distinguish contribution to BSBS from speckles (regime (a)) and CBSBS (regime (b)) we propose to look at angular divergence of BSBS. In general one expects gain narrowing of the scattered light: the modes close to the most unstable mode, with gain rate , dominate. Here . Fig. 3 shows from CBSBS as a function of laser intensity above CBSBS threshold at propagation distance . is chosen from the physical condition that there is sufficient convective CBSBS gain, to amplify the energy of thermal acoustic fluctuations at wavenumber to have reflectivity , and for fusion plasma this is typically (see e.g. FroulaPRL2007 ()), where is the power convective gain exponent. Then is conventionally defined by half width at half maximum: . Important feature of CBSBS seen in Fig. 3 is that at threshold with and near threshold. Fig. 3 should be compared with from speckle-dominated backscatter. Previous work DivolMounaixPRE1998 () suggested that speckles can also cause below top-hat width, , for very intense speckle backscatter. We estimate based on Refs. RosePhysPlasm1995 (); GarnierPhysPlasm1999 ()) that for nominal ICF plasma ( speckle volumes), most intense speckle is which gives near CBSBS threshold, where is the the gain over speckle with the average intensity We performed direct simulations of backscatter from speckle and found that which means that asymptotic DivolMounaixPRE1998 () is still not applicable. In other words, finite size plasma effects dominates over asymptotic theory of infinite plasma. We conclude that regime (a) can be easily distinguished from CBSBS regime (b): near CBSBS threshold with condition (7) satisfied one should see backscattered light spectrum with essential peak whose width is given by Fig. 3 and wide weak background determined by speckles.
In summary, we found a novel coherence time regime in which is too large for applicability of well-known statistical theories (RPA) but rather an intermediate regime, is small enough to suppress speckle BSBS. Unlike coherent beam CBSBS has threshold typically much larger than that determined by damping while for laser intensity many times above convective instability threshold for incoherent beam, the coherent theory is recovered.
We acknowledge helpful discussions with B. Afeyan, R. Berger, L. Divol, D. Froula, and N. Meezan. This work was carried out under the auspices of the NNSA of the DOE at LANL under Contract No. DE-AC52-06NA25396
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