Crossed beam energy transfer, CBET, is investigated by taking into account the speckle structure of smoothed laser beams that overlap in a plasma with an inhomogeneous flow profile. Using the two-dimensional simulation code Harmony, it is shown how ponderomotive self-focusing of laser speckles in crossing beams can significantly affect the transfer of energy from one beam to the other. The role of plasma flow in speckle self-focusing is investigated and revisited, in particular its consequences in terms of redirection and increasing angular spread of the laser beams due to beam bending and plasma-induced smoothing, respectively. In close-to-sonic flow, the onset of self-focusing in the beam speckle structure occurs at considerably lower beam intensities than expected for the case without flow. CBET and speckle self-focusing can, hence, occur together when two crossed beams with equal frequency resonantly exchange energy via their ponderomotively driven density perturbations flowing with sound speed. From the simulations, it is found that consequences of ponderomotive self-focusing can be expected above an average intensity threshold scaling as $IL\u223c2\xd71014\u2009W\u2009cm\u22122(\lambda 0/1\u2009\mu m)\u22122(Te/$ keV $)$, with an impact on the spatial and temporal coherence of the transmitted light. The density perturbations due to the ponderomotive force of the crossing beams can locally be enhanced in self-focusing speckles, partly leading to shock-like structures. These structures eventually increase the effect of plasma-induced smoothing and are at the origin of the stronger angular spread.

## I. INTRODUCTION

In the two approaches to laser-driven inertial confinement fusion (ICF) experiments, namely, the direct-drive^{1–5} and the indirect-drive,^{6–8} crossed beam energy transfer (CBET) is of prime importance since it governs the coupling of laser energy to plasma. The two schemes of ICF involve multiple laser beams crossing each other at different angles and directions. For indirect drive, beams cross in the low density plasma of the laser entrance hole while propagating toward the hohlraum wall; in direct drive, they cross in the coronal plasma of the fuel capsule at a considerably higher density. In both approaches, and especially for the direct drive ICF, plasma flow plays an important role in defining resonance conditions and plasma response during the CBET. In addition to the context of ICF, CBET is also the principal mechanism for the amplification of a laser pulse of ps duration by a pump laser pulse^{9,10} in recent pump-probe plasma diagnostic experiments^{11} and theory.^{12} In the laser pulse compression and amplification schemes, the energy transfer is devised to occur in pre-formed (mostly gas-jet) plasmas to obtain spatiotemporal growth of the probe.

The laser-plasma configurations in ICF experiments of concern for CBET involve two laser beams with wave vectors and frequencies $(k\u21921,\omega 1)$ and $(k\u21922,\omega 2)$, crossing at an angle *θ* and leading to induced^{13–15} or stimulated Brillouin scattering (SBS) of one beam into the other.^{16–19} The laser light beams scatter off the grating of ion acoustic waves (IAWs) produced by the ponderomotive force of the two beams. In most of the experimental configurations, plasmas are inhomogeneous and flow with a velocity $v\u2192p$. Therefore, the CBET requires that the three-wave SBS resonance conditions are fulfilled for wave vectors (momentum) and frequencies (energy): $k\u2192s\u2261k\u21921\u2212k\u21922$ and $\omega 1\u2212\omega 2\u2261\omega s+k\u2192s\xb7v\u2192p$, respectively, where *ω _{s}* and $ks\u22432|k1|\u2009sin\u2009(\theta /2)$ are the IAW frequency and wave number for light beams crossing at angle

*θ*. As a result of CBET, the power distribution in the two beams is changed; this can seriously affect the laser energy coupling to plasmas in both the indirect and direct drive ICF schemes.

Our study in this article is focused on two important aspects that prove to be important for CBET, namely,

the speckle (or hot spot) structure of the laser beams and

the role of plasma flow.

Currently, all ICF experiments are carried out with “smoothed” laser beams resulting from spatial and/or spatiotemporal smoothing techniques; in particular, in the case of spatial smoothing only, random phase plates (RPPs)^{20,21} induce spatial incoherence in the laser beams. On a coarse scale, smoothed beams show a smooth average intensity profile in their cross section, while on the fine scale of the laser wavelength $(\lambda 0)$, they have a speckle structure with a known statistical distribution of the speckle peak intensity $Isp$.^{20–22} The goal of using smoothed laser beams is to control the onset of self-focusing in speckles, so as to restrict it to an energetically unimportant, small percentage of intense laser speckles.

While the theory of crossed beam power transfer between speckle beams has been developed in Refs. 23 and 24, most of the current modelings of CBET between multiple RPP beams in ICF experiments are described by averaging over the realistic beam speckle structure.^{6,25,26} Studies on the role of speckles and their self-focusing in CBET in the presence of plasma flow are relatively recent.^{4,16,27} In the regime of moderate laser intensity of the crossing beams, i.e., when no self-focusing in laser speckles arises, recent studies^{19,28} have shown that the role of speckles in the energy transfer is merely of statistical nature: the deviation in the energy transfer arising from different RPP realizations decreases with the number of speckles in the crossing volume, and the expectation value of the transfer corresponds to the value obtained when assuming the average intensity of each beam.

The onset of ponderomotive self-focusing (PSF) in speckles arises if the power in a speckle, *P*, exceeds the critical power for PSF, *P _{c}*. For direct-drive configurations,

^{2–4}beams essentially cross in counterpropagating geometry so that energy exchange can be computed on the basis of standard 1D models for backward SBS.

^{29,30}PSF in such configurations appears in the individual beams and on a longer time scale than CBET.

The scenario is different for indirect drive ICF where PSF and CBET occur on a similar time scale. Furthermore, the zone of sonic flow, in the vicinity of which efficient CBET takes place between the crossing beams, is relatively large. It is known that the transverse plasma flow reduces the threshold for PSF in the sub-sonic regime.^{31–33} In the vicinity of the spatial domain where the plasma flow is sonic, the so-called effect of beam-bending occurs^{23,33–35} where a beam is redirected into a direction different from its incident direction. The latter has consequences to intense speckles in smoothed laser beams: speckles located in the region of overlapping beams and close to the region of sonic flow will eventually be redirected toward the direction of the other beam.

On the other hand, beam bending slows down the plasma flow by momentum conservation and can locally lead to density profile steepening. Density and velocity perturbations in the plasma that are enhanced by the PSF are carried by the plasma flow away from the localized ponderomotive force of the crossing beams and can eventually develop shock-like structures characterized by steep wave fronts.^{36–38} These structures will scatter electromagnetic waves, enhance levels of ion acoustic fluctuations over a wide range in the wave-vector spectrum, and contribute to enhanced plasma-induced smoothing of the transmitted light. In the transmitted light beams, the latter leads to broadening in the angular aperture and introduces (or increases) temporal incoherence, resulting in non-negligible temporal bandwidth.

Two smoothed beams crossing in an expanding inhomogeneous plasma, as sketched in Fig. 1, are relevant to indirect-drive ICF experiments, where the plasma at the laser entrance hole is weakly inhomogeneous, in both density and flow.^{39–44} Experimental studies with similar configurations have been undertaken recently,^{45,46} but also at smaller angles^{47,48} or partially at larger angles.^{49}

In this article, we show how important ponderomotive self-focusing (PSF) in laser speckles becomes when plasma flow is present. For the case of two crossing RPP beams, we show that in inhomogeneous plasmas,^{16} speckle self-focusing and the deflection of speckles lead to a significant effect on the CBET, which increases with the laser beam intensity. We have not considered spatiotemporal smoothing as “smoothing by spectral dispersion” (SSD) in the current study, but we discuss the impact of temporal incoherence on our results.

This article is organized as follows: the model used for describing CBET for beams with a speckle structure in a plasma, with the details concerning the fundamental equations used in our modeling with the code Harmony, is presented in Sec. II. Particular attention is paid to the ponderomotive coupling in Sec. III. We also recall the essential theory for beam-to-beam CBET in Sec. III A. In Sec. III B, we will develop a model to explain ponderomotively induced density perturbations that are seen in speckle beams. The simulation results for several laser plasma parameters as obtained from the code Harmony are presented in Sec. IV. In this section, we also discuss the role played by the laser speckle structure, ponderomotive self-focusing, and density shock structure in CBET. Conclusions are presented in Sec. V.

## II. MODELING CBET BETWEEN TWO BEAMS IN A FLOWING PLASMA

We will describe in the following the interaction between two laser beams crossing at the angle *θ* in an inhomogeneous plasma. Figure 1 shows the particular configuration that corresponds to two “s”-polarized beams crossing at a relatively small angle *θ* and having their common wave vector component along the positive *x* direction, while the ponderomotively generated ion acoustic waves (IAWs) propagate along the *y* direction. Such a configuration of crossing beams may be relevant to the basic element of the geometry of many crossing beams at laser entrance holes (“LEHs”) in the indirect drive ICF experiments. We have also chosen an inhomogeneous plasma flow profile, $vp,y(y)e\u2192y$, with the dominating direction of flow along the *y*-axis. This is one of several possible situations that may be encountered in LEH flow profiles,^{50} and it is the one with the strongest possible exchange between beams that cross at a small angle. For two “s”-polarized beams, with wave vectors and frequencies $(k\u21921,\omega 1)$ and $(k\u21922,\omega 2)$, the SBS matching conditions are satisfied when $vp,y(y)/cs=(\omega 1\u2212\omega 2\u2212\sigma \omega s)/(csks)$, where *σ* is the sign of $\omega 1\u2212\omega 2\u2212(k\u21921\u2212k\u21922)\xb7\u2009v\u2192p$ and where $cs\u2261[(cse2/(1+ks2\lambda De2)+3vi2]1/2$ is the IAW velocity, with $cse\u2261(ZTe/mi)1/2$, where *T _{e}* is the electron temperature,

*λ*the Debye length,

_{De}*v*the ion thermal velocity, and

_{i}*m*and

_{i}*Z*are the ion mass and charge number, respectively.

### A. Model equations

In the following, we express the complex electric field envelope as the superposition of two beams (j = 1, 2) incident at the angles $\xb1\theta /2$ to the x-axis, where $k\u2192j\u2225$ and $k\u2192j,y$ are the parallel and transverse components of the wave vectors, respectively, with $|k\u2192j\u2225|=|k\u2192j|\u2009cos\u2009(\theta /2),\u2009|k\u2192j,y|=|k\u2192j|\u2009sin\u2009(\theta /2)$ and $|k\u2192j|=(\omega j2\u2212\omega p2)1/2/c$. The electric field can then be written as

where *a*_{01} and *a*_{02} are the electric field amplitudes of both beams normalized to the field strength $E\u0302$. For not too large angles *θ* between the two beams, the paraxial approximation can be used, and for $|\omega 1\u2212\omega 2|\u22610$ and $k\u21922,y=\u2212k\u21921,y$, the propagation of the incoming beams can be described by paraxial equations for $a(x\u2192,t)\u2261$$a01\u2009exp\u2009{ik\u21921,y\xb7y\u2192}+\u2009a02\u2009exp\u2009{\u2212ik\u21921,y\xb7y\u2192}$ or, alternatively, individually for *a*_{01} and *a*_{02}, coupled to the plasma density perturbations,^{51}

where $vgx\u2261vg\u2009cos\u2009(\theta /2)$; *ω _{p}*$=(nee2/\epsilon 0me)1/2$ is the electron plasma frequency; $nc=\epsilon 0me\omega 02/e2$ denotes the critical density, with

*m*and

_{e}*e*being the plasma electron mass and charge, respectively, and $\delta n=(n\u2212ne)$ is the density perturbation about the equilibrium density

*n*. Note that Eq. (2) describes the evolution of the electromagnetic wave amplitudes in the paraxial approximation on the scale of hydrodynamic evolution and long wavelength IAW response of $\delta n/ne$. The high-frequency response for IAWs due to backscattered SBS is treated in Harmony via a harmonic decomposition,

_{e}^{52}which one has to consider for large angles $\theta \u223c$ 180°. In Harmony, we solve Eq. (2) by imposing a boundary condition for the entering laser light at $x=$ 0 for $a(x=0,y,t)$.

The plasma dynamics is described by the standard hydrodynamic equations in the isothermal approximation,

where *U* stands for the ponderomotive potential and *ν _{s}* for the ion acoustic damping. In our simulations, we assume for

*ν*a linear wave-number dependence, and $n\nu s\upsilon \u2192$ is computed in Fourier space with $\nu s(ks)=\nu \u0302\u2009\omega s(ks)$, accounting for Landau damping.

_{s}^{51,52}

For the electric field resulting from the superposition of two beams of equal frequency, $a(x\u2192,t)\u2261$ $a01\u2009exp\u2009{ik\u21921,y\xb7y\u2192}+\u2009a02\u2009exp\u2009{\u2212ik\u21921,y\xb7y\u2192}$, the ponderomotive force $\u2207U\u221d\u2207|a(x\u2192,t)|2$ can be subdivided into two separate contributions, provided that the central wave vector component in *y* for each beam, $|k\u21921,y|$ and $|k\u21922,y|$, exceeds the wave number spread $\u25b3k$ related to the angular aperture of each RPP beam; the latter is a function of the focusing *f*-number, namely,^{21} $\u25b3k\u2261|k\u21921|/[1+4f2]1/2\u223c|k1|/(2f)$.

Keeping in mind the condition, for two separate fields, the wave number separation needs to be greater than the angular aperture of the fields, $|k\u21922,y\u2212k\u21921,y|>\u25b3k$, and the ponderomotive force in the right-hand side of the equation of motion can be expressed in two distinct terms, namely, $\u2207U=Te\Gamma \u2207|a(x\u2192,t)|2\u2261$ $\u2207Ucross+\u2207Uself$, given by

where $\Gamma =\u27e8vosc2\u27e9/vth2=vosc2/(2vth2)$ is the coupling coefficient, which involves the thermal velocity $vth=(Te/me)1/2$ and the electron quiver velocity $vosc=eE\u0302/(me\omega )$ of the field $E\u0302$ to which *a*_{01} and *a*_{02} are normalized. In practical units, the coefficient is given by $\Gamma =$ 0.09 $I0\lambda 02$ (10^{15} W *μ*m^{2}/cm^{2})/*T _{e}* (keV), with

*I*

_{0}denoting the beam average intensity.

The ponderomotive force contribution $\u2207Ucross$ acts essentially on the plasma fluid due to the beating between the two waves *a*_{01} and *a*_{02} in SBS. This term alone cannot account for self-interaction occurring in an individual beam. The ponderomotive force contribution denoted by $\u2207Uself$ is therefore the one accounting for ponderomotive self-interaction in the paraxial approximation for each individual beam. This self-interaction can be associated with self-focusing and with forward-SBS inside each beam. Note, that in contrast to the case of large angles, both contributions to the ponderomotive force have major components along the *y*-direction, i.e., across the main common propagation axis *x*.

In most of the studies on crossed-beam coupling, only the $\u2207Ucross$ term was considered in the description of CBET. This term is responsible for the coupling between the average beams, which we will denote as “beam-to-beam CBET” later on in Sec. III A. The term $\u2207Uself$ was mostly neglected in the context of CBET because self-focusing effects are expected to occur for laser intensities that are above those considered in laser fusion configurations. This argument has to be revisited in plasmas with the transversal flow, as it is done further on in Sec. III B. We will draw attention to important work that has been done in the past by considering plasmas with flow even in the presence of a single laser beam.^{31–33}

### B. The simulation configuration in Harmony

In the two dimensional (2D) simulations with our code Harmony,^{51,52} we have chosen a crossing angle of $\theta =$ 20°. The plasma flow profile follows a linear ramp in the *y*-direction, as defined by $vp,y(y)/cs=(y\u2212Ly/2+Lv)/Lv$, with *L _{v}* (=200

*λ*

_{0}in the simulations) being the gradient length,

^{19}so that, assuming equal frequencies for both beams, $\omega 1=\omega 2=\omega 0$, SBS matching occurs at $y=Ly/2$ in the center. We have performed simulations for the case when both entering beams have the same average intensity, $I02=I01$. For this case, $I02/I0=I01/I0\u22611$, the reference intensity,

*I*

_{0}, corresponds, for $\lambda 0=$ 0.35

*μ*m light, to an average laser intensity of $I0=IL\u2243$ 0.9× 10

^{15}W/cm

^{2}at $Te=$ 3 keV.

The laser wave amplitude, $a(x\u2192,t)$ in Eq. (2), at the laser entrance boundary is generated via a Fourier series for two separate wave-fields for *a*_{01} and *a*_{02}. In Fourier space, each of the wave-field contributions is centered around the wave vector components $|k\u21921,y|=\u2212|k\u21921|\u2009sin\u2009(\theta /2)$ and $|k\u21922,y|=|k\u21921|\u2009sin\u2009(\theta /2)$, respectively. For a RPP with *i *=* *1,…, $NRPP$ elements in each component *j *=* *1, 2, with random phases $\varphi j,i$, this reads $a0j(y)=eik\u2192j,y\xb7y\u2192\u2211ki=\u2212\u25b3k\u25b3k|a\u0302j,i|eikiy+i\varphi j,i$, where the *i*th element has the wave number *k _{i}* and amplitude $a\u0302j,i$, with the spacing between them, $ki+1\u2212ki=2\u25b3k/NRPP$. The width in

*k*is given by

_{y}^{21}$\u25b3k\u2261|k\u21921|/[1+4f2]1/2\u223ck1/2f$. The total field with the

*a*

_{01}and

*a*

_{02}components has zero elements in the Fourier series for $|ky|<|k\u2192j,y|\u2212\u25b3k$, which corresponds to the angular interval around $\theta =0\xb0$. For an angular separation of

*a*

_{01}and

*a*

_{02}, the condition $\u25b3k<|k\u21921,y|$ has to be fulfilled.

Note that we have also performed simulations in which a single field array was used instead of separating the fields into two arrays. In this configuration, the ponderomotive force is given simply by $\u2207U=Te\Gamma \u2207|a(x\u2192,t)|2$. We have verified that the simulations of both types of configurations show the same results when both terms $\u2207Ucross$ and $\u2207Uself$ are taken into account in the configuration with two field arrays.

## III. LASER PLASMA COUPLING IN THE PRESENCE OF FLOW AND OF TWO CROSSING LASER BEAMS

The standard criterion for the ponderomotive self-focusing of laser speckles in RPP beams, derived in previous work (see, e.g., Ref. 54), reads

with $Isp$ being the peak intensity of a laser speckle given in the units of 10^{14} W/cm^{2}, *λ*_{0} in *μ*m, *T _{e}* in keV, and

*f*as the speckle f-number;

*η*is a numerical factor of the order of unity, being $\eta =1.23$ in 2D geometry.

^{54}

However, it is important to note that this criterion cannot be applied to flowing plasmas, when CBET between laser beams of equal frequency occurs at close-to-sonic flow.

We will continue to use $P\u0302$ in the text to indicate beam power in a speckle, even if the ponderomotive self-focusing instability may occur with flow already for values $P\u0302<$ 1.

For crossing beams with a speckle structure, like RPP beams, the consequences of the onset of PSF in the presence of flow are manyfold: speckles located in the vicinity of sonic flow will self-focus and grow in amplitude at an angle different from their incidence angle,^{31} a phenomenon denoted as beam bending.^{23,33–35} The latter may eventually lead to the deviation of the speckle into the direction of the other beam, thus corresponding to beam bending CBET; also, the density perturbations ponderomotively induced in the plasma by the numerous speckles may develop shock-like structures characterized by steep wave fronts in the presence of flow, as shown later on.

These effects become more and more pronounced with increasing laser intensity. At beam intensities slightly above the moderate intensity values considered in our previous work in Refs. 19 and 28, the difference in the description of CBET with and without the speckle structure becomes already striking.

To illustrate this, we have carried out simulations with two types of beams, namely, optically smoothed beams with a speckle structure and beams with a flat profile without speckles, with both types of beams having the same average intensity. For the current study, we use RPP beams as the prototype for speckle beams.

We have not considered beams with temporal incoherence. They usually are specific to the laser facility and in most cases introduce temporal variations that, as we will demonstrate below, involve longer time scales than plasma induced correlation times of crossing RPP beams. Here, we concentrate on the essential effects of PSF in speckles on CBET. We call the beams without the speckle structure “regular beams.” The properties of such “regular” beams are reminiscent of plane waves, i.e., with constant, steady wave fronts, with a unique phase constant and without any randomness. Such regular laser beams are used in most of the theoretical models describing CBET in large scale laser facilities.^{6,55} We use those beams for the comparison with the RPP beam, in order to illustrate effects with and without the speckle structure.

Figure 2 illustrates the difference between the two types of beams for crossing beams of equal intensities. Shown are snapshots of intensity profiles in *x* and *y* in the interaction region, for the case of RPP smoothed beams [with speckles, left column, (a)–(c)] and “regular” beams without the speckle structure [right column (d)–(f), see Sec. IV for more details on simulation results].

In Subsections III A–III C, we discuss the basic processes that come into play and that are seen in our simulations for the case of single speckle dynamics and for SBS modeling. We describe in Sec. III A the spatiotemporal evolution of the forward SBS responsible for CBET, in Sec. III B, the modifications of the filamentation threshold due to transverse plasma flow, and in Sec. III C, the linear plasma response to the ponderomotive force of PSF in the flowing plasmas, leading to beam bending and steepening of the density perturbations. Although most of these results have been discussed separately before, we found this summary useful as it will inform our interpretation of the RPP crossed-beam coupling and energy transfer in large scale Harmony simulations.

### A. Beam-to-beam CBET

We will recall here results of CBET theory describing two crossing structure-less beams and refer to Ref. 19 or earlier work^{53} for the details of the derivations. The equations describing the evolution of the laser field amplitudes *a*_{01} and *a*_{02} and propagating by the angle $\theta /2$ with respect to their common axis *x* take the following form:

$\nu 1,2$ stand for collisional damping (they can be neglected for the ICF plasma conditions), $\u2111(1/ds)$ denotes the imaginary part of the resonance denominator $1/ds$, where $ds=(\omega \u2032)2+2i\omega \u2032s\nu s\u2212\omega s2$, with $\omega \u2032s=\omega 1\u2212\omega 2\u2212k\u2192s.v\u2192p$ describes resonant coupling between the crossing laser waves and the ion acoustic wave, $\omega s2\u2261ks2cs2$, and *ν _{s}* is the acoustic damping.

After a transient period of non-stationary energy exchange between the beams, Eqs. (6) can be reduced to a set of stationary equations for the beam intensities $|a01|2$ and $|a02|2$. An efficient way of expressing this coupling is realized by introducing oblique, non-orthogonal coordinates *η* and *ξ*, $x=\eta e\u2192x.e\u2192\eta +\xi e\u2192x.e\u2192\xi $ and $y=\eta e\u2192y.e\u2192\eta +\xi e\u2192y.e\u2192\xi $, with $e\u2192y.e\u2192\eta =\u2212sin\u2009(\varphi \u2212\theta /2)$ and $e\u2192x.e\u2192\xi =\u2212sin\u2009(\varphi +\theta /2)$. Choosing $\varphi =0$ as the angle between $v\u2192p$ and $k\u2192s=k\u21921\u2212k\u21922$ leads to $e\u2192y.e\u2192\eta =sin\u2009(\theta /2)$ and $e\u2192x.e\u2192\xi =\u2212sin\u2009(\theta /2)$ in the two-dimensional geometry (see Fig. 1) of Harmony simulations (cf. Fig. 2). For arbitrary ratios between $|a01|2$ and $|a02|2$, for which one cannot neglect depletion of either beam 1 or 2, the set of equations to solve reads

which simplifies to first-order differential equations when the depletion of one of the beams can be neglected, as for $|a01|2/|a02|2\u226b$ 1 (or, $\u226a$ 1). The function $\beta (\xi ,\eta )$ accounts for the geometry of the rhombus-like shape of the crossing zone,

It depends on the spatial growth rate $\gamma 02/(\nu scs)$ for SBS ($vg,2\u2261cs$), where *γ*_{0} is the temporal growth rate, $\gamma 02\u2261(ne/nc)(\omega 1/\omega s)ks2cs2vosc2/(4vth2)$$=(ne/nc)(\omega 1/\omega s)ks2cs2\Gamma /2$, with the coupling coefficient Γ from Eqs. (4a) and (4b) involving the quiver velocity $vosc$ corresponding to the laser pump field to which *a*_{01} and *a*_{02} are normalized. The auxiliary functions,

result, with $vp,y(y)/cs=(y\u2212Ly/2+Lv)/Lv$, in

The integration over the domain of interest, namely, the beam width *D*, see Ref. 19 and Fig. 1, yields a spatial amplification described by the gain *G* coefficient given by

where $Linh=\pi (Lv\nu s/\omega s)/|\u2009cos\u2009\varphi \u2009sin\u2009[\varphi \u2212\theta /2]|$ denotes the inhomogeneity length, being $Linh=\pi (Lv\nu s/\omega s)/\u2009sin\u2009(\theta /2)$ in our case. For the case $D/(2\u2009sin\u2009\theta )>Linh$, the resulting gain is equivalent to the “Rosenbluth” gain coefficient.^{56}

### B. The role of plasma flow and of speckles in CBET

For the case of non-flowing plasmas, laser speckles are expected to self-focus with speckle power following the criterion Eq. (5). In terms of the speckle peak intensity, *I _{sp}*, it reads in practical units $(Isp/I0)\u223c$ 3.1 $(nc/10ne)(f/8)\u22122Te$ (keV). It yields that speckles self-focus when $Isp>(f/8)\u22122$ 9 $\xd71015$ W/cm

^{2}at $\lambda 0=$ 0.351

*μ*m for $ne/nc=$ 0.1, $Te=$ 3keV, and $I0=$ 10

^{15}W/cm

^{2}. For typical ICF conditions, only a very few speckles would hence undergo PSF.

As already mentioned, this criterion cannot be applied in the presence of plasma flow. The criterion for the onset of the filamentation instability has to be derived from the set of Eqs. (2), (3a), (3b), and (4b), taking into account the advective terms $vp\u2192\xb7\u2207v\u2192$ and $\u2207(nvp\u2192)$ due to transverse plasma flow $v\u2192y$. Assuming for unstable modes, the dependence $\u221d\u2009exp\u2009(qx+ikyy)$ with respect to the *x* and *y* axis and with flow along the *y* direction, one obtains a criterion for the filamentation instability in terms of the spatial growth rate *q*.

The resulting relation reads^{31} (for $|q|2\u226ak02$)

where $M=vp,y/cs$ is the Mach number of the transversal flow, the damping coefficient $\nu \u0302=\nu s(k)/\omega s(k)$, and $\epsilon =1\u2212ne/nc$ is the dielectric constant in the plasma. Note that when considering Gaussian beams or RPP speckles, *k _{y}* should be larger than the minimum $ky,min=\Delta k\u2243k0/2f$ related to their focal width. The resulting threshold for the instability, $\u211cq>0$, depends on transverse flow via a resonance denominator, in contrast to the standard criterion without flow, i.e., $M=vp,y/cs\u2261$ 0. For small damping, $M\nu \u0302\u226a|1\u2212M2|$, Eq. (12) predicts growth of the filamentation instability at wave vectors,

*k*, and $\u27e8vosc2\u27e9$ satisfying threshold condition, $(\u27e8vosc2\u27e9/vth2)(ne/\epsilon nc)(k02/ky2)\u22611\u2212M2$ for subsonic flow, $M2<1$. This clearly indicates that the onset of PSF is altered with flow. For the particular case without flow, $M=$ 0, this criterion is equivalent to the onset of PSF, Eq. (5), for speckles, by associating $Pc\u223cvth2nc/n$ with the critical power and $Psp\u223cvosc2(k02/ky2)$ with the speckle power, assuming $k0/ky=k0/\Delta k\u22432f$. There is, however, no instability for supersonic flow $|M|=|vp,y/cs|>1$. In detail, this has been worked out in Ref. 32 for the case of RPP smoothed laser beam. This work clearly shows via numerical simulations that PSF growth is enhanced for subsonic flow with respect to the case without flow $vp,y\u2261$ 0.

_{y}We have solved Eq. (12) as a function of *k _{y}* for different values of the Mach number

*M*for subsonic flow and for two cases with intensities $IL\lambda 02=$ 10

^{14}W/cm

^{2}

*μ*m

^{2}and 2.5 × 10

^{14}W/cm

^{2}

*μ*m

^{2}, respectively (and for $ne/nc=$ 0.1, $Te=$ 3 keV, and $\nu \u0302=$ 0.05). The unstable solutions found are shown in Fig. 3. For the case without flow, $M=$ 0, one observes a clear cutoff $kcutoff$ in

*k*above which, $ky>kcutoff$, no unstable solutions exist. For the two cases shown, this cutoff corresponds to the threshold criteria for speckles with $Isp=IL$ and $Psp/Pc=$ 0.2 and 0.5, respectively, for $f=$ 8. In contrast to this, one cannot observe a cutoff for the subsonic flow cases, 0 $<|M|<$ 1. It is furthermore evident from the spatial growth values $q(ky)$ obtained that for subsonic flow, the ponderomotive modifications expected from PSF are stronger than those for $ky<kcutoff$ in the case without flow. The linear growth rate of the filamentation instability Eq. (12) has been discussed in Ref. 33, where it is shown that the enhanced density perturbations due to the instability contribute to the laser beam deflection. Such a beam bending has been observed in simulation results

_{y}^{34,35}and can contribute to CBET.

Beam bending enhanced by the PSF of a single Gaussian beam is illustrated in Fig. 4. In the case shown, the beam is originally focused at $x=$ 1000*λ*_{0}. The plasma flow transverse to the propagation direction of the entering beam is at $M=$ 0.96. For the beam intensity chosen, the criterion according to Eq. (5) yields $Psp/Pc=$ 0.5, and it is thus below the onset value for PSF. What can be seen shows, however, a clear onset of PSF and beam bending: the beam is deflected into a direction oblique with respect to its incidence, and the beam remains trapped in its own plasma density channel for distances considerably longer than the Rayleigh length. We have furthermore examined cases with still higher intensity values of single beams (not shown), for which strong non-stationary behavior can appear together with flow, driving density channels in the plasma that eventually propagates freely, leading to shock-like structures in the plasma density.

For crossing beams, it follows that the processes of beam bending and speckle self-focusing lead to an effective energy transfer into the other beam. For the described scenario of transfer between speckles and the other beam at close-to-sonic flow, one again has to consider the basic theory for CBET, as outlined in Eqs. (7)–(11). Two differences essentially emerge as compared to “beam-to-beam” CBET, namely,

the speckle width, $f\lambda 0\u223cD$, is most likely smaller than the inhomogeneity length, yielding $f\lambda 0/2\u2009sin\u2009\theta <Linh$ and

the local intensity inside a speckle can be several times (say up to 6–9 times) higher than the average beam intensity. One can approximate $|a01|2|a02|2$ in Eqs. (7) by $\u27e8I\u27e9Isp/\u27e8I\u27e92$.

Consequently, the approximation for the gain describing the transfer between speckles and the other beam results in $G\u2243f\lambda 0\u2009sin\u2009\theta \gamma 0,sp2/(\nu scs)$ in which $\gamma 0,sp$ is the SBS growth rate evaluated for the speckle intensity $Isp$. Compared to the gain values expected for “beam-to-beam” CBET, this gain value is by the factor $f\lambda 0/min{D/(2\u2009sin\u2009\theta ),Linh}$ different. The width of a single speckle, $f\lambda 0$, is generally much smaller than the width of a RPP laser beam or the width of the interaction region in an inhomogeneous profile, whereas the factor $\gamma 0,sp2$ can assume values up to 6–9 times higher than $\gamma 0,2$ for the average laser beam.

Amplification of individual speckles by crossing overall beams has also been seen in experiments.^{45} However, this transfer process into intense speckles remains transient as has been demonstrated in the experiments in Ref. 45 involving relatively short laser pulses of 2–4 ps duration. For longer laser pulse durations, the transfer from one to the other beam affects the whole beam. Transfer from speckles to the crossing overall beam is, on the other hand, very likely to happen and is also seen in our simulations. The consequences of the latter process, as will be discussed and illustrated later on in Sec. IV B, lead mostly to enhancement in the angular spread of the beam receiving energy from speckles.

### C. Density perturbation and beam bending

Although the steepening of the ion acoustic waves is a nonlinear process, the linear approximation of the plasma response in the presence of background flow already clearly points to the shock formation, which is well reproduced in our simulations. It is helpful to recall a model of the plasma response to the stationary ponderomotive potential in the presence of transverse flow. The linear response of the plasma density is described by the wave equation obtained from linearizing Eqs. (3a) and (3b),

where $\delta n/ne$ is the density perturbation; *ν _{s}* is the IAW damping;

*U*denotes in this section the normalized ponderomotive potential, $U\u2261Uself/Te$. A convenient way to introduce a transverse flow into Eq. (13) is to consider a moving beam in a stationary plasma

^{35}that is equivalent to the flowing plasma in the

*y*-direction and a stationary laser beam or crossed beams giving rise to the ponderomotive potential. In this new frame of reference, $U=Uself(y+vp,y0t)$ describes the ponderomotive potential moving to the left with a uniform plasma flow velocity $vy0$. The analytical solution to Eq. (13) can be obtained using the procedure of Ref. 57, where we introduced the Laplace transform of $\delta n/ne$ and of

*U*, ${\delta N\u0303,U\u0303}(y,s)=\u222b0\u221edt\u2009e\u2212st{\delta n/ne,U}(y,s)$; $\delta N\u0303$ satisfies the wave equation,

for $cs\u2243cse$ with $\beta 2=(s2+2\nu ss)/cs2$, having the solution

After integration by parts and calculating the inverse Laplace transform, $\delta N(t)=1/(2\pi i)\u222b\u2212i\u221e+\sigma i\u221e+\sigma ds\u2009est\delta N\u0303(s)$, we obtain the following time dependent solutions for the density perturbation in the frame of the moving plasma and for the stationary ponderomotive potential: for plasma flow different from the speed of sound, i.e., for $M\u2261vy0/cs\u22601$, one obtains

with $I(\xb1)=\u222b0cstd\xi e\u2212\nu s\xi /csU(y\xb1\xi \u2212M\xi )$ and with the approximation $\beta \u2248(s+\nu s)/cs$. For sonic plasma flow, *M *=* *1, the density response from Eq. (15) yields the expression

In Figs. 5 and 6, we show the density perturbation response to the ponderomotive force for the case of a speckle in the *y*-direction transverse to the laser propagation, close to focus, taken at late times $\nu st\u226b$ 1. Figure 5 illustrates, for different Mach numbers, $M=$ 0.9, 1, and 1.1, the linear response computed from model Eq. (16) to a Gaussian-shaped speckle, $U(y)=U0\u2009exp\u2009{\u2212y2/(f\lambda 0)2}$, namely, $(n\u2212n0)/U0=\delta N(y,t)/U0$, where *n*_{0} corresponds to the equilibrium density and *U*_{0} to the peak value of the ponderomotive force in the center of a laser speckle. The solid and dashed curves correspond to the damping rates *ν _{s}* with $(\nu s/cs)(f\lambda 0)=$ 0.15 and 0.05, respectively. For the lower damping rate, the asymmetry predicted from Eq. (16) is more pronounced. Note that for the sonic case $M=$ 1, for which the 2nd term $\u223c\u2202yU(y)$ in Eq. (17) dominates (because of $(cs/\nu s)f\lambda 0>$ 1), the density perturbations reach the highest amplitudes and display the steepest gradients. The asymmetry between maximum and minimum of density response contributes to the beam bending

^{33,35}and modifies CBET in our simulations.

Figure 6 shows the response obtained from simulations, for $M=$ 0.9, 1, and 1.1, accounting also for non-linearity in Eqs. (3a) and (3b). The gray curve shows the case when the speckle power is well below PSF critical power [according to Eq. (5)], $P/Pc=$ 0.05 for $M=$ 1; the other curves show the case with $P/Pc=$ 1, where a departure from the linear response becomes visible mostly in the loss of symmetry between $n(y)\u2212n0\u2009>$ 0 and the density depression $n(y)\u2212n0\u2009<$ 0, which is deeper but also more localized. The corresponding spectra of $n\u2212n0$ as a function of *k _{y}* are shown in Fig. 7: the Gaussian-shaped form as a function of

*k*is preserved only in the linear regime, with a peak around $ky\u2243k0/f$. For the nonlinear case, here with $P/Pc=$ 1, the peak in the spectrum is found at lower

_{y}*k*values, almost $ky\u2243k0/(2f)$ with a linear (Lorentzian-type) exponential decrease. We shall see later that these spectra help to partially interpret the ion density perturbations in a multi-speckle environment of crossed beams.

_{y}## IV. ANALYSIS OF THE SIMULATION RESULTS

In the present study, for all simulations in Figs. 2 and 8, 9–11, we have chosen a domain of $4500\lambda 0$ in length and $2300\lambda 0$ in width, i.e., along the *x* and the *y* axes, respectively; the beams have a common wave vector component along *x* and opposite wave vector components along *y*. In this chosen geometry, the gradients of the plasma profiles in density and velocity point predominantly along the *y*-direction. The density profile is parabolic around the center, $y=Ly/2$, given by $ne(y)=$ 0.1*n _{c}* $exp\u2009\u2212[(y\u2212Ly/2)/1615\lambda 0]2$. We apply a linear density ramp starting at $x=$ 0 over 500

*λ*

_{0}along

*x*in order to avoid boundary effects at the laser entry. As already mentioned in Sec. II, we apply a linear flow ramp with sonic flow in the center of the crossing beams, i.e., $vp,y(y=Ly/2)=+cs$, and a flow gradient $Lv=$ 200

*λ*

_{0}. We have focused our study on the case when both beams have equal intensity $I01=I02$ at the entrance

*x*=

*0, and both beams have the same focusing $f\u2212$ number, namely, $f=$ 6 for RPP beams. Here, the coefficient $\nu \u0302$ takes the value of $\nu \u0302=$ 0.1, except in the study examining the sensitivity of CBET in the ion acoustic wave damping, see Sec. IV B. In plasmas with inhomogeneous flow, CBET occurs when the effective beam width, $Lbeam=D/(2\u2009sin\u2009\theta $), is larger than the interaction length $Linh=\pi [\nu (ks)/\omega s][Lv/\u2009sin\u2009(\theta /2)]$, where D is the beam diameter.*

^{19}This is equivalent to $D/Lv>4\pi \u2009(\nu /\omega s)\u2009sin\u2009(\theta /2)$, in practical units, $D/Lv\u223c.2(\theta /20\xb0)(\nu /0.1\omega s)$ for small

*θ*.

### A. The role of speckles and ponderomotive self-focusing in CBET in RPP and regular beams

In order to illustrate the role of laser speckles and of ponderomotive self-focusing in CBET in the presence of a flow, we compare simulation results between the cases of (i) two crossed RPP beams and of (ii) two “regular” beams. The regular beams have the same average intensities and envelope shapes as the RPP beams.

In the following, we illustrate our results in two sub-sections: in the first one, we show the results of interacting laser beams based on a single realization of a RPP, in the second subsection, Sec. IV A 2, and Figs. 8 and 11, we show the results based on the ensemble average over eight realizations of RPP. No ensemble averaging is necessary for “regular” beams.

#### 1. Results from a single realization of RPP beams

All snapshots shown in Figs. 2(a)–2(c) are based on a single RPP realization for each beam; they illustrate the local dynamics arising due to the speckles of this realization. Figures 2(d)–2(f) are based on regular beams. Figures 2(a) and 2(d) show the initial (t = 0) beam geometry before interaction, for the cases of two crossed RPP and of two “regular” beams, respectively. One may again consult Fig. 1 for the general geometry of the simulations. We display the particular case where the average beam intensities are $I01=I02=$ 6*I*_{0}. The interaction region of the two beams for both cases forms a rhombus-like shape in the center of the simulation box. In our configuration, the plasma flow profile is chosen in such a way that sonic flow appears along the major diagonal of the rhombus, which is parallel to the longitudinal direction *x*. CBET is hence excited around $y=Ly/2$ with a plasma flow gradient $\u221d1/Lv$ along the *y*-axis.

##### a. Crossing speckle beams: PSF and flow

Figure 2(b) shows that in the presence of self-interaction, the two crossed RPP beams undergo significant deflections with respect to the initial beam directions [see Fig. 2(a)]; also the angular aperture of each beam is broader than that initially, while without the $\u2207Uself$ force, see Fig. 2(c), the beams transfer energy without significant deflection or broadening. Figure 2(b) also exhibits features of plasma induced smoothing^{54} and moving filaments^{58} at the rear of the simulation box. The two terms $\u2207Uself$ and $\u2207Ucross$ in Eq. (3b) are responsible for this additional spatial and temporal incoherence in the transmitted light. In Fig. 2(b), an enhanced transfer of energy from beams 1 to 2 (as will be shown later) and a strong angular spread in the presence of self-interaction are observed. These processes are due to the fact that for $I01=I02=$ 6*I*_{0}, a significant population of speckles has sufficiently high power to be unstable with respect to PSF. As elaborated in Sec. III B, due to flow, PSF in speckles occurs already at intensities lower than that indicated by the standard expression Eq. (5) for $Psp/Pc$.

Our simulation results demonstrate the importance of the plasma flow. For an inhomogeneous flow profile, as in our simulations, resonant coupling between the crossing beams takes place around the region where the plasma velocity is close to the sound velocity, $vp,y\u223ccs$. This is where one can see in Fig. 2(e) that beam bending arises so that some speckles are redirected toward the other beam and effectively contribute to CBET.

##### b. Crossing regular beams: PSF and flow

Regular beams, because of the flat, almost plane wave-type wave fronts, can be unstable to PSF and to the filamentation instability for the intensity range considered, so that any perturbation in the beam structure or in the plasma density will trigger the onset of filamentation instability. Such initial perturbations in regular beams are produced by the ponderomotive force of crossing beams. They can further develop and lead to filamentation in the simulations when the $\u2207Uself$ term is taken into account. The interaction of two crossed regular beams illustrated in Fig. 2(e) (with both the $\u2207Uself$ and the $\u2207Ucross$ terms taken into account) results in the transfer of energy into a beam that propagates along a common axis. The filamentary structure develops in the overlapping regular beams where they interact with the density modulations due to the ponderomotive forces induced by the crossing beams. The regular beam filaments also undergo beam bending as seen in Fig. 2(e) in near to sonic flow, $vp=\xb1cs$.^{34,59} This contributes to the beam component propagating along the x-axis and partially to angular broadening of the transmitted light. We have also carried out a simulation without applying the $\u2207Uself$ term, see Fig. 2(f). For this case, perturbations in both beams arise inside the (rhombus-shaped) resonant zone for CBET, and—in contrast to the case with self-interaction—no further filamentation develops in the small beamlets. Note that the structures of regular beams induced by CBET and by filamentation point mostly along the common axis between both beams, a feature that is clearly not observed in the RPP beams with a speckle structure.

##### c. Induced density perturbations

For the case of crossing RPP beams, significantly different density perturbations are excited in the plasma when taking into account both the effects of ponderomotive self-interaction and crossed-beam coupling together [“*self + cross*” in Fig. 9(a)], only a single one of these effects, i.e., only crossed-beam coupling [“*cross only*” in Fig. 9(b)], or only beam self-interaction (“*self only*”). The latter is illustrated via the density perturbations defined as $\delta n\u2261n(x,y,t)\u2212ne$, with $ne\u2261n(x,y,t=0)$, taken at $x\u22431500\lambda 0$ in the front part of the rhombus, and shown in the three subplots of Fig. 9, also indicating the flow profile $vp,y(y)/cs$, with sonic flow at $y/\lambda 0=$ 1100.

##### d. Angular broadening of the beams

In addition, and in order to illustrate the effect of deflection and angular broadening observed for crossed RPP beams in Fig. 2(b) (in the presence of beam self-interaction) and 2(c) (in the absence of beam self-interaction), we plot the temporal evolution of the angular spectrum of the transmitted light (detected at the rear of the simulation box) in Figs. 10(a) and 10(b), respectively. The light signals appearing in the upper right corner of the simulation box between times $2k1cst=$ 150 and 200 in Fig. 10(a) show that in the presence of self-interaction, beam 2—initially propagating at an angle of $10\xb0\xb1$ 3.5°—has components up to large angles of $\u223c25\xb0$, with a central direction at ∼11° (width ±7°), while in the absence of the $\u2207Uself$ term, Fig. 10(b), beam 2 does not undergo strong deflection: it is simply characterized by an asymmetric angular spread around ∼11.5 $\xb0\u2009(+$ 5°/−4°), at $t=$ 200. Similarly, beam 1 is characterized by an enhanced angular broadening around −10° (±5.5°) in the case with beam self-interaction [Fig. 10(a)] in contrast to the case without beam self-interaction [Fig. 10(b)], showing an asymmetry around −12° (−5°/+4°); the latter is interpreted to be due to pump depletion.

To summarize these results, as seen in Figs. 2 and 10, one can characterize the role of the speckle structure in CBET as follows: (i) the importance of the speckle structure for CBET increases with beam intensity due to the increasing number of speckles undergoing PSF; (ii) the onset of PSF in subsonic flow regions occurs in speckles with peak intensities lower than that predicted by the standard criterion, Eq. (5), which eventually increases considerably the number of speckles affected by PSF; (iii) at sonic flow, speckle beams are deviated by beam bending, which can lead to a net transfer into the other beam; and (iv) particularly, striking is the onset of temporal incoherence of the beams for the case with self-interaction around +10°, resulting in fluctuations with relatively short correlation times of $2k1cstcorr\u223c$ 20 ($tcorr\u223cps$ in real units), which can be associated with the effect of plasma-induced smoothing.

#### 2. Ensemble averaging over multiple RPP realizations

In the following, we will revisit the previous results and analyze them further by both varying the beam intensity and examining the angular aperture of the transmitted beams. To do so, we have averaged results of simulations over different RPP realizations. In Fig. 8, we have summarized our results from a series of simulations with RPP beams; we have averaged over eight realizations. Shown is the energy transfer as a function $I02/I0$, which clearly shows that the role of RPP speckles in CBET can no longer be disregarded above the reference intensity, $I0\lambda 02$ = 10^{14} W/cm $2\mu $ m^{2} at 3 keV. To illustrate this, we plot the transfer from beam 1 (downwards propagating) to beam 2 by increasing the incoming intensity values of both *I*_{01} and *I*_{02}, while keeping their ratio $I01/I02=1$ constant. We compare the power gained by beam 2 for the RPP [subplot(a), solid curves] and regular [subplot(b), dashed curves] beam cases in the presence (blue curves) and absence (red curves) of the $\u2207Uself$ terms. To do so in the simulations with Harmony, we switched on and off this term on the rhs of Eq. 3(b). The power transfer ratio is defined as

where $E(ky)$ is the 1D Fourier transform of $a(x,y,t)$ in *y*.

Note that Fig. 8(b) also displays a curve (in green color) that corresponds to the results in Ref. 19 for the interval 0 $<I02\u2272$ 0.75 *I*_{0}, for the same geometry, also obtained using our code Harmony. In this interval, no significant differences in the transfer between the beams were seen when comparing regular and RPP beams, even when accounting for self-interaction. The differences seen between the results for different RPP realizations are merely due to speckle statistics.^{28}

From our new results, differences due to PSF in speckles appear for $I02\u2273$ 0.75 *I*_{0}, when one enters in the regime where ponderomotive effects such as self-channeling, deflection via beam bending in a flowing plasma, as well as plasma-induced smoothing occur. For RPP beams, all these processes depend on the laser speckle distribution.

We should mention here that experimental studies with crossed RPP beams^{17} have reported both on angular broadening and on spectral broadening in an intensity regime between 2.4 and 8 × 10^{13} W/cm^{2} at $\lambda 0=$ 1 *μ*m, however, at lower temperature (0.5 keV) and higher density (0.3*n _{c}*) than those considered in this article, with effects of collisional absorption present.

##### a. The role of self-focusing in speckles

Our simulation results summarized in Fig. 8 show as a function the beam intensity and those for $I02/I0>0.75$; the power transfer to beam 2 first increases for the case of RPP beams with self-interaction [Fig. 8(a) solid blue curve], then reaches a maximum around $I02=I01=3I0$, and eventually decreases for still higher intensities. There is a striking difference between the cases with and without the effect of self-interaction: when neglecting the self-interaction effects, no significant increase in power transfer as a function of *I*_{02} is seen, and the onset of the decrease in the power transfer occurs already for $I02=I01\u22731.5I0$. Although the standard threshold criterion for PSF in Eq. (5) would indicate that only a few extreme speckles can have $Psp/Pc>$ 1, it is the presence of flow that changes considerably the PSF in speckles in the beam overlapping region, both for sonic and subsonic flows. Consequently, the light is deflected toward the direction of beam 2, which is a net contribution to CBET for RPP beams in the intermediate regime $1\u2009\u2272\u2009I02/I0\u2009\u2272\u20093$. Also, for $I02/I0>0.75$, comparing the blue and red curves in Fig. 8(a), with and without the self-interaction effects, respectively—as expected—it can be clearly seen that the power transfer is systematically stronger when the contribution of the self-interaction effects in the ponderomotive force is taken into account. The relevant regime corresponds, in practical units, and for the plasma conditions considered here, to laser fluxes from $I02\lambda 02>$ 10^{14} W $\mu m2$/cm^{2}. This means that the onset of PSF effects in speckles arises still in an intensity regime that is very relevant with respect to current ICF parameters.

The power transfer between regular beams, shown in Fig. 8(b) for $I02/I0>0.75$, is also systematically lower than that for the power transfer between RPP beams. For regular beams (dashed curves), without speckles, the self-focusing and forward SBS play only a limited role, in spite of the differences in the angular spectra observed in Figs. 11(c) and 11(d) for the cases with (c) and without (d) self-interaction effects. For high beam intensities, $I02/I0>$ 2, one reaches a maximum power transfer between the beams, beyond which the power transfer decreases as a function of *I*_{02}. This is due to the non-linearities of the ion acoustic wave perturbations (see Sec. IV B). Nonlinear sound waves enhance forward scattering and diffraction to a broader angular spread of the transmitted beams, which yields asymptotically an effective equilibration between the angular spectra of both beams toward $\u222bky>0|a\u0302(k,x=Lx)|2dk\u2243$ $\u222bky\u22640|a\u0302(k,x=Lx)|2dk$.

##### b. Angular broadening as a function of intensity

The observed broadening of the angular spectrum, as displayed as a function of time in Fig. 10 for the single example of a RPP beam with $I01=I02=6I0$, is summarized as a function of $I02/I0$ in Fig. 11. The values shown are taken late in time, namely, $2k1cst=$ 200 when the CBET processes have reached a quasi-stationary regime. The comparison between the angular distribution of the transmitted light of the two crossed beams is shown for four different cases: Figs. 11(a) and 11(b) show the cases of RPP beams, in the presence and in the absence of self-interaction, respectively; Figs. 11(c) and 11(d) show the angular distributions of the transmitted light for regular beams. Generally, RPP beams clearly exhibit an increasing angular spread of transmitted light with increasing intensity; in addition, this angular spread is enhanced by self-interaction. The results for regular beams [see Figs. 11(c) and 11(d)] are dominated by the strong central beam structure close to $\theta \u223c0$. The angular width of this central beam increases with $I02/I0$.

Two distinct beams can always be identified in the RPP case; only for the case of RPP beams with self-interaction, a weak beam arises around $\theta \u223c0$. The resulting angular spectra for RPP and regular beams start to broaden for *I*_{02} > *I*_{0}, shown in Fig. 11(a) with self-interaction. The case without self-interaction is shown in Fig. 11(b) with less pronounced broadening. Furthermore, the onset of angular spread contributes to the increase in spatial and temporal incoherence [also see Figs. 2(b) and 10(b)].

### B. Nonlinear sound wave perturbations with self-interaction and their role in CBET

As already mentioned, the energy transfer between the crossing beams decreases according to Fig. 8 as a function of the beam intensity for I $02/I0>$ 3. This behavior is correlated with both (i) the increase in the angular spread as a function of intensity, associated with enhanced spatial and temporal incoherence in each beam with increasing beam power and (ii) the onset of non-linearities in the IAW perturbations. Thus, it is important to analyze the role of nonlinear density perturbations in the plasma during CBET. For this purpose, we present in Figs. 12–14, a set of simulation results for the same laser-plasma parameters as used in Fig. 2, however, now with different realizations and a smaller simulation domain [with lengths of $3500\lambda 0$ and $1200\lambda 0$ along the *x*(longitudinal) and *y*(transverse) axes, respectively]. Figures 12(a) and 12(b) show the intensity profile and the IAW perturbations in the plasma, respectively, zoomed in the regions of overlap of the two crossed RPP beams. For the case shown, both self-interaction and crossed-beam coupling are taken into account. Figures 12(c) and 12(d) show the same quantities as in subplots in 12(a) and 12(b) for the case where only crossed-beam coupling is taken into account, while self-interaction is switched off. Comparing Figs. 12(a) and 12(c), we clearly see strong deflection of the RPP beams in the presence of the self-interaction process, while no deflections are observed in the absence of self-interaction. Also, comparing the corresponding density perturbations in Figs. 12(b) and 12(d), one can observe nonlinear density perturbations (in terms of deep density channels) only for the case in the presence of self-interaction. In order to demonstrate this effect in detail, we also plot the lineouts of the intensities of the two crossed RPP beams and of the plasma density perturbations for the three different cases in the set of Figs. 12(e) and 12(f). These lineouts are taken at $x\u22481525\lambda 0$ in the longitudinal direction and along the transverse direction *y* from the fields shown in Figs. 12(a) and 12(c). In these figures of line outs, the blue and green curves distinguish the intensity profiles of the two crossed RPP beams, while the orange curve shows the density profile. In Fig. 12(e), we can observe the dominant short wavelength plasma density perturbations (orange curve), having the wavelength $\lambda cbet=2\pi /|k1,y|$. The oscillations are seen in a wide zone around the resonant sonic flow region due to the IAWs in the pure CBET-SBS case; the blue and orange curves show the redistribution of energy between the two RPP beams. The low amplitude oscillations in density [orange curve in Fig. 12(e)] correspond to CBET between the average RPP beams, similar to what would be seen in regular beams. We can also see regions with locally higher oscillation amplitudes corresponding to CBET where the exchange is enhanced between the average beam and intense laser speckles in the other beam. Figure 12(f) clearly shows the impact of the PSF process in speckles along with the short-wavelength CBET-SBS-driven IAWs. From the figure, we see that redistribution of energy between the two beams under the combined effect of self-interaction and crossed-beam coupling leads to significantly higher intensity peaks as compared to the case with only crossed-beam coupling [Fig. 12(e)].

For the case without the CBET-SBS process, as shown in Fig. 12(g), short-wavelength components are absent and the density perturbations are merely due to the imprint of the ponderomotive force from the speckles. In the set of Fig. 12, sonic flow, i.e., *M *=* *1, is located in the vicinity of $y=$ 655*λ*_{0} for $x>$ 600*λ*_{0}, where one can observe a phase shift between the induced density minimum and the intensity maxima. Outside the region of sonic flow, for $y<$ 645*λ*_{0} and $y>$ 665*λ*_{0}, one can systematically observe that significant intensity maxima coincide spatially with density minima, as expected for ponderomotively induced density perturbations.

The two ponderomotive force terms on the rhs of Eq. (3b) can be strong enough to induce nonlinear density perturbations [as seen in Figs. 12(b) and 12(f)] in plasma with steepened wave fronts, as discussed earlier in Sec. III B. Also, the plasma has a flow, which affects the evolution of the density channels due to the PSF and the IAW perturbations due to the CBET-SBS process and evolves in shock-like structures, characterized here by ponderomotively driven density perturbations that develop steep wave fronts when propagating in the plasma. The time evolution of such shock-like structures is illustrated in Fig. 13(a). Plasma density perturbations are recorded along the same lineout [as in Figs. 12(e) and 12(f)] for different time instants and for a simulation case where only the $\u2207Uself$ term was retained. Also, Fig. 13(b) shows correspondingly the Fourier transform of the plasma density perturbations at the same time instants. In the lineouts as a function of *y*, the plasma flow is in three different regimes along the transverse direction: the flow is sub sonic for $y<$ 1110*λ*_{0}, it is sonic around ($y\u2243$ 1110*λ*_{0}), and supersonic for $y>$ 1110*λ*_{0}.

In Fig. 13(a), during the initial stage of interaction ($2k1cst$ = 70, magenta curve), we only see small density perturbations in the three regions; however, the density perturbations in the respective regions increase with ongoing time with the development of a wave train close to the sonic region, along with a steeping in the density perturbation in the sonic region. In Fig. 13, the observed shift in the position of the shock-like structure around the sonic point ($vp\u2261cs$, at y = 1110*λ*_{0}) with time can be attributed to the plasma flow. The corresponding Fourier analysis of the density perturbations in Fig. 13(b) shows that as the shock structure becomes more prominent with time, the spectra develop a plateau in the region $0.6\u2009\u2272\u2009k/k1\u22641$, a feature characteristic of shock generation. Also, as seen in Figs. 12(f), 12(g), and 13(a), the typical size of the non-linear structures in the density perturbation corresponds to the size of laser speckles, and the perturbations are strongest in the vicinity where the plasma flow is sonic.

In the set of Fig. 14, we present and compare Fourier transforms of the nonlinear density perturbations for the three cases shown in Figs. 12(e)–12(g). In the same figures, we also compare the wave number spectra with the change in damping coefficient $\nu \u0302$ (accounting for the both collisional and Landau damping). Figure 14(a) shows the case with crossed-beam coupling only. The spectrum peaks around the value expected for CBET-SBS at $k1,y/k1=2\u2009sin\u2009(\theta /2)\u223c$ 0.35, followed along *k* by other equally spaced peaks, corresponding to the second and third harmonics. This means that the CBET-SBS-driven density perturbations can evolve into non-linear wave train structures when propagating through the plasma. These higher harmonics to the density perturbations associated with CBET-SBS decrease as a function of the IAW damping. In the spectrum corresponding to the case with self-interaction only, in Fig. 14(c), the contribution at the CBET-SBS wave number is evidently absent. More characteristic for this case is the plateau region in *k* associated with the formation of isolated shock-like structures. The wave number spectra for the case with both self-interaction and crossed-beam coupling, in Fig. 14(b), combine the features originating from both processes. For all the three cases, we observe that the magnitude of the wave number components is reduced (note the log scale in Fig. 14) with the increase in the damping $\nu \u0302$.

## V. CONCLUSION

We have investigated the role of the laser beam speckle structure in crossed beam energy transfer. It is an aspect of CBET that has been considered only in recent studies. We have demonstrated that the speckle structure plays an important role in laser beams crossing in a plasma with a flow when both the self-focusing of intense laser speckles and forward SBS in RPP beams come into play. This can be expected for laser fluxes above $IL\lambda 02>$ 10^{14} W $\mu m2$/cm^{2}, i.e., for an intensity regime that is relevant for current ICF parameters.

For plasmas with inhomogeneous flow, where CBET occurs around sonic surfaces, the onset of self-focusing instability in speckles is enhanced, leading to a significant beam deflection and resulting in broadening of the angular light distribution in the transmitted laser beams. A regime with a maximum in the net transfer rate is attained for the intensity range of 1.5 $<\u2009IL\lambda 02/($ 10^{14} W $\mu m2$/cm $2)<$ 3 with an observable deflection of the amplified beam to higher angles and an increase in the angular width. Beam deflection and angular broadening may have a significant impact on ICF laser energy deposition. Broadening of the angular distribution in both beams can be attributed to the plasma-induced smoothing and scattering of nonlinear IAW density perturbations.

The temporal incoherence due to plasma-induced smoothing observed in our simulations with RPP beams, see Fig. 10(a), corresponds to correlation times of the order of $2k1cstcorr\u223c20$ being equivalent to a short ps time scale for $\lambda 0=$ 0.35 *μ*m, $Te\u223c$ 3 keV, and angles $\theta \u223c$ 20°. While we have not considered spatiotemporal smoothing, such as SSD, in our study, the result indicates that the effect of SSD with a bandwidth that is smaller than 50 GHz should be quite marginal for the effects described in this study. Furthermore, it is known that the available SSD bandwidth on the National Ignition Facility has not prevented the onset of CBET. A higher SSD bandwidth may be available, e.g., on the Omega laser facility at LLE Rochester, as this may be necessary for the direct drive ICF experiments.

Our study also shows that the presence of self-interaction results in shock-like structures with steepened wave fronts in the plasma density perturbation, which can further lead to deflections of RPP beams in inhomogeneous plasmas.

## ACKNOWLEDGMENTS

This work was carried out within the framework of the EUROfusion Consortium and received funding from the Euratom Research and Training Programme 2014–2018 and 2019–2020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. We would like to thank the CPHT computer support team. W.R. was partially supported by the DOE Fusion Energy Sciences grant, user No. FWP100182.