Towards complete suppression of parametric instabilities with decoupled broadband lasers in plasma
A theoretical analysis for the stimulated Raman scattering (SRS) instability driven by two laser beams with certain frequency difference is presented. It is found that strong coupling and enhanced SRS take place only when the unstable regions for each beam are overlapped in the wavenumber space. Hence a threshold of the beam frequency difference for their decoupling is found as a function of their intensity and plasma density. Based upon this, a strategy to suppress the SRS instability with decoupled broadband lasers (DBLs) is proposed. A DBL can be composed of tens or even hundreds of beamlets, where the beamlets are distributed uniformly in a broad spectrum range such as over 10% of the central frequency. Decoupling among the beamlets is found due to the limited beamlet energy and suitable frequency difference between neighboring beamlets. Particle-in-cell simulations demonstrate that SRS can be almost completely suppressed with DBLs under the laser intensity W/cm. DBLs can be attractive for driving inertial confined fusion.
Campaigns to achieve ignition on National Ignition Facility (NIF) yielded significant insights of inertial confinement fusion (ICF) Betti and Hurricane (2016); Lindl et al. (2014), including reaching the milestone of fuel gain exceeding unity Hurricane et al. (2014). Meanwhile, a few critical challenges to further enhance the laser-target energy coupling efficiency have been revealed. Currently there are no clear paths to ignition on NIF or similar-sized facility. Exploring alternative approaches is necessary. Laser plasma instabilities (LPI) are among the major obstacles to both direct- and indirect-drive schemes, causing asymmetric Town et al. (2014); Moody et al. (2014a) and insufficient drive Moody et al. (2014b); Igumenshchev (2012) and preheating Regan et al. (2010); Smalyuk et al. (2008); Sangster et al. (2008). A few ideas have been proposed to suppress LPI by use of various beam smoothing techniques Skupsky et al. (1989); Lehmberg and Obenschain (1983); Froula et al. (2010); Moody et al. (2001), temporal profile shaping Albright et al. (2014), laser beams with broadband width Thomson and Karush (1974); Eimerl et al. (1992), enhanced plasma damping Turner et al. (1985); Craxton et al. (2015), etc. However, it is not possible to suppress LPI completely.
In this work, we present a theory, backed by particle-in-cell (PIC) simulations, that a new type of lasers called decoupled broadband lasers (DBLs) can completely suppress stimulated Raman scattering (SRS), a major concern to both direct- and indirect-drive ICF Strozzi et al. (2017). A DBL is composed of many beamlets, which may have different frequencies among beamlets within certain range. A related idea is the Coherent Amplification Network (CAN) Mourou et al. (2013). Different from the CAN scheme, here the required laser power of DBLs for fusion application is much lower and there is no requirement for the phase lock between DBL beamlets, which is called incoherent combination Andrusyak et al. (2009); Hamilton et al. (2004); Farmer et al. (1999); Benedetti et al. (2014). More recently, a concept of broadband laser driver called StarDriver was proposed for ICF application to control both hydrodynamic and laser-plasma instabilities Eimerl et al. (2016), where a laser driver is consisted of many beamlets at an aperture. Physically, it is not clear so far whether and how a broadband laser driver may suppress the laser plasma instabilities. In this work, we will clarify the mechanism and conditions on DBLs for almost complete suppression of the SRS instability based upon theoretical and numerical studies.
Ii Model of two beam coupling and decoupling
We first introduce a model for DBLs. The temporal part of such light can be written as , where is the normalized amplitude of the -th beamlet with a carrier frequency , is a random phase between , and is the number of beamlets typically around a few hundreds. The beamlets are nearly uniformly distributed in the total frequency spectrum bandwidth . Here the amplitude is related to the light intensity given by . Before the study of a DBL propagation in plasma, we first consider the coupling of two light components with or 2, where and are the laser wavenumber and frequency, respectively.
Let and , where and are the center frequency and center wavenumber, respectively, and is the frequency difference between them. Under the condition , we can write and , where , and is the electron plasma frequency. The coupled fluid equations for SRS backscattering are
where and are respectively the vector potential of backscattering light and plasma-density perturbations Kruer (1988). For simplicity, we consider a cold plasma, so that the Bohm-Gross frequency for the electron plasma wave , where is the thermal velocity. The characteristic time for SRS development is defined as the reciprocal of growth rate when the instability has developed to a considerable level. Without loss of generality, in the strong coupling regime, the perturbation of resonance system can be treated as a quasi-static process when . Therefore, the dispersion relation of SRS for the two coupled beamlets in the one-dimension (1D) geometry is then obtained as
where . If there is , we have the dispersion relation in the decoupling regime as
Note that Eqs. (3) and (4) are good approximations in the time scale . An explicit threshold condition for Eq. (4) will be given later.
The growth rate of SRS is found by solving Eq. (3) or (4) with the imaginary part of , i.e., , and the area where is the instability region. Here, taking an example, let us consider the case for the laser amplitudes with frequency difference and . We take the plasma density , where is the critical density. The numerical solutions of the dispersion relation Eqs. (3) and (4) with different are plotted in Fig. 1(a). When , it is found that these the SRS instability regions in the space for the two laser beams overlap to form a single instability region. This implies that two laser pulses are coupled in developing the SRS instability. The wavenumber of the maximum growth rate is . However, when the frequency difference between the two lasers are increased to , the instability regions are separated, each of them will develop independently. Note that the maximum growth rate of the coupled case is much higher than the decoupled one.
To validate the coupling of two lasers, we have carried out PIC simulations by use of klap code Chen et al. (2008). We have taken a homogeneous plasma slab in one-dimension. The length of the simulation box is where , and the plasma occupies a region from to with plasma density . The initial temperature is = 100eV. The ions are stationary with a charge . We have taken 100 cells per wavelength and 50 particles per cell. The wavenumber distributions of Langmuir wave are plotted in Fig. 1(b) for with two different frequency gaps. Only one peak can be found at when the frequency difference . When increases to , the strength of Langmuir wave is greatly reduced and two independent peaks can be found at and . This is quite similar as Fig. 1(a). Note that since , the strength of the mode with higher is slightly larger. As a result, one can conclude that when the difference of the two laser beams is small enough, they can be coupled with the same plasma wave with a much higher instability growth rate than that corresponding to two individual laser beams.
In the following, we derive a general condition for the decoupling between two lasers. Defining as the width of the instability region for the light with and amplitude . By letting the growth rate according to Eq. (4) in underdense plasma , one finds
where . Solutions of obtained from Eqs. (4) and (5) are compared as shown in Figs. 1(c) and 1(d). One can find that is strictly proportional to the laser amplitude from Fig. 1(c). Based on Fig. 1(d) we know that is also proportional to the plasma density. Generally Eq. (5) fits well with the numerical results of Eq. (4) in the low density regime. In the derivation of Eq. (5), we have assumed that , therefore the theoretical value is smaller than the numerical solution at . The above results indicate that for a given density profile, we can reduce the laser amplitude to shrink the instability region of backward SRS.
In the case of two incident lasers with , the plasma wavenumber changes with frequency according to . Therefore the condition for decoupling between the two laser beamlets given above in cold plasma can be obtained as , where these two instability regions have no intersections in the wavenumber space. When , this simply corresponds to
where is the linear growth rate of stimulated Raman backscattering for a single beamlet with zero bandwidth. Equation (6) defines the required frequency difference for the decoupling of two laser beamlets under the same amplitude . In this case, the growth rate is determined by a single beamlet even if the whole laser beam is composed of many beamlets. In this way, the instability of the whole laser beam will be controlled provided the instability of a single beamlet is controlled. This is relatively easy to realize since the energy of a single beamlet can be limited to a low level by increasing the number of beamlets. Note that previous theoretical models only suggest that the instability growth rate is modified by a laser with finite bandwidth by . In this case, the linear growth rate is reduced provided Thomson and Karush (1974); Kruer (1988), which can be understood as a destruction of the resonant system. However, this does not imply an effective suppression of the instability when the driving laser energy is high enough.
Iii PIC simulations of interactions between DBLs and plasmas
iii.1 In homogeneous plasma
Following above discussions, we can now extend the coupling of two lasers to a large number of beamlets, such as the DBLs model introduced at the beginning. Here we take the amplitude of each laser beam to be a constant. In the following we compare two cases of DBLs each composed of beamlets under different energy. The frequency difference is fixed as . The overall bandwidth of the DBL can be obtained according to . At the plasma density and this bandwidth, Eq. (6) suggests that the amplitude for each beamlet must be less than in order to suppress the coupling between neighboring beamlets and overall development of SRS. We will consider two cases with the amplitude and . The total light energy of the 100 beams is the same as the single beam with the amplitude and , respectively. For , the maximum growth rate , and its character time is with the laser oscillation period. The coupling between neighboring beamlets will lead to a high SRS level. On the other hand, when the laser amplitude is reduced to , the neighboring laser beamlets will not be coupled any more.
To validate the above theoretical prediction, a series of 1D PIC simulations have been performed for the interactions between DBLs and homogeneous plasma. The plasma length is and two vacuum regions are set at the two side of the plasma. The initial electron temperature is = 100eV. First, we take the case of the DBL with relatively high intensity, i.e., the DBL composed of 100 beamlets each with . Figure 2(a) gives an example of the temporal structure when taking , , and . It shows that there are some fluctuations in the envelope profile. But overall the amplitude appears around . It is to be compared with the single coherent laser beam with the same amplitude , . To see the development of SRS excitation, we diagnose the backscattering light at . The temporal envelop of the backscattered light is compared between the normal laser and DBL as shown in Fig. 2(b). In the case of the DBL with the bandwidth 12%, the growth rate of backscattered light is considerably reduced. However, after certain time about , the backscattered light starts to grow quickly. Finally at , the scattered light saturates at the same level as produced by normal coherent lasers. As discussed above, the character time for an individual beam to develop SRS is , therefore the beams are coupled by the Langmuir wave. From these results, one concludes that instability can grow to a high level due to the coupling between neighboring beamlets as long as Eq. (6) is not satisfied.
Now we consider the opposite case when Eq. (6) is satisfied with a lower laser amplitude and the same beam number . At this amplitude, the decoupling threshold for the bandwidth is about or according to Eq. (6). When the threshold is satisfied, the growth of backscattering light is greatly reduced, as shown in Fig. 3(a), where a comparison between two different bandwidth cases and is made. The result changes slightly when the bandwidth is larger than the threshold. The maximum amplitude for the DBL with at is , which is much smaller than the case for the normal laser beam. Therefore, the electron heating is almost completely suppressed at . Simulation of case is also performed up to , during which nonlinear saturation of SRS is not found. On the contrary, for the case with a normal laser light, the backscattering light reaches to a high saturation level quickly. Correspondingly, hot electrons with temperature around keV are generated, which corresponds to electron heating by the large amplitude of Langmuir wave with a phase velocity about . These simulation results imply that DBLs can overcome the two major problems (laser energy loss and hot electron production) in laser plasma interactions.
When light energy and bandwidth are fixed, the right-side of Eq. (6) will be divided by , which indicates that beam number can reduce the threshold. Figures 3(c) and 3(d) present the Langmuir wave in the space with different beam number , under the same light energy and bandwidth. Considering the incident light frequency changes in [0.94,1.06], one finds that the corresponding ranges in [1.49,1.74], taking . The intensity of comb-spectrum for is much weaker than the case, as shown in the comparison between Fig. 3(c) and 3(d). The latter one breaks the threshold, and the beamlets coupled together to develop SRS, which leads to the production of hot electrons as shown in Fig. 3(b).
iii.2 Effect of nonuniform plasma density
The above theory and simulation are developed for homogeneous plasma. It is expected that the SRS suppression with DBLs is also effective in inhomogeneous plasma. Assuming an inhomogeneous plasma density profile , where mm inside a Hohlraum target for indirect-drive ICF Myatt et al. (2014). For an inhomogeneous plasma, the coupling of each beams will be reduced when their resonant region decreased, where , and are the group velocity of scattered light and Langmuir wave, respectively Liu et al. (1974). When approaches to infinite, the situation transits to homogeneous case. Therefore, the convective instability can be more easily suppressed when or is reduced. An upper-limit threshold is provided by Eq. (6) for inhomogeneous plasma. For NIF with the peak laser intensity and laser wavelength m, the corresponding laser amplitude is . 1D PIC simulations were performed in inhomogeneous plasma with , , and the plasma density linearly ranges in [0.07,0.09]. The initial electron temperature is eV. To compare with the above simulation, here we take and .
We diagnose the energy of the Langmuir wave which is a direct estimation of the strength of SRS. Figure 4 shows the evolutions of for coherent laser and DBL. When , the condition is close to the NIF situations. For the light with , grows linearly with a very small growth rate. On the contrary, increases exponentially at when , and large numbers of hot electrons are produced at . These results are similar to the homogeneous case. If decreased to , the coupling of DBLs is reduced due to the resonant region becomes narrow, which leads to the complete suppression of SRS as shown in Fig. 4.
In passing, we mention that, even though we have only shown the effectiveness of SRS suppression with DBLs with 1D simulation, it is also true in multi-dimensional cases. This is because typically the backscattering has the highest growth rate than the side scattering. Once the backscattering is suppressed, side scattering will be controlled as demonstrated by our 2D simulation Zhao et al. (2015).
Iv Summary and discussion
In conclusion, we have proposed a strategy to suppress SRS significantly by use of so called DBLs with certain bandwidth. It is based upon a model of the coupling between two laser beams with slightly different frequencies. It is found that the couple of the two laser beams in the excitation of SRS is weak as long as their frequency difference is larger than the 70% width of instability region for an individual beam. The latter is proportional to the laser amplitude. Therefore, with a DBL composed of many beamlets (such as 100) with certain frequency difference between individual beamlets (such as 0.12%), SRS could be dramatically suppressed due to the decoupling of the beamlets. It is expected that the DBLs may also be applied to suppress other parametric instabilities for ICF applications.
If a DBL beam is adopted, which is composed of 100 beamlets with the frequency difference between neighboring beamlets, the SRS excitation can be largely avoided. Similar ideas may be utilized to suppress the stimulate Brillouin scattering and two-plasmon decay (TPD) instabilities. Generally, the laser technology for DBLs still needs to be developed. Note that the comb-like spectrum for DBLs can be produced with different schemes Ho and Kahn (1993); Yu et al. (2016). Also the gain bandwidth of lasers over 10% can be realised via parametric amplification in nonlinear crystals Dabu (2010). Therefore, in principle it is possible to build a high power laser system for DBLs.
This work was supported by National Science Foundation of China (Grant No. 11421064).
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