Laser plasma instabilities (LPIs),1–3 in the form, for example, of stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), or two-plasmon decay (TPD), are fundamental limiters of fusion performance for all approaches to laser-driven inertial confinement fusion (ICF),4–6 because they may cause significant laser energy loss, generate undesirable hot electrons,5–13 and seriously influence the drive symmetry on the target. Thus, mitigation of LPI effects is crucial for achieving predictable and reproducible fusion at high gain in the progress toward the practical realization of inertial fusion energy. To suppress LPI for ICF, it has been proposed that the instability growth rate in the plasma could be reduced by either temporal or spatial incoherence of the driving laser. This is because the incoherence of a laser can be understood as a superimposition of temporal and spatial modes, which changes the amplitude a or intensity a² of the laser in time or space. We know that the total laser energy can be written as ∝∫a²dt, while the total instability growth in the linear region can be written as ∝∫adt. Thus, for a given total laser energy, a laser with a changing amplitude exhibits less total instability growth. The effect of incoherence in suppressing LPI can also be explained in another way. For two given stimulated waves, such as a plasma wave and a scattering wave, only one mode of the driving laser can couple with them exactly resonantly, while the others contribute less to the instability growth rate. Therefore, for a given average laser intensity, an incoherent laser has a lower instability growth rate. To date, many methods have been proposed to suppress LPI using spatial and temporal incoherence in the driving lasers, through, for example, laser smoothing techniques14–18 and broadband laser technology.19–23 For the latter in particular, both theoretical and experimental results indicate that the linear growth of LPI can be controlled well when the laser bandwidth is much larger than the growth rate.23,24 Meanwhile, decoupled broadband lasers13,25–27 or polychromatic lights composed of multiple colors of beamlets10,28 show better inhibitory effects than continuous broadband lasers.

Even with these methods, the SRS of the inner laser beams at the National Ignition Facility (NIF)1,12,29–31 remains a major problem for high-convergence implosion with a CH capsule inside a gas-filled hohlraum driven by a long laser pulse.32–34 The inner beams of the NIF hohlraums have a long propagation path and encounter plasma blowoff from the high-Z walls, low-Z plasma ablated from the capsule, and any filling gas within the hohlraum, which results eventually in unpredictable and serious SRS. With the aim of achieving a low level of LPI, attention has turned to the use of ablators made of high-density carbon (HDC), which has the advantage of high density, allowing the use of thinner capsules and shorter laser pulses and thus leading to a low level of LPI and better symmetry control in low gas-filled hohlraums. With an HDC capsule, the recent tremendous breakthrough of 3.15 MJ fusion yield from 2.05 MJ energy input on the NIF has provided a successful demonstration of ignition in ICF.35 Nevertheless, the NIF results demonstrate that there is a limit on the degree to which LPI can be reduced via both temporal incoherence and spatial incoherence, and suppression of SRS is still a crucial task if the NIF is to be able to achieve a predictable and reproducible fusion gain via a low-entropy and high-convergence implosion driven by a long laser pulse.

Given that the temporal and spatial dimensions are related to energy (frequency) and momentum, while the angular dimension is related to angular momentum, it should be possible to suppress LPI via angular incoherence. The spatial coherence area is usually expressed as the product of two spatial lengths, that is, Δs = Δy · Δz for a laser propagating in the x direction. We can rewrite this as Δs = rΔrΔφ. In this way, Δr is related to radial coherence and Δφ to angular coherence. Taking SRS as an example, we know that SRS in plasma is a three-wave coupling process related to the decay of a driving laser into an electron plasma wave and a scattering wave, in which the following frequency and wave number matching corresponding to energy and momentum conservation must be satisfied: ω_L = ω₁ + ω₂ and k_L = k₁ + k₂. In addition to energy and momentum, the driving laser may also carry angular momentum, in which case angular momentum conservation must also be satisfied for the LPI, that is, L_L = L₁ + L₂. In fact, the presence of orbital angular momentum in the driving laser beam (such as in a Laguerre–Gaussian beam) has been used to study Raman scattering in laser–plasma interaction.36–39 In this paper, we propose for the first time to suppress LPI with angularly incoherent light and pay particular attention to the effect of angular incoherence on the instability growth rate, which should pave the way towards a low-LPI laser system with a super light spring of incoherence in all dimensions of time, space, and angle. We prove analytically and with three-dimensional (3D) particle-in-cell (PIC) simulations that angular incoherence provides suppression of the instability growth rate that is additional to and much stronger than that provided by temporal incoherence.

A good description of the radial and angular momenta of the driving laser can be given using the Laguerre–Gaussian modesaLx,t=an⁡expiωLt−ikLx+ilLφ+ϕ,(1)where a_n is the transverse profile, ω_L is the laser frequency, k_L = 2π/λ_L is the wave number, l_L is the topological charge, φ = tan⁻¹(z/y), and ϕ is the original phase at the waist plane. The transverse profile a_n is given byan=a0(−1)pCplw(x)2rw(x)l⁡exp−r2w(x)2Lpl2r2w(x)2,where a₀ is the normalized amplitude, C_pl is a normalization constant, wx=w01+x2/xR2 [where x_R is the Rayleigh length Rx=(x2+xR2)/x and w₀ is the beam-waist radius], r=y2+z2, is the associated Laguerre polynomial, and p is the number of radial nodes in the intensity distribution. In this paper, we concentrate on studying the effect of angular incoherence, and therefore we set p ≡ 0.

Here, we consider a specific case. The driving laser is a superimposition of N modes of different frequency ω_L and topological charge l_L. The laser amplitude is written asaLx,t=∑n=0N−1an⁡expiωLnt−ikLnx+ilLnφ+ϕn.(2)For simplicity, we assume a_n ≡ a₀. The frequencies ω_Ln = 1 + nɛ₁ω_L0 of the modes have the same interval ɛ₁ω_L0, and the total bandwidth is Δω = (N − 1)ɛ₁ω_L0, where ω_L0 is the frequency of the first mode and ɛ₁ is a constant that indicates the bandwidth gap. Usually, Δω/ω_L0 ≪ 1. The charges l_Ln = 1 + nɛ₂l_L0 and the total topological charge spread Δl = (N − 1)ɛ₂l_L0, where l_L0 is the topological charge corresponding to the first mode and ɛ₂ is a constant that indicates the topological charge gap. Typically, ɛ₂l_L0 is an integer. If ϕ_n ≡ 0, the structure of such a laser is called a light spring.40,41 Thus, the distribution of the highest intensities, that is, the strong point, resembles a spring. In this paper, we call a light spring with random phases ϕ_n a super light spring. Hereinafter, for simplicity, a laser with Laguerre–Gaussian modes is referred to as an LG laser, and a laser with a light spring or super light spring structure is referred to as an LS or super LS laser. Owing to superimposition, the amplitude for ϕ_n ≡ 0 can be written asaLx,r,φ,t=a0rsinNε1ωL0t−ε1kL0x−ε2lL0φ/2sin(ε1ωL0t−ε1kL0x−ε2lL0φ)/2.(3)This implies that the angular spread is reduced to 2π/N, and the angular position of the highest intensity can be written as φ_max = ɛ₁ω_L0t/(ɛ₂l_L0). By changing φ from 0 to 2π, we can obtain the pitch of an LS laser.

SRS of a nonrelativistic laser can be described by42,43∂2∂t2−c2∇2+ωpe2ã=−ñeaL,(4)∂2∂t2+ωpe2−3ve2∇2ñe=n0∇2aL⋅ã,(5)where ω_pe is the electron plasma frequency, a_L is the vector potential of the driving laser pulse, ã is the vector potential of the backscattered wave, ñe is the plasma density perturbation, and n₀ = ωpe2/4πe. We neglect the radial gradient (this is reasonable for a sufficiently large focal spot) and consider that a wave of the form ∼exp(iωt − ikx + ilφ) is excited. For the scattered wave of lower frequency, we haveω2−ωl2=kl2c2ωpe2a024×∑n=0N−11(ω−ωLn)2−k−kLn2c2−c2(l−lLn)2/r2−ωpe2,(6)where ωl=ωpe2+3ve2k2+l2/r21/2 is the Langmuir wave frequency and kl2=k2+l2/r2. Note that the dispersion relation depends on the radial position r. For simplicity, we discuss the dispersion relations at radius R of the highest intensity for the LG and LS pulses.

Note that only one laser mode is exactly resonant. Denoting the resonant mode by (ω_L0, k_L0, l_L0), we obtainωl−ωL02−k−kL02c2−l−lL02c2/R2−ωpe2=0.(7)We write ω = ω_l + δω = ω_l + iγ_s, with δω ≪ ω_l. For simplicity, we ignore the effect of wave vector matching.

First, we discuss the case of ɛ₂ = 0. We haveγs2≈kl2c2a0216ωpe2ωlωs0∑n=0N−111+nε1ωL02/γs2,(8)where ω_s0 = ω_L0 − ω_l is the frequency of the backward SRS beam. When nɛ₁ω_L0 ≪ γ, we obtain the following expression for the instability growth rate γ_s:γs≈klca04ωpe2ωlωs01/2N1−Nε1ωL023γs21/2≈γ01−Δω26γs2,(9)where γ0=N(klca0/4)(ωpe2/ωlωs0)1/2 is the growth rate without spreading. Equation (9) means that a larger bandwidth is better for suppressing SRS.

We now discuss the case of ɛ₁ = 0. We assume that orbital angular momentum is not transferred to the electron plasma wave and that nε2lL02c2/R2≪ωlωs0. We haveγs2≈kl2c2a0216ωpe2ωlωs0∑n=0N−111+nε2lL02c2/ωs0γsR22.(10)When nε2lL02c2/R2≪ωs0γs, we obtain the following expression for the instability growth rate γ_s:γs≈klca04ωpe2ωlωs01/2N1−13Nε2lL02c2ωs0γsR221/2≈γ01−16lL0Δlc2ωs0γsR22.(11)Here, we assume that Δω/ω_L0 ≪ 1 and Δl/l_L0 ≪ 1, which are reasonable assumptions to obtain the approximate growth rate.

For an actual superimposed laser pulse, the peak laser amplitude is Na₀. Therefore, the present formulas for the growth rate is the same as before except for the correction term lL0Δlc22/6(ωs0γsR2)2. According to Eqs. (9) and (11), we can see that there are two correction terms that can reduce the instability growth rate. One is the total bandwidth Δω, which has been widely investigated in ICF. The other is the total topological charge spread Δl. Note that at Δl>1/lL0(ωs0/ωL0)2πR/λL02(Δω/ωL0), the Δl term will dominate the correction term, and hence the spread of angular momentum plays a more important role than the bandwidth in suppressing γ_s. Because Δω/ω_L0 ≪ 1, it is easy to realize Δl>1/lL0(ωs0/ωL0)2πR/λL02(Δω/ωL0) for a real superimposed laser pulse.

To confirm the above analysis, we perform 3D PIC simulations using the POCH code.44 In all simulations, we set ɛ₂l_L0 = 1. To reduce the simulation time, we take l_L0 = 3 and N=7,resulting in an average topological charge l̄=6. Note that a much larger l_L0 can be used in a real experiment. For simplicity, we consider the same transverse profile for each mode in Eq. (2):an≡a0=a002rw0lL0⁡exp−r2w02.To further reduce the simulation time, we choose a large amplitude a₀₀ = 0.06, w₀ = 10 µm. The central laser wavelength is λ_Lc = 0.8 µm and the corresponding central frequency is ω_Lc = 3.75 × 10¹⁴ Hz. The frequency of the first mode is ω_L0 = 3.4125 × 10¹⁴ Hz (ω_L0 = 0.91ω_Lc).

Therefore, the power of the LS pulse is P = 0.16 TW. The frequency separation ɛ₁ω_L0 = 0.03ω_Lc = 0.033ω_L0 (ɛ₁ = 0.033), and thus the total frequency spread is N−1ε1ωL0=0.198ωL0. With this, we obtain a pitch Δx = 2πc/(ɛ₁ω_L0) ≈ 27 µm for the LS pulse. The laser has a constant amplitude in time, with a duration of about 93.3 fs. The plasma density in the simulation is n_e = 1.7 × 10²⁰ cm⁻³. The size of the simulation box is 60 µm (x) × 80 µm (y) × 80 µm (z), corresponding to a moving window of 600 × 800 × 800 cells, with one particle per cell. The plasma occupies the 15 µm < x < 800 µm region in the direction of the laser pulse propagation, and −75 µm < y < 75 µm, −75 µm < z < 75 µm. It should be noted here that we consider a plasma with a relatively high density to ensure that the scattered wave can be separated from the driving wave completely in spectral space and then filtered out for mathematical analysis. Actually, the angular coherence as well as the temporal and spatial coherences are characteristics of the light, and the suppression of SRS for an LS laser with angular incoherence also works in a low-density plasma.

Figure 1(a) shows the 2D k-space distribution of the driving laser pulse after propagation for 240 µm. As can be seen, the LS pulse is composed of seven separated frequencies, and the weak signal on the left side of the driving laser is Raman scattering. For comparison, we consider a broadband LG laser pulse, which is a superimposition of N modes with different frequencies but the same topological charge l̄, that is, with a spread of ɛ₂l_L0 = 0 for the angular momentum. The total power and other parameters remain the same as those for the above LS pulse. Figure 1(b) shows the 2D k-space distribution for the LG case, again after the pulse has propagated 240 µm. Clearly, there is a much stronger SRS signal in the LG case, which confirms the above theoretical expectation that the Δl incoherence dominates the suppression of instabilities. We have also performed the simulations for a laser composed of Gaussian pulses at multiple frequencies, and the results are similar to those in the case of a broadband LG pulse and there is less of a suppressive effect than in the case of an LS pulse with Δl incoherence.

To clearly observe the growth process of SRS due to Δl, we plot in Fig. 2 the maximum field ratio between the SRS and the driving pulses along the laser propagation direction for different cases of Δω and Δl. The maximum field ratio is denoted by η_S = E_s/E_L, where E_s and E_L are the maximum fields of the SRS and driving pulses, respectively. From Fig. 2, we can observe the following. First, η_S is highest for ɛ₁ = 0 and ɛ₂l_L0 = 0, indicating that this case has the most serious SRS for an LG pulse with a single frequency and a single topological charge. Second, at ɛ₂l_L0 = 0, i.e., with a single topological charge, η_S at 200 μm decreases from 0.36 in the narrowband case (ε1=0 to 0.2 in the broadband case (ε1=0.033. This means that, as is well known, the use of a broadband pulse can help to suppress SRS. Third, keeping the same frequency band, we compare the LS case (ε2lL0=1 and the LG case (ɛ₂l_L0 = 0). As can be seen, in the narrowband case (ɛ₁ = 0), η_S at 200 μm drops from 0.36 for the LG pulse to 0.15 for the LS pulse; and for the broadband case (ε1=0.033, η_S drops from 0.2 for the LG pulse to 0.05 for the LS pulse with random phase. These results indicate that the angular spread Δl can strongly suppress SRS growth.

It should be noted here that if a laser is composed of several beamlets with the same frequency but different topological charges, such as in the case of ɛ₁ = 0 and ɛ₂l_L0 = 1, then the pitch of the LS pulse is infinite. This implies that the strong point with the highest intensity remains unchanged in the transverse direction. However, as shown in Fig. 2, SRS can still be suppressed to some degree, even though the peak intensity is much larger than that for an LG pulse. This strongly supports our idea that the spread of the angular momentum of the laser plays an important role in suppressing SRS. In addition, from Eq. (11), we can see that the suppressive effect is positively correlated with Δl. This implies that the suppressive effect is stronger for larger Δl. It should be noted that a large Δl means that the (average) topological charge of the LS beam should also be large, which means that the transverse spot of the driving beam must be sufficiently large. However, the focal spot of the driving beam in ICF is usually several hundred micrometers in size, which is large enough compared with the 10 µm spot that we considered above. Therefore, taking account of the practical manipulations involved in ICF, the results in the present paper will be applicable at larger Δl.

One concern is that the superimposition of the LG beamlets produces a strong point in space and time, which may produce extra hot electrons and generate unacceptable shock preheating and entropy in the ICF fuel. Thus, it is necessary to investigate whether an LS pulse may produce more hot electrons and a higher electron temperature than an LG pulse. Figure 3 compares the electron distributions in the plasma driven by LG and LS pulses, respectively. Here, the plasma wave has still not completely decayed. It can clearly be seen that the LS pulse does not produce a high temperature or more hot electrons. Thus, the spread of the angular momentum of the LG pulse does not give rise to extra hot electrons. It should also be noted that SRS is a relatively long-term behavior in laser–plasma interaction. Here, we have considered relatively short pulses of 93.3 fs to save computational time. To avoid artificial results, we have carried out an additional PIC simulation for a longer duration of 186.6 fs, and this also shows a clear suppressive effect and a similar growth rate.

In a laser facility, several laser beamlets are usually clustered into a laser bundle, which is used to drive a target. This makes it possible to change the beamlets from the usual Gaussian pulses to LG pulses with different topological charges and combine them into a super LS pulse with a spread topological charge. As shown in Fig. 4, an LS pulse can be produced by clustering obliquely incident LG beams of different frequencies ω_L and topological charges l_L owing to the changing thickness of the phase plate azimuthally. The incident angles θ of each incident LG beam relative to the x axis as shown in Fig. 4 differ slightly, owing to the small differences between the distances from the beamlets to the center of the source field plane. The pitch of the LS pulse is Δx = 2πc/(ω_L0ɛ₁), with ω_L0ɛ₁ = Δω/Δl. For large laser facilities, it is at present still very difficult to control the relative phases of the beamlets. This implies that ϕ_n is not the same for different LG pulses. Therefore, we usually obtain a super LS because of the random phase. Although this has several strong points instead of a single perfect one, the spread of the angular momentum remains. The lowest growth rate is obtained in this case owing to the additional spatial incoherence, as shown in Fig. 2.

In a laser facility, we can create super LS bundles with incoherence in all dimensions of time, space, and angle by combining LG beamlets. By controlling the phases and frequencies of these beamlets before their combination, we can finally obtain the following three cases of the LS bundle, as shown in Fig. 4: (a) LS light with long pitch, which is narrowband, with one strong point of time-independent intensity; (b) LS light of short pitch, which is broadband, with one strong point of time-dependent intensity; (c) super LS light, with several strong points of time-dependent intensity owing to the random phase. Our results for these three cases confirm that angular incoherence clearly reduces the instability growth rate, with the suppressive effect of the super LS light being the most significant and even much stronger than that of temporal incoherence.

In summary, using both theoretical analysis and 3D PIC simulations, we have demonstrated the ability of an LS pulse with angular incoherence to suppress SRS in a plasma, which is of great significance for laser-driven ICF. According to our analytical study and simulations, angular incoherence has a much stronger suppressive effect on the instability growth rate than the temporal incoherence that is typically used. In particular, it is the angular momentum spread that plays an important role instead of the topological charge in suppressing instabilities. In other words, little suppression of SRS is achievable by only increasing the topological charge. In addition, it is interesting to note that the LS pulse does not generate additional hot electrons. It should be noted that in the simulations in this paper, we have considered the suppression of SRS as a specific example, but our conclusions can be applied to the suppression of SBS and other parametric instabilities. In a laser system, laser bundles of a super LS pulse can be generated by combining LG laser beamlets with different frequencies, different topological charges, and slightly different incidence angles, which can be produced from Gaussian beamlets using different phase plates. Our work has revealed a novel way to suppress LPI using light with angular incoherence, and it should be feasible to construct a low-LPI laser system by using super LS pulses with incoherence in all dimensions of time, space, and angle corresponding respectively to energy (frequency), momentum, and angular momentum. This may pave the way toward a low-LPI laser system for achieving predictable and reproducible fusion at high gain and may open the door to the use of longer-wavelength lasers for inertial fusion energy.

Acknowledgment. This work was supported by the National Key R&D Program of China (Grant No. 2018YFA0404803) and the National Natural Science Foundation of China (Grant Nos. 11922515, 11935008, 11335013, and 12035002).