Mitigation of laser plasma parametric instabilities with broadband lasers

Abstract

Laser plasma instabilities (LPIs) cause laser energy loss, asymmetric and insufficient compression, and target preheating, thus are assumed to be among the major concerns of inertial confinement fusion research. Mitigation of LPIs can enhance the laser–target coupling efficiency and optimize the target compression dynamics, which is critical for the realization of robust and high-efficiency fusion ignition. Broadband lasers with polychromatic components or random phases have been investigated for decades as an effective alternative to mitigate LPIs. Here, we present a brief overview on the progress of broadband LPIs, including the models of broadband lasers, the involved physics, the conditions for effective suppression of LPIs, and some schemes to produce broadband lasers.

1 Introduction

Large-scale plasmas are produced by laser irradiation in most of the inertial confinement fusion (ICF) schemes (Froula et al. 2010; Pesme et al. 2002; Hinkel et al. 2004; Tikhonchuk et al. 2019), where a target containing deuterium and tritium fuel is directly (direct drive) or indirectly (indirect drive) compressed by high-power laser beams (Craxton et al. 2015; Lindl et al. 2014; Hurricane et al. 2014; Betti and Hurricane 2016; He et al. 2016; Lan et al. 2017; Zhang et al. 2020a, b). During the interactions of high-power lasers with large-scale plasmas, laser plasma instabilities (LPIs) can be fully developed at the 10 picosecond time scale (Kruer 1988). The LPIs, such as stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), and two-plasmon decay (TPD), are among the major obstacles to achieving fusion ignition, as they can cause significant laser energy loss (Ping et al. 2019; Rosenberg et al. 2018), asymmetric and insufficient compression (Igumenshchev et al. 2012; Strozzi et al. 2017; Moody et al. 2012), and target preheating (Dewald et al. 2010; Smalyuk et al. 2008; Christopherson et al. 2021; Montgomery 2016). Cross-beam energy transfer (CBET) may significantly affect the implosion velocity and ablation pressure in ICF typically involving many laser beams (Radha et al. 2016), where one of the pump beams is a seed light to drive SBS via beating with the other one. Even though the nonlinear processes of LPIs and their mitigation have been investigated for five decades (Kaw 2017), these problems have not yet been completely understood and solved, partially due to the parameter sensitivity and complicated coupling mechanism at large temporal and spatial scales. For the mitigation of LPIs, a few ideas have been proposed, such as the use of spike trains of uneven duration and delay (Albright et al. 2014), second- and third-harmonic mixing (Feng et al. 2020), application of strong DC magnetic fields (Liu et al. 2018; Zhou et al. 2021), polarization rotation (Barth and Fisch 2016; Zhong et al. 2021), as well as broadband lasers (Zhao et al. 2017b; Follett et al. 2019).

The recent achievement of the fusion energy yield of 1.3MJ at the National Ignition Facility (NIF) (Abu-Shawareb et al. 2022) indicates that the enhancement of the laser–target coupling efficiency remains a critical issue for achieving robust high-efficiency ignition. Compared to the compromise scheme of near-vacuum hohlraum (Hopkins et al. 2015), both the energy loss and the dynamic behavior of gas-filled hohlraum could be controlled using the broadband lasers, which may improve the system stability. The experiment result about shock ignition also indicates that a good control of both filamentation and SRS remains a key factor for the success of the shock ignition scheme (Cristoforetti et al. 2021). The basic physics associated with the mitigation of LPIs is to break the phase-matching conditions so that the instabilities effectively grow within limited time and space. Currently, broadband lasers are considered to be promising for future ICF drivers. In this paper, we present an overview on relevant progress made on this topic.

Mitigation of LPIs via broadband lasers has been investigated from the early 1970s. The early broadband laser model was based upon the Kubo-Anderson process, where the laser phase jumps randomly within a range $[0, 2 \pi ]$ following a Poisson process (Laval et al. 1977; Lu 1988). The mean jump time $\Delta t_{\text{c}}$ (coherence time) of the random phase leads to a bandwidth $\sim 1/\Delta t_{\text{c}}$. A simplification of the random-phase equations was usually made according to the pump frame transformation, where the group velocity of incident laser $v_0$ is assumed to be infinite while comparing to that of daughter waves near the quarter-critical density or other special configurations (Lu 1989; Divol 2007). The solutions of linear stochastic differential equations can be expressed by Greens function, and the growth rate is obtained from the imaginary part of the pole (Brissaud and Frisch 1974; Thomson 1975; Laval et al. 1977). The random-phase model was also used in the description of LPIs driven by smoothing lasers which introduce a small bandwidth though frequency modulation (Guzdar et al. 1991a; Divol 2007). Different from the random-phase model, low-coherence lasers are generated by a fundamental seed pulse with a large bandwidth (such as 50 nm), which have a broad spectrum at any time within the pulse laser duration (Gao et al. 2020). These kinds of lasers are often considered to be composed of many different-color components in numerical simulations (Estabrook and Kruer 1983; Zhao et al. 2017a). A sunlight-like model was proposed recently, which can describe a broadband laser with continuous frequency spectrum. Such lasers satisfy the Maxwells equations and can be easily implemented in simulation codes (Ma et al. 2021).

Thomson and Karush (1974) investigated the mitigation of LPIs via a broadband laser with random phases in homogeneous plasmas, and found that the parametric instabilities can be reduced when the laser bandwidth is much larger than the linear growth rate. It was also found that the threshold of convective modes in inhomogeneous plasmas is enhanced by a large bandwidth (Thomson 1975). The reflectivity and nonlinear evolution of instabilities driven by broadband pumps were investigated via particle simulations, which show an enhancement of forward SRS via wave beating of different-color components in tenuous plasmas (Kruer et al. 1973; Estabrook et al. 1980). A considerable reduction of reflection was found when the bandwidth is ten times larger than the growth rate (Estabrook et al. 1981; Estabrook and Kruer 1983). The absolute and convective instability thresholds for incoherent pump were studied based on the envelop equations (Laval et al. 1976, 1977), and later the validity regions for the statistical description were established (Pesme et al. 2007). Bonnaud and Reisse (1986) found that the forward SRS is less sensitive to laser bandwidth than backward SRS, and the former is weakened only at large detunings. An effective Hamiltonian method was developed to describe the LPIs in the presence of space-time fluctuations in homogeneous and inhomogeneous plasmas, whose applications in the absolute instabilities near the quarter-critical density were presented in detail (Lu 1988, 1989). Guzdar et al. (1991b) found that the interaction region of the daughter waves is extended by the bandwidth in inhomogeneous plasmas, which can compensate the reduction of temporal growth rate. DuBois et al. (1992) introduced the finite bandwidth description into the multibeam LPIs model. Control of SRS and hot-electron generation in the interactions of chirped pulse and low-density plasmas was found at the time scale less than one picosecond (Dodd and Umstadter 2001). However, the mitigation of SRS via chirped pulse fails at the nanosecond-scale LPIs (Zhong et al. 2021; Wu et al. 2001). The general dispersion relations of partially coherent light in a uniform plasma were obtained according to a generalized Wigner–Moyal statistical theory, which were then used to analyze the broadband effect on SRS and SBS (Santos et al. 2007; Brandão et al. 2021). Follett et al. (2018) concluded from three-dimensional (3D) simulations that 0.7% frequency detuning is sufficient to eliminate the hot electrons generated via TPD under the OMEGA laser conditions. They later evaluated the absolute instability threshold for SRS and TPD driven by a broadband laser numerically (Follett et al. 2019). Palastro et al. (2018) found that laser bandwidth has little effect on resonance absorption in the linear regime. Numerical simulations showed a significant reduction in CBET with bandwidth of 0.2–0.6% (Bates et al. 2018), which was confirmed in the experiments (Marozas et al. 2018). Control of CBET can significantly enhance the beam–target coupling efficiency (Döppner et al. 2020). Zhao et al. (2017b, 2019, 2021) proposed the decoupling conditions of polychromatic lights in both homogeneous and inhomogeneous plasmas under single and multibeam configurations. Such polychromatic beams behave well both in the linear and nonlinear regimes of LPIs under weak decoupling conditions.

Some early experiments with low-intensity drive laser were reported to be consistent with the random-phase theoretical predictions (Obenschain et al. 1976). Mitigation of LPIs via temporal smoothing lasers or combining with polarization smoothing (PS) has been extensively studied in the experiments. The laser bandwidth introduced via smoothing by spectral dispersion (SSD) causes the speckle structure at the focal plane to vary in time due to the angular dispersion. For indirect-drive scheme, the experiments using 527 nm drive laser with a bandwidth of 0.1% indicate that the SRS levels in low-density region can be dramatically reduced (Montgomery et al. 1996). Recently, Seaton et al. (2022) found with particle simulation that the nonlinear ion behaviors can be suppressed by the introduced spectral dispersion. Beam smoothing suppresses the filamentation instability, which indirectly reduces the SRS backscatter. High-efficient laser coupling was found for a 527 nm drive beam smoothed with combined SSD and PS at laser intensity $I_0\le 7\times 10^{14}$ W/cm$^2$. Without SSD and PS the coupling is efficient only at lower intensities $I_0\le 3\times 10^{14}$ W/cm$^2$ (Moody et al. 2009). The smoothing time scale of SSD is the decorrelation time of a speckle, whereas it is instantaneous for PS. Therefore, application of both 250 GHz of SSD and PS in combination produces a significantly reduction in the total backscattering from a long scale-length gasbag plasma (Moody et al. 2001). For laser and plasma scales relevant to NIF experiments, it was found that PS or SSD alone does not always effectively control LPIs. Theoretical work indicated that a small amount of laser bandwidth can strongly reduce the reflectivity from a spatially smoothed laser beam, and the effective bandwidth is doubled using PS (Divol 2007). Mitigation of LPIs via SSD was also found for direct drive. Shock ignition experiments performed on LMJ-PETAL showed the considerable decrease of SBS and SRS as well as hot-electron generation via SSD (Baton et al. 2020). Stimulated Raman side-scatter was found to be reduced by introducing one-dimensional (1D) SSD (Kang et al. 2021).

Based on the mitigation effect of SSD, a broadband laser driver scheme StarDriver containing a large number of beamlets was proposed (Eimerl et al. 2016). Each beamlet has a two-frequency two-dimensional (2D) SSD, with a total phase bandwidth up to 1.5 THz at 351 nm wavelength. As a result, the ensemble lasers have a bandwidth 2–10%. However, it was proposed that as many as 15,000 monochromatic beamlets are required to reach 1% asymptotic smoothness, which makes the system complicated. Some new smoothing schemes were proposed to reduce the amplitude modulation of SSD using multi-color spectral distribution (Fusaro et al. 2021) and to generate the light field with time-dependent polarization rotation (Zhong et al. 2021). Low-coherence green laser with energy over 400 J and bandwidth $\sim$ 0.6%$\omega _0$ has been produced in KUNWU (Gao et al. 2020). The fourth generation laser for ultrabroadband experiments (FLUX) is under construction, which aims to generate 150 J and 1.5 ns ultraviolet incoherent pulses with fractional bandwidth over 1.5% (Dorrer and Spilatro 2022).

Significant efforts have been paid to produce the ultraviolet broadband lasers, and many new technologies have been proposed, such as stimulated rotational Raman scattering (Weaver et al. 2017), excimer lasers (Obenschain et al. 2015) and sum-frequency generation scheme (Dorrer et al. 2021; Dorrer and Spilatro 2022). However, no practical scheme has been demonstrated up to now for efficient third-harmonic generation (THG) of broadband beams (Dorrer et al. 2021). Similar to the control of LPIs, the phase-matching conditions required for the frequency conversion of broadband lasers are also hard to be satisfied, which lead to the inefficient conversion of near-infrared broadband pulses to ultraviolet. Zhao et al. (2022) proposed the polychromatic driver scheme with significant improvement of the conversion efficiency, where high-power polychromatic beamlets are generated via optical parametric amplification (OPA) (Cui et al. 2016; Eimerl et al. 1995; Banks et al. 2002; Haynam et al. 2007; Baisden et al. 2016; Mima 2010), and each beamlet is triple-frequency converted independently. The polychromatic scheme are based on the matured technologies, and both single-beam and collective LPIs are effectively mitigated, thus may pave the way towards realization of robust and high-efficiency fusion ignition.

In this paper, we present a review on the recent progress of LPI mitigation with broadband lasers. First, some models on broadband lasers are introduced in Sect. 2. The mitigation of LPIs by broadband lasers in homogeneous and inhomogeneous plasmas is discussed in Sects. 3 and 4, respectively. In Sect. 5, we introduce some broadband technologies for high-power laser systems, especially the polychromatic driver scheme. A summary and some discussions on the interactions of broadband lasers with plasma are given in Sect. 6.

2 Description of broadband lasers

Five kinds of broadband lasers are mainly discussed here, i.e., chirped pulse, modulated pulse, polychromatic light, low-coherence laser and sunlight-like laser. We first investigate the chirped pulse with central frequency $\omega _0=2\pi /\tau$, which can be described as

$$\begin{aligned} a_{{\text{c}}}=A(t)\exp (-i\omega _0t)\exp \left[ -\frac{1+i\alpha }{2}\left( \frac{t}{T}\right) ^{2m}\right] . \end{aligned}$$

(1)

Here, A(t), $\tau$, $\alpha$ and T are the pulse envelop, light period in vacuum, chirp rate and pulse duration, respectively. The pulse envelop can be flattened by increasing the parameter m, and the full-width at half-maximum (FWHM) of spectrum is $\sqrt{\ln 2(1+\alpha ^2)}/\pi T$ for $m=1$. Note that the frequency of the chirped pulse varies with time, i.e., its spectrum is an instantaneously narrow band. For example, the pulse bandwidth is very small $\sim 0.05\%\omega _0$ within the LPI developing time scale $\sim 10$ ps regarding for a nanosecond pulse with a whole bandwidth even up to $\Delta \omega _0=5\%\omega _0$. Therefore, chirped pulse may mitigate the convective SBS in a large-scale plasma, but has little effect on the long-time behavior of SRS.

Sinusoidal frequency modulation laser is considered here as an example of modulated pulse, which can be described by

$$\begin{aligned} a_{f}=a_0\cos [\omega _0t+m\cos (\omega _0f_mt)]. \end{aligned}$$

(2)

The bandwidth of $a_{f}$ is approximately given by $\Delta \omega _0/\omega _0\approx (2m+1)f_m$, and $f_m$ is the frequency difference between two adjacent frequencies. The temporal part of the laser phase $\phi (t,y)=\omega _0t+m\sin (\omega _0f_mt+\kappa y)$ at the final focusing lens via SSD is the same to Eq. (2), where $\kappa y$ is related to the phase shift across the lens due to grating dispersion (Moody et al. 2001). Note that Eq. (2) can be expanded to

$$\begin{aligned} a_{f}=a_0J_0(m)\cos (\omega _0t)+\sum _{j=1}a_j\{\cos [(1+f_mj)\omega _0t]-\cos [(1-f_mj)\omega _0t]\}, \end{aligned}$$

(3)

where $a_j=a_0J_j(m)$, and $J_j(m)$ is the j-th order Bessel function of the first kind. For a finite beam number, the general expression of Eq. (3) is the polychromatic light model given by

$$\begin{aligned} a_{{\text{pl}}}=\sum _{j=1}^Na_j\cos (k_jx-\omega _jt+\phi _{j}), \end{aligned}$$

(4)

where $\phi _{j}$ is a random phase between $[-\pi , \pi ]$, $\omega _j$ and $k_j$ denote the frequency and wavenumber of j-th beam, respectively. The spectrum is truncated by the number of frequencies N for a finite bandwidth $\Delta \omega _0=\omega _N-\omega _1$. The polychromatic model Eq. (4) has been widely used in the studies of broadband effects on LPI mitigation. Different power spectra of Eq. (4) can be applied such as Gaussian, Lorentzian, and flat profiles (Zhao et al. 2017b; Follett et al. 2019; Luo et al. 2022). For the flat spectrum, the amplitude of each color component is $a_j=a_0/\sqrt{N}$ under the same incident energy of a monochromatic laser (Zhao et al. 2017b, 2019). Low-coherence lasers are generated by a fundamental seed pulse with a large bandwidth, which have a broad spectrum at any time within the pulse duration (Gao et al. 2020). Therefore, it is always described by the polychromatic model with a continuous spectrum.

The above polychromatic light model can be extended to a sunlight-like laser with continuous spectra (Ma et al. 2021). If one introduces the amplitude frequency spectrum of a broadband laser light $f(\omega )$ and the phase frequency spectrum $\phi (\omega )$, where $f(\omega )$ is a predetermined positive function and $\phi (\omega )$ is a random function of the frequency changing within $[0, 2\pi ]$, then the overall amplitude–phase frequency spectrum can be written as

$$\begin{aligned} E(\omega )=f(\omega )\exp [i\phi (\omega )]. \end{aligned}$$

(5)

The electric field E(t) of a broadband laser in the time domain can then be obtained by the inverse Fourier transform of the amplitude–phase frequency spectrum $E(t)=\mathcal{F}^{-1}[E(\omega )]$, where $\mathcal{F}^{-1}$ denotes the inverse Fourier transform. The bandwidth of the sunlight-like laser is determined by the frequency range of $f(\omega )\ne 0$. According to Eq. (5), the amplitude-phase frequency spectrum $E(\omega )$ changes with different random-phase spectra. Therefore, when two linearly polarized broadband beams with different amplitude or phase spectra are synthesized in two orthogonal polarization directions respectively, one can obtain a new kind of laser with randomly changed polarization in time (Ma et al. 2021). This kind of light is named as sunlight-like laser, which may be generated by dividing a single low-coherence laser evenly into two pulses in time and then synthesizing them in two orthogonal polarization directions. Comparing to the typical PS method, the sunlight-like laser actually realizes temporal random polarization.

Coherence time is a good metric to describe the broadband laser with continuous spectra, which is defined as $\Delta t_{\text{c}}\equiv \int _{-\infty }^{-\infty }|f(\tau _0)|^2{\text{d}}\tau _0$ with $f(\tau _0)\equiv \langle E^*(t)E(t+\tau _0)\rangle /\langle |E(t)|^2\rangle$ being the correlation function (Goodman 2015; Follett et al. 2019). Here, the angular brackets represent the time average. The bandwidth scale is usually estimated by the reciprocal of autocorrelation or coherent time (i.e., $\Delta \omega _0\sim 1/\Delta t_{\text{c}}$) in the broadband LPI models with random-phase modulation (Thomson and Karush 1974; Lu 1988).

3 Mitigation of LPIs in homogeneous plasmas with broadband lasers

3.1 Random phase model of broadband lasers

Absolute instabilities are the major concern in homogeneous plasmas, which grows with time exponentially. The fundamental requirements for the development of LPIs are the wave-vector matching condition ${\vec{k}}_0={\vec{k}}_{\text{p}}+{\vec{k}}_{\text{s}}$ and the frequency matching condition $\omega _0=\omega _{\text{p}}+\omega _{\text{s}}$, where ${\vec{k}}_0$, ${\vec{k}}_{\text{p}}$ and ${\vec{k}}_{\text{s}}$ are the wave vectors of incident light, plasma wave and scattering light, and $\omega _{\text{p}}$ and $\omega _{\text{s}}$ are the frequencies of plasma wave and scattering light, respectively. In general, broadband laser mitigates LPIs by detuning the matching conditions.

Based on the three-wave coupling equations, Thomson and Karush (1974) studied the dynamic behavior of LPIs driven by a broadband laser $a_{f}=a_0\cos [\omega _0t+m(t)\cos (\omega _0f_mt)]$, where m(t) is a stochastic component, and its autocorrelation time $\Delta t_{\text{c}}\sim 1/\Delta \omega _0$ is much shorter than the characteristic growth time of the instability mode $\Delta t_{\text{g}}=1/\Gamma$. The linear growth rate $\Gamma$ is the imaginary part of plasma-wave frequency. When $\Delta \omega _0\gg \Gamma$, the instability only effectively grows within time $\Delta t_{\text{c}}$. Therefore, the growth rate is reduced by the factor $\Delta t_{\text{c}}/\Delta t_{\text{g}}=\Gamma /\Delta \omega _0$ to be

$$\begin{aligned} \Gamma _m=\frac{\Gamma ^2}{\Delta \omega _0}. \end{aligned}$$

(6)

The reduction of growth rate can affect the threshold of LPIs. Under the dipolar case $v_0=\infty$, Laval et al. (1976) obtained the modified threshold based on the envelop equations with stochastic phase. When the damping rates of plasma wave $\nu _{\text{p}}$ and scattering light $\nu _{\text{s}}$ are approximately equal $\nu _{\text{p}}\sim \nu _{\text{s}}$, and $\Delta \omega _0>\nu _{\text{p}}|v_{\text{s}}/v_{\text{p}}|$, the threshold of backscattering is given by $\Gamma ^2/\Delta \omega _0>\nu _{\text{p}}$. Otherwise the small bandwidth $\Delta \omega _0|v_{\text{p}}/v_{\text{s}}|\ll \nu _{\text{p}}$ is just a perturbation of the usual coherent threshold. The corresponding threshold is estimated by

$$\begin{aligned} \Gamma >\frac{1}{2}\sqrt{|v_{\text{s}}/v_{\text{p}}|}(\nu _{\text{p}}+\Delta \omega _0|v_{\text{p}}/v_{\text{s}}|), \end{aligned}$$

(7)

where $v_{\text{s}}$ and $v_{\text{p}}$ are the group velocities of scattering light and plasma wave, respectively.

Analogous to the mitigation condition of temporal mode growth, Montgomery et al. (1996) gave the condition to control the filamentation by adding beam smoothing bandwidth, which is estimated by taking the coherence length of laser less than the characteristic growth scale of filamentation

$$\begin{aligned} \frac{\Delta \omega _0}{\omega _0}>4.7\times 10^{-3}H\left( \frac{I_0}{2\times 10^{15}\,{{\text{W/cm}}}^2}\right) \left( \frac{1.5\,{{\text{keV}}}}{T_{\text{e}}}\right) \left( \frac{n_{\text{e}}}{10^{21}\,{{\text{cm}}}^{-3}}\right) \left( \frac{\lambda }{0.527\,\upmu {{\text{m}}}}\right) ^4, \end{aligned}$$

(8)

where H is a factor to account for thermal effects, $T_{\text{e}}$ is the electron temperature, and $\lambda$ is the light wavelength. The relation between normalized amplitude $a_0$ and laser intensity $I_0$ is $a_0=\sqrt{I_0({{\text{W}}}/{{\text{cm}}}^2)[\lambda (\upmu {{\text{m}}})]^2/1.37\times 10^{18}}$. For example, a bandwidth of $\Delta \omega _0/\omega _0\sim 4\times 10^{-3}$ is needed for $H=2$ at the intensity $\sim 10^{15}\,{{\text{W}}}/{{\text{cm}}}^2$. The further simulation results suggest more optimistic effect than that estimated by Eq. (8). Therefore, the theoretical estimate for filamentation mitigation is $\Delta \omega _0/\omega _0\sim 10^{-3}$, which supports the experimental observation using 527 nm drive laser (Montgomery et al. 1996; Moody et al. 2009).

Santos et al. (2007) and Brandão et al. (2021) obtained the general dispersion relations for SRS and SBS driven by a low-coherence pump according to a generalized photon kinetics model

$$\begin{aligned} \frac{\omega _{\text{p}}^2}{2}\left( \frac{k^2c^2}{D_{\text{p}}}-1\right) \int \rho _0({\vec{k}}_0) \left( \frac{1}{D({\vec{k}}_0)_+}+\frac{1}{D({\vec{k}}_0)_-}\right) {\text{d}}{\vec{k}}_0=1, \end{aligned}$$

(9)

where $D(\mathbf {k_0})_{\pm }=[\omega _0({\vec{k}}_0)\pm \omega ]^2-({\vec{k}}_0\pm {\vec{k}})^2c^2-\omega _{\text{p}}^2$ and $\rho _0({\vec{k}}_0)$ is the zero-order photon distribution function. For SRS, $\omega _{\text{p}}=\omega _{\text{pe}}$ and $D_{\text{p}}=\omega ^2-\omega _{\text{pe}}^2$, and for SBS, $\omega _{\text{p}}=\omega _{\text{pi}}=\omega _{\text{pe}}\sqrt{Zm_{\text{e}}/m_{\text{i}}}$ and $D_{\text{p}}=\omega ^2-k_{\text{pi}}^2c_{\text{s}}^2$. Here, Z is the ion charge, $k_{\text{pi}}$ and $c_{\text{s}}$ are the wavenumber and velocity of ion-acoustic wave, $m_{\text{e}}$ and $m_{\text{i}}$ are the electron mass and ion mass, respectively. In general, $k^2c^2/D_{\text{p}}\gg 1$. The one-dimensional water bag distribution was used for the zero-order photon distribution function. It was found that the conclusion that forward SRS is less sensitive to broadband fields than backward one, and the range of unstable wave number is increased by the bandwidth (Santos et al. 2007). Analogously, the backscattering growth rate of SBS is found to decrease with the increase of bandwidth in the weak coupling regime (Brandão et al. 2021), and at the large bandwidth limit, it is

$$\begin{aligned} \Gamma _{{{\text{SBBS}}}}^{{{\text{sat}}}}=\frac{\pi a_0^2\omega _{\text{pi}}^2}{16k_0c_{\text{s}}} \end{aligned}$$

(10)

due to the widened unstable wave number. Actually, Eq. (10) can be reduced to $\Gamma _{{{\text{SBBS}}}}^{{{\text{sat}}}}=\Gamma _{{{\text{SBBS}}}}^2/(4k_0c/\pi )$ with $\Gamma _{{{\text{SBBS}}}}$ being the growth rate of backward SBS driven by a monochromatic laser, which corresponds to an ultra-broad bandwidth according to the work of Thomson and Karush (1974).

Fig. 1

[Reprinted from Zhao et al. (2017a), with the permission of AIP Publishing.]

a Temporal profile of the backscattered light waves. The polychromatic light and sinusoidal-frequency-modulation light are denoted as inc and sin, respectively. Both of them have the same bandwidth $\Delta \omega _0=15\%\omega _0$. b Temporal evolutions of the electron temperature for the incident lights with different bandwidths or polarizations. Linear polarization and random polarization are denoted as lp and rp, respectively. The random polarization light is described by $\textbf{a}_{rp}=\Sigma _{j=1}^Na_j[\cos (2\pi p_j)\textbf{e}_y+\sin (2\pi p_j)\textbf{e}_z]\cos (\omega _jt)$, where $p_j$ is a random number in [0, 1], and $\textbf{e}_y$ and $\textbf{e}_z$ are the unit vectors in the transverse directions

Kruer et al. (1973) validated the mitigation of reflected light by random phase lasers with finite bandwidths using particle simulations. Zhao et al. (2017a) performed a series of particle-in-cell (PIC) simulations with different bandwidth models as shown in Figs. 1 and 2. The electron temperature and plasma density respectively are $T_{\text{e}}=100$ eV and $n_{\text{e}}=0.08n_{\text{c}}$, i.e., plasma is initially in the fluid regime, and finally is heated to the kinetic regime. The amplitude of the monochromatic laser is $a_0=0.04$. It is found that the broadband laser only reduces the linear growth of SRS, but has little effect on the saturation level and electron heating in the nonlinear stage even though the bandwidth $\Delta \omega _0>10\%\omega _0$. The comparison of backscattering lights between sinusoidal-frequency-modulation laser and polychromatic light exhibited in Fig. 1a indicates that the mitigation effects can be improved via dispersing the energy of the peak spectrum. Figure 1b shows that random polarizations have no significant improvement in the SRS mitigation, owning to the small changes of ponderomotive force though an average of large-scale space and time. The wavenumber–frequency distributions of Langmuir wave $(k_{\text{L}}, \omega _{\text{L}})$ displayed in Fig. 2 show a much wider wavenumber–frequency spectrum of the broadband light than that of the monochromatic laser, which indicates a broad range of the Langmuir-wave velocity for the broadband case.

Fig. 2

[Reprinted from Zhao et al. (2017a), with the permission of AIP Publishing.]

Phase-space distributions of the Langmuir waves driven by a monochromatic light (a) and a polychromatic light (b), which are obtained for the time windows [301, 600]$\tau$ and [601, 1000]$\tau$, respectively

The linear bandwidth theory is based on the fluid model, which cannot be directly used to describe the nonlinear process, since the latter involves the interaction between electrostatic fields and particles. Zhou et al. (2018) numerically studied the coupling of broadband lasers with the frequency shifted plasma wave in the deep nonlinear stage. According to the simulation results presented in Fig. 3, they found that much more hot electrons are produced by the broadband laser than the monochromatic laser at $t=2750\tau$ due to more intense frequency shift of Langmuir wave of the former. The inflationary spectrum coupling is also found in the broadband laser interaction with inhomogeneous plasmas, owning to the natural broadband of the Langmuir waves in inhomogeneous plasmas. The broad range of the plasma wavenumber may also lead to a significant level of wave turbulence (Zhao et al. 2017a; Brandão et al. 2021]). Actually, the amplitude modulation of low-coherence laser always leads to a peak intensity around twice higher of the average (Gao et al. 2020), which may eliminate the reduction effect of bandwidth by enhancing the temporal growth rate and developing nonlinear intermittent SRS (Liu et al. 2022).

Fig. 3

[Reprinted from Zhou et al. (2018), with the permission of AIP Publishing.]

a Electron energy distribution at $t=2750\tau$. b Spatial frequency spectrum of Langmuir wave at $t=2750\tau$. The intensity of incident monochromatic laser is $I_0=2.8\times 10^{15}$ W/cm$^2$. The plasma density is $n_{\text{e}}=0.128n_{\text{c}}$ with electron temperature $T_{\text{e}}=3$ keV

The results shown above suggest that the broad bandwidth laser can still develop intense LPIs due to the coupling of different-color components, which indicate that the phase-matching conditions are satisfied in a finite spectrum range. The coupling of different-color components makes the LPI mitigation with broadband lasers ineffective at large scales. Therefore, the decoupling conditions of beamlets with different frequencies are critical to the effective applications of broadband lasers.

3.2 Decoupling conditions for the polychromatic light

For polychromatic lights, Estabrook et al. (1980, 1981); Estabrook and Kruer (1983) found that each spectrum line acts independently if the frequency difference $\delta \omega _0=\Delta \omega _0/(N-1)$ of adjacent lines is much larger than the growth rate. The spatial gain was found to be proportional to $\Gamma /\Delta \omega _0$ for $\Delta \omega _0/\Gamma >10$. A large laser bandwidth $\Delta \omega _0/\omega _0=10$–$36\%$ was used in their simulations, and filamentation and SRS are observed to be enhanced by the beating of discrete lines when the beat frequency is equal to $\omega _{\text{pe}}$. Bonnaud and Reisse (1986) found that forward SRS is insensitive to $\Delta \omega _0$, and it is weakened only at a large frequency difference. However, the ambiguity condition $\delta \omega _0\gg \Gamma$ fails to give an effective bandwidth model for the suppression of LPIs. Zhao et al. (2017b) first obtained the exact decoupling threshold based on the two-color laser model, which illustrates a physical picture of bandwidth coupling mechanism. As an example, we consider the development of SRS by two-color components (Zhao et al. 2021), which can be described by the following fluid equations

$$\begin{aligned} \left( \partial _{tt}-3v_{{\text{th}}}^2\nabla ^2+\omega _{\text{pe}}^2\right) {\tilde{n}}_{\text{e}}= & {} \frac{\omega _{\text{pe}}^2}{4\pi ec}\nabla ^2({\vec {\tilde{{A}}}}_{\text{s}}\cdot \textbf{a}_{0}), \end{aligned}$$

(11)

$$\begin{aligned} \left( \partial _{tt}-c^2\nabla ^2+\omega _{\text{pe}}^2\right) \vec {{\tilde{A}}}_{\text{s}}= & {} -4\pi ec{\tilde{n}}_{\text{e}}\textbf{a}_{0}, \end{aligned}$$

(12)

where $\textbf{a}_0=\textbf{a}_1+\textbf{a}_2$ is the amplitude of incident light composed of two-frequency components $\omega _1$ and $\omega _2$, $v_{{\text{th}}}$ is the electron thermal velocity, $\vec {{\tilde{A}}}_{\text{s}}=\vec {{\tilde{A}}}_{{\text{s1}}}+\vec {{\tilde{A}}}_{{\text{s2}}}$ and ${\tilde{n}}_{\text{e}}$ are the perturbations of scattering light and electron density, respectively. In the strongly coupled regime, the scattering light developed by one of the incident lights can be shared by the other pump beam. Without loss of generality, the frequency coupling term of $\textbf{a}_1\cdot \vec {{\tilde{A}}}_{{\text{s2}}}$ is $\cos [(\omega _1-\omega _{s2})t]=\cos [(\delta \omega _0+\omega _{\text{L}})t]$, which can resonate with the Langmuir wave $\omega _{\text{L}}=\sqrt{\omega _{\text{pe}}^2+3k_{\text{L}}^2v_{{\text{th}}}^2}$ at $1/\delta \omega _0\gg 1/\Gamma$. Here $\delta \omega _0=|\omega _1-\omega _2|$. Therefore, the Fourier analysis of Eqs. (11) and (12) in the strongly coupled regime leads to the dispersion relation

$$\begin{aligned} \begin{aligned}&\omega ^2-\omega _{\text{pe}}^2-3k^2v_{{\text{th}}}^2=\frac{\omega _{\text{pe}}^2k^2c^2}{4} \left[ a_1^2\left( \frac{1}{D_{{\text{e}}1-}}+\frac{1}{D_{{\text{e}}1+}}\right) +a_2^2\left( \frac{1}{D_{{\text{e}}2-}}+\frac{1}{D_{{\text{e}}2+}}\right) \right] \\&+\frac{\omega _{\text{pe}}^2k^2c^2}{4}\left[ a_1a_2\left( \frac{1}{D_{{\text{e}}1-}} +\frac{1}{D_{{\text{e}}1+}}+\frac{1}{D_{{\text{e}}2-}}+\frac{1}{D_{{\text{e}}2+}}\right) \right] , \end{aligned} \end{aligned}$$

(13)

where $D_{{\text{e}}1\pm }=(\omega \pm \omega _1)^2-({\vec{k}}\pm {\vec{k}}_1)^2c^2 -\omega _{\text{pe}}^2=\omega _{1\pm }^2-{\vec{k}}_{1\pm }^2c^2-\omega _{\text{pe}}^2$, $D_{{\text{e}}2\pm }=(\omega \pm \omega _2)^2-({\vec{k}}\pm {\vec{k}}_2)^2c^2 -\omega _{\text{pe}}^2=\omega _{2\pm }^2-{\vec{k}}_{2\pm }^2c^2-\omega _{\text{pe}}^2$. We have $D_{{\text{e}}2-}\approx D_{{\text{e}}1-}+2(\delta \omega _0\omega _{1-}-\delta {\vec{k}}_0\cdot {\vec{k}}_{1-})$ with $\delta {\vec{k}}_0=\omega _1{\vec{k}}_1\delta \omega _0/(\omega _1^2-\omega _{\text{pe}}^2)$. The modulus of $D_{{\text{e}}1-}$ satisfies $|D_{{\text{e}}1-}|\sim \Gamma$, and $|\delta \omega _0\omega _{1-}+\delta {\vec{k}}_0\cdot {\vec{k}}_{1-}|\sim \delta \omega _0$. When $\Gamma \gg \delta \omega _0$, the backscattering mode $1/D_{{\text{e}}2-}$ is approximately equal to $(1-\delta _1)/D_{{\text{e}}1-}$ with $\delta _1=2(\delta \omega _0\omega _{1-}-\delta {\vec{k}}_0\cdot {\vec{k}}_{1-})/D_{{\text{e}}1-}$. Therefore, the two beams are strongly coupled under a small frequency difference $\delta \omega _0\ll \Gamma$. On the contrary, $2(\delta \omega _0\omega _{1-}-\delta {\vec{k}}_0\cdot {\vec{k}}_{1-})$ is the dominant term of $D_{{\text{e}}2-}$ for a large frequency difference $\delta \omega _0\gg \Gamma$, and correspondingly the two beams are decoupled. Here, one comes to the same conclusion of Estabrook et al. (1981); Estabrook and Kruer (1983).

Based on the photon kinetics model (Santos et al. 2007), a similar dispersion relation can be obtained from Eq. (9) for two-photon beams $\rho _0({\vec{k}})=a_1^2\delta ({\vec{k}}-{\vec{k}}_1)+a_2^2\delta ({\vec{k}}-{\vec{k}}_2)$ under $a_1\approx a_2$, i.e., $a_1^2(1/D_{{\text{e}}1-}+1/D_{{\text{e}}1+})+a_2^2(1/D_{{\text{e}}2-}+1/D_{{\text{e}}2+})\approx a_1a_2(1/D_{{\text{e}}1-}+1/D_{{\text{e}}1+}+1/D_{{\text{e}}2-}+1/D_{{\text{e}}2+})$, where the two beams are strongly coupled. Therefore, Eq. (9) describes the dispersion relation driven by a finite-bandwidth laser with a continuous and slowly varying spectrum.

In the decoupling regime $\delta \omega _0\gg \Gamma$, different-color beams develop Langmuir waves without sharing the scattering light. Therefore, the off-resonant terms $\textbf{a}_1\cdot \vec {{\tilde{A}}}_{s2}$ and $\textbf{a}_2\cdot \vec {{\tilde{A}}}_{s1}$ are neglected, and the dispersion relation is reduced to

$$\begin{aligned} \omega ^2-\omega _{\text{pe}}^2-3k^2v_{{\text{th}}}^2=\frac{\omega _{\text{pe}}^2k^2c^2}{4} \left[ a_1^2\left( \frac{1}{D_{{\text{e}}1-}}+\frac{1}{D_{{\text{e}}1+}}\right) +a_2^2 \left( \frac{1}{D_{{\text{e}}2-}}+\frac{1}{D_{{\text{e}}2+}}\right) \right] . \end{aligned}$$

(14)

Fig. 4

[Reprinted from Zhao et al. (2017b), with the permission of AIP Publishing.]

a The growth rate $\Gamma$ of backward SRS versus $k_{\text{L}}$ for $n_{\text{e}}=0.08n_{\text{c}}$, and $a_1=a_2=0.02$ under different $\delta \omega _0$. b The corresponding phase plot of Langmuir wave at $t=340\tau$ from 1D PIC simulations, where the laser amplitudes and plasma density are the same to a. c, d show the comparison between analytical results of Eq. (15) and numerical solutions of SRS dispersion relation regarding the backscattering mode

For a broadband laser, the different-color beamlets propagate in the same direction. Therefore, Eqs. (13) and (14) can be solved in 1D regarding to the backward SRS. As exhibited in Fig. 4a, the two-color beams act as one monochromatic beam when their frequency difference is small enough $\delta \omega _0=0.15\%\omega _0$. With the bandwidth increasing to $\delta \omega _0=1\%\omega _0$, the instability regions are separated from each other, and each beam develops SRS independently. The numerical results are validated by the PIC simulations as illustrated in Fig. 4b. Note that the intensity of Langmuir wave is significantly reduced by the phase decoupling. Based on the phase plots, the major parameter for the coupling of different-color beams is the width of instability region, which also indicates a finite valid range for the phase-matching conditions. One notes that the LPI intensity of long-wavelength pulse is stronger than that of short one due to the relatively higher density of the former.

Equation (14) can also describe the collective SRS driven by decoupled multi-color beams with a certain intersecting angle (Zhao et al. 2021). In 2D geometry, both lasers with s-polarization (electric field components perpendicular to the incident plane) are mainly considered, which can develop more intense SRS than the p-polarized case (electric field components within the incident plane). According to the 2D phase plot displayed in Fig. 5a, the width of instability region for the j-th beam

$$\begin{aligned} \Delta k_{{\text{L}}j}=a_jk_{{\text{L}}}\sqrt{\frac{\omega _{\text{pe}}(\omega _j-\omega _{\text{pe}})}{\omega _j^2-2\omega _j\omega _{\text{pe}}}} \end{aligned}$$

(15)

gradually decreases from the backscattering mode to the forward-scattering mode along the linear dispersion relation

$$\begin{aligned} \begin{aligned}&\left[ k_{{\text{L}}x}c-k_{jx}c(1+\eta )\right] ^2+\left[ k_{{\text{L}}y}c-k_{jy}c(1+\eta )\right] ^2\\&\quad =\omega _j^2-2\omega _j\omega _{\text{pe}}+\eta (k_{\text{j}}^2c^2+\omega _j^2-2\omega _j\omega _{\text{pe}}), \end{aligned} \end{aligned}$$

(16)

where $\eta =3v_{{\text{th}}}^2/c^2(1-\omega _j/\omega _{\text{pe}})$ is the thermal perturbation correction. Figure 5b indicates that the scattering light is decoupled from the other beam when the wavenumber difference of scattering lights $\delta k_{\text{s}}=|k_{s1}-k_{s2}|$ is larger than the corresponding instability width.

One finds from Fig. 4c, d that Eq. (15) is validated at $n_{\text{e}}\le 0.23n_{\text{c}}$, since SRS backscatter is coupled with the forward scattering at $n_{\text{e}}\approx 0.25n_{\text{c}}$. As an example, the instability width of backward SRS $\Delta k_{{\text{LB}}}$ can be obtained by substituting $k_{{\text{L}}}=k_0+c^{-1}\sqrt{\omega _0^2-2\omega _0\omega _{\text{pe}}}$ into Eq. (15). The frequency detuning shifts the backward SRS wavenumber as ${\text{d}}k_{{\text{L}}}/{\text{d}}\omega _0=c^{-1}\omega _0(\omega _0^2-\omega _{\text{pe}}^2)^{-1/2}+c^{-1} (\omega _0-\omega _{\text{pe}})(\omega _0^2-2\omega _0\omega _{\text{pe}})^{-1/2}$. Therefore, the decoupling condition for SRS backscattering is $\delta \omega _0({\text{d}}k_{{\text{L}}}/{\text{d}}\omega _0)>\sqrt{2}\Delta k_{{\text{LB}}}$, where their instability regions are no longer overlapped in the phase space. In the underdense plasma with $\omega _{\text{pe}}\ll \omega _0$, the decoupling threshold of backward SRS can be reduced to

$$\begin{aligned} \delta \omega _0/\omega _0>a_0\sqrt{2\omega _{\text{pe}}/\omega _0}=2\sqrt{2}\Gamma _{{\text{SBRS}}}/\omega _0, \end{aligned}$$

(17)

where $\Gamma _{{\text{SBRS}}}=a_0\sqrt{\omega _0\omega _{\text{pe}}}/2$ is the growth rate of backward SRS developed by a monochromatic laser.

Fig. 5

[From original figure in Ref. Zhao et al. (2022), $\copyright$ IOP Publishing. Reproduced with permission. All rights reserved.]

Distributions of the SRS growth rate $\Gamma$ driven by two decoupled beams in the wavenumber space. The plasma density is $n_{\text{e}}=0.2n_{{\text{c}}1}$, and the electron temperature is $T_{\text{e}}=2$ keV, where $n_{{\text{c}}1}$ is the critical density for beam 1. The amplitude of each beam is $a_1=a_2=0.01$ with frequency difference $\delta \omega _0=\omega _1-\omega _2=2\%\omega _1$. a The two beams propagate in the same direction, where the locations for stimulated forward (SFRS) and backward (SBRS) Raman scattering are shown. b The angle between the two beams is 16$^\circ$

Fig. 6

Large-scale plasmas with a whole length of 1 mm are set in the simulations to include the convective effects. The polychromatic light contains four-color beamlets, and the electron temperature is 2.5 keV [From original figure in Ref. Zhao et al. (2022), $\copyright$ IOP Publishing. Reproduced with permission. All rights reserved.]

a Transmission rate of incident energies at different plasma densities. b Electron energy distributions heated by monochromatic laser and polychromatic light at $t=15000\tau$ under $n_{\text{e}}=0.07n_{\text{c}}.$

Note that the frequency difference not only reduces the radius of the wavenumber circle $\delta r_j=(\omega _j-\omega _{\text{pe}})\delta \omega _0/c\sqrt{\omega _j^2-2\omega _j\omega _{\text{pe}}}$, but also shifts the center to the left ${\vec{k}}_j(1-\omega _j\delta \omega _0/k_j^2c^2)$ as shown in Fig. 5a. Therefore, the mitigation condition of SRS forward scattering $\delta r_j-\omega _j\delta \omega _0/k_jc^2>\Delta k_{{\text{LF}}}$ is larger than that of SRS backscattering Eq. (17), where $\Delta k_{{\text{LF}}}$ can be obtained by taking the wavenumber of forward SRS mode into Eq. (15). Even though forward SRS is insensitive to a small bandwidth (Bonnaud and Reisse 1986; Santos et al. 2007), frequency difference $\delta \omega _0\gtrsim 1\%\omega _0$ is sufficient to mitigate forward SRS and decouple the scattering lights under $a_j<0.01$.

Considering for the filled gas in hohlraum, the developing conditions of SBS always satisfy $a_0\sim 10^{-2}$ and $n_{\text{e}}\lesssim 0.1n_{\text{c}}$ (Zhao et al. 2017b). In analogy to the SRS discussed above, the phase decoupling condition for backward SBS is

$$\begin{aligned} \delta \omega _0/\omega _0>2a_0\frac{\omega _{\text{pi}}}{\omega _0} \sqrt{\frac{\omega _0}{k_0c_{\text{s}}}}=4\sqrt{2}\Gamma _{{{\text{SBBS}}}}/\omega _0, \end{aligned}$$

(18)

where $\Gamma _{{{\text{SBBS}}}}=a_0k_0c\omega _{\text{pi}}/\sqrt{8\omega _0k_0c_{\text{s}}}$ is the backward SBS growth rate developed by a monochromatic laser. In general, $\Gamma _{{\text{SBRS}}}/\Gamma _{{{\text{SBBS}}}}>2$ under $c_{\text{s}}/c\sim 10^{-3}$; therefore, both SRS and SBS are effectively mitigated as long as Eq. (17) is satisfied. The speckles developed via beam smoothing, such as continuous phase plate (CPP), lead to the growth of transverse density perturbations that results in laser filamentation due to the nonuniform pressure. The growth of filamentation can be described by the same dispersion relation of SBS (Kruer 1988). Analogous to the two-dimensional dispersion relation of SRS discussed in the above, the decoupling condition of the laser bandwidth for filamentation is in the same order of that for backward SBS.

In the linear growth stage, the plasma instability level driven by the decoupled beamlets at different frequencies can be estimated as $\Delta n_{j}=N\delta n\exp (\Gamma _jt)$, where N is the beamlet number, $\delta n$ is the initial perturbation and $\Gamma _j$ is the growth rate of j-th beam. Analogously, the instability level developed by a single monochromatic beam is $\Delta n_{0}=\delta n\exp (\Gamma _0t)$. One always finds $\Delta n_{j}/\Delta n_{0}>1$ at the time $t<(1+1/\sqrt{N})\ln (N)/\Gamma _0$ due to $a_j=a_0/\sqrt{N}$. For a large enough beamlet number $N\gg 1$, the instability level of decoupled beamlets is higher than that of the monochromatic beam if the latter is saturated before $\ln (N)/\Gamma _0$ but the former still grows. Therefore, there exists a maximum beamlet number N to mitigate LPIs in the fluid regime. However, the Langmuir wave is heavily damped under the ICF conditions. In hot plasmas without considering the damping of scattering light, the threshold for developing absolute instability is $\Gamma _j>\nu _{\text{p}}\sqrt{|v_{s}/v_{\text{p}}|}/2$ (Liu et al. 2019). Therefore, LPIs can be effectively mitigated by a polychromatic light when each beamlet intensity of the polychromatic light is below the developing threshold. According to the 1D simulation results shown in Fig. 6a,  the polychromatic light can save at least 8.8% of the incident energy from LPIs in the hot plasmas compared to the monochromatic laser. A hot electron tail with 52.2 keV can be found for the monochromatic laser at 22.3 ps in Fig. 6b. However, the electrons have not been obviously heated at this time for the polychromatic light. 

3.3 SRS mitigation by broadband sunlight-like lasers

Fig. 7

[Reprinted from Ma et al. (2021), with the permission of AIP Publishing.]

Temporal profiles of electric field components (a) $\textbf{E}_y$ and (b) $\textbf{E}_z$. c The amplitude frequency spectrum $f(\omega )$ and phase frequency spectrum $\phi (\omega )$ of $\textbf{E}_y$. d Stokes parameters of the sunlight-like laser, where I denotes the intensity regardless of polarization, Q the linear polarization along the y ($+$) or z (−) axes, U the linear polarization at $+45^{\circ }$ ($+$) or $-45^{\circ }$ (−) from the y axis, V the right-handed ($+$) or left-handed (−) circular polarization

Fig. 8

[Reprinted from Ma et al. (2021), with the permission of AIP Publishing.]

a The average SRS reflectivity within $6000\tau$ obtained from 1D PIC simulations as a function of laser intensity for monochromatic laser (MCL), broadband laser (BBL) with bandwidths $\Delta \omega _0=1\%\omega _0$ and $2\%\omega _0$, and sunlight-like laser (SLL) with bandwidth $\Delta \omega _0=1\%\omega _0$, respectively. The purple dashed line denotes $R_\texttt {SRS}=1\%$. b The saturation time of the backward reflected light as a function of laser intensity for three kinds of laser beams. c The temporal evolution of the backscattering light. d The electron distributions for monochromatic laser, broadband laser and sunlight-like laser at $6000\tau$. e, f show the temporal evolution of $|E_x|$ for broadband laser and sunlight-like laser, respectively. All the incident lasers have the same intensity scale $\sim 10^{15}$ W/cm$^2$, and the broadband laser and sunlight-like laser have the same bandwidth $\Delta \omega _0=1\%\omega _0$. The plasma density is $n_{\text{e}}=0.128n_{\text{c}}$ and initial electron temperature is 3 keV

The electric fields of two broadband laser ($\textbf{E}_y$ and $\textbf{E}_z$) for constructing a sunlight-like beam are displayed in Fig. 7a, b, where both of them have the same amplitude frequency spectrum but different phase frequency spectrum. The amplitude and phase frequency spectrum of $\textbf{E}_y$ are displayed in Fig. 7c. The sunlight-like laser, synthesized by $\textbf{E}_y$ and $\textbf{E}_z$, has a random polarization with time, which is validated by the Stokes parameters shown in Fig. 7d. To some extent, a sunlight-like laser can be considered as a broadband elliptically polarized light whose polarization orientation, axial ratio and handedness changes randomly with time. For some relevant studies, it was found that the parametric backscattering can be reduced significantly when the polarization of incident laser light rotates slowly (Barth and Fisch 2016). Zhong et al. (2021) also found the time-dependent polarization rotation of incident laser can greatly suppress LPIs, since the ponderomotive force is reduced by the polarization difference between incident laser and scattering light in certain areas.

To verify the advantages of sunlight-like lasers in mitigating LPIs, Ma et al. (2021) performed a series of 1D PIC simulations. For a better comparison, three kinds of laser beams including monochromatic laser, conventional broadband laser, and sunlight-like laser were utilized in the simulations. Ions are immobile to evaluate the effect of SRS only in the simulations. The SRS reflectivity $R_\texttt {SRS}$ shown in Fig. 8a indicates that the reflectivity steeply increases at a certain intensities and then slowly saturates (Montgomery et al. 2002; Yin et al. 2006). The intensity threshold of SRS is defined according to $R_\texttt {SRS}=1\%$. Under this criterion, one finds that the SRS intensity threshold is enhanced from $7.6\times 10^{14}$W/cm$^2$ to $1.12\times 10^{15}$W/cm$^2$ for a conventional broadband laser with $\Delta \omega _0=1\%\omega _0$. Note that the threshold is further enhanced to $1.42\times 10^{15}$ W/cm$^2$ by the sunlight-like laser with the same bandwidth, which is about double of the monochromatic beam. The conventional broadband laser requires a nearly doubled bandwidth $\Delta \omega _0=2\%\omega _0$ to achieve the comparable mitigation effect of the sunlight-like laser. One finds from Fig. 8b that the sunlight-like laser as well as conventional broadband laser can delay the saturation time of the SRS, which is consistent with the studies of Zhao et al. (2015).

The instantaneous intensities of scattered lights developed by the conventional broadband laser and sunlight-like laser are found to be stronger than that of the monochromatic laser based on Fig. 8c. However, the average reflectivity $R_{\texttt {SRS}}$ are 20.6%, 17.2% and 10.7% for the monochromatic, conventional broadband and sunlight-like laser, respectively. The electron energy distributions shown in Fig. 8d indicate that a significant reduction of hot electron generation can be found for the sunlight-like laser. Therefore, the sunlight-like laser may mitigate LPI further compared to the usual broadband laser due to the time-dependent random polarization, which will be experimentally investigated in the near future (Li et al. 2022).

4 Mitigation of LPIs in inhomogeneous plasmas with broadband lasers

4.1 Convective amplification of broadband lasers

Frequency bandwidth may partially compensate the wavenumber mismatch in inhomogeneous plasmas. Lu (1988) found that the inhomogeneous feature is significantly reduced at $\Delta \omega _0/K_0\gg 1$, where $K_0=k_0-k_{\text{p}}-k_{\text{s}}$ is the wavenumber mismatch. Guzdar et al. (1991b) found that even though the temporal growth rate is reduced by the bandwidth

$$\begin{aligned} \Gamma _m=\frac{2\Gamma ^2}{\Delta \omega _0(1+|v_{\text{s}}/v_0|)}, \end{aligned}$$

(19)

the interaction time is extended

$$\begin{aligned} \Delta t_{\text{i}}=\frac{2\pi \omega _{\text{s}}L\Delta \omega _0}{\omega _{\text{pe}}k_{\text{s}}c^2}(1+|v_{\text{s}}/v_0|), \end{aligned}$$

(20)

where L is the density scale length. Note that the explicit value of the correlation function has been used to obtain Eq. (19) with an effective bandwidth $\Delta \omega _0(1+|v_{\text{s}}/v_0|)$, which can be reduced to Eq. (6) at $|v_{\text{s}}/v_0|\approx 1$. Finally, the overall amplification $G\sim \Gamma _m\Delta t_{\text{i}}$ becomes independent of bandwidth $\Delta \omega _0$. As a special case of sinusoidal phase modulation with finite bandwidth, Wen et al. (2021) found that the resonance location has a nonzero velocity, and the scattered light is amplified continuously to achieve the maximum gain when its group velocity is equal to the velocity of the resonance location. Therefore, the maximum-gain condition is met when the indicator $\eta \equiv 2\Delta \omega _0f_mL/\omega _{\text{pe}}c$ satisfies $\eta =1$. The simulation results displayed in Fig. 9 demonstrate that the SRS gain peaks at $\eta =1.16$, which is independent of the density scale length.

Fig. 9

[Reprinted from Wen et al. (2021), with the permission of AIP Publishing.]

The mean convective SRS gains, normalized by the Rosenbluth gain $G_{\text{r}}$, as a function of the maximum-gain indicator $\eta$ for density scale length $L=200$ $\upmu$m (squares), 330 $\upmu$m (circles), and 660 $\upmu$m (crosses). The plasma density linearly ranges from 0.09$n_{\text{c}}$ to 0.14$n_{\text{c}}$ with electron temperature $T_{\text{e}}=0.1$ keV. The intensity of monochromatic laser is $I_0=5\times 10^{14}$ W/cm$^2$

Different from the above random-phase model, Zhao et al. (2019) studied the polychromatic LPIs based on the Rosenbluth convective model (Rosenbluth 1972), where the linear wavenumber mismatch is introduced into the phase. Actually, the frequency difference $\delta \omega _0\sim 10^{-2}\omega _0$ has little effect on the group velocity of scattering light. For example, $\delta v_{\text{s}}/v_{\text{s}}\sim 10^{-3}$ at $n_{\text{e}}=0.1n_{\text{c}}$. When the frequency difference is much larger than the corresponding growth rate, LPIs driven by a two-color beam in a local region with phase coupling can be described by

$$\begin{aligned} \nu _{\text{s}}a_{\text{s}}-v_{\text{s}}\partial _xa_{\text{s}}= & {} \frac{\Gamma _m}{\sqrt{2}}a_{\text{p}}\left[ \exp \left( i\int _0^x K_1{\text{d}}x'\right) +\exp \left( i\int _0^x K_2{\text{d}}x'\right) \right] , \end{aligned}$$

(21)

$$\begin{aligned} \nu _{\text{p}}a_{\text{p}}+v_{\text{p}}\partial _xa_{\text{p}}= & {} \frac{\Gamma _m}{\sqrt{2}}a_{\text{s}}\left[ \exp \left( -i\int _0^x K_1{\text{d}}x'\right) +\exp \left( -i\int _0^x K_2{\text{d}}x'\right) \right] , \end{aligned}$$

(22)

where the wavenumber mismatches are $K_1=k_1-k_{\text{s}}-k_{\text{p}}=k_0-\delta k_0/2-k_{\text{s}}-k_{\text{p}}=K_0-\delta k_0/2 \quad {\text{and}} \quad K_2=k_2-k_{\text{s}}-k_{\text{p}}=k_0+\delta k_0/2-k_{\text{s}}-k_{\text{p}}=K_0+\delta k_0/2$, and the temporal effect of broadband lasers is included in the modified growth rate $\Gamma _m=\Gamma ^2/\Delta \omega _0$. The above Eqs. (21) and (22) can be reduced to

$$\begin{aligned} \nu _{\text{s}}a_{\text{s}}-v_{\text{s}}\partial _xa_{\text{s}}= & {} \sqrt{2}\Gamma _ma_{\text{p}}\exp (-iK_0'x^2/2)\cos (\delta k_0'x^2/4), \end{aligned}$$

(23)

$$\begin{aligned} \nu _{\text{p}}a_{\text{p}}+v_{\text{p}}\partial _xa_{\text{p}}= & {} \sqrt{2}\Gamma _ma_{\text{s}}\exp (iK_0'x^2/2)\cos (\delta k_0'x^2/4), \end{aligned}$$

(24)

where $K_0'={\text{d}}K_0/{\text{d}}x$ and $\delta k_0'={\text{d}}\delta k_0/{\text{d}}x$. Comparing to the monochromatic case, the drive terms of Eqs. (23) and (24) are reduced by a factor $\cos (\delta k_0'x^2/4)$. In the heavy damping plasmas, the saturation coefficient is obtained to the first order

$$\begin{aligned} G=\frac{2\pi \Gamma _m^2}{v_{\text{s}}v_{\text{p}}K_0'}\left[ 1+\cos \left( \frac{2\nu _{\text{p}}^2\delta k_0'}{K_0'^2v_{\text{p}}^2}\right) \right] . \end{aligned}$$

(25)

Equation (25) indicates that the two-color beams are coupled to develop convective instability in a common resonant region. However, the saturation level is decreased by the phase detuning as compared to the Rosenbluth gain saturation coefficient. Especially, for the SRS in an inhomogeneous plasma with density profile $n_{\text{e}}=n_0(1+x/L)$, Eq. (25) can be reduced to

$$\begin{aligned} G_{{{\text{SRS}}}}\approx \frac{2\pi \Gamma _m^2}{v_{\text{s}}v_{\text{p}}K_0'}\left[ 1+\cos \left( \frac{4L\omega _0\nu _{\text{p}}^2\omega _{\text{L}}^2}{n_0c(\omega _0^2-\omega _{\text{pe}}^2)^{3/2}}\delta \omega _0\right) \right] . \end{aligned}$$

(26)

According to Eq. (26), the two-color beams are weakly coupled when

$$\begin{aligned} \delta \omega _0\gtrsim \frac{\pi n_0c(\omega _0^2-\omega _{\text{pe}}^2)^{3/2}}{8L\omega _0\omega _{\text{L}}^2\nu _{\text{p}}^2}. \end{aligned}$$

(27)

As an example, for a plasma with $n_0=0.08n_{\text{c}}$, $T_{\text{e}}=3$keV and $L=3000\lambda$, we have the threshold $\delta \omega _0\approx 2.4\%\omega _0$. Therefore, for effective mitigation of SRS, the frequency difference between two adjacent components is in the order of $10^{-2}\omega _0$.

Fig. 10

[From original figure in Ref. Zhao et al. (2019), $\copyright$ IOP Publishing. Reproduced with permission. All rights reserved.]

Schematic diagram as an example for the secondary amplification of backscattering light developed by two-color beamlets with frequencies $\omega _1=1.02\omega _0$ and $\omega _2=0.98\omega _0$. The backscattering light developed at $x_2$ will be amplified again at $x_1$

Note that the above linear model is suitable for describing the behavior of daughter waves in a finite density range. However, situations are different for the scattering light propagation in a large-scale plasma, because a scattering light produced by a certain color beam will be amplified again as a seed mode in the subsequent parametric excitation region where its frequency is equal to the scattering light developed by another low-frequency beam. An example is shown in Fig. 10, the scattering light developed by the laser with $\omega _1=1.02\omega _0$ at $0.1n_{\text{c}}$ will be amplified again by the other color beam $\omega _2=0.98\omega _0$ as a seed mode at $0.078n_{\text{c}}$, since both of the scattering lights have an equal frequency $\omega _{\text{s}}=0.7\omega _0$. The amplification coefficient is almost twice of Eq. (26), which counteracts the mitigation of broadband lasers. Therefore, the decoupling condition of SRS scattering lights in a finite inhomogeneous plasma with density range $[n_{\text{e}}(x_1), n_{\text{e}}(x_2)]$ (the corresponding Langmuir-wave frequency is $[\omega _{L1}, \omega _{L2}]$) is $\delta \omega _0>\omega _{L2}-\omega _{L1}$, which is always larger than the threshold of Eq. (27) in a large-scale plasma.

Fig. 11

[From original figure in Ref. Zhao et al. (2019), $\copyright$ IOP Publishing. Reproduced with permission. All rights reserved.]

a Temporal evolution of electrostatic energy driven by incident lights with different bandwidths. b Electron energy distributions for monochromatic laser and polychromatic light with different beam number N under the same energy at $t=5000\tau$. Spectra of the backscattering lights for the polychromatic lights composed of c $N=2$ and d $N=20$ beamlets under the same energy. The density scaling length is $L=1000\lambda$, and the density range is $[0.08, 0.12]n_{\text{c}}$. The initial electron temperature is $T_{{\text{e}}}=2$ keV. The incident monochromatic laser has a semi-infinite uniform amplitude $a_0=0.014$ (the corresponding intensity is $I_0=2.5\times 10^{15}\,{{\text{W}}}/{{\text{cm}}}^2$ with $\lambda =0.33\,\upmu m$)

The simulation result displayed in Fig. 11a shows that the polychromatic light with a finite bandwidth $\Delta \omega _0=2\%\omega _0$ can significantly reduce the strength of Langmuir waves in inhomogeneous plasmas, and the saturation level is further reduced by increasing the bandwidth $\Delta \omega _0$ to $4\%\omega _0$. However, most of the beamlets are still coupled at the large beamlet number $N=20$ according to Eq. (26). The reduction of Langmuir wave leads to the suppression of hot-electron tail as shown in Fig. 11b. One notes that the frequency range of Langmuir wave is $[0.33, 0.38]\omega _0$ in this simulation example. Considering the case with $\Delta \omega _0=2\%\omega _0$ and $N=2$, the spectra of the two scattering lights, respectively, are $\omega _{s1}=[0.61, 0.66]\omega _0$ and $\omega _{s2}=[0.63, 0.68]\omega _0$, where an overlapping frequency range $[0.63, 0.66]\omega _0$ can be found for them. The overlapping spectra are much higher than the sidebands since they are amplified twice as demonstrated in Fig. 11c. The sharing parts are shrunk with the increase of the bandwidth, and totally are separated at $\Delta \omega _0=6\%\omega _0$. Based on the secondary amplification mechanism, the scattering lights are strongly coupled for $N=20$ even at $\Delta \omega _0=6\%\omega _0$ as presented in Fig. 11d.

Different from SRS, the secondary amplification mechanism of scattering lights is the major concern for the mitigation of SBS and CBET (Zhao et al. 2019). The bandwidth of SBS scattering light can be estimated as $\Delta \omega _{\text{s}}\lesssim 2\Delta (k_{\text{pi}}c_{\text{s}})\sim 0.004\omega _0$ with including the effect of plasma flow, where $\Delta k_{\text{pi}}\sim 2\omega _0/c$ is the scale of ion-acoustic wavenumber, and $c_{\text{s}}\sim 10^{-3}c$ is the ion-acoustic velocity. In the linear regime, SBS and CBET can be effectively mitigated by the frequency detuning $\delta \omega _0\gtrsim 0.5\%\omega _0$ between two-color beams. Numerical simulations performed by Seaton et al. (2022) indicate that laser bandwidth becomes highly effective in reducing the CBET gain exponent if it satisfies $\Delta \omega _0>\omega _{\text{pi}}$. Beams with sufficiently large bandwidth can induce both forwards (pump to seed) and reverse (seed to pump) energy transfers, and the net CBET is totally suppressed when the opposing transfers are balanced at a certain bandwidth. Beams with Gaussian and Lorentzian spectral profiles are close to optimal, producing a $\Delta \omega _0^{-3}$ scaling of the CBET gain exponent, where CBET is insensitive to the damping rate of the ion-acoustic waves. However, the efficacy of bandwidth is not guaranteed in the nonlinear stage. (Bates et al. 2020) found that a bandwidth of about 8THz is required to eliminate CBET under the realistic laser features and the plasma conditions similar to those on the NIF.

Fig. 12

[From original figure in Ref. Zhao et al. (2021), $\copyright$ IOP Publishing. Reproduced with permission. All rights reserved.]

Wavenumber distributions of Langmuir wave driven by a two monochromatic lasers at $t=1500\tau$ and b two polychromatic lights with bandwidth $\Delta \omega _0=3\%\omega _1$ at $t=3000\tau$. c Temporal growth of electrostatic energies developed by pump beams with different bandwidths. d Electron energy distributions under different bandwidths at $t=6000\tau$. The incident angles of beam 1 and beam 2 in vacuum are $\theta _{v1}=10^\circ$ and $\theta _{v2}=-10^\circ$, respectively. Plasma density of the beam overlapping region is $n_{\text{e}}\approx 0.188n_{\text{c}}$, and the density scaling length is $L=1000\lambda$. The initial electron temperature is $T_{\text{e}}=2$ keV. The incident monochromatic beams have an equal amplitude $a_1=a_2=0.01$ and same frequency $\omega _1=\omega _2$

In the multibeam configuration, considering for the case of two pumps with an incident angle $\theta _v$ in vacuum, the common Langmuir wave shared by the monochromatic lasers at $n_{\text{e}}/n_{\text{c}}\le \cos ^4\theta _v/4$ is divided into multiple weakly coupled modes by the polychromatic lights with a broad spectrum $\Delta \omega _0\gtrsim 3\%\omega _0$. The saturation coefficient of Langmuir waves is reduced by the wave-vector detuning $G_m=G_0[1+\cos (\delta \omega _{{\text{LB}}}\beta )]$ according to Eq. (26), where $G_0=\pi La_j^2k_{L}^2/2k_{\text{s}}$ and $\beta =2L\omega _0\nu _{\text{p}}^2\omega _{L}^3/3k_{L}v_{{\text{th}}}^2n_0(\omega _0^2-\omega _{\text{pe}}^2)^{3/2}\cos \theta _v$ for low-density plasmas (Zhao et al. 2021). Without loss of generality, assume that each polychromatic light has N beamlets, and every beamlet shares different common modes with all other polychromatic beamlets, i.e., there are $N^2$ backscattered common waves. For weakly coupled modes, the amplification coefficient for the polychromatic lights can be estimated as $\alpha _{\text{p}}=N^2\exp (1)$, and that for the monochromatic lasers with the same energy is $\alpha _{n}=\exp (N)$ based on $a_j=a_0/\sqrt{N}$. The amplification ratio $\alpha _{\text{p}}/\alpha _{n}$ is found to be less than one when $N>3$, and the energy fraction $\alpha _{\text{p}}^2/\alpha _{n}^2<1$ is always satisfied for any $N>1$. The scattering lights of different-color beamlets are decoupled from each other when Eq. (27) is satisfied, i.e., $\delta \omega _0\gtrsim 1\%\omega _0$, and the secondary amplification is mitigated by the refraction of scattering light in 2D geometry.

One obtains the common wavevectors of forward and backward SRS as $|{\vec{k}}_{{\text{LF}}}c|=\sqrt{\omega _1^2\cos ^2\theta _v-\omega _{\text{pe}}^2}-\sqrt{\omega _1^2\cos ^2\theta _v-2\omega _1\omega _{\text{pe}}}$ and $|{\vec{k}}_{{\text{LB}}}c|=\sqrt{\omega _1^2\cos ^2\theta _v-\omega _{\text{pe}}^2}+\sqrt{\omega _1^2\cos ^2\theta _v-2\omega _1\omega _{\text{pe}}}$ from Eq. (16) at $k_{Ly}=0$ and $v_{{\text{th}}}=0$. The wave-vector of beam 1 presented in Fig. 12a is $O_1(0.87, 0.17)\omega _1/c$, and the common SRS backscatter can be found at $|{\vec{k}}_{{\text{LB}}}c|\approx 1.17\omega _1$, which is the most intensity mode at $n_{\text{e}}\sim 0.188n_{{\text{c}}1}$. The wave-vector of the scattering light developed by beam 1 is ${\vec{k}}_{s1}c={\vec{k}}_{1}c-{\vec{k}}_{{\text{LB}}}c\approx (-0.27, 0.18)\omega _1$, and the wave-vector of the other scattering light is ${\vec{k}}_{s2}c={\vec{k}}_{2}c-{\vec{k}}_{{\text{LB}}}c\approx (-0.27, -0.18)\omega _1$. One notes that two concomitant electrostatic modes ${\vec{k}}_{L1}c={\vec{k}}_{1}c-{\vec{k}}_{s2}c\approx (1.17, 0.36)\omega _1$ and ${\vec{k}}_{L2}c={\vec{k}}_{2}c-{\vec{k}}_{s1}c\approx (1.17, -0.36)\omega _1$ are induced by the scattering light from the other beam. The intensities of the common Langmuir wave and the concomitant electrostatic field are found to be significantly reduced by the polychromatic lights with bandwidth $\Delta \omega _0=3\%\omega _1$ according to Fig. 12b. The polychromatic light is insufficient to drive intense plasma waves if the beamlets are weakly coupled, and therefore the nonlinear process is mitigated. The temporal growths of the electrostatic energies shown in Fig. 12c indicate that the mitigation of multibeam SRS by polychromatic lights can also be found in the nonlinear regime. The electron energy distributions in Fig. 12d also demonstrate that the polychromatic lights with a bandwidth $\Delta \omega _0=3\%\omega _1$ can effectively reduce the electron hot tail in the fully nonlinear regime.

4.2 Mitigation of absolute instabilities by broadband lasers

Follett et al. (2018) performed a series of 3D simulations with two different multi-color configurations, one is the polychromatic light and another is the tiled configuration (each beam is monochromatic, but the different beams have different frequencies). The incident beams were simulated with phase plates and polarization smoothing. The density scale length is $L=211\,\upmu {{\text{m}}}$ at 0.25 $n_{\text{c}}$, and the electron and ion temperatures are $T_{\text{e}}=2.6$ keV and $T_{\text{i}}=1$ keV, respectively. They found that a polychromatic light with three colors and a 0.7%$\omega _0$ (1.76 nm) detuning is sufficient to eliminate TPD-driven hot-electron generation at $I_0=7\times 10^{14}$ W/${{\text{cm}}}^2$. However, hot-electron generation cannot be obviously eliminated in the tiled configuration at the same laser intensity. The 2D simulations with normally incident plane waves indicated that the amplitude modulation of broadband lasers has a significant impact on the TPD threshold. When speckled beams are used, the threshold is always enhanced by increasing the beamlet number.

Lu (1988) found an effective Hamiltonian $H_{{\text{eff}}}=H_0+\Delta \omega _0H_1$ of the linear stochastic equation, where $H_0$ is the primary part describing the coherent effects, and $\Delta \omega _0H_1$ is the finite bandwidth modification. The imaginary part of the eigenvalue of $H_{{\text{eff}}}$ gives the growth rate near 0.25$n_{\text{c}}$ in the pump frame, where $v_0=\infty$ was assumed to eliminate the spatial variable x in the random phase term. A series of scaling results were found for SRS and TPD. As shown in Table 1, Follett et al. (2019) numerically evaluated the threshold of the absolute instabilities including SRS and TPD, and compared them to the theoretical predictions of Lu (1989).

Table 1 SRS and TPD absolute threshold power-law dependence on $\Delta \omega _0$, L, $\lambda$, and $T_{\text{e}}$ from the analytical monochromatic result ($I_{{{\text{thr,srs}}}}$ and $I_{\text{thr,tpd}}$, respectively), the analytical result with bandwidth ($I_{\text{Lu,srs}}$ and $I_{\text{Lu,tpd}}$, respectively), and the fits to the result of laser-plasma simulation environment code ($I_{\text{thr}}$)

They finally found a qualitative match of the parameter scalings between them with significant quantitative differences. Lorentzian power spectrum of broadband lasers used in the simulations has an equal bandwidth to the assumption of the Kubo–Anderson process in the theoretical model, i.e., $\Delta \omega _0=2/\Delta t_{\text{c}}$. Approximate scaling laws for the SRS and TPD thresholds with a modest bandwidth were obtained based on a combination of the simulation results and the analytical expressions

$$\begin{aligned}{} & {} a_{{{\text{SRS}}}}^2\approx 0.067\frac{\lambda }{L}\left( \frac{\tau }{\Delta t_{\text{c}}}\right) ^{1/3}, \end{aligned}$$

(28)

$$\begin{aligned}{} & {} a_{{{\text{TPD}}}}^2\approx 2.1\left( \frac{\lambda }{L}\right) ^{2/3}\left( \frac{v_{{\text{th}}}}{c}\right) ^{2/3}\left( \frac{\tau }{\Delta t_{\text{c}}}\right) ^{1/2}. \end{aligned}$$

(29)

At a large bandwidth, the scaling of $\tau /\Delta t_{\text{c}}$ becomes linear.

For the polychromatic light, two absolute mechanisms are considered including the $90^\circ$ SRS side scattering and the degenerate modes around 0.25 $n_{\text{c}}$. The dispersion relation Eq. (14) provides the maximum decoupling condition for absolute side scattering in inhomogeneous plasmas due to the detuning of Langmuir waves. When the threshold Eq. (27) is satisfied, the convective couplings of sidescattered lights are detuned. Therefore, the effective mitigation of $90^\circ$ side-scattering lights can be found at $\delta \omega _0\gtrsim 1\%\omega _0$ according to Eqs. (14) and (27). Under the multibeam configuration, the frequency difference of pump beams can shift the absolute region $|{\vec{k}}_{{\text{LF}}}c|=|{\vec{k}}_{{\text{LB}}}c|$ to

$$\begin{aligned} n_{{\text{e}}c}/n_{{\text{c1}}}=\frac{\cos ^4\theta _v}{4}\left( 1-\frac{\delta \omega _0}{\omega _{1}}\right) , \end{aligned}$$

(30)

which introduces a wavenumber detuning into the SRS side scattering.

Fig. 13

Schematic diagram of the absolute instability regions in inhomogeneous plasmas for SRS and TPD driven by two decoupled beamlets with frequency $\omega _1$ and $\omega _2$, respectively. The instability regions for the two decoupled beamlets are marked in different colors, which correspond to different plasma density regions $[(1/2-a_1k_1c/4\omega _1)^2n_{c1}, (1/2+a_1k_1c/4\omega _1)^2n_{c1}]$ and $[(\omega_2/2\omega_1-a_2k_2c/4\omega_1)^2n_{c1}, (\omega_2/2\omega_1+a_2k_2c/4\omega_1)^2n_{c1}]$, respectively

Both of the absolute SRS and TPD are locally developed around 0.25$n_{\text{c}}$, which is the critical factor for their decouplings. The absolute instability region $[\omega _1/2-a_1k_1c/4, \omega _1/2+a_1k_1c/4]$ of beam 1 can be obtained from the SRS and TPD dispersion relations in cold plasmas (Zhao et al. 2019, 2020), which is the maximum valid range for the development of absolute instabilities in inhomogeneous plasmas. The polychromatic beamlets are detuned from each other at $\omega _1/2-a_1k_1c/4>\omega _2/2+a_2k_2c/4$, i.e.,

$$\begin{aligned} \delta \omega _0\gtrsim 0.433(a_1+a_2). \end{aligned}$$

(31)

When the decoupling condition Eq. (31) is satisfied, different-color beamlets drive their respective absolute modes in non-overlapping density regions $[(\omega_j/2\omega_1-a_jk_jc/4\omega_1)^2n_{c1}, (\omega_j/2\omega_1+a_jk_jc/4\omega_1)^2n_{c1}]$ as shown in Fig. 13. For example, compared to the same energy of monochromatic laser with intensity $I_0\sim 10^{15}\,{{\text{W}}}/{{\text{cm}}}^2$, absolute instabilities are developed independently by the polychromatic beamlets under beamlet number $N=5$ and bandwidth $\Delta \omega _0=(N-1)\delta \omega _0=1.7\%\omega _0$. According to the intensity threshold of absolute SRS in inhomogeneous plasmas, the whole level of absolute SRS is considerably mitigated by the decoupled beamlets when $a_j\lesssim 0.323(L/\lambda _j)^{-2/3}$ (Liu et al. 1974). For a large enough bandwidth, the above results indicate that the drive threshold is enhanced by the increase of beamlet number, which have been observed in the simulations performed by Follett et al. (2019).

Fig. 14

[From original figure in Ref. Zhao et al. (2019), $\copyright$ IOP Publishing. Reproduced with permission. All rights reserved.]

Spatial Fourier transform of the electric field at $t=800\tau$ under a $\Delta \omega _0=0$, b $\Delta \omega _0=2\%\omega _0$ and c $\Delta \omega _0=4\%\omega _0$, where $N=20$. The white line denotes the expected wavenumber distribution for plasmons occurring in the TPD instability. d Energy distributions of electrons for different bandwidth at $t=1200\tau$. The density scaling length is $L=660\lambda$, and the initial electron temperature is $T_{{\text{e}}}=2$ keV. The amplitude of incident monochromatic laser is $a_0=0.014$

As can be seen from Fig. 14a, b, the strength of TPD is reduced by the bandwidth $\Delta \omega _0=2\%\omega _0$. However, TPD is still intense enough to produce abundant electrons as presented in Fig. 14d. When the decoupling condition is completely satisfied $\Delta \omega _0=4\%\omega _0>2.7\%\omega _0$, TPD is almost totally suppressed as shown in Fig. 14c, d.

Fig. 15

[From original figure in Ref. Zhao et al. (2021), $\copyright$ IOP Publishing. Reproduced with permission. All rights reserved.]

a Temporal growth of electrostatic energies developed by two pumps with different bandwidth. b Electron energy distributions under different bandwidth or beam number at $t=3000\tau$. The incident angles of beam 1 and beam 2 in vacuum are $\theta _{v1}=28^\circ$ and $\theta _{v2}=-28^\circ$, respectively. Plasma density of the beam overlapping region is $n_{\text{e}}\approx 0.243nc$. The density scaling length is $L=1000\lambda$, and the initial electron temperature is $T_{{\text{e}}}=3$keV. The amplitude of incident monochromatic lasers have an equal amplitude $a_1=a_2=0.01$

The temporal growth of the electrostatic energies exhibited in Fig. 15a validates that the saturation level and nonlinear evolution of absolute SRS can be reduced by a polychromatic light with large enough bandwidths. The electron energies are diagnosed in the fully nonlinear regime $t=3000\tau$, and the distributions are shown in Fig. 15b. The SRS intensity developed by two polychromatic beams with bandwidth $\Delta \omega _0=3\%\omega _1$ is comparable to that of one monochromatic laser. Single-beam SRS can be considerably suppressed by a polychromatic light with $\Delta \omega _0=3\%\omega _1$, and the same mitigation effect is found at a larger bandwidth $\Delta \omega _0=5\%\omega _1$ for two polychromatic beams, where their distributions are almost overlapped without hot-electron generations.

5 Schemes for broadband laser drivers

There have been a few schemes for the construction of broadband laser drivers. Several proposed schemes are discussed briefly in this section. Different from the StarDriver scheme (Eimerl et al. 2016), which requires a large number of relatively narrowband lasers, the FLUX concept is aimed to produce broadband single beam. Sum-frequency generation of broadband optical pulses is use of a narrowband pulse at 526.5 nm, where the combination of angular dispersion and noncollinearity cancels out the wave-vector mismatch and its frequency deviation (Dorrer et al. 2021). The shaping approaches for the control of the spectral density and temporal shape of nanosecond spectrally incoherent pulses was proposed, which are applicable to high-energy facilities (Dorrer and Spilatro 2022).

Low-coherence green laser with energy over 400 J and bandwidth $\sim$0.6%$\omega _0$ has been produced at the KUNWU (Gao et al. 2020), which may provide a platform to test the broadband theoretical model. Based on the broadband nature of second-harmonic generation (SHG) in Deuterated potassium dihydrogen phosphate (DKDP) crystals, low-coherence 529 nm lasers have been generated with conversion efficiency around 63% (Gao et al. 2020). The spectrum of the high-power broadband laser is a triangular profile with the FWHM $\sim 3.2$ nm. An efficient smoothing effect, combining a CPP and the induced spatial incoherence (ISI), was found for the low-coherence pulse (Gao et al. 2020). The rms nonuniformity is reduced from 40.7 to 14.7%, and the smoothed focal spot has a shorter longitudinal speckle length. However, no practical scheme has been demonstrated up to now for efficient third-harmonic generation (THG) of broadband beams (Dorrer et al. 2021). None of the proposed technologies, such as stimulated rotational Raman scattering and excimer lasers, performs well in both of the sufficient bandwidth and high conversion efficiency (Weaver et al. 2017; Obenschain et al. 2015; Dorrer et al. 2021). Moreover, the commercial availability of diffraction gratings and nonlinear crystal lead to the extremely complicated and enormous cost of conventional broadband laser drivers. The amplitude modulation is a universal phenomenon of broadband lasers, which is one of the technical challenges for the optical damage threshold. The system simplification of broadband driver needs the innovations of optical technologies. The concept of polychromatic drivers is proposed as a feasible and effective way to mitigate the LPIs compared to the conventional broadband driver. The basic mechanism and preliminary design of decoupled polychromatic drive beams was first proposed in 2017 (Zhao et al. 2017b), and a scheme to realize such drivers is put forward recently (Zhao et al. 2022).

Fig. 16

Optical design diagram of a two-color beamlet generator

A typical polychromatic driver scheme was proposed based on the decoupling conditions of multi-color beams and the complexity of the project, where the full spectrum width is $\Delta \omega _0=7.5\%\omega _0$ with total 16 colors of monochromatic beamlets, and the difference between every two adjacent frequencies is $\delta \omega _0=0.5\%\omega _0$. The four-color beamlets with adjacent frequency difference $\delta \omega _0=1\%\omega _0$ are synthesized to form a single polychromatic beam with a spectrum width $\Delta \omega _0=3\%\omega _0$. The frequency difference between any two beamlets of a polychromatic light is at least $1\%\omega _0$, which is sufficient to decouple the single-beam LPI submodes. The collective LPIs are also mitigated by the adjacent polychromatic beams due to the large enough beamlet number $N=4$ and the minimum frequency detuning $\delta \omega _0=0.5\%\omega _0$. With the same energy of a third-harmonic laser $I_0=1.1\times 10^{15}$ W/cm$^2$, the beamlet intensity of a polychromatic light is significantly reduced to $I_j=2.75\times 10^{14}$ W/cm$^2$. Therefore, the whole LPI intensity is effectively mitigated when the beamlets are weakly coupled.

In general, a polychromatic driver is able to generate numerous polychromatic beamlets with whole spectrum width larger than $10\%\omega _0$ (Cui et al. 2016). Therefore, the parameter range is large enough for design of the optimal scheme according to the drive configuration and facility cost. The two-color light generator, including an Nd:glass laser (1053 nm) and the other color signal light, is a unit of the polychromatic system as shown in Fig. 16. The second harmonic of Nd:glass laser with wavelength 526.5 nm is the pump source to amplify the signal light and simultaneously produce the idler beam via OPA. The generated two-color beams are then second-harmonic and triple-frequency converted independently. The other polychromatic beamlets are produced in the same way, and parts of the beamlets are synthesized and focused via the dichroic mirror and off-axis parabolic mirror, respectively.

DKDP is one of the optimal nonlinear crystals currently used in the high-power laser systems due to its high optical quality for kilojoule pulses. The lengths of DKDP in the processes of OPA and SHG are determined by the intensity fraction of different-color beams. The total energy conversion efficiency of polychromatic ultraviolet light is about 60%, where additional 15% of energy is lost in the OPA process with comparing to the general monochromatic laser. The conversion efficiency of a fundamental laser to the third-harmonic laser with bandwidth 1.5%$\omega _0$ as reported is only a third of the efficiency of the polychromatic light with a spectrum width 3%$\omega _0$. The suppression effects of such polychromatic lights have been investigated via large-scale PIC simulations, which indicate that more than 35% of the incident energy can be saved from the LPIs compared to monochromatic lasers for the direct-drive scheme, or high-density filled target for the indirect-drive scheme. The project cost and construction difficulty of the proposed scheme could be controlled, as moderate changes are required in the whole system. The conversion efficiency and beam collimation of the drive system could be enhanced by simplifying the beam synthesis, which will be investigated in the near future.

6 Discussion and summary

The interactions of broadband lasers with plasma at the nanosecond time and millimeter spatial scale may deserve further investigations, as such interactions may involve new physics associated with LPIs. For example, the self-adaptive coupling between the plasma evolution and broadband LPIs may have considerable impact on implosion dynamics and final yields. The introduced bandwidth can modify the properties of plasma in a long-time scale, including the energy deposition rate and hot electron generation, which in turn affect the subsequent LPI processes.

The production of hot electrons by broadband lasers has been comprehensively studied for $n_{\text{e}}\le 0.25n_{\text{c}}$ under the same initial conditions of plasmas. The broadband lasers with coupled beamlets or continuous spectra can still produce a large number of hot electrons, especially near the quarter-critical density (Zhao et al. 2015, 2017a). The growth rates of forward and side-scatter SRS are found to be insensitive to the bandwidth (Santos et al. 2007; Zhao et al. 2022), which may produce a large number of hot electrons in the kinetic regime. The large frequency shift of Langmuir wave can compensate the three-wave mismatch introduced by the bandwidth (Zhou et al. 2018), which along with the broad range of plasma wavenumber could lead to a significant level of wave turbulence (Zhao et al. 2017a; Brandão et al. 2021). The amplitude modulation of broadband lasers has a significant impact on the LPI threshold, which may cause a series of nonlinearly collective bursts (Follett et al. 2018; Liu et al. 2022). However, the effect of amplitude modulation can be reduced by phase smoothing of incident lasers, and the TPD-driven hot-electron generation is greatly reduced by the smoothed laser with a small bandwidth 0.7%$\omega _0$ (Follett et al. 2018). The decoupling threshold is critical to the effective suppression of hot electrons by broadband lasers. Actually, when the beamlets of a polychromatic light are decoupled, each beamlet is insufficient to drive intense plasma waves. Therefore, the nonlinear processes are suppressed effectively as shown in Figs. 6, 11, 12, 13 and 14.

The kinetic behaviors of broadband lasers near the critical density need to be explored further. Palastro et al. (2018) studied the linear and nonlinear resonance absorption of a broadband pulse with a fluid code. In the linear regime, the modified fractional absorption is $f_{{\text{rm}}}\approx f_{\text{r}}(\omega _0)\left[ 1+(1-12q^{3/2}+8q^3)\left( \Delta \omega _0/\omega _0\right) \right]$, where $f_{\text{r}}(\omega _0)$ is the fractional absorption of an electromagnetic wave with frequency $\omega _0$, and $q=(2\pi L/\lambda )^{2/3}\sin ^2\theta _v$. For a broad bandwidth $\Delta \omega _0=10\%\omega _0$, the correction is only 2% near the peak absorption where $q=0.5$. Therefore, laser bandwidth fails to mitigate resonance absorption in the linear regime, where the ponderomotive response of the ions is neglected. In the nonlinear stage, the enhancement of absorption caused by the steep ion-density profile and electromagnetic decay instability can be suppressed by the broadband lasers. Broadband lasers can save considerable energies from SRS, TPD and SBS in the plasma $n_{\text{e}}<n_{\text{c}}$. The hot electrons generated by parametric decay instabilities around $n_{\text{c}}$ cannot be ignored in the case of the broadband lasers, since more energies could be deposited at the critical density than the monochromatic light.

In summary, mitigation of LPIs with broadband lasers can contribute to the enhancement of beam–target coupling and robust ignition performance. Decoupling conditions of polychromatic beams in homogeneous and inhomogeneous plasmas are identified and tested by numerical simulation. It is found that a laser bandwidth larger than 1% is typically required to mitigate LPIs effectively. This is applied to the design of broadband laser drivers. Up to now, no practical scheme has been demonstrated for efficient THG of broadband lasers. A new design of polychromatic drivers is proposed to effectively mitigate both the single and multibeam LPIs. It is based on the matured technologies and, thus, shows high feasibility and cost-effectiveness.