Suppression of parametric instabilities in inhomogeneous plasma with multi-frequency light

Here we consider the spatial amplification of the instability modes in a plasma with density profile n_e = n₀(1 + x/L), where L is the density scale length and x is the longitudinal axis. The driving laser beam is composed of many beamlets with different frequencies

$$\begin{eqnarray}&&a=\displaystyle \sum _{i=1}^{N}{a}_{i}\cos ({\omega }_{i}t+{\phi }_{i}),\end{eqnarray} \tag{ 1 }$$

where a_i is the normalized amplitude of ith beamlet with a carrier frequency ω_i and a random phase ϕ_i, and N is the number of beamlets. The relation between a_i and laser intensity I_i is given by ${a}_{i}=\sqrt{{I}_{i}({\rm{W}}\,{\mathrm{cm}}^{-2}){\left[\lambda (\mu {\rm{m}})\right]}^{2}/1.37\times {10}^{18}}$. To simplify the problem, we first study the convective instability developed by two light beamlets with different frequencies, i.e. N = 2. Assuming the two lights have an equal amplitude ${a}_{1}\,={a}_{2}={a}_{0}/\sqrt{2}$, and different frequencies ω₁ = ω₀ − δω₀/2 and ω₂ = ω₀ + δω₀/2, where ω₀ is the central frequency, and δω₀ is the light bandwidth. We have an approximation for δk₀ =k₂ − k₁ ≈ ω₀δω₀/k₀c² with k₀ the wavenumber of central frequency. According to previous studies on the effect of laser bandwidth on the parametric instability in homogeneous plasma [17], when δω₀ ≫ Γ₀, a modified temporal growth rate is given by ${{\rm{\Gamma }}}_{m}={{\rm{\Gamma }}}_{0}^{2}/\delta {\omega }_{0}$ for the whole incident light. In the following, we study the convective process by using the modified Γ_m

$$\begin{eqnarray}\begin{array}{rcl}{\nu }_{s}{a}_{s}+{v}_{s}{\partial }_{x}{a}_{s} & = & \displaystyle \frac{{{\rm{\Gamma }}}_{m}}{\sqrt{2}}{a}_{p}\left[\exp \left(i\displaystyle \int {K}_{1}{dx}\right)\right.\\ & & \left.+\exp \left(i\displaystyle \int {K}_{2}{dx}\right)\right],\end{array}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\nu }_{p}{a}_{p}-{v}_{p}{\partial }_{x}{a}_{p} & = & \displaystyle \frac{{{\rm{\Gamma }}}_{m}}{\sqrt{2}}{a}_{s}\left[\exp \left(-i\displaystyle \int {K}_{1}{dx}\right)\right.\\ & & \left.+\exp \left(-i\displaystyle \int {K}_{2}{dx}\right)\right],\end{array}\end{eqnarray} \tag{ 3 }$$

where a_s and a_p respectively are the normalized amplitude of the scattered light and plasma wave, ν_s and ν_p are the damping for the scattered light and the plasma wave, respectively. Wavenumber mismatch K₁ = k₁ − k_s − k_p = k₀ − δk₀/2 −k_s − k_p and K₂ = k₂ − k_s − k_p = k₀ + δk₀/2 − k_s − k_p, where k_s and k_p are the wavenumber for the scattered light and the plasma wave, respectively. The above equations (2) and (3) can be reduced to

$$\begin{eqnarray}&&{\nu }_{s}{a}_{s}+{v}_{s}{\partial }_{x}{a}_{s}=\sqrt{2}{{\rm{\Gamma }}}_{m}{a}_{p}\exp ({{iK}}_{0}^{{\prime} }{x}^{2}/2)\cos (\delta {k}_{0}^{{\prime} }{x}^{2}/4),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\nu }_{p}{a}_{p}-{v}_{p}{\partial }_{x}{a}_{p}=\sqrt{2}{{\rm{\Gamma }}}_{m}{a}_{s}\exp (-{{iK}}_{0}^{{\prime} }{x}^{2}/2)\cos (\delta {k}_{0}^{{\prime} }{x}^{2}/4),\end{eqnarray} \tag{ 5 }$$

where K₀ = k₀ − k_s − k_p, ${K}_{0}^{{\prime} }={{dK}}_{0}/{dx}$, and $\delta {k}_{0}^{{\prime} }=d\delta {k}_{0}/{dx}$. Considering a heavy damping for the plasma wave a_p, the saturation coefficient is obtained to the first order

$$\begin{eqnarray}&&G=\displaystyle \frac{2\pi {{\rm{\Gamma }}}_{m}^{2}}{{v}_{s}{v}_{p}{K}_{0}^{{\prime} }}\left[1+\cos \left(\displaystyle \frac{2{\nu }_{p}^{2}\delta {k}_{0}^{{\prime} }}{{{K}_{0}^{{\prime} }}^{2}{v}_{p}^{2}}\right)\right].\end{eqnarray} \tag{ 6 }$$

Equation (6) indicates that the two different frequency beams can be coupled to develop convective instability in a same resonant region. However, the saturation level is lowered by the bandwidth as compared to the Rosenbluth gain saturation coefficient. For SRS instability, equation (6) can be reduced to

$$\begin{eqnarray}&&{G}_{{\rm{SRS}}}\approx \displaystyle \frac{2\pi {{\rm{\Gamma }}}_{m}^{2}}{{v}_{s}{v}_{p}{K}_{0}^{{\prime} }}\left[1+\cos \left(\displaystyle \frac{4L{\omega }_{0}{\nu }_{p}^{2}{\omega }_{L}^{2}}{{n}_{0}c{\left({\omega }_{0}^{2}-{\omega }_{{pe}}^{2}\right)}^{3/2}}\delta {\omega }_{0}\right)\right],\end{eqnarray} \tag{ 7 }$$

where ${\omega }_{L}=\sqrt{{\omega }_{{pe}}^{2}\,+\,3{k}_{L}^{2}{v}_{{\rm{th}}}^{2}}$ is the frequency of Langmuir wave with v_th being the electron thermal velocity. An approximation can be made based on equation (7) that the two beamlets are mutually independent when

$$\begin{eqnarray}&&\delta {\omega }_{0}\gtrsim \displaystyle \frac{\pi {n}_{0}c{\left({\omega }_{0}^{2}-{\omega }_{{pe}}^{2}\right)}^{3/2}}{8L{\omega }_{0}{\omega }_{L}^{2}{\nu }_{p}^{2}}.\end{eqnarray} \tag{ 8 }$$

Considering a plasma with n₀ = 0.08n_c, T_e = 3 keV and L = 3000λ with λ being the central light wavelength in vacuum, and the corresponding Landau damping is ν_p ≈ 0.055ω₀ [24], we have the threshold δω₀ ≈ 2.4%ω₀. Consequently, for suppressing SRS, bandwidth between any two beamlets is in the order of 10⁻²ω₀. For the case of N beamlets, the frequency difference between two neighboring beamlets is δω₀ = Δω₀/(N − 1), where Δω₀ is the total bandwidth of the multi-frequency light. Therefore, a small beamlet number N is more suitable for the suppression of SRS when the total bandwidth is in the order of Δω₀ ∼ 10⁻²ω₀. Different from the instability region-coupling in homogeneous plasma, the threshold equation (8) is independent of laser amplitude due to the wavenumber detuning out of the local resonant region, which we will discuss in detail in the next section.

Situations are rather complicated for the scattering light propagating in a large scale inhomogeneous plasma. A scattering light produced by one incident light can be amplified again as a seed mode in the subsequent parametric excitation process in a region where its frequency is equal to the scattering light developed by another light. Therefore, the above linear model is suitable for describing the behavior of Langmuir waves, due to its linear propagation in the whole inhomogeneous plasma. Briefly, the bandwidth weakens the strength of longitudinal electrostatic field and therefore the production of hot electrons, however, a part of the scattering light produced by one incident light may be magnified by the other light when the frequencies of the two scattering lights have a overlap. For example, as shown in figure 1, the scattering light developed by ω₁ = 1.02ω₀ at 0.1n_c has frequency ω_s1 = 0.7ω₀, and it will be amplified again by the other incident light ω₂ = 0.98ω₀ as a seed mode at 0.078n_c, where the frequency of the scattering light is also ω_s2 = 0.7ω₀. The amplification coefficient can be estimated by

$$\begin{eqnarray}&&{G}_{{\rm{SL}}}=2\pi \left(\displaystyle \frac{{{\rm{\Gamma }}}_{1}^{2}}{{v}_{s1}{v}_{p1}{K}_{01}^{{\prime} }}+\displaystyle \frac{{{\rm{\Gamma }}}_{2}^{2}}{{v}_{s2}{v}_{p2}{K}_{02}^{{\prime} }}\right),\end{eqnarray} \tag{ 9 }$$

where the subscripts 1 and 2 refer to the parameters at x₁ and x₂, respectively. Therefore, the saturation level of this part of scattered lights has not been greatly reduced due to the secondary amplification. For a finite inhomogeneous plasma with Langmuir wave frequency range [ω_L1, ω_L2], the two scattering lights developed by SRS are amplified independently in the propagation when ω₂ − ω_L2 ≳ ω₁ − ω_L1, i.e.

$$\begin{eqnarray}&&\delta {\omega }_{0}\gt {\omega }_{L2}-{\omega }_{L1}.\end{eqnarray} \tag{ 10 }$$

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Considering an inhomogeneous plasma with a density range [0.08, 0.1]n_c, the threshold for suppression of the secondary amplification of scattering light is δω₀ > 3.3%ω₀. By comparing with the linear threshold equations (8), (10) is relatively larger for a large scale inhomogeneous plasma. Therefore, both the hot electron production and the reflectivity are well-controlled when equation (10) is satisfied.

Different from SRS, the frequency of SBS backscattering light changes little with the plasma density. For an inhomogeneous plasma with density [n₁, n₂], the frequency range of the scattering light is $| \delta {\omega }_{s}| \approx 2{c}_{s}{\omega }_{{pe}}(\sqrt{{n}_{2}}-\sqrt{{n}_{1}})/{k}_{0}{c}^{2}\,\lt 2{\omega }_{0}{c}_{s}/c\sim {10}^{-3}{\omega }_{0}$. Therefore, the suppression threshold for SBS in finite plasma is

$$\begin{eqnarray}&&\delta {\omega }_{0}\gt 2{c}_{s}{\omega }_{{pe}}(\sqrt{{n}_{2}}-\sqrt{{n}_{1}})/{k}_{0}{c}^{2}\sim {10}^{-4}{\omega }_{0}.\end{eqnarray} \tag{ 11 }$$

Note that a small frequency difference δω₀ ∼ 10⁻³ω₀ is sufficient for the effective suppression on SBS. The beamlet number is in the order of N ∼ 10 for a multi-frequency light with total bandwidth Δω₀ = (N − 1)δω₀ ∼ 10⁻²ω₀.

According to the TPD dispersion relation in cold homogeneous plasmas [24]

$$\begin{eqnarray}&&\left({\omega }^{2}-{\omega }_{{pe}}^{2}\right)\left[{\left(\omega -{\omega }_{0}\right)}^{2}-{\omega }_{{pe}}^{2}\right]={{\rm{\Gamma }}}_{\mathrm{TPD}}^{2},\end{eqnarray} \tag{ 12 }$$

where Γ_TPD is the temporal growth rate of TPD, we know that TPD is a local instability, and always happens in a narrow region [0.5ω₀ − Γ_TPD, 0.5ω₀ + Γ_TPD] with Γ_TPD ≈ k₀ca₀/4. Therefore, the frequency deviation between different beamlets can separate their developing region, and each beam will be independent when

$$\begin{eqnarray}&&\delta {\omega }_{0}/{\omega }_{0}\gt \displaystyle \frac{{k}_{0}{{ca}}_{i}}{2{\omega }_{0}}\sim 0.433{a}_{i}.\end{eqnarray} \tag{ 13 }$$

Note that equation (13) has not included the temperature effects. For a beamlet with a_i ∼ 10⁻³, the threshold for suppressing TPD is around δω₀ ∼ 10⁻³ω₀. Therefore, the beamlet number is in the order of N ∼ 10 for a multi-frequency light with total bandwidth Δω₀ ∼ 10⁻²ω₀.

In summary, we have presented the required frequency difference between two laser beams for their decoupling. Once they are decoupled, one can simply control the parametric instabilities by controlling a single beamlet. With this, one can design the beam structure for the driving light under a given intensity and bandwidth. In the following section, we will carry out numerical simulation to test the theory predictions found in this section.

The space and time given in the following are normalized by the laser wavelength in vacuum λ and the laser period τ. The length of the simulation box is 600λ, where the plasma occupies a region from 50λ to 550λ with density profile n_e(x) = 0.08[1 + (x − 50)/1000]n_c, i.e. the density range for this finite inhomogeneous plasma is [0.08,0.12]n_c. The initial electron temperature is T_e0 = 2 keV. Here in this subsection, we only consider the SRS effects, therefore the ions are immobile with a charge Z = 1. A linearly-polarized semi-infinite pump lasers with a uniform amplitude a₀ = 0.014 (the corresponding intensity is I₀ = 2.5 × 10¹⁵ W cm⁻² with λ = 0.33 μm) is incident from the left boundary of the simulation box. We have taken 100 cells per wavelength and 50 particles per cell. In the simulation, we change the number of laser beamlets N while keeping the total incident beam energy and bandwidth conserved, and therefore the beamlet amplitude is ${a}_{i}={a}_{0}/\sqrt{N}$, i.e. a_i = 0.01 for N = 2 and a_i = 0.0031 for N = 20.

As we can see from figure 2(a), comparing to the case with a single frequency beam, the beam with finite bandwidth Δω₀ = 2%ω₀ can significantly reduce the strength of the Langmuir wave in the inhomogeneous plasma. The saturation level is lowered further when the bandwidth Δω₀ increases to 4%ω₀. Therefore, with the beam structure proposed in the last section, the bandwidth of incident light can reduce the saturation level of SRS. Comparing two cases in figure 2(a), one finds that the suppression effect is weakened when increasing the beamlet number N under a same bandwidth. In homogeneous plasma, the instability region of each beamlet is shrunk by reducing the amplitude ${a}_{i}={a}_{0}/\sqrt{N}$ when the total energy is unchanged. Therefore, the beamlets are gradually decoupled with the increase of N. Different from this, the suppression condition equation (8) in inhomogeneous plasma is independent of the laser amplitude, due to the wavenumber mismatch outside the resonant region. Therefore, the beamlets are still coupled when the beamlet number increased, which will be proved by the following two phase plots.

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The linear relation between the frequency difference of two beamlets δω₀ and wavenumber difference of Langmuir wave δk_L is

$$\begin{eqnarray}\begin{array}{rcl}\delta {k}_{L} & = & ({{dk}}_{L}/d{\omega }_{0})\delta {\omega }_{0}\\ & = & \left(\displaystyle \frac{{\omega }_{0}}{\sqrt{{\omega }_{0}^{2}-{\omega }_{{pe}}^{2}}}+\displaystyle \frac{{\omega }_{0}-{\omega }_{{pe}}}{\sqrt{{\omega }_{0}^{2}-2{\omega }_{{pe}}{\omega }_{0}}}\right)\delta {\omega }_{0}/c.\end{array}\end{eqnarray} \tag{ 14 }$$

For a case with N = 2 and δω₀ = Δω₀ = 4%ω₀, we have δk_Lc ∼ 0.087ω₀ at n_e = 0.1n_c. The phase plot presented in figure 2(b) indicates that the instability region has already been separated by the frequency difference δω₀ = 4%ω₀ when N = 2. Under the same conditions, the phases are still coupled for N = 20 with a relatively small amplitude a_i =0.0031 and with δω₀ = Δω₀/(N − 1) ≈ 0.2%ω₀. Therefore, the phase coupling of incident beamlets has no relations to their amplitude in inhomogeneous plasmas.

As shown in figure 2(c), the electron temperature is reduced by the light with Δω₀ = 2%ω₀, due to the suppression of Langmuir wave discussed above. Otherwise, one finds a long hot electron tail heated by absolute SRS via SRS rescattering at Δω₀ = 0 [26]. With the multi-frequency component laser beam, the first-order SRS is reduced to a level below the required threshold for developing SRS rescattering. The electron temperature for N = 2 case is slightly lower than the one N = 20, which is in agreement with the Langmuir wave strength in figure 2(a). Further reduction can be found for a larger bandwidth Δω₀ = 6%ω₀ with N = 20.

Figure 2(d) shows that the reflectivity of backscattering light decreases with the increase of bandwidth. Note that the reflectivity is still large at Δω₀ = 2%ω₀, even though the Langmuir wave has been greatly reduced under the same condition as shown in figure 2(a). This is mainly because of the secondary amplification of the scattering lights as discussed in section 2.2. In this example, the range of Langmuir wave frequency is [0.33,0.38]ω₀, the threshold for suppressing secondary amplification is δω₀ ∼ 5%ω₀ according to equation (10). Considering the case with Δω₀ = 2%ω₀ and N = 2, the spectra of the two scattering light respectively are ω_s1 = [0.61, 0.66]ω₀ and ω_s2 = [0.63, 0.68]ω₀, where an overlapping frequency range [0.63, 0.66]ω₀ can be found. Therefore, the scattering light ω_s1 = [0.63, 0.66]ω₀ will be amplified again as a seed mode when it propagates into the resonant region of ω_s2 = [0.63, 0.66]ω₀. From figure 2(e) we know that the sharing parts are shrunk with the increase of the bandwidth, and beams are totally separated until Δω₀ =6%ω₀. The scattering lights are strongly coupled for N = 20 even at Δω₀ = 6%ω₀ as presented in figure 2(f). This further proved that the threshold for decoupling of incident beamlets are independent of light amplitude in inhomogeneous plasma.

Generally speaking, the decoupled laser beam structure with certain bandwidth not only reduces the reflectivity and the SRS rescattering process, but also suppresses the Langmuir wave amplitude. As a result, the hot electron production is also significantly reduced.

To further validate the suppression effects of bandwidth on SBS, we performed a series of 1D PIC simulations with mobile ions. The mass of ion is m_i = 1836m_e with a charge Z = 1. We set the ion temperature T_i0 = 0 in our simulations, where SBS usually develops more easily than the case with finite ion temperature. If SBS can be suppressed by a multi-frequency laser beam at T_i0 = 0, then one expects it will be suppressed with finite T_i0. The other parameters are the same as the above simulations.

Different from SRS, a very small bandwidth is sufficient to effectively suppress SBS according to equation (11). Therefore, the suppression effect is better for the light with larger beam number N = 20 under a fixed bandwidth. The wavenumber distributions of IAW is presented in figure 3(a) for two cases with N = 1 and N = 2. The instability regions are separated when Δω₀ = 2%ω₀. Even though the amplitude of each beamlet for N = 2 is smaller than the N = 1 case, the intensity is still large enough to excite intense SBS. Therefore, a large amount of lights are scattered out of the plasma as shown in figure 3(b). Note that SRS (the corresponding spectrum is around ω_s ∼ 0.62ω₀) has been greatly suppressed by SBS under this condition [27, 28]. Figure 3(c) indicates that SBS is significantly suppressed by increasing the beamlet number N to 20 under the same bandwidth Δω₀ = 2%ω₀ with comparing to figure 3(b). The frequency difference between two neighbouring beamlets is δω₀ ≈ 10⁻³ω₀ > δω_s ∼10⁻⁴ω₀, therefore beamlets are decoupled and each beamlet is too weak to develop intense SBS. In the meanwhile, SRS is the dominant instability when SBS is suppressed, which indicates that the optimum inhibition parameters are different for SRS and SBS. From figure 3(d), we can see that the reflectivity is finally maintained at 30% for N = 2, and is well below 4% for N = 20.

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Simulation for $0.3{n}_{c}\leqslant {n}_{e}\leqslant 0.45{n}_{c}$ with T_i0 = 1 keV and Δω₀ = 4%ω₀ has also been performed, and the results are similar to the above examples. One finds that reflectivity can be reduced around six times when N = 20 with comparing to N = 2. Therefore, multiple frequency components with N ≳ 20 is better for SBS suppression.

Different from the above two stimulated scattering instabilities, TPD always happens in a local region near 0.25n_c. To validate the suppression effects on TPD, we have performed several two-dimensional (2D) simulations. The length of the simulation box is 600λ, where the plasma occupies a region from 30λ to 180λ with the density profile n_e(x) = 0.22[1 +(x − 30)/660]n_c. Here we mainly consider the TPD instability, therefore the ions are immobile with a charge Z = 1. The initial electron temperature is T_e0 = 2keV. A p-polarized (electric field of light is parallel to the simulation plane) semi-infinite pump lasers with a uniform amplitude a₀ = 0.014 is incident from the left boundary of the simulation box.

Similar to SBS, the frequency difference equation (13) for suppressing TPD is in the order of ∼10⁻³. Therefore, larger beamlet number is better for instability inhibition under a same bandwidth. Without the loss of generality, here we take N = 20 for the broad bandwidth light, i.e. a_i = 0.0031 and the threshold of bandwidth is Δω₀ ≳ 2.6%ω₀ according to equation (13). As can be seen from figures 4(a) and (b), the strength of TPD is reduced by the bandwidth Δω₀ = 2%ω₀. However, TPD is still intense enough to heat abundant electrons as presented in figure 4(d). When the threshold is completely satisfied Δω₀ = 4%ω₀ > 2.6%ω₀, TPD is almost totally suppressed, and only a weak SRS mode can be found in figure 4(c) [29]. From figure 4(d), we know that hot electrons are greatly suppressed under Δω₀ = 4%ω₀. The results are consistent with the previous fluid simulations about TPD suppression [30].

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Based on the above discussions, we know that SRS can be greatly suppressed by a multi-frequency light with a broad bandwidth and small beamlet number N≲10. On the contrary, SBS and TPD can be effectively inhibited by a small bandwidth and large beamlet number N ≳ 20. Generally, both of TPD near n_e ∼ 0.25n_c and SBS in n_e > 0.25n_c exist in the direct-drive, therefore multi-frequency light with N ≳ 20 and total bandwidth Δω₀ ≳ 4%ω₀ is better for direct-drive scheme. However, a tradeoff should be made for indirect-drive according to the relative intensity between SRS and SBS. As one knows, the amplitude of SBS backscattering light is mainly determined by the parameter ZT_e0/T_i0. Therefore, a critical value of ZT_e0/T_i0 should be found for different optimum inhibition parameters.

Simulations for initial ion temperature T_i0 = 300 eV and bandwidth Δω₀ = 4%ω₀ have been performed with other parameters unchanged, and the corresponding ZT_e0/T_i0 is about 6.67. The spectra of the backscattering light for different beamlet number N is displayed in figure 5(a). The intensity of SRS spectrum around ω_s = 0.65ω₀ is relatively lower for N = 2. However, the corresponding SBS spectrum is much larger than the one of N = 20. We find that their reflectivities are both nearly equal to 3.1%. Therefore, large beamlet number N ≳ 20 is better for the regime ZT_e0/T_i0 ≳6.67. Reflectivity of backscattering light for incident light with different bandwidth and beamlet number under ZT_e0/T_i0 = 2 is plotted in figure 5(b). In this regime, SRS is the dominant mode, therefore small beamlet number N = 2 is better than N = 20, which is similar to figure 2(d).

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