Theoretical model

In our model, we consider a one-dimensional static plasma slab which locates between x = 0 and x = L. The pump laser pulse is supposed to incident into the plasma from the left side (x = 0), while the scattered light reflects from the opposite boundary (x = L). Only backward scattering is dealt with in this article. Customarily, we define I _i as the intensity of light with the frequency ω_i and the wave number k _i, where the subscript i = 0, s for incident and backscattered light, respectively. To describe the broadband scattered light, the intensity is defined as ${I}_{\rm s}\lpar x\rpar = \int {{i}_{\rm s}\lpar {{\rm \omega }}_{\rm s}\comma\; x\rpar } \, d{{\rm \omega }}_{\rm s}$, where i _s is the spectral density (intensity per angular frequency). Following the DEPLETE model (Strozzi et al., Reference Strozzi, Williams, Hinkel, Froula, London and Callahan2008), as well as Gong et al. (Reference Gong, Li, Zhao, Hu and Zheng2013), the coupled intensity equations between the pump and scattered light for the steady state in strong damping limit can be described by

(1)$${\partial {I}_{0}\over \partial x} = -{{{\rm \omega}}_{0}\over {{\rm \omega}}_{\rm s}} \Gamma {I}_{0}{I}_{\rm s}\comma $$

(2)$${\partial {I}_{\rm s}\over \partial x} = -\Gamma {I}_{0}{I}_{\rm s}\comma $$

where the coupling coefficient Γ is

(3)$$\Gamma = {{e}^{2}\over 2{{\rm \varepsilon}}_{0}{m}_{\rm e}^{2}{c}^{4}} {{k}^{2}\over {{\rm \omega}}_{0}{k}_{0}{k}_{\rm s}} {\rm Im}\left[{{{\rm \chi}}_{\rm e}\lpar 1 + {{\rm \chi}}_{i}\rpar \over 1 + {{\rm \chi}}_{\rm e} + {{\rm \chi}}_{i}} \right],$$

where e, ɛ₀, m _e, c denote the electron charge, vacuum permittivity, electron mass, and speed of light, respectively. k is the wave number of the Langmuir wave for the SRS and the ion acoustic wave for the SBS, respectively. χ_e(χ_i) is the electron (ion) susceptibility. For a Maxwellian distribution, the electron susceptibility χ_e is given by

(4)$${{\rm \chi}}_{\rm e} = -{1\over 2{k}^{2}{{\rm \lambda}}_{{\rm De}}^{2}} Z'\lpar {{\rm \xi}}_{\rm e}\rpar \comma $$

where λ_De = υ_te/ω_pe denotes the Debye length of electrons, ${\rm \upsilon }_{\rm te} = \sqrt {T_{\rm e}/m_{\rm e}}$ is the electron thermal velocity, ${\rm \omega }_{\rm pe} = \sqrt {n_{\rm e} e^2/{\rm \varepsilon }_0 m_{\rm e}}$ means the electron plasma frequency. T _e and n _e are the electron temperature and density, respectively. The plasma dispersion function (Fried and Conte, Reference Fried and Conte1961) is

(5)$$Z\lpar {{\rm \xi}}_{\rm e}\rpar = i{{\rm \pi}}^{1/2}{e}^{-{{{\rm \xi}}}_{\rm e}^{2}}\, {\rm erfc}\lpar -i{{\rm \xi}}_{\rm e}\rpar \comma $$

where erfc is the complimentary error function (Abramowitz and Stegun, Reference Abramowitz and Stegun1970). ${\rm \xi }_{\rm e} = {\rm \omega }'/\sqrt {2}k{\rm \upsilon }_{\rm te}$, and ${\rm \omega }' = {\rm \omega }- \vec {k}\cdot \vec {u}$ is the frequency of the Doppler-shifted Langmuir wave due to the plasma speed $\vec {u}$. The total ion susceptibility χ_i is

(6)$${\rm \chi}_i = \sum _{\,j}{{{\rm \chi}}_{\,j}} \comma\; $$

where the susceptibility for the ion species j is given by

(7)$${\rm \chi}_j = -{1\over 2{k}^{2}{{\rm \lambda}}_{{\rm D}j}^{2}} {Z}'\lpar {{\rm \xi}}_{\,j}\rpar \comma $$

where λ_Dj = υ_tj/ω_pj, ${\rm \upsilon }_{{\rm t}j} = \sqrt {T_j/m_j}$, ${\rm \omega }_{{\rm p}j} = \sqrt {n_j{Z}_{j}^{2}e^2/{\rm \varepsilon }_0m_j}$. T _j, n _j, and Z _j are the temperature, density, and charge state of ion species j, respectively. ${\rm \xi }_j = {\rm \omega }'/\sqrt {2}k{\rm \upsilon }_{{\rm t}j}$, and ${\rm \omega }' = {\rm \omega }-\vec {k}\cdot \vec {u}$ is the frequency of the Doppler-shifted ion acoustic wave.

Based on Eqs (1) and (2), the intensity of the scattered light at the left boundary (x = 0) is derived as follows:

(8)$$I_{\rm s}\lpar 0\rpar = {I}_{\rm s}\lpar L\rpar \times {e}^{G}\comma $$

where the linear gain exponent G is defined as

(9)$$G = \int _{0}^{L}{\Gamma} {I}_{0}\, dx.$$

Replacing the Γ in Eq. (9) via Eq. (3) yields

(10)$$G = \int_{0}^{L}{I_0 {e}^{2}\over 2{{\rm \varepsilon}}_{0}{m}_{\rm e}^{2}{c}^{4}} {{k}^{2}\over {{\rm \omega}}_{0}{k}_{0}{k}_{\rm s}} {\rm Im}\left[{{{\rm \chi}}_{\rm e}\lpar 1 + {{\rm \chi}}_{i}\rpar \over 1 + {{\rm \chi}}_{\rm e} + {{\rm \chi} }_{i}} \right]{d}x.$$

Applying Tang's model (Tang, Reference Tang1966) directly, the reflectivity R of SRS or SBS, as a function of the gain G, can be obtained by

(11)$${\rm \varepsilon} = {R\lpar 1-R\rpar \over e^{G\lpar 1-R\rpar }-R}\comma $$

with an initial seed value for the backscattered light of ɛ = 1 × 10⁻⁹.

Discretized algorithm

The system of equations from Eqs (1) to (11) will be discretized in one-dimensional space using a staggered grid with N cells having N + 1 boundaries (see Fig. 1). Cell j is located between interface j and j + 1, while interface j separates cell j − 1 from cell j. In general, G, T _e, n _e, T _j, n _j, and Z _j are defined at cell centers, I ₀, I _s, R, and x are defined at interface.

Fig. 1. Schematic of the model and staggered grid.

Different from Strozzi et al. (Reference Strozzi, Williams, Hinkel, Froula, London and Callahan2008), who solved the equation system via a shooting method, we propose a novel “predictor-corrector” method in our model. The algorithms of the model proceed in two steps: firstly, the “prediction” step. Based on the plasma parameters (T _e, n _e, T _j, n _j, Z _j, and I ₀), the gain exponent G in each cell is calculated by Eq. (10). The total gain G _all is obtained by adding contributions from all the cells:

(12)$$G_{{\rm all}} = \sum_{\,j = 1}^{N}{G_j}.$$

The total reflectivity of the backscattered light (the reflectivity at the first interface R ₁) can be derived from Eq. (11). Then, we define the gain deviation of the backscattered light as

(13)$$\Delta G = \lpar \ln {R_1}- \ln {{\rm \varepsilon}}\rpar -G_{{\rm all}}.$$

Secondly, the “corrector” step. We revise the gain exponent G in each cell by

(14)$$G_j' = G_j + {G_j\over G_{{\rm all}}}\Delta G.$$

The reflectivity of the backscattered light at each interface R _j can be derived from

(15)$$\ln R_j - \ln R_{\,j + 1} = G_j'\comma\; $$

marching from the right to left boundary. The above calculation is suitable for both SRS and SBS. For the pump laser intensity at each interface I _0j, the energy conservation reads

(16)$$1-R_1^{{\rm SRS}}-R_1^{{\rm SBS}} = I_{0j}-R_j^{{\rm SRS}}-R_j^{{\rm SBS}}.$$

Now, the system of equations is closed, and all the quantities are solved. One can prove that, for the homogeneous plasma, the reflectivity derived by the “predictor-corrector” method will be the same as the Tang's model:

(17)$$\ln R_1' = G_1' + \cdots + G_N' + \ln {\rm \varepsilon} = G_{{\rm all}} + \Delta G + \ln{\rm \varepsilon} = \ln R_1.$$

This proves the stability of our method clearly. Compared with the traditional Tang's model, the inhomogeneous plasma case can be also dealt with in our model. Besides, our model exhibits much higher efficiency than the DEPLETE model (Strozzi et al., Reference Strozzi, Williams, Hinkel, Froula, London and Callahan2008) since the solving of a two-point boundary value problem via a shooting method is avoided.

Numerical results

In order to demonstrate the validity of our model, the typical numerical results are presented in this section. The plasma and laser parameters in experiments conducted by Montgomery et al. (Reference Montgomery, Afeyan, Cobble, Fernandez, Wilke, Glenzer, Kirkwood, MacGowan, Moody, Lindman, Munro, Wilde, Rose, Dubois, Bezzerides and Vu1998) are considered. A uniform 1 mm plasma, consisting of 50% C₃H₈ and 50% C₅H₁₂, is generated before the incidence of pump laser (I ₀ = 2 × 10¹⁵ W/cm², λ ₀ = 351 nm). The electron temperature is heated to T _e ≈ 3 keV, while the ion temperature keeps as T _i = T _e/3. The electron density of the plasma varies from 0.05 to 0.15n _c for different calculating cases, where $n_{\rm c} = {\rm \omega }_{0}^2 {\rm \varepsilon }_0 m_{\rm e} / e^2$ denotes the critical density of the pump laser. Hereafter, we choose the above plasma and laser parameters as default input condition to calculate the linear gain exponent, reflectivity, and distribution of backscattered light.

Figure 2 shows the gain exponent of SRS as a function of the scattered light frequency for different electron densities. One should notice that we take the above method in steady state to apply independently at each scattered frequency. This may be viewed as a “completely incoherent” treatment of the scattered light at different frequencies. The maximum gain of SRS appears at the frequency where the three-wave-coupling condition is exactly satisfied. Besides, the maximum gain of SRS increases with the electron density, which is consistent with the traditional linear theory (Froula et al., Reference Froula, Divol, London, Berger, Doppner, Meezan, Ralph, Ross, Suter and Glenzer2010). Furthermore, with the increase of the electron density, the gain peak moves to lower frequencies, and narrows. The narrowing in frequency with increased electron density is mainly due to the enhancement of the three-wave-coupling.

Fig. 2. The gain exponent of SRS as a function of the scattered light frequency (normalized by the pump laser frequency ω₀) for n _e/n _c = 0.05 (black dashed), n _e/n _c = 0.1 (blue dotted), and n _e/n _c = 0.15 (red solid), respectively.

The gain exponent of SBS as a function of the scattered light frequency is displayed in Figure 3. Compared with that of the SRS, the general SBS gain peak is much narrower (note that the range of x coordinator axis is much smaller than that in Fig. 2), and the maximum also increases with the electron density as well. Because of the small frequency of ion acoustic wave, the spectral width of SBS gain keeps almost the same. For the low-density case (n _e/n _c = 0.05), the maximum SBS gain (16.5) is higher than the SRS (8.1). However, with the increase of electron density, the maximum SRS gain rises rapidly and will become five times of the SBS for the high-density case (n _e/n _c = 0.15).

Fig. 3. The gain exponent of SBS as a function of the scattered light frequency (normalized by the pump laser frequency ω₀) for n _e/n _c = 0.05 (black dashed), n _e/n _c = 0.1 (blue dotted), and n _e/n _c = 0.15 (red solid), respectively.

In order to investigate the dependence of the gains on the electron density quantitatively, we focus on the maximum gain. Figure 4 presents the maximum gain of both SRS and SBS versus the electron density with electron temperature T _e = 3 keV. The maximum SBS gain exhibits linear dependence on the electron density, while the maximum SRS gain increases exponentially with the electron density. The maximum SRS gain exceeds the SBS at the density of n _e/n _c = 0.11. Physically, for the SRS, the Landau damping of the Langmuir wave is very sensitive to the electron density and decreases significantly with the increase of electron density. However, for the SBS, the Landau damping of the ion acoustic wave keeps almost the same, so the maximum SBS gain scales linearly with the electron density. Our calculation is consistent with the results given by Gong et al. (Reference Gong, Li, Zhao, Hu and Zheng2013).

Fig. 4. Maximum gain of both SRS (red solid) and SBS (blue dashed) versus the electron density with electron temperature T _e = 3 keV.

The dependence of the maximum gain on the electron temperature with n _e/n _c = 0.1 is exhibited in Figure 5. For both SRS and SBS, the maximum gain reduce noticeably with the electron temperature. Compared with the SBS, the maximum gain of SRS drops much more intensely. Above the temperature of T _e = 2.5 keV, the maximum gain of SRS becomes smaller than the SBS. For the SBS, the Landau damping of the ion acoustic wave rises with the electron temperature, which results in the deduction of the maximum gain. While, for the SRS, the Landau damping of the Langmuir wave is mainly determined by the Debye length of electrons λ_De = υ_te/ω_pe. With higher electron temperature, the Debye length of electrons becomes larger. Thus, the maximum gain of SRS decreases with electron temperature significantly. This indicates that by increasing the electron temperature, one can effectively suppress the gain of both SRS and SBS in ICF.

Fig. 5. Maximum gain of both SRS (red solid) and SBS (blue dashed) versus the electron temperature with n _e/n _c = 0.1.

The spatial intensity distributions of pump laser and the backscattered light of both SRS and SBS are shown in Figure 6. Two different electron density cases are compared with each other: (a) the low-density case (n _e/n _c = 0.05) and (b) the high-density case (n _e/n _c = 0.15). The scattered intensities are integrated by all frequencies of the backscattered light. For the low-density case (see Fig. 6a), the intensity of pump laser (green solid) keeps almost constant all over the whole interaction region. While both SRS (red dashed) and SBS (blue dotted) are driven slightly since the gains of both SRS and SBS are small as shown in Figures 2 and 3. The SBS dominates the scattering process in the low-density case because the maximum SBS gain is nearly twice the value of SRS as mentioned above. For the high-density case (see Fig. 6b), the pump laser is depleted obviously because of the large linear gains of SRS and SBS. Different from the low-density case, the SRS exhibits much higher reflectivity than the SBS due to its higher gain. Furthermore, the gain of SRS is more sensitive to electron density than that of the SBS, and the ratio of SRS intensity to SBS intensity increases drastically with electron density. One should notice that the pump depletion is self-consistently included in our model. Thus, considering the coexistence of the SRS and SBS processes, the denser the plasma is, the more significantly the SBS is suppressed.

Fig. 6. Spatial intensity distributions of pump laser (green solid) and the backscattered light of both SRS (red dashed) and SBS (blue dotted) for two different electron densities: (a) n _e/n _c = 0.05 and (b) n _e/n _c = 0.15. The intensities are normalized by the intensity of pump laser at the left boundary.

Nonlinear saturation and competition between SRS and SBS

Our model has the capability to calculate the reflectivities of both SRS and SBS simultaneously. However, the reflectivity is calculated based on the linear gain exponent by Eq. (11), and nonlinear saturation mechanisms, such as decay instability of daughter waves (Labaune et al., Reference Labaune, Baldis, Bauer, Tikhonchuk and Laval1998; Depierreux et al., Reference Depierreux, Labaune, Fuchs, Pesme, Tikhonchuk and Baldis2002), wave breaking (Forslund et al., Reference Forslund, Kindel and Lindman1975), and nonlinear frequency shift due to particle trapping (Morales and O'Neil, Reference Morales and O'Neil1972; Divol et al., Reference Divol, Berger, Cohen, Williams, Langdon, Lasinski, Froula and Glenzer2003), are not included in the model. A detailed discussion of the nonlinear saturation mechanisms is beyond the scope of this paper. Whereas, neglecting of the nonlinear saturation mechanisms may lead to an overestimate of the final reflectivity. In order to make the model be able to calculate the reflectivity reasonably, a phenomenological method is proposed to include the nonlinear saturation mechanisms in our model. The nonlinear saturation mechanisms mentioned above mitigate the growth of backscattered light by saturating its gain exponent, so that a threshold for the gain exponent G _th is introduced in our model. According to the relevant results of experiments (Lindl et al., Reference Lindl, Amendt, Berger, Glendinning, Glenzer, Haan, Kauffman, Landen and Suter2004), we set G _th = 30, then the gain exponent G is revised as

(18)$$G' = {G G_{{\rm th}}\over \sqrt\lsqb 10\rsqb {G^{10} + G_{{\rm th}}^{10}}}.$$

This revision form guarantees that for small G (G ≤ 10), the revision is negligible, while for large G (G ≥ 30), the gain exponent is restrict to G _th. Traditional theory shows that the backscattered light extracts energy from the pump laser and grows quickly from the noise level up to its maximum (Kruer, Reference Kruer1988; Lindl et al., Reference Lindl, Amendt, Berger, Glendinning, Glenzer, Haan, Kauffman, Landen and Suter2004). Although the pump depletion is self-consistently included in our model, the competition between SRS and SBS should be also considered in the model. Thus, the gain exponents of SRS and SBS are further revised as

(19)$$G_{{\rm SRS}}' = {G_{{\rm SRS}} G_{{\rm th}}\over \sqrt\lsqb 10\rsqb {G_{{\rm SRS}}^{10} + G_{{\rm SBS}}^{10} + G_{{\rm th}}^{10}}}\comma\; $$

(20)$$G_{{\rm SBS}}' = {G_{{\rm SBS}} G_{{\rm th}}\over \sqrt\lsqb 10\rsqb {G_{{\rm SRS}}^{10} + G_{{\rm SBS}}^{10} + G_{{\rm th}}^{10}}}\comma\; $$

respectively. G _SRS and G _SBS denote the initial gain exponents derived from Eq. (10). For small gain exponents, the above revisions are negligible. However, when the gain exponents of SRS and SBS become noticeable, the larger one will suppress the smaller one significantly according to Eqs (19) and (20).

To illustrate the influences of nonlinear saturation and competition effect on the SRS and SBS processes, reflectivities of both SRS and SBS versus normalized electron density are presented in Figure 7. Parameters of the pump laser and the plasma are the same as that used in Figure 4. It is seen that, without the nonlinear saturation effect, the sum of the SRS (red dotted) and SBS (blue dotted-dashed) reflectivities exceeds the intensity of pump laser beyond the intensity of n _e/n _c = 0.13, which is unphysical. Obviously, the nonlinear saturation effect suppresses the gain of the SRS and SBS processes significantly; therefore, the sum of the reflectivities of SRS (red solid) and SBS (blue dashed) are restricted to a reasonable value. Furthermore, without the nonlinear saturation and competition effect, the reflectivities of both SRS and SBS increase with the electron density monotonically. However, considering the nonlinear saturation and competition effect, totally different trend appears. On the one hand, the SRS reflectivity (red solid) tends to be saturated when the electron density becomes higher than n _e/n _c = 0.115. On the other hand, the SBS reflectivity does not continuously grow with the electron density any more. Instead, it keeps increasing with the electron density until n _e/n _c = 0.95, then it starts to decrease with the electron density. Compared with the SRS, the SBS reflectivity is negligible for a high-density case. The typical anti-correlation between SRS and SBS versus electron density observed in experiments (Montgomery et al., Reference Montgomery, Afeyan, Cobble, Fernandez, Wilke, Glenzer, Kirkwood, MacGowan, Moody, Lindman, Munro, Wilde, Rose, Dubois, Bezzerides and Vu1998) is reproduced in our model, which proves the effectiveness of the phenomenological method utilized in our model.

Fig. 7. Reflectivities of both SRS (red) and SBS (blue) versus normalized electron density with (solid, dashed) and without (dotted, dotted-dashed) nonlinear saturation and competition effect.

In order to investigate the efficiency and accuracy of our model, two typical experimental results from the OMEGA laser facility (Soures et al., Reference Soures, McCrory, Verdon, Babushkin, Bahr, Boehly, Boni, Bradley, Brown and Craxton1996) are chosen to be compared with our model for SRS and SBS, respectively. Figure 8 presents the theoretical SRS reflectivity as a function of the gain exponent by our model, compared with the experimental and simulation results from Froula et al. (Reference Froula, Divol, London, Berger, Doppner, Meezan, Ralph, Ross, Suter and Glenzer2010). In the experiments, the electron density is scaled in the 2-mm-long target platform from 0.11 to 0.13n _c, maintaining the electron temperature above 2.5 keV. The reflectivity is calculated after the rise of the heating laser beams (1.2 ns, 1 × 10¹⁵ W/cm², 351 nm) by averaging over a 100 ps range. Hydrodynamic simulations by using the pf3d code (Berger et al., Reference Berger, Still, Williams and Langdon1998) are also performed. Generally, our calculation is in agreement with the results of the experiments and simulations, which indicates that our model can qualitatively reveal the SRS reflectivity. However, the overestimate of the reflectivity for the high gains reflects the lack of the multidimensionality of the real instability process in our model.

Fig. 8. Theoretical SRS reflectivity as a function of the gain exponent, compared with the experimental and simulation results from Froula et al. (Reference Froula, Divol, London, Berger, Doppner, Meezan, Ralph, Ross, Suter and Glenzer2010) for different electron densitites.

Comparison of our model calculation with the experimental results by Berger et al. (Reference Berger, Suter, Divol, London, Chapman, Froula, Meezan, Neumayer and Glenzer2015) for the SBS is exhibited in Figure 9. In the experiments, the laser intensity, approximately constant with a 1ns pulse length, is varied between 2.2 × 10¹⁴ and 8.8 × 10¹⁴ W/cm². The 2-mm-long hohlraum is filled with 1 atm of CO₂, which yields an electron density of n _e/n_c = 0.06. The electron temperature reaches the peak value of 3.5 keV. The ion Landau damping rate is varied by adding hydrogen to the CO₂ hohlraum gas fill. As shown in Figure 9, the symbols with different colors represent the SBS reflectivities measured for different Landau damping rates of the ion acoustic wave. The black solid line is the calculation result of our model. Obviously, the experimental results are qualitatively reproduced by our model. Besides, the result of experiments shows that increasing ion Landau damping can strongly suppress the SBS. However, due to the strong damping limit of our model, deviation from the experimental results exists in the small damping cases, which means that our model is not valid under the small damping condition.

Fig. 9. Theoretical SBS reflectivity as a function of the gain exponent, compared with the experimental data from Berger et al. (Reference Berger, Suter, Divol, London, Chapman, Froula, Meezan, Neumayer and Glenzer2015) for various Landau damping rates of ion acoustic wave (υ/ω).