Accounting for speckle-scale beam bending in classical ray tracing schemes for propagating realistic pulses in indirect drive ignition conditions

Ruyer C.; Loiseau P.; Riazuelo G.; Riquier R.; Debayle A.; Masson-Laborde P. E.; Morice O.

doi:10.1063/5.0124360

Volume 8 Issue 2

Mar. 2023

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Ruyer C., Loiseau P., Riazuelo G., Riquier R., Debayle A., Masson-Laborde P. E., Morice O.. Accounting for speckle-scale beam bending in classical ray tracing schemes for propagating realistic pulses in indirect drive ignition conditions[J]. Matter and Radiation at Extremes, 2023, 8(2): 025901. doi: 10.1063/5.0124360

Citation:

Ruyer C., Loiseau P., Riazuelo G., Riquier R., Debayle A., Masson-Laborde P. E., Morice O.. Accounting for speckle-scale beam bending in classical ray tracing schemes for propagating realistic pulses in indirect drive ignition conditions[J]. Matter and Radiation at Extremes, 2023, 8(2): 025901. doi: 10.1063/5.0124360

Citation:

Ruyer C., Loiseau P., Riazuelo G., Riquier R., Debayle A., Masson-Laborde P. E., Morice O.. Accounting for speckle-scale beam bending in classical ray tracing schemes for propagating realistic pulses in indirect drive ignition conditions[J]. Matter and Radiation at Extremes, 2023, 8(2): 025901. doi: 10.1063/5.0124360

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Accounting for speckle-scale beam bending in classical ray tracing schemes for propagating realistic pulses in indirect drive ignition conditions

doi: 10.1063/5.0124360

Ruyer C.^{1, 2
,
,
,},
Loiseau P.^{1, 2
,},
Riazuelo G.^{1, 2},
Riquier R.^1
,,
Debayle A.^{1, 2
,},
Masson-Laborde P. E.^{1, 2
,},
Morice O.¹

1.
CEA, DAM, DIF, F-91297 Arpajon, France
2.
CEA, LMCE, Université Paris-Saclay, 91680 Bruyères-le-Châtel, France

More Information

Corresponding author: a)Author to whom correspondence should be addressed: charles.ruyer@cea.fr
Received Date: 2022-09-05
Accepted Date: 2022-12-24

Available Online: 2023-03-01

Publish Date: 2023-03-01

Abstract

Abstract

We propose a semi-analytical modeling of smoothed laser beam deviation induced by plasma flows. Based on a Gaussian description of speckles, the model includes spatial, temporal, and polarization smoothing techniques, through fits coming from hydrodynamic simulations with a paraxial description of electromagnetic waves. This beam bending model is then incorporated into a ray tracing algorithm and carefully validated. When applied as a post-process to the propagation of the inner cone in a full-scale simulation of a National Ignition Facility (NIF) experiment, the beam bending along the path of the laser affects the refraction conditions inside the hohlraum and the energy deposition, and could explain some anomalous refraction measurements, namely, the so-called glint observed in some NIF experiments.

Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
C. Ruyer: Conceptualization (lead); Investigation (lead); Methodology (lead); Software (equal); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal). P. Loiseau: Conceptualization (supporting); Methodology (supporting); Software (equal); Writing – review & editing (equal). G. Riazuelo: Software (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal). R. Riquier: Software (equal); Writing – original draft (supporting); Writing – review & editing (supporting). A. Debayle: Conceptualization (supporting); Methodology (supporting); Writing – review & editing (equal). P. E. Masson-Laborde: Conceptualization (supporting); Methodology (supporting); Writing – review & editing (supporting). O. Morice: Writing – review & editing (supporting).
The data that support the findings of this study are available from the corresponding author upon reasonable request.
To validate the predictions of Sec. II C, five 3D Hera simulations of Gaussian beams have been performed with a wavelength of λ₀ = 0.35 μm, a maximum intensity of I₀ = 10¹⁵ W/cm² and an f-number f_♯ = 8. The homogeneous fully ionized carbon plasma at 10% of the critical density has a drift velocity along the y direction at a Mach number M₀ = 0.4, 0.8, 1, 1.2, or 1.5, with (T_e, T_i) = (2.5, 1) keV. The size of the simulation domain is L_x × L_y × L_z = 100 µm × (40 µm)² and is composed of 50 × 512² meshes. The laser is injected from the left boundary at x = 0, and its focal spot is located at (x_foc, y_foc, z_foc) = (50 µm, 0, 0). The laser has a constant temporal profile preceded by a 1 ps-long linear rise. Additionally, the Landau damping operator is calculated in the Fourier space transverse to the main laser direction,^50,68,69 and the thermal correction of Eq. (10) (while setting Ω = 1) is accounted for.
The resulting intensity profiles at t = 10 ps, illustrated in Fig. 6, reveal a deviation of the beam toward the flow (y) direction. The theoretical predictions of Sec. II C, which we aim at validating here, gather all the x dependence of the bending rate into the fitting factor S. Hence, a comparison of the 3D Hera predictions with Eq. (19) requires that we remove from the simulation results the influence of the x-Gaussian laser profile. We thus start by isolating the x-dependent factors (waist and intensity), leading in 3D todθdx≃d2⁡ydx2=1[1+(x−xfoc)2/zc2]3/2dθ0dx,where zc=πf♯2λ0, and dθ₀/dx is the part of the beam bending rate that is independent of x, thus corresponding to Eq. (16). Hence, for dθ/dx(x = 0) = y(x = 0) = 0, the deviation is related to dθ₀/dx throughy≃1[1+(Lx−xfoc)2/zc2]1/2Lx22dθ0dx.The value S = 1, imposed in Sec. II C, corresponds to a deviation that reads y0=0.5Lx2dθ0/dx. In summary, to extract the relevant time-averaged centroid deviation from our 3D Hera simulation, we first compute the centroid deviation ⟨y⟩_⊥ at the exit plane [using Eq. (2)], divide it by the factor [1+(Lx−xfoc)2/zc2]−1/2≃0.81, and average the results between t₀ and t₀ + 7 ps. We use t₀ = 1.3 ps here to account for the time required for the laser to reach the exit plane of the simulation (∼0.3 ps) and the 1 ps linear rise of the laser time envelope.
Equation (B2) has been derived using the small-amplitude density and temperature hypothesis, namely, dn_e/n_e ≪ 1 and dT_e/T_e ≪ 1, and the quasistationary temperature hypothesis, namely, dt(T_e) < ν_eiI₀/cn_c. In our simulations, the laser propagates into a 2 mm helium plasma that is assumed to be uniform with density n_e(t = 0) = 0.1n_c, T_e = 2 keV, and T_i = 0.5 keV. The Landau damping rate has been determined by a kinetic dispersion solver, and its value is γ₀ = 0.031. The laser’s average intensity is ⟨I₀⟩ = 2 × 10¹⁴ W/cm². This relatively small value has been chosen to limit the angular spread of the beam. The transverse and longitudinal mesh sizes are dy = dz = 0.75λ₀ (for λ₀.35 µm) and dx = 1.6 µm, respectively.
In the case of PS smoothing, the code models the propagation of two independent electromagnetic waves (the first polarized along the y axis and the second along the z axis), but the source term in Eq. (B2) is computed with the contributions of both electromagnetic waves.
The Parax code simulates the propagation of electromagnetic waves in a plasma.⁸⁸ The propagation of one fixed polarization electromagnetic wave is modeled by a single generalized scalar paraxial equation (B1) for the electric field amplitude E and a wave equation (B2) for the plasma response in the perpendicular (y, z) plane:2iω0c2∂t+2ik0η∂x+i∂xk0η+∇⊥2−ω02c2ne−ne0nc−νeiω0ne0c2ncE=0.Here, ν_ei is the electron–ion collision frequency. Stimulated Raman and Brillouin backscattering do not occur, owing to the presence of a single paraxial incident wave. However, both filamentation and forward stimulated Brillouin scattering (FSBS) can grow and interact. The plasma density is modeled using a fluid description, where expansion to second order in the field perturbation leads to an ion-acoustic wave driven by ponderomotive and thermal effects:⁸⁸,⁸⁹(∂t+ν+vy∂y)2−cs2∇⊥2lognene0=Zicminc∇⊥2(AtI).Here, v_y represents a transverse plasma drift assumed to be along the y axis, and ν is the ion-acoustic wave damping rate. The logarithm saturates the density response and thus impedes the blowup of the self-focusing process that would otherwise be induced by the cubically nonlinear Schrödinger equation derived from Eq. (B1). Equation (B2) uses the acoustic type of plasma response and accounts for the plasma heating using a nonlocal electron transport model according to Ref. 59. The operator A_t in the source term is applied to the laser intensity; it accounts for the inverse bremsstrahlung heating, the ponderomotive effect, and the nonlocal transport. Its spectrum in Fourier space for the transverse spatial coordinates (y, z) uses the fit introduced in Ref. 59.
Troll is a 3D arbitrary Lagrangian–Eulerian (ALE) radiative hydrodynamic code with unstructured mesh.³⁸ The simulation presented in Sec. IV B was performed in an axisymmetric 2D geometry. We used the latest CEA tabulated equations for state and opacities, with a model for NLTE correction of the emissivity. The radiation transport was solved using an implicit Monte Carlo method and the heat flux using the Spitzer–Härm model, limited to 10% of the freestreaming flux. The lasers were simulated using the classical 3D ray tracing method (without the beam bending model), with inverse bremsstrahlung absorption, corrected to account for the Langdon effect. Both Raman and Brillouin backscattering were measured to be just a few percent, and therefore were not taken into account. No multipliers were used on the incident laser power.
In order to facilitate the implementation of the present beam bending model in a hydrodynamic code, we introduce here a fit of the function ⟨G⟩τSSD [Eq. (23)] that appears in the ray tracing model of Eq. (24). We propose to use the following function defined in the domain M₀ > 0, with k_c = 2^1/2/f_♯λ₀, G3=Akc2aM0+bM021+(M0/c)4where a, b and c are polynomial functions of τ_SSDc_s/f_♯λ₀ and γ₀. They follow,a=∑n=03∑m=04an,mγ0nτSSDcsf♯λ0m,b=∑n=03∑m=04bn,mγ0nτSSDcsf♯λ0m,c=∑n=03∑m=04cn,mγ0nτSSDcsf♯λ0m.The value of the coefficients a_n,m, b_n,m, c_n,m are given in Table I. We retain the predictions of the above fit when G3(M0)>0 and set it to zero elsewhere. We also use G3(−M0)=−G3(M0) in the negative Mach number domain. As shown in Fig. 7, this fit works fairly correctly in the domain 0.007 ≤ γ₀ ≤ 0.05, |M₀| < 2.53 and 0.1 ≤ τ_SSDc_s/f_♯λ₀ ≤ 2 in a mono-species plasma, as shown in Figs. 7(a)–7(d).

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