New laser energy deposition algorithm for the RALEF-2D code∗

S. Faik1, An. Tauschwitz1, J. A. Maruhn1, and M. M. Basko2

1Goethe University, Frankfurt am Main, Germany;2KIAM, Moscow, Russia

Correct modeling of the laser beam evolution and powerdeposition on unstructured grids is a computationally andnumerically challenging task. The first goal for a new al-gorithm for the radiation-hydrodynamics code RALEF-2D[1] is the calculation of the refracted laser light distribu-tion with minimal numerical diffusion. Therefore a longcharacteristics approach is applied. The incoming laserbeam intensity is spatially discretized into single rays be-ing traced through the computational grid. On the basisof the eikonal equation in geometrical optics [2] the equa-tion of motion of a ray [3] in the undercritical regime,ne ≪ nc, becomesd2~x/dt2 = −(c2/2)(~∇ne/nc). Con-tinuous transitions of the ray trajectories up to the first spa-tial derivative between the numerical cells are guaranteedby the division of the original two-dimensional quadrilat-eral grid into triangles. Within each triangle the free elec-tron densityne(~x) then is uniquely defined piecewise linearby the vertex-centered valuesne and the gradients~∇ne. In-side each triangle a ray with incoming powerP0 depositsthe powerPdep = P0 (1− e−κ) given the optical depthκ = ∆s Im (σ). Here∆s is the length of the ray segment,∆s =

∫ ∆t

0 c√

1− ne [~x (t)] /nc dt, ∆t the correspond-ing transition time, andσ the complex refractive index (perunit length) of the associated quadrilateral cell calculatedby Kramers’ inverse bremsstrahlung formula.

Figure 1 shows the mesh setup and two simulated ray tra-jectories for a quadratic density trough test case [3]. Withthe density profilene(y) = (2nc/l2y)(y2− lyy + l2y/2), themesh heigthly = 100 mm, the temperature dependenceT (y) ∝ (ne(y)/nc)2/3, and a fixed Coloumb logarithmand ionization, analytical solutions for the cosine-shapedtrajectories and their optical depth exist. The dimensionlesstrajectory errorǫT at the mesh exit point and the relative er-ror ǫP in the summed-up deposited powers as functions ofthe number of quadrilateral cells in they-directionNy fora quarter cosine-wavelength are shown in Figure 2.

The second goal for the new algorithm is the calculationof the deposited and reflected powers and the angular dis-tribution of the reflected laser light close to and above thecritical free electron density (ne & nc), e.g. at the sur-face of a solid metal. Therefore the raytracing algorithm isaugmented by an one-dimensional wave optics solver forthe wave propagation and energy deposition in a stratifiedmedium [2]. A geometrical optics ray can split up into an”overcritical” wave optics ray propagating perpendiculartothe critical surface and a reflected geometrical optics ray.Further test simulations will be conducted soon.

∗Work supported by BMBF (Project 05P12RFFTR), HIC for FAIR

Figure 1: Mesh setup and two simulated ray trajectories fora quadratic density trough test case [3] with mesh heigthly = 100 mm. Trajectory entry points:ya = 20, 80 mm;analytical trajectory exit point:yan

e = 50 mm.

Figure 2: Dimensionless trajectory errorǫT at the meshexit point and relative errorǫP in the summed-up depositedpowers as functions of the number of quadrilateral cells inthey-directionNy for a quarter cosine-wavelength.

References

[1] M. M. Basko et al., GSI Report 2010-1 410.

[2] M. Born, E. Wolf, ”Principles of Optics”, 7th Ed., 2005.

[3] T. B. Kaiser, Phys. Rev. E 61 (2000) 895.

GSI SCIENTIFIC REPORT 2014 APPA-MML-PP-13

DOI:10.15120/GR-2015-1-APPA-MML-PP-13 287