1. Laser package for CRASH Igor Sokolov, Ben Torralva and CRASH team

2. Page 2 Outline Solar physics background Statement of the problem, main parameters Ray tracing and energy deposition Test Results

3. Page 3 In application to solar physics the synthetic radio-image is the origin point for a ray. The rays refract and reflect while approaching the critical density. Emissivity should be integrated over the ray, to obtain the intensity in the given pixel. Quite analogous problems should be solved by the laser package, to find the laser beam attenuation and the energy deposition. Thus, to solve the CRASH- related problem we re-use the tools available in the SWMF

4. Page 4 Geometry of a target for OMEGA. Wavelength: 0.353 μm Beam radius: 438 μm. Geometrical optics. Strong refraction. Beam diameter Density profile in The laser corona Ray trajectory

5. Page 5 The concept of ray. Ray is directed along the gradient of eikonal Ray curvature is the transverse relative gradient of the refraction index. Density profile in The laser corona Ray trajectory

6. Page 6 Ray Tracing. The number of rays should be close to the number of cells intersected by the “critical density surface”. The simplest choice is a single ray per each cell per beam at the left boundary of the computational domain, x=-x0. At each time step for each of the rays we trace it by solving numerically the following equation: The refraction index is to be calculated from the electron density distribution.

7. Page 7 Laser Energy Deposition The absorption is calculated as inverse Bremsstrahlung, with an effective imaginary frequency component The energy deposition is peaked at high density and also sparse. Therefore: At each time step the laser energy deposition is added to right-hand- side of the electron heat conduction equation to be solved implicitly. The implementation of the delta-function: distribute energy between the nearest cells with the total of the interpolation coefficients to be equal to one.

8. Test result Test problem using axially directed beams pointed at different locations (thus, not realistic for CRASH)

9. Page 9 Test run parameters: Laser energy: 3.8 kJ Pulse duration: 1.1 ns = 0.1 ns raise time + 0.9 ns constant power + 0.1 ns decay time Hydro: multi-group radiation diffusion, RZ geometry. 3 beams with circular-ring cross-section: #BEAM:10.0 Slope 0.0 yCr 1.0 Amplitude #BEAM: 20.0 Slope 200.0 yCr 0.7 Amplitude #BEAM: 30.0 Slope 400.0 yCr 0.7 Amplitude yCr R Z Slope

10. Page 10 Typical ray geometry (0.2 ns) Density profile in the laser corona: Z-dependence for two somewhat different R Ray trajectory Interface between Be and “vacuum” 350 km/s*0.2 ns incident 0.55 n cr

11. Page 11 “Challenging” ray geometry (0.2 ns) Ray trajectory incident Density profile in the laser corona: Z-dependence for two somewhat different R 0.55 n cr

12. Page 12 All Rays (Color Shows the Plasma Density)

13. Supplementary materials

14. Page 14 Ray Equations for Isotropic Media The equation for the ray trajectory is derived from Fermat’s principle − a ray joining two points P 1 and P 2 will choose the path which minimizes the integral of the refractive index. Therefore, the variation of the integral about this path is zero: (1) The integral (1) is transformed into a Lagrangian form by exchanging s for a time-like independent variable τ. If, then : (2)

15. Page 15 Ray Equations for Isotropic Media (continued) Substitution of (2) into (1) gives,, with We apply the Euler method to the Lagrangian to find the vector differential equation of the curve satisfying Fermat’s principal. Upon multiplying by dτ/ds, the result is:. With the use of the vector identities the equation takes the form:. (3)

16. Page 16 Ray Equations for Isotropic Media (continued) Utilizing the direction vector, v = dr/ds, the system (3) is transformed into the set of six first-order differential equations (4) Note that the independent variable, s, is the ray arc length so the solution to (4) is a naturally parameterized curve. This infers the “conservation law”

17. Page 17 Ray Tracing. The number of rays should be close to the number of cells intersected by the “critical density surface”. The simplest choice is a single ray per each cell per beam at the left boundary of the computational domain, x=-x0. At each time step for each of the ray we trace it by solving numerically the following equation: The refraction index is to be calculated from the electron density distribution (in the same way as we do this while calculating synthetic radio-images of the Sun).

18. Page 18 We already have the ray-tracing algorithm ready-to-use with the laser package. Why? Stream line and field line tracing is a common problem in space physics. Tracing a line is an inherently serial procedure. One needs to solve the equation: –Vector field is distributed over many processors –Collecting the vector field onto one one processor may be too slow and it requires a lot of memory. –To overcome these problems, the task to trace the ray trajectory is delegated to the coupling toolkit of the SWMF (an alternative way is implemented in BATSRUS – see the supplementary materials)

19. Page 19 Laser Absorption in Collisional Plasmas Electron-ion collisions modify the laser-plasma interaction causing laser energy to be deposited into the plasma. The dielectric permittivity becomes: Where the effective electron-ion collision frequency is

20. Page 20 Energy Tracing We consider the energy emitted by the laser within the time step, as being distributed over the rays in accordance with the beam cross-section areas related to each of the rays and local intensity distribution over the focal spot. While propagating along the ray, the energy is gradually absorbed. The absorption over the arc length interval, ds, gives us the local point-wise energy deposition, E ijk. The algorithm to find the coordinates of the point sets along the rays and the local energy deposition is thus described. The advantage in this algorithm is the exact energy conservation: the total of the deposited energy is equal to the total of the absorbed laser energy.

21. Page 21 Laser Energy Deposition Physically, the laser energy deposition is peaked near the “critical density” surface. For “blue light” (the third harmonic of the Nd-glass laser) the “critical electron density” is 10 22 cm -3 For the chosen algorithm (ray-tracing for finite number of discrete rays) the energy deposition is not only peaked, but it also sparse. Therefore: At each time step while advancing the system of the motion equation for the laser-produced plasma the laser energy deposition (in Joules) is added to right-hand-side of the electron heat conduction equation to be solved implicitly. The implementation of the delta-function: distribute energy between the nearest cells with the total of the interpolation coefficients to be equal to one.

22. Page 22 Ray Refraction in Isotropic Plasmas The electromagnetic ray trajectory is treated as a curve with position r in three-dimensional space for any starting point r 0 and initial direction v 0. The ray trajectory may be unambiguously determined as long as the index of refraction, n(r,ω), is known for a given laser frequency ω. For the natural parameterization of the curve, r(s), where s is its arc length, the direction vector v(s) is defined as the first derivative of r, v(s) = dr/ds. The components of v(s) are the direction cosines, and |v(s)| = 1 at any point of the curve

23. Page 23 Ray Refraction in Isotropic Plasmas (Continued) The index of refraction n(r,ω) is the square root of the dielectric permittivity,, and is a function of the plasma density distribution, ρ(r) = m p n e. n e is the electron number density. For isotropic collisionless plasmas,, where is the plasma frequency. Therefore,, where. The relative gradient of the refractive index is:

24. Page 24 Parallel Algorithm without Interpolation (Gabor Toth). PE 1 PE 3 PE 2 PE 4 Stages in the algorithm.

25. Page 25 Laser Absorption in Collisional Plasmas (cont.) The absorption coefficient can then be found to be: The laser electric field will then propagate through the plasma as: Where the complex wave vector is, and

26. Page 26 Algorithm Description & Numerical Implementation The integration of the system of equations (4) is carried out using the Boris (1970) CYLRAD algorithm which guarantees. The algorithm is implemented with an adaptive step size mechanism to efficiently handle the large changes in the plasma density. Over a single integration step each ray is tested for precision and correctness, and to ensure that rays do not penetrate the region with ρ > ρ cr.

27. Page 27 Flexibility: we may vary pulse shape and beams #. For UQ studies we can switch off any beam. #LASERPULSE T UseLaserPackage 3.8e12 IrradianceSI 1.1e-9 tPulse 1.0e-10 tRaise 1.0e-10 tDecay #BEAM 10.0 SlopeDeg 0.0 yCrMuM 1.0 AmplitudeRel #BEAM …

28. Page 28 x / y/ t = 190T Our experience in LP Interactions x / P x,e /m e c Rad. losses: 43% Electrons: 5.7% x / P x,e /m e c w/o rad. losses t = 100T N. Naumova, et al. PRL 102, 025002 (2009) a 0 = 100 circ, n 0 = 10n c, m i = 2m p a 0 = 100 circ, n 0 = [5-100]n c, m i = 2m p PIC simulations of the laser-plasma interaction: Intensities ; Account for radiation back-reaction: (Sokolov et al, JETP 2009; PoP 2009; Phys. Rev. E 2010, PRL 2010). PIC may be embedded to the hydro code via the framework (SWMF). This is NOT what we are doing or planning to do for CRASH T. Schlegel, et al. PoP 16, 083103 (2009)