Pan, Y.L.

Shay, H.D.

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1 his is a preprint of a paper intended for publication in .» journal or proceedings. Since changes may be made bWurr publication, this preprint is made available with tin- imriVrtit^nihn^that it will not be citeri or reproduced without the permission of the author. UCRL - 75882 PREPRINT LAWRENCE UVERMORE LABORATORY Jntversity of Ca'ifornia/Livermore. California LASER-PLASMA CALCULATIONS WITH REFRACTION Yu-Li Pan and Henry D. Shay October 28, 1974 - NOTICE-This leport was prepared as an account of woik sponsored by the United States Government. Neither the United States nor the United States Energy Research and Development Administration, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal EabUity or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disposed, or represents that its use would not infringe privately owned rights. A»W This Paper Was Prepared for the 16th Annual Meeting of the Division of Plasma Physics of the American Physical Society October 28-31, 1974 Albuquerque, New Mexico Laser-Plasma Calculat ions w i th Refract ion Yu-Li Pan and Henry D. Shay Univers i ty of Ca l i f o rn ia Lawrence Livermore Laboratory Livermore, Ca l i f o rn ia The de ta i l s of the ray t rac ing LASNEX has j u s t been presented. Before using t h i s code in actual l ase r - ta rge t s imulat ions ca l cu la t i ons , we made a number of checks to understand be t te r the charac te r i s t i cs of the code. Some of these tes t problems have ana ly t i c so lu t i ons . There­f o r e , the performance of the code ca lcu la t ion can be r igorous ly checked. We discuss here several of these comparison problems, most o f which demonstrate the a b i l i t y of the code to pred ic t c o r r e c t l y ray t r a j e c t i o n . The f i r s t o f these t e s t problems involves a l i g h t ray enter ing a medium w i t h a l i n e a r densi ty gradient . The ana ly t i c so lu t ion f o r 2 t h i s problem has been given by Shearer . The f i r s t f i g u r e (F ig . 1) shows t h i s r e s u l t . Figure 2 shows the comparison between the ana ly t i c so lu t ion (dot ted curve) and the code ca lcu la t ions . The second problem involves l i g h t propagation in a medium w i th a quadrat ic densi ty gradient transverse to the d i r ec t i on of propagation. The ana ly t i c so lu t ion f o r t h i s problem i s presented on the next f i g u r e ( F i g . 3) . Note the paraxia l ray approximation. The value of A has 2 2 been chosen as H /2500 (1/micron ) so tha t the maximum and minimum of •Research performed under the auspices of the U. S. Atomic Energy ••Commission. 'G. B. Zimmerman, LLL Rpt. UCRL-74811, Oct. 1973. J . W. Shearer, Phys. of F lu ids H 183, 1971. - 2 -the sinusoidal oscillation will appear every 25 micron. Further, to make the comparison simpler, we check the solution one term at a time. That is, we set R , and R ' to zero in sequence. We will discuss our results for positive A first. Fiqure 4 shows the quadratic gradient used in these calculations. The next fiaure (Fig. 5) shows the paths of the rays calculated by the code. The rays enter the problem from the 150 micron side with R ' equal to zero. The box on the figure shows the zone size used in the calculation, ( 1 x 2 micron). Next, we set R equal to zero and let R ' be 5, 10 and 15 degrees. The paths of the rays calculated by the code are shown on the next figure (Fig. 6). Notice that the ray which is entering at 5 degrees disagrees badly with the analytic solution. This is espe­cially disturbing because this ray is closer to the axis than the other ray in the problem. The reason for this disagreement is due to the zoniny. We recall that the slope of the density is flat at the center. Thus, the change in the slope is maximum at this point. In addition, we note that the 5-degree ray traverses only about 1-1/2 zones in the R direction. (Can also point out again the way the density is calculated in the code.) The next figure (Fig. 7) shows the code calculation for different zone sizes. The + curve is the same one shown in the preceeding figure. For negative values of A, the ray should diverge. Fiqure 8 shows the result of this calculation. The effect of zone size is again obvious. - 3 -A third test problem is the propagation of light ray inside a spherical plasma with a density profile which increases with r. Ginzbui-g"* presents an analytical description of the trajectory of such a ray—as seen on Fig. 9. LASNEX predicts the correct re­flection point for a ray initially at 45° to a radius vector. In addition to verifying that trajectories are properly com­puted, test problems also show that the spatial variation of intensity and inverse bremsstrahlung absorption rate are, as well, correctly predicted. One important caveat must be imposed on the use of geometrical ray optics: the neglect of diffraction can, in certain cases, be a severe oversight, especially in cases of thermal self-focusing. As discussed by Perkins and Valeo diffraction dominates in those cases 2 where D = k, L/2k is much greater than one. In such cases involving focusing to small diameter over long distances, geometrically optics predicts unphys 1 Co 1 focusing. With the proper attention to the regime of application and the appropriate zoning of the Lagrangian mesh, this ray tracing scheme is quite accurate. 3 V. L. Ginzburg, "The Propagation of Electromagnetic Waves in Plasmas", (Addison-Wesley, Reading, Mass., 1964), p. 378. 4 F. W. Perkins and E. J. Valeo, Phys. Rev. Letters, 3£, (1974), 1234. I Vacuum u Solid ISL = m N cos 20 n c 0 13 S3 m 1.0 0.5 -^ T X \ 1 1 1 1 1 1 Hydrogen T = 10 keV linear gradient \ c o s 2 0 O \ \ \ \ \ \ J I L J I I L 0 20 40 60 80 0 o deg Physical cases: (A) Propagation of Gaussian-like intensity profile laser beam in a slightly absorbing medium. (B) Absorption of pump light in solid laser rods. (C) Dielectric and optical waveguides; light pipes. N(R) - N 0 M - ^ AR 2 For paraxial rays: D 2R + A R = 0 DZ 2 For positive A: R(Z) = R 0 cos A 1 / 2 Z + R; A"1 / 2sin A 1 / 2 Z For negative A: R(Z) = R 0 cosh A 1 / 2 Z + R^A^sinh A 1 / 2 Z U 73 m in 20 R, micron L5 c TO rrl C • o v. U 1 0 - 1 - 2 - 3 - 4 —5 - • 25 50 Hydrogen T = 10 keV n » « • . . . • • 75 Z, micron 100 125 150 L5 cz 3D 4 3 2 c 1 s u E 0 DC - 1 - 2 - 3 - 4 " "~ I I I • *. Hydrogen T= 10keV • . • • • • . . L • • - • • • • • • : • • • • • • • 1 1 • • • . • • — • • _ • • • • -* • • • * • * • • • • • • • • - • . • • -• • • • • • . • • » • • • • • • -. . -1 1 1 -L_ I 0 25 50 75 Z, micron 100 125 150 L2 - + + + Hydrogen T = 10 keV . + • v • a T | O £7) k. a u 73 m E 03 «, DC 1 E 0 1 -+ + + • . • +. + + • + + + + • + +' X 25 50 75 Z, micron 100 125 150 20 1 1 T7 I — = — 1 .u c o | 10 A / / , / RQ coshv'AZ X / / / S' +•• ' ' I I I 1 I I L 10 Z, micron 20 TEST OF RAY TRACE IN SPHERICAL PLASMA L9 n(r) r sin0(r) = constant Expect at turning point n(r)r = n(r Q)r 0 sin0(rQ) Ray trajectory Electron density profile 500 400 300 200 R, jum 100 100 200 300 400 500 R.jum 

Laser--plasma calculations with refraction

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Laser--plasma calculations with refraction

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