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Energy content of target and electron flow in femtosecond laser target interactions

Published online by Cambridge University Press:  08 April 2015

Eran Nardi*
Faculty of Physics Weizmann Institute of Science, Rehovot, Israel
Zeev Zinamon
Faculty of Physics Weizmann Institute of Science, Rehovot, Israel
Yitzhak Maron
Faculty of Physics Weizmann Institute of Science, Rehovot, Israel
Address correspondence and reprint requests to: Eran Nardi, Faculty of Physics Weizmann Institute of Science, Rehovot, Israel. E-mail:


The heating of the titanium foil in a recent femtosecond laser plasma experiment is investigated theoretically in two different ways. In the first, the energy content and thus the heating efficiency of the central volume of the foil is derived by integrating the transverse temperature profiles obtained in this experiment, using specific heats based on the average atom model. In the second approach target heating by the fast electrons, both by direct energy deposition and by resistive heating is investigated. The latter approach makes use of a specially devised electron flow model which includes a simplified quantitative treatment of multi-refluxing as a crucial component. In all, the calculated results of electron beam heating are consistent with experiment within the limitations of the modeling. Finally, a prediction for the temporal dependence of the pulse from the central volume of the foil based on our electron flow model is given.

Research Article
Copyright © Cambridge University Press 2015 

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Angulo Gareta, J.J. & Riley, D. (2006). Prospects for the diagnosis of electron–ion temperature equilibration rates of warm dense matter by ultra-short pulse hard X-ray diffraction with an X-ray free electron laser. High Energy Density Phys. 2, 8389.CrossRefGoogle Scholar
Berger, M.J., Coursey, J.S., Zucker, M.A. & Chang, J. (2005). “Stopping power and range tables for electrons protons and Helium ions”. Scholar
Bell, A.R. & Kingham, R.J. (2003). Resistive collimation of electron beams in laser-produced plasmas. Phys. Rev. Lett. 91, 035003.CrossRefGoogle ScholarPubMed
Chen, C.D., Patel, P.K., Hey, D.S., Mackinnon, A.J., Key, M.H., Akli, K.U., Bartal, T., Beg, F.N., Chawla, S., Chen, H., Freeman, R.R., Higginson, D.P., Link, A., Ma, Y.T., MacPhee, A.G., Stephens, R.B., Van Woerkom, L.D., Westover, B. & Porkolab, M. (2009). Bremsstrahlung and Kα fluorescence measurements for inferring conversion efficiencies into fast ignition relevant hot electrons. Phys. Plasmas 16, 082705.CrossRefGoogle Scholar
Davies, J.R. (2003). Electric and magnetic field generation and target heating by laser-generated fast electrons. Phys. Rev. E 68, 056404.CrossRefGoogle ScholarPubMed
Fill, E.E. (2011). Relativistic electron beams in conducting solids and dense plasmas: approximate analytical theory. Phys. Plasmas 8, 14411444.CrossRefGoogle Scholar
Green, J.S., Ovchinnikov, V.M., Evans, R.G., Akli, K.U., Azechi, H., Beg, F.N., Bellei, C., Freeman, R.R., Habara, H., Heathcote, R., Key, M.H., King, J.A., Lancaster, K.L., Lopes, N.C., Ma, T., MacKinnon, A.J., Markey, K., McPhee, A., Najmudin, Z., Nilson, P., Onofrei, R., Stephens, R., Takeda, K., Tanaka, K.A., Theobald, W., Tanimoto, T., Waugh, J., VanWoerkom, L., Woolsey, N.C., Zepf, M., Davies, J.R. & Norreys, P.A. (2008). Effect of laser intensity on fast-electron-beam divergence in solid-density plasmas. Phys. Rev. Lett. 100, 015003.CrossRefGoogle ScholarPubMed
Gremillet, L., Bonnaud, G. & Amiranoff, F. (2002). Filamented transport of laser-generated relativistic electrons penetrating a solid target. Phys. Plasmas 9, 941948.CrossRefGoogle Scholar
Hatchett, S.P., Brown, C.G., Cowan, T., Henry, E.A., Johnson, J.S., Key, M.H., Koch, J.A., Langdon, A.B., Lasinski, B.F., Lee, W., Mackinnon, A.J., Pennington, D.M., Perry, M.D., Phillips, T.W., Roth, M., Sangster, T.C., Singh, M.S., Snavely, R.A., Stoyer, M.A., Wilks, S.C. & Yashuika, K. (2000). Electron, photon, and ion beams from the relativistic interaction of Petawatt laser pulses with solid targets. Phys. Plasmas 7, 2076.CrossRefGoogle Scholar
Honrubia, J.J., Antonicci, A.A. & Moreno, D. (2004). Hybrid simulations of fast electron transport in conducting media. Laser Part. Beams 22, 125135.CrossRefGoogle Scholar
Latter, R. (1955). Temperature behavior of the Thomas–Fermi statistical model for atoms. Phys. Rev. 99, 18541870.CrossRefGoogle Scholar
Langhoff, H., Bowes, B.T., Downer, M.C., Hou, B. & Nees, J.A. (2009). Surface energy transport following relativistic laser-solid interaction. Phy. Plasmas 16, 072702.CrossRefGoogle Scholar
Lee, Y.T. & More, R.M. (1983). An electron conductivity model for dense plasma. Phys. Fluids 27, 12731286.CrossRefGoogle Scholar
Liberman, D.A. (1979). Self consistent field model for condensed matter. Phys. Rev. B 20, 49814989.CrossRefGoogle Scholar
Makita, M., Nersisyan, G., McKeever, K., Dzelzainis, S., White, S., Kettle, B., Dromey, B., Doria, D., Zepf, M., Lewis, C.L.S., Robinson, A.P.J., Hansen, S.B. & Riley, D. (2014). Fast electron propagation in Ti foils irradiated with sub-picosecond laser pulses at I λ2>10 18 Wcm−2μm 2. Phys. Plasmas 21, 023113 111.CrossRefGoogle Scholar
Margenau, H. & Murphy, G.M. (1955). The Mathematics of Physics and Chemistry. New York: Van Nostrand.Google Scholar
More, R.M., Warren, K.H., Young, D.A. & Zimmerman, G.B. (1988a). A new quotidian equation of state (QEOS) for hot dense matter. Phys. Fluids 31, 3059.CrossRefGoogle Scholar
More, R.M., Zinamon, Z., Warren, K.H., Falcone, R. & Murname, M. (1988b). Heating of solids with ultra-short laser pulses. J. Phys. Colloq. C7, supplement 12 Tome 49, 4351.Google Scholar
Nakatsutsumi, M., Davies, J.R., Kodama, R., Green, J.S., Lancaster, K.L., Akli, K.U., Beg, F.N., Chen, S.N., Clark, D., Freeman, R.R., Gregory, C.D., Habara, H., Heathcote, H., Hey, D.S., Highbarger, K., Jaanimagi, P., Key, M.H., Krushelnick, K., Ma, T., MacPhee, A., MacKinnon, A.J., Nakamura, H., Stephens, R.B., Storm, M., Tampo, M., Theobald, W., Van Woerkom, L., Weber, R.L., Wei, M.S., Woolsey, N.C. & Norreys, P.A. (2008). Space and time resolved measurements of the heating of solids to ten million kelvin by a petawatt laser. New J. Phys. 10, 043046 113.CrossRefGoogle Scholar
Nardi, E. & Zinamon, Z. (1978). Energy deposition by relativistic electrons in high temperature targets. Phys. Rev. A 18, 12461249.CrossRefGoogle Scholar
Neumayer, P., Aurand, B., Basko, M., Ecker, B., Gibbon, P., Hochhaus, D.C., Karmakar, A., Kazakov, E., Kühl, T., Labaune, C., Rosmej, O., Tauschwitz, An., Zielbauer, B. & Zimmer, D. (2010). The role of hot electron refluxing in laser-generated K-alpha sources. Phys. Plasmas 17, 103103.CrossRefGoogle Scholar
Nilson, P.M., Davies, J.R., Theobald, W., Jaanimagi, P.A., Mileham, C., Jungquist, R.K., Stoeckl, C., Begishev, I.A., Solodov, A.A., Myatt, J.F., Zuegel, J.D., Sangster, C., Betti, R. & Meyerhofer, D.D. (2012). Time-resolved measurements of hot-electron equilibration dynamics in high-intensity laser interactions with thin-foil solid targets. Phys. Rev. Lett. 108, 085002.CrossRefGoogle ScholarPubMed
Ovchinnikov, V.M., Kemp, E., Schumacher, D.W., Freeman, R.R. & VanWoerkom, L. (2011). How well do time-integrated Ka images represent hot electron spatial distributions? Phys. Plasmas 18, 072704.CrossRefGoogle Scholar
Passoni, M. & Lontano, M. (2004). One-dimensional model of the electrostatic ion acceleration in the ultraintense laser-solid interaction. Laser Part. Beams 22, 163169.CrossRefGoogle Scholar
Passoni, M., Tikhonchuk, V.T., Lontano, M. & Bychenkov, V.Yu. (2004). Charge separation effects in solid targets and ion acceleration with a two-temperature electron distribution. Phys. Rev. E 69, 026411.CrossRefGoogle ScholarPubMed
Patoary, M.A.R., Alfazuddin, M., Haque, A.K.F., Basak, A.K., Taulukder, M.R., Karim, M.R. & Saha, C. (2008). Electron impact K-shell ionization cross sections of atoms at relativistic energies. Int. J. Quantum Chem., 108, 10231035.CrossRefGoogle Scholar
Ping, Y., Shepherd, R., Lasinski, B.F., Tabak, M., Chen, H., Chung, H.K., Fournier, K.B., Hansen, S.B., Kemp, A., Liedahl, D.A., Widmann, K., Wilks, S.C., Rozmus, W. & Sherlock, M. (2008). Absorption of short laser pulses on solid targets in the ultrarelativistic regime. Phys. Rev. Lett. 100, 085004.CrossRefGoogle ScholarPubMed
Quinn, M.N., Yuan, X.H., Lin, X.X., Carroll, D.C., Tresca, O.R., Gray, J., Coury, M., Li, C., Li, Y.T., Brenner, C.M., Robinson, A.P.L., Neely, D., Zielbauer, B., Aurand, B., Fils, J., Kuehl, T. & McKenna, P. (2011). Refluxing of fast electrons in solid targets irradiated by intense, picosecond laser pulses. Plasma Phys. Control. Fusion 53, 025007.CrossRefGoogle Scholar
Romagnani, L., Fuchs, J., Borghesi, M., Antici, P., Audebert, P., Ceccherini, F., Cowan, T., Grismayer, T., Kar, S., Macchi, A., Mora, P., Pretzler, G., Schiavi, A., Toncian, T. & Willi, O. (2005). Dynamics of electric fields driving the laser acceleration of multi-MeV protons. Phys. Rev. Lett. 95, 195001.CrossRefGoogle ScholarPubMed
Sandhu, A.S., Dharmadhikani, A.K. & Kumar, G.R. (2005). Time resolved evolution of structural, electrical, and thermal properties of copper irradiated by an intense ultrashort laser pulse. J. Appl. Phys. 97, 023526.CrossRefGoogle Scholar
Seely, J.F., Szabo, C.I., Audebert, P. & Brambrink, E. (2011). Energetic electron propagation in solid targets driven by the intense electric fields of femtosecond laser pulses. Phys. Plasmas 18, 062702.CrossRefGoogle Scholar
Stambulchik, E., Kroupp, E., Maron, Y., Zastrau, U., Uschmann, I. & Paulus, G.G. (2014). Absorption-aided x-ray tomography of planar targets. Phys. Plasmas 21, 033303.CrossRefGoogle Scholar
Vauzour, B., Debayle, A., Vaisseau, X., Hulin, S., Schlenvoigt, H.-P., Batani, D., Baton, S.D., Honrubia, J.J., Nicolai, Ph., Beg, F.N., Benocci, R., Chawla, S., Coury, F., Dorchies, F., Fourment, C., d'Humières, E., Jarrot, L.C., McKenna, P., Rhee, Y.J., Tikhonchuk, V.T., Volpe, L., Yahia, V. & Santos, J.J. (2014). 1–15 Unraveling resistive versus collisional contributions to relativistic electron beam stopping power in cold-solid and in warm-dense plasmas. Phys. Plasmas 21, 033101.CrossRefGoogle Scholar
Westover, B., Chen, C.D., Patel, P.K., Key, M.H., McLean, H., Stephens, R. & Beg, F.N. (2011). Fast electron temperature and conversion efficiency measurements in laser-irradiated foil targets using a bremsstrahlung x-ray detector. Phys. Plasmas 18, 06310.CrossRefGoogle Scholar
Zastrau, U., Audebert, P., Bernshtam, V., Brambrink, E., Kampfer, T., Kroupp, E., Loetzsch, R., Maron, Y., Ralchenko, Yu., Reinholz, H., Ropke, G., Sengebusch, A., Stambulchik, E., Uschmann, I. & Forster, E. (2010). Temperature and Kα yield radial distributions in laser-produced solid-density plasmas imaged with ultrahigh-resolution x-ray spectroscopy. Phys. Rev E 81, 026406.CrossRefGoogle ScholarPubMed
Zastrau, U., Burian, T., Chalupsky, J., Döppner, T., Dzelzainis, T.W.J., Fäustlin, R.R. & Förster, E. (2012a). XUV spectroscopic characterization of warm dense aluminum plasmas generated by the free-electron-laser FLASH. Laser Part. Beams 30(01), 4556.CrossRefGoogle Scholar
Zastrau, U., Sengebusch, A., Audebert, P., Brambrink, E., Fäustlin, R.R., Kämpfer, T., Kroupp, E., Loetzsch, R., Maron, R., Reinholz, H., Röpke, G., Stambulchik, E., Uschmann, I. & Förster, E. (2012b). High-resolution radial Kα spectra obtained from a multi-keV electron distribution in solid-density titanium foils generated by relativistic laser matter interaction. High Energy Density Phys. 7, 4753.CrossRefGoogle Scholar
Zeldovich, Ya.B. & Raizer, V. (1967). Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. New York: Academic Press.Google Scholar