ILIAS Ion and Laser Beam Interaction and Application Studies

Report 2008 - 05 April

Editor: Peter Mulser

Gesellschaft für Schwerionenforschung mbH Planckstraße 1 · D-64291 Darmstadt · Germany Postfach 11 05 52 · D-64220 Darmstadt · Germany

ILIAS

Ion and Laser Beam Interaction and Application Studies

Progress Report No. 3 of the PHELIX theory group

Editor: Peter Mulser

Contents 1

The objectives of ILIAS 3

The scientific potential of PHELIX 5

Achievements of ILIAS in 2005 –2007 9

List of ILIAS seminars 11

Participating Institutions 13

Collaborators of the ILIAS study group 15

Progress reports 17

List of ILIAS-related publications in 2007 63

Refereed publications 63

Conference Proceedings 64

Invited talks 65

The objectives of ILIAS

The ILIAS study group was founded in January 2005 at GSI/Darmstadt aiming at

(i) preserving the expertise accumulated in the past by theoretical groups associated to GSI through projects combining intense heavy-ion and high-energy laser beams;

(ii) focusing this expertise to the new goal of establishing a GSI-specific program involving the PHELIX facility in combination with the GSI heavy-ion accelerators.

The two objectives may be achieved by

• exploring the potential of intense heavy-ion and laser beams interacting with bulk matter, with mass-limited systems, and with single particles (clusters, mesoscopic systems, ions, and nuclei);

• offering theoretical assistance to the PHELIX* project in planning relevant experiments and discussing them in detail with the experimentalists involved

• creating a platform for activating synergies by interacting with researchers from German and European institutions:

Paul Gibbon/Jülich Hartmut Ruhl/Bochum Oswald Willi/Düsseldorf Hans-Jörg Kull, Thomas Pesch/Aachen Christoph Keitel, Dieter Bauer, Antonino Di Piazza/MPK Heidelberg Maurizio Lontano/CNR Milano Dimitri Batani /Univ. Milano Bicocca Anna Antonicci /Univ. Milano Bicocca

To disseminate the expertise in plasma, atomic, relativistic, and collective physics among PHELIX a weekly workshop has been launched successfully at GSI in January 2005.

______* The acronym PHELIX as used in the context here stands for all high-power laser facilities, short pulse, long pulse, and all kinds of intense laser-matter interaction research planned at GSI.

The scientific potential of PHELIX

There are no limits, horizons only Gerard A. Mourou, DPG Meeting in Düsseldorf, 2007

Currently there exist about 50 Petawatt laser installations worldwide and their number is continously growing. Several of them are ready to produce energy flux densities I 22 (“intensities”) up to 10 W/cm². The electric field amplitude E0 associated with this value 14 amounts to 2.75×10 V/m. The concomitant dimensionless vector potential a = eE0/mecω = 68 indicates that a free electron exposed to such a field at Ti:Sa laser frequency of ω =2.36×1015 −1 1/2 s is highly relativistic, for its mean oscillation energy is (γ − 1) mec² = mec² [(1 + a ²/2) − 1] = 24 MeV. Hence Petawatt laser systems offer unique possibililties to study light –matter interactions in the relativistic, sub- and ultra-relativistic domain. In the following a representative selection of studies in a variety of fields are listed.

Bright radiation sources Intense continous sub-ps X ray flashes and ps X ray line bursts for novel applications (diagnostics, ultrashort times) originate from the spectrum of hot electrons exceeding several times the mean electron oscillation energy. Of particular interest is the generation of coherent high order harmonics from gas targets. Here generally phase matching with the incident laser radiation is required. Recently however, quasi-phasematching has been shown to provide a promising alternative (231 maximum harmonic order detected, > 360 eV from argon). Much higher order harmonics (3.8 keV, beyond order 3200 so far detected at Rutherford-Appleton labs) originate from the γ spikes of the solid taget surface oscillating ultrarelativistically in the laser field. A combination of efficiency, scaling favorably with increasing intensity and with pulses on the order of attoseconds, and perhaps even zeptoseconds, unprecedented X ray intensities (>> 1025 W/cm²) are theoretically expected in the coherent harmonic focus. One of the attractive applications of atto- and zeptosecond pulses is in the field of scanning molecular and atomic, perhaps even nuclear, dynanics.

Acceleration of electrons and ions In the wake of an ultrashort intense laser pulse propagating through an underdense plasma bunches of electrons are efficiently accelerated to reach GeV energies in 1 mm to 3 cm long gas-filled capillary discharge waveguides. This kind of table-top accelerators are capable of producing 1012 V/m fields, hence exceeding conventional magnetic devices by factors of 104 − 105. When the resonantly produced hot electrons from solid targets reach the surface opposite to the impinging laser a space charge is created in which monolayers of ions applied there are accelerated to nearly monochromatic energies of the order of 100 MeV. Similar acceleration effects are reached from ultrashort laser pulses interacting with gas jets. Not only are acceleration studies of interest from a fundamental perspective, high energy ions have many potential applications such as compact ion sources for radiotherapy, production of isotopes for positron emission tomography, short pulse neutron spallation sources, proton radiography, proton deflectometry for diagnosing electric and magnetic fields, fast ignition of inertial fusion pellets and for technical processes in the industrial application. Laser plasma accelerated ion beams have a number of favorable characteristics: small emittance, effective source size, short temporal scale, high luminosity and good collimation.

Fast ignition of fusion pellets Inertial fusion with lasers requires heating of compressed matter (a thousand times solid DT) to an ion temperature of the order of 5 − 10 keV keV during a time interval of a few tens of ps. There is a world-wide effort to reach this ambitious goal with superintense laser beams of intensities ranging from I = 1020 to 1021 W/cm². Even taking the relativistic increase of the critical density into account at which conversion of the laser energy into electron heat flow occurs, with present high-power lasers deep penetration of the laser beam into the dense pellet is not possible (although it would solve a lot of problems). From this fact the challenging enterprise starts of how such intense energy flows can reach the dense interior from the deposition corona in due time. Two-stream type instabilities, and magnetic fields owing to imperfect electric current neutralization may inhibit the energy transfer. It is for the first time that such extreme energy flux densities are encountered in nature. Here, basic investigations with PHELIX type lasers can make fundamental contributions to successfully solving a major problem in this field of alternative energy research.

High-power laser absorption mechanism and tailoring of the electron spectra The three foregoing subjects have one aspect in common: Is it possible by appropriately design targets to tailor the fast electron spectrum evolving from the interaction with the laser beam in such a way as to make the process under consideration, i.e. x ray emission, electron and ion acceleration, energy transport, most efficient? The question primarily addresses the absorption mechanisms of intense, long wavelength radiation in matter over a range of electron densities of several orders of magnitude, with the emphasis on the processes of collisionless photon deposition. By means of particle-in-cell simulations, with the necessary atomic physics implemented (collisonal and field ionization, collisonal heating), absorption and resulting electron spectra can be determined to some degree of agreement with experiments, however, the underlying physical processes, the key to modelling, are not well understood. Still after more than two decades of related experiments there is a huge demand for basic experiments with PHELIX-type laser systems on absorption, electron spectra and their evolution in time.

Magnetic reconnection This complex and not well understood phenomenon plays an important role in extraterrestrial, solar and cosmic physics and, perhaps, in inertial confinement fusion. It is a subject of intense research activity. Its theory has several unanswered questions. Laboratory experiments provide important benchmarking data and by applying appropriate scaling laws they may be used to study cosmic jet formation. First experiments with two 1 ns laser beams of 200 J each focused on an Al target surface at a distance to create a magnetic reconnection geometry showed the formation of two neighboring jets emanating from the reconnection region and a rise of temperature and X ray emission as a consequence of the magnetic field collapse. Such experiments are of extensive interest to the physics community as a whole.

Relativistic wave propagation At high intensities electromagnetic waves behave strongly nonlinear (snl), at a degree which goes far beyond the familiar nonlinear (nl) optics. The origin of the drastically increased nonlinearity lies in the self-generated sudden variation of the local electron density due to field ionization (i), the ponderomotive (or wave pressure) effects (ii) and, at relativistic intensities, the coupling of the transverse field to the lonitudinal plasma wave (iii). Whereas in standard nl optics the number of wave crests of a pulse is invariant, as a consequence of (i) it can vary: after transiting or reflection e.g. five crests become six or seven crests. Ponderomotive effects (ii) lead to a large number of instabilities, formation of irregular structures, solitons, cavitons, wave breaking and chaos. In addition, due to (iii) all effects

may even become stronger. So far, relativistic wave propagation has been studied extensively only with waves of constant phase velocity, and recently stationary quasiperiodic solutions have also been presented. Under the influence of (ii) the plasma medium is expected to become heavily deformed and even self-quenching of the electromagnetic mode may occur. As at the extreme intensities to be expected in the near future (I > 1023 W/cm²) hot dense matter at solid and above solid density may be produced by relativistic overcritical light penetration the study of intense wave propagation in dense matter is of vital importance, in itself as well as for applications (equation of state, fast ignition, astrophysics). The beauty in this context is that great portion of related experiments can already be performed, by analogy, at nonrelativistic and close to relativistic intensities in sub- and super-critical foams and porous media with lasers like PHELIX.

Laser-induced nuclear transitions and vacuum nonlinearities With incresing light intensity laser-based nuclear physics will become a reality. In analogy to atomic systems excitation, stimulated deexcitation and triggering of single decay processes may be controlled by laser. Already with the next generation of photon flux densities (> 1023, perhaps >> 1023 W/cm²) the direct excitation of nuclear levels and the triggered decay of isomeric nuclei can be studied, provided the photon energy is high enough (order of keVs, free electron laser systems). Alternatively, the aim may be reached by the coherent superposition of extremely high order harmonics generated from solid targets. Such pulses will be used also to study the nuclear Stark effect and fine tuning of nuclear levels for various applications and diagnostics. All kinds of vacuum nonlinearities have their origin in the separation (slang: “creation”) of electron-positron or more massive pairs by corresponding energy supply. The critical field for the lightest pair production is E = 1.3×1018 V/m, corresponding to the laser intensity I = 2.3×1029 W/cm². Their existence becomes noticeable already through their virtual appearance at much lower energies (and fortunately, after all “there are no limits” of I to make them real). Nonperturbative vacuum nonlinearities manifest themselves in a variety of processes and effects of the quantum vacuum exposed to superintense photon fields: generation of harmonics, photon splitting, light by light scattering, vacuum polarization. Two favored candidates, accessible to an upgraded PHELIX sytem, are the merging of two photons in the laser field interacting with TeV protons and the dramatic increase of electron bremsstrahlung from a nucleus in the intense radiation field by many orders of magnitude. Worth mentioning because accessible already to experimental verification with the existing PHELIX facility is the effect of electron scattering in a Coulomb collision in a laser field of not yet relativistic intensity maifesting itself in additional focusing (attracting potential) and defocusing (repulsivc potential) of the scattered particles.

Related references

Bright radiation sources G. Korn et al., Ultrashort 1-kHz laser plasma hard x-ray source, Optics Lett. 27, 866 (2002). Teodora Baeva, S. Gordienko, and A. Pukhov, Theory of high-order harmonic generation in relativistic laser interaction with overdense plasma, Phys. Rev. E 74, 046404 (2006). B. Dromey et al., Bright multi-keV harmonic generation from relativistically oscillating plasma surfaces, Phys.Rev. Lett. 99, 085001 (2007).

Acceleration of electrons and ions K. Nakamura et al., GeV electron beams from cm-scale channel guided laser wakefield

accelerator, Phys. Plasmas 14, 056708 (2007). S. Karsch et al., GeV-scale electron acceleration in a gas-filled capillary discharge waveguide, New J. Phys.9, 415 (2007). H. Habara et al., Fast ion acceleration in ultraintense laser interactions with an overdense plasma, Phys. Rev. E 69, 036407 (2004). Fast ignition of fusion pellets S. Hain and P. Mulser, Fast ignition without hole boring, Phys. Rev. Lett. 86, 015 (2001). P. Mulser and D. Bauer, Fast ignition of fusion pellets with superintense lasers: Concepts, problems, and prospectives, Laser&Particle Beams 22, 5 (2004). P. Mulser and R. Schneider, On the inefficiency of hole boring in fast ignition, Laser&Particle Beams 22, 157 (2004). R. Kodama et al., Cone-guided fast ignition, Nature 412, 798 ; 418, 933 (2000).

High-power laser absorption mechanism and tailoring of the electron spectra P. Gibbon and A. R. Bell, Cossisionless absorption in sharp-edged plasmas, Phys. Rev. Lett. 68, 1635 (1992). H. Ruhl and P. Mulser, Relativistic Vlasov simulation of intense fs laser pulse-matter inteeraction, Physics Lett. A 205, 388 (1995). Mirela Cerchez et al., Absorption of ultrashort laser pulses in strongly overdense plasmas, ILIAS Report No. 3, this issue. P. Mulser and H. Ruhl, Collisionless high-power laser beam absorption: a NO GO theorem, ILIAS Progress Report No. 2, GSI Rep. 2007-2 (2007).

Magnetic reconnection Luise Willingale, Ion acceleration from high intensity laser plasma interactions: Measurements and applications, PhD thesis, Plasma Physics Group, Imperial College, University of London, 2007. N. A. Bobrova et al., Force-free equilibria and reconnection of the magnetic field lines incollisionless plasma configurations, Phys. Plasmas 8, 759 (2001).

Relativistic wave propagation P. Mulser, F. Cornolti, and D. Bauer, Modelling field ionization in an energy conserving form and resulting nonstandard fluid dynamics, Phys. Plasmas 5, 4466 (1998). T. Pesch and H.-J. Kull, Waves with constant phase velocity in relativistic plasmas, ILIAS Progress Report No. 2, GSI Rep. 2007-2 (2007). T. C. Pesch and H.-J. Kull, Quasiperiodic waves in relativistic plasmas, ILIAS Progress Report No. 3 (present issue). P. Mulser, H. Ruhl, M. Lontano, and R. Schneider, Relativistic critical density increase in a linearly polarized laser beam, ILIAS Progress Report No. 3 (present issue).

Laser-induced nuclear transitions and vacuum nonlinlinearities H. Schwörer et al. (Eds.), Lasers and Nuclei (Springer, Heidelberg, 2006). C. Müller et al., Coupling of nuclear matter to intense photon fields, A. Di Piazza et al., QED effects in strong laser fields, T. Ristow and H.-J. Kull, Coulomb focusing in electron-ion collisions in a strong laser field, in ILIAS Progress Report No. 3 (present issue). A. Di Piazza, C. Müller et al., QED, nuclear and high-energy processes in extremely strong laser pulses, ILIAS Progress Report No. 2, GSI Rep. 2007-2 (2007). Ch. Keitel, Jörg Everts, and D. Habs, Symposium Kernphysik mit starken Laserfeldern (SYKL), in Verhandlungen der DPG Frühjahrstagung, Darmstadt, 10. – 14. 03. 2008, p. 317 – 319.

Achievements of ILIAS in 2005 – 2007

A scientific community of theoreticians from German and European Reseach Institutes has been established to bring in their expertise in

High intensity laser-plasma interaction (N. Tahir, P. Gibbon, H. Ruhl, P. Mulser) Electromagnetic wave propagation and absorption (H.-J. Kull, M. Lontano, H. Ruhl) Laser interaction with foams and porous media (P. Gibbon) Electron beam-plasma interaction (Anna Antonicci) Electron acceleration and relativistic heavy ion physics (W. Scheid) Hohlraum radiation transport for warm dense matter studies (T. Schlegel) Direct interaction of nuclei with supereintense laser fields (C. Müller) Laser-vacuum QED (A. Di Piazza) Particle-in-cell and hydro code development (J. Maruhn, H. Ruhl, T. Schlegel).

It is the hope to preserve this know-how during the following years and to make sucessful use of it during the forthcoming period of planning new experiments with the PHELIX laser system.

With 104 sessions of the ILIAS weekly workshop a platform was created during the past three years to acquire and enlarge the expertise in the above-mentioned disciplines among PHELIX-oriented theoreticians and to disseminate this knowledge among the experimental plasma physics groups at GSI. Should the workshop be continued once again in the future much effort will be put on intensifying dissemination and knowledge transfer from theory to the experimental groups. Approximately 20 % of time has been dedicated to reporting on international conferences and meetings and to analyzing critically scientific progress and its significance for PHELIX.

List of ILIAS workshop/seminar presentations in 2007

The workshop is a joint venture of the ILIAS study group, PHELIX laser project, and groups from the Plasma and Atomic Physics divisions of GSI. Organisation by P. Mulser

The weekly ILIAS seminar was started on Tuesday, January 11th, 2005. The presentations, in total 105, took place on Tuesdays at 13:15 Place: GSI, Main Building, 1st floor: Atomic Physics Seminar Room; beginning: 13:15

09.01. Seminarvorbesprechung 16.01. Theo Schlegel, GSI, Homogenes Plasmaheizen mit Hohlraumstrahlung, Teil I 23.01 Theo Schlegel, GSI, Homogenes Plasmaheizen mit Hohlraumstrahlung, Teil II 30.01. Seminar entfällt wegen Hirschegg-Meeting 06.02 V. Efremov, IHED-RAS, Moskau, Investigation of porous matter under intense energy fluxes 13.02 Diskussionsthema I: Strahlungstransport und Raytracing; Strahlungsfelder. Diskussionsthema II: PHELIX-relevante Anregungen von der Hirscheggtagung. 20.02. ILIAS-Seminar entfällt (Karneval) 27.02 V. Basko, ITEP Moskau, zur Zeit GSI, Self-consistent equations of radiative hydrodynamics 06.03. Seminar entfällt (Konferenzbesuch) 13.03. A. Blazevic, Arbeitsgruppe Prof. Hoffmann/GSI, Ionenstoppen in Laser- geheizten Targets 20.03. ILIAS-Seminar entfällt wegen DPG-Tagung in Düsseldorf (19. – 23.03.) 27.03 ILIAS-Veranstaltung zu Mikhail Baskos 60.: Einführung durch Theo Schlegel: Würdigung des Jubilars. M. Basko, Eigene neuere Beiträge zu Ionenstoppen. 03.04 P. Mulser, Stoßabsorption von Laserstrahlung im Plasma mit Bezug zu Ionenstoppen: Modelle, Ergebnisse, offene Probleme 10.04. frei wegen wichtiger Terminüberschreitungen 17.04. Teodora Baeva, Arbeitsgruppe Prof. A. Pukhov, Univ. Düsseldorf, High harmonic generation from plasma: The theory of relativistic spikes and its experimental confirmation 24.04. Theo Schlegel, Theorie GSI, Bericht über Dream beams symposium at MPQ Garching. Workshop on fast ignitor physics in Madrid; Teil I 01.05. frei (1. Mai) 08.05. Marius Schollmeier, Arbeitsgruppe Prof. M. Roth, Laser- Protonenbeschleunigung mit Linienfoki 15.05. 15.05. N. E. Andreev, Joint Institute for High Temperatures, Russian Academy of Sciences, z.Z. LULI, Paris, Nonideal femtosecond laser plasmas: modeling and pump- probe experiments 22.05. H. Ruhl, H. Krull, Inst. Theoretische Physik, Ruhr-Universität Bochum, Modell zu Laser-Absorptionsphysik an oszillierenden Oberflächen 29.05. P. Mulser, TQE – TU Darmstadt, Laser-erzeugte stoßfreie elektrostatische Schocks und Ionenbeschleunigung 05.06. Natalia Borisenko, Lebedev Physical Institute, Moscow, spricht über Application of plastic aerogels for undercritical density laser experiments. Diagnostics and experimental results. 12.06. P. Mulser, TQE, über Thomson-Streudiagnostik I: Einführung in die Lichtstreuung;

19.06. P. Mulser, TQE, über Lichtstreuung im Plasma II: Qunatitative Analyse der spontanen Streuung; relevante Anwendungen 26.06. Theo Schlegel, Theorie-Abteilung Prof. KH Langanke, Bericht über Dream beams symposium at MPQ Garching und Workshop on fast ignitor phsics in Madrid; Teil II. 03.07. P. Mulser, TQE, über Dopplereffekt, Lichtdruck und Laser-Plasmainstabilitäten 10.07. Fortsetzung, Teil II: Stimulierte Streuung – Lichtdruck-getriebene prametrische Laser- Plasmainstabilitäten. 17.07. Prof. Dr. A. G. Ramm, Mathematics Department, Kansas State University, z.Z. Fachbereich Mathematik der TUD, über Wave scattering by small impedance particles in a medium 24.07. Toma Toncian, Institut für Laser und Plasma Physik, Heinrich-Heine-Universität Düsseldorf, über Laser Triggered Micro Lens for Energy Selection and Focusing of MeV protons

SOMMERPAUSE

18.09. P. Mulser, TQE – TU Darmstadt, über Bericht von der 5. IFSA2007, 9. – 14.9. in Kobe mit Schwerpunkt auf schnelle Zündung von Fusionspellets (Journal Club) 25.09. M. Schollmeier, Arbeitsgruppe Prof. M. Roth, über Teilchenbeschleunigung mit Lasern: Bericht von der IFSA2007, 9. – 14.9. in Kobe (Journal Club) 02.10. entfällt wegen ULIS-Koferenz in Bordeaux, 1. – 5- Oktober 09.10. P. Mulser, TQE-TUD, über Trends I in ULIS (Journal Club) 16.10. entfällt wegen Workshop Theory of Short Pulse Petawatt Laser Plasm Interaction (TSPPLPI), 14. –17. Oktober, Org. Theo Schlegel 23.10. Nachlese TSPPLPI und ULIS II (Journal Club) 30.10. K. Witte, GSI, über PHELIX Science, Status and Roadmap 06.11. Seminar muss wegen Überlapp mit Helmholtzveranstaltung entfallen 13.11. Mirela Cerchez, Institut für Laser- und Plasmaphysik, Heinrich-Heine-Universität Düsseldorf, über Absorption of extremely short laser pulses in strongly overdense targets (ganz neue Messergebnisse) 20.11. Anna Tauschwitz. GSI, über Dynamics of tensile pressure in ion-beam heated matter applied to a liquid lithium production target for the Super- FRS at FAIR 27.11. K. Witte, GSI & TUD, über Experimente mit PHELIX 04.12. P. Mulser, TQE-TUD, über Resonanzabsorption in steilen Dichteprofilen, MULTI und Gültigkeitsbereich der Fresnelformeln (Journal Club zur Arbeit von Mirela Cerchez [s. Vortrag vom 13.11.]) 11.12. H.-J. Kull, RWTH Aachen, über Korrelierte Elektron-Ion-Stöße in starken Laserfeldern. Beginn 13:00 Uhr

Series of ILIAS seminars terminated (34 sessions in 2007) ______

Participating Institutions

GSI, Darmstadt

Theoretical Quantum Electronics (TQE), Technical University of Darmstadt

Institute for Theoretical Physics, University of Frankfurt

Theory Division, Max Planck Institute of Nuclear Physics (MPK), Heidelberg

Institute for Theoretical Physics, University of Giessen

Institute of Laser and Plasma Physics and Transregio 18: Relaticistic Laser Plasma Dynamics,

University of Düsseldorf

Institute for Theoretical Physics I; University of Bochum,

Laser Physics, Theoretical Physics A, RWTH Aachen

Complex Atomic Modelling and Simulation, Central Institute for Applied Mathematics,

Research Centre Jülich

Istituto di Fisica del Plasma Piero Caldirola, CNR, Milano

Dipartimento di Fisica G. Occhialini, Università Milano Bicocca

Collaborators of the ILIAS study group

Rudolf Bock, Heavy ion beam fusion, Organizer of Plasma Physics Seminar GSI, [email protected]

Thomas Kühl, Spectroscopy, Atomic Physics Division, GSI, [email protected]

Bärbel Rethfeld, Ultrafast dynamics of laser-irradiated solids, GSI, [email protected]

Olga Rosmej, X ray spectroscopy, Plasma Physics Division, GIS, [email protected]

Theodor Schlegel, High-power laser-matter interaction, GSI, [email protected]

Anna Tauschwitz, Numerical hydro-simulations, Frankfurt (Prof. Maruhn), [email protected]

Marco Tomaselli, Atomic physics, GSI, [email protected]

Peter Mulser, High-power laser-matter interaction, organizer of ILIAS workshop, TQE, [email protected]

Joachim Maruhn, Computational physics, numerical hydro- simulation [email protected]

Christoph H. Keitel, Relativistic laser-matter interaction, Heidelberg [email protected]

Dieter Bauer, Multiphoton physics with lasers, Heidelberg [email protected]

Antonino Di Piazza, Vacuum QED in relativistic laser fields, Heidelberg [email protected]

Carsten Müller, Dynamics of mattere in superstrong laser fields, Heidelberg [email protected]

Werner Scheid, Atomic physics and electron acceleration with lasers [email protected]

Nicolas Kowarsch, Electron acceleration with lasers, [email protected]

Oswald Willi, Intense fs laser-matter interaction experiments, [email protected]

Hartmut Ruhl, Modelling and simulation of laser plasma interaction, Bochum

[email protected]

Paul Gibbon, Mesh-free plasma simulation, Supercomputing, [email protected]

Hans-Jörg Kull, Laser plasmas, wave-plasma interactions, RWTH Aachen, [email protected]

Thomas Pesch, Relativistic electromagnetic waves, RWTH Aachen, [email protected]

Torsten Ristow, External field effects on collisions, RWTH Aachen, [email protected]

Maurizio Lontano, Relativistic solitary waves and wave pressure, Milano, [email protected]

Dimitri Batani, Fast electron transport, shock waves and ponderomotive pressure, Milano, [email protected]

Anna Antonicci, Hydro-PIC code plasma simulations, Univ. Milano Bicocca, [email protected]

Progress reports

PHELIX – Achievements in 2007 K. Witte et al.

Coupling of nuclear matter to intense photon fields C. Müller, A. Palffy, A. Ipp, A. Shahbaz, A. Di Piazza, T. J. Bürvenich, J. Evers, and C. H. Keitel

QED effects in strong laser fields Di Piazza, E. Lötstedt, K. Z. Hatsagortsyan, U. D. Jentschura, and C. H. Keitel

Relativistic critical density increase in a linearly polarized laser beam P. Mulser, H. Ruhl, M. Lontano, and R. Schneider

Absorption of ultrashort laser pulses in strongly overdense targets M. Cerchez, R. Jung, J. Osterholz, T. Toncian, and O. Willi, P. Mulser, and H. Ruhl

Coulomb focusing in electron-ion collisions in a strong laser field T. Ristow and H.-J. Kull

Quasiperiodic waves in relativistic plasmas T. C. Pesch and H.-J. Kull

An overview of theoretical work on high energy density physics using intense particle beams N.A. Tahir, A.R. Piriz, A. Shutov, V. Kim, I.V. Lomonosov, D.H.H. Hoffmann, and C. Deutsch

Heavy ions in a high-power laser beam D.B. Blaschke, A.V. Filatov, A.V. Prozorkevich, and D.S. Shkirmanov

Monte-Carlo study of electron dynamics in silicon during irradiation with ultrashort VUV laser pulses N. Medvedev, and B. Rethfeld

18 PHELIX – Achievements in 2007 K. Witte1, V. Bagnoud1, A. Blazevic1, S. Borneis1, C. Bruske1, J. Caird2, S. Calderon3, U. Eisen- barth1, J. Fils1,4, S. Götte1, T. Hahn1, H.-M. Heuck1,5, D. H. H. Hoffmann4, D. Javorkova1, G. Klap- pich1, F. Knobloch1, Th. Kühl1,6, M. Kugler1,7, S. Kunzer1, M. Kreutz1, B. LeGarrec8, T. Merz- Mantwill1, E. Onkels1, S. Radau9, M. Rebscher1, D. Reemts1, R. M. Richard10, M. Roth1,4, A. Rous- sel8, Andreas Tauschwitz1, Anna Tauschwitz11, R. Thiel1, U. Thiemer1, D. Ursescu1, U. Wittrock5, B. Zielbauer1,12 , D. Zimmer1,6 1GSI Darmstadt; 2Lawrence Livermore National Laboratory/USA; 3Pegasus Design, Inc., Pleasan- ton, Cal./USA; 4Technische Universität Darmstadt; 5University of Applied Sciences Münster; 6Johannes Gutenberg-Universität Mainz; 7University of Applied Sciences Darmstadt; 8Commissariat à l’Energie Atomique at Le Barp; 9ROM Engineering, Inc., Tucson, Arizona/USA; 10College of Optical Sciences, University of Arizona/USA; 11University of Frankfurt; 12Max-Born- Institut Berlin

A large variety of fundamental issues in plasma and stretched to 1 ns. The expected linear increase of Eout with atomic physics can be ideally experimentally investigated Ein proves that the amplification is entirely in the small- by the combination of highly charged heavy-ion and pow- signal gain regime up to the projected output energy of erful laser beams. The GSI is in the worldwide unique 500 J. Any deviation from linearity would indicate the position to have both types of beams at its disposal, the unwanted appearance of saturation and/or parasitic oscil- two ion beams UNILAC and SIS-18 and the PHELIX lations. The double-pass gain of ~100 is consistent with (Petawatt High-Energy Laser for Heavy-Ion Experi- the previously measured single-pass gain of ~10. ments) beam. At the experimental area Z6, the UNILAC ion and PHELIX beams will enable the study of the inter- 504 J action of MeV ions in hot strongly ionized matter with 500 • respect to the energy loss of the ions and their charge- state changes. At the measuring site HHT, SIS-18 and • PHELIX will be combined to study equation-of-state and 400 phase transitions in yet unexplored regions of warm dense matter. As to atomic physics, QED in strong fields be- Eout/J • comes accessible by the excitation of hydrogen-like ura- 300 • nium ions provided by the SIS-18 beam with a PHELIX • • driven x-ray laser. 200 In 2007 PHELIX has become available for laser-plasma • experiments in the laser bay. Our major achievements • include (a) the demonstration of the projected output energy of 500 J for pulses stretched to 1 ns, (b) the commis- 100 • sioning of the 31.5-cm aperture Faraday isolator (FI), (c) setup of the compressor in the laser bay, (d) improved • serrated apertures enabling sharp-edged circular and ellip- 0 2 4 6 tical fluence profiles of high energy loading, (e) a new 0 E /J stretcher with 1740 grooves/mm adapted to the new MLD in (multi-layer dielectric) gratings and allowing a precise and easy control of the pulse duration, (f) larger band- Figure 1: Output energy Eout of the main amplifier versus width of the stretched pulse enabling shorter durations of its input energy Ein the compressed pulse and thereby increasing its power, (g) further improvements of the PHELIX Control System (b) The 31.5-cm aperture FI serves for the protection of (PCS), and (h) the x-ray laser campaign using the com- PHELIX from light back-reflected from the target by plete facility that has confirmed the suitability of PHELIX normal scattering, stimulated Brillouin and/or Raman for state-of-the-art laser-plasma interaction experiments. scattering. The FI consists of two polarizers rotated by 45° versus each other and in-between the 19-mm thick (a) Fig. 1 shows the main amplifier (MA) output energy Faraday glass encapsulated by the coil that generates the Eout versus its input energy Ein at constant stored energy in required magnetic field for about 0.5 milliseconds (pulsed the MA for pulses from the femtosecond front end operation). The FI was first commissioned electrically,

19 then synchronized to the laser pulse, and finally optimized Properly designed Lyot filters can enlarge the spectral with respect to the contrast ratio. The maximal energy of width by loss modulation. When the pulse passes the fil- the back-reflected light tolerated to leak through the FI is ter, the centre of the spectrum is more suppressed than the 0.1 J. When this value is exceeded, optical components wings. Such filters are best incorporated in the low-power upstream the MA will be damaged. Assuming a pulse section of an amplifier chain where the energy loss is with 500 J and a back-reflection from the target not larger easiest to compensate. In PHELIX a suitable place is in- than 10%, the minimally allowable contrast ratio is between the two regenerative amplifiers of the front end. 50/0.1=500. The measured value of 1000 provides some The Lyot filter increases the bandwidth from 5.5 to 12 margin for the planned installation of the booster ampli- nm. Due to gain narrowing, the width of 12 nm shrinks to fier. A sensitive element in the FI is the Faraday glass 4.9 nm at the exit of the main amplifier at typical whose damage threshold limits the throughput energy to PHELIX operating conditions. Without the Lyot filter, the 500 J for a pulse of 1-ns duration and 25-cm diameter bandwidth amounts 2.8 nm only. Assuming a constant while 10-ns pulses allow the operation at 1 kJ. spectral phase, the Fourier transform of the broader spectrum yields the pulse duration of 340 fs instead of 550 fs. (c) The compressor has been first commissioned with the multi-layer dielectric (MLD) gratings from Horiba- Jobin-Yvon which span 47cm in the horizontal direction (dispersion) and 33 cm in the vertical direction. The larger MLD gratings from LLNL (80 cm by 40 cm) will be used later. The incidence angle of 72° results in lower align- ment and pointing sensitivity. With the small gratings in place, the incident beam profile is horizontally limited to 12 cm. In the vertical direction, the currently used beam extension is 24 cm so that the beam is elliptical. The compressor output energy depends on the pulse duration. The limiting mechanism is the damage threshold of the final grating. For 10- to 50-ps pulses, the maximal throughput energy is 200 J. For the shortest pulses of ~350-fs duration, it is reduced to 120 J.

(d) Serrated apertures (SA) combined with spatial fil- tering are well proven for the generation of beams with rectangular, squared, circular, or elliptical cross sections. The SA in the PHELIX beam line located between the front end and the preamplifier (see Fig. 2) was formerly fabricated using micro-machining techniques, offering a Figure 2: Schematics of PHELIX and the target area. relatively poor control over the tooth shape. By using NF near field fluence pattern, FF far field fluence pattern, micro-lithograhically fabricated SAs, a better control over WF wave front. the tooth shape was achieved. This resulted in steep- edged beam profiles with high-energy loading. The fill (g) As the first user of the control system (CS) frame- factors go up to 0.8. The 12 by 24 cm² elliptical beam work, PHELIX employs the version 3.0 in which the could be amplified without any signs of distortion. event mechanism is based on DIM (Distributed Informa- tion Management) [2]. This eased the PHELIX control (e) The new stretcher was built following the Banks de- system (PCS) that is currently running distributed on 14 sign [1], however with some modifications. The essential nodes. About 300 instances of nearly 50 different classes new feature is the adjustability of the stretched-pulse du- are used within the PCS, and roughly 10,000 variables are ration thereby allowing the operation of the compressor handled to ensure a safe shot procedure. Incorporation of with gratings at a fixed distance. This way the pulse dura- the beam lines to the experimental areas Z6 and HHT will tion can be easily changed from its minimal value of ~400 double the size of the PCS that is supposed to go to ver- fs to 20 ps with either positive or negative chirp. The op- sion 3.1 soon thereby benefiting from the developments tion to choose the sign of the chirp also helps to improve available nowadays [3]. The internal distribution of tim- the recompression fidelity by the extinction of unwanted ing signals were renewed in 2007. The hardware includes phase contributions occurring upon amplification. an interlock in case of a malfunction of the large-aperture FI. The synchronisation with the external experiments (f) Gain narrowing is the main mechanism in reducing will be the next step. The work on the safety system of Z6 the spectral width of the pulse and thereby also its power. made significant progress in 2007. Its connection to the

PHELIX interlock system (PILS) which embodies an im- mirror towers were completed. Particularly noteworthy is portant milestone will be realized soon. the confirmation that the two periscope mirrors for the (h) In December 2007 the PHELIX facility was used in laser bay mirror tower which have a six-point whiffletree a campaign of 24 full-system shots for the investigation of back-support meet all the requirements imposed on them x-ray laser emission at 6.8 and 7.3 nm from plasma of as to wave front distortion (<60 nm peak-to-valley) and nickel-like samarium ions employing a new simplified thermo-mechanical stability. pumping scheme [4] that is particularly suited for the high pump energies needed for x-ray lasers emitting below 10 Theory nm. The scheme was realized by placing a Mach-Zehnder To support the experimental program of PHELIX, the interferometer into the beam line downstream the hydro-code CAVEAT is currently extended to include stretcher (see Fig. 2). It generates two pulses, the pre- non-linear heat conduction and radiation transport. The pulse and the pump pulse, whose separation in time and new code CAVEAT-TR will enable the in-house simula- intensity ratio can be chosen according to the experimen- tion of laser heated matter including hohlraum targets that tal needs. Typical values are 1 ns and 1:20, respectively. will be used for energy loss experiments in the Z6 target These two pulses were amplified to energies of ~5 J and area. While the implementation of heat conduction is fin- ~100 J, respectively, recompressed to the duration of 50 ished and successfully tested [5], the implementation of to 100 ps, and transported to the target chamber. A low- the radiation transport will be completed in time to aid the cost 30-cm diameter, metallic off-axis parabola of good experimental program at Z6 in 2008. optical quality turned the beam by 90° and directed it to a 20-cm diameter spherical mirror. These two mirrors generated a homogeneously illuminated line focus of 8 mm References by 100 µm on the samarium target producing a travelling [1] Aberration-free, all reflective laser pulse stretcher, US wave excitation. This pumping scheme ensures perfect Patent 5960016, Publication date 28 Sept. 1999. on-target overlap of the two pulses which is mandatory for successful x-ray laser experiments and is rather diffi- [2] Enhancing the CS Framework for Distributed Control cult to achieve by beam splitting and subsequent overlap- Systems, D. Beck, H. Brand, and S. Götte, GSI Scien- ping of the pulses. In addition, any change of the experi- tific Report 2006, page 214. mental setup needs the realignment of a single beam only. [3] Improving the usability of the CS framework, D. The two samarium lines mentioned above could be Beck, H. Brand, M. Feldmann, M. Kugler, and A. clearly identified thus verifying the suitability of the new Schwinn, GSI Scientific Report 2007 (this publica- pumping scheme for x-ray lasers below 10 nm and, in the tion). future, also in the water window. Furthermore, this measuring campaign qualified PHELIX for state-of-the-art [4] A 180-eV X-Ray Laser Pumped by PHELIX, J. Habib laser-plasma interaction experiments. et al., GSI Scientific Report 2007 (this publication).

Beam Line to Z6 [5] CAVEAT-TR: A 2D hydrodynamic code with heat The main optical and opto-mechanical components of conduction and radiation transport. Implementation of the Z6 10°-beam line were ordered and delivered in 2007. the SSI method for heat conduction, M. Basko, J. Ma- The setup of the relay telescope and the north and south ruhn, A. Tauschwitz, GSI Report 2007 -05

21 Coupling of nuclear matter to intense photon ﬁelds

C. Müller, A. Pálffy, A. Ipp, A. Shahbaz, A. Di Piazza, T. J. Bürvenich∗, J. Evers, and C. H. Keitel Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany ∗Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438 Frankfurt, Germany

The interaction of intense laser pulses and high- tives for studying indirect coupling schemes via electrons energy photons with atomic nuclei and quarks is or other particles. Until these new machines start oper- theoretically studied. The processes under investi- ation, it is a major task for theory to develop reliable gation comprise direct laser-nucleus coupling, de- predictions for future experimental investigations on the excitation of isomeric nuclear levels triggered by influence of laser light on nuclear degrees of freedom. It photon absorption or electron capture, isotope ef- is also of importance to compare the laser-induced nuclear fects in the high-harmonic response from strongly excitation channel with competing mechanisms involving, laser-driven muonic atoms, and the production of e.g., bound electron transitions or free electron capture. polarized photons by an anisotropic quark-gluon Against this background, we consider in the present plasma. The prospects for experimental realiza- contribution selected phenomena from the field of laser- tion of these phenomena by upcoming intense high- nuclear physics. Section 2 is devoted to the resonant frequency radiation sources are discussed. single-photon excitation of nuclear ground states by intense laser radiation. Particular emphasis is placed upon 1 Introduction electric dipole-forbidden transitions in the few-keV range. We show that the observation of this direct laser-nucleus The efficient coupling of light to a quantized matter system coupling will be rendered feasible at the upcoming x-ray typically requires photon energies ~ω corresponding to a free-electron laser (XFEL) facilities. In Sect. 3 we study characteristic transition energy ∆E within the system. For the deexcitation of nuclear isomers via triggering to short- transitions between low-lying levels in atomic nuclei a rep- lived higher levels. Conceivable schemes to trigger the de- resentative value is ∆E 10–100 keV, whereas ~ω 1 eV cay employ photo-absorption or electron capture, where ∼ ∼ for optical lasers. This huge energetic discrepancy has the latter leads to more efficient triggering as will be ruled out the observation of direct laser-nucleus interac- shown. Section 4 considers muonic atoms exposed to in- tions so far. Instead, laser-nuclear physics has relied on tense laser pulses. Finite nuclear mass and size effects secondary particles mediating the interaction [1, 2]. Cele- are demonstrated in the laser-induced harmonic radiation brated experiments on laser-induced nuclear fusion [3], fis- emitted by muonic hydrogen isotopes. This result indi- sion [4] and neutron production [5] were enabled by laser- cates that the traditional spectroscopy of muonic transi- driven electrons which produce γ-ray bremsstrahlung or tions between stationary states could be extended to dy- lead to ion acceleration. This way, secondary particles namic, time-dependent studies in the presence of strong have formed until now a bridge between the widely di- external fields. Similar effects can, in principle, be in- verging laser and nuclear energy scales. vestigated in (ordinary) highly-charged ions. In Sect. 5, With the advent of coherent radiation sources in the light-matter interaction at extreme conditions is extended VUV and x-ray frequency domain (~ω 100eV–10keV) to the study of the quark-gluon plasma (QGP), which is ∼ the situation is presently changing, though. At the mo- one of the main objectives of the FAIR project at GSI ment, photon energies of about 90 eV at intensities up to [10]. Photons are a primary probe for the QGP dynamics, 1016 W/cm2 are provided by the FLASH facility at DESY as they are likely to leave the plasma without further in- (Hamburg, Germany) which is based on a free-electron teraction. Particularly, it is shown that a possible quark laser (FEL) [6]. A similar regime of interaction will be ac- polarization is transfered efficiently to the polarization of cessible by the XRL operation mode of the PHELIX laser emitted photons. These photons can be used to estimate at GSI (Darmstadt, Germany) [7]. There exist a few iso- the global quark polarization in the QGP. We finish with topes with very low-lying first excited levels (e.g., 235U a conclusion in Sect. 6. where ∆E 77 eV) which could already be reached by ≈ single-photon absorption from these VUV devices. Near- 2 Laser-driven electric dipole-forbidden future upgrades of FEL-based machines are envisaged to nuclear transitions produce hard x-ray beams with peak intensities close to 1020 W/cm2 [8]. Even higher intensities at comparable Recently, direct laser-nucleus interactions have been in- frequencies might be attainable via plasma surface har- vestigated, motivated by the advent of novel super-intense monics [9]. The laser frequency seen by the nucleus can coherent light sources both in the visible and in the x-ray be further enhanced by virtue of the Doppler effect, when frequency regime. X-ray laser facilities, especially together a relativistic ion beam instead of a fixed target is applied. with an acceleration of the nuclei to match photon and All these novel “light” sources offer clear prospects for re- transition frequency, do allow for resonant laser-excitation alizing direct interactions of coherent photon beams with of nuclei [11], whereas laser sources in the visible range may nuclear matter. They also provide alternative perspec- induce off-resonant laser-nucleus interaction [12]. While in

22 atomic systems typically electric dipole transitions dominate the light-matter interaction because of both their relevant transition frequencies and their larger dipole mo- ments, in the case of nuclei, this issue is less obvious. The spectrum of E1 transitions is limited to few low-lying nuclear excited states with small reduced transition probabilities, and giant resonances, at energies of several MeV, which are not directly accessible nowadays with the laser. We therefore investigate dipole-forbidden transitions of nuclei interacting with super-intense laser fields, considering several stable nuclei with suitable first excited states. Since in most cases near-resonant intermediate states are not available and the present laser intensities suppress non-resonant multiphoton transitions, our study is devoted to one-photon transitions. Among the most promising candidates for fourth-generation light sources, which would operate at higher power, brilliance and temporal and transverse coherence at low wavelengths, the Euro- pean XFEL [8] is designed to reach 12.4 keV photon energy with an average spectral brightness of 1.6 1025 photons/(sec mrad2 mm2 0.1%bandwidth) in pulses× of 100 fs at an average· · repetition· rate of 40 kHz. We consider laser-driven transitions between the ground and first excited state of suitable stable or long-lived isotopes, including all the magnetic sublevels in a multilevel model of the nuclear system interacting with the laser field [13]. In order to account for dipole-forbidden transitions, the plane waves describing the electromagnetic field are expanded in spherical waves characterized by well-defined multipolarity and parity. The nuclear interaction matrix element is expressed with the help of the reduced transition probabilities, using experimental values for the latter. Figure 1: Number of signal photons per nucleus per laser The European XFEL design report parameters [8] are used pulse N for several isotopes with first excited states to numerically estimate the possible magnitude of direct SIGNAL below 12.4 keV (upper figure) and above 12.4 keV (lower laser-nucleus interaction. figure). The green squares denote E1 transitions, the blue The interaction matrix elements and population in- circles E2 transitions and the red crosses M1 transitions. version for electric and magnetic dipole and electric The results are plotted versus the transition energy E. quadrupole transitions are compared for several stable or long-lived nuclei with low-lying first excited states. This comparison includes both cases in which the direct laser- 12.4 keV, M1 transitions are the most promising for laser nucleus interaction is by itself possible (the energy of the excitation. This is in contrast to atomic quantum optics, first excited state allows for direct resonant interaction, where higher multipole transitions are strongly suppressed. En < 12.4 keV), as well as nuclear systems for which mod- The equivalence of different multipole transitions in direct erate acceleration of the target nuclei is required to match nucleus-laser interactions considerably enhances the range nuclear transition and photon energies (En > 12.4 keV). of suitable isotopes. Assuming that all excited nuclei decay radiatively, the population of the excited state after one pulse is equal to the 3 Isomer triggering via nuclear excitation signal photon rate per nucleus per pulse NSIGNAL for the by electron capture driven nuclear transition. We present in Fig. 1 the signal photon rate for several isotopes for nuclear transitions The search for practical methods to change the internal with energies above and below 12.4 keV using the Euro- state of atomic nuclei has been the subject of a number pean XFEL laser parameters. The highest signal rate is of investigations in the last years. In particular, isomer −10 NSIGNAL = 3 10 photons per nucleus per pulse for the triggering refers to the possibility to excite the long-lived × 153 case of the E1 transition of the 62 Sm isotope, followed by isomeric nuclear state to a higher level which is associated 165 the one for the M1 transition of 67 Ho which is smaller with freely radiating states and therefore releases the en- only by a factor of two. ergy of the metastable state. Isomers are of interest in From our analysis, we find that M1 transitions are different contexts, for example, due to fascinating poten- prospective candidates for resonant coherent driving of tial applications related to the controlled release of nuclear nuclei due to their large transition probabilities and suit- energy on demand, such as in nuclear batteries, or moti- able energies. In particular for excited levels lying below vated by the fundamental challenge to understand the for-

23 mation of isomers and their role in the evolution of the universe [14]. Isomer triggering is schematically presented Table 1: Total resonance strengths S (in b eV) for NEEC in Fig. 2 for the case of 93Mo, and can occur via a number and x-ray triggering of isomers. EI and ET are the iso- of nuclear excitation mechanisms such as photoabsorption, meric state and triggering level energies, respectively. A I→F I→F Coulomb excitation or coupling to the atomic shells. Z X EI (keV) ET (keV) SNEEC Sx−ray Motivated by the need to identify efficient triggering 93 −6 −8 42Mo 2424.89 2429.69 9.1 10 1.4 10 mechanisms, the possible isomer activation by coupling to 152 × −4 × −5 63 Eu 45.599 65.296 3.4 10 6.5 10 the atomic shell in the process of nuclear excitation by 178 × −7 × −8 72 Hf 2446.05 2573.5 2.0 10 5.4 10 electron capture (NEEC) is investigated. NEEC is a reso- 189 × −3 × −2 76 Os 30.812 216.661 1.2 10 2.2 10 nant process in which the collision of a highly-charged ion 204 × −5 × −6 82 Pb 2185.79 2264.33 4.9 10 8.7 10 with a free electron with matching kinetic energy leads to 235 × −1 × −2 92 U 0.076 51.709 1.3 10 1.3 10 a resonant capture into an atomic orbital with simultane- 242 × −3 × −8 95 Am 48.60 52.70 3.6 10 2.4 10 ous excitation of the nucleus. We envisage NEEC isomer × × triggering via the nearest triggering level lying above the metastable state that has been either experimentally con- ratios in photoexcitation and NEEC triggering can be dif- firmed or for which the corresponding transition is theoret- ferent depending on the initial electronic configuration of ically predicted. Low-lying triggering levels are desirable the isomer. for obtaining high energy gain and facilitating the excita- The values in Table 1 show that for the considered tion to the triggering level. low-lying triggering levels, the NEEC nuclear excitation The total resonance strength, which is the integral of the mechanism is more efficient than the photoabsorption one, total cross section σ over the continuum electron energy, 189 with the exception of 76 Os. For this isomer, the photoex- for NEEC followed by the decay of the triggering level to citation rate is higher than the NEEC rate because of the the state F is given (in atomic units) by [15] large transition energy (185 keV). A further comparison with nuclear excitation by electron transition and Coulomb 2π2 242 SI→F = Y I→T BT→F , (1) excitation cross sections from Ref. [17] for 95 Am proves NEEC p2 that NEEC is the most efficient triggering mechanism for this isotope. where p denotes the momentum of the continuum electron, I→T T→F For practical purposes, the feasibility of isomer trig- Y is the NEEC rate and B is the branching ratio gering schemes is still a debated issue, especially in view for the decay of the triggering level T avoiding the iso- of several photoexcitation triggering experiments [18]. In meric state. The NEEC rate is calculated using a rigorous contrast to other processes involving coupling of nuclei to treatment of the electron-nucleus interaction following the the atomic shell, NEEC has not been observed until now, formalism developed in [16]. The values for the NEEC res- mostly due to the strong atomic background and the nar- onance strength for triggering of a number of isomers with row nuclear state widths. An important new aspect when low-lying triggering levels are presented in Table 1. As considering isomer triggering via NEEC is that the ener- a comparison, photoexcitation resonance strengths using gies of the excitation from the isomeric state and of the the same triggering levels are also presented. The pho- I→F signal photons emitted in the decay of the triggering level toexcitation resonance strength Sx−ray has an expression are different, thus reducing substantially the experimental equivalent to the one in Eq. (1), involving the photon mo- background due to radiative recombination. Using real- mentum and nuclear photoexcitation rate. The branching istic parameters of the future GSI facility in Darmstadt, we obtain a reaction rate of 6.5 10−2 s−1 for the NEEC 235m × −3 −1 17/2+ triggering of U and of 1.1 10 s for the case of Triggering level T 189mOs [15]. × 2429.69 keV In conclusion, our theoretical calculations show that E2 E2 among the investigated isomer triggering mechanisms, the 21/2+ NEEC Isomeric state I coupling to the atomic shell via the process of nuclear excitation by electron capture is the most efficient one. An E4 2424.89 keV 13/2+ experimental verification of our findings at the borderline First decay state of atomic and nuclear physics may be provided by upcom- of cascade F ing ion storage ring facilities and ion beam traps which decay cascade 2161.83 keV will commence operation in the near future. 5/2+ Ground state GS 4 Isotope effects in high-harmonic spectra 0.00 keV from compact atomic systems 93 Figure 2: Partial level scheme of 42Mo. The isomeric state 4.1 Muonic atoms (I) can be excited to the triggering level (T ) which subse- quently decays back to I or to a level F , initiating a cas- Muonic atoms represent traditional tools for nuclear spec- cade via different intermediate states (dashed line) to the troscopy by employing atomic physics techniques [19]. Due ground state (GS). The direct I F decay is a strongly to the large muon mass mµ 200me as compared to the hindered E4 transition. → electron mass and the correspondingly≈ small Bohr radius,

24 -4 the muonic wave function has a large overlap with the nu- 10 -7 cleus. Precise measurements of x-ray transitions between 1×10

-8 stationary muonic states are therefore sensitive to nuclear 5×10 structure features such as finite size, deformation, surface 0 thickness, or polarization. When a muonic atom is sub- 1056 1058 1060 -8 ject to a strong laser field, the muon becomes a dynamic 10 nuclear probe which might provide additional information on nuclear properties as compared to the usual field-free muonic atom spectroscopy. Against this background, we have studied the harmonic response of muonic hydrogen Harmonic Signal [arb. u.] isotopes in intense VUV laser fields [20]. Below we demon- -12 10 strate signatures of the finite nuclear mass and size in the radiation emitted by the muonic charge cloud which oscil- 0 200 400 600 800 1000 lates across the nucleus under the influence of the external Harmonic Order field. Our calculation is perfomed within the nonrelativistic Figure 3: Finite nuclear size effects in the high-harmonic framework of the Schrödinger theory. Since the nuclear response from different muonic hydrogen isotopes. The mass mn cannot be assumed as infinite in muonic atoms, black line shows the spectrum for muonic hydrogen at the the motion of both binding partners must be taken into ac- laser parameters I(H) = 8.5 1022 W/cm2 and ℏω(H) = count. The corresponding two-particle Schrödinger equa- 59 eV. The red line gives the× spectrum for muonic deu- tion separates into center-of-mass and relative coordinates terium at the scaled laser parameters I(D) = 1.05 when the laser field is treated in dipole approximation. 1023 W/cm2 and ℏω(D) = 62 eV (see text for explanation).× The higher harmonics are generated by the relative mo- The inset shows a blow-up of the cutoff region on a linear tion which is nonlinear and governed by scale. ~2 2 ~ ∂ ∂ i ψ (x, t) = 2 exF (t) + V (x) ψ (x, t) (2) driven by the laser field into opposite directions along the ∂t −2mr ∂x − polarization axis. Upon recombination their kinetic ener- where e is the muon charge, F (t) = F0 sin(ωt) the laser gies sum up as indicated on the right-hand side of Eq. (3). field, V (x) the nuclear Coulomb potential, and mr = Within this picture, the larger cutoff energy for muonic hy- mµmn/(mµ + mn) the reduced mass. By applying a stan- drogen results from the fact that due to its smaller mass, dard scaling transformation: t = ρt′ and x = ρx′, with the proton is more strongly accelerated by the laser field ρ = me/mµ, one can recast Eq.(2) into a form that essen- than the deuteron. tially describes an ordinary hydrogen atom in a laser field In order to reveal the effects arising from the finite nu- ′ ′ 2 of the scaled parameters ω = ρω, F0 = ρ F0. The scal- clear size, we employ the nuclear drop model and consider ing procedure does not account for the finite nuclear size, the nucleus as a sphere of uniform charge density within though. But it shows that high laser frequencies and in- the nuclear radius R. The proton and deuteron charge tensities are required for an efficient laser-muon coupling, radii amount to Rp 0.875 fm and Rd 2.139fm, respec- which is also evident by the large binding energies and tively. The nuclear potential≈ accordingly≈ reads Coulombic field strengths in muonic atoms. In fact, in 2 2 the ground state of muonic hydrogen the muon is bound e 3 x R 2 2R2 for x R, by 2.5keV at a Bohr radius of 280fm and experiences an V (x) = − 2 − | | ≤ (4)  e for x > R. electric field intensity of 4.2 1025 W/cm2. − |x| | | The Schrödinger equation× (2) is numerically solved and Figure 3 shows the harmonic spectra resulting from this the harmonic spectrum obtained from the resulting dipole  potential for muonic hydrogen versus muonic deuterium. acceleration via a Fourier transformation. The main in- The above-mentioned nuclear mass effect was compensated fluence of the nuclear mass on the spectrum can be in- for by applying properly scaled frequencies and intensities fered directly from Eq. (2). The harmonic cutoff position with ω m and F m2 so that the cutoff positions is determined by the formula ~ω = ǫ + 3.17U , with r 0 r max b p N =∝ω /ω of both∝ spectra coincide. We see that the binding potential ǫ and the ponderomotive energy U max max b p the harmonic signal from muonic hydrogen is larger (by (see, e.g., [21]). In the present case, the latter amounts to about 50 % in the cutoff region) than that from muonic 2 2 2 2 deuterium. The reason is that a smaller nuclear radius e F0 e F0 1 1 Up = 2 = 2 + (3) steepens the potential near the origin, which leads to more 4ω mr 4ω mµ mn violent acceleration and enhanced harmonic emission. The and is, thus, the larger the smaller the reduced mass is. finite nuclear size has therefore an impact on the plateau Consequently, in an intense laser field with Up ǫb, height of high-harmonic spectra from muonic atoms. ≫ muonic hydrogen gives rise to a larger cutoff energy than Our results demonstrate that muonic atoms in high- muonic deuterium; the relative difference is about 5% ac- intensity, high-frequency laser fields can, in principle, be (H) (D) (D) (H) cording to ωmax/ωmax mr /mr 1.05. The nuclear utilized to dynamically gain structure information on nu- mass effect can also be≈ explained by≈ the separate motion clear ground states. Moreover, the considered setup offers of the atomic binding partners. The muon and nucleus are prospects for pump-probe experiments on excited nuclear

25 levels: the periodically driven muon can ﬁrst excite the nu- a) b γ x y ) z x cleus and then probe the excited state and its deexcitation φ mechanism. ion

b 4.2 Highly-charged ions z y ˆsrest Finite nuclear size effects also appear in high-precision ion spectroscopic studies of electron transitions in ordinary highly charged ions such as hydrogenlike uranium where the K-shell radius is about 500 fm and the nuclear size Figure 4: (a) Heavy ion collisions with non-zero impact pa- R 7 fm. In the harmonic signal produced by such sys- rameter b may produce a global spin polarization of quarks tems≈ in the presence of a superintense laser wave, nuclear along yˆ as proposed in Ref. [25]. (b) Such a global quark signatures could be visible, too. It is interesting to com- spin ˆs = yˆ could be detected through the azimuthal pare the expected effects with those found for laser-driven rest φ-dependence of photons emitted from the plasma [24]. muonic atoms via a simple analysis based on nonrelativistic estimates [22]. Note that our approach to muonic atoms which employs Schrödinger theory is not applica- and RHIC and will soon be accessible also at the LHC. In ble to highly-charged ions where the bound electron moves order to learn more about the QGP, one would like to have relativistically. a primary probe directly from its interior, which is difficult In order to draw a comparison, we assume an electronic due to its very short lifetime τ 5 fm/c. Such a probe and a muonic hydrogenlike system of nuclear charge num- is given by direct photons that, when∼ produced, typically bers Z1 and Z2, respectively, and employ the mass scaling leave the plasma without further interaction. Recently, we parameter ρ (see above) with ρ1 = 1 for an electronic ion have suggested that a possible quark spin polarization in a and ρ2 1/200 for a muonic atom. The K-shell Bohr ra- QGP can be detected through circularly polarized photons ≈ dius, binding energy, and electric field strength amount to emitted from the plasma [24]. 2 aK (Z, ρ) = a0ρ/Z, ǫK (Z, ρ) = ǫ0Z /ρ, and FK (Z, ρ) = A global polarization of quarks can be produced in non- 3 2 F0Z /ρ , respectively, where a0, ǫ0 and F0 denote the central heavy-ion collisions through spin-orbit coupling corresponding values for ordinary hydrogen. The nuclear [25] (see Fig. 4). The polarization can be transferred to radius is approximately given by R(Z) 1.2(2Z)1/3 fm massive particles like Λ-hyperons. Experimentally, an up- ≈ and typically varies among different isotopes by a few per- per limit of the polarization P ¯ . 0.02 has been ob- | Λ,Λ| cent (except for hydrogen vs. deuterium). Similarly pro- tained [26], while recent calculations suggest PΛ 0.05 nounced nuclear size effects in the harmonic spectra can [27]. But hyperon polarization is affected by all stages≈ − of be expected when R(Z1)/aK(Z1, ρ1) R(Z2)/aK(Z2, ρ2), the collision to an unknown degree [28]. In the following, ≈ i.e., when the ratio between the nuclear and the atomic we estimate to what degree a global quark polarization radius which is proportional to Z4/3/ρ, has a similar value would be transferred to the polarization of photons. for both atomic systems. This holds, e.g., for electronic Direct photons can be produced in the QGP through 91+ 2+ U (where Z1 = 92, ρ1 = 1) and muonic He (where Compton scattering of (anti-)quarks and gluons, qg qγ → Z2 = 2, ρ2 = 1/200). Consequently, within this simple (¯qg qγ¯ ), and annihilation of quarks and antiquarks, qualitative consideration we find that the importance of qq¯ →gγ. The calculation of the production rate of photons finite nuclear size effects in very heavy electronic systems with→ momentum q = (E, q) requires a separation into hard and light muonic atoms should be comparable. Proposed ( T ) and soft ( gT ) momentum transfer contributions studies on the harmonic radiation emitted by strongly [29],∼ with g being∼ the strong coupling constant. As in laser-driven highly-charged ions (e.g. [21, 23]) could there- [30], we restrict ourselves to hard 2 2 particle processes fore aim for detection of nuclear signatures, as well. and corresponding soft processes. For↔ hard momenta, we One should observe, however, that the binding energy obtain the rate contributions from Compton scattering Rc and electric field strength in a highly-charged ion are sub- or annihilation Ra through (~ = c = kB = 1) stantially larger than in a muonic atom when both have 4/3 dR the same ratio of Z /ρ. I.e., the laser frequency and in- E c = 20π f F (q )f B(q ) S 2 f F (q ) 2 , d3q ξ 1 ξ 3 |Mc | − ξ 2 |Mc| tensity that must be applied to electronic ions in order to Zδ reveal nuclear size effects in the harmonic response need to dR 320π E a = f F (q )f F (q ) 1 + f B(q ) 2,(5) be larger than for muonic atoms. In this respect, muonic d3q 3 ξ 1 ξ 2 ξ 3 |Ma| Zδ atoms are more favorable systems to study the influence of 3 3 3 9 (4) the nuclear size on the high-harmonic generation process. with d q1d q2d q3/(2π) δ (q1 + q2 q3 δ ≡ − − q)/(8E1E2E3) (the momenta qj are defined in Fig. 5), R R F/B 5 Photon polarization as a probe for and fermionic or bosonic distribution functions fξ . The quark-gluon plasma dynamics numerical prefactors include summing over Nc = 3 col- ors and up and down quarks with current quark mass The study of the quark-gluon plasma (QGP) is one of the m, for which the ultrarelativistic limit (T m) ap- cornerstones at the future GSI-FAIR facility [10]. This new plies. Pauli blocking is taken into account by≫ including state of matter exists at temperatures above T & 170 MeV. the scattering Compton matrix element either polarized It has been studied in heavy-ion collisions at CERN/SPS ( ), or summed over outgoing quark polarization ( S). Mc Mc 26 a) b) c) In a heavy-ion collision, an expansion of the QGP along q1 q3 q the beam axis may induce momentum space anisotropy q ρ that is described by stretching the isotropic distribution + q k F/B F/B F/B ρ k q1 q2 functions fiso according to fξ ( ) = fiso (kξ), with − q3 ρ+ ρ+ q + k 2 2 2 2 ρ+ kξ = kx + ky + (1 + ξ)kz , and the anisotropy parame- q2 ter ξ > 1 [30]. We use the temperature T as hard energy scale,− even though it is properly defined only in the isotropic case. Figure 5: Feynman diagrams corresponding to (a) Comp- For an ensemble of quarks, states with partial polar- ton scattering, (b) quark-antiquark annihilation, and (c) ization can be expressed in the ultrarelativistic limit us- photon self-energy (straight, curly, and wiggly lines repre- ⊥ ing the density matrices ρ = κ/ 1 pkγ5 + p⊥γ5/sˆ /2, sent quarks, gluons, and photons). Arrows indicate where ± ∓ where the upper (lower) sign refers to particles (antipar- polarized states are used in the calculation. The gray filled ticles) [31], and pk and p⊥ denote the longitudinal and circle indicates a soft propagator dressed by the plasma transverse components of the spin s with respect to the medium. quark momentum κ [32]. In accordance with global polarization proposed for non-central collisions [25, 27] (see

D unpolarized

Fig. 4), we assume that the spin of each quark in its rest 4 0.00015 frame is aligned along the same direction yˆ and that par- - ticles with same energy κ0 share the same degree of po- GeV 4

∗ - larization prest(κ0) =p ¯restΘ(κ0 k ), with global degree 0.00010

− fm @ of polarizationp ¯rest and the Heavyside Θ-function. The 2 left right ∗ T energy threshold k is chosen equal to the separation scale q 3 0.00005 between soft and hard modes. For the Compton scattering d diagram in Fig. 5(a) with electron charge e we obtain EdR 2 2 2 e g k k q q1 q q2 0.00000 c = 1 + p (q1)p (q2) · + · 0 0.5Π Π 1.5Π 2Π |M | 12 q q2 q q1 · · Φ k k q q2 q q1 ∗ + p (q1) + p (q2) · · idet ǫǫ qˆ q q1 − q q2 | | · · ⊥ ⊥ ⊥ ⊥ 1 1 Figure 6: Photon rate for E/T = 5 and ξ = 10 for full 2p (q1)p (q2) q sˆ q sˆ − · 1 · 2 q q − q q quark polarizationp ¯rest = 1, separated in left and right · 1 · 2 ⊥ ⊥ ⊥ ⊥ circularly polarized and unpolarized photons, as a function q sˆ q1 sˆ q sˆ q2 sˆ +ˆs⊥ sˆ⊥ · 1 · 2 · 2 · 1 (6) of the azimuthal angle φ in the x-y plane. The shaded 1 · 2 − q q − q q · 1 · 2 #) areas indicate the error estimate from varying the hard- soft separation scale by a factor 2 up (delimited by dashed ⊥ ⊥ ⊥ with transverse spin directionss ˆ1 =s ˆ (q1) ands ˆ2 = line) or down (delimited by dotted line) around its central ⊥ S 2 sˆ (q2). For unpolarized outgoing quarks one has c = value (thick line). 2 2, with pk(q ) = p⊥(q ) = 0. The scattering|M | ma- |Mc| 2 2 trix a for the annihilation diagram in Fig. 5(b) fol- lowsM from Eq. (6) by crossing symmetry. The pieces cular polarization with i det ǫǫ∗qˆ = +( )1. The pho- k k 3 | | − quadratic in the polarization, proportional to p (q1)p (q2) ton rate EdR/d q is finally given by adding the results of and p⊥(q )p⊥(q ), vanish independently in the infrared Eqs. (5) and (7), with an intermediate cutoff k∗ √gT 1 2 ∼ (IR) limit through the angular integrations in Eq. (5), but using Monte Carlo integration [30]. they may still contribute at larger momentum transfers as Figure 6 shows the photon rate for moderate anisotropy we find numerically. Only the piece linear in the polar- ξ = 10 as a function of the angle φ in the x-y plane (see k k ization, proportional to p (q1) + p (q2), survives in the IR Fig. 4). For the QGP where the hard energy scale exceeds limit. This piece has a matching ultraviolet contribution T & 200 MeV, the ratio E/T = 5 corresponds to photon from the soft sector. Its result can be obtained from the energies of 1GeV or higher. One observes a maximum for imaginary part of the photon self-energy in Fig. 5(c). One left circular polarized photons along the direction of the finds within the Hard Loop approximation (which assumes global spin. This is in accordance with high-energy Comp- gT k q T ) [29] ton scattering, where helicity states couple to circularly ∼ ≪ ∼ polarized photons. Results for other ξ values look quali- dR 1 E soft = 1 + pk(q)idet ǫǫ∗qˆ tatively similar. The maximum degree of circular polar- d3q −(2π)3 | | ization decreases for smaller anisotropies, but increases for 5f F (q) d3k larger energies. For energies E/T & 4 the result appears e2 ξ Im q S⋆ (k) (7) × 3 q (2π)3 · R to be rather insensitive to the choice of the intermediate | | Z cutoff scale k∗. Thus, the quark polarization is imprinted ⋆ with the retarded dressed fermionic propagator SR(k) [33]. on emitted photons, and photon polarization can be used The determinant in Eqs. (6) and (7) vanishes for linear to measure QGP spin polarization, provided one can de- polarization, but contributes for right-(left)-handed cir- tect the polarization state of high-energy photons. Since

27 unpolarized photons are routinely observed in high-energy [10] W. Henning, Nucl. Phys. A 734, 654 (2004); D. collisions [34], the main experimental challenge is to de- H. H. Hoffmann et al., Laser Part. Beams 23, 47 tect circularly polarized photons in the GeV energy range (2005); see also the homepage of the FAIR project [35]. One could use a single aligned crystal to convert cir- http://www.gsi.de/fair . cular polarization into linear polarization that can then [11] T. J. Bürvenich, J. Evers, and C. H. Keitel, Phys. be analyzed by means of another aligned crystal through Rev. Lett. 96, 142501 (2006). coherent electron-positron pair production [35, 36]. [12] T. J. Bürvenich, J. Evers, and C. H. Keitel, Phys. Rev. C 74, 044601 (2006). 6 Conclusion [13] A. Pálffy, J. Evers, and C. H. Keitel, arXiv:0711.0015 [nucl-th]. Almost 50 years after the invention of the laser and its [14] P. M. Walker and G. D. Dracoulis, Nature 399, 35 manifold applications in atomic, molecular, and solid state (1999); A. Aprahamian and Y. Sun, Nature Phys. 1, physics, the door is now opening towards the field of laser- 81 (2005). nuclear physics. We have shown that direct excitation of [15] A. Pálffy, J. Evers, and C. H. Keitel, Phys. Rev. Lett. nuclei by x-ray radiation sources will become feasible. The 99, 172502 (2007). transitions of interest are not limited to electric dipole- [16] A. Pálffy, W. Scheid, and Z. Harman, Phys. Rev. A allowed ones, but also comprise electric quadrupole and 73, 012715 (2006). magnetic dipole transitions. Moreover, the triggered decay [17] A. A. Zadernovsky and J. J. Carroll, Hyperfine Inter- of isomeric nuclear levels has been considered, which lib- act. 143, 153 (2002). erates the energy stored in these metastable states. It was [18] C. B. Collins et al., Phys. Rev. Lett. 82, 695 (1999); found that isomer triggering via nuclear excitation by elec- Europhys. Lett. 57, 677 (2002); Laser Phys. Lett. tron capture is more efficient than alternative mechanisms 2, 162 (2005); I. Ahmad et al., Phys. Rev. Lett. such as resonant photo-absorption. Furthermore, nuclear 87, 072503 (2001); Phys. Rev. C 67, 041305 (2003); signatures were demonstrated in the high-harmonic radi- Phys. Rev. C 71, 024311 (2005). ation emitted by muonic hydrogen isotopes in very strong [19] E. Borie and G. A. Rinker, Rev. Mod. Phys. 54, 67 VUV laser fields. The laser-driven muon is, in principle, (1982). also capable of nuclear excitation and subsequent probing of the excited level. Analogous studies are conceivable [20] A. Shahbaz, C. Müller, A. Staudt, T. J. Bürvenich, 98 with electrons bound to highly-charged ions like uranium. and C. H. Keitel, Phys. Rev. Lett. , 263901 (2007). Finally, we have studied the polarization of direct photons [21] Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and emitted from the QGP as a probe of its dynamics. It was C. H. Keitel, Phys. Rep. 427, 41 (2006). found that global quark spin is effectively transfered to cir- [22] A. Shahbaz, C. Müller, T. J. Bürvenich, and C. H. cular polarization of photons. The detection of circularly Keitel, in preparation. polarized photons from the bulk of the QGP would there- [23] M. Casu and C. H. Keitel, Europhys. Lett. 58, 496 fore indicate to what extent quarks within the plasma are (2002); S. X. Hu, A. F. Starace, W. Becker, W. Sand- on average polarized. ner, and D. B. Milosevic, J. Phys. B 35, 627 (2002); C. C. Chirilˇa, C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, Phys. Rev. Lett. 93, 243603 (2004). References [24] A. Ipp, A. Di Piazza, J. Evers, and C. H. Keitel, [1] H. Schwoerer, J. Magill, and B. Beleites (Eds.), Lasers arXiv:0710.5700 [hep-ph]. and Nuclei (Springer, Heidelberg, 2006). [25] Z.-T. Liang and X.-N. Wang, Phys. Rev. Lett. 94, [2] S. Matinynan, Phys. Rep. 298, 199 (1998). 102301 (2005). [26] I. Selyuzhenkov, J. Phys. G 32, S557 (2006). [3] T. Ditmire et al., Nature (London) 386, 54 (1997); et al. ibid. 398, 489 (1999). [27] J.-H. Gao , arXiv:0710.2943 [nucl-th]. [28] B. Betz, M. Gyulassy, and G. Torrieri, Phys. Rev. C [4] K. W. D. Ledingham et al., Phys. Rev. Lett. 84, 899 76, 044901 (2007). (2000); T. E. Cowan et al., ibid. 84, 903 (2000). [29] J. I. Kapusta and C. Gale, Finite-Temperature Field [5] D. Umstadter, J. Phys. D 36, R151 (2003); K. W. D. Theory (Cambridge University Press, 2006). Ledingham, P. McKenna, and R. P. Singhal, Science [30] B. Schenke and M. Strickland, Phys. Rev. D 76, 300 , 1107 (2003). 025023 (2007). [6] A. A. Sorokin et al., Phys. Rev. Lett. 99, 213002 [31] J. D. Bjorken and S. D. Drell, Relativistic Quantum (2007); for current information on the FLASH facility Mechanics (Boston, McGraw-Hill College, 1965). see also http://www-hasylab.desy.de/facility/fel . [32] H. A. Olsen, P. Osland, and I. Overbo, Nucl. Phys. B [7] P. Neumayer et al., Appl. Phys. B 78, 957 (2004). 171, 209 (1980). [8] M. Altarelli et al., XFEL: The European X-Ray [33] B. Schenke and M. Strickland, Phys. Rev. D 74, Free-Electron Laser. Technical Design Report, DESY, 065004 (2006). Hamburg 2006, The European X-Ray Laser project [34] S. S. Adler et al., Phys. Rev. Lett. 94, 232301 (2005). homepage, http://www.xfel.net . [35] U. I. Uggerhoj, Rev. Mod. Phys. 77, 1131 (2005). [9] G. D. Tsakiris et al., New J. Phys. 8, 19 (2006); A. [36] N. Cabibbo et al., Phys. Rev. Lett. 9, 435 (1962). Pukhov, Nature Physics 2, 439 (2006).

28 QED effects in strong laser fields A. Di Piazza, E. Lötstedt, K. Z. Hatsagortsyan, U. D. Jentschura, and C. H. Keitel Max-Planck-Institut fur¨ Kernphysik, D-69117 Heidelberg, Germany

Quantum electrodynamical processes occurring Such strong magnetic fields cannot be created in labora- in the presence of a strong laser field are dis- tory and this is why vacuum effects in the presence of cussed. We review the processes of vacuum high- strong magnetic fields can significantly occur only in as- harmonic generation and light-by-light diffraction trophysical environments like around highly magnetized that we have studied recently as a means of test- neutron stars [16, 17, 18]. ing vacuum polarization effects in a strong laser The rapid development of laser technology allows for field. Then, we investigate in detail two processes the use of intense laser fields even to probe QED in the which are more feasible experimentally: laser pho- presence of a strong wave field. Table-top multiterawatt ton merging in laser-proton collisions and laser- lasers which are already available are employed to cre- assisted bremsstrahlung. ate new x-ray and γ-ray radiation sources [19], to ac- celerate electrons, protons and ions to high energies [20] 1 Introduction and even to prime nuclear fusion reactions [21]. More- over, theoretical proposals have been put forward to reach Quantum Electrodynamics (QED) has been tested exper- the critical electric field by focusing the high-order har- imentally under very different conditions. Its predictions monics generated in the reflection of a strong laser beam especially in atomic physics have been confirmed by exper- by a plasma surface [22]. A laser field with an elec- iments with very high accuracy like those on the anoma- tric field amplitude Ecr would have an intensity of Icr = 2 29 2 lous magnetic moment of the electron or the Lamb shift Ecr/8π = 2.3 × 10 W/cm . The most intense laser field (see the recent reviews [1, 2] and the references therein). ever produced in a laboratory has an intensity of “only” However, many of these tests relate to perturbative QED 7 × 1021 W/cm2 [23]. Numerous Petawatt laser systems in which the theoretical predictions are extracted starting are under construction in different laboratories, as e. g. from the Dyson expansion of the S-matrix with respect to at GSI [24] and at Jena [25], capable, in principle, to at- the fine-structure constant α = e2/4π ≈ 1/137 [3]. Here, tain an intensity of about 1023 W/cm2 [26]. Also, the we have introduced the electron charge −e < 0 and we Extreme Light Infrastructure (ELI) is expected to reach have used, as throughout this Report, natural units with unprecedented intensities of about 1025-1026 W/cm2 [27]. ~ = c = 1. Experimental tests of QED in the presence Although spontaneous pair creation in vacuum is expo- of classical strong electromagnetic fields have so far been nentially suppressed at fields below the critical field, vac- successful mostly in the case of strong Coulomb fields, i. e. uum manifests nonlinear properties due to the presence the fields created by highly charged nuclei. In the context of virtual electron-positron pairs (see the recent reviews of QED the expression “highly charged nuclei” indicates [28, 29, 30]). However, the only successful experiment on nuclei with a charge number Z . 1/α ≈ 137. In fact, vacuum nonlinearities induced by intense laser beams, so these nuclei produce an electric field of the order of the far was performed at SLAC where electron-positron pairs 2 16 “critical” electric field Ecr = m /e = 1.3 × 10 V/cm (m have been produced in the collision of a high-energy elec- is the electron mass) at the typical QED length given by tron beam with a strong laser pulse [31]. the Compton length λc = 1/m. In turn, an electric field of In the following, we first review a few processes that have the order of the critical field Ecr is able to create electron- been considered in order to measure nonlinear vacuum po- positron pairs in vacuum [4]. The Coulomb field produced larization effects. Then, two examples which indicate the by highly charged nuclei is so strong that higher orders in possibility of testing QED in the presence of strong laser Zα contribute significantly to the QED predictions at the beams are reported. In the first example we consider the current level of experimental accuracy, even for low nu- possibility of laser photons merging when they interact clear charge number (see the recent Refs. [5, 6, 7, 8, 9]). with the electromagnetic field of a high-energy proton [32]. In some cases, as for the vacuum polarization effects in- In the second one we study laser-assisted bremsstrahlung duced by strong classical electromagnetic fields, even all [33, 34]. orders in Zα have to be included because nonperturbative effects cannot be neglected. This has been experimentally 2 Vacuum nonlinearities in strong laser tested in the case of the Delbruc¨ k scattering [10], i. e. the fields scattering of a photon by the Coulomb field of a heavy nucleus and of the photon splitting in a strong Coulomb In this Section we shortly review a few processes that we field [11]. The theoretical predictions developed in [12, 13] have recently studied to detect the nonlinear properties of have been confirmed. quantum vacuum in the presence of strong laser beams ex- Vacuum polarization effects in strong constant and uni- perimentally. In [35] we have investigated the possibility of form magnetic fields have also been studied theoretically. observing high-harmonic generation in the collision in vac- These effects become apparent in the presence of magnetic uum of two ultra-strong laser beams. We have found that fields with an amplitude of the order of the so-called crit- nowadays only the scattering of two photons is experimen- 2 13 ical magnetic field Bcr = m /e = 4.41 × 10 G [14, 15]. tally feasible by making three lasers collide (laser-assisted

29 and frequency ω0 such that the parameter ξ = eE0/mω0 is much larger than unity. We assume that the laser beam propagates along the positive y direction and that it is linearly polarized along the z direction. The proton moves with velocity β along the negative y direction. If ϑ is the angle between the outgoing photon and the y direction, it can be shown that the differential rate dR2n/dϑ is given by Figure 1: Feynman diagram corresponding to the process 3 4 3 2 2 of laser photon merging in a proton field induced by vac- dR2n α (1 + β)m sin ϑ |c1,2n| + |c2,2n| = 2 3 4 3 . uum polarization effects. dϑ 64π ω0 (1 − cos ϑ) n (1) In this expression we have introduced the coefficients cj,2n photon-photon scattering). This process seems to be the 1 ∞ most promising to detect for the first time vacuum nonlin- −iπ/3 dλ − exp(iπ/3)λ−x2n cj,2n = e dv e bj,2n (2) earities in a strong laser field and could, in principle, be 0 0 λ observed with the next generation of Petawatt laser sys- Z Z 2 3 2 2 tems. In [36] we have investigated how a strong optical where j ∈ {1, 2}, x2n = χ2nλ (1 − v ) /96 and where standing wave can “diffract” an x-ray probe beam that 4/j passes through it. The diffraction affects the polarization 2 2 1 − v 0 In(x2n) bj,2n = jχ2nλ [In(x2n)−In(x2n)]+ , (3) of the probe beam of an amount large enough that it could 16 λ be measurable in the near future. Finally, in [37] we stud- with In(x) being the modified Bessel function of order n ied the process of photon splitting in a strong laser field and I0 (x) its derivative. As it is clear, the rate depends of arbitrary shape and polarization. However, the exper- n only on the parameter χ2n which is given by imental observation of this process is more problematic; more details can be found in [37]. I 2n(1 + β)ω 1 − cos ϑ χ = 0 0 . (4) 2n I m 1 + β cos ϑ r cr 3 QED effects in laser-proton collisions We consider below a numerical example illustrating the In this section we investigate the vacuum polarization ef- possibility of observing the process of photon merging fects arising from the head-on collision of a high-energy experimentally. We consider the interaction between a proton and a strong laser beam. The proton has unique proton bunch with parameters available at the LHC and features that allow, as we will see, the detection of these a Petawatt laser pulse. We use the following laser pa- effects. In fact, on the one hand, the proton is light enough rameters [39, 27]: a pulse energy of 5 J, a pulse dura- to be accelerated to very high energies like up to 7 TeV at tion of 5 fs at 10 Hz repetition rate and an intensity of 22 2 the Large Hadron Collider (LHC) [38]. This implies that I0 = 5 × 10 W/cm . The main parameters of the proton the laser field in the rest frame of the proton is enhanced bunch are [38]: a proton energy of 7 TeV, a number of pro- by a large factor compared to its value in the laboratory tons per bunch of 11.5 × 1010, a bunch transversal radius frame. On the other hand, the proton is heavy enough that of 16.6 µm, a bunch length of 7.55 cm. As we have men- the multiphoton Thomson scattering of the laser photons tioned, the (2n)-photon Thomson scattering of the laser by the proton is negligible. This feature is very impor- photons by the proton beam is a competing process of tant because, in general, multiphoton Thomson scattering (2n)-photon merging. In fact, it can be seen that the en- represents a background of our process. The Feynman ergies of the photons produced via (2n)-photon merging diagram of the photon merging process is represented in and via (2n)-photon Thomson scattering are equal. Then, Fig. 1. The diagram shows that the proton is consid- the total photon rate has to be calculated by summing up ered, due to its large energy, as an external field. The the amplitudes of the two processes. In Fig. 2 we compare thick electron-positron loop in Fig. 1 indicates that the the differential rate dW (2)/dϑ of the photons emitted only propagators are calculated by exactly taking into account via 2-photon Thomson scattering with the total differen- the presence of the laser field. From a physical point of tial rate dT (2)/dϑ which also includes the photons result- view, the merging of the laser photons is mediated by the ing from the merging (continuous line). The contribution virtual electron-positron pair that absorbs a certain num- of the vacuum effect is rather large. This becomes more ber of laser photons and emits only one. From the Furry apparent if we compare the total rate of photons emitted theorem it can be inferred that only an even number of only via 2-photon Thomson scattering with the total rate photons can be merged in the process [3]. The calcula- of the two processes together. In fact, it can be shown that tion of the rate R2n of photons resulting from the merging approximately 888 events per hour are obtained in the first of 2n (n ≥ 1) laser photons follows the usual steps. One case while about 1850 events are predicted per hour in the calculates the amplitude of the process according to the second case. The total rate of photons resulting only from Feynman rules and then applies the Fermi golden rule [3]. 2-photon merging is about 1140 photons per hour. This The details of the calculations can be found in [32]. Be- implies a small, destructive interference effect between the low, we consider only the case of an ultra-relativistic op- two processes. Finally, it is evident in Fig. 2 that the 2 tical laser field with amplitude E0, intensity I0 = E0 /8π values of the parameter χ2 in the relevant region of the

30 qf qf kb q

p˜f kb p˜i

q qi qi

Figure 3: Feynman diagrams describing the process of laser-assisted bremsstrahlung. Thick lines indicate that Figure 2: The rate per unit angle ϑ of photons emitted only Volkov wave functions and Volkov propagators are used via 2-photon Thomson scattering (dashed line) and via for the electron, thus treating the electron-laser interaction both 2-photon Thomson scattering and 2-photon merging nonperturbatively. The effective four-momentum of the (continuous line). The upper horizontal axis shows the initial (final) electron is qi (qf ) and the four-momentum parameter χ2 as a function of ϑ [see also Eq. (4)]. of the intermediate electron state is denoted by p˜i,f . The virtual Coulomb photon with three-momentum q is drawn spectrum are of the order of unity: the perturbative ap- as a dashed line, and the emitted photon with momentum proach, which is valid as χ2 1, does not apply. Finally, kb as a wavy line. one can also show that the rate of photons resulting from the merging of four laser photons amounts to 19.5 events per hour. In this case it can be seen that the 4-photon squared effective electron mass, ωb is the frequency of the bremsstrahlung photon, p˜i,f = qi,f − sk0 + kb is the inter- Thomson scattering is safely negligible due to its scaling s,n 8 −2 mediate four-momentum, ` is a screening length, and A as ξp , with ξp = (m/M)ξ ≈ 8.3 × 10 (here M indicates i,f the proton mass). are certain functions depending on the laser parameters. In a strong laser field, photon emission can occur even 4 Laser-assisted bremsstrahlung without a Coulomb field through laser-induced Compton scattering. This means that for values of ωb satisfying 2 2 As we have mentioned in the Introduction, one of the goals the modified Compton formula, we may have p˜i,f = m∗, of theoretical high-energy laser physics is to study the in- with a formally diverging matrix element as a result. This fluence of a strong laser field on fundamental processes of divergence can be rendered finite by the inclusion of an QED. However, since the strong coupling to the laser field imaginary contribution to the electron energy, represented does not allow for the perturbation theory to be used, the by the term iMi,f in the matrix element (5). We note formulas resulting from the theory tend to be complex and that other methods, such as taking into account a finite difficult to follow. Here, numerical work becomes impor- space-time duration of the laser or energy spread of the tant. Bremsstrahlung is the process of light emission from initial electron also would produce finite results, and that an electron scattering at the Coulomb potential of a nu- our method is valid in the limit of long laser pulses. An cleus. In the absence of the laser, the problem in the first example of a photon spectrum resulting from the matrix Born approximation was solved by Bethe and Heitler. In element (5), characterized by a large number of peaks, is [33, 34] we have studied and numerically evaluated the ef- shown in Fig. 4. This kind of resonances are absent in the fect of an intense laser on the process of bremsstrahlung nonrelativistic treatment of the problem [40] where the in a Coulomb field. The second order Feynman diagrams dipole approximation for the laser is employed. for this process are displayed in Fig. 3 and the resulting transition matrix element Sfi takes the form of a double 5 Conclusion sum over photon orders: ∞ In this Report we have considered two examples in which δ(Q − Q − nω + ω ) S = f i 0 b QED can be tested in the presence of strong laser fields. fi q2 + `−2 n,s=−∞ We have seen that even non-perturbative vacuum polar- X (5) ization effects can be detected in the collision of high- As,n s,n f Ai energy protons and strong laser fields. Also, the dramatic × 2 2 + 2 2 , p˜f − m∗ + iMf p˜i − m∗ + iMi ! influence of the presence of a strong laser beam on the bremsstrahlung process has been pointed out. µ µ µ 2 2 where qi,f = pi,f + k0 m ξ /(4k0 · pi,f ) is the effective four- momentum of the electron in the laser field (pi,f is the elec- References tron four-momentum outside the laser field), where k0 = k 0 (ω0, 0) is the wave four-vector of the laser, Qi,f = qi,f [1] M. I. Eides, H. Grotch and V. A. Shelyuto, Phys. Rep. 2 2 2 2 is the effective energy, m∗ = qi,f = m (1 + ξ /2) is the 342, 63 (2001).

31 25 [15] W. Dittrich and H. Gies, Probing the Quantum Vac- 10 uum (Springer-Verlag, Berlin, 2000). × 20 2 15 I0 = 5 10 W/cm [16] S. L. Adler et al., Phys. Rev. Lett. 25, 1061 (1970). 10 [17] Z. Bialynicka-Birula and I. Bialynicki-Birula, Phys. 2 5 Rev. D , 2341 (1970). [18] S. L. Adler, Ann. Phys. (N.Y.) 67, 599 (1971). 3 10 I0 = 0 et al. m [19] C. Rischel , Nature 390, 490 (1997); Ph. Zeitoun

)] −5 et al. et al. b 10 , Nature 431, 426 (2004); J. Seres , Nature ω 433 et al. d 0 10000 20000 30000 39000 , 596 (2004); A. L. Cavalieri , Phys. Rev. Lett. b 30 94, 114801 (2005).

(dΩ 10 [20] G. Mourou and D. Umstadter, Sci. Am. 286, 80 / et al. 298 σ 20 (1997); V. Malka , Science , 1596 (2002); Th.

[d 10 Katsouleas, Nature 431, 515 (2004); P. D. Mangles et al., Nature 431, 535 (2004); C. G. R. Geddes et al., Na- 10 10 ture 431, 538 (2004); J. Faure et al., Nature 431, 541 PSfrag replacements (2004); J. Fuchs et al., Nature Phys. 2, 48 (2006); B. M. 0 10 Hegelich et al., Nature 439, 441 (2006); H. Schwoerer et al., Nature 439, 445 (2006). 0 10 20 30 [21] M. Tabak et al., Phys. Plasmas 1, 1626 (1994); P. ωb/ω0 Mulser and D. Bauer, Laser Part. Beams 22, 5 (2004); S. Atzeni, J. Meyer-ter-Vehn, The Physics of Inertial Figure 4: The cross section differential in the solid angle Fusion (Clarendon Press, Oxford, 2004). Ωb and the frequency ωb of the emitted photon, scaled with [22] T. Tajima and G. Mourou, Phys. Rev. ST Accel. the cubed electron mass m3. The calculation is performed Beams 5, 031301 (2002); S. Gordienko, A. Pukhov, O. for an initial electron with an energy Ei = 5.11 MeV, Shorokhov, and T. Baeva, Phys. Rev. Lett. 94, 103903 counterpropagating with a linearly polarized laser beam of (2005). 20 2 frequency ω0 = 1 eV and intensity I0 = 5 × 10 W/cm . [23] S.-W. Bahk et al., Opt. Lett. 29, 2837 (2004). ◦ The photon is emitted at an angle of 1 with respect to the [24] See the Phelix project homepage at direction of the initial electron. The black line corresponds http://www.gsi.de /forschung/phelix/. to the case without laser. [25] See the Polaris project homepage at http:// www.physik.uni-jena.de/qe/Forschung/F-Deutsch/ [2] S. G. Karshenboim, Phys. Rep. 422, 1 (2005). Petawatt/FP-Petawatt.html (in German). [3] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, [26] J. Norby, Laser Focus World, January 1 (2005). Quantum Electrodynamics - 2nd Ed. (Elsevier, Oxford, [27] See the ELI proposal at http://www.extreme-light- 1982). infrastructure.eu/. [4] W. Greiner, B. Muller,¨ and J. Rafelski, Quantum [28] Y. I. Salamin et al., Phys. Rep. 427, 41 (2006). electrodynamics of strong fields: with an introduction [29] G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. into modern relativistic quantum mechanics (Springer, Mod. Phys. 78, 309 (2006). Berlin, 1985). [30] M. Marklund and P. K. Shukla, Rev. Mod. Phys. 78, [5] A. Czarnecki, U. D. Jentschura, and K. Pachucki, 591 (2006). Phys. Rev. Lett. 95, 180404 (2005). [31] D. L. Burke et al., Phys. Rev. Lett. 79, 1626 (1997). [6] U. D. Jentschura, A. Czarnecki, and K. Pachucki, [32] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. A 72, 062102 (2005). Phys. Rev. Lett. (in press). See also arXiv:0708.0475 [7] U. D. Jentschura et al., Phys. Rev. Lett. 95, 163003 [hep-ph]. (2005). [33] E. Lötstedt, U. D. Jentschura, and C. H. Keitel, Phys. [8] U. D. Jentschura and V. A. Yerokhin, Phys. Rev. A Rev. Lett. 98, 043002 (2007). 73, 062503 (2006). [34] S. Schnez, E. Lötstedt, U. D. Jentschura, and C. H. [9] U. D. Jentschura, Phys. Rev. A 74, 062517 (2006). Keitel, Phys. Rev. A 75, 053412 (2007). [10] Sh. Zh. Akhmadaliev, et al., Phys. Rev. C 58, (1998) [35] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, 2844. Phys. Rev. D 72, 085005 (2005). [11] Sh. Zh. Akhmadaliev et al., Phys. Rev. Lett. 89, [36] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, (2002) 061802. Phys. Rev. Lett. 97, 083603 (2006). [12] R. N. Lee, A. I. Milstein, V. M. Strakhovenko, Zh. [37] A. Di Piazza, A. I. Milstein, and C. H. Keitel, Phys. Eksp. Teor. Fiz. 112, 1921 (1997) [JETP 85, 1049 Rev. A 76, 032103 (2007). (1997)]; Phys. Rev. A 57, 2325 (1998); Phys. Rev. A [38] W.-M. Yao et al., J. Phys. G 33, 1 (2006). 58, 1757 (1998). [39] See the petawatt field synthesizer homepage at [13] R. N. Lee et al., Phys. Rep. 373, 213 (2003). http://www.attoworld.de/research/PFS.html. [14] W. Dittrich and M. Reuter, Effective Lagrangians [40] M. Dondera and V. Florescu, Rad. Phys. Chem. 75, in Quantum Electrodynamics (Springer-Verlag, Berlin, 1380 (2006). 1985).

32 Relativistic critical density increase in a linearly polarized laser beam P. Mulser1, H. Ruhl2, M. Lontano3, and R. Schneider1 1 TQE: Theoretical Quantum Electronics, Tech. Univ. Darmstadt, Hochschulstr. 6, D-64289 Darm- stadt, Germany, 2 Institute of Theoretical Physics I, Ruhr-Universität Bochum, D-44780 Bochum, Germany, 3 Istituto di Fisica del Plasma , CNR, Via Roberto Cozzi, 53 20125 Milano, Italy

Abstract formation (e and me electron charge and mass, εo int. sys- A high critical density is important for laser beam propa- tem of units). At low, nonrelativistic laser intensities it is gation into high density samples for uniform heating and certainly true that the fully ionized plasma can be treated efficient energy tramsmission and transport. Its increase as a monofluid for electromagnetic wave propagation in under irradiation of intense linearly polarized laser light the homogeneous medium with an invariant refractive 1/2 1/2 is analyzed and effects contributing are addressed. It is index η = (1 − ωp²/ω²) = (1 − ne/nc) . Only the laser shown “experimentally” that the concept of a cycle- frequency changes according to the relativistic Doppler averaged critical density is meaningful at relativistic in- formula when switching from one inertial system to an- tensities and the corresponding overall Lorentz factor γ is other. At high laser intensities the ions can still be treated determined and compared with analytical expressions. as immobile while the electrons move at relativistic ve- The relevance of critical density increase to the concept of locity. Thus, in any reference system one has to deal with fast ignition is illustrated by numerical examples. a two-fluid model and the problem has to be faced what the implications of the transition from a monofluid to a 1 Introduction two-fluid model are. In addition, the question arises whether the concept of a critical density remains still Currently high power lasers emit in the infrared (Nd valid when at relativistic intensities highly nonlinear den- YAG, Ti:Sa), or in the near UV if for convenience a KrF sity structures and cavitons may form [7]. In what follows system can be used. This means that the emitted laser both problems, the existence of a critical density and its beams cannot penetrate matter of solid density with elec- 23 24 -3 correct relativistic increase in a highly overdense plasma tron densities of typically 10 – 10 cm . On the other will be clarified for the first time “experimentally” and on hand it is of extreme interest to heat such dense, and even a broader theoretical basis, i.e., by particle-in-cell (PIC) denser, plasmas uniformly and isochorically for studies of simulations and fitted analytical expressions. Finally, as a equations of state, bright X ray sources, generation of specific example the implications of laser radiation cou- high harmonics and for fast ignition of inertial fusion pel- pling to matter under fast ignition conditions are dis- lets [1]. In the latter case, with a laser beam emitting in cussed on the basis of the PIC results obtained. the soft X ray range several problems of radiation-pellet coupling of traditional lasers would be eliminated. It is a fortune that at intensities at which the electrons gyrate 2 Relativistic critical density increase relativistically in a circularly polarized beam an increase Only under the assumption of an invariant electron den- of the critical electron density is obtained owing to the sity an intense circularly polarized electromagnetic wave electron mass increase by the relativistic Lorentz factor γ can penetrate considerably deeper into a fully ionized = (1 − v²/c²)−1/2, v gyrovelocity, c speed of light in vacuum plasma owing to the alredy mentioned relativistic electron [2]. Perhaps in analogy to this result of Akhiezer and mass increase by the Lorentz factor γ, Polovin, an increase of the critical density by a Lorentz 1/2 factor γ is assumed for a linearly polarized wave where γ = (1 + a²) , a = eA0/mec, the gyrovelocity is replaced by the averaged oscillatory A = − (i/ω)E, (A, E)(x,t) = (A0, E0)( x,t) exp(− iω t); (1) velocity [3, 4, 5]. By means of numerical studies the authors of [4] found “anomalous penetration” i.e., consid- A, A0, E, E0 vector potential and electric field with asso- erably weaker penetration than corresponding to their ciated amplitudes [2]. The refractive index is η = [1 − 1/2 formula for relativistic mass increase. This and analogu- ne/γnc] , thus showing that the relativistic critical density ous results indicate that for reasons which will become for circular polarization is ncr = γ nc. As the electron clear in this contribution, the general problem of penetra- mass increase in a linearly polarized wave is by γ = (1 + 1/2 tion is not yet solved completely a²/2) , under the assumption of ne = const this would be In linear polarization, and in principle also at circular the true γ factor we are looking for. It seems, however, polarization, a problem arises because the critical density that this problem has never been considered with rigor depends on the ratio of electron mass me to electron den- explicitly in the literature and considerations seemed to be sity ne. As a consequence of the Lorentz contraction ne devoted to the increase of the elctron mass [3 – 5, 8, 9]. If also transforms by the same factor γ [6] so that one is left it is correct that ne does not transform with γ by Lorentz with the well-known fact that the plasma frequency ωp = contraction according to ne = γ neo, neo electron density at 1/2 (nee²/εome) is invariant with respect to a Lorentz trans- rest, there must exist a physical reason for it.

33 It is a matter of fact that in a divergence-free flow v 2.2 Quasiperiodic and chaotic electron orbits (e.g. v constant in space, v = v0(t)) from dne/dt = − ne∇v In the critical region the phenomena addressed above lead follows ne = γneo when it is set into motion by the laser to a strongly fluctuating potential φ which in combination field. From this one is lead to the unavoidable conclus- with the laser field may transform the regular 8-shape sion that the electron motion induced by the laser must be electron motion into quasiperiodic and chaotic orbits with highly divergent because otherwise in an originally qua- unknown Lorentz factor γ . To illustrate the situation a sineutral plasma the Lorentz contraction would lead to a p time-independent typical selfgenerated potential strong electrostatic imbalance induced by the electron φ = m c²a[1 + κ²(x/λ)²]1/2 with the free paramter κ and the charge density ρ = − e(γ n − n ) = − e(γ − 1)n which e e eo io e laser wavelength λ is used. The beam propagates in x di- tries to restore quasineutrality. It depends on the available rection. The laser field strength is held fixed at a = 1 and time interval to what degree charge neutralization hap- pens. In the circularly polarized wave the available time is κ is varied (see Fig.1). The first picture, obtained with κ = the pulse length extending over many oscillations and 6.2, shows a stable configuration of the electron orbiting around a vacuum-like 8-shape trajectory. Only a slight hence, n ≅ n . In a linearly polarized wave the available e eo asymmetry in diagonal direction is observable. At κ = 6.5 time is half a laser cycle π/ω long only and restoring of the electron manages it to escape from the stable configu- quasineutrality is expected to be incomplete. In a tenuous ration after several turns in the direction y of the laser plasma (ω << ω) perturbation theory yields n = (1 + p cr field, and eventually it comes back again (picture at 3a²/8)1/2 n , with 3/8 < 1/2 for this reason [10, 11]. c RHS). A further increase of κ by only 8 % to κ = 7.0 a very complex orbit results (3rd picture). By plotting the 2.1 Lorentz contraction and quasineutrality time history of the transverse motion ([y,t] plot in 4th pi- The dispersion of propagating plane waves of arbitrary cure) it becomes clear that during the registered 100 laser strength and linear polarization in constant plasma density cycles the elctron crosses 9 attractors. In a sense, the tra- under Lorentzian gauge is governed by the set of equa- jectory is a repeated composition of orbits from the 2nd tions picture. The contribution of such orbits to the relativistic increase of the critical density must be left to future inves- α A = −J/ε0c² , ∂αA = 0, A = (A, φ /c), J = (j, cρe), tigations.

ρ e = e(ni0 − ne) = εo(vϕ²/c² − 1)∂²φ /∂x², 0.15 0.9 ∂φ/∂x = (c²/ vϕ) ∂Ax /∂x, (2) 0.8 0.1 0.7 0.6 j = − (e²n /γm ) A − ε [ev (1 − c²/ v ²) ∂²A / ∂x²] A /γm . 0.05 y e0 e y o ϕ ϕ x y e 0.5 0.4 0 0.3 0.2 -0.05 φ is the scalar potential and vϕ the constant phase veloc- 0.1 -0.1 0 ity. The second of these relations is the Lorentz gauge. -0.1 -0.15 -0.2 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 The current density jy follows from the canonical momentum conservation γmevy= eAy for a cold electron fluid 3.5 3.5 initially at rest. In circular polarization the second term 3 3 2.5 2.5 vanishes owing to Ax = φ /c = 0. The dispersion of waves 2 2 described by (2) has been extensively studied and dis- 1.5 1.5 1 1 cussed recently also close to cut-ff and it has been found 0.5 0.5 that the majority of electromagnetic modes exhibits the 0 0 -0.5 -0.5 typical eight-like electron motion as in the tenuous -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0 10 20 30 40 50 60 70 80 90 100 plasma [11]. Besides, a purely electrostatic mode with circular electron motion in the plane of incidence is also Figure 1: Relativistic motion of an electron in a plane possible, in accordance with the fact that the B-field an electromagnetic wave of strength a = 1 under the influ- 2 2 2 1/2 electromagnetic wave near cut-off behaves like B0η → 0 ence of a static potential V = mec a[1 + κ x ] . The wave [11]. Unfortunately, light coupling to high density matter, propagates in x-direction, y is the direction of the electric which is the typical fast ignition-relevant situation, is ac- field; λ wavelength. κ is increased from the value 6.2 (1st companied by plasma density profile steepening, partial picture) through 6.5 (2nd picture) to 7.0 leading to increas- reflection and extremely nonuniform electron fluid ex- ing complexity of the orbit. pansion [12] and recession during one cycle, with concomitant laser frequency Doppler shift in the frame of co- moving critical layer. Owing to such complexity and, 2.3 Thermal Lorentz factor γth eventually, due to caviton formation [13], a reliable de- Due to inherent stochasticity anharmonic resonance, by termination of the critical density increase is accessible which collisionless absorption of intense laser beams is only to particle-in-cell (PIC) or similar simulation proce- accomplished [15], is accompanied by a heating effect. In dures (e.g. Vlasov). Here we use the 3D PSC PIC code the rest frame of an electron fluid element the bare elec- [14] to give the promised “experimental” answer. tron mass me turns into the “dressed” thermal mass meth = γthme, with γth to be determined kinetically.

34 2.4 “Experimental” γ factor tal”) relativistic critical density ncr and its relativistic increase γ = n /n respectively. The values obtained at the Lorentz contraction has an attenuating influence on the γ cr c, 5 intensities are indicated by the first two colums for per- factor of ncr, by γth the electron becomes heavier, thus pendicular and 45° laser incidence in Fig. 3. For laser acting in favour of a high γ value, γp can act in both direc- intensities I ≥ 1020 W/cm² higher γ values result at oblique tions and has not been studied so far. As the γ values as- incidence than at normal irradiation. The hump in |E|² sociated with these various effects interact in a complex between x = 4.6 and x = 5.0 is due to the superposition of nonlinear manner with the effect of the relativistic in- higher harmonics [12]. crease of the bare electron mass, the correct value of the For supporting the numerical results and perhaps gain- resulting γ actually is determined in the most reliable ing some physical insight, in the absence of any knowl- manner by analyzing 1D PIC runs at a sequence of laser edge on phenomenon (II) , we start from the assumption intensities. Furthermore, only such an “experimental” of regular vacuum-like 8-shape orbits and determine a procedure can reveal to what extent the existence of a “theoretical” γ factor from the formula critical density and a well defined critical point are secure or whether they have to be given up owing to irregular γ = [1 + (1 + r)²a²/2]1/2, (3) density fluctuations and periodic recessions at extreme laser intensities. with ra the reflected normalized wave amplitude. Note that this expression is invariant with respect to a rotation

by an angle of incidence α. This can be recognized by ne transforming to the boosted reference system in which M jE both quantities E0 and ω are multiplied by cosα and A0 is their ratio [see (1)]. The magnitude of r is obtained from |E|2 max |E|² = M and min |E|² = m closest to the cut off according to

r = (√M − √m)²/(M − m) (4)

m These γ factors are visualuzed in Fig. 3 by columns 3 and 4 for normal and 45° incidence, respectively. All values, theoretical and experimental, are taken at t = 80 fs. Althoug at some intensities, in particlur at Iλ² = 21 10 W/cm²µm², the γ values for 90° and 45° from PIC 22 −3 Figure 2: Plane target of density n0 = 9 × 10 cm irradi- differ considerably from each other, and so does γ ob- 21 2 2 ated by INd = 10 W/cm µm in p-polarization under 45° tained by means of (3) for 90° and 45° at all intensities Iλ² after 80 fs. Lorentz boost (density upshift). Quantities ≥ 1020 W/cm²µm², in first approximation the results look averaged over two cycles as functions of space coordinate like as if produced by the relativistic mass increase only x: laser field |E|2 bold with M maximum, m minimum, and almost no relativistic influence stemming from electron density ne solid, absorption jE oscillating around changes of ne and from chaotic orbits. The latter are re- 0.0 dashed and dotted. vealed by test particle injection in the PSC simulations and following their motion. The impression of dominating A fully ionized target of 100 times critical density with mass increase is reinforced if a fit to the averages of each of the fife quadruples is done. For this we found, black heavy ions (to suppress hole boring, of no interest in the context) is exposed to linearly polarized laser irradiance solid line, of Iλ² = 10s W/cm²µm², s = 18 − 22, under perpendicular 18 1/2 and p-45° incidence and, when the interaction has be- γ = {1 + Iλ² [W/cm²µm²]/10 } . (5) come stationary, all field quantities are averaged over two laser cycles. A typical result for Iλ² = 5×1021 Wcm−²µm² The weak relativistic contribution from ne to γ has a quali- under 45° on target is presented in Fig. 2 for t = 80 fs. In tative explanation, a posteriori. By the radiation pressure contrast to strongly fluctuating pictures taken at single strong electron density profile steepening sets in over a time intervals, after sufficiently averaging quiescent scale length which is a small fraction of the laser wave- shapes in all field quantities are obtained. Hence, a first length λ only. As a consequence the laser wave acts result is that the concept of a cycle averaged, not instanta- mostly on a low density shelf with ne << ncr and, in the neous, critical density makes sense. It is to be expected at evanescent region, on ne >> ncr (see Fig. 2). For ne << ncr, 3 1 the point of zero curvature of the evanescent branch of γ is close to the vacuum value ( /8 instead of /2). At ne ≅ |E|² occuring close to half the maximum. In nearly all no charge neutralization is up to 20 times faster than 2π/ω. cases this position coincides with the maximum of ab- We conclude that up to the intensities considered an av- sorption jE, a fact which is of great help in the analysis. erage critical density makes sense and its approximate Thus the density at this point is taken as the (“experimen-

35 relativistic increase is given by a simple formula which 3.1 Ballistic model reflects essentially the increase of the bare electron mass. Inspired by [1] and [21] we consider a model in which the hot electrons basically propagate ballistically through the 110 critical and intermediate densities and interact collision-

100 ally in the compressed pellet core. The minimum flux 21 90 density of hot electrons q0 = 10 W/cm² is estimated to PIC 90° rd 80 be sufficient when generated by the 3 harmonic of Nd PIC 45° 70 Theo. 90° laser frequency Theo. 45° 15 −1 60 analytic ω = 1.78×10 s . Assuming R= 0.5 for the laser follow γ 22 −3 50 = 6.7 and ncr = 6.7×10 cm , the latter exceeding solid 40 22 −3 DT density nDT = 5×10 cm . In order to transport the 30 absorbed flux density q0 with electron velocity u for the 20 mean energy of the hot electrons must hold 10 0 1/2 1,00E+18 1,00E+19 1,00E+20 1,00E+21 1,00E+22 n u = q , u = c(1− 1/γ) , γ = 1 + /m c² (6) cr 0 e

Figure 3: Critical density increase by factor γ as a func- From (6) the mean energy results as low as = 0.37 2 st MeV. An upper limit for is obtained by making the tion of Iλ at normal and 45° incidence, 1 two columns very reasonable assumption that it may not be much PIC, next two columns “theoretical” γ factors; black line higher than the mean oscillatory electron energy which in Eq. (5). the specific case with a = 4.3 is Eos = 1.1 MeV, i.e. consistency is guaranteed. 3 Implications for fast ignition by laser The advantages of fast pellet ignition with powerful lasers 3.2 Diffusive model for inertial confinement fusion are well known: Decoup- If already in the critcal and medium density domain ling of compression from ignition phase, less require- strong thermalization occurs a diffusive model applies. In ments on pulse shaping of the main compression pulse, [17] it was found on the basis of [16] that the minimum lowering of symmetry constraints on peak compression, 21 required flux density has to be close to q0 = 10 W/cm² explicitly shown in [16]. Extensive computer studies with because most of the energy supplied is diffusively spread flux limited Spitzer heat transport have shown that a typi- all over the compressed pellet and there in particular all cal ignition energy is 70 kJ and, by optimizing the proc- 20 over its low density corona. In addition the deposition ess, 50 kJ at a laser intensity not lower than I = 10 density should not be inferior to ρ = (4 – 5) g/cm³ DT; W/cm² may represent the minimum energy and intensity dep below ρ = 1 g/cm³ no ignition was possible [16]. With requirements [17]. Under the condition that such an dep this value for n = 5×1022 cm−3 and a safety factor s = (4 amount of energy is deposited in fast electrons the density dep – 5) the condition n = n s = γ(λ)n (λ) together with I(λ) of the deposition zone had not to be less than 4 – 5 gcm−3 cr dep c = 2 q for R = 0.5 must be fulfilled. For Nd holds n (λ) = DT; below 1 gcm−3 in no run a burn wave evolved. As the 0 c laser energy cannot be deposited beyond the critical den- ncNd λNd²/λ². Hence, from (5) is recovered sity in standard coronal ignition the flux of the energetic 18 1/2 λ/λNd = (ncNd / ndep s)(2 q0 /10 ) = 0.9/s electrons has to travel a long distance up to the com- 18 1/2 pressed core thereby undergoing sensitive diffusive at- ⇒ ω = 1.1ωNd/s ; γ(λ) = (2 q0 /10 ) λ/λNd = 40/s. (7) tenuation. For comparison, simulations without flux lim- 21 iter in [16] show that the energy needed for the pure igni- At I = 2×10 W/cm² the maximum wavelength for tion process (the “free ignition energy”) is typically 15 kJ, achieving ignition is close to λNd if s = 1 is set. For the in agreement with simpler models of direct energy depo- fundamental of the Ti:Sa laser the condition is fulfilled. It sition in the most favorable region [18, 19]. Originally is obvious that the consistency condition (6) is fulfilled hole boring was intended as reducing noticeably the dis- with a large safety factor. As = 0.95 MeV and u = tance between laser deposition region and compressed 0.87c consistency is secured easily also for s = 5, i.e. the core. Unfortunately hole boring has proven not to be effi- fifth harmonic of ωNd because of Eos = 3.5 MeV > . cient enough for this purpose [20]. Rather is cone guided It can be concluded that at high laser intensities the rela- fast ignition more advantageous to bring the two regions tivistic increase of the critical density is favorable to closer together and, in addtion, to provide for better cou- achieving uniform heating of condensed matter and to pling of the laser beam [1]. However, also in this scheme facilitate energy coupling in fast ignition. It rises the there is the constraint on laser energy conversion into question whether by other means, e.g. geometrical effects kinetic energy of the electrons not beyond the critical further increase can be achieved. density.

36 References [12] T. Baeva, S. Gordienko, and A. Pukhov, Phys. Rev. E 74, 046404 (2006). [1] R. Kodama et al., Nature 412, 798 (2001). R. Kodama [13] S. Weber, M. Lontano, M. Passoni, et al., Phys. et al., Nature 418, 933 (2002). Plasmas 12, 112107(?) (2005). [2] A. I. Akhiezer and R.V. Polovin, Sov. Phys. JETP 3, [14] H. Ruhl, in M. Bonitz et al. (eds) Introduction to 696 (1956) Computational Mehods in Many [3].W. L. Kruer, The Physics of Laser Plasma Interac- Body Physics, ISBN 1-58949-009-6, (2006). tions, Addison Wesley Pub. (1988), p. [15] P. Mulser, D. Bauer, and H. Ruhl, Collisionless ab- 42. sorption of intense laser beams by [4] H. Sakagami and K. Mima, Phys. Rev. E 54, 1870 anharmonic resonance, sumitted for publication. (1996). [16] S. Hain and P. Mulser, Phys. Rev. Lett. 86, 015 [5] B. Hafizi et al., Phys. Plasmas 10,1483 (2003). (2001). [6] M. Lontano, S. V. Bulanov, and J. Koga, Phys. Plas- [17] P. Mulser and D. Bauer, Laser&Particle Beams 22, 5 mas 8, 5113 (2002) (2004). [7] N. J. Sircombe, T. D. Arber, and R. O. Dendy, Phys. [18] S. Atzeni, Phys. Plasmas 6, 3316 (1999). Plasmas 12, 012303 (2005). [19] S. Atzeni, M. Temporal, and J.J. Honrubia, Nucl. [8] K. Nagashima, Y. Kishimoto, and H. Takuma, Phys. Fusion 42, L1 (2002). Rev. E 58, 4937 (1998). [20] P. Mulser and R. Schneider, Laser&Particle Beams [9] P. Jha et al., Phys. Plasmas 11, 3259 (2004). 22, 157 (204). [10] P. Kaw and J. Dawson, Phys. Fluids 13, 472 (1970); [21] Y. Sentoku, B. Chrisman, A. Kemp, and T.E. C. J. McKinstrie and D. W. Cowan, Hot electron energy coupling in Forslund, Phys. Fluids 30, 904 (1987). cone-guided fast ignition, Paper Tu07.6 at IFSA [11] T. C. Pesch and H.-J. Kull, Phys. Plasmas 14, 2007, Kobe, Sept. 9 – 14, 2007. 083103 (2007).

37 Cerchez et al.

Absorption of ultrashort laser pulses in strongly overdense targets

M. Cerchez,∗ R. Jung, J. Osterholz, T. Toncian, and O. Willi Heinrich-Heine Universität Düsseldorf, Universitätsstr.1, 40225 Düsseldorf, Germany

P. Mulser Theoretical Quantum Electronics (TQE), Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany

H. Ruhl Institute for Theoretical Physics I, Ruhr-Universit¨at Bochum, 44797 Bochum, Germany (Dated: April 18, 2008) Absorption measurements on solid conducting targets have been performed in s- and p- polarization with ultrashort, high-contrast Ti:Sa laser pulses at intensities up to 5 × 1016 W/cm2 and pulse duration of 8 fs. The particular relevance of the reported absorption measurements lies in the fact that the extremely short laser pulse interacts with matter close to solid density during the entire pulse duration. A pronounced increase of absorption for p-polarization at increasing angles is observed reaching 77% for an incidence angle of 80◦. Simulations performed using a 2-D Particle-In-Cell code combined with alternative models allow to isolate a high collisionless absorption component in steep plasma proﬁles of relative scale lengths L/λ ≈ 0.01. In interpreting the underlying mechanism of this kind of interaction it is found that the model of anharmonic resonance is a favorable candidate.

PACS numbers: 73.23.-b,75.70.Cn

In this report, we present experimental investigations of linear resonance absorption with the increasing scale of laser energy absorption of high-contrast, sub-10 fs length is well reproduced. Now, we are left with the laser pulses by a conducting target. The laser pulse exciting question whether (i) the impressive preponder- parameters (duration and high-contrast) allow, for the ance of p-absorption on s-absorption and (ii)itshigh first time, to study the absorption under conditions value for p-polarization is preserved or even reinforced where the pulse energy is basically directly transfered to when the laser-target interaction occurs in very steep the solid matter. plasma profile at density close to solid state and missing Absorption of intense laser pulses in the ps [1] and undercritical density shelf. The results presented in the sub-ps regimes [2–5] has been measured in several ex- following will confirm both conjectures. perimentswithsolidtargetsinthepast,inanintensity The experiments have been carried out employing a range which is of interest to the present report. In Ti:Sa laser system described in [10] operating in CPA all measurements, except [4] (no s-polarization inves- mode. Under experimental conditions, the laser system tigated), absorption has prevailed considerably under delivers linearly polarized pulses of 100-120 μJ at 790 p-polarization relative to s-polarization. When the laser nm (central wavelength) and 8 fs duration on target. pulse is typically longer than 100 fs, and/or a prepulse The pulse contrast was experimentally determined is present, a pre-plasma is formed in front of the target using a high dynamic range third-order auto-correlator and undergoes hydrodynamic expansion. In these (Sequoia). The diagnosis reveals a contrast ratio better situations the absorption process has a characteristic than 105 for times larger than 1 ps before the main dependence on the polarization and, for p-polarization, pulse and better than 108 for the Amplified Spontaneous is most naturally attributed to the mechanism of linear Emission prepulse. The laser pulse was focused in resonance absorption. Moreover, in presence of an un- vacuum onto target by an f/2.8 off-axis parabola of 108 dercritical shelf it has been shown that linear resonance mm effective focal length to a spot diameter of ≈ 3.2 absorption increases with shortening of the critical μm (FWHM), giving at normal incidence, an average density scale length L and simultaneous increasing intensity of (4 − 5) × 1016 W/cm2. The absorbed energy angle of incidence [6]. Under large angle of incidence fraction was experimentally determined as a function of and non-relativistic laser intensities, for p-polarized the incidence angle θ and the polarization of the laser laser pulses, absorption levels as high as 80% have radiation. The experimental investigations cover a wide been measured [2]. Particle-In-Cell (PIC) [4, 7, 8] and range of incident angles (10◦ -80◦) and over 4 orders Vlasov simulations [9] are in qualitative agreement with of magnitude of the laser intensity (5 × 1012 W/cm2 experiments and the characteristic angular distribution -5× 1016 W/cm2.The targets consisted of mirror-flat aluminium layers with a thickness of ≈ 300 nm and a roughness of less than 5 nm, deposited on planar silicon substrates. The target was placed at the center of an ∗Electronic address: [email protected] integrating Ulbricht sphere of 10 cm in diameter. The

38 2

1.0 1.0 experiment, p-polarization 0.3 experiment, s-polarization s - polarization p - polarization 0.8 MULTI-fs, p-polarization 0.8 0.2 45° incidence angle MULTI-fs, s-polarization 0.1 0.6 0.6 Absorption 0 1013 1014 1015 1016 1017 2 0.4 0.4 I L [W/cm[W/cm ] Absorption Absorption

0.2 0.2

0.0 0.0 10 20 30 40 50 60 70 80 90 13 14 17 10 10 1015 1016 10 Incidence angle θ [°] 2 IL [W/cm[W/cm ] FIG. 1: Experimental angular dependence of the absorption FIG. 2: Absorption of sub-10 fs, 790 nm laser pulses depen- of 8 fs, 790 nm laser pulses by an aluminium target, s-(circle θ ◦ filled symbols) and p-polarized (squared filled symbols) at an dence of the laser intensity at the incidence angle =45, 16 2 for an aluminium target. In the main frame, the absorbed average intensity of 5 · 10 W/cm .MULTI-fshydrocode results under the experimental conditions, for a plasma profile fraction of the p-polarized beam is shown and in the inset the of L=2 nm are shown with open symbols. same dependence for a s-polarized laser pulse. amount of laser light collected by the sphere after being point represents an average value over 10 - 20 shots. The reflected by the target was measured with a high-speed error bars shown on the graphs indicate the standard photodetector coupled to the sphere via an optical fiber deviations and are in the range of 5 − 12% except for bundle. For our energy range, we carefully checked the θ ≤ 15◦. These experimental data points have been linearity of the photodiode and the optical bundle prior recorded in the best focal position of the target. To prove to the measurements. that the interaction occurs close to the surface of the Thelaserwasoperatedinsingleshotmodeandthe target, in addition to the above mentioned good contrast fluctuations of the pulse energy varied by less than 5% ratio, are (i) the missing higher order transitions in the with respect to the average value over tens of shots. X-ray spectra[11] and (ii) the absence of a preplasma in Between the shots, the target was moved in order to the observation of the ionization front propagation in provide a fresh target surface. The laser pulse was gaseous targets [14]. Regarding the dependence of the linearly polarized in the horizontal direction and the absorbed fraction versus the laser intensity IL, plotted rotation axis of the target was vertical or horizontal for in Fig. 2 for an incidence angle of θ =45◦ for both p- p- and s-polarization measurements, respectively. The and s-polarization, the following observations are made: intensity of the laser beam was varied by moving the (1) In the case of s-polarization, starting from the low target out of focus along the laser propagation direction. intensity regime of ≈ 5 · 1012 W/cm2, the absorption The signal at the photodetector was proportional to the is approximately constant (10%) over 2 orders of mag- fraction R of the laser energy reflected (specular and nitude of the laser intensity. It starts to increase to scattered) from the target. Previous experimental works about 20% at the average intensity of 5 · 1016 W/cm2(2) (see, for example, [5, 12]) proved that the contribution P- absorption starts to increase significantly at the of the diffusely scattered light is less than 2%. Also, threshold 1014 W/cm2 andis5timesasstrongwhen the backscatter laser energy represents less than 4% for approaching 5 · 1016 W/cm2.(3)Intheintensityrange incidence angles larger than 10◦ and thus, is negligible between 5 · 1014 W/cm2 and 5 · 1016 W/cm2 the absorbed [13]. The experimental investigations addressed here do fraction of the p-polarized beam Ap scales with intensity . ± . not include the measurement of the backscattered light. IL and wavelength as Ap ∝ (IL · λ2)0 12 0 02.Assuming The absorbed fraction A is given by A =1− R. collisional absorption for s-polarization of the laser, In Fig.1, the experimental results of the angular the contribution due to collisionless absorption can dependence of the absorbed fraction for both s- and be obtained with some uncertainty by subtracting the p-polarized laser pulses incident on aluminium targets absorbed fraction As from the corresponding fraction are presented. For s-polarization, while the angle of Ap of the p-polarized beam. The collisionless absorption . ± . incidence is increasing, the absorption drops from 19% then scales as (Ap − As) ∝ (IL · λ2)0 10 0 05. at θ =15◦ to 6% at θ =70◦. Absorption of the Now, let us proceed to the difficult enterprise of theo- p-polarized laser light increases for larger angles and retical interpretation. Simulations have been performed reaches its maximum value of 77% at 80◦. Each data using the 2-dimensional Particle-In-Cell (PIC) Plasma 39 3

1.0 surface. The simulations were performed for p- and PIC, L/λ = 0.625 s-polarization of the incident laser pulse including the PIC, L/λ = 0.250 binary collisions in the code. The absorption profiles PIC, L/λ = 0.025 0.8 PIC, L/λ = 0.012 were also subtracted from each other in order to remove PIC, L/λ = 0.002 the effect of collisions. When the collisional module is λ 0.6 MULTI-fs, L/ = 0.002 switched off the results differ by less than 2 %. The experimental data computational and experimental results at an averange

focal intensity of 2 · 1016 W/cm2 are shown in Fig. 3. 0.4 The experimental data are well reproduced for proﬁles in the range of 10-20 nm. For longer proﬁles, L=200 nm and L=500 nm, the well known linear resonance 0.2 absorption behavior is reproduced with an optimum

Collisionless absorption, Ap-As Collisionless absorption, absorption angle at intermediate values. 0.0 Alternatively, hydrodynamic simulations have been 20 40 60 80 carried out using the widely tested and frequently Incidence angle θ [°] employed 1-dimensional MULTI-fs code [18]. The simulations were motivated by the fact that MULTI-fs FIG. 3: Contribution of the collisionless absorption on p- can account, in addition to resonance absorption, for polarization expressed as difference between absorbed frac- collisional ionization over a wide range of temperatures tion on p- and s-polarization at average laser intensity I = from cold solid to hot plasma. The code has been widely 16 2 2 · 10 W/cm . The filled symbols represents the experimen- used to analyze the interaction of ultra short laser pulses tal data. PIC simulations for the same laser condition and with steep density gradient plasmas. Here the absorption L/λ different pre-plasma scale lengths are shown by open was calculated for different preplasma scale lengths and symbols. MULTI-fs code results for a pre-plasma profile of for the laser parameters of our experiment. In order to L=2 nm are indicated by asterisk symbols. take into account the transversal laser profile in the 1-D calculations, the simulations were carried out with an Simulation Code (PSC) [15]. A temporally Gaussian intensity averaged over the beam cross section [12]. The simulation results for a preplasma scale length of 2 nm shaped laser pulse with a duration of 10 fs (FWHM) was focused inside a simulation box of 20 μm×20 are added in Fig.1. The calculated angular dependence μm. In the focus, the laser pulse achieved an intensity of the absorption reproduces the experimental data for 16 2 both p- and s-polarization, with a maximum of 78% at of 2 × 10 W/cm (cycle averaged). A rectangular ◦ aluminium target of 15 μm×1 μm was placed at the 80 in p-polarization; only for the small angles, there are best focal position in the center of the simulation box. slight deviation. The electron density profile at the target boundary was A third model is stimulated by re-examining papers −1 studying purely collisional absorption under conditions given by ne (x) /ne0 =(1+exp(−2x/L)) with respect − in which Fresnel’s formulas should apply [19], [20]. to the target normal, where L = |∇n/n| 1 is the density To compare with the PIC simulations from Fig.3 we scale length. For the simulations reported here a grid calculated the Fresnel absorption (Ap − As)fromaflat resolution of 40 cells per μm was chosen with 4 particles surface of refractive index squared η2 = −22 + 46i.Itis per cell and species. The target was rotated in steps of ◦ the refractive index of a free electron gas (fully ionized 10 around the position of the maximum density slope − plasma) of density 1.15 × 1023cm 3 in 3.5 times ionized located at the best focal position (i.e. corresponding to n /n / L solid Al and a postulated electron-ion collision frequency e e0 =1 2). Various density scale lengths of 2 nm, ν =2× ω =4.7 × 1015s−1. The result is shown in 10 nm, 20 nm, 200 nm and 500 nm were used. Additional Fig.4 together with the experimental data. Again, good simulations have been performed for higher resolutions agreement is obtained. For reasonable temperatures the up to 120 cells per μm, but the results have not been ion density profile length after 10 fs is definitely less sensitive to the increased resolution. To account for the than λ/100 so that Fresnel formulas for step profiles high contrast-ratio of the laser pulse, it was assumed apply [21], [22]. that the target is primarily ionized by optical field We are faced with three different models mutually ionization at the moment the 10-fs pulse interacts with excluding each other (collisionless kinetic, fluid based the aluminium slab. Therefore, in the simulation, the resonance absorption combined with collisions, and aluminium was ionized from the ground state according purely collisional linear optics). Let us start the inquiry to the ADK model [16] included in the code. The by excluding linear resonance absorption. It has to be maximum field-induced ionization state observed in this excluded for two reasons. First, experiments indicate scenario was about 3.5, in agreement with analytical and as well as the models under examination show the estimates [17]. The fraction of absorption of the pulse maximum of p absorption shifted to very high angles energy during the interaction with the aluminium slab of incidence. This is a safe indicator for absorption was then calculated by analyzing the Poynting flux of occurring in a density profile of scale length not longer the pulse before and after being reflected by the target 40 4 than L =0.01λ. In such a plasma the plasma frequency effect [33]. A careful inspection, accompanied by accu- ωp is everywhere much higher than the laser frequency rate test calculations [33] shows that most models do not ω. The second reason is based on the high intensity lead to absorption higher than 10 - 15 %, coupling to regime. Imposing a value for the relative amplitude of the target is very delicate in [28] and has never been suc- the electron plasma wave, = n1/ne, this translates into cessfully realized in an experiment. It is fair to say that an upper limit for the exciting laser intensity I [23], for more than two decades searching for the physical nature of collisionless absorption was not successful. Only very recently a real breakthrough in understanding has 1 4 me I ≤ · 2/3 · 2 · c8 · ω7 1/3 · · L1/3 2 been achieved [34]. The model is conceptually simple ( π ) ( e ) ( ) ( ) (1) 2 3 and robust, i.e. it works under all conditions as soon a 2 ⇒ I[W/cm2]=2.9 × 1018 · · (L[m])1/3 (2) certain level of laser intensity is reached. The interaction region can be thought as divided into a high number With =0.1 and L = 10 nm linear resonance absorp- of plasma layers of thickness d, oriented parallel to the tion is for I =6· 1013 W/cm2 at its limit. The L2/3 surface. Each layer represents an oscillator in which the dependence reflects the property of the resonance width electron fluid oscillates against the attracting immobile increasing with L1/3. Such a low limit has severe conse- ion background. For low excitation, i.e. small amplitude ξ ω quences for the use of MULTI-fs. Plotting the absorbed of displacement , its eigenfrequency o is the well known ω power j ·E from Poynting’s theorem in space reveals that plasma frequency p which, at solid or even higher den- almost all absorption MULTI from Figs.1 and 3 is due to sity is well out of resonance and does not absorb energy linear resonance (MULTI-fs can only handle linear reso- from the laser. However, at large displacements when the nance absorption). At intensities well above the limit (2) oscillator is driven into the nonlinear regime the restor- wave breaking sets in. The latter is out of the possibil- ing force becomes weaker as is typical for Coulomb type ity of a fluid description. To bypass the phenomenon an interactions. As a consequence, in the nonlinear domain artificial viscosity is introduced which leads to artificial of excitation an elementary calculation shows that the heating in our context because the collision frequency in eigenfrequency of the oscillator decreases according to the critical domain is at most 1% of the laser frequency, 2 1/2 −1/2 ωo =(π/4)(ωpod) ξ , (3) thus excluding hydrodynamic shock formation. For these two severe shortcomings the simulations by MULTI, al- When ωo equals the driving frequency ω of the laser, reso- though reproducing best the experimental results, have nance occurs and the oscillator takes on energy by orders to be discarded. We do not consider them any further. of magnitude higher than out of resonance. In the res- The purely collisional model is able to explain Ap absorp- onance zone the oscillator undergoes a phase shift by π. tion with sufficient accuracy by properly choosing the This is the phase shift needed for energy coupling to the electron density, ionization degree and average electron plasma, searching for which in the absence of collisions energy. A thoroughly prepared map of complex refrac- took more than two decades. The phase shift is easily un- tive indexes in Drude approximation (corresponding to derstood. An eigenfrequency higher than the driver fre- the fully ionized classical plasma model) in a wide range quency means that the oscillator is stiff and therefore it of these parameters shows that from Frenel’s formula for follows in phase the applied driver. As soon as resonance Ap reasonable agreement is is not achieved with collision occurs ωo sinks below ω, the restoring force is weaker frequencies below ν/ω = 2. On the other hand, assum- than the driver, with the consequence that the oscillator ing an average laser intensity of I =2× 1016 W/cm2 the moves dephased by π with respect to the driver. It can be average oscillation energy of the free electron amounts to shown analytically, and has been extensively tested nu- 5 keV. There are several formulas for ν available taking merically, that as soon as the oscillator leaves the linear both temperature and oscillatory energy into account, regime, very little additional energy is needed to drive it e.g. [24]. All of them yield values ν/ω not exceeding the into resonance. Careful statistics of the orbits of all accel- value 0.2 even if the ionization degree of Al is taken as erated (”heated”) electrons in PIC flying into the target high as Z = 7. In addition, the above mentioned thresh- revealed that 95 % of all particles obtain their energy by old behavior is every unlike to be produced in collisional undergoing resonance. We give it the name anharmonic processes. because its eigenfrequency is a function of the degree of Concluding, the most likely absorption mechanism is excitation, in contrast to the harmonic oscillator whose of collisionless nature, as the PSC PIC simulations con- frequency is a constant. Anharmonic resonance is the firm. PIC simulations do not tell what the the underlying standard phenomenon in nature because all potentials at physics leading to absorption in the absence of collisions large displadements widen more than the parabola as a in a highly overdense plasma is. Although there exist nu- consequence of the Coulomb attraction varying like 1/r at merous models which are based on independent effects, large distances. Anharmonic resonance is a possible sce- like sheath inverse bremsstrahlung [26], anomalous skin nario for catastrophes, e.g. collaps of suspension bridges, layer absorption [27], vacuum heating [4, 7], excitation of forming of cracks under oscillating strain, and sheds new surface plasmons [28], j × B heating [29], Landau damp- light on the not well understood pheenomenon of wave- ing [30], laser dephasing hearting [31] and the Brunel breaking in the plasma. Anharmonic resonance exhibits 41 5 threshold behaviour. For the single oscillator, the thresh- been found for the first time in the experiment for the old for starting resonance (absorption) is easily deter- onset of p-absorption at the intensity 1−2×1014 W/cm2. mined as [34] Present analysis let us hope to bring the two values closer together and to find analytic laws for the amount of ab- 2 E = meωpod/(4e), (4) sorption for the first time. A detailed partial analysis will be the subject of a separate paper. where e is the electron charge. Using d = 0.12 nm In conclusion, we report on the first absorption ex- ω × 16 −1 and po =2 10 s , the required laser intensity is periments of sub-10 fs high-contrast Ti:Sa laser pulses 15 / 2 10 W cm . This is a theoretical threshold for the sin- incident on solid targets. The particular high absorption gle layer. Unfortunately one is not able to calculate an- values at high angles in p-polarization show that a very alytically realistic thresholds yet because when one layer steep profile L/λ ≈ 1% is produced and the pulse inter- undergoes resonance it crosses the neighboring oscillators acts basically with the matter close to solid density. The π owing to the phase shift by . The only statement we can PSC PIC simulations, eventually with a smaller contri- make at the moment is that crossing, i.e. wavebreaking bution from collisions clearly confirm the observations. leads inherently to a lowering of the eigenfrequencies of The high value of measured absorption Ap − As, confirm the oscillators in the neighborhood. This means that (3) by collisionless simulations, as well as the existence of a represents an upper limit. Wave breaking is also the rea- threshold for its onset is a strong indication for the an- son that no closed formula can be given yet for the shape harmonic resonance mechanism. Our analysis has also of the electron energy spectrum and the degree of absorp- shown that quantitative agreement between the experi- tion. At the moment such values must be calculated by ment and a theoretical model, MULTI-fs in the actual PIC simulations. However, we understand the physics of case, does not necessarily prove the latter. collisionless absorption now and there is the legitimate hope that the model of anharmonic resonance will bring We would like to thank K. Eidmann and B. Hidding for new dynamics into the field of collisonless absorption, valuable discussions concerning the MULTI-fs code. This stagnating after the presentation of Brunel’s model in work has been performed within the SFB/Transregio 1987. The latter has been the best qualitative approach TR 18 and the ILIAS contract with GSI/Darmstadt. so far according to our analysis [33, 34]. A threshold has

[1] J. C. Kieﬀer et al., Phys. Rev. Lett. 62, 760 (1989); D. mas (Plenum Press, New York, 1994), p.425-426 Meyerhofer et al., Phys. Fluids B 5, 2584 (1993). [18] K. Eidmann, J. Meyer-ter-Vehn, T. Schlegel, S. Huller , [2] R. Sauerbrey et al., Phys. Plasmas 1, 1635 (1994). Phys. Rev. E 62, 1202 (2000). [3] T. Feurer et al., Phys. Rev. E 56, 4608 (1997); C. T. [19] O. L. Landen, D. G. Stearns, and E. M. Campbell, Phys. Hansen, S. C. Wilks and, P. E. Young , Phys. Rev. Lett. Rev. Lett. 63, 1475 (1989). 83, 5019 (1999). [20] R. Fedosejevs et al., Appl. Phys. B 50, 79 (1990). [4] L. M. Chen et al., Phys. Plasmas 8, 2925 (2001). [21] K. Rawer: Ann.Phys. 35,385 (1939); 42, 294 (1942). [5] D. F. Price et al., Phys. Rev. Lett. 75, 252 (1995). [22] See [17] page 395 [6] H.-J. Kull, Phys. Plasmas 26, 1881 (1983). [23] A. Bergmann, S. H¨uller, P. Mulser, and H. Schnabl, Eu- [7] P. Gibbon and A. R. Bell, Phys. Rev. Lett. 68, 1535 rophys. Lett. 14, 661 (1991). (1992). [24] P. Mulser et at., Phys. Rev. E 63, 016406 (2000). [8] S. C. Wilks, W. L. Kruer, M. Tabak and, A. B. Langdon, [25] D. W. Forslung et al., Phys. Rev. Lett. 39, 284 (1977). Phys. Rev. Lett. 69, 1383 (1992); S. C. Wilks, Phys. [26] P. J. Catto and R. M. More, Phys. Fluids 20, 704 (1977). Fluids B 5, 2603 (1993); P. Gibbon, Phys. Rev. Lett. 73, [27] T. Y. Brian Yang et al., Phys. Plasmas 2, 3146 (1995). 664 (1994); H. Ruhl, A. Macchi, P. Mulser, F. Cornolti, [28] A. Macchi et al., Phys. Rev. Lett. 87, 205004 (2001). S. Hain, Phys. Rev. Lett. 82, 2095 (1999). [29] W. L. Kruer and K. Estabrook, Phys. Fluids 28, 430 [9] H. Ruhl and P. Mulser, Phys. Lett. A 205, 388 (1995). (1985). [10] M. Hentschel et al., Appl. Phys. B 70, 161 (2000). [30] D. F. Zaretsky, J.Phys. B37, 4817 (2004). [11] J. Osterholz et al., Phys. Rev. Lett. 96, 085002 (2006). [31] C. Jungreuthmayer et al., Phys. Rev. Lett 92, 133401 [12] K. Eidmann et al., Europhys. Lett. 55, 334 (2001). (2004) [13] M. Borghesi, A. J. Mackinnon, R. Gaillard, O. Willi, and [32] F. Brunel, Phys. Rev. Lett. 59, 52 ((1987); F. Brunel, D.Riley,Phys.Rev.E60, 7374 (1999). Phys. Fluids 59, 2714 (1988). [14] R. Jung et al., to be published. [33] D. Bauer and P. Mulser, Phys. Plasmas 14, 023301 [15] H. Ruhl, in M. Bonitz and D. Semkat, Introduction to (2007). Computational Methods in Many Particle Body Physics [34] P. Mulser, D. Bauer, and H. Ruhl, Anharmonic reso- (Rinton Press, Paramus, New Jersey, 2006). nance absorption of high-power laser beams, submitted [16] M. V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. for publication. Phys. JETP 64, 1191 (1986). [17] P. Mulser, Laser Interaction with Atoms, Solids and Plas-

42 Coulomb focusing in electron-ion collisions in a strong laser ﬁeld T. Ristow and H.-J. Kull Theoretical Physics A, RWTH Aachen University, Germany

The effect of Coulomb focusing on electrons scattered by an ion in a strong laser field is investigated. Coulomb focusing is explained quantum- mechanically by the interference between an incoming plane wave and an outgoing spherical wave. In the near-field behind the scattering center, the interference pattern shows focusing or defocusing of the electron density depending on the sign of the ion charge. A strong laser-field polarized along the beam direction can reverse the direction of motion of the electron after its first collision with the ion. For such post-collision interactions, Coulomb focusing leads to strongly enhanced scattering rates. In the present work a full quantum-mechanical calculation of electron-ion scattering in a strong laser field is presented and the effect of Coulomb focusing is thereby demonstrated[1]. Electron-ion scattering in strong laser fields is of basic importance for inverse bremsstrahlung absorption (IBA) of laser light in plasmas and also for strong-field above threshold ionization (ATI) of atoms. In the past, substan- tial efforts have been devoted to the study of the field- dependent collision frequency and the energy distribution of the scattered electrons. Some of the major theoretical approaches have been based either on binary collision models [2, 3, 4, 5] or on classical [6, 7, 8, 9] and quantum- mechanical [10, 11, 12, 13] many-particle theories. In high-frequency fields, where the light frequency is much larger than the plasma frequency, dynamic screening of the ion can often be neglected and then the single- electron approximation becomes applicable. However, even in the single-electron approximation, the spatial probability density of the electron becomes distorted in the Figure 1: Spatial probability density of a Gaussian vicinity of the ion, leading to Coulomb focusing for an wavepacket with momentum k0 = 0.7 au scattered by an attractive and Coulomb defocusing for a repulsive po- attractive (upper graph) and a repulsive (lower graph) tential. Local enhancement of the electron density by Coulomb potential. The interference maxima indicate Coulomb focusing is known to play an important role in Coulomb focusing in the attractive and defocusing in the ion-atom collisions[14] as well as in intense field atomic repulsive case. processes[15]. It is also noted that an analogous intensity enhancement exists in the near field of optical scattering experiments[16]. written as a superposition of an incoming plane wave and Coulomb focusing is most easily explained in the clas- an outgoing spherical wave sical framework by the attraction or deflection of parti- f cle trajectories near the scattering center. In quantum ψ = eikz + eikr, f = |f|eiϕ . r mechanics the interpretation is less obvious. No particle trajectories exist and moreover the charge state Z of the The phase ϕ of the scattering amplitude f depends on ion enters the Coulomb cross-section only as Z2. Scatter- the sign of Z, e.g. in the Born approximation ϕ = 0 for ing rates based on the Coulomb cross section are therefore Z = +1 and ϕ = π for Z = −1. The probability density equal for attractive and repulsive potentials. This indi- |ψ|2 shows interference maxima at positions that follow cates that the effect of Coulomb focusing is not related to from the condition of constructive interference the magnitude but to the phase of the scattering ampli- k(z − r) = 2πn + ϕ , n = 0, 1, 2, 3, ··· . tude. According to time-independent scattering theory, the between the incoming and outgoing wave amplitudes. wavefunction of an electron scattered by an ion can be Using cylindrical coordinates z, % one obtains a set of

43 parabolic interference fringes, a spectral method that is better suited to evaluate total energies[1]. An example of the momentum distribution 2 cn % 2πn + ϕ |ψ |2 is represented in Fig.2. It shows a red maximum for zn = − + , cn = k 2 2cn k elastic scattering, yellow energy shells for diﬀerent multiphoton transitions as well as the presence of side-lobes in For ϕ = 0 the positive z axis corresponds to a maximum the angular distribution of a given photon-order (intensity and for ϕ = π to a minimum of |ψ|2 in agreement with modulation along energy shells). The side-lobes can best forward focusing for Z = +1 and defocusing for Z = −1. be seen at the highest photon orders. The interference pattern has been studied in more detail by numerical solutions of the time-dependent Schr¨odinger equation. The incoming electron beam is represented by a Gaussian wavepacket

2 2 2 ik0z −[(z+0.5L) +% ]/d ψ0 = e e 0 with initial position z0 = −0.5L and width d0 = 0.4|z0|. It is propagated up to the time, when the free wavepacket has reached the final position zf = 0.5L. The propagation distance L is chosen sufficiently large such that the wavepacket is asymptotically free at the beginning and at the end of the calculation (L ≈ 100 au). The momentum k0 is varied between 0.7 au and 2.0 au and all quantities are expressed in atomic units (au). In Fig. 1, the final probability density of the scattered wavepacket is represented for an attractive (Z = +1) and a repulsive (Z = −1) potential. It can be noticed that the interference pattern is sensitive to the sign of the ion Figure 2: Momentum distribution for scattering of elec- charge, indicating focusing of the wavepacket towards the z trons with initial velocity k = 2 at a repulsive potential, axis in the attractive and defocusing from the z axis in the 0 Z = −1, in the presence the laser field. The red maxi- repulsive case. The interference fringes are parabolic with mum corresponds to elastic scattering, the yellow rings to a curvature that is proportional to the local wavenumber p 2 different multi-photon orders. k = k0 + Z/r. In the attractive case k is larger than in the repulsive one. To study the effect of Coulomb focusing on electron-ion In Fig. 3, the energy spectra are represented for a beam scattering in a laser field, we have calculated the electron of fast electrons (k0 = 2 au, upper graph) and for a beam energy distributions for scattering from both attractive of slow electrons (k0 = 0.7 au, lower graph). The highest and repulsive potentials under otherwise identical condi- peak corresponds to elastic scattering at the beam energy tions. The difference in these spectra then is attributed 2 k = 2 to the different post-collision interactions for focused and E = 0.5k2 = 0 . 0 0 0.245 k = 0.7 defocused density distributions. The laser field is repre- 0 sented in the dipole approximation by a time-dependent The spectra show the familiar ATI-features with a se- electric field quence of peaks at multiples of the photon energy. The envelope of these peaks has the shape of a scattering plateau E(t) = E0 sin(ωt), E0 = 0.3, ω = 0.3 that extends up to the classical cut-off energy, 8 k = 2 polarized along the axis of the incident electron beam. E = 0.5(k + 2v )2 = 0 . max 0 0 3.6 k = 0.7 The numerical values correspond to a quiver velocity v0 = 0 E0/ω = 1 au. The maximum energies of the scattered elec- For the beam velocity k0 = 2, there is no significant differ- trons are much larger than the photon energy such that the ence between the attractive and repulsive potential. This scattering process is a high-order multi-photon transition. result can be explained by the fact, that the quiver veloc- The energy spectra have been calculated from the spatial ity cannot reverse the direction of the incident electron for Fourier-transform ψk of the wavefunction at the final time k0 > v0 and there is effectively no post-collision interac- according to the definition tion in the laserfield. In contrast, for the beam velocity 1 k0 = 0.7 au, being smaller than v0 = 1 au, there is a sig- P (E) = hψ|δ( k2 − E)|ψi . nificant enhancement of the scattering probabilities for the 2 attractive potential with respect to the repulsive case. It is It is noted, that this energy distribution is a distribution therefore concluded, that the laser-induced post-collision of kinetic energy. However, since the major fraction of interaction is strongly enhanced by Coulomb focusing in the final wavefunction is localized in a region far from the an attractive potential. scattering center, the kinetic energy distribution is actu- In summary, electron-ion scattering in a strong laser ally a good approximation to the total energy distribution. field has been treated by a time-dependent wavepacket ap- This agreement has been confirmed by comparison with proach. The evolution of the wavepacket is described in

44 [4] P. Mulser, F. Cornolti, E. B´esuelle, and R. Schneider, Phys. Rev. E 63, 016406 (2000). 0 Z=+1 10 Z=-1 [5] J. G¨orlinger, H.-J. Kull, and V. T. Tikonchuk, Laser Physics 15, 245 (2005). -3 10 [6] V. P. Silin, Sov. Phys. JETP 20, 1510 (1965). [7] J. Dawson and C. Oberman, Phys. Fluids 5, 517

p(E) (1962). -6 10 [8] C. D. Decker, W. B. Mori, J. M. Dawson, and T. Kat- souleas, Phys. Plasmas 1, 4043 (1994). -9 [9] B. U. Felderhof and T. Vehns, J. Plasma Physics 72, 10 k =2 0 85 (2006). ´ 012345 6 7 8 9 10 11 [10] V. L. Perel’ and G. M. Eliashberg, Sov. Phys. JETP E [au] 14, 633 (1962). [11] H. Reinholz, R. Redmer, G. R¨opke, and A. Wierling, Phys. Rev. E 62, 5648 (2000). 0 Z=+1 10 Z=-1 [12] T. Bornath, M. Schlanges, P. Hilse, and D. Kremp, Phys. Rev. E 64, 026414 (2001). -3 10 [13] H.-J. Kull and L. Plagne, Phys. of Plasmas 8, 5244 (2001).

p(E) [14] J. K. Swenson, C. C. Havener, N. Stolterfoht, K. Som- -6 10 mer, and F. W. Meyer, Phys. Rev. Lett. 63, 35 (1989). [15] T. Brabec, M. Y. Ivanov, and P. B. Corkum, Phys. -9 Rev. A 54, R2551 (1996). 10 k =0.7 0 [16] S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandogh- 012345 6 7 dar, Phys. Rev. Lett. 97, 017402 (pages 4) (2006). E [au] [17] G. M. Fraiman, V. A. Mironov, and A. A. Balakin, Phys. Rev. Lett. 82, 319 (1999). Figure 3: Comparison of the energy distributions for at- [18] A. Brantov, W. Rozmus, R. Sydora, C. E. Capjack, tractive and repulsive potentials. The parameters are V. Yu. Bychenkov, and V. T. Tikhonchuk, Phys. of v0 = 1 and k0 = 2 (upper graph) and k0 = 0.7 (lower Plasmas 10, 3385 (2003). graph). For k0 < v0 the laser ﬁeld can reverse the direction of motion of the incident electron and then the attractive potential leads to enhanced scattering rates due to Coulomb focusing.

the framework of the time-dependent Schrödinger equation and complete numerical solutions have been obtained both for the spatial probability distribution and the energy distribution of the scattered wavepacket. A comparison of the results for attractive and repulsive potentials shows the importance of Coulomb focusing of the wavefunction in strong laser fields. Laser-induced post-collision interactions of the focused wavefunction can lead to strongly enhanced scattering rates. It is also noted that classical particle simulations indicate analogous enhancement effects due to multiple correlated collisions at the same ion[17, 18].

[1] G. Rascol, H. Bachau, V. T. Tikhonchuk, H.-J. Kull, and T. Ristow, Phys. of Plasmas 13, 103108 (pages 9) (2006). [2] N. M. Kroll and K. M. Watson, Phys. Rev. A 8, 804 (1973). [3] L. Schlessinger and J. Wright, Phys. Rev. A 20, 1934 (1979).

45 Quasiperiodic waves in relativistic plasmas T. C. Pesch and H.-J. Kull Theoretical Physics A, RWTH Aachen University, Germany

In this work, we analytically study quasiperiodic plane linearly polarized electromagnetic waves in cold plasmas at relativistic intensities. Electro- magnetic and electrostatic waves are separated for small plasma densities by a two-time-scale-method. With this method the basic system of equations for the wave potentials is solved up to fourth order in the plasma density. The solution completely accounts for the eﬀects of amplitude and phase self- modulation that can be related by an adiabatic invariant. A comparison with numerical solutions shows excellent agreement.

Quasiperiodic waves are the most general stationary so- Figure 1: The periodic solution corresponds in the vacuum lutions of the problem of plane electromagnetic waves in or for a small plasma density to a closed eight-like trajec- cold plasmas at relativistic intensities [1, 2, 3]. These tory. (inner trajectory hγi = 1.06, i.e. A0 = 0.5 , outer 3 0 0 waves result from the coupling of a periodic electro- trajectory hγi = A0 = 10 ; ξ = x ω/c, η = y ω/c). magnetic wave to an electrostatic plasma oscillation. Quasiperiodic waves are a generalization of periodic electromagnetic waves that have been studied extensively in the past [1 - 11]. For periodic solutions the trajectory re- sembles at small plasma densities the famous closed eight- like vacuum trajectory (Fig.1). Periodic solutions exist only for properly adjusted initial conditions (Fig.3). Elec- trostatic oscillations can be excited by the light pressure or the Lorentz-force of the laserbeam. The result is a quasiperiodic wave. For quasiperiodic solutions the orbit is in general non-closed [12, 13] (Fig.2). The nonlinearity of the coupling leads also to amplitude and phase self-modulation. The goal of this work is to separate the quickly varying electromagnetic and the slowly vary- Figure 2: In plasmas an electrostatic oscillation is ex- ing electrostatic timescales of the quasiperiodic wave for a cited by the light pressure. The trajectory is non-closed small plasma density using a two-time-scale-method and to (quasiperiodic solution) and electromagnetic and electro- solve the basic equations for the case of linear polarization. static waves are nonlinearly coupled. (ξ = x0ω/c, η = This approach is based on previous work by Borovsky et al. y0ω/c). [12] that can be extended to fully consistent higher-order solutions in accordance with numerical results. Here σ2 = ω2v /(v2 − 1), ω is the plasma frequency For analytical investigations of the propagation of plane √ p ph ph p electromagnetic waves in cold plasmas the basic equations and γ = 1/ 1 − v2 denotes the relativistic factor. Dimen- have been introduced by Akhiezer and Polovin [1]. In this sionless quantities are used throughout the paper. They model the electrons are treated relativistically, the ions are are connected with the dimensional quantities (dashed) assumed stationary, and thermal effects are neglected. The by t = ωt0, x = x0ω/c, p = p0/(mc), A = eA0/(mc2) 0 equations follow from Maxwell equations and the relativis- and n = n /n0, where n0 is the density of the ions at tic equations of motion. Under the assumption of plane rest. Eqs. (1) describe the full relativistic coupling be- waves with wavenumber k and a constant phase velocity tween the transverse electromagnetic and the longitudinal vph = ω/k propagating in the x-direction all quantities electrostatic oscillations. ˆ depend only on the phase φ = t − x/vph. It allows one The equations for the momenta are equivalent to a sim- to transform the partial differential equations into ordi- pler system of equations for the Lagrangian coodinates nary differential equations. Akhiezer and Polovin use the η, ζ, ξ of the particles (as a function of proper time) [2, 3]. momenta p as independent variables [1, 4], 2 2 Another equivalent system of equations uses the poten- 00 σ vph p⊥ + p⊥ = 0, (1a) tials A and ϕ as variables [9 - 12]. For the investigation of vphγ − px quasiperiodic waves this system of equations is advanta- 2 2 00 ωpvph geous, since in the longitudinal variable ϕ the separation (vphpx − γ) + px = 0. (1b) vphγ − px of the two time-scales is more distinct than in ξ or px.

46 15 10 5 d d 0 = ² = µ∂ + ²∂ , (6a) -5 dφ ds f s 0 100 200 300 d2 4 = µ2∂ + ² (2µ∂ + µ ∂ ) + ²2∂ . (6b) ϕ dφ2 ff fs s f ss 0 A -4 Furthermore, it is assumed that the potentials can be 0 100 200 300 expressed as a series in ² for small plasma densities 10 2 5 A(s, f) = A0(s, f) + ²A1(s, f) + ² A2(s, f) + ...,(7a) 2 0 ϕ(s, f) = ϕ0(s, f) + ²ϕ1(s, f) + ² ϕ2(s, f) + ...(7b) . -5 0 100φ 200 300 In contrast to Borovsky et al. [12], it is also assumed that µ is a series in ²,

2 Figure 3: For arbitrary initial conditions the solution µ(s) = µ0(s) + ²µ1(s) + ² µ2(s) + ... . (8a) is in general quasiperiodic (above, ϕ(0) = 1.2; below, This considerably simplifies the elimination of secular ϕ(0) = 10.0). To obtain a periodic solution the initial terms in higher orders and is an important assumption condition has to be adjusted properly. (central, ² = 0.1, to obtain a consistent solution. A(0) = −5.0, ϕ(0) = 3.6668, The oscillation in ϕ ∝ With this ansatz we solve the basic set of equations (3) cos(2f) is no longer within the accuracy of the graphical order by order in ². Together with the condition of con- representation). sistency (elimination of secular terms) there are sufficient equations to calculate ϕn(s, f), An(s, f) and µn(s) in ev- For the same reason, we restrict attention to small plasma ery order. Without loss of generality the initial condition A = 0 at f = s = 0 is used. For simplicity the initial densities, where the plasma frequency ωp is considerably f smaller than the electromagnetic frequency ω. In this case condition is already completely satisfied in the lowest or- the phase velocity is slightly greater than one. Following der, i.e. ϕn, ϕn, s and An are set equal to zero at φ = 0 [12] we introduce the small parameter ², for n ≥ 1. The lowest order result for ϕ is q 2 ² = vph − 1. (2) ϕ (s, f) = ϕ (s) > 0, (9a) 0 0µ ¶ 1 −2 1 2 −3/2 Futhermore it is convenient to define a new phase φ = ϕ0(s)ss + 1 − ϕ − g ϕ = 0. (9b) √ 2 0 2 0 0 ˆ 2 2 2 2 αφ, where α = σ vph = ωp(1+² )/² . With this notation the basic set of equations for the potentials becomes The differential equation can be integrated once, ³ ´ 00 2 −1 2 −1/2 A + F A = 0, (3a) ϕ0(s)s + ϕ0 + ϕ0 + g0ϕ0 = const. = E0. (10a) ϕ00 + (F ϕ − 1) = 0, (3b) s Our calculation finally yields the fully consistent solution 1 + ²2 up to order ²4, F = . (3c) ϕ2 + ²2(1 + A2) ϕ1(s, f) = 0, (11a) In the subsequent analysis we use a two-time-scale- 1 2 −1/2 ϕ2(s, f) = − g0ϕ0 cos(2f) + ϕ2R(s), (11b) method to solve these equations for the case of linear 16 µ ¶ ϕ 3 1 polarization (A = Ay, Az = 0). In this framework we 2R 2 1/2 −3/2 ϕ2R(s)ss + 1 + g ϕ − g0g2ϕ (11c) now assume that all quantities depend on a slowly varying ϕ3 8 0 0 2 0 µ0 phase s = ²φ and a quickly varying phase f separately. 1 3 − 1 + g2ϕ−3/2 + 2ϕ−2 + g2E ϕ−5/2 The first one describes the time dependence of the electro- 8 0 0 0 16 0 0 0 ¶ static oscillation and the last one the time dependence of 15 45 the electromagnetic wave. The relation between f and s − g4ϕ−3 − g2ϕ−7/2 − 3ϕ−4 = 0, 16 0 0 16 0 0 0 is described by an unkown nonlinear function µ(s), 3 2 ϕ0 s ϕ3(s, f) = − g0 sin(2f), (11d) df(s) 1 64 ϕ0 = µ(s), µ(s) > 0, (4) ds ² 19 4 −2 ϕ4(s, f) = 13 g0ϕ0 cos(4f) (11e) 2 µ so that Z 1 1 ³ 15 1 s − g g 3ϕ−1/2 − E ϕ−3/2 f(s) = µ(s) ds. (5) 8 0 16 0 0 4 0 0 ² 0 49 3 ´ 1 √ √ + g2ϕ−2 + ϕ−5/2 − g ϕ ϕ−3/2 (s = αsˆ, f = fˆ, µ =µ/ ˆ α). The derivative with respect 0 0 0 0 2R 0 16 ¶ 4 4 to the phase φ can now be expressed by partial derivatives −1/2 +g2ϕ cos(2f) + ϕ4R(s), with respect to f and s (indicated by the indices f and s): 0

47 1e+06

A0(s, f) = −a0(s) cos(f), (12a) 1000 1/4 a0(s) = g0ϕ0 , (12b) 1 A1(s, f) = 0, (12c) 0 0.1 0.2 0.3 0.4 0.5

3 3 −5/4 A2(s, f) = g0ϕ0 cos(3f) − a2(s) cos(f) (12d) 1e+06 A 256 µ ϕ 1 a (s) = g 3E ϕ−3/4 + g2ϕ−5/4 (12e) 2 64 0 0 0 0 0 1000 ¶ −7/4 1 −3/4 1/4 +3ϕ + g ϕ ϕ + g ϕ , 1 0 4 0 2R 0 2 0 0 0.1 0.2 0.3 0.4 0.5 frequency 21 ϕ A (s, f) = g3 0 s sin(3f) − b (s) sin(f), (12f) 3 2048 0 7/4 3 ϕ0 Figure 4: Spectra for the periodic (above) and quasiperi- 85 odic case (below). The relative magnitude is plotted A (s, f) = − g5ϕ−11/4 cos(5f) + ... , (12g) 4 217 0 0 against the normalized frequency. In the quasiperiodic case the higher harmonics of the quickly varying timescale are superimposed by the slowly varying contribution. (above ² = 0.1, A(0) = −5.0, ϕ(0) = 3.6668, below r ϕ(0) = 10.0, compare Fig.3). 1 µ0(s) = , (13a) ϕ0 µ1(s) = 0, (13b) the momenta px (py = A) and the energy γ, ϕ 9 3 √ p µ2(s) 2R −2 0 2 2 = − + − ϕ0 (13c) Ex = ϕ α/ 1 + ² ,Bz = 0 (14a) µ0 ϕ0 16 16 √ p 0 2 3 3 Ey = −A α, Bz = Ey/ 1 + ² , (14b) − E ϕ−1 − g2ϕ−3/2, 16 0 0 32 0 0 1 n − 1 = 2 (1 − F ϕ) , (14c) µ3(s) = 0. (13d) ² 1 ³p p ´ p = ϕ2 + ²2(1 + A2) − 1 + ²2ϕ (14d) x ²2 This result extends the results of Borovsky et al. [12] by 1 ³p p ´ three orders in ². The expansion up to order ²4 is neces- γ = 1 + ²2 ϕ2 + ²2(1 + A2) − ϕ (14e) ²2 sary to obtain the equation of motion for ϕ2R(s) (lowest non-trivial correction to the electrostatic oscillation The physical explanation for the amplitude and phase ϕ0(s)). For the same reason it would be necessary to cal- self-modulation is the following one. The electrostatic culate up to order ²6 to obtain the equation of motion for oscillation provokes an additional density modulation in ϕ4R(s). Furthermore the prefactors of the higher harmon- comparison with the periodic case. This density modula- ics of the order ²2 differ from the expressions calculated in tion causes due to the coupling [14]. The solution (11)-(13) completely accounts for the n ¤A + ω2 A = 0, (15) effect of phase modulation (µ(s)) and amplitude modula- p γ tion (a(s)). In all expressions the electrostatic and electromagnetic timescales are strictly separated. The coupling a frequency- and therefore also a phase modulation of the between the electrostatic and the electromagnetic part is electromagnetic wave, expressed by the constants gi. For gi = 0 the differential r 1 2 equation for ϕi corresponds to the famous equation of mo- µ = ²fs = + O(² ). (16) tion for the relativistic electrostatic oscillation (expanded ϕ0 in ²). In addition the solution contains the harmonics gen- The phase modulation, in turn, yields the amplitude mod- eration in A as well as in ϕ. ulation, because according to Eqs. (12b), (13a) the ex- The special case of periodic solutions is included in (11)- pression a2µ is invariant up to the order ²2,

(13) if one demands ϕ0 = const. and ϕnR = 0 for n ≥ 1. 2 2 2 2 2 For quasiperiodic and periodic solutions the highest har- a µ = a0²µ0 + O(² ) = g0 + O(² ). (17) monics expressions coincide in each order. The difference This invariant is the well-known adiabatic invariant of is that the coefficients are slowly time dependent for the an oscillator with a slowly time varying frequency µ, quasiperiodic case, ϕ0 = ϕ0(s). However, in the spectra I = E/µ ∝ a2µ. The whole modulation behaviour is de- of the quasiperiodic solution dominate the slowly varying picted in Fig. 5. A large longitudinal momentum (caused contributions with peaks at multiples of the plasma fre- by the electrostatic oscillation) provokes a density bunch- quency Ωp (including relativistic corrections) (Fig.4). ing. Thereby the electromagnetic frequency in the labora- From the solution (11)-(13) one can deduce other quan- tory system is increased. This in turn, leads according to tities like the electric and magnetic fields, the density n, Eq. (17) to a decrease of the amplitude.

48 10 ϕ A n 5 n/γ 5

ϕ 4

-5 0 100 0 100 200 300 25 50 75 φ φ

Figure 7: Comparison of the quickly varying higher har- Figure 5: The electrostatic oscillation provokes an addi- monics in ϕ. (order ²0 (blue), ²2(black) and numerical tional density modulation (bunching). This yields a fre- (red), initial condition ² = 0.3, A(0) = −5.0, ϕ(0) = 3.0). quency modulation of the electromagnetic wave. Because of the adiabatic invariant I = E/Ω ∝ a2Ω the frequency modulation, in turn, yields the amplitude modulation. ϕ. These harmonics become visible when the amplitude of (² = 0.1, A(0) = −5.0, ϕ(0) = 10.0). the electrostatic oscillation is small or the plasma density is large (Fig.7). Because ϕ0 does not contain any harmon- The amplitude and phase modulation diﬀers from the ics it is essential to consider the correction ϕ2. Again, the coincidence between the numerical result and the result of behaviour observed by Bardsley et al. [15] in a simpliﬁed 2 system (linear restoring force, no gamma factor). In this order ² is very good. Finally we compare the results for system a large longitudinal momentum causes a decrease of the electromagnetic frequency in the laboratory system. A A +ε2A In the last part of this work we compare the analytic 5 0 2 A0 solution with the numerical solution of the basic equations a0 (3), calculated with a fourth-order Runge-Kutta-method. Figure 6 depicts the analytical results of order ²0 and order 0

10 -5 0 50 100 150

2 Ai-A 5 ϕ ϕ +ε2ϕ 0 ϕ0 2 0 -2 A 0 50 100 150 0 0 50 100 150 φ 1 ϕ ϕ i- 0 Figure 8: Comparison of the results for the vector potential A. In the lower part the differences to the numerical result -1 are again presented. (² = 0.2, A(0) = −5.0, ϕ(0) = 10.0). 0 50 100 150 φ the vector potential. Fig. 8 shows the solutions of order ²0 and order ²2 (including the phase correction) together Figure 6: Comparison of the anayltical solution of order ²0 with the numerical solution. Without the phase correc- 2 and of order ² with the numerical result for the scalar po- tion µ2 the analytical and numerical solution gets out of tential ϕ. In the lower part the differences to the numerical phase in a short period of time. The corrections of order result are presented. (² = 0.2, A(0) = −5.0, ϕ(0) = 10.0). ²2 compensates for this. In the upper part of the figure the difference between the result of order ²2 and the numerical ²2 for the scalar potential ϕ together with the numerical re- result is not visible in the graphical representation. sult. For better visual convenience the differences between In summary, we have studied the most general class of the analytical and the numerical result are presented. No- stationary linearly polarized electromagnetic waves in cold tice that the analytical solution ϕ0 contains an accumu- plasmas at relativistic intensities. While previous work lating error. However, the inclusion of the correction of was often concerned with periodic waves (eight-like tra- order ²2 yields a very good agreement. The next com- jectories), the present treatment applies to quasiperiodic parison concerns the quickly varying higher harmonics in waves as well. Using the basic set of equations for the

49 potentials, the slowly varying electrostatic and the quickly varying electromagnetic timescales have been separated for small plasma densities. With the two-timescale-method a fully consistent solution up to order ²4 has been derived. This result extends the results of Borovsky et al. [12] by three orders in ² and completely accounts for the effect of phase and amplitude self-modulation. The interplay between amplitude and phase self-modulation can be explained by the adiabatic invariant following from the wave equation. A comparison with a numerical solution demon- strates that it is essential to include the higher-order corrections. Finally it is mentioned that the effects of phase and amplitude modulation do also accure for circular polarization. Besides the missing of the higher harmonics, the analytical results for this case show only little modifications.

[1] A. I. Akhiezer and R. V. Polovin, Sov. Phys. JETP 3, 696 (1956). [2] T. C. Pesch and H.-J. Kull, Waves with constant phase velocity in relativistic plasmas, GSI Report 2007-03, ILIAS Ion and Laser beam Interaction and Application Studies (2007). [3] T. C. Pesch and H.-J. Kull, Physics of Plasmas 14, 3103 (2007). [4] P. Kaw and J. Dawson, Phys. Fluids 13, 472 (1970). [5] W. L¨unow, Plasma Phys. 10, 879 (1968). [6] C. Max and F. Perkins, Phys. Rev. Lett. 27, 1342 (1971). [7] A. C.-L. Chian and P. C. Clemmow, Journal of Plasma Physics 14, 505 (1975). [8] A. Decoster, Phys. Rep. 47, 285 (1978). [9] P. Sprangle, E. Esarey, and A. Ting, Phys. Rev. Lett. 64, 2011 (1990). [10] W. Mori, C. Decker, and W. Leemans, IEEE Trans. Plasma Sci. 21, 110 (1993). [11] A. I. Zhmoginov and G. M. Fraiman, JETP 100, 895 (2005). [12] A. V. Borovsky, A. L. Galkin, V. V. Korobkin, and O. B. Shiryaev, Phys. Rev. E 59, 2253 (1999). [13] P. K. Kaw, A. Sen, and E. J. Valeo, Physica D Non- linear Phenomena 9, 96 (1983). [14] A. V. Borovsky et al., Laser Physics at Relativistic Intensities (Springer, 2003). [15] J. N. Bardsley, B. M. Penetrante, and M. H. Mittle- man, Phys. Rev. A 40, 3823 (1989).

50 An Overview of Theoretical Work on High Energy Density Physics Using Intense Particle Beams∗

N.A. Tahir1, A.R. Piriz2, A. Shutov3, V. Kim3, I.V. Lomonosov3, D.H.H. Hoffmann4, and C. Deutsch5 1GSI, Darmstadt, Germany; 2UCLM, Ciudad Real, Spain; 3IPCP, Chernogolovka, Russia; 4TU Darmstadt, Germany; 5LPGP, Orsay, France An overview of theoretical work carried out during the past years in different FAIR [Facility for Antiprotons and Ion Research] related research areas is presented in this contribution.

1. High Energy Density States In Matter: The HEDgeHOB Collaboration Three different experimental schemes have been pro- Figure 1: Cylindrical HIHEX: solid metallic target en- posed to study High Energy Density (HED) states in matter at FAIR within the framework of the HEDgeHOB collabo- closed within a cylindrical wall of a strong material like ration. sapphire or LiF which is transparent to IR, visible and VUV. Beam is incident on one face of the cylinder, beam radius is bigger than the cylinder radius, particle range (a) HIHEX [Heavy Ion Heating and Expansion]: larger than the target length that ensures uniform energy Proposed in 2002 [1], the HIHEX scheme exploits the spe- deposition in radial and longitudinal direction. Heated ma- cial features of ”isochoric” and ”uniform” heating of solid terial expands in the gap and thermalizes within the cav- matter by an intense ion beam that generates states of high ity. Physical properties of matter like density, temperature, entropy and high pressure in the sample. The heated sam- pressure and sound speed is directly measured to determine ple expands isentropically and transforms into one of the the EOS of the sample. Final density is controlled by vary- High Energy Density (HED) states. These include an ex- ing the width of the gap between the wall and the target. panded hot liquid phase, two-phase liquid-gas state, critical point parameters and a strongly coupled plasma state. Meticulous numerical simulations were carried out to imploded in a multi-layered cylindrical target using a mul- study design of HIHEX scheme using solid (Fig. 1) as well tiple shock reflection scheme. Two possible experimental as porous (Fig. 2) metallic targets [2, 3, 4]. configurations are shown in Figs. 3 and 4. In Figure 3 one These extensive theoretical studies have shown that us- uses hollow beam with an annular focal spot so that the ing the beam parameters that will be available at FAIR, sample is not directly heated by the beam and only a part of namely, a uranium beam with a particle energy of 1 GeV/u, the surrounding high-Z shell is irradiated and heated by the maximum intensity = 5 1011 and a bunch length of 50 - projectile particles, thereby generating a cylindrical shell 100 ns, employing the HIHEX× technique, one can access of high pressure around the test material. the entire phase diagram for all the materials of interest. It The high pressure generates a strong shock that is is to be noted that the critical temperatures(CT) for most of launched radially inwards that reverberates between the the metals are very high and can not be accessed by tradi- axis of the cylindrical sample and the boundary between tional methods of shock waves and therefore are not exper- the outer shell and the sample material. This scheme leads imentally measured and we only have estimated values of to ultra-high densities, very high pressures but relatively CT for these materials. Using the HIHEX technique, this low temperatures. This design was proposed in 2001 [5] should not be a problem as one can achieve the required and first calculations showed that using the FAIR beam pa- temperatures by depositing the appropriate amount of spe- rameters, one would achieve a density of 1–2 g/cm3, pres- cific power. Moreover,in case of HIHEX, one is not limited sure of 10–20 Mbar while the temperature will be a few to a specific type of material, but one can study all type of kK. These are theoretically predicted physical conditions materials of interest. for hydrogen metallization and this design is therefore very suitable to study this important problem. (b) LAPLAS [Laboratory Planetary Science]: In Analytic studies showed [6] that a LAPLAS implosion this proposed experimental scheme one achieves a low- is very robust as the results are very insensitive to large entropy compression of a test material like hydrogen that is variations in beam and target parameters, provided the specific energy deposition is kept constant. Moreover, it was ∗ Work supported by BMBF shown [3] that a LAPLAS target will be very stable to hy-

51 order of 1 GHz will be required to have 1 % asymmetry in the driving pressure that is needed for a successful implosion. From the technological point of view, this rotation frequency is possible achieve. Figure 4 shows a LAPLAS scheme that uses a circular focal spot instead of an annular one. Calculations have shown that in this case the hydrogenis also strongly heated, but the pressure in the surrounding high-Z shell is still orders of magnitude higher than that in the hydrogen. As a result, one still gets very high compression with a density of 1.2 g/cm3, a pressure of 11 Mbar and a temperature of a few eV. These are the physical conditions that are expected to exist in the interiors of the giant planets. For details see [3, 11]. Therefore using this scheme one can study the planet interiors in the laboratory.

(c) Ramp Compression: Figure 5 shows a schematic Figure 2: Cylindrical HIHEX: porous metallic target en- diagram of this proposed experiment which consists of a closed within a cylindrical wall of a strong material like cylindrical disc of high-Z reservoir followed by the sam- sapphire or LiF which is transparent to IR, visible and ple material and the two are enclosed in a strong cylindri- VUV, without a gap between the two. A beam blocker is cal casing. The ion beam is incident on the reservoir and used to avoid damage to the wall by the ion beam. Grains the ions are completely stopped in the material. The high heated by the beam, ﬁll the pours and uniform physical pressure due to the Bragg peak launches a shock in the lon- conditions are achieved in the sample. gitudinal direction that releases material when it arrives at the reservoir boundary. The expanding material piles up against the sample and pressure builds up slowly that drives a shockless compression of the sample material. Simula- tion show a 60 % compression of an Al sample while the temperature and pressure are of the order of 800 K and 1 Mbar respectively. This scheme is suitable to study material properties like shear modulus and yield strength under dynamic conditions. This will be a very valuable contribution to this ﬁeld because currently this data is available only at room temperature and pressure.

Figure 3: Beam-target conﬁguration of LAPLAS scheme using an annular focal spot. drodynamic instabilities including the Rayleigh-Taylor (R- T) and Richtmyer-Meshkov (R-M) instabilities. Further, analytic and numerical analysis of the the R-T and R-M instabilities including elastic and plastic effects has been performed [7, 8, 9]. It has been proposed that an annular focal spot can be achieved by using an rf wobbler that will rotate the beam at a very high frequency. Analytic and simulation work has shown [10] that a rotation frequency of the

Au Circular Beam Figure 5: Proposed conﬁguration for studies of material properties under dynamic conditions.

H 2. Production Target for Super-FRS at FAIR For production and separation of rare radioactive iso- Figure 4: Beam-target conﬁguration of LAPLAS scheme topes at FAIR, a Superconducting Fragment Separator using an circular focal spot. (Super-FRS) is being designed. Unlike the targets used in

52 HED experiments, the Super-FRS production target should survive during the course of experiments performed at a repetition rate of 1 Hz over an extended period of time. However, with the FAIR beam intensities, due to the high specific energy deposition, a solid target may be destroyed in a single experiment in a fast extraction mode. In principle, the specific energy can be reduced to an acceptable level by increasing the focal spot size, but there is an upper limit for the focal spot size due to the issues of isotope resolution and transmission of the secondary particles through the fragment separator system. We have carried out extensive theoretical work to analyze these problems in detail [12, 13]. Survival of the production target depends on values of two parameters. First, the temperature should always remain well below the melting or sublimation temperature of the material. Second, the level of the thermal stress induced by the beam should remain below the yield strength of the target material. Solid graphite is very attractive because it has a high sublimation temperature of the order of 4000 K, a shear modulus of 12 GPa and a yield strength of 65 MPa. Figure 6: Cross sectional view of a solid graphite Super- It is to be noted that resolving power of a fragment sepa- FRS production target: Von Mises parameter at t = 1 µs rator is inversely proportional to the x-dimension whereas after irradiation. the transmission is affected by the y-dimension of the focal spot only. It has been shown [12] that in case of the fast extraction scheme, an elliptic focal spot with up to σX = 2.0 mm and σY = 12.0 mm, will fulfill the conditions of good resolution and acceptable level of transmission while at the same time it will lead to low enough specific energy deposition for many beam intensities of interest.It has been shown [13] that the thermal stress induced in a solid graphite target by a uranium beam having an intensity of up 1011 ions per bunch, will be below the critical value and the target will survive. This is shown in Fig. 6 where we plot the Von Mises parameter at t = 1 s (Von Mises param- µ Figure 7: Proposed design for an antiproton production tar- eter less than 1 means that the material remains in elastic get at FAIR. regime and the target will be restored to its original state after irradiation). For higher beam intensities, the solid target will be destroyed even if one uses the maximum allowable References focal spot size and it is necessary to develop an alternate [1] D.H.H. Hoffmann et al.,Phys. Plasmas 9 (2002) 3651. target concept, for example, a liquid metal jet target. De- tailed 3D numerical simulations of a liquid Li target for the [2] N.A. Tahir et al.,Phys. Rev. Lett. 95 (2005) 035001. full intensity of the FAIR beam are reported in [14]. [3] N.A. Tahir et al., Phys. High Energy Density 2 (2006) 21. [4] N.A. Tahir et al., Nucl. Instr. Meth. A 577 (2007) 262. 3. Production Target for Antiproton Generation [5] N.A. Tahir et al., Phys. Rev. E 62 (2001) 016402. at FAIR [6] A.R. Piriz et al., Phys. Rev. E 66 (2002) 056403. [7] A.R. Piriz et al., Phys. Rev. E 74 (2006) 037301. Another very important experiment at FAIR will be the [8] A.R. Piriz et al., Laser Part. Beams 24 (2006) 275. production and collection of antiprotons. A bunched beam of 2 1013 29 GeV protons will be smashed onto a pro- [9] J.J. Lopez Cela et al., Laser Part. Beams 24 (2006) 427. duction× target for this purpose with a repetition rate of 0.2 [10] A.R. Piriz et al., Plasma Phys. Controlled Fusion 45 (2003) Hz. Like the Super-FRS target, the antiproton production 1733. target should survive over an extended period of time. A [11] N.A. Tahir et al., Nulc. Instr. Meth. A 577 (2007) 238. proposed target design is shown in Fig. 7 and first numer- [12] N.A. Tahir et al., J. Phys D: Appl. Phys. 38 (2005) 1828. ical simulations have been carried out to study the target [13] N.A. Tahir et al., Laser Part. beams 26 (2008) Accepted. behavior. Further work is being done to achieve a final de- [14] N.A. Tahir et al., Laser Part. beams 25 (2007) 295. sign for a viable target for this experiment.

53 Heavy ions in a high-power laser beam D.B. Blaschke1,2,3, A.V. Filatov4, A.V. Prozorkevich4, D.S. Shkirmanov4 1 University of Wroclaw, Poland; 2 Joint Institute for Nuclear Research Dubna, Russia; 3 Rostock University, Germany; 4 Saratov State University, Russia

The investigations of heavy ion acceleration in plasma matter can cause acceleration of particles in itself laser beams reported in the previous report is- even without participation of the laser field as was recently sue are continued for different field configurations. demonstrated in the SLAC experiment [15]. The correct The main observation is the existence of a big va- description of ion - laser beam interaction at the intensity riety of acceleration modes due to many fitting pa- above 1022 W/cm2 must account probably for non-linear rameters even for only one Gaussian beam and for vacuum effects as the plasma-beam selfinteraction. crossed ones even more so. An essentially non- One beam scheme. Following [11, 16] we consider monotonic dependence of energy gain on relevant the heavy ion acceleration as the motion of classical point variables such as initial velocity or pulse duration particles in the prescribed electromagnetic field is found which makes the search for the most effective acceleration modus very complex. There is p˙ = e E + v × B , (1) a threshold level for the intensity (∼ 1024 W/cm2) S E = − [(x + y) sin φk + (x − y) sin (φk + φ )], (2) when the ion moves in the capture mode in one z kw2 p direction. The crossed beam scheme is at least w0S three times more effective than one beam scheme Ex,y = [ cos φ⊥ cos (φ⊥ + φp)], (3) within the considered range of parameters. How- 2w ever, such a scheme works only for definite phasing S B = [(x − y) sin φk − (x + y) sin (φk + φ )], (4) of the beams which is difficult to provide at such z kw2 p field intensity. Moreover, the other non-linear effects as pair creation and vacuum polarization can Bx = −Ey, By = Ex, (5) also be active. x2 + y2 S = E exp − (2t/τ)4 exp − , (6) 0 w2 The idea of particle acceleration by means of laser fields n o first proposed in [1, 2] was investigated carefully ever since where E0 is the field amplitude, φk and φ⊥ are the phases with the main attention on methods using plasma-beam in- of longitudinal and transversal field components teractions. The laser wakefield accelerator concept which z was introduced in Ref. [3] and became widespread in the φk = φ⊥ + arctan , (7) z recent years is now close to practical realization [4, 5]. c z z x2 + y2 Some variant of this method is the interaction of high- ⊥ φ = φ0 + η + arctan − 2 , (8) intensity laser pulses with solid targets which has proven zc zc w to be effective for applications such as proton radiography 2 or isotope production and nuclear activation [6]. with η = ωt − kz and zc = kw0/2 being the diffraction length, The expected fast development of “table-top” high- intensity lasers [7, 8] opens the possibility of additional 2 2 2 w = w0[1 + (z/zc) ], (9) vacuum acceleration of heavy ion beams for the upgrade of existing accelerators such as the Dubna nuclotron [9], w0 is the minimum spot size of the laser beam at focus, φ0 for example. In a previous work [10] we demonstrated that is a constant and a parameter φp fixes the polarization of such a scheme should be realizable at laser intensities of the field (φp = 0: linear and φp = π/2: circular). the order of 1025 − 1026 W/cm2. The presence of many 2 fitting parameters, on which the result depends essentially I, W/cm λ, nm τ, T φ0 φp w0, λ non-monotonic, opens the possibility that a more effective 5 · 1025 1000 20 0 0 100 acceleration mode could be found. The authors of Ref. e M, Gev v0, c t0, τ z0, λ [11] have concluded that the circular polarization is the 79 179 0.968 -1 -100 most effective one for electron acceleration. On the contrary, we have found that the linear polarization is more Table 1: Basic set of parameters preferable for the heavy ion acceleration. On the other hand, the vacuum pair creation mechanism could be initi- ated at such field intensities provided its appropriate con- It is supposed that at t = t0 the ion is injected on the figuration [12, 13, 14]. This would change considerably the beam axis at the distance z0 from the focus with an ini- conditions for motion of the ion beam. Since the direct cal- tial velocity v0 along axis. The Cauchy problem for the culation of the creation rate for the Gaussian field form is equation (1) x(t0) = (0, 0, z0), v(t0) = (0, 0, v0) is solved not possible we shall perform some oblique estimate with numerically using the standard Runge-Kutta method with help of the field invariants. The presence of short-living basic parameters set as it is specified in the Table 1.

10 ] ] 14

n n φ

=0 o o p e e l l 12 8 c c φ = π/4 u u p n n / / φ = π/2 V V 10 p 6 e e G G

[ [

8 n y

i 4 g a r g e

n 6 y g E 2 r e

4 n E 0 -10 0 10 0 1 2 3 4 5 6 φ [ radian ] Time [ T ] 0

Figure 1: The time dependence of the energy of fully ion- Figure 2: The dependence of the ion energy gain ∆ on the 79+ ized gold ions Au for laser fields the different polariza- initial phase φ0 for different polarizations, φp = 0 - solid tions. line, φp = 1 - dashed line and φp = 0.5 - dotted line.

The main characteristics of interest is the ion energy gain ∆ during the time of the ﬁeld action. The choice of 79+ ]

the ion type (fully ionized gold Au ) corresponds to the

n 15 φ o =0 conditions at the Dubna nuclotron [9] where the ion beam p e l is assumed to have the ﬁnal energy 4 GeV/nucleon. c φ = π/4 u 12 p n A typical picture for the time dependence of the ion / φ = π/2 V p energy is shown in Fig. 1. The monotonic increase of en- e 9 G

[ ergy means that the heavy ion is captured by the laser n i

ﬁeld. Under these conditions, the phase relations have a a 6 g

crucial inﬂuence on the acceleration mode. This is demon- y g strated for example by the dependence of the energy gain r

e 3 n

on the initial phase φ0 in Fig. 2. The most pronounced E influence of φ is observed for linearly polarized field and 0 0 500 750 1000 1250 1500 the least one for the circular polarization. The permanent character of acceleration is confirmed also by Fig. 3 where Wave length [ nm ] the dependence of the energy gain on the wavelength of the laser field is presented. The larger the wavelength the more energy is gained by the ion because of the increase Figure 3: The energy gain ∆ vs. the wave length of laser of the interaction length. Analogous is the dependence of field. The 1053 nm Phelix laser [17] is three times more the energy gain on the intensity of the laser field, Fig. 4 effective than the 527 nm Astra laser [18] for the same In this sense it is surprising that the dependence of intensity. energy gain on the pulse duration τ has essentially non- monotonic character in both, the small time scales of order ing point of two laser beams which create some longitu- of field period for linear polarization (see Fig. 5) as well dinally pulling electric field. The resulting field is defined as the large ones for circular polarization, see Fig. 6. The mainly by the phase relations of the colliding laser beams curve in Fig. 6 represents simultaneously the envelope of the ensuring of which can be a hard technical problem in fast energy oscillations for the linear polarization case. the case of high power systems [24]. The two-laser method A similar behavior shows the energy as a function of is applied successfully in plasma accelerators too [25]. the initial ion velocity, see Fig. 7. The position of a local Let us consider two laser beams propagated according extremum on this curve can be changed by varying pa- to the model [20], Fig. 8: rameters such as pulse duration or field intensity. This property can in principle be used to decrease the veloci- Ex = (Ex1 + Ex2) cos θ + (Ez1 − Ez2) sin θ, ties dispersion (”cooling”) of the ion beam: in order to fit the falling region of curve at the mean velocity of particle Ez = −(Ex1 − Ex2) sin θ + (Ez1 + Ez2) cos θ, beam, its the velocity dispersion must be decreased. x1,2 = x cos θ ∓ z sin θ,

Two crossed beams scheme. The application of spe- z1,2 = x sin θ + z cos θ, (10) cial geometrical schemes for increasing the laser acceleration efficiency was discussed by many authors [19]-[23]. where E1(x1, t) and E2(x2, t) are defined by Eqs. (2)- The basic idea is to send the charged particle in the cross- (6). To become the pure axial field along the z axis it

]

n 6 n o 16 φ =0 o e

l p e φ

l = 1 c p c 5 u φ = π/4 u

n p / n / V 12 φ = π/2 V

e p 4 e G

G [

[

3 i n i

a 8 a g

y 2 y g r g r e e n 4 1 n E E 0 3,0x1025 6,0x1025 9,0x1025 0 20 40 60 80 100 2 Field intensity [ W/cm ] τ [ T ]

Figure 4: The energy gain ∆ vs. the intensity of laser ﬁeld Figure 6: The energy gain ∆ vs. the laser pulse duration of the diﬀerent polarizations with λ= 1053 nm correspond- for large time scales. ing to Phelix laser [17].

is suﬃcient to set E01 = −E02 as it was made in [20] ] 10 providing the 1D-motion of the charge. n

o φ =0

e p The main characteristics of this scheme is the angle de- l c φ π

u 8 = /2 pendence of the energy gain presented in Fig. 9 for the p n / basic set of parameters. This picture depends essentially V e 6 G on other parameters such as the initial position z0 or the [

n pulse duration τ. For example, the curve ∆(θ) becomes i 4 a more smooth by decreasing |z |. In any case, the optimal g 0

◦ y value of the crossing angle is of the order ∼ 20 , which g r 2 e

conﬁrms well the result of Ref. [23]. The eﬃciency of n the crossing angle scheme is about three times higher than E 0 that of the single beam one in the considered range of pa- 0,85 0,90 0,95 1,00 rameters. v [ c ] 0

Figure 7: The energy gain ∆ vs. the initial velocity of the ion. ] 10

n o e l 8 c Unfortunately the direct calculation of pair creation in a u n / Gaussian shaped ﬁeld is beyond the reach at present. We

V 6 e can only make some estimates via the ﬁeld invariants G

[ 4 2 2 n

i E − B , EB, (11) a g

y 2 supposing that the deviation of these quantities from zero g r

e indicates the “creative potential” of the electromagnetic n 0 E field. This trick bases on the analogy with the known 19,0 19,5 20,0 20,5 property of a plane wave where the pair creation is absent. τ [ T ] In this sense, the one Gaussian beam is close to the plane wave since the longitudinal component is suppressed by a factor λ/w0 1 so Figure 5: The energy gain ∆ vs. the laser pulse duration E2 − B2 λ 2 for small time scales of the order of the laser period. ξ = . . (12) E2 w 0 0 Pair creation. It is known that a homogeneous elec- Hence it is natural to suppose that the creation rate is neg- tric field creates a sufficiently dense quasiparticle plasma ligible for such a field. On the contrary, the crossed beams already at an intensity of the order of 1022 W/cm2 [14]. can produce a field of electric type locally by appropri- Therefore it is necessary to account for the possible influ- ate geometry and phasing, e.g., in the case of counter- ence of this effect on the process of heavy ion acceleration. propagating beams when the standing waves are formed.

56 eam -st b B2 f 1 n o ctio dire

direction of 2-nd beam

y Figure 10: The shape of invariant ξ(x, z) at t = 0 for linearly polarized crossed beams with θ = 20◦. The beam Figure 8: The geometry of the crossed beam scheme ac- ”waist” w0 is decreased here to 10λ for sake of presentation cording to [20] for the linear polarization case. convenience.

[7] G. Mourou, T. Tajima, S.V. Bulanov, Rev. Mod. Phys. 78, 309 (2006). [8] J.Ren, W. Cheng, S. Li and S. Suckewer, Nature ] 30 φ =0 p n Physics 3, 732 (2007). o φ π e 25 = /2 l p [9] http://nucloserv.jinr.ru/index.htm. c u n

/ 20 [10] D.B. Blaschke, A.V. Prozorkevich, S.A. Smolyansky, V

e D.S. Shkirmanov, M. Chubaryan, GSI Report 2007- 15 G

[ 03, ILIAS, Ion and Laser Beam Interaction and Ap-

n i 10 plication Studies, Eds. P. Mulser and T. Schlegel, a g p. 34. y 5 g r [11] A. Bahari, V.D. Taranukhin, Quantum Electronics e n 0 34

E , 129 (2004). 0 5 10 15 20 [12] A. Ringwald, Phys. Lett. B 510, 107 (2001). Half crossing angle [ grad ] [13] C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, Phys. Rev. Lett. 89, 153901 (2002). [14] D.B Blaschke, A.V. Prozorkevich, C.D. Roberts, S.M. Figure 9: The energy gain ∆ vs. the half crossing angle. Schmidt, and S.A. Smolyansky, Phys. Rev. Lett. 96, 140402 (2006). [15] I. Blumenfeld et al. Nature 445, 741 (2007). The arbitrary crossing angle scheme can be estimated nu- [16] Y.I. Salamin, S.X. Hu, K.Z. Hatsagortsyan, and C.H. merically only, see Fig. 10. We observe that the regions Keitel, Phys. Rep. 427, 41 (2006). with sufficiently high value of ξ occupy an appreciable part [17] http://www.gsi.de/forschung/phelix/ of field domain and one has to expect that the pair creation mechanism should work. Most likely, the presence [18] http://www.clf.rl.ac.uk / Facilities/ AstraWeb/ As- of electron-positron pairs influences positively on the effi- traGeminiHome.htm ciency of ion acceleration but this question must be con- [19] C.M. Haaland, Opt. Comm. 114, 280 (1995). sidered separately. [20] E. Esarey, R. Sprangle, J. Krall, Phys. Rev. E 52, 5443 (1995). References [21] Y.C. Huang, D. Zheng, W.M. Tulloch, and R.L. Byer, Appl. Phys. Lett. 68, 753 (1996). [1] K. Shimoda, Appl. Opt. 1, 33 (1962). [22] Y.I. Salamin, C.H. Keitel, Appl. Phys. Lett. 77, 1082 [2] A. Ashkin, Phys. Rev. Lett. 24, 156 (1970); 25, 1321 (2000). (1970). [23] Y.I. Salamin, G.R. Mocken, C.H. Keitel, Phys. Rev. [3] T. Tajima and J.M. Dawson, Phys. Rev. Lett. 43, 267 E 67, 016501 (2003). (1979). [24] A. Aiello and H. Woerdman, arxiv:0710.1643 . [4] V. Malka et al., Science 298, 1596 (2002). [25] J. Faure, C. Rechatin, A. Norlin, A. Lifschitz, Y. Glinec and V. Malka, Nature 444, 737 (2006). [5] N. Patel, Nature 449, 133 (2007). [6] S. Atzeni, M. Temporal, and J. J. Honrubia, Nucl. Fusion 42, L1 (2002).

57 References [13] S. Weber, M. Lontano, M. Passoni, et al., Phys. Plasmas 12, 112107(?) (2005). [1] R. Kodama et al., Nature 412, 798 (2001). R. Kodama [14] H. Ruhl, in M. Bonitz et al. (eds) Introduction to et al., Nature 418, 933 (2002). Computational Mehods in Many [2] A. I. Akhiezer and R.V. Polovin, Sov. Phys. JETP 3, Body Physics, ISBN 1-58949-009-6, (2006). 696 (1956) [15] P. Mulser, D. Bauer, and H. Ruhl, Collisionless ab- [3].W. L. Kruer, The Physics of Laser Plasma Interac- sorption of intense laser beams by tions, Addison Wesley Pub. (1988), p. anharmonic resonance, sumitted for publication. 42. [16] S. Hain and P. Mulser, Phys. Rev. Lett. 86, 015 [4] H. Sakagami and K. Mima, Phys. Rev. E 54, 1870 (2001). (1996). [17] P. Mulser and D. Bauer, Laser&Particle Beams 22, 5 [5] B. Hafizi et al., Phys. Plasmas 10,1483 (2003). (2004). [6] M. Lontano, S. V. Bulanov, and J. Koga, Phys. Plas- [18] S. Atzeni, Phys. Plasmas 6, 3316 (1999). mas 8, 5113 (2002) [19] S. Atzeni, M. Temporal, and J.J. Honrubia, Nucl. [7] N. J. Sircombe, T. D. Arber, and R. O. Dendy, Phys. Fusion 42, L1 (2002). Plasmas 12, 012303 (2005). [20] P. Mulser and R. Schneider, Laser&Particle Beams [8] K. Nagashima, Y. Kishimoto, and H. Takuma, Phys. 22, 157 (204). Rev. E 58, 4937 (1998). [21] Y. Sentoku, B. Chrisman, A. Kemp, and T.E. [9] P. Jha et al., Phys. Plasmas 11, 3259 (2004). Cowan, Hot electron energy coupling in [10] P. Kaw and J. Dawson, Phys. Fluids 13, 472 (1970); cone-guided fast ignition, Paper Tu07.6 at IFSA C. J. McKinstrie and D. W. 2007, Kobe, Sept. 9 – 14, 2007. Forslund, Phys. Fluids 30, 904 (1987).

[11] T. C. Pesch and H.-J. Kull, Phys. Plasmas 14,

083103 (2007).

[12] T. Baeva, S. Gordienko, and A. Pukhov, Phys. Rev.

E 74, 046404 (2006).

58 Monte-Carlo study of electron dynamics in silicon during irradiation with an ultrashort VUV laser pulse

N. Medvedev∗ and B. Rethfeld Technische Universit¨atKaiserslautern, Erwin-Schr¨odinger-Str. 46, D-67663 Kaiserslautern, Germany (Dated: April 17, 2008)

PACS numbers: 05.10.Ln, 52.65.Pp, 72.20.Jv, 78.20.Bh, 78.30.Fs

Monte-Carlo simulation was applied for de- system by a laser pulse as well as subsequent excitation scription of excitation and ionization of the elec- of secondary generations of free-electrons need some fem- tronic subsystem within a solid silicon target ir- toseconds ( 10−15 s) [8]; (2) energy exchange with the radiated with a femtosecond laser pulse (25 fs, ~ω lattice and∼ related to these processes cooling down of = 38 eV). The Asymptotical Trajectory Monte excited electrons takes the time interval from 10−13 Carlo method was extended in order to take into s up to 10−11 s [9–12]. For practical applications∼ account the electronic band structure and Pauli’s and to control∼ the laser induced damage, these processes principle for electrons excited into the conduc- play a fundamental role. The first processes in the elec- tion band. Secondary excitation and ionization tronic subsystem provide the initial conditions for the processes were included and simulated event by second ones. Therefore, the temporal evolution of the event as well. free-electron density and their energy distribution in a The temporal distribution of the density of ex- semiconductor during ultrashort laser pulse irradiation ited and ionized electrons, the energy of these are of essential interest. electrons and their energy distribution function In this work Asymptotical Trajectory Monte-Carlo were calculated. It is demonstrated that the final method (ATMC) with binary collision approximation kinetic energy of free electrons is much less than (BCA) [13, 14] was used and modified, where necessary, the total energy provided by the laser pulse, due for description of the kinetics of electronic excitation in a to the energy spent to overcome ionization po- semiconductor irradiated with a femtosecond laser pulse. tentials. It was found that the final total number The number of exited and ionized electrons, the energy of electrons excited or ionized by a single pho- of these free electrons and the energy distribution func- ton is significantly less than ne ~ω/Egap. We tion were calculated for a 25 fs VUV laser pulse in solid introduce the concept of an ‘effective∼ band gap’ Si. Laser parameters and a target were chosen according for collective electronic excitation, which can be to experiments made with a new light source FLASH at applied to estimate the free electron density after DESY in Hamburg [15], which can provide a new kind of high-intensity XUV laser pulse irradiation. ultrashort high-intensity VUV laser pulses.

I. INTRODUCTION II. MODEL

A. Target and photo-ionization processes Interaction of ultrashort laser pulses with semiconduc- tors produce a nano- or micrometric excitation of matter. With the invention of femtosecond lasers, a new regime The ATMC and BCA methods, when excited free elec- of laser-matter interaction was opened where the pulse trons are considered between collisions as point-like par- duration is comparable with characteristic times of pro- ticles having well deﬁned trajectories, are applied. A cesses in the electronic subsystem of material. Nanomet- solid target was treated as a homogeneous isotropic ideal ric spatial and femtosecond temporal scales give new abil- gas having the solid-like density [13, 16]. Interaction of ities for nanotechnologies, micromachining and medical lattice atoms with excited electrons and energy transfer surgery[1–3]. High potential of femtosecond laser pulses to the atomic subsystem were not taken into account on in applications stimulate fundamental theoretical and the investigated timescale ( 10−13 s), which is assumed ≪ experimental investigations of strongly non-equilibrium to be too short for signiﬁcant electron-phonon coupling states of matter [4–7]. [17, 18]. As well, in this research we did not analyze the The kinetics of excitations and relaxations of the tar- energy transformations resulting from radiative decay of get can be divided into a number of processes well sep- electronic vacancies, Auger processes and electron-hole arated temporally: (1) excitation of the electronic sub- recombinations because they usually need times longer than timescales investigated here [19, 20]. In the present work, solid Si was chosen as a target 28 −3 with the density nat = 4.5 10 m . The temporal ∗ · Electronic address: [email protected] distribution of intensity of the laser pulse was chosen in

59 Gaussian shape. Parameters of the laser pulse were taken B. Impact ionizations by exited and ionized in order to fit experimental data [15]: energy of photons electrons is equal to 38 eV, total duration of laser pulse is 25 fs, and during the pulse only 52 photons are absorbed by The conduction band was divided into small intervals ∼ Si cube of 10 10 10 nm: as well. During the simulation only ionizations or exci- × × tations into energy intervals containing still free places, (t 12.5 fs)2 according to DOS-function and Pauli principle, were con- F = 0.059 exp − (1) sidered as possible ones. In the case when there are no · − 25 fs µ ¶ free places for electrons, this event was considered as impossible and was not taken into account. The following where t is the time measured in femtoseconds, function algorithm for the determination of the interaction pa- F is number of photons per femtosecond. rameters realizing the fixed collision of an exited elec- Periodical boundary conditions for the cube were used. tron with an atomic electron was used: first, the path At the initial time, target’s electron distributions (for lengths between all possible next collisions of the electron both, bounded and free electrons) were regarded accord- with bounded electrons in different energetic states were ing to band structure [21, 22]. Electrons with energies be- determined. Then, according to shortest realized path low the bottom of the valence band were treated as ones length, the bounded electron being ionized was chosen. occupying the atomic levels. These ionization potentials And finally, the transferred energy was determined. As were taken from online database on spectral properties the result, the time of movement and the energy loss of of atoms and ions [23]. All electrons upper than final electron were obtained. ‘outermost’ state of the conduction band are treated as In the framework of gaseous model, the distribution e−e free electrons (i.e. belonging to continuum). All elec- of the path lengths lα between series of collisions with tronic effective masses are chosen equal to free electron bounded electron in α-th energetic state can be obtained e−e α e−e α mass. The energy scale of the kinetic energy of delocal- in the exponential form fp(lα ) = 1/l0 exp( lα /l0 ), α − ized electrons is chosen as starting from the bottom of where l0 is the mean free path collisions with an α- the conduction band. Interactions of every single photon electron [13]. According to the Monte-Carlo method, the with bounded electrons were simulated event by event. probability of the electron collision after the path length e−e The probability of interaction with every single electron lα corresponds to the generated random value γα, uni- is determined from the relative cross-sections taking into formly distributed in [0, 1): account the concentrations of electrons with different en- e−e l ergy in a material: α γ = f (x) dx = 1 exp( le−e/lα) (3) α p · − − α 0 C σ Z0 w = α α (2) α C σ β β Due to uniform distributions in [0, 1), (1 γα) and γα β can be considered as equivalent resulting in− the following P e−e dependence of the realized path length lα on γα: Here wα is the probability of interaction with α-s elec- le−e = ln(γ ) lα (4) tron, Cα is the concentration of electrons at α-s level, σα α − α · 0 is the cross-section of interaction of photon with an α-s The mean free path can be written as lα = (n σ k)−1, electron. For simplification of calculation and saving of 0 α α· the computational time, the pure cross-section of photo- where nα is the volume density of α-electrons, σα is the total cross-section of impact ionization of α-particle ionizations σα of electrons from different energetic levels were considered as equal. The differences were intro- determined by the following equation obtained by M. duced only by different number of electrons in different Gryziński[24]: levels, according to DOS-function. 2 3/2 The valence band was divided into small intervals α-s 2 Ry Iα Ee Iα for numerical calculation ( 0.01 eV) and each interval σe−e = 4πa0 − Iα · Ee · Ee + Iα × was normalized according to∼ number of electrons in cube µ ¶ µ ¶ 3 2 I E of 1000 nm . That gave us the relative values of con- 1 + 1 α ln 2.7 + e 1 (5) centration of electrons C in different energetic intervals × 3 · − 2E I − α Ã µ e ¶ Ã r α !! α-s. The ionization by photon of an ‘α’-s electron was fixed and coefficient k is responsible for free places in conduction band, related to full probability (w fv(1 fc)σ) by a randomly given value γ, uniformly distributed in ∼ c.b.− c.b. α−1 α for numerical calculation written in form k = Nα /Ntot , c.b. [0, 1) when γ failed into to the range wβ, wβ , where Nα is the number of free places in α-interval of c.b. "β=1 β=1 ! the conduction band, Ntot is the total number of elec- where wβ are defined by Eq.(2). P P tronic places in α-interval of the conduction band in our

60 3 cube of 1000 nm , Iα is the ionization potential of α- total number of electrons electron, Ee is the energy of the free electron, Ry = 13.6 impact e-e ionized electrons ph

photoionized electrons eV is the Rydberg constant. So, after calculations of the 8 /N e total cross sections for all possible excitation and ion- 7 N ization processes of the traced free electron the realized e−e 6 free paths lengths lα were calculated by Eq.(4). The number of electron α was determined by the shortest free 5

path length. Written in this form mean free path length Laser pulse automatically take into account the Pauli’s principle: if 3 there is no free places in α-interval such free path becomes inﬁnite, so, such collision becomes impossible. 2

The energy transferred to the bounded electron during 1 the interaction with the traced electron was fixed by the Numberof electrons 0 per exciting photon per random value ranging in [Iα,Ee]. Subsequent propaga- 0 5 10 15 20 25 tion of secondary electrons produced by the first genera- Time, [fs] tion of the free electrons and their interactions with the target were also taken into account. FIG. 1: Number of free electrons in Si target during the laser It should be noted that no large errors related to com- pulse irradiation. putations and the used approximations for specific interaction cross-sections appeared because the investigated times are very short [13, 16].

IV. CONCLUSIONS III. RESULTS AND DISCUSSION

In Fig.1 the temporal evolution of the number of free Initial kinetics of ionization and redistribution of free electrons ionized by direct photon absorption and by elec- electrons in semiconductor (Si) during ultrashort XUV tron impact are presented together with the total number laser pulse irradiation were simulated with Monte-Carlo of free electrons. The time-scale in Fig.1 starts from be- method. The method was extended in order to take into ginning of irradiation with laser pulse which is also shown account the band structure of material. The temporal in Fig.1. distributions of the number of free electrons and their The time between two events of impact ionizations is energy distribution at times up to 25 femtoseconds were 10−16 s, which is much shorter than characteristic time obtained and analyzed. ∼of the problem ( 10−15 s). Therefore, the number of It is demonstrated that the band structure influences electrons increase∼ very fast during the laser pulse. The significantly the excitation of electrons. It was found maximum increase occurs directly when laser intensity that an essential part of the total energy provided by and thus the photoionization probability have their max- the laser pulse was accumulated as the potential energy ima. The electron-electron impact ionization is the dom- ( 90%). The final number of excited electrons is much ∼ inating process for the free electron generation (in con- less than estimated by the expression ~ω/Egap. To trast to the case of dielectric irradiation with visible light use this simplest estimation one must replace∼ the actual [6, 7]). As one can see from Fig.1, finally, due to these band gap of the material with an effective value, which secondary processes, each photon excites about Nper ph depends on the properties of target and laser pulse. = 7.6 electrons. A simple estimation of the number of free electrons, given by Nest = ~ω/Egap = 32.6 electrons per photon (where ~ω is the photon energy and Egap is the energy V. ACKNOWLEDGEMENTS gap of material), obviously overestimates the number of excited electrons. The actual final number of free elec- The authors thank to Dr. K. Sokolowski-Tinten (Uni- trons is much less than Nest, because electrons are located not only in upper state of valence band and being versittätDuisburg-Essen, Duisburg, Germany) for bring- ionized not only into lower state of conduction band. So, ing their interest to this topic. Dr. A. Volkov (RRC Kur- for the estimation of the number of ionized electrons by chatov Institute, Moscow, Russia) and Dr. D. Ivanov an expression of this kind, one should replace the real (Technische UniversitätKaiserslautern, Kaiserslautern, energy gap with an effective one, which depends on laser Germany) are acknowledged for fruitful discussions. pulse energy as well as on DOS of target. In the presented The research was supported by German grants BMBF ∗ case the effective band gap is Egap = 5 eV. FSP 301 FLASH, and DFG RE 1141/11-1.

61 [1] A. Vogel, J. Noack, G. Hüttman,G. Paltauf, Appl. Phys. 2002 B 81, 1015 (2005) [15] J. Chalupsky, L. Juha , J. Kuba, J. Cihelka, V. Hajkova, [2] C. L. Arnold, A. Heisterkamp, W. Ertmer, H. Lu- S. Koptyaev, J. Krasa, M. Bergh, C. Caleman, J. Ha- batschowski, Appl. Phys. B 80, 247 (2005) jdu, R. M. Bionta, H. Chapman, A. Velyhan, R. A. Lon- [3] F. Korte, J. Koch, and B. N. Chichkov, Appl. Phys. A don, M. Jurek, J. Krzywinski, R. Nietubyc, J. B. Pelka, 79, 879 (2004) S. P. Hau-Riege, R. Sobierajski, J. Meyer-ter-Vehn, A. [4] A. A. Manenkov and A. M. Prokhorov. Usp. Fiz. Nauk Tronnier, K. Sokolowski-Tinten, K. Tiedtke, S. Toleikis, 29, 179 (1986) [Sov. Phys. Usp. 29, 104 (1986)] T. Tschentscher, H. Wabnitz, U. Zastrau, N. Stojanovic. [5] B. Rethfeld. Phys. Rev. B 73, 035101 (2006) OPTICS EXPRESS 6036, Vol. 15, No. 10, 14 May 2007 [6] B. Rethfeld. Phys. Rev. Lett. 92, 187401 (2004) [16] B. Gervais, S. Bouffard. Nucl. Instr. and Meth. in Phys. [7] A. Kaiser, B. Rethfeld, M. Vicanek, G. Simon. Phys. Research B 88, 355 (1994) Rev. B 61, 11437 (2000) [17] A. E. Volkov, V. A. Borodin. Nucl. Instr. and Meth. in [8] B. Rethfeld, A. Kaiser, M. Vicanek, G. Simon. Phys. Phys. Research B 146, 137 (1998) Rev. B 65, 214303 (2002) [18] L. Van Hove. Phys. Rev. 95, 249 (1954) [9] M. Kaganov, I. Lifshitz, and L. Tanatarov, Zh. Eksp. [19] R. L. Kauffman, K. A. Jamison, T. J. Gray, P. Richard, Teor. Fiz. 31, 232 (1956), [Sov. Phys. JETP bf 4:173, Phys. Rev. A 3, 872 (1975) 1957] [20] J. A. Demarest, R. L. Watson. Phys. Rev. A 17, 1302 [10] S. Anisimov, B. Kapeliovich, and T. Perelman, Zh. (1978) Eksp. Teor. Fiz. 66, 776 (1974), [Sov. Phys. JETP 39:375 [21] D. A. Papaconstantopolous, Handbook of the Band (1974)] Structure of Elemental Solids, Plenum, New York, 1986 [11] H. Elsayed-Ali, T. Norris, M. Pessot, and G. Mourou, [22] L. E. Ramos, L. K. Teles, L. M. R. Scolfaro, J. L. P. Phys.Rev.Lett. 58, 1212 (1987) Castineira, A. L. Rosa, J. R. Leite, Phys. Rev. B 63, [12] P. Lorazo, L. J. Lewis, M. Meunier. Phys. Rev. B 73, 165210 (2000) 134108 (2006) [23] Online database on spectral properties of atoms and ions: [13] W. Eckstein. Computer Simulations of Ion-Solid Inter- http://spectr-w3.snz.ru actions Springer-Verlag, New York, 1991 [24] M. Gryziński, Phys. Rev. A 138, 302 (1965), Phys. Rev. [14] J. M. Hernández-Mangas,J. Arias, L. Bailón,M. Jara´ız, A 138, 322 (1965) Phys. Rev. A 138, 336 (1965) J. Barbolla. J. Appl. Phys. 73, Vol. 91, No. 2, 15 January

List of ILIAS-related publications in 2007

Publications

D. Bauer and P. Mulser, Vacuum heating versus skin layer absorption of intense femtosecond laser pulses, Phys. Plasmas 14, 023301 (2007).

E. Brambrink, T. Schlegel, G. Malka et al., Direct evidence of strongly inhomogeneous energy deposition in target heating with laser- produced ion beams, Phys.Rev.E 75, 065401(R) (2007).

A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Enhancement of vacuum polarization effects in a plasma, Phys. Plasmas 14, 032102 (2007).

A. Di Piazza, A. I. Milstein, and C. H. Keitel, Photon splitting in a laser field, Phys. Rev. A 76, 032103 (2007).

P. Gibbon et al., Stability of nanostructure targets irradiated by high intensity laser pulses, Plasma Phys. Control. Fusion 49, 1873 (2007).

D.H.H. Hoffmann, A. Blazevic, S. Korosity, P. Ni, S.A. Pikuz et al., Inertial fusion energy issues of intense heavy ion and laser beams interacting with ionized matter studied at GSI- Darmstad, Nucl. Inst. Meth. A 577, 8 (2007).

I.V. Lomonosov and N.A. Tahir, Theoretical investigation of shock wave stability in metals, Appl. Phys. Lett. 92, 101905 (2008).

P. Mulser, H. Ruhl, R. Schneider, and D. Batani, Coronal fast ignition by laser: relativistic critical density increase and constraints on maximum laser wavelength, accepted by Journal of Physics.

A. Palffy, J. Evers, and C. H. Keitel, Isomer triggering via nuclear excitation by electron capture, Phys. Rev. Lett. 99, 172502 (2007).

J. Peatross, C. Müller, and C. H. Keitel, Electron wave-packet dynamics in a relativistic electromagnetic field: 3-D analytical approximation, Opt. Express 15, 6053 (2007).

T.C. Pesch and H.-J. Kull, Linearly polarized waves with constant phase velocity in relativisitc plasmas, Phys. Plasmas 14, 083103 (2007).

A.R. Piriz, J.J. Lopez Cela, M.C. Serna Moreno et al., A new approach to Rayleigh Taylor instability: Application to accelerated elastic solids, Nucl. Inst. Meth. A 577, 250 (2007).

A.R. Piriz, N.A. Tahir, J.J. Lopez Cela, O.D. Cortazar, M.C. Serna Moreno et al., Analytic models for the design of the LAPLAS target, , Contrib. Plasma Phys. 47, 213 (2007).

G. Rascol, H. Bachau, V. T. Tikhonchuk, H.-J. Kull, T. Ristow, Quantum calculations of correlated electron-ion collisions in a strong laser field, Phys. Plasmas 13, 103108 (2006).

A. P. L Robinson and P. Gibbon, Production of proton beams with narrow-band energy spectra from laser-irradiated ultrathin foils, Phys. Rev. E 75, 015401R (2007).

A. Shahbaz, C. Müller, A. Staudt, T. J. Bürvenich, and C. H. Keitel, Nuclear signatures in high-order harmonic generation from laser-driven muonic atoms, Phys. Rev. Lett. 98, 263901 (2007).

N.A. Tahir, P. Spiller, A. Shutov, I.V. Lomonosov, V. Gryaznov et al., HEDgeHOB: High energy density matter generated by heavy ion beams at the future facility for antiprotons and on Research, Nucl. Inst. Meth. A 577, 238 (2007).

N.A. Tahir, A.R. Piriz, A. Shutov, I.V. Lomonosov, G. Wouchuk et al., A survey of theoretical work for design of HIHEX and LAPLAS experiments for the HEDgeHOB Collaboration, Contrib. Plasma Phys. 47, 223 (2007).

N.A. Tahir, V. Kim, A. Matveichev, A. Ostrik, I.V. Lomonosov et al., Numerical modeling of heavy ion induced stress waves in solid targets, Laser Part. Beams 25 523 (2007).

N.A. Tahir, P. Spiller, A.R. Piriz, A. Shutov, I.V. Lomonosov et al., Studies of high energy density states using isochoric heating of matter by intense heavy ion beams: HEDgeHOB Collaboration, Physica Scripta (2008); accepted.

S. Udrea, V. Ternovoi, N. Shilkin, A. Fertman, V.E. Fortov et al., Measurement of electrical resistivity of heavy ion beam produced high energy density matter: Latest results for lead and tungsten, Nucl. Inst. Meth. A 577, 257 (2007).

D. Varentsov, V.Ya. Ternovoi, M. Kulish, A. Fertman, D. Fernengel et al., High energy density physics experiments with intense heavy ion beams, Nucl. Inst. Meth. A 577, 262 (2007).

Conference Proceedings

A. Di Piazza, K. Z. Hatsagortsyan, J. Evers, and C. H. Keitel, Vacuum fluctuations and nuclear quantum optics in strong laser pulses, Proc. SPIE 6603, 660305 (2007).

A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Nonlinear interaction of strong laser fields in vacuum, Laser Phys. 17, 345 (2007).

C. Müller, G. R. Mocken, K. Z. Hatsagortsyan, and C. H. Keitel High-energy quantum dynamics in ultraiIntense laser pulses, AIP Conf. Proc. 963, 552 (2007).

N. A. Tahir, I.V. Lomonosov, A. Shutov, V. Kim, V.E. Forotv et al., High energy density matter research using intense heavy ion beams at the future FAIR Facility at Darmstadt, J. Phys.: Conf. Series (2008); accepted.

Reports

P. Mulser and T. Schlegel (Eds.), ILIAS: Ion and Laser beam Interaction and Application Studies, Progress Report No. 2 of the PHELIX theory group, GSI Report 2007- 03, February 2007.

Invited talks

●3rd International Conference on the Frontiers of Plasma Physics and Technology, Bangkok, Thailand, 5 – 9 March 2007 P. Mulser, D. Bauer, and H. Ruhl, Collisionless high-power laser beam absorption: A NO GO theorem and its relativistic implications for energy transport and fast ignition.

●Centre de Physique Théorique de l'École Polytechnique, Palaiceau, 15/03/2007 A. Di Piazza, Enhancement of vacuum polarization effects in a plasma.

●Frühjahrstagung der Deutschen Physikalischen Gesellschaft, Düsseldorf, 19 – 23 March 2007 P. Mulser and D. Bauer, The physics of collisionless absorption of intense laser pulses.

●16th International Laser Physics Workshop (LPHYS'07), Leon, Mexico, 20-24 August 2007 C. Müller, A. Di Piazza, J. Evers, K. Z. Hatsagortsyan, and C. H. Keitel, High-energy, nuclear and QED processes in strong laser fields.

●5th International Conference on Inertial Fusion Sciences and Applications (IFSA 2007), Kobe, Japan, 9 – 14 Septembere, 2007 P. Mulser, Anna Antonicci, and H. Ruhl, Coronal fast ignition by laser: Constraints on energy fluxes and maximum laser wavelength.

N. A. Tahir, V. Kim, A. Shutov, L. V. Lomonosov, A. R. Piriz, G. Wounchuk, J. J. Lopez Cela, D. H. H. Hoffmann, and C. Deutsch, Studies of high energy density states in matter using intense heavy ion beams: The HEDgeHOB collaboration.

● International Conference of Computational Methods in Sciences and Engineering (ICCMSE 2007), Corfu, Greece, 25-30 September 2007 C. Müller, G. R. Mocken, K. Z. Hatsagortsyan, and C. H. Keitel High-energy Quantum Dynamics in Ultra-Intense Laser Pulses.

●1st International Conference on Ultra-Intense Laser Interaction Sciences (ULIS 2007), Bordeaux, France, 1-5 October 2007 A. Di Piuazza, K. Z. Hatsagortsyan, A. I. Milstein, E. Lötstedt, U. D. Jentschura, and C. H. Keitel, QED effects in strong laser fields.

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Particle and nuclear physics in strong laser fields.

P. Mulser, Anna Antonicci, D. Batani, , H. Ruhl, and T. Schlegel, Relativistic increase of critical density and its implications for fast ignition.

●FIAS Seminar Frankfurt Institute for Advanced Studies, Frankfurt, 15 November 2007 C. Müller, Photon physics then and today: From Einstein's photoelectric effect to modern laser applications in physics and medicine.

●University of Umeå, Umeå, Sweden, 04/12/2007 A. Di Piazza, Vacuum polarization effects in strong laser pulses.

●University of Pisa, Pisa, Dec. 2007 A. Di Piazza, Vacuum nonlinearities in strong laser pulses.

Posters

1st International Conference on Ultra-Intense Laser Interaction Sciences (ULIS 2007), Bordeaux, France, 1-5 October 2007 Mirela Cerchez, R. Jung, P. Mulser, H. Ruhl, T. Toncian, and O. Willi, Absorption of ultrashort laser pulses onto solid targets.

Acknowledgement

I would like to thank Mrs. Karin Weyrich for help and advice in editing this report.