copertina dossier pubblicazioni 2014 new · chvykov, a maksimchuk, v yanovsky, z h he, b x hou, j a...

Publications of the ILIL group 20012 – 2013(14)

1. P. Ferrara, M. Ciofini, L. Esposito, J. Hostaša, L. Labate, A. Lapucci, A. Pirri, G. Toci, M. Vannini, and L. A. Gizzi 3-D numerical simulation of Yb:YAG active slabs with longitudinal doping gradient for thermal load effects assessment Optics Express 22, 5375–5386 (2014)

2. T. Pisarczyk, S. Yu. Gus'kov, Z. Kalinowska, J. Badziak, D. Batani, L. Antonelli, G. Folpini, Y. Maheut, F.Baffigi, S. Borodziuk, T. Chodukowski, G. Cristoforetti, N. N. Demchenko, L. A. Gizzi, A. Kasperczuk, P.Koester, E.Krousky, L. Labate, P. Parys, M. Pfeifer, O. Renner, M. Smid, M. Rosinski, J. Skala, R. Dudzak, J. Ullschmied, and P. Pisarczyk. Pre-plasma effect on energy transfer from laser beam to shock wave generated in solid target Physics of Plasmas Phys. Plasmas 21, 012708 (2014).

3. M.Ferrario et al., (see manuscript for full list) IRIDE: Interdisciplinary research infrastructure based on dual electron linacs and lasers Nuclear Instruments and Methods in Physics Research A740, 138-146 (2014)

4. G. Grittani, M.P. Anania, G. Gatti, D. Giulietti, M. Kando, M. Krus, L. Labate, T. Levato, P. Londrillo, F. Rossi, L.A. Gizzi High energy electrons from interaction with a structured gas-jet at FLAME Nuclear Instruments and Methods in Physics Research A740, 257-265 (2014)

5. P Koester, L Antonelli, S Atzeni, J Badziak, F Baffigi, D Batani, C A Cecchetti, T Chodukowski, F Consoli, G Cristoforetti, R De Angelis, G Folpini, L A Gizzi, Z Kalinowska, E Krousky, M Kucharik, L Labate, T Levato, R Liska, G Malka, Y Maheut, A Marocchino, P Nicolai, T O’Dell, P Parys, T Pisarczyk, P Raczka, O Renner, Y J Rhee, X Ribeyre, M Richetta, M Rosinski, L Ryc, J Skala, A Schiavi, G Schurtz,MSmid, C Spindloe, J Ullschmied,, JWolowski and A Zaras Recent results from experimental studies on laser–plasma coupling in a shock ignition relevant regime Plasma Phys. Control. Fusion 55 (2013) 124045

6. G Sarri, W Schumaker, A Di Piazza, K Poder, J M Cole, M Vargas, D Doria, S Kushel, B Dromey, G Grittani, L Gizzi, M E Dieckmann, A Green, V Chvykov, A Maksimchuk, V Yanovsky, Z H He, B X Hou, J A Nees, S Kar, Z Najmudin, A G R Thomas, C H Keitel, K Krushelnick and M Zepf Laser-driven generation of collimated ultra-relativistic positron beams Plasma Phys. Control. Fusion 55 (2013) 124017

7. G. Cristoforetti, E. Tognoni and L.A. Gizzi Thermodynamic equilibrium states in laser-induced plasmas: From the general case to laser-induced breakdown spectroscopy plasmas Spectrochimica Acta Part B 90 (2013) 1-22

8. T. Ceccotti, V. Floquet, A. Sgattoni, A. Bigongiari, O. Klimo, M. Raynaud, C. Riconda, A. Heron, F. Baffigi, L. Labate, L. A. Gizzi, L. Vassura, J. Fuchs, M. Passoni, M. Kveton, F. Novotny, M. Possolt, J. Prokupek, J. Proska, J. Psikal, L. Stolcova, A. Velyhan, M. Bougeard, P. D’Oliveira, O. Tcherbakoff, F. Reau, P. Martin, and A. Macchi Evidence of Resonant Surface-Wave Excitation in the Relativistic Regime through Measurements of Proton Acceleration from Grating Targets Phys. Rev. Lett. 111, 185001 (2013)

9. V. Floquet, O. Klimo, J. Psikal, A. Velyhan, J. Limpouch, J. Proska, F. Novotny, L. Stolcova, A. Macchi, A. Sgattoni, L. Vassura, L. Labate, F. Baffigi, L. A. Gizzi, Ph. Martin and T. Ceccotti Micro-sphere layered targets efficiency in laser driven proton acceleration J. Appl. Phys. 114, 083305 (2013)

10. A. Giulietti, A. André, S. Dobosz Dufrénoy, D. Giulietti, T. Hosokai, P. Koester, H. Kotaki, L. Labate, T. Levato, R. Nuter, N. C. Pathak, P. Monot and L. A. Gizzi Space- and time-resolved observation of extreme laser frequency upshifting during ultrafast-ionization Phys. Plasmas 20, 082307 (2013).

11. A. Sgattoni, T. Ceccotti, V. Floquet, A. Bigongiari, M. Raynaud, C. Riconda, F. Baffigi, L. Labate, L. A. Gizzi, L. Vassura, J. Fuchs, O. Klimo, M. Kveton, F. Novotny, M. Possolt, J. Prokupek, J. Proska, J. Psikal, L. Stolcova, A. Velyhan, M. Bougeard, P. Martin, I. Prencipe, A. Zani, D. Dellasega, A. Macchi, M. Passoni Laser plasma proton acceleration experiments using foam-covered and grating targets Proc. SPIE 8779, Laser Acceleration of Electrons, Protons, and Ions II; and Medical Applications of Laser-Generated Beams of Particles II; and Harnessing Relativistic Plasma Waves III, 87790L (May 7, 2013); doi:10.1117/12.2017449.

12. L.Labate, M. G. Andreassi, F. Baffigi, G. Basta, R. Bizzarri, A. Borghini, G. Candiano, C. Casarino, M. Cresci, F. Di Martino, L. Fulgentini, F.Ghetti, M. C. Gilardi, A. Giulietti, P. Koester, F. Lenci, T. Levato, Y. Oishi, G. Russo, A. Sgarbossa, C. Traino, L. A. Gizzi Small-scale laser based electron accelerators for biology and medicine: a comparative study of the biological effectiveness Proc. SPIE 8779, Laser Acceleration of Electrons, Protons, and Ions II; and Medical Applications of Laser-Generated Beams of Particles II; and Harnessing Relativistic Plasma Waves III, 87790O (May 7, 2013); doi:10.1117/12.2019689..

13. G. M. Grittani, M. P. Anania, G. Gatti, D. Giulietti, M. Kando, M. Krus, L. Labate, T. Levato, Y. Oishi, F. Rossi, L. A. Gizzi High energy electrons from interaction with a 10 mm gas-jet at FLAME Proc. SPIE 8779, Laser Acceleration of Electrons, Protons, and Ions II; and Medical Applications of Laser-Generated Beams of Particles II; and Harnessing Relativistic Plasma Waves III, 87791B (May 7, 2013); doi:10.1117/12.2027030.

14. Antonio Lapucci, Marco Ciofini, Laura Esposito, Paolo Ferrara, Leonida A. Gizzi, Jan Hostaša, Luca Labate, Angela Pirri, Guido Toci,Matteo Vannini Characterization of Yb:YAG active slab media based on a layered structure with different doping Proc. SPIE 8780, 87800J-1 (2013); doi:10.1117/12.2017380 Online Publication Date: May 22, 2013.

15. Leonida Antonio Gizzi, Carlo Benedetti, Carlo Alberto Cecchetti, Giampiero Di Pirro, Andrea Gamucci, Giancarlo Gatti, Antonio Giulietti, Danilo Giulietti, Petra Koester, Luca Labate, Tadzio Levato, Naveen Pathak and Francesco Piastra Laser-Plasma Acceleration with FLAME and ILIL Ultraintense Lasers Appl. Sci. 2013, 3, 559-580; doi:10.3390/app3030559

16. M. Ferrario, D. Alesini, M. Anania, A. Bacci, M. Bellaveglia, O. Bogdanov, R. Boni , M. Castellano, E. Chiadroni, A. Cianchi, S.B. Dabagov, C. De Martinis, D. Di Giovenale, G. Di Pirro, U. Dosselli, A. Drago, A. Esposito, R. Faccini, A. Gallo, M. Gambaccini, C. Gatti, G Gatti, A. Ghigo, D. Giulietti, A. Ligidov, P. Londrillo, S. Lupi, A. Mostacci, E. Pace, L. Palumbo, V. Petrillo, R. Pompili, A.R. Rossi, L. Serafini, B. Spataro, P. Tomassini, G. Turchetti, C. Vaccarezza, F. Villa, G. Dattoli, E. Di Palma, L. Giannessi, A. Petralia, C. Ronsivalle, I. Spassovsky, V. Surrenti, L. Gizzi, L. Labate, T. Levato, J.V. Rau SPARC_LAB present and future Nuclear Instruments and Methods in Physics Research B 309, 183-188 (2013)

17. L.A. Gizzi, M.P. Anania, G. Gatti,D. Giulietti, G. Grittani, M. Kando, M. Krus, L. Labate, T. Levato, Y. Oishi, F. Rossi Acceleration with self-injection for an all-optical radiation source at LNF Nuclear Instruments and Methods in Physics Research B 309, 202-209 (2013)

18. G C Bussolino, A Faenov, A Giuliett1, D Giulietti, P Koeste1, L Labate, T Levato, T Pikuz and L A Gizzi Electron radiography using a table-top laser-cluster plasma accelerator J. Phys. D. Appl. Phys. 46 245501 (2013)

19. T. Levato, M. Calvetti, F. Anelli, D. Batani, R. Benocci, L. Cacciotti, C.A. Cecchetti, O. Cerafogli, P. Chimenti, A. Clozza, N. Drenska, a, A. Esposito, R. Faccini, a, S. Fioravanti, A. Gamucci, C. Gatti, A. Giulietti, D. Giulietti, L. Labate, V. Lollo, S. Martellotti, M. Monteduro, E. Pace, N.C. Pathak, L. Pellegrino, F. Piastra, M. Pistoni, G. Di Pirro, R. Di Raddo, U. Rotundo, R. Ricci, M. Richetta, C. Vaccarezza, P. Valente, a, L.A. Gizzi First electrons from the new 220 TW Frascati Laser for Acceleration

and Multidisciplinary Experiments (FLAME) at Frascati National Laboratories (LNF) Nuclear Instruments and Methods in Physics Research A720, 95-99 (2013)

20. ILIL Group Laser-plasma acceleration at LNF-Frascati – a selection from the LNF internal report contributions on the subject from 2003 to 2011 INO Internal Report,Prot. N.3156 del 04/04/2013 (2013)

21. T. Yabuuchi, R. Mishra, C.McGuffey, B. Qiao, M. S Wei, H. Sawada, Y. Sentoku, T. Ma, D. .P Higginson, .K U. Akli, D Batani, H Chen, L A Gizzi, M. H. Key, A. J. Mackinnon, H. S. McLean , P. A. Norreys, P. K. Patel 4, R. B. Stephens, Y. Ping, W Theobald, C Stoeckl, and F. N. Beg Impact of extended preplasma on energy coupling in kilojoule energy relativistic laser interaction with cone wire targets relevant to fast ignition New Journal of Physics, 15 015020 (2013)

22. G. Cristoforetti, M. P. Anania, A. Ya. Faenov, A. Giulietti, D. Giulietti, S. B. Hansen, P. Koester, L. Labate, T. Levato, T. A. Pikuz, L.A. Gizzi Spatially resolved analysis of Kα x-ray emission from plasmas induced by a femtosecond weakly relativistic laser pulse at various polarizations Physical Review E 87 023103 (2013)

23. D. Jovanovic, R. Fedele, F. Tanjia, S. De Nicola, and L.A. Gizzi Nonlocal effects in the self-consistent nonlinear 3D propagation of an ultrastrong, femtosecond laser pulse in plasmas Eur. Phys. J. D 66, 328 (2012)

24. G. Sarri, A. Macchi, C. A. Cecchetti, S. Kar, T. V. Liseykin, X. H. Yang, M. E. Dieckmann, J. Fuchs, M. Galimberti, L. A. Gizzi, R. Jung, I. Kourakis, J. Osterholz, F. Pegoraro, A. P. L. Robinson, L. Romagnani, O. Willi, and M. Borghesi Dynamics of Self-Generated, Large Amplitude Magnetic Fields Following High-Intensity Laser Matter Interaction Phys. Rev. Lett. 109, 205002 (2012)

25. L. Labate, P. Koester, T. Levato, L.A. Gizzi A novel technique for single-shot energy-resolved 2D X-ray imaging of plasmas relevant for the Inertial Confinement Fusion Rev. Sci. Instrum. 83, 103504 (2012)

26. N.C. Pathak, G. Bussolino, C.A. Cecchetti, A. Giulietti, D. Giulietti, P. Köster, T. Levato, L. Labate, L.A. Gizzi, Frequency shift of an intense laser pulse induced by plasma wave Nuclear Instruments and Methods in Physics Research A680 (2012) 103–107

27. G. Cristoforetti, M. Tiberi, A. Simonelli P. Marsili, and F. Giammanco Toward the optimization of double-pulse LIBS underwater: effects of experimental parameters on the reproducibility and dynamics of laser-induced cavitation bubble Applied Optics 51, (2012)

28. G. Cristoforetti, E. Pitzalis, R. Spiniello, R. Ishakc, F. Giammanco, M. Muniz-Mirandae, S. Caporali Physico-chemical properties of Pd nanoparticles produced by Pulsed Laser Ablation in different organic solvents Applied Surface Science 258 3289– 3297 (2012)

29. T.Levato, C.A. Cecchetti, N.Drenska, R.Faccini, C.Gatti, A.Giulietti, D.Giulietti, L.Labate, , V.Lollo, S.Martellotti, N.C.Pathak, F.Piastra, M.Richetta, F.Tani, C.Vaccarezza,P.Valente, L.A. Gizzi, Preliminary results of the self injection test experiment (SITE) at FLAME. in Proceedings of the International School of Physics «Enrico Fermi» Edited by F. Ferroni, A. Gizzi Leonida, R. Faccini (2012)

30. LA.Giulietti et al., Towards Laser driven Mini-Linac's for biomedical uses. in Proceedings of the International School of Physics «Enrico Fermi» Edited by F. Ferroni, L. A. Gizzi, R. Faccini (2012)

31. L.A.Gizzi Laser-Plasma Diagnosstics in Proceedings of the International School of Physics «Enrico Fermi» Edited by F. Ferroni, L. A. Gizzi, R. Faccini (2012)

3-D numerical simulation of Yb:YAG active slabs with longitudinal doping gradient for

thermal load effects assessment P. Ferrara,1 M. Ciofini,2 L. Esposito,3 J. Hostaša,3,4 L. Labate,1,5,* A. Lapucci,2 A. Pirri,6

G. Toci,6 M. Vannini,6 and L. A. Gizzi1,5 1Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Via G. Moruzzi, 1 - I-56124 Pisa, Italy

2Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Largo Enrico Fermi 6, I-50125 Firenze, Italy 3Istituto di Scienza e Tecnologia dei Materiali Ceramici, Consiglio Nazionale delle Ricerche, Via Granarolo 64,

48018 Faenza, Italy 4Department of Glass and Ceramics, ICT Prague, Technická 5, 166 28 Prague, Czech Republic

5Istituto Nazionale di Fisica Nucleare, Sezione di Pisa and Laboratori Nazionali di Frascati, Italy 6Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, I-50019 Sesto

Fiorentino (FI), Italy *[email protected]

Abstract: We present a study of Yb:YAG active media slabs, based on a ceramic layered structure with different doping levels. We developed a procedure allowing 3D numerical analysis of the slab optical properties as a consequence of the thermal load induced by the pump process. The simulations are compared with a set of experimental results in order to validate the procedure. These structured ceramics appear promising in appropriate geometrical configurations, and thus are intended to be applied in the construction of High Energy Diode Pumped Solid State Laser (DPSSL) systems working in high repetition-rate pulsed regimes. © 2014 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (140.3580) Lasers, solid-state; (140.3480) Lasers, diode-pumped; (140.6810) Thermal effects; (140.3615) Lasers, ytterbium.

References and links 1. W. Koechner, Solid-State Laser Engineering, 2nd ed. (Springer, 1988). 2. A. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” IEEE J. Quantum

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#202559 - $15.00 USD Received 4 Dec 2013; accepted 27 Jan 2014; published 28 Feb 2014(C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005375 | OPTICS EXPRESS 5375

14. A. Lapucci, M. Ciofini, M. Vannoni, and A. Sordini, “High efficiency, diode pumped Nd:YAG ceramics slab laser with 230 W continuous-wave output power,” Appl. Opt. 51(18), 4224–4231 (2012).

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24. A. Lapucci, M. Ciofini, L. Esposito, P. Ferrara, L. A. Gizzi, J. Hostaša, L. Labate, A. Pirri, G. Toci, and M. Vannini, “Characterization of Yb:YAG active slab media based on a layered structure with different doping,” Proc. SPIE 8780, 87800J1 (2013).

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1. Introduction

In Diode-Pumped Solid State Lasers (DPSSLs) with end-pumped geometry, excellent efficiencies can be obtained along with a good modal selection, due to the optimal spatial matching of the pumping beam with the fundamental mode of the laser resonator [1,2]. In such geometries, power extraction is ultimately limited by consistent thermal distortions arising in the active medium even for relatively small pumping powers [3]. Thermal distortions of the active medium produce deleterious effects such as thermal lensing, depolarization and bifocusing. Thus, carefully optimized designs are needed to combine the high laser efficiency with maximum power extraction capabilities [4]. The impact of pump heating is considerably reduced choosing a gain medium with a favourable energy level structure, such as ytterbium-doped materials [1,5]. In Yb:YAG active medium the quantum defect is small (less than 10%). Moreover, the absence of energy levels other than those concerned with pump and laser radiation, prevents the occurrence of parasitic processes (excited-state absorption, up-conversion, self quenching), making thermal load further reduced. Yb:YAG is also an optimal candidate for the attainment of High Energy DPSSLs working in high repetition rate pulsed regimes [6] thanks to its relatively long upper state lifetime. However, owing to the quasi-three-level scheme, high pumping intensities are required to deplete the ground state population.


Specific architectures, such as the slab laser [7] or the disk laser [8], have been adopted in the past to overcome the limits imposed by thermal distortions of the active medium.

Recently, ceramics have been produced and tested in laser systems [9–11]. Good mechanical properties, more uniform composition, shorter production time, larger slabs size, superior micro-hardness and fracture toughness, together with higher doping concentration of laser active ions, have been demonstrated [12,13]. Nd:YAG and Yb:YAG ceramics have been successfully employed in high power lasers, with slab or disk geometry [14,15]. Most importantly, ceramic materials present the further advantage of allowing an easy production of composite structures with different doping levels. These structured active media have been proposed [16–18] as a possible solution to the thermal problem. Multi-layer active media can be produced with a combination of tape casting and hot-isostatic pressing (HIP) technology. The biggest drawback of HIP method is the high fabrication cost that it implies. Recently, it has been found that simple vacuum sintering technology with optimized fabrication parameters can be used in place of HIP, which can significantly limit the production costs [19,20].

Thermal effects in solid state lasers have been widely investigated from the theoretical point of view. The early works by Koechner [21,22] were devoted to address thermal effects in laser rods. The modelling was done in a two-dimensional approximation, either with the idealization of an infinitely long laser rod subject to uniform pumping, or utilizing the approximations of two-dimensional temperature variations. A thorough review of the studies on diode-pumped lasers with special reference to ytterbium-doped materials has been presented in a recent paper [23]. In this work the authors provide an interesting discussion about the main approximations adopted for modelling thermal effects in end-pumped lasers, namely, the plain strain approximation (long rods) and the plain stress approximation (thin discs). In both cases the three dimensional problem can be reduced to a two dimensional one, either for translational invariance (plain strain) or for small longitudinal variations (plain stress).

A non-uniform doping distribution leads to an absence of any translational symmetry, and thus quests for a full three-dimensional model. In our work, we adopt a numerical approach based on Finite Element Mesh (FEM) analysis, allowing a longitudinal variation of the parameters. This enables us to estimate thermal distributions and thermo-mechanical stresses in spatially structured active media. The calculation of optical wavefront aberrations generated inside the distorted active media permits to evaluate their different behaviours in terms of thermal lens. The work is aimed at optimizing the longitudinal doping profile in a ceramic Yb:YAG specimen. Experimental measurements of thermal lens effective focal lengths (EFLs) are used to validate our numerical approach. We also show that optimal doping distribution is strongly related to the pumping and cooling geometry. This complex numerical analysis is necessary in order to assess the advantages or disadvantages of a specific doping distribution design in terms of thermal effects.

In this paper we report on the thermal effects analysis on structured samples referring to a cooling and pumping geometry available in our lab [24]. Consistently with the experimental observations our simulations show that a longitudinal variation of the Yb doping does not produce significant advantages in terms of thermal effects. This appears to be true as long as the cooling direction is substantially transverse. For comparison we show some results in case of a longitudinal cooling geometry, typical of active-mirror or disk lasers. In this case longitudinally structured samples do show advantages in terms of thermal lens reduction. This result appears more relevant for systems with a high aspect-ratio (d/t), that is the ratio between the pumped volume diameter (d) and the sample thickness (t).


2. Description of the simulation model

2.1 FEM model

In this subsection the methodology is described, adopted to provide a numerical solution to the thermo-mechanical problem of a laser irradiated slab with longitudinal doping gradient.

The Finite Element Method (FEM) software COSMOS, which provides a powerful numerical solver [25], has been used to perform this analysis.

The simulated slabs have all a cylindrical shape with a 10 mm diameter and 3 mm thickness. They are pumped on a 0.5 mm diameter region on one side. As it will be discussed later, we simulated 2 different cooling geometries: in the first one the slab is cooled on a ring surface external to a 5 mm diameter on the opposite side with respect to the pump, see Fig. 1(a); in the second one the cooling is applied to the whole surface, see Fig. 1(b). The first cooling geometry exactly models our experimental test bed [24–26] and is used for the model validation. The second cooling scheme models the one-sided face-cooling characterizing disk or active-mirror lasers [27]. To compare the results in the two different situations we have always modelled the pumping as a single passage pump-beam scheme.

The typical computational grid is shown in Fig. 1(c). The central region shows a finer mesh which corresponds to the pumped region. The outer region, where a coarser mesh is used, is where the heat flows through the slab away from the pumped volume.

Each slab is modelled as a stack of 5 equally thick layers, each layer having its own doping. Different combinations of the 5 layer dopings were simulated, thus corresponding to different longitudinal absorption profiles. In order to perform a significant comparison, the incident pump power is adjusted so as to obtain the same optical absorption in each sample. Such absorption was set to 25 W, which, in the specific case of Yb:YAG, corresponds to a total thermal load (i.e. the power to be dissipated) of about 2.6 W.

The pump absorption in each layer was calculated using the absorption coefficient %Aα δ= ⋅ , where %δ is the Ytterbium doping percentage in the chosen layer and

A = 0.5 cm−1 [28,29]. Each layer is heated with a power density that can be calculated as the power absorbed by

the single layer divided by its volume. On the surface in contact with copper heat sink the temperature is set to REFT = 20°C; the initial condition for the simulation is INT = 20°C in the whole slab.

a b c

Fig. 1. Pictorial view of the simulated geometries with COSMOS software package. (a): annular cooling; (b): cooling on the whole surface; (c): finite element grid of the computational domain.

2.2 Calculation of induced wavefront aberrations

In a slab active medium, the induced optical path difference OPD(r) at different transverse positions with respect to an unperturbed beam is related to 3 contributions [22,23]:


• the surface bulging due to thermal expansion,

• the refractive index variations with temperature,

• the stress induced birefringence due to the photo-elastic effect. Hence OPD(r) can be calculated by using this formula (we assume here a cylindrical

symmetry):

( ) ( ) ( )( )

( )( )( )

( )0, ,0 0

1L L r L L r

REF jj r z j T

n nOPD r n L r T r T dz r dzT ϑε

εε

+Δ +Δ

=

∂ ∂ = − ⋅ Δ + ⋅ − + ⋅ ∂ ∂

(1) where L is the slab thickness without pumping, 0n is the refractive index without

deformations ( 0 1.82n = for Yb:YAG), REFT is the temperature of the cooling system ( 20REFT C= ° ), ΔL(r) is the surface deformation at radial position r , jε is a diagonal component of the strain tensor, and z is the direction parallel to the cylindrical axis. Variables (r, θ) refer to the polar coordinate system in the plane transverse to propagation.

The FEM thermo-mechanical analysis described in the previous sub-section provides a group of data for each node of the computational grid:

• a scalar temperature variation field ( )REFT T TΔ = − , • the 3 components of the displacements,

• the 6 components of the stress tensor (3 for iσ and 3 for ijτ ) [23]. Strains ( ijε ) can be easily calculated from the stress and temperature fields using the

conventional constitutive equations for linear elastic deformations in a material:

( ) ( )

( ) ( )

( ) ( )

1

1 ;

1

r r z REF

r z REF

z z r REF

T TE

T TE

T TE

φ

φ φ

φ

ε σ ν σ σ α

ε σ ν σ σ α

ε σ ν σ σ α

= − + + − = − + + − = − + + −

(2)

1

1 ;

1

r r

rz rz

z z

E

E

E

φ φ

φ φ

νε τ

νε τ

νε τ

+ = + =

+ =

(3)

where, considering our host material (YAG), 112.8 10E Pa= ⋅ is the Young modulus, 0.28ν = is the Poisson coefficient, 67.9 10 / Kα −= ⋅ is the thermal expansion coefficient.

The first two contributions to the OPD (thermal and surface contribution) can be directly calculated from the output of the thermo-mechanical simulations. To determine the third contribution we need the relation between the refractive index (for a non-polarized beam) and the strain tensor. This relation is in general expressed by a variation of the dielectric impermeability 2nd rank tensor Bij as a function of strains [22], (Bij is defined as the reciprocal of the permeability 2nd rank tensor [30]). We thus can write:


,ij ijkl klB p εΔ = (4)

where ijklp is the 4th rank elasto-optical tensor. Equation (4) is valid only in a coordinate system with the axes parallel to the crystal axes (for the sake of simplicity, we will call this system “crystal oriented coordinate system”). The components of the impermeability 2nd rank tensor (Bij) are the coefficients of an ellipsoid, called “indicatrix”, that, in the principal directions (indicated with the “ ” symbol), can be expressed by this formula:

( )21,2,3

1,i ii

B x=

= (5)

where:

( )0 21 ,i ii

B B Bn

= + Δ = (6)

for a diagonal dielectric impermeability tensor. The index of refraction can be written as:

( )0, ,

,ii jj r z j T

nn n r

ϑε

ε=

∂ = + ⋅ ∂

(7)

where: in is the refractive index along one of the principal directions of the stress-perturbed

indicatrix [22], iB is the dielectric impermeability tensor along one of the principal directions, 0B is the initial value of this tensor in such directions: ( ) 0IN ijijB B δ= , where:

0 20

1 0.30Bn

= .

Equation (4), involving the 4th rank tensor, reduces to a much simpler formulation in the case of a cubic crystal with [111] orientation [22,23]. Moreover Khazanov et al. [31] demonstrated that the YAG ceramic behaviour is similar to that of the [111] oriented single crystal, given the averaging effect of the different grains orientations. This fact was also confirmed by experimental birefringence measurements in Nd:YAG ceramics [32]. Under these conditions the impermeability tensor can be written in the crystal oriented coordinate system as:

( )( )

( )( )

11 1 12 2 3 44 6 44 5

44 6 11 2 12 1 3 44 4

44 5 44 4 11 3 12 1 2

', ', ' ,p p p p

B x y z p p p pp p p p

ε ε ε ε εε ε ε ε εε ε ε ε ε

+ +Δ = + +

+ +(8)

where [22,23]:

11

12

44

0.02900.0091.0.0615

ppp

= − = + = −

In our numerical simulations the values of jε in (8) are obtained by COSMOS FEM calculations referring to the “crystal oriented” system. After calculating BΔ for each node using Eq. (8), the corresponding values of BΔ are transformed back to the slab coordinate system. Afterwards, the B tensor is diagonalized, in order to get the indicatrix coefficients of Eq. (5). From these coefficients the refractive indices are directly found.


The propagated beam wavefront is thus calculated from the nodal refractive index values and fitted using Zernike polynomials [30].

As it is well known, any wavefront distribution ( ),f r ϑ in the transverse plane can be described in terms of Zernike polinomials:

( ) ( )( )0

, , ,n

mnm n

n m nf r a Z rϑ ϑ

∞

= =−

= (9)

where the coefficients nma , giving the magnitude of the different aberrations, can be calculated inverting the previous equation:

( ) ( ), , .mnm ncircle

a f r Z r rdrdϑ ϑ ϑ= (10)

The wavefront focal length is given in our case by the following equation:

2

20

,4

MAXrfa

⋅

(11)

where MAXr is the radius of the pumped region and 20a is the defocus coefficient.

3. Discussion of the simulation results

3.1 Ring Cooled Geometry

In this paragraph we show the results obtained from our numerical approach for two different cooling geometries, as described in section 2.1. In the first one, called “RCG” (i.e. “Ring Cooling Geometry”), the slab is cooled only on a ring shaped surface on the non-pumped side. In the second one, called “FCG” (i.e. “Face Cooling Geometry”), the slab is cooled on the whole surface on the non-pumped side.

Table 1. Doping levels of the uniform/structured samples studied in our numerical simulations.

Sample # Sample name Yb percentage in different layers (%) L5 L4 L3 L2 L1 1 Uniform 5 5 5 5 5 2 Graded A 7 5 5 3 3 3 Graded B 7 7 5 3 1 4 Stepped 10 10 10 0 0 5 Capped 0 10 10 10 0

Each geometry has been analyzed considering 5 different Yb doping distributions, as shown in Table 1, where the first doping value (L1) refers to the layer on the pumped side of the sample, whereas the last one (L5) refers to the layer on the cooled side. They are shown in a reverse order in Table 1, so as to mimic the pumping direction as in Figs. 1(a) and 1(b).

Figure 2 shows the thermal distributions on an r-z section of the cylindrical slabs obtained from our simulations for the five samples defined in Table 1. In the mentioned figure, cooled sides are on the left and pumped sides are on the right.

The isothermal surfaces show that, given the ring shaped cooling surface, heat flows out of the pumped volume in a substantially radial way. We can also see that, due to the more uniform pump absorption, graded doping can make the temperature distribution smoother than all the other cases, including the case of uniform doping. Thus also the maximum internal temperature results reduced in this case. However, we will see in what follows that this does not lead to a substantial reduction of the thermal lens. In fact the z-oriented beam propagation determines an averaging of the refractive index variations cancelling the effect of a more uniform pump power absorption.


Fig. 2. Temperature distribution in an axial cross-section for the 5 cases in RCG. Maximum temperatures are for the five samples Tmax(1) = 71°C, Tmax(2) = 65°C, Tmax(3) = 71°C, Tmax(4) = 81°C, Tmax(5) = 81°C.

Fig. 3. Surface deformation in an axial cross-section for the 5 cases in RCG.

Figure 3 shows the slab deformations in the 5 analyzed cases. Here it is clear that smaller surface deformations are obtained when there is a low doping level close to the external surfaces. The surface deformations for the five cases are listed in Table 2.

Table 2. Maximum temperature, maximum surface deformations and maximum Von Mises stresses for the 5 analyzed cases in RCG.

Sample # Max. Temp. [°C] ΔLp (pump side) [x10−8 m] ΔLc (cooler side)

[x10−8 m] Max. Von Mises

stress [x107 N/m2] 1 71 3.567 1.940 3.1 2 65 2.476 2.980 2.9 3 71 1.295 3.200 3.35 4 81 0.467 3.420 3.7 5 81 1.315 0.720 3.6

Figure 4 shows the Von Mises stress values for the 5 cases analyzed in RCG configuration. Here it is clear that the “stepped” and “capped” cases, that show an advantage in terms of surface deformations, present higher internal stresses. This is due to the fact that a higher doping has been used in the internal layers, in order to reach the same total pump absorption. Thus the graded sample (# 2) is expected to show the highest fracture-limit power absorption, while the stepped sample (# 4) is expected to show the lowest one.


Fig. 4. Von Mises stress distribution in an axial cross-section for the 5 cases in RCG.

In Table 3 the global thermal lens e.f.l.s are given for the 5 sample distributions defined in Table 1, in the RCG case. The maximum focal length deviation from the average value is 3.3%, for this set of samples.

Table 3. Effective focal length (e.f.l.) calculated for the 5 analyzed samples in the RCG configuration (thermal power loading equal to 2.6 W).

Sample # e.f.l. (cm) 1 8.60 2 8.68 3 8.87 4 8.80 5 9.11

For the sake of a comparison and validation of our modelling, Fig. 5 shows the radial profile of the wavefront obtained with the numerical procedure described in the previous section, compared to the one measured on a real sample using a Shack-Hartman sensor in our lab [24–26].

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0-2.0

-1.5

-1.0

-0.5

0.0

O.P

.D. (

μm)

radial position (mm)

F.E.M. Calculations experimental (x) experimental (y)

Fig. 5. Comparison of the calculated OPD radial profile and that experimentally measured with a Shack-Hartmann on a 2 mm diameter aperture of our sample, data reported here refer to the uniform doping distribution (sample #1).

The aberrated wave-front is correctly reconstructed both inside and outside the pumped area (0.5 mm diameter). Limiting the aperture to the pumped region the wavefront shape is substantially parabolic (see Fig. 6). This allows us to describe the behaviour of different


samples in terms of thermal lens Dioptric Power. Interestingly the experiments performed on a few selected cases [24] confirm that, in this geometry, the thermal Dioptric Power is substantially independent from the longitudinal doping distribution. This validates our modelling results. A more detailed description of the experimental tests is discussed in a different paper [26], both in terms of thermal lens and in terms of thermally induced depolarization.

Fig. 6. OPD approximated using Zernike polynomials (sample n.1).

3.2 Face Cooled Geometry

The situation is quite different in the case of a mainly longitudinal heat flow, as exemplified in the FCG configuration of our analysis.

1 2 3 4 5

Fig. 7. Temperature distribution in an axial cross-section for the 5 doping cases of Table 1, adopting a Face Cooling Geometry (FCG) with 3 mm thickness and 0.5 mm pump volume diameter.

Figure 7 shows the temperature distribution on axial sections (r-z) in FCG configuration referring to the same 5 samples studied in the RCG case. Here we can observe that the heat flux is no longer purely radial, as in RCG configuration, but it shows a relevant longitudinal component. Of course the relative relevance of the longitudinal flux is higher for systems with higher aspect-ratios d/t, where d is the pumped volume base diameter and t the sample thickness. Thus we have analyzed the thermal lens behaviour for the five sample doping distributions in the FCG configuration for different aspect ratios. Diameters and thicknesses


have been chosen in such a way as to maintain the same pumping power densities in all cases (i.e. scaling the square root of the pumping volume diameter with the inverse of the thickness). Figure 8 summarizes the results of our analysis in terms of thermal lens Dioptric Power. Our numerical simulations allow to evaluate the different contributions to the thermal lens effect described in Eq. (1), namely direct thermal refractive index dependence, photo-elastic contribution and surface deformations. In Fig. 8 the three contributions to the Dioptric Power are separately indicated.

0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

12

Elastic

Surface

Thermal

1-uniform 2-graded A 3-graded B 4-stepped 5-capped

Ther

mal

Len

s D

iopt

ric p

ower

(m-1)

Thickness (mm)

TOTAL

Fig. 8. Thermal lens Dioptric Power versus thickness for the five analyzed doping distributions in case of Face Cooling Geometry.

The relative weight to the total thermal lens in the analyzed cases is reported in Table 4. The table shows the minimum and maximum share of each type of effect.

Table 4. Minimum and maximum relative contributions to the thermal lens Dioptric Power.

Cooling Thickness [mm] Thermal [%] Surface [%] Elastic [%] RCG 3.0 84 – 88 5 – 12 3.6 - 6.8 FCG 3.0 86 – 91 5 – 10 3.6 - 5.3 FCG 1.5 75 – 80 19 – 22 3.1 - 3.4 FCG 0.75 66 – 68 30 – 32 1.7 - 1.9 FCG 0.5 60 - 61 38 - 39 1.0 - 1.1

Some important considerations can be drawn from this analysis. First of all one can see that the maximum relative focal length deviation from the average value grows from 4% in the case of 3 mm thick samples to 18% in the case of 0.5 mm thick samples. This fact confirms that varying the doping distribution results in a more significant outcome in systems with a strongly longitudinal heat flux. Moreover it can be noted that the thermal and elastic contributions to the thermal lens dioptric power scale linearly with the thickness while the


surface contribution does not. Thus the relative importance of the different contributions changes with the aspect-ratio as clearly evidenced in Table 4.

In general the lowest direct thermal effect is obtained when maximizing the doping close to the cooling surface. The lowest photo-elastic effect is obtained in systems with less steep doping gradients. Finally, smaller surface deformations are obtained in systems with small doping close to the external faces.

4. Conclusions

In this paper we have reported on a numerical simulation procedure suited to analyze the thermo-mechanical behaviour of Yb:YAG ceramic slabs with longitudinal doping gradient, along with the corresponding optical effects. Our work was motivated by the availability of structured slabs with longitudinal doping gradient given by the ceramic sintering technology.

The first set of simulations confirms our experimental observations. They show that a longitudinally structured material is ineffective for the reduction of thermal effects in end-pumped systems with radial cooling. On the contrary longitudinal doping gradients are useful in case of a parallel cooling geometry, such as in the case of disk or active-mirror lasers. This effect is more pronounced in high aspect-ratio systems.

Our numerical simulations also allow to evaluate the relevance of the different contributions to the thermal lens effect, namely direct thermal refractive index dependence, photo-elastic contribution and surface deformations. The future steps will be the study of specific laser designs, seeking ways to optimize the pump energy deposition profile in order to achieve a better thermal management and a mitigation of deleterious optical effects. The final aim is the scaling of DPSSLs based on ceramic doping gradient media to higher peak and average powers.

Acknowledgments

We acknowledge support from the EC initiative “LASERLAB-EUROPE” (EC contract no. 284464) - Joint Research Activity WP33 - “European Research Objectives on Lasers for Industry, Technology and Energy (EURO-LITE)” and the HiPER (ESFRI-FP7) project. J.H. would also like to acknowledge the financial support from a specific university research grant (MSMT No 20/2013); P.F would also like to acknowledge the financial support from regione Toscana through the project “R&D of innovative wavefront sensors and Adaptive Optics for laser-driven Radiological Devices – AdOpRad” (protocollo ISTI-CNR No 0000745, 09/03/2012).


Pre-plasma effect on energy transfer from laser beam to shock wavegenerated in solid target

T. Pisarczyk,1 S. Yu. Gus’kov,2 Z. Kalinowska,1 J. Badziak,1 D. Batani,3 L. Antonelli,3

G. Folpini,3 Y. Maheut,3 F. Baffigi,4 S. Borodziuk,1 T. Chodukowski,1 G. Cristoforetti,4

N. N. Demchenko,2 L. A. Gizzi,4 A. Kasperczuk,1 P. Koester,4 E. Krousky,5 L. Labate,4

P. Parys,1 M. Pfeifer,5 O. Renner,6 M. Smid,6 M. Rosinski,1 J. Skala,5 R. Dudzak,5

J. Ullschmied,5 and P. Pisarczyk71Institute of Plasma Physics and Laser Microfusion, Warsaw, Poland2P.N. Lebedev Physical Institute of RAS, 53 Leninsky Ave., 119 991 Moscow, Russia3Universit�e Bordeaux, CNRS, CEA, CELIA (Centre Lasers Intenses et Applications), UMR 5107,Talence, France4Intense Laser Irradiation Laboratory at INO-CNR, Pisa, Italy5Institute of Plasma Physics ASCR, v.v.i., ZaSlovankou 3, 182 00 Prague 8, Czech Republic6Institute of Physics ASCR, v.v.i., Na Slovance 2, 182 21 Prague 8, Czech Republic7Warsaw University of Technology, ICS, 15/19 Nowowiejska St., 00-665 Warsaw, Poland

(Received 17 October 2013; accepted 8 January 2014; published online 23 January 2014)

Efficiency of the laser radiation energy transport into the shock wave generated in layered planar

targets (consisting of massive Cu over coated by thin CH layer) was investigated. The targets were

irradiated using two laser pulses. The 1x pulse with the energy of �50 J produced a pre-plasma,imitating the corona of the pre-compressed inertial confinement fusion target. The second main

pulse used the 1x or 3x laser harmonics with the energy of �200 J. The influence of the pre-plasma on parameters of the shock wave was determined from the crater volume measurements

and from the electron density distribution measured by 3-frame interferometry. The experimental

results show that the energy transport by fast electrons provides a definite contribution to the

dynamics of the ablative process, to the shock wave generation, and to the ablation pressure in

dependence on the target irradiation conditions. The strong influence of the pre-plasma on the

investigated process was observed in the 1x case. Theoretical analysis supports the explanation ofexperimental results. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4862784]

I. INTRODUCTION

One of the main research aims related to the shock igni-

tion concept (SI)1–3 is the investigation of the ablation pres-

sure mechanism due to the laser spike at the intensity of

1–50 PW/cm2 and duration of several-hundred-ps, assuming

that the main part of the absorbed laser energy is converted

to fast electrons under the presence of the pre-plasma. The

energy transfer by fast electrons into the plasma with super-

critical density can provide an ablation pressure of several

hundreds of Mbar, which is necessary for generating the

igniting shock.4–6 Recent experiments with OMEGA laser7,8

seem to suggest an increasing efficiency of the energy trans-

fer to both planar and spherical targets, resulting from the

contribution of fast electrons generated due to stimulated

Raman scattering and two-plasmon decay in an extended

pre-plasma.

This work extends our previous research9–11 on the role

of fast electrons in the laser energy conversion to shock

waves performed with the PALS iodine laser, delivering a

300 ps duration pulse at intensities of 1–50 PW/cm2 using

the first (1x, 1314 nm) and third (3x, 438 nm) harmonicsradiation. In those experiments, massive targets of Al and Cu

have been irradiated at various focal spot radii of the laser

beam, RL, to identify the mechanisms of laser absorption and

to determine their influence on the absorbed energy transfer

to the target. The mass of the ablated solid material as well

as the fraction of the laser energy deposited in the plasma

have been determined by using the 3-frame interferometer

and by measuring the volume of the crater created on the

solid surface. Each of the interferometric channels is

equipped with own independent interferometer system of the

wave type.12 Interferometer is irradiated by a part of the 1xmain beam subsequently converted to the second harmonic.

Interferograms provided by these interferometers are

obtained by separation, inversion, and folding of the front

face of the probing wave.

The experiments have shown a strong influence of the

wavelength and the intensity of the laser beam on the effi-

ciency of the laser energy transfer to the massive target, in-

dependent from its material. 2D numerical simulations,

including fast electrons transport10,13 as well as theoretical

analysis based on an analytical model,11 fully confirmed the

experimental results and demonstrated conclusively that in

the case of 1x, intensities of 10–50 PW/cm2 and withoutpre-plasma on the target surface, the dominant ablation

mechanism is heating by fast electrons generated at the reso-

nant absorption. For the maximum laser energy of 580 J and

intensity of 50 PW/cm2, the ablative pressure reaches about

180 Mbar in spite of two-dimensional expansion of the target

corona.9,10 However, for 3x, the ablation pressure originatesfrom the thermal electron conductivity heating, and its value

of about 50 Mbar is several times lower in comparison with

the 1x case.9,10

1070-664X/2014/21(1)/012708/7/$30.00 VC 2014 AIP Publishing LLC21, 012708-1

PHYSICS OF PLASMAS 21, 012708 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

193.205.148.189 On: Wed, 09 Apr 2014 07:10:42

http://dx.doi.org/10.1063/1.4862784http://dx.doi.org/10.1063/1.4862784http://dx.doi.org/10.1063/1.4862784http://crossmark.crossref.org/dialog/?doi=10.1063/1.4862784&domain=pdf&date_stamp=2014-01-23

The results of the next step experiment directed to the

study of the pre-plasma effect on the ablation and the energy

conversion efficiency to the shock wave are reported below.

II. EXPERIMENTAL CONDITIONS AND RESULTS

As shown in Fig. 1, this investigation used planar lay-

ered targets consisting of massive Cu and a 25-lm-thicklayer of light plastic (CH) material. Accordingly to the SI

concept, targets were irradiated using two laser pulses. The

first 1x beam with energy of �50 J produced a pre-plasmaimitating the corona of the pre-compressed inertial confine-

ment fusion (ICF) target that is a spherical target destined for

creation an inertially confined thermonuclear plasma under

an action of pulsed energy driver. The spike-driven shock

wave was generated by the main pulse with energy of

�200 J (either at 1x or 3x). The influence of the pre-plasmaon the shock wave parameters was determined from meas-

urements of the crater volume and the electron density distri-

butions measured by 3-frame interferometry.

Interferograms were registered 2 ns after the second

laser pulse maximum. This registration time seems to be op-

timum, since the processes of the absorption of the laser radi-

ation in the plasma plume terminated already. The delay

between the main and the auxiliary laser beams was kept

fixed to Dt¼ 1.2 ns. Interferograms were registered for dif-ferent focal spot radii RL in the range of 40–160 lm. Typicalinterferograms obtained at 1x or 3x, with and withoutpre-plasma, are shown in Figs. 2 and 3.

In the case of the pre-plasma absence, Fig. 2, the raw

interferograms display very high axial symmetry, both for

1x and 3x. In contrast, this symmetry is partially disturbedby the 1x beam which produces the pre-plasma (see Fig. 3).The phase distribution results (calculated on the basis of the

shifts of interference fringes) indicate that in the case of 1xthe axial asymmetry does not exceed 15% and slightly more

only for 3x. To solve the Abel equation, average values ofthe phase corresponding to the top and the bottom halves of

the interferogram were taken into account. The FFT (Fast

Fourier Transform)14 was applied for the electron density

determinations. Because the inaccuracy of determination of

the shifts of the fringes (the method of the maximumfringe15) is relatively high (several percent only), the accu-racy of the electron density determination results from the

degree of the axial symmetry of the investigated plasma.

That is why outside of the opacity zone, in the case of no

pre-plasma, the density error is small—at the level of 10%.

For the pre-plasma case, it is somewhat larger but still does

not exceed 30%.

An example of plasma density profile obtained from the

interferograms is shown in Fig. 4.

The density profile on axis follows an exponential decay

to a good approximation ne(z)¼ n0e�z/L. The parameters ofthis function determine the maximum electron density gradi-

ent in the opacity zone: [dne/dz]z¼ 0¼ n0/L, where L is thescalelength of the density gradient and n0 is the maximum

electron density.

From the 2D density profile, one can calculate Ne, the

total electron number in the plasma plume, assuming cylin-

drical symmetry. (A condition which is already applied in

order to use Abel inversion for the determination of density

from the interferogram.)

Another quantity, which was measured in the experi-

ment, is the crater volume Vcr. In order to obtain information

about the shape and dimensions of the craters, we used their

replicas made of cellulose acetate (see Fig. 5).

FIG. 1. Scheme of the layered target and its irradiation by laser beams.

FIG. 2. Raw interferograms and their reconstructions for the case without

pre-plasma and: (a) 1x and (b) 3x radiation.

012708-2 Pisarczyk et al. Phys. Plasmas 21, 012708 (2014)


193.205.148.189 On: Wed, 09 Apr 2014 07:10:42

To reconstruct quantitatively the crater shape, micropho-

tography was employed. The crater shape in a chosen cross-

section was digitized to provide data for subsequent calcula-

tions. As some craters are not quite symmetrical and their

shapes are irregular, photographs of their replica were made

in two mutually perpendicular directions. Therefore, the vol-

umes of the craters were determined by splitting them into

quarters prior to calculations, as shown in Fig. 5. Each quar-

ter of the crater volume was then evaluated independently.

Finally, summation of these intermediate results provided

the total crater volume.

Fig. 6 presents a comparison of the crater volumes Vcrand the Ne/Vcr ratios obtained for the two wavelengths (1xand 3x) without and with the pre-plasma (Ne is the totalelectron number in the plasma plume). The importance of

Ne/Vcr parameter is described in Ref. 9.

Since the laser energy is constant (E¼ 200 J), by varyingthe focal spot radius, we also vary the laser intensity on tar-

get. Typically, when we vary the focal spot radius between

40 and 160 lm, the laser intensity changes between 1.6 �1016 W/cm2 and 1.0 � 1015 W/cm2. As shown in Fig. 6(a),without pre-plasma, the dependences of Vcr and Ne/Vcr on

RL are similar to both Al and Cu targets presented in Refs. 9

and 10. In the 3x case with the predominant inverse brems-strahlung absorption,11 the efficiency of the crater creation

FIG. 3. Raw interferograms and their reconstructions for the case with pre-

plasma and: (a) 1x and (b) 3x.

FIG. 4. Determination of the maximum electron density gradients: (a) exper-

imentally obtained density distribution and (b) density profile on axis.

FIG. 5. Crater volume determination.

FIG. 6. Dependence of the crater volume and the Ne/Vcr ratio on the focal

spot radius of the main laser beam in the case of (a) without and (b) with the

pre-plasma. The data correspond to target presented in Fig. 1, laser energy

E¼ 200 J, and delay Dt¼ 1.2 ns in all shots.



193.205.148.189 On: Wed, 09 Apr 2014 07:10:42

reduces with decreasing RL due to the lateral expansion

effect. In contrast, for 1x, the efficiency of the crater crea-tion increases with the decreasing RL, which according to

numerical simulations directly corresponds to the energy

transfer to the target by fast electrons generated due to reso-

nant absorption.

The strong influence of the pre-plasma on the crater crea-

tion process in the case of 1x is clearly seen in Fig. 6(b). Thecrater volume decreases by more than one order of magnitude

and simultaneously, Ne/Vcr increases by the same rate. It can

be assumed that in the presence of pre-plasma the fast electron

energy transfer effect becomes insignificant in 1x case. In thecase of 3x, the pre-plasma influences the crater formation pro-cess only weakly, and both Vcr and Ne/Vcr values remain at

the same level as in the case without pre-plasma. Under these

conditions, the fast electron effect remains insignificant, in

analogy to the case without pre-plasma.

The electron density distributions obtained for 1x and3x in the cases with and without pre-plasma at differentfocal spot radii (RL) are presented in Fig. 7. These data

clearly demonstrate an effect of the pre-plasma on the main

laser beam interaction with the target. Without the

pre-plasma and in the 1x case, Fig. 7(a), the plasma is moreexpanded in the radial direction and has a characteristic min-

imum on the z-axis.

This is clearly visible from the density distributions for

RL¼ 40 lm. In contrast with 1x, for the case of 3x, theplasma expansion is directed along the z-axis. Both the axial

range and the axial density grow up with the increasing RL.

The presence of the pre-plasma, Fig. 7(b), results in limited

radial expansion of the central plasma produced by the main

beams (1x and 3x). In the case of 3x, the radial limitationof the central plasma expansion part is larger, i.e., the expan-

sion character is more axial.

The dependencies of the maximum density gradient, the

maximum electron density, and the scalelength on RL for the

above-mentioned conditions of the target irradiation are

shown in Fig. 8.

Without pre-plasma and using 1x, Fig. 8(a), the densitygradient increases strongly with the decreasing RL even for

the radii smaller than 80 lm. Conversely, in the case of 3x,the density gradient decreases with the decreasing focal spot

radius. For RL< 80 lm, the density gradient scalelength ofthe ablative plasma created by means of the 1x beam isabout 200 lm, which is almost twice smaller in comparisonwith the plasma generated by 3x.

Without pre-plasma, both the crater volume and the den-

sity gradient decrease with the decreasing beam radius at 3x,and, conversely, increase (particularly strongly at the small

radii of 40 and 80 lm) at 1x. As previously shown,9,10 theenergy transfer by fast electrons is responsible for the growth

of the crater volume with the decreasing beam radius (i.e.,

increasing laser intensity) in the 1x case. Below, we demon-strate that the same effect gives rise to the observed depend-

ence of the density gradient. The presence of the pre-plasma

obviously leads to the minimum growth of these characteris-

tics with the decreasing beam radius. This means that under

the presence of the pre-plasma, the effect of the energy trans-

fer by fast electrons is insignificant.

III. DISCUSSION OF THE EXPERIMENTAL RESULTS

The essence of the energy transfer by fast electrons

without pre-plasma can be explained by means of simple

relations. The crater volume is determined by the fraction

of the laser-produced plasma (plasma plume) energy Epwhich is transmitted to the energy Es of the shock wavepropagating in the solid part of the target and creating the

crater, Vcr¼KprpEL/aeq, where Kp¼Ep/EL is the couplingefficiency, rp¼Es/Ep is the ablation loading efficiency, e isthe specific energy required to vaporize a unit mass of the

material, the parameter a�1 is the fraction of thermalenergy corresponding to shock adiabate of the material,

and q is density of solid material. In the approximation ofthe planar expansion, the energies Ep and Es are deter-mined by the values Pawa and Psws, respectively. Here, thepressure behind the shock wave Ps is close to the ablationpressure Pa (the pressure at the ablation front, i.e., at theborder between the plasma plume and the target), and the

velocities behind the ablation and shock wave fronts are wa

FIG. 7. Electron densities observed for the 1x and 3x radiation without (a)and with (b) the pre-plasma at different RL.



193.205.148.189 On: Wed, 09 Apr 2014 07:10:42

/ (Pa/qa)1/2 and ws / (Ps/q)1/2, respectively (qa is the abla-tion density, i.e., the density at the ablation front). This is in

agreement with the well-known result rp / (qa/q)1/2, seeRef. 16. The ablation density is uniquely expressed through

the areal mass of the evaporated part of the target l,qa/ l/Lp, where Lp¼ (Is3/l)1/2 is the size of the plasmaplume, I¼KpIL, and s is the duration of the laser pulse.Therefore, qa / l3/2/I1/2s3/2. The analytical model of axiallysymmetric lateral plasma expansion due to the pulse ablation

of the plane target by laser beam with a given radius RLwas developed in Ref. 11 on the basis of self-similar solu-

tion,17 describing the isothermal expansion of a portion of

the material with a given mass. The model consists in a

combination of self-similar solutions for the plane and

spherical geometries in one solution, being a function of the

parameter Lp/RL. The expressions obtained for qa and rp are

qa ¼3

ð2pÞ1=2l3=2

I1=2s3=2W�2; (1)

rp ¼21=2l3=4

2pð Þ3=4 cþ 1ð Þ1=2I1=4s3=4q1=2W1=2

2W� 1ð Þ3=2; (2)

where the factor of the lateral expansion is defined by

W ¼ 1þ 23=4

3

LpRL

!

and c is the adiabatic index.The areal evaporated mass l is determined either by the

depth of the matter heating by thermal conductivity wave

during the period of laser pulse action

lc � j1=3I2=3s2=3C�7=6V (3)

or by the range of fast electrons10,11

le ¼E20 � A � mp4pe4 � Z � K ; E0ðkeVÞ � 8 ILðpwÞk

2l

� �2=3: (4)

Here, j(erg� cm�1� s�1� keV�7/2)¼ 8.4� 1019/(Zþ 3.3) is electron conductivity coefficient (at the value ofCoulomb logarithm K¼ 5, which was chosen as a scale forthe ranges of plasma temperature and density of 0.5–2 keV

and 0.05–0.2 g/cm3, respectively), CV¼ZkB/Amp(c � 1) isthe specific heat, Z and A are the charge and atomic numberof the plasma ions, kB is the Boltzmann constant, mp is theproton mass, E0 is the energy of fast electrons, e is the elec-tron charge, and kl is the laser radiation wave length meas-ured in lm.

In the case of two-layer target, the scenario of crater cre-

ation in Cu-bulk is the following. Ablation pressure of the

laser-produced plastic plasma excites and supports the shock

wave in a solid plastic, during the period of the laser pulse.

This shock wave propagates first in the relatively thin plastic

layer and then in the Cu-bulk, where it produces a crater

with the depth of several hundred lm (significantly largerthan plastic layer thickness) during the period of several tens

of ns (significantly longer than laser pulse). The main energy

loses of the shock wave in thin plastic layer occur due to its

reflection from the boundary between the plastic layer and

the Cu-bulk. Under the approximation of strong wave, and

assuming the density of plastic q1 significantly smaller thanthe density of copper, q2 the ratio of the energy of shockpassing into Cu-bulk related to the energy of incident wave

is well known to be rs� [(c1þ 1)/(c2þ 1)1/2(q1/q2)]1/2,where c1 and c2 are the adiabatic exponents of plastic andcopper, respectively. So, the expression for the crater volume

created in the Cu-bulk of the two-layer target is

Vcr ¼KprprsEL

aeq2; (5)

where rs� 1/3 under the neglecting its the weak dependenceon c1 and c2, and ae� 2.3� 103 J/g for copper.

Consequently, the dependence of the crater volume on

the beam radius is determined by the ablation loading effi-

ciency for the laser-produced plasma of the plastic ablator.

Under conditions of the considered experiments, the expres-

sions (3) and (4) show the following relationship between the

ablated masses of plastic lc and le. In the 3x case, lc> le forall radii; in the 1x case, lc>le for the radii RL¼ 160 and120 lm but lc< le for the radii RL¼ 80 and 40 lm. Theenergy transfer by fast electrons provides a stronger growth of

the areal mass, and consequently a stronger growth of the

ablation density of the flat layer with the increasing laser in-

tensity than the thermal conductivity wave: qa(e) / I3/2, whileqa(c) / I1/2. At a given laser intensity, this means that qa(e) /R�3L , while qa(c) / R�1L . At the beam radii of RL¼ 160 and120 lm, for both harmonics, the ablation process is inducedby the thermal conductivity wave and the lateral expansion

effect is negligible, i.e., LP

/ r(c) / R�1/2L . Consequently, for large beam radii at bothharmonics, the crater volume depends weakly on the radius.

For small beam radii of RL¼ 80 and 40 lm, the lateral expan-sion effect is strong, Lp�RL, and therefore, q/qa / (RL/L)2/ RL2 l/I . From this, we get q(e)/qa(e) / RL4/3 and q(c)/qa(c)/ RL8/3. As a result, for the 3x case when the ablation is dueto the thermal conductivity wave we have Vcr / r(c) / RL5/6 .For the 1x when the ablation is provided by the fast electronheating we have Vcr / r(e) / RL�5/6.

Let us evaluate, as an example, the crater characteristics

in the case of large radii. The estimations based on (2) and

(3) give for ablation loading efficiency the values of

rs� 2.4� 10�2 and rs� 6.5� 10�2, respectively, for 1xand 3x cases. Substitution of these values as well as the val-ues of rs¼ 0.33, ae¼ 2.3� 103 J/g, q¼ 8.9 g/cm3, EL¼ 200 Jinto expression (5) gives the best agreement with experimen-

tal results at coupling efficiencies of Kp¼ 0.2 and Kp¼ 0.6in the 1x and 3x cases, which correspond to calculatedvalues of the crater volume Vcr� 2� 10�5 cm�3 andVcr� 1.5� 10�4 cm�3, respectively. Under laser interactionwith low-Z material, such as plastic, the energy losses due to

thermal radiation are small and coupling efficiency is close

to the absorption coefficient. The absorption coefficient of

0.2 in the 1x case and 0.6 in the 3x case is in a good agree-ment with numerical simulations of PALS laser interaction

with low-Z material using the 2D ATLANT-HE code.9,10

For example, the volume Vcr� 1.5� 10�4 cm�3 ofsemi-spherical crater corresponds to the crater radius equal,

approximately, to 300 lm. In this case, the evaluation of theaverage shock wave velocity for 3x irradiation gives thevalue of 3� 105 cm/s and the period of the crater creation isabout 100 ns. Note that the crater scaling relations in the 3xcase give results close to observed ones in the all radii range.

The crater scaling relations in the 1x case give results closeto observed ones at the large radii of 120 and 160 lm only.For small radii, the scaling relations give values of the crater

volume smaller than observed ones. The reason for this con-

sists in the fact that a fraction of fast electrons of high energy

tail produces the crater by the direct heating of solid that is

more efficient than the crater production by the shock wave.

Thus, in the 1x case, the effect of the energy transfer inthe dense plasma by fast electrons whose energy increases

with the decreasing radius of the laser beam (with the

increasing intensity) is stronger than the effect of the lateral

expansion, whereby the efficiency of the energy transmission

to the shock wave and also the crater volume increase with

the decreasing beam radius. In the 3x case, the effect of thethermal conductivity heating is weaker than the effect of the

lateral expansion. Consequently, the efficiency of the energy

transmission to the shock wave and the crater volume

decrease with the decreasing radius of the beam.

The dependencies of the density gradient on the beam

radius at different harmonics are also determined by the

peculiarities of the ablation density formation at different

mechanisms of the energy transfer by fast electrons and the

thermal conductivity wave. Bearing in mind that after the

laser pulse termination the plasma plume expands adiabati-

cally, the density gradient at the time of the interferometry

measurement ti can be defined as

G � dndz� C Zq

Amp

qLi

LpLi; (6)

where the density q is given by (1), Li� tiLp/s is the size ofthe plasma plume in the time ti, Zq is the charge of the ionstaking into account the quenching effect, and C is a constant.At large radii, G / Kp1/3RL�2/3 for both harmonics. At smallradii, G / R2L for thermal conductivity (the 3x case) and G/ Kp�1RL�2 for the energy transfer by fast electrons (1xcase). Theoretical dependencies of the density gradient cal-

culated by the formulas (6), (3), and (4) for time ti¼ 3.45 ns(ti¼ 14s) are compared to the experimental values in thecase without pre-plasma in Fig. 9.

The value Zq/A¼ 0.4 was used on the basis of the quench-ing effect estimates for time ti¼ 3.45 ns, the values of the cou-pling efficiency Kp were 0.2 and 0.6 for 1x and 3x cases, i.e.,the same as in above mentioned estimates. The best fit to the

experimental data is provided by the constant C¼ 1.8.A few comments should be added concerning the rele-

vance of our results for the shock ignition approach to ICF.

Of course, the described experiments are quite far from

«realistic» ICF conditions. Nevertheless, we notice that simula-

tions18 predict that for Laser Megajoule conditions, the distance

between the ablation front (indentified by the foot of the

electron temperature profile), the critical density nc occurs

around L1� 150lm and the distance between nc and nc/4 isL2� 350lm. These values are indeed comparable to resultsshown at the bottom of Fig. 8(b), although the pre-plasma is

much colder due to the limited energy available for its creation.

Again, in connection to the relevance of these results to

ICF, we should comment on impact of parametric instabil-

ities in our experimental conditions. Indeed, such instabil-

ities (stimulated Raman scattering (SRS), stimulated

Brillouin scattering (SBS), and TPD) can, in principle, relate

to a considerable amount of energy and therefore strongly

affect the laser-target coupling conditions. In principle, the

laser pulse duration of 300 ps used in PALS experiments is

compatible with the well-developed parametric instabilities.

Therefore, in order to provide a realistic estimate of the laser

energy deposited on the target, accurate backscattering meas-

urements are highly desired (see Refs. 19 and 20). On the

other hand, we have measured SRS and TPD spectra as well

as the total energy reflected within the lens cone due to SBS

and SRS. In the same time, the energy reflected outside the

lens cone has been estimated by using a few mini-

calorimeters placed inside the interaction chamber. The

overall conclusion of the measurement is that in our

FIG. 9. Comparison of the maximum electron density gradients for both har-

monics in the case without pre-plasma: (a) experimental and (b) theoretical data.



193.205.148.189 On: Wed, 09 Apr 2014 07:10:42

experimental situation, the fraction of incident laser light

reflected due to parametric instabilities is always less than

10%.

Therefore, we can conclude that in our case the laser

coupling efficiency on target is not deteriorated due to para-

metric instabilities.

IV. CONCLUSIONS

Two-beam experiments directed to imitate the spike-

laser interaction with the pre-produced plasma have shown

the significantly decreasing efficiency of the 1x radiationenergy transmission to the solid part of the target in compari-

son with the case without pre-plasma. Experiments with the

single 1x beam demonstrated an enhanced efficiency ofenergy transfer to shock wave associated with the fast elec-

trons energy transfer to the dense plasma region.

The presence of the pre-plasma creates poor conditions

for resonant absorption and, therefore, for the laser energy

conversion to fast electrons. This results in suppression of

the fast electrons contribution to the ablation process. The

significantly smaller effectiveness of the energy transfer to

dense plasma regions under the pre-plasma created by the

high-intensity 1x pulse is clearly seen from both the cratersvolumes and the density gradient data. On the other hand,

two-beam experiments have not provided data indicating

alteration of the fast electrons generation due to resonant

absorption by any other mechanism connected with paramet-

ric plasma instabilities in the pre-plasma.

ACKNOWLEDGMENTS

This work was supported in part by the Access to

Research Infrastructure activity in the 7th Framework

Program of the EU Contract No. 284464, Laserlab Europe

III, by the Czech Republic’s Ministry of Education, Youth

and Sports under PALS RI project (No. LM2010014), by

National Centre for Science (NCN), Poland under Grant No.

2012/04/M/ST2/00452 and within the HiPER project under

Grant Agreement No. 211737. The participation of S. Yu.

Gus’kov and N. N. Demchenko in this work was supported

by RFBR projects Nos. 14-02-00010 and 13-02-00295. The

participation of O. Renner was supported by the Czech

Science Foundation project No. CZ.1.07/2.3.00/20.0279 and

by the ELI Project No. CZ.1.05/1.1.00/02.0061.

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IRIDE: Interdisciplinary research infrastructure based on dual electronlinacs and lasers

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