Abstract

Simulations based on the coupled relativistic equations of motion show that protons stemming from laser-plasma processes can be efficiently post-accelerated employing single and crossed pulsed laser beams focused to spot radii on the order of the laser wavelength. We demonstrate that the crossed beams produce quasi-monoenergetic accelerated protons with kinetic energies exceeding 200 MeV, small energy spreads of about 1% and high densities as required for hadron cancer therapy. To our knowledge, this is the first scheme allowing for this important application based on an all-optical set-up.

© 2010 Optical Society of America

1. Introduction

Accelerating charged particles is of paramount importance in a wide variety of fields, ranging from medicine [1, 2] and material science [3] to being used to resolve the smallest structures of our universe [4, 5]. During the past decade, the rapid development of high-intensity laser systems which are likely to exceed in the near future 1025 W cm−2 [6, 7] rendered particle beam creation by laser-matter interaction feasible. Laser-driven accelerators offer the unique feature of ultra-high electric field gradients of several TV m−1, outperforming those in conventional accelerators by more than six orders of magnitude and thus offering the possibility of compact and low-cost devices [8]. Electron [9] and proton beams have recently been generated by focusing high-intensity laser light onto solid targets. This mechanism of target normal sheath acceleration (TNSA) [10, 11, 12, 13] is realized by the strong quasi-static electric field induced by the ionization and acceleration of electrons by the intense laser field. A further laser-plasma-interaction process, the skin-layer ponderomotive acceleration (S-LPA) resulting from the huge electric potential gradient of the plasma leads to ion beams of high density [14]. To date the energy resolution and the ion kinetic energy of the generated beams merely reach the parameters required for skin-deep neoplasm, i.e. excluding deep-seated tumors. Furthermore there is still controversy [12] which beam quality, i.e. which total particle number, kinetic energy and energy spread will be accessible with the forthcoming multi petawatt-class laser systems [6, 7].

In this paper we investigate the vacuum post-acceleration of plasma-generated ions by means of lasers in a single or crossed-beams configuration. The latter scheme of laser particle acceleration was first proposed for electrons [15, 16, 17]. The basic idea is to send the protons originating from a TNSA or S-LPA experiment through the crossing point of two laser beams at a half-angle θ with respect to the z-axis (see Fig. 1b for a scheme and a coordinate system). Within the simple plane-wave picture and assuming the laser fields to have the same amplitude, frequency, phase and to be polarized as shown in Fig. 1b, the resultant electric field component along the symmetry axis of the set-up vanishes for all points on the x-axis. At the same time, the x-component, increased by constructive interference, violently accelerates the particles. Such a coherent combination of intense beams is experimentally feasible (see p. 52 of Ref. [6]). Subsequent motion of the charged particle ejected from the focal region and no longer interacting with the laser pulse may be taken as linear. The results of our theoretical simulations demonstrate that the protons gain kinetic energies larger than 200 MeV (employing two crossed beams each with a peak intensity of 1.9×1024 W cm−2) with an energy spread of roughly 1%. Furthermore (assuming a realistic repetition rate of 10 Hz [18, 6]), the total number of generated protons reaches 1010 min−1. For the first time, all requirements are fulfilled for broader radio-oncological use [1, 2] based on an optical accelerator.

 

Fig. 1 (Color online) (a) The protons, produced by laser-plasma interaction, are injected with the angle θi with respect to the propagation direction of a pulsed beam through its focus. The laser field polarization is denoted by E and the propagation direction is given by k. The protons are ejected out of the focus in the polarization direction E. (b) Here, the protons are injected through the intersection point of two pulsed beams with crossing half-angle θ. The laser field polarizations are denoted by E1 and E2 and their propagation directions are given by k1 and k2. The protons are ejected in direction of the resulting electric field E.

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2. Model and results

In order to generate ultra-strong accelerating fields [6, 7] of 1024 W cm−2, one needs to focus the laser field to a beam waist radius on the order of the laser wavelength [19], which also ensures that the Lawson-Woodward theorem is violated. This necessitates an accurate description of the fields beyond the widely-used paraxial approximation. The parameters of a linearly polarized Gaussian beam which propagates in the z-direction and is polarized in the x-direction and subsequently rotated by θi (Fig. 1a) will be used to model the fields, i.e. the beam waist radius w0, the Rayleigh length zr=πw02/λ, where λ is the laser’s wavelength, and the diffraction angle is ε = w0/zr = λ/(πw0). The expressions giving the Cartesian field components Ex, Ey, Ez, Bx, By, Bz, as well as the expression for the power of the fields to order ε11 in the diffraction angle can be found in Ref. [20, 21]. For the intensity profile of the employed Gaussian beams see Fig. 2.

 

Fig. 2 (Color online) (a) Intensity profile of two crossed Gaussian beams in the propagation plane y = 0. For visualization purpose the crossing-half angle is set to θ = 35°. The brighter areas correspond to higher field intensities. (b) Vector field plot of the polarization plane z = 0 at t = 0.2 fs for the single beam scheme (blue arrows) with power P and crossed beam scheme (red arrows) with power P/2 for each beam. The constructive interference of the crossed beams results in a higher electric field strength in the intersection volume. The background of the graph shows a density map of the electric field strength |E| of two crossed beams in the polarization plane.

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High-intensity laser systems provide their energy in short pulses which are already sufficient to accelerate particles to high velocities [22]. Employing pulsed fields also ensures that the particles injected into the focus get captured rather than reflected. To the lowest order in time, this can be described by multiplying the electromagnetic field components with a Gaussian temporal envelope factor,

Eexp((tz/c)22Δt2)E,Bexp((tz/c)22Δt2)B,
with Δt defined via the Full Width at Half Maximum (FWHM) pulse duration ΔtFWHM=22log2Δt. This approximation is valid for TtFWHM ≪ 1, with T being the laser period. For the titanium-sapphire laser with wavelength λ = 0.8 μm (T = 2.65 fs) and pulse durations of Δt ≳ 10 fs used in our simulations, this turns out to be an adequate description. Hence, further temporal corrections [23] which describe the field solutions as a dual power series in the diffraction angle ε and in the small ratio T/2πΔt can be neglected.

The motion of an ensemble of N identical particles of mass m and charge q in the electric and magnetic fields E and B, respectively, of a laser beam is considered classically, with randomized initial distributions. The use of laser systems of high intensity (exceeding 1024 W cm−2 for protons) requires a relativistic treatment of particle motion. Thus, the dynamics is governed by the coupled Newton-Lorentz (or energy-momentum transfer) equations (given in SI units):

dpjdt=q(E(rj)+Ejint.+cβj×(B(rj)+Bjint.)),djdt=qcβj(E(rj)+Ejint.).
The relativistic energy and momentum of a given particle labeled with j are denoted here by ℰj = γjmc2 and pj = γjmcβj, respectively, with βj = vj/c its velocity scaled by c, and γj=(1βj2)1/2 its Lorentz factor. The fields mediating inter-ionic interaction are modeled by Ejint.=Σkj(ϕjktAjk) and Bjint.=Σkj(×Ajk) with j,k ∈ {1, 2,...,N}. The interaction potentials read
ϕjk=q4πɛ01|rjrk|,
Ajk=q8πɛ0c2|rjk|(vk+rjk(vkrjk)|rjk|),
with the relative displacement rjk = rjrk and ε0 being the vacuum permittivity. Eq. (3) is the scalar part of the interaction given by the Coulomb potential, whereas relativistic effects such as retardation and current-current interaction are included in the Darwin vector potential up to 𝒪(β2) [24] in Eq. (4). Typical kinetic energies of the accelerated protons are of about 200 MeV (cf. Tab. 1), which corresponds to β2 ≈ 0.3. Consequently, the truncation of the interaction up to 𝒪 (β2) is justified and higher-order contributions such as those up to 𝒪 (β4) in Ref. [25] will not be taken into account.

To obtain the kinetic energy gained by interaction with a laser beam, numerical solutions of the equations of motions will be sought. A numerical integration of Eq. (2) yields βjfin and, hence, γjfin at a later final time tfin taken equal to many laser field cycles. Finally, one calculates the final kinetic energy of the particle from Kjfin=γjfinmc2.

Foremost, we carry out simulations based on the coupled equations of motion Eq. (2) for an ensemble of 50 particles at later used particle densities in order to determine the dominant nearest-neighbor contribution of proton-proton interaction effects on the resulting particle beam. Due to the dominating ponderomotive laser forces which lead to a fast drifting apart of the ensemble’s particles, for relativistic laser intensities, i.e. > 1024 W/cm2 for protons, it turns out that the dynamics, the energy gain and its spread are influenced negligibly by inter-ionic interaction. This is in contrast to the non-relativistic laser regime. Here, the repulsive Coulomb interaction is the prevailing part of the interaction, which is in this case non-negligible compared to the electromagnetic fields of the laser. As a consequence, the accelerated ions occupy a larger phase space volume. From radio-oncological point of view we are interested in proton beams of fully relativistic energies, hence for further calculations at these energies and densities it is sufficient to study the uncoupled equations of motions only, i.e. we set Bjint.=Ejinc.=0 in Eq. (2).

The definition of a coordinate system for the crossed beams set-up depicted in Fig. 1b is given by the coordinate transformations x1 = x cosθzsinθ, y1 = y, z1 = x sinθ + zcosθ for the first beam and x2 = x cosθ + zsinθ, y2 = y, z2 = −xsinθ + zcosθ for the second beam, respectively. The resultant field components which enter Eq. (2) are

Ex=(E1x+E2x)cosθ+(E1zE2z)sinθ,Ey=E1y+E2y,Ez=(E1x+E2x)sinθ+(E1z+E2z)cosθ,Bx=(B1zB2z)sinθ,By=B1y+B2y,Bz=(B1z+B2z)cosθ.
Choosing a small crossing half-angle θ leads to constructive addition of the dominating x-components of the electric fields in Eq. (5). The adding of further laser beams would further increase the laser intensity and hence the exit kinetic energy of the accelerated particles. With respect to the energy spread we could not achieve substantial improvement, however. For all subsequent simulations we restrict our analysis to the case of two crossed beams and set θ = 3°.

In order to simulate a realistic particle injection into the focal point of the laser pulses, we consider an ensemble of 5000 particles initially randomly distributed in a micron-scale volume Vfocus oriented along the z-axis. Initially, we have to ensure that the ensemble is not already exposed to the laser fields. This is realized by starting the simulations at time t ≤ −5Δt. At these initial times the electromagnetic fields are damped by the Gaussian envelope factor such that the particles’ motion is only negligibly influenced by the external laser fields. Fig. 3 compares the motion of a proton starting at the same spatial point but having two different initial times. One can see that in the unphysical case when the particle is directly exposed to the laser fields (dashed line), it immediately gains energy and is ejected out of the focus after one laser cycle. In a realistic setting, we choose an initial time of t = −10Δt: the particle slowly starts to oscillate and then gets captured and accelerated by the approaching pulse (full line). Choosing the initial time to be t = 0 would lead to an energy gain overestimation by a factor of approximately three.

 

Fig. 3 Kinetic energy K for one proton initially being at rest and located at x = z = λ/30 and y = 0. The initial time is ti = 0 (dashed line) and ti = −10Δt (full line). The single proton dynamics is compared at same laser parameters.

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The volume initially containing the particle ensemble has a length of the order of the laser wavelength and a radius of tens of nanometers, dependent on the focus diameter of the applied laser system, ensuring that all protons are exposed to a homogeneous field. The particles will be assumed to possess initial kinetic energies distributed normally around a mean value and having a spread ΔK. As a source we take protons originating from laser-plasma interactions, such as the S-LPA mechanism, with = 17 keV [14] and assuming a large energy spread of ΔK = 100% or from the TNSA mechanism, with = 1.2 MeV and ΔK = 25% [10]. The total power and the beam pulse duration are varied. Note that the peak intensity of one linearly polarized 10 PW laser beam focused to w0 = 1λ is already I ∼ 9.6 × 1023 W cm−2 [6, 7].

Our main interest is in the energy gain, or exit kinetic energy, of the nuclei, their trajectories and, hence, the aspects that determine the quality of an accelerated beam of such nuclei. In Tab. 1, simulation results for the laser acceleration are summarized. The single and crossed beams scheme are compared at same total laser power P, pulse duration Δt and focus radius w0. The energy gain is ranging from 59 MeV to 233 MeV in case of the crossed beams setup and from 28 MeV to 113 MeV for injection of the ensemble into the focus of only one beam, respectively. Looking at the energy spread one can see that for the crossed beams it is always 1% and a little higher for the case of the single beam setup.

Moreover, one can see from Tab. 1 that for the systems studied here, the average exit kinetic energy obeys for constant injection angle and injection energy the rough scaling behavior K¯IP/w02. This behavior results from the fact that the optimal acceleration regime depends strongly on the pulse duration Δt rather than only on the electric field strength. It is achieved for the laser-particle interaction length being of the same order of the Rayleigh length. In Ref. [26] the same scaling behavior was derived for electrons. Therefore, in order to maximize the energy gain of the accelerated protons we first choose the laser power P and the focus radius w0, and then adjust to the optimal pulse duration Δt.

Tables Icon

Table 1. Average particle kinetic energy and its percentual spread for different laser system parameters. Ni = ni · Vfocus is the number of ions one can accelerate as one bunch with niSLPA1021cm3 and niTNSA1019cm3 is the ion density of the source used and Vfocus denotes the volume initially containing all ions. The crossing half-angle is θ = 3°. The optimal particle injection angle for the single beam set-up is θi = 3° for the S-LPA source and θi = 50° in case of the TNSA source, respectively. For two crossed beams the particles are injected with an angle θc with respect to the symmetry axis (z-axis) of the laser beam configuration. In case of the S-LPA source we have θc = 0° and for the TNSA source θc = 50°.

A further scaling law can be derived for the particle number, which is proportional to the focus volume. Using Vfocusw02zr and the definition of the Rayleigh length zrw02, one obtains Vfocusw04. The typical particle number needed for ion cancer treatment is 106–1010 per shot with a repetition rate on the order of 5 Hz [27], depending on the ionic species. For a typical interaction volume, one needs at least an ion density of ni = 106/Vfocus ≈ 1020 cm−3 in order to render our scheme feasible for medical applications. Using plasma-generated protons one obtains with the TNSA mechanism an ion density up to the order of niTNSA1019cm3 and the S-LPA ion source generates a density of niSLPA1021cm3 [14]. The latter yields particle numbers of 107 per laser shot. Combined with lasers operated at 10 Hz repetition rate [6, 18] this ion number is sufficient for cancer therapy while for the TNSA mechanism a modest improvement would still be necessary.

Such improvement can e.g. be achieved by substituting the assumed titanium-sapphire laser by a super-intense CO2 laser with a typical wave length of λ = 10.6 μm. Using the fact that the waist radius w0λ and hence the Rayleigh length zrλ, the focal region thus increases by three orders of magnitude. Consequently, the ion density needed decreases by three orders of magnitude and the needed laser intensity by two orders of magnitude. In the near future the use of high repetition rate laser systems [6, 18] will further decrease the number of protons needed per shot.

The combination of the laser-plasma mechanism as a proton source and the post acceleration process by means of single or crossed laser beams places laser acceleration of particles on the cusp of medical feasibility utilizing present-day or near-future laser technology [6, 7, 18], while anticipated costs are presently on the scale of those for current synchrotron facilities (cf. [28] and [29]). The rapid advancement of laser technology renders significant reduction likely for the near future. All requirements needed may be achieved: sufficient proton density and exit kinetic energies, and sharp energy spread of approximately 1 %. The scheme that we introduce in the present work calls for tight focusing mechanisms [19] and relies on results of laser-plasma-interaction research [10, 11, 12, 13].

Acknowledgments

ZH acknowledges insightful conversations with Y. I. Salamin. Supported by Helmholtz Alliance HA216/EMMI.

References and links

1. S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009). [CrossRef]   [PubMed]  

2. O. Jäkel, M. Krämer, C. P. Karger, and J. Debus, “Treatment planning for heavy ion radiotherapy: clinical implementation and application,” Phys. Med. Biol. 46, 1101–1116 (2001). [CrossRef]   [PubMed]  

3. J. A. van Kan, A. A. Bettiol, and F. Watt, “Three-dimensional nanolithography using proton beam writing,” Appl. Phys. Lett. 83, 1629–1631 (2003). [CrossRef]  

4. K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for nuclear phenomena generated by ultra-intense lasers,” Science 300, 1107–1111 (2003). [CrossRef]   [PubMed]  

5. “LHC–The Large Hadron Collider,” http://lhc.web.cern.ch/lhc/.

6. “The Extreme Light Infrastructure European Project (ELI). Scientific Case,” (2007). http://www.extreme-light-infrastructure.eu/pictures/ELI-scientific-case-id17.pdf.

7. “High Power Laser Energy Research (HiPER). HiPER technical background and conceptual design report,” http://www.hiperlaser.org/docs/tdr/HiPERTDR2.pdf.

8. M. Dunne, “Laser-driven particle accelerators,” Science 312, 374–376 (2006). [CrossRef]   [PubMed]  

9. V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002). [CrossRef]   [PubMed]  

10. H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006). [CrossRef]   [PubMed]  

11. J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006). [CrossRef]  

12. L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007). [CrossRef]  

13. A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001). [CrossRef]   [PubMed]  

14. J. Badziak, “Laser-driven generation of fast particles,” Opto-Electr. Review 15, 1–12 (2007). [CrossRef]  

15. C. M. Haaland, “Laser electron acceleration in vacuum,” Opt. Commun. 114, 280–284 (1995). [CrossRef]  

16. E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E 52, 5443–5453 (1995). [CrossRef]  

17. Y. I. Salamin and C. H. Keitel, “Subcycle high electron acceleration by crossed laser beams,” Appl. Phys. Lett. 77, 1082–1084 (2000). [CrossRef]  

18. Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

19. S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (1022 W/cm2),” Opt. Lett. 29, 2837–2839 (2004). [CrossRef]  

20. Y. I. Salamin, “Fields of a Gaussian beam beyond paraxial approximation,” Appl. Phys. B 86, 319–326 (2007). [CrossRef]  

21. Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008). [CrossRef]   [PubMed]  

22. J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473–7478 (1999). [CrossRef]  

23. Z. Yan, Y. K. Ho, P. X. Wang, J. F. Hua, Z. Chen, and L. Wu, “Accurate description of ultra-short tightly focused Gaussian laser pulses and vacuum laser acceleration,” Appl. Phys. B 81, 813–819 (2005). [CrossRef]  

24. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), 3rd ed.

25. X. Jaén, J. Llosa, and A. Molina, “A reduction of order two for infinite-order Lagrangians,” Phys. Rev. D 34, 2302–2311 (1986). [CrossRef]  

26. G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86, 5274–5277 (2001). [CrossRef]   [PubMed]  

27. H. Eickhoff, D. Böhne, T. Haberer, B. Schlitt, P. Spiller, J. Debus, and A. Dolinskii, “HIT–Heidelberg Ion beam Therapy. Scientific Case.” http://www-aix.gsi.de/~spiller/facilit_ep00.ps.

28. “HIT–Heidelberg Ion beam Therapy. Facts in short.” http://www.klinikum.uni-heidelberg.de/HIT-Facts-in-short.117995.0.html?&L=en.

29. E. Gerstner, “Laser physics: extreme light,” Nature 446, 16–18 (2007). [CrossRef]   [PubMed]  

References

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  1. S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
    [Crossref] [PubMed]
  2. O. Jäkel, M. Krämer, C. P. Karger, and J. Debus, “Treatment planning for heavy ion radiotherapy: clinical implementation and application,” Phys. Med. Biol. 46, 1101–1116 (2001).
    [Crossref] [PubMed]
  3. J. A. van Kan, A. A. Bettiol, and F. Watt, “Three-dimensional nanolithography using proton beam writing,” Appl. Phys. Lett. 83, 1629–1631 (2003).
    [Crossref]
  4. K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for nuclear phenomena generated by ultra-intense lasers,” Science 300, 1107–1111 (2003).
    [Crossref] [PubMed]
  5. “LHC–The Large Hadron Collider,” http://lhc.web.cern.ch/lhc/ .
  6. “The Extreme Light Infrastructure European Project (ELI). Scientific Case,” (2007). http://www.extreme-light-infrastructure.eu/pictures/ELI-scientific-case-id17.pdf .
  7. “High Power Laser Energy Research (HiPER). HiPER technical background and conceptual design report,” http://www.hiperlaser.org/docs/tdr/HiPERTDR2.pdf .
  8. M. Dunne, “Laser-driven particle accelerators,” Science 312, 374–376 (2006).
    [Crossref] [PubMed]
  9. V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
    [Crossref] [PubMed]
  10. H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
    [Crossref] [PubMed]
  11. J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
    [Crossref]
  12. L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
    [Crossref]
  13. A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
    [Crossref] [PubMed]
  14. J. Badziak, “Laser-driven generation of fast particles,” Opto-Electr. Review 15, 1–12 (2007).
    [Crossref]
  15. C. M. Haaland, “Laser electron acceleration in vacuum,” Opt. Commun. 114, 280–284 (1995).
    [Crossref]
  16. E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E 52, 5443–5453 (1995).
    [Crossref]
  17. Y. I. Salamin and C. H. Keitel, “Subcycle high electron acceleration by crossed laser beams,” Appl. Phys. Lett. 77, 1082–1084 (2000).
    [Crossref]
  18. Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).
  19. S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (1022 W/cm2),” Opt. Lett. 29, 2837–2839 (2004).
    [Crossref]
  20. Y. I. Salamin, “Fields of a Gaussian beam beyond paraxial approximation,” Appl. Phys. B 86, 319–326 (2007).
    [Crossref]
  21. Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
    [Crossref] [PubMed]
  22. J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473–7478 (1999).
    [Crossref]
  23. Z. Yan, Y. K. Ho, P. X. Wang, J. F. Hua, Z. Chen, and L. Wu, “Accurate description of ultra-short tightly focused Gaussian laser pulses and vacuum laser acceleration,” Appl. Phys. B 81, 813–819 (2005).
    [Crossref]
  24. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), 3rd ed.
  25. X. Jaén, J. Llosa, and A. Molina, “A reduction of order two for infinite-order Lagrangians,” Phys. Rev. D 34, 2302–2311 (1986).
    [Crossref]
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2009 (2)

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

2008 (1)

Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
[Crossref] [PubMed]

2007 (4)

E. Gerstner, “Laser physics: extreme light,” Nature 446, 16–18 (2007).
[Crossref] [PubMed]

Y. I. Salamin, “Fields of a Gaussian beam beyond paraxial approximation,” Appl. Phys. B 86, 319–326 (2007).
[Crossref]

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

J. Badziak, “Laser-driven generation of fast particles,” Opto-Electr. Review 15, 1–12 (2007).
[Crossref]

2006 (3)

M. Dunne, “Laser-driven particle accelerators,” Science 312, 374–376 (2006).
[Crossref] [PubMed]

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
[Crossref] [PubMed]

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

2005 (1)

Z. Yan, Y. K. Ho, P. X. Wang, J. F. Hua, Z. Chen, and L. Wu, “Accurate description of ultra-short tightly focused Gaussian laser pulses and vacuum laser acceleration,” Appl. Phys. B 81, 813–819 (2005).
[Crossref]

2004 (1)

2003 (2)

J. A. van Kan, A. A. Bettiol, and F. Watt, “Three-dimensional nanolithography using proton beam writing,” Appl. Phys. Lett. 83, 1629–1631 (2003).
[Crossref]

K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for nuclear phenomena generated by ultra-intense lasers,” Science 300, 1107–1111 (2003).
[Crossref] [PubMed]

2002 (1)

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

2001 (3)

O. Jäkel, M. Krämer, C. P. Karger, and J. Debus, “Treatment planning for heavy ion radiotherapy: clinical implementation and application,” Phys. Med. Biol. 46, 1101–1116 (2001).
[Crossref] [PubMed]

A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
[Crossref] [PubMed]

G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86, 5274–5277 (2001).
[Crossref] [PubMed]

2000 (1)

Y. I. Salamin and C. H. Keitel, “Subcycle high electron acceleration by crossed laser beams,” Appl. Phys. Lett. 77, 1082–1084 (2000).
[Crossref]

1999 (1)

J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473–7478 (1999).
[Crossref]

1995 (2)

C. M. Haaland, “Laser electron acceleration in vacuum,” Opt. Commun. 114, 280–284 (1995).
[Crossref]

E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E 52, 5443–5453 (1995).
[Crossref]

1986 (1)

X. Jaén, J. Llosa, and A. Molina, “A reduction of order two for infinite-order Lagrangians,” Phys. Rev. D 34, 2302–2311 (1986).
[Crossref]

Ahmad, I.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Aleonard, M.-M.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Amthor, K.-U.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
[Crossref] [PubMed]

Antici, P.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Apolonski, A.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Audebert, P.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Badziak, J.

J. Badziak, “Laser-driven generation of fast particles,” Opto-Electr. Review 15, 1–12 (2007).
[Crossref]

Bahk, S.-W.

Bettiol, A. A.

J. A. van Kan, A. A. Bettiol, and F. Watt, “Three-dimensional nanolithography using proton beam writing,” Appl. Phys. Lett. 83, 1629–1631 (2003).
[Crossref]

Borghesi, M.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
[Crossref] [PubMed]

Brambrink, E.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Burgy, F.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Campbell, H.

A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
[Crossref] [PubMed]

Cecchetti, C. A.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Chambaret, J.-P.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Chemin, J.-F.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Chen, Z.

Z. Yan, Y. K. Ho, P. X. Wang, J. F. Hua, Z. Chen, and L. Wu, “Accurate description of ultra-short tightly focused Gaussian laser pulses and vacuum laser acceleration,” Appl. Phys. B 81, 813–819 (2005).
[Crossref]

Chvykov, V.

Clarke, R. J.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

Combs, S. E.

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

d’Humières, E.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Dangor, A. E.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Debus, J.

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

O. Jäkel, M. Krämer, C. P. Karger, and J. Debus, “Treatment planning for heavy ion radiotherapy: clinical implementation and application,” Phys. Med. Biol. 46, 1101–1116 (2001).
[Crossref] [PubMed]

Dunne, M.

M. Dunne, “Laser-driven particle accelerators,” Science 312, 374–376 (2006).
[Crossref] [PubMed]

Esarey, E.

E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E 52, 5443–5453 (1995).
[Crossref]

Esirkepov, T.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
[Crossref] [PubMed]

Feng, L.

J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473–7478 (1999).
[Crossref]

Fritzler, S.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Fuchs, J.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Fülöp, J. A.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Gerstner, E.

E. Gerstner, “Laser physics: extreme light,” Nature 446, 16–18 (2007).
[Crossref] [PubMed]

Haaland, C. M.

C. M. Haaland, “Laser electron acceleration in vacuum,” Opt. Commun. 114, 280–284 (1995).
[Crossref]

Haberer, T.

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

Harman, Z.

Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
[Crossref] [PubMed]

Hatchett, S.

A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
[Crossref] [PubMed]

Hein, J.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Henig, A.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Ho, Y. K.

Z. Yan, Y. K. Ho, P. X. Wang, J. F. Hua, Z. Chen, and L. Wu, “Accurate description of ultra-short tightly focused Gaussian laser pulses and vacuum laser acceleration,” Appl. Phys. B 81, 813–819 (2005).
[Crossref]

J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473–7478 (1999).
[Crossref]

Hörlein, R.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Hua, J. F.

Z. Yan, Y. K. Ho, P. X. Wang, J. F. Hua, Z. Chen, and L. Wu, “Accurate description of ultra-short tightly focused Gaussian laser pulses and vacuum laser acceleration,” Appl. Phys. B 81, 813–819 (2005).
[Crossref]

Huber, P. E.

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

Jäckel, O.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
[Crossref] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), 3rd ed.

Jaekel, O.

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

Jaén, X.

X. Jaén, J. Llosa, and A. Molina, “A reduction of order two for infinite-order Lagrangians,” Phys. Rev. D 34, 2302–2311 (1986).
[Crossref]

Jäkel, O.

O. Jäkel, M. Krämer, C. P. Karger, and J. Debus, “Treatment planning for heavy ion radiotherapy: clinical implementation and application,” Phys. Med. Biol. 46, 1101–1116 (2001).
[Crossref] [PubMed]

Kalintchenko, G.

Kaluza, M.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Karger, C. P.

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

O. Jäkel, M. Krämer, C. P. Karger, and J. Debus, “Treatment planning for heavy ion radiotherapy: clinical implementation and application,” Phys. Med. Biol. 46, 1101–1116 (2001).
[Crossref] [PubMed]

Karsch, S.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Keitel, C. H.

Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
[Crossref] [PubMed]

Y. I. Salamin and C. H. Keitel, “Subcycle high electron acceleration by crossed laser beams,” Appl. Phys. Lett. 77, 1082–1084 (2000).
[Crossref]

Key, M. H.

A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
[Crossref] [PubMed]

Klingebiel, S.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Kong, Q.

J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473–7478 (1999).
[Crossref]

Krall, J.

E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E 52, 5443–5453 (1995).
[Crossref]

Krämer, M.

O. Jäkel, M. Krämer, C. P. Karger, and J. Debus, “Treatment planning for heavy ion radiotherapy: clinical implementation and application,” Phys. Med. Biol. 46, 1101–1116 (2001).
[Crossref] [PubMed]

Krausz, F.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Kruber, S.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Krushelnick, K.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Ledingham, K. W. D.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
[Crossref] [PubMed]

K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for nuclear phenomena generated by ultra-intense lasers,” Science 300, 1107–1111 (2003).
[Crossref] [PubMed]

Lefebvre, E.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Liesfeld, B.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
[Crossref] [PubMed]

Lindau, F.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

Llosa, J.

X. Jaén, J. Llosa, and A. Molina, “A reduction of order two for infinite-order Lagrangians,” Phys. Rev. D 34, 2302–2311 (1986).
[Crossref]

Lundh, O.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

Mackinnon, A. J.

A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
[Crossref] [PubMed]

Major, Z.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Maksimchuk, A.

Malka, G.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Malka, V.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Manclossi, M.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Mangles, S. P. D.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

McCanny, T.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

McKenna, P.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for nuclear phenomena generated by ultra-intense lasers,” Science 300, 1107–1111 (2003).
[Crossref] [PubMed]

Meyroneinc, S.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Molina, A.

X. Jaén, J. Llosa, and A. Molina, “A reduction of order two for infinite-order Lagrangians,” Phys. Rev. D 34, 2302–2311 (1986).
[Crossref]

Mora, P.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Mourou, G. A.

Münter, M. W.

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

Najmudin, Z.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Neely, D.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

Nikoghosyan, A.

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

Osterhoff, J.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Patel, P. K.

A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
[Crossref] [PubMed]

Pépin, H.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Pervak, V.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Pfotenhauer, S.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
[Crossref] [PubMed]

Pittman, M.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Planchon, T. A.

Popp, A.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Robson, L.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

Rousseau, J.-P.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Rousseau, P.

Salamin, Y. I.

Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
[Crossref] [PubMed]

Y. I. Salamin, “Fields of a Gaussian beam beyond paraxial approximation,” Appl. Phys. B 86, 319–326 (2007).
[Crossref]

Y. I. Salamin and C. H. Keitel, “Subcycle high electron acceleration by crossed laser beams,” Appl. Phys. Lett. 77, 1082–1084 (2000).
[Crossref]

Sauerbrey, R.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
[Crossref] [PubMed]

Scheid, W.

J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473–7478 (1999).
[Crossref]

Scheurer, J.-N.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Schiavi, A.

A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
[Crossref] [PubMed]

Schreiber, J.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Schulz-Ertner, D.

S. E. Combs, A. Nikoghosyan, O. Jaekel, C. P. Karger, T. Haberer, M. W. Münter, P. E. Huber, J. Debus, and D. Schulz-Ertner, “Carbon ion radiotherapy for pediatric patients and young adults treated for tumors of the skull base,” Cancer 115, 1348–1355 (2009).
[Crossref] [PubMed]

Schwoerer, H.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, “Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets,” Nature 439, 445–448 (2006).
[Crossref] [PubMed]

Siebold, M.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Simpson, P. T.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

Singhal, R. P.

K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for nuclear phenomena generated by ultra-intense lasers,” Science 300, 1107–1111 (2003).
[Crossref] [PubMed]

Snavely, R.

A. J. Mackinnon, M. Borghesi, S. Hatchett, M. H. Key, P. K. Patel, H. Campbell, A. Schiavi, R. Snavely, S. C. Wilks, and O. Willi, “Effect of plasma scale length on multi-MeV proton production by intense laser pulses,” Phys. Rev. Lett. 86, 1769–1772 (2001).
[Crossref] [PubMed]

Sprangle, P.

E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E 52, 5443–5453 (1995).
[Crossref]

Stupakov, G. V.

G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86, 5274–5277 (2001).
[Crossref] [PubMed]

Toncian, T.

J. Fuchs, P. Antici, E. d’Humières, E. Lefebvre, M. Borghesi, E. Brambrink, C. A. Cecchetti, M. Kaluza, V. Malka, M. Manclossi, S. Meyroneinc, P. Mora, J. Schreiber, T. Toncian, H. Pépin, and P. Audebert, “Laser-driven proton scaling laws and new paths towards energy increase,” Nature Physics 2, 48–54 (2006).
[Crossref]

Trushin, S.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

van Kan, J. A.

J. A. van Kan, A. A. Bettiol, and F. Watt, “Three-dimensional nanolithography using proton beam writing,” Appl. Phys. Lett. 83, 1629–1631 (2003).
[Crossref]

Wahlström, C.-G.

L. Robson, P. T. Simpson, R. J. Clarke, K. W. D. Ledingham, F. Lindau, O. Lundh, T. McCanny, P. Mora, D. Neely, C.-G. Wahlström, M. Zepf, and P. McKenna, “Scaling of proton acceleration driven by petawatt-laser-plasma interactions,” Nature Physics 3, 58–62 (2007).
[Crossref]

Walton, B.

V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. P. D. Mangles, Z. Najmudin, M. Pittman, J.-P. Rousseau, J.-N. Scheurer, B. Walton, and A. E. Dangor, “Electron acceleration by a wake field forced by an intense ultrashort laser pulse,” Science 298, 1596–1600 (2002).
[Crossref] [PubMed]

Wandt, C.

Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

Wang, J. X.

J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473–7478 (1999).
[Crossref]

Wang, P. X.

Z. Yan, Y. K. Ho, P. X. Wang, J. F. Hua, Z. Chen, and L. Wu, “Accurate description of ultra-short tightly focused Gaussian laser pulses and vacuum laser acceleration,” Appl. Phys. B 81, 813–819 (2005).
[Crossref]

J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473–7478 (1999).
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Wang, T.-J.

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Z. Major, S. Trushin, I. Ahmad, M. Siebold, C. Wandt, S. Klingebiel, T.-J. Wang, J. A. Fülöp, A. Henig, S. Kruber, R. Weingartner, A. Popp, J. Osterhoff, R. Hörlein, J. Hein, V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, “Basic concepts and current status of the petawatt field synthesizer – a new approach to ultrahigh field generation,” The Review of Laser Engineering 37, 431–436 (2009).

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Figures (3)

Fig. 1
Fig. 1 (Color online) (a) The protons, produced by laser-plasma interaction, are injected with the angle θi with respect to the propagation direction of a pulsed beam through its focus. The laser field polarization is denoted by E and the propagation direction is given by k. The protons are ejected out of the focus in the polarization direction E. (b) Here, the protons are injected through the intersection point of two pulsed beams with crossing half-angle θ. The laser field polarizations are denoted by E1 and E2 and their propagation directions are given by k1 and k2. The protons are ejected in direction of the resulting electric field E.
Fig. 2
Fig. 2 (Color online) (a) Intensity profile of two crossed Gaussian beams in the propagation plane y = 0. For visualization purpose the crossing-half angle is set to θ = 35°. The brighter areas correspond to higher field intensities. (b) Vector field plot of the polarization plane z = 0 at t = 0.2 fs for the single beam scheme (blue arrows) with power P and crossed beam scheme (red arrows) with power P/2 for each beam. The constructive interference of the crossed beams results in a higher electric field strength in the intersection volume. The background of the graph shows a density map of the electric field strength |E| of two crossed beams in the polarization plane.
Fig. 3
Fig. 3 Kinetic energy K for one proton initially being at rest and located at x = z = λ/30 and y = 0. The initial time is ti = 0 (dashed line) and ti = −10Δt (full line). The single proton dynamics is compared at same laser parameters.

Tables (1)

Tables Icon

Table 1 Average particle kinetic energy and its percentual spread for different laser system parameters. Ni = ni · Vfocus is the number of ions one can accelerate as one bunch with n i S LPA 10 21 cm 3 and n i TNSA 10 19 cm 3 is the ion density of the source used and Vfocus denotes the volume initially containing all ions. The crossing half-angle is θ = 3°. The optimal particle injection angle for the single beam set-up is θi = 3° for the S-LPA source and θi = 50° in case of the TNSA source, respectively. For two crossed beams the particles are injected with an angle θc with respect to the symmetry axis (z-axis) of the laser beam configuration. In case of the S-LPA source we have θc = 0° and for the TNSA source θc = 50°.

Equations (5)

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E exp ( ( t z / c ) 2 2 Δ t 2 ) E , B exp ( ( t z / c ) 2 2 Δ t 2 ) B ,
d p j d t = q ( E ( r j ) + E j int . + c β j × ( B ( r j ) + B j int . ) ) , d j d t = qc β j ( E ( r j ) + E j int . ) .
ϕ j k = q 4 π ɛ 0 1 | r j r k | ,
A j k = q 8 π ɛ 0 c 2 | r j k | ( v k + r j k ( v k r j k ) | r j k | ) ,
E x = ( E 1 x + E 2 x ) cos θ + ( E 1 z E 2 z ) sin θ , E y = E 1 y + E 2 y , E z = ( E 1 x + E 2 x ) sin θ + ( E 1 z + E 2 z ) cos θ , B x = ( B 1 z B 2 z ) sin θ , B y = B 1 y + B 2 y , B z = ( B 1 z + B 2 z ) cos θ .

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