Abstract

We study the direct acceleration of a free electron in infinite vacuum along the axis of a pulsed radially-polarized laser beam. We find that net energy transfer from laser pulse to electron is maximized with the tightest focusing. We show that the net energy gain of an electron initially moving at a relativistic velocity may exceed more than half the theoretical limit of energy transfer, which is not possible with an initially stationary electron in the parameter space studied. We determine and analyze the power scaling of maximum energy gain, extending our study to include a relatively unexplored regime of low powers and revealing that substantial acceleration is already possible without the use of petawatt peak-power laser technology.

© 2010 OSA

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  3. A. H. Sari, F. Osman, K. R. Doolan, M. Ghoranneviss, H. Hora, R. Hopfl, G. Benstetter, and M. H. Hantehzadeh, “Application of laser driven fast high density plasma blocks for ion implantation,” Laser Part. Beams 23(04), 467–473 (2005).
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  7. E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81(3), 1229–1285 (2009).
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  13. F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).
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  21. Y. C. Huang, D. Zheng, W. M. Tulloch, and R. L. Byer, “Proposed structure for a crossed-laser beam, GeV per meter gradient, vacuum electron linear accelerator,” Appl. Phys. Lett. 68(6), 753–755 (1996).
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  25. P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially-polarized laser beams: Scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
    [CrossRef]
  26. C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
    [CrossRef] [PubMed]
  27. A. Karmakar and A. Pukhov, “Collimated attosecond GeV electron bunches from ionization of high-Z material by radially polarized ultra-relativistic laser pulses,” Laser Part. Beams 25(03), 371–377 (2007).
    [CrossRef]
  28. E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46(10), 6640–6653 (1992).
    [CrossRef] [PubMed]
  29. W. S. Graves, W. Brown, F. X. Kärtner, and D. E. Moncton, “MIT inverse Compton source concept,” Nucl. Instr. Meth. A 608(1), S103–S105 (2009).
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    [CrossRef]

2010 (1)

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially-polarized laser beams: Scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
[CrossRef]

2009 (3)

R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. 5(1), 64–67 (2009).
[CrossRef]

W. S. Graves, W. Brown, F. X. Kärtner, and D. E. Moncton, “MIT inverse Compton source concept,” Nucl. Instr. Meth. A 608(1), S103–S105 (2009).
[CrossRef]

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81(3), 1229–1285 (2009).
[CrossRef]

2008 (1)

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

2007 (2)

Y. I. Salamin, “Mono-energetic GeV electrons from ionization in a radially polarized laser beam,” Opt. Lett. 32(1), 90–92 (2007).
[CrossRef]

A. Karmakar and A. Pukhov, “Collimated attosecond GeV electron bunches from ionization of high-Z material by radially polarized ultra-relativistic laser pulses,” Laser Part. Beams 25(03), 371–377 (2007).
[CrossRef]

2006 (3)

C. Varin, M. Piché, and M. A. Porras, “Analytical calculation of the strong axial longitudinal electric field resulting from the tight focusing of an ultrafast transverse magnetic pulsed beam in free space,” J. Opt. Soc. Am. A 23, 2027–2038 (2006).
[CrossRef]

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
[CrossRef]

Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73(4), 043402 (2006).
[CrossRef]

2005 (3)

T. Plettner, R. L. Byer, E. Colby, B. Cowan, C. M. S. Sears, J. E. Spencer, and R. H. Siemann, “Visible-laser acceleration of relativistic electrons in a semi-infinite vacuum,” Phys. Rev. Lett. 95(13), 134801 (2005).
[CrossRef] [PubMed]

A. H. Sari, F. Osman, K. R. Doolan, M. Ghoranneviss, H. Hora, R. Hopfl, G. Benstetter, and M. H. Hantehzadeh, “Application of laser driven fast high density plasma blocks for ion implantation,” Laser Part. Beams 23(04), 467–473 (2005).
[CrossRef]

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
[CrossRef] [PubMed]

2003 (1)

S. V. Bulanov, T. Esirkepov, and T. Tajima, “Light intensification towards the Schwinger limit,” Phys. Rev. Lett. 91(8), 085001 (2003).
[CrossRef] [PubMed]

2002 (2)

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

S. X. Hu and A. F. Starace, “GeV electrons from ultraintense laser interaction with highly charged ions,” Phys. Rev. Lett. 88(24), 245003 (2002).
[CrossRef] [PubMed]

2001 (3)

G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86(23), 5274–5277 (2001).
[CrossRef] [PubMed]

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253 (2001).
[CrossRef]

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

1999 (1)

1998 (1)

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 58(3), 3719–3732 (1998).

1997 (1)

G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997).
[CrossRef]

1996 (1)

Y. C. Huang, D. Zheng, W. M. Tulloch, and R. L. Byer, “Proposed structure for a crossed-laser beam, GeV per meter gradient, vacuum electron linear accelerator,” Appl. Phys. Lett. 68(6), 753–755 (1996).
[CrossRef]

1995 (2)

E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 52(5), 5443–5453 (1995).
[CrossRef] [PubMed]

F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

1992 (1)

E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46(10), 6640–6653 (1992).
[CrossRef] [PubMed]

1991 (1)

M. O. Scully and M. S. Zubairy, “Simple laser accelerator: Optics and particle dynamics,” Phys. Rev. A 44(4), 2656–2663 (1991).
[CrossRef] [PubMed]

1988 (1)

H. Hora, “Particle acceleration by superposition of frequency-controlled laser pulses,” Nature 333(6171), 337–338 (1988).
[CrossRef]

1985 (2)

E. D. Courant, C. Pellegrini, and W. Zakowicz, “High-energy inverse free-electron-laser accelerator,” Phys. Rev. A 32(5), 2813–2823 (1985).
[CrossRef] [PubMed]

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[CrossRef]

Babzien, M.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Benstetter, G.

A. H. Sari, F. Osman, K. R. Doolan, M. Ghoranneviss, H. Hora, R. Hopfl, G. Benstetter, and M. H. Hantehzadeh, “Application of laser driven fast high density plasma blocks for ion implantation,” Laser Part. Beams 23(04), 467–473 (2005).
[CrossRef]

Ben-Zvi, I.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Bochove, E. J.

E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46(10), 6640–6653 (1992).
[CrossRef] [PubMed]

Britten, J. A.

Brown, C.

Brown, W.

W. S. Graves, W. Brown, F. X. Kärtner, and D. E. Moncton, “MIT inverse Compton source concept,” Nucl. Instr. Meth. A 608(1), S103–S105 (2009).
[CrossRef]

Bulanov, S. V.

S. V. Bulanov, T. Esirkepov, and T. Tajima, “Light intensification towards the Schwinger limit,” Phys. Rev. Lett. 91(8), 085001 (2003).
[CrossRef] [PubMed]

Byer, R. L.

T. Plettner, R. L. Byer, E. Colby, B. Cowan, C. M. S. Sears, J. E. Spencer, and R. H. Siemann, “Visible-laser acceleration of relativistic electrons in a semi-infinite vacuum,” Phys. Rev. Lett. 95(13), 134801 (2005).
[CrossRef] [PubMed]

Y. C. Huang, D. Zheng, W. M. Tulloch, and R. L. Byer, “Proposed structure for a crossed-laser beam, GeV per meter gradient, vacuum electron linear accelerator,” Appl. Phys. Lett. 68(6), 753–755 (1996).
[CrossRef]

Campbell, L. P.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Cao, N.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253 (2001).
[CrossRef]

Chen, Y. J.

F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

Cline, D. B.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Colby, E.

T. Plettner, R. L. Byer, E. Colby, B. Cowan, C. M. S. Sears, J. E. Spencer, and R. H. Siemann, “Visible-laser acceleration of relativistic electrons in a semi-infinite vacuum,” Phys. Rev. Lett. 95(13), 134801 (2005).
[CrossRef] [PubMed]

Courant, E. D.

E. D. Courant, C. Pellegrini, and W. Zakowicz, “High-energy inverse free-electron-laser accelerator,” Phys. Rev. A 32(5), 2813–2823 (1985).
[CrossRef] [PubMed]

Cowan, B.

T. Plettner, R. L. Byer, E. Colby, B. Cowan, C. M. S. Sears, J. E. Spencer, and R. H. Siemann, “Visible-laser acceleration of relativistic electrons in a semi-infinite vacuum,” Phys. Rev. Lett. 95(13), 134801 (2005).
[CrossRef] [PubMed]

Dilley, C. E.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Doolan, K. R.

A. H. Sari, F. Osman, K. R. Doolan, M. Ghoranneviss, H. Hora, R. Hopfl, G. Benstetter, and M. H. Hantehzadeh, “Application of laser driven fast high density plasma blocks for ion implantation,” Laser Part. Beams 23(04), 467–473 (2005).
[CrossRef]

Esarey, E.

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81(3), 1229–1285 (2009).
[CrossRef]

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253 (2001).
[CrossRef]

E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 52(5), 5443–5453 (1995).
[CrossRef] [PubMed]

Esirkepov, T.

S. V. Bulanov, T. Esirkepov, and T. Tajima, “Light intensification towards the Schwinger limit,” Phys. Rev. Lett. 91(8), 085001 (2003).
[CrossRef] [PubMed]

Faure, J.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

Fochs, S. N.

F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

Fortin, P.-L.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially-polarized laser beams: Scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
[CrossRef]

Fung, R.

R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. 5(1), 64–67 (2009).
[CrossRef]

Gallardo, J. C.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Gauduel, Y. A.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

Ghoranneviss, M.

A. H. Sari, F. Osman, K. R. Doolan, M. Ghoranneviss, H. Hora, R. Hopfl, G. Benstetter, and M. H. Hantehzadeh, “Application of laser driven fast high density plasma blocks for ion implantation,” Laser Part. Beams 23(04), 467–473 (2005).
[CrossRef]

Golick, B.

Gottschalk, S. C.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Graves, W. S.

W. S. Graves, W. Brown, F. X. Kärtner, and D. E. Moncton, “MIT inverse Compton source concept,” Nucl. Instr. Meth. A 608(1), S103–S105 (2009).
[CrossRef]

Hantehzadeh, M. H.

A. H. Sari, F. Osman, K. R. Doolan, M. Ghoranneviss, H. Hora, R. Hopfl, G. Benstetter, and M. H. Hantehzadeh, “Application of laser driven fast high density plasma blocks for ion implantation,” Laser Part. Beams 23(04), 467–473 (2005).
[CrossRef]

Hartemann, F. V.

F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

Hatsagortsyan, K. Z.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
[CrossRef]

He, P.

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Y. C. Huang, D. Zheng, W. M. Tulloch, and R. L. Byer, “Proposed structure for a crossed-laser beam, GeV per meter gradient, vacuum electron linear accelerator,” Appl. Phys. Lett. 68(6), 753–755 (1996).
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Keitel, C. H.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
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F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

Kimura, W. D.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
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P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253 (2001).
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E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 52(5), 5443–5453 (1995).
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Kusche, K. P.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
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F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

Leemans, W. P.

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V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
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W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
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F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

Malka, G.

G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997).
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Malka, V.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
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Miquel, J. L.

G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997).
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Moncton, D. E.

W. S. Graves, W. Brown, F. X. Kärtner, and D. E. Moncton, “MIT inverse Compton source concept,” Nucl. Instr. Meth. A 608(1), S103–S105 (2009).
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Moore, G. T.

E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46(10), 6640–6653 (1992).
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Mora, P.

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 58(3), 3719–3732 (1998).

Mourou, G.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
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P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253 (2001).
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Osman, F.

A. H. Sari, F. Osman, K. R. Doolan, M. Ghoranneviss, H. Hora, R. Hopfl, G. Benstetter, and M. H. Hantehzadeh, “Application of laser driven fast high density plasma blocks for ion implantation,” Laser Part. Beams 23(04), 467–473 (2005).
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R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. 5(1), 64–67 (2009).
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Pantell, R. H.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
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Pellegrini, C.

E. D. Courant, C. Pellegrini, and W. Zakowicz, “High-energy inverse free-electron-laser accelerator,” Phys. Rev. A 32(5), 2813–2823 (1985).
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Pennington, D.

Perry, M. D.

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F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

Phuoc, K. T.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
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Piché, M.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially-polarized laser beams: Scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
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C. Varin, M. Piché, and M. A. Porras, “Analytical calculation of the strong axial longitudinal electric field resulting from the tight focusing of an ultrafast transverse magnetic pulsed beam in free space,” J. Opt. Soc. Am. A 23, 2027–2038 (2006).
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C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
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Plettner, T.

T. Plettner, R. L. Byer, E. Colby, B. Cowan, C. M. S. Sears, J. E. Spencer, and R. H. Siemann, “Visible-laser acceleration of relativistic electrons in a semi-infinite vacuum,” Phys. Rev. Lett. 95(13), 134801 (2005).
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Pogorelsky, I. V.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
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Porras, M. A.

C. Varin, M. Piché, and M. A. Porras, “Analytical calculation of the strong axial longitudinal electric field resulting from the tight focusing of an ultrafast transverse magnetic pulsed beam in free space,” J. Opt. Soc. Am. A 23, 2027–2038 (2006).
[CrossRef]

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
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Powell, H. T.

Pukhov, A.

A. Karmakar and A. Pukhov, “Collimated attosecond GeV electron bunches from ionization of high-Z material by radially polarized ultra-relativistic laser pulses,” Laser Part. Beams 25(03), 371–377 (2007).
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B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 58(3), 3719–3732 (1998).

Quimby, D. C.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
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Rousse, A.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
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Y. I. Salamin, “Mono-energetic GeV electrons from ionization in a radially polarized laser beam,” Opt. Lett. 32(1), 90–92 (2007).
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Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
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Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73(4), 043402 (2006).
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Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

Saldin, D. K.

R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. 5(1), 64–67 (2009).
[CrossRef]

Sari, A. H.

A. H. Sari, F. Osman, K. R. Doolan, M. Ghoranneviss, H. Hora, R. Hopfl, G. Benstetter, and M. H. Hantehzadeh, “Application of laser driven fast high density plasma blocks for ion implantation,” Laser Part. Beams 23(04), 467–473 (2005).
[CrossRef]

Schroeder, C. B.

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81(3), 1229–1285 (2009).
[CrossRef]

Scully, M. O.

E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46(10), 6640–6653 (1992).
[CrossRef] [PubMed]

M. O. Scully and M. S. Zubairy, “Simple laser accelerator: Optics and particle dynamics,” Phys. Rev. A 44(4), 2656–2663 (1991).
[CrossRef] [PubMed]

Sears, C. M. S.

T. Plettner, R. L. Byer, E. Colby, B. Cowan, C. M. S. Sears, J. E. Spencer, and R. H. Siemann, “Visible-laser acceleration of relativistic electrons in a semi-infinite vacuum,” Phys. Rev. Lett. 95(13), 134801 (2005).
[CrossRef] [PubMed]

Sessler, A. M.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253 (2001).
[CrossRef]

Shneerson, V.

R. Fung, V. Shneerson, D. K. Saldin, and A. Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. 5(1), 64–67 (2009).
[CrossRef]

Siemann, R. H.

T. Plettner, R. L. Byer, E. Colby, B. Cowan, C. M. S. Sears, J. E. Spencer, and R. H. Siemann, “Visible-laser acceleration of relativistic electrons in a semi-infinite vacuum,” Phys. Rev. Lett. 95(13), 134801 (2005).
[CrossRef] [PubMed]

Skaritka, J.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Spencer, J. E.

T. Plettner, R. L. Byer, E. Colby, B. Cowan, C. M. S. Sears, J. E. Spencer, and R. H. Siemann, “Visible-laser acceleration of relativistic electrons in a semi-infinite vacuum,” Phys. Rev. Lett. 95(13), 134801 (2005).
[CrossRef] [PubMed]

Sprangle, P.

E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 52(5), 5443–5453 (1995).
[CrossRef] [PubMed]

Starace, A. F.

S. X. Hu and A. F. Starace, “GeV electrons from ultraintense laser interaction with highly charged ions,” Phys. Rev. Lett. 88(24), 245003 (2002).
[CrossRef] [PubMed]

Steinhauer, L. C.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Strickland, D.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[CrossRef]

Stuart, B. C.

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(23), 5274–5277 (2001).
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S. V. Bulanov, T. Esirkepov, and T. Tajima, “Light intensification towards the Schwinger limit,” Phys. Rev. Lett. 91(8), 085001 (2003).
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Tulloch, W. M.

Y. C. Huang, D. Zheng, W. M. Tulloch, and R. L. Byer, “Proposed structure for a crossed-laser beam, GeV per meter gradient, vacuum electron linear accelerator,” Appl. Phys. Lett. 68(6), 753–755 (1996).
[CrossRef]

van Steenbergen, A.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
[CrossRef] [PubMed]

Varin, C.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially-polarized laser beams: Scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
[CrossRef]

C. Varin, M. Piché, and M. A. Porras, “Analytical calculation of the strong axial longitudinal electric field resulting from the tight focusing of an ultrafast transverse magnetic pulsed beam in free space,” J. Opt. Soc. Am. A 23, 2027–2038 (2006).
[CrossRef]

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005).
[CrossRef] [PubMed]

Vergino, M.

Wang, P. X.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253 (2001).
[CrossRef]

Woodworth, J. G.

F. V. Hartemann, S. N. Fochs, N. C. Luhmann, J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, A. K. Kerman, and G. P. Le Sage, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field at focus,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51(5), 4833–4843 (1995).

Yakimenko, V.

W. D. Kimura, A. van Steenbergen, M. Babzien, I. Ben-Zvi, L. P. Campbell, D. B. Cline, C. E. Dilley, J. C. Gallardo, S. C. Gottschalk, P. He, K. P. Kusche, Y. Liu, R. H. Pantell, I. V. Pogorelsky, D. C. Quimby, J. Skaritka, L. C. Steinhauer, and V. Yakimenko, “First staging of two laser accelerators,” Phys. Rev. Lett. 86(18), 4041–4043 (2001).
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Yanovsky, V.

Yuan, X. Q.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253 (2001).
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Zakowicz, W.

E. D. Courant, C. Pellegrini, and W. Zakowicz, “High-energy inverse free-electron-laser accelerator,” Phys. Rev. A 32(5), 2813–2823 (1985).
[CrossRef] [PubMed]

Zheng, D.

Y. C. Huang, D. Zheng, W. M. Tulloch, and R. L. Byer, “Proposed structure for a crossed-laser beam, GeV per meter gradient, vacuum electron linear accelerator,” Appl. Phys. Lett. 68(6), 753–755 (1996).
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Zolotorev, M. S.

G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86(23), 5274–5277 (2001).
[CrossRef] [PubMed]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, “Simple laser accelerator: Optics and particle dynamics,” Phys. Rev. A 44(4), 2656–2663 (1991).
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Appl. Phys. Lett. (2)

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253 (2001).
[CrossRef]

Y. C. Huang, D. Zheng, W. M. Tulloch, and R. L. Byer, “Proposed structure for a crossed-laser beam, GeV per meter gradient, vacuum electron linear accelerator,” Appl. Phys. Lett. 68(6), 753–755 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. At. Mol. Opt. Phys. (1)

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially-polarized laser beams: Scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
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Laser Part. Beams (2)

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Figures (10)

Fig. 1
Fig. 1

Schematic of simulations at initial time.

Fig. 2
Fig. 2

(Color online) (a) Maximum energy gain and (b) corresponding optimal beam waist vs. power P from 0.1 to 40 PW for various τ. All solid lines correspond to z ( 0 ) = 0 . Dashed lines correspond to optimal z ( 0 ) for w 0 = 2 μm (optimal waist). All cases shown correspond to forward scattering of the electron.

Fig. 3
Fig. 3

(Color online) Maximum energy gain vs. normalized z ( 0 ) for P = 1 PW, τ = 10 fs and various w 0 . All cases shown correspond to forward scattering of the electron.

Fig. 4
Fig. 4

(Color online) (a) Normalized maximum energy gain and (b) corresponding normalized optimal initial position vs. P from 5 TW to 40 PW for various w 0 andτ. All cases shown correspond to forward scattering of the electron. Cases of very non-relativistic final kinetic energy are not plotted to reduce clutter.

Fig. 5
Fig. 5

(Color online) Close-up of plot of maximum energy gain vs. P for various w 0 andτ.

Fig. 6
Fig. 6

(Color online) Normalized maximum energy gain and corresponding normalized optimal D vs. P from 5 TW to 40 PW for various w 0 and E K ( 0 ) : (a), (b) non-relativistic E K ( 0 ) ; (c), (d) marginally-relativistic E K ( 0 ) ; and (e), (f) relativistic E K ( 0 ) . τ = 7.5 fs. All cases shown correspond to forward scattering of the electron. Cases of very non-relativistic final kinetic energy are not plotted to reduce clutter.

Fig. 7
Fig. 7

(Color online) Normalized maximum energy gain and corresponding normalized optimal D vs. P from 5 TW to 40 PW for various w 0 and E K ( 0 ) : (a), (b) non-relativistic E K ( 0 ) ; (c), (d) marginally-relativistic E K ( 0 ) ; and (e), (f) very relativistic E K ( 0 ) . τ = 15 fs. All cases shown correspond to forward scattering of the electron. Cases of very non-relativistic final kinetic energy are not plotted to reduce clutter.

Fig. 8
Fig. 8

(Color online) Normalized maximum energy gain and corresponding normalized optimal D vs. E K ( 0 ) from 0.01 to 20 MeV for various w 0 and P: (a), (b) P = 10 TW; (c), (d) P = 100 TW; (e), (f) P = 1 PW; and (g), (h) P = 10 PW. τ = 10 fs. All cases shown correspond to forward scattering of the electron.

Fig. 9
Fig. 9

(Color online) Maximum energy gain vs. normalized D for τ = 7.5 fs, w 0 = 2 μm and E K ( 0 ) = 10 MeV for variousP. All cases shown correspond to forward scattering of the electron.

Fig. 10
Fig. 10

(Color online) Electron kinetic energy vs. normalized position for P = 17.3 TW, w 0 = 2 μm, τ = 7.5 fs, E K ( 0 ) = 10 MeV, and optimal ψ 0 and D. Inset “close-up 2” zooms into the point at which the electron enters its effectively final accelerating cycle.

Equations (13)

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E ( r , z , t ) = Re { E ˜ ( r , z ) e j ( ξ + ψ 0 ) sech ( ξ + k z i ξ 0 ) } , B ( r , z , t ) = ϕ ^ 1 c E ( r , z , t ) r ^ ,
E ˜ ( r , z ) f 2 ρ e f ρ 2 8 η 0 P π w 0 2 [ r ^ z ^ 2 j k r ( 1 f ρ 2 ) ] , B ˜ ( r , z ) ϕ ^ 1 c E ˜ ( r , z ) r ^ ,
P = 1 2 μ 0 0 d r 2 π r Re { E ˜ ( r , 0 ) × B ˜ * ( r , 0 ) z }
E p u l s e = 1 μ 0 d t 0 d r 2 π r E ( r , 0 , t ) × B ( r , 0 , t ) z P d t sech 2 ( ω t + k z i ξ 0 ) = P 2 ξ 0 ω
τ = ξ 0 ω sech 1 ( exp ( 1 ) ) .
d p d t = d ( γ m v ) d t = e ( E + v × B ) ,
E z = [ 1 / z 0 1 + ( z / z 0 ) 2 8 η 0 P π ] sin ( ω t k z + 2 tan 1 ( z z 0 ) + ψ 0 ) sech ( ω t k ( z z i ) ξ 0 )
d β d t = e E z γ 3 m c , d z d t = v = c β .
d β d T = e γ 3 m c 2 [ 1 / κ 1 + ς 2 8 η 0 P π ] sin ( T κ ς + 2 tan 1 ( ς ) + ψ 0 ) sech ( T κ ( ς ς i ) ξ 0 ) , d ς d T = β κ
Δ E lim = 0 d z [ e / z 0 1 + ( z / z 0 ) 2 8 η 0 P π ] sin ( 2 tan 1 ( z / z 0 ) ) = e 8 η 0 P π P [ P W ] [ G e V ]
a 0 e m c ω z 0 8 η 0 P t h π 1
v ( 0 ) D z ( 0 ) c D z i D z ( 0 ) β ( 0 ) z i 1 β ( 0 )
Δ E v = c = d z ( e E z ) d z α [ e / z 0 1 + ( z / z 0 ) 2 8 η 0 P π ] sin ( 2 tan 1 ( z / z 0 ) + ψ 0 ) = 0

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