Abstract

We study the nonlinear refraction of X-rays in highly ionized condensed matter by using a classical model of a cold electron plasma in a lattice of still ions coupled with Maxwell equations. We discuss the existence and stability of nonlinear waves. As a real-world example, we consider beam self-defocusing in crystalline materials (B, C, Li, Na). We predict that nonlinear processes become comparable to the linear ones for focused beams with powers of the order of mc 3/ro (≈10 GW), the classical electron power. As a consequence, nonlinear phenomena are expected in currently exploited X-ray Free-Electron Lasers and in their future developments.

© 2008 Optical Society of America

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    [CrossRef]
  3. Techical Design Report of the European XFEL, DESY (2006-097). http://xfel.desy.de/tdr/tdr.
  4. H. Hora, Physics of Laser Driven Plasmas (Wiley-Interscience, New York, 1981).
  5. R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, "Pair Creation and an X-Ray Free Electron Laser, " Phys. Rev. Lett. 87, 193902-(4) (2001).
    [CrossRef] [PubMed]
  6. R. Moshammer Y. H. Jiang, L. Foucar, A. Rudenko, Th. Ergler, C. D. Schrter, S. Ldemann, K. Zrost, D. Fischer, J. Titze, T. Jahnke, M. Schffler, T. Weber, R. Drner, T. J. M. Zouros, A. Dorn, T. Ferger, K. U. Khnel, S. Dsterer, R. Treusch, P. Radcliffe, E. Plnjes, and J. Ullrich, " Few-Photon Multiple Ionization of Ne and Ar by Strong Free-Electron-Laser Pulses," Phys. Rev. Lett. 98, 203001-(4) (2007).
    [CrossRef] [PubMed]
  7. H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005).
    [CrossRef] [PubMed]
  8. Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, " Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy," Phys. Rev. Lett. 94, 043001-(4) (2005).
    [CrossRef] [PubMed]
  9. T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005).
    [CrossRef]
  10. E. P. Benis, D. Charalambidis, T. N. Kitsopoulos, G. D. Tsakiris, and P. Tzallas, " Two-photon double ionization of rare gases by a superposition of harmonics," Phys. Rev. A 74, 051402(R)-(4) (2006).
    [CrossRef]
  11. T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, "Nonlinear optics in the extreme ultraviolet," Nature (London) 432,605-608 (2004).
    [CrossRef] [PubMed]
  12. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, "Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry, " Phys. Rev. 174,813-822 (1968).
    [CrossRef]
  13. S. S. Jha, and C. S. Warke, "Nonlinear electromagnetic response of Bloch electrons in a magnetic field," Nuovo Cimento 53B,120-135 (1968).
  14. A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, "Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals," Phys. Rev. A 67, 041804-(4) (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. R. W. Boyd, Nonlinear Optics (Academic Press, New York, 2002).
  18. R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature (London) 406,752-757 (2000).
    [CrossRef] [PubMed]
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    [CrossRef]

2006 (1)

T. Pfeifer, C. Spielmann, and G. Gerber, "Femtosecond x-ray science," Rep. Prog. Phys. 69,443-505 (2006).
[CrossRef]

2004 (1)

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, "Nonlinear optics in the extreme ultraviolet," Nature (London) 432,605-608 (2004).
[CrossRef] [PubMed]

2000 (1)

R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature (London) 406,752-757 (2000).
[CrossRef] [PubMed]

1972 (1)

S. S. Jha, and J. W. F. Woo, "Nonlinear Response of Bound Electrons to X Rays," Phys. Rev. B 5,4210-4212 (1972).
[CrossRef]

1968 (2)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, "Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry, " Phys. Rev. 174,813-822 (1968).
[CrossRef]

S. S. Jha, and C. S. Warke, "Nonlinear electromagnetic response of Bloch electrons in a magnetic field," Nuovo Cimento 53B,120-135 (1968).

1963 (1)

N. Bloembergen, "Wave propagation in nonlinear electromagnetic media," Proceedings of the IEEE 51,124-131 (1963).
[CrossRef]

Alkofer, R.

R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, "Pair Creation and an X-Ray Free Electron Laser, " Phys. Rev. Lett. 87, 193902-(4) (2001).
[CrossRef] [PubMed]

Benis, E. P.

E. P. Benis, D. Charalambidis, T. N. Kitsopoulos, G. D. Tsakiris, and P. Tzallas, " Two-photon double ionization of rare gases by a superposition of harmonics," Phys. Rev. A 74, 051402(R)-(4) (2006).
[CrossRef]

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, "Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry, " Phys. Rev. 174,813-822 (1968).
[CrossRef]

N. Bloembergen, "Wave propagation in nonlinear electromagnetic media," Proceedings of the IEEE 51,124-131 (1963).
[CrossRef]

Chang, R. K.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, "Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry, " Phys. Rev. 174,813-822 (1968).
[CrossRef]

Charalambidis, D.

E. P. Benis, D. Charalambidis, T. N. Kitsopoulos, G. D. Tsakiris, and P. Tzallas, " Two-photon double ionization of rare gases by a superposition of harmonics," Phys. Rev. A 74, 051402(R)-(4) (2006).
[CrossRef]

de Castro, A. R. B.

T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005).
[CrossRef]

H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005).
[CrossRef] [PubMed]

Frster, E.

A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, "Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals," Phys. Rev. A 67, 041804-(4) (2003).
[CrossRef]

Gerber, G.

T. Pfeifer, C. Spielmann, and G. Gerber, "Femtosecond x-ray science," Rep. Prog. Phys. 69,443-505 (2006).
[CrossRef]

Grtler, P.

H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005).
[CrossRef] [PubMed]

T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005).
[CrossRef]

Hajdu, J.

R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature (London) 406,752-757 (2000).
[CrossRef] [PubMed]

Hasegawa, H.

Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, " Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy," Phys. Rev. Lett. 94, 043001-(4) (2005).
[CrossRef] [PubMed]

Hecht, M. B.

R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, "Pair Creation and an X-Ray Free Electron Laser, " Phys. Rev. Lett. 87, 193902-(4) (2001).
[CrossRef] [PubMed]

Jha, S. S.

S. S. Jha, and J. W. F. Woo, "Nonlinear Response of Bound Electrons to X Rays," Phys. Rev. B 5,4210-4212 (1972).
[CrossRef]

S. S. Jha, and C. S. Warke, "Nonlinear electromagnetic response of Bloch electrons in a magnetic field," Nuovo Cimento 53B,120-135 (1968).

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, "Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry, " Phys. Rev. 174,813-822 (1968).
[CrossRef]

Kanai, T.

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, "Nonlinear optics in the extreme ultraviolet," Nature (London) 432,605-608 (2004).
[CrossRef] [PubMed]

Kitsopoulos, T. N.

E. P. Benis, D. Charalambidis, T. N. Kitsopoulos, G. D. Tsakiris, and P. Tzallas, " Two-photon double ionization of rare gases by a superposition of harmonics," Phys. Rev. A 74, 051402(R)-(4) (2006).
[CrossRef]

Kosuge, A.

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, "Nonlinear optics in the extreme ultraviolet," Nature (London) 432,605-608 (2004).
[CrossRef] [PubMed]

Laarmann, T.

T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005).
[CrossRef]

H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005).
[CrossRef] [PubMed]

Laasch, W.

H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005).
[CrossRef] [PubMed]

T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005).
[CrossRef]

Lee, C. H.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, "Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry, " Phys. Rev. 174,813-822 (1968).
[CrossRef]

Midorikawa, K.

Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, " Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy," Phys. Rev. Lett. 94, 043001-(4) (2005).
[CrossRef] [PubMed]

Mller, T.

H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005).
[CrossRef] [PubMed]

T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005).
[CrossRef]

Nabekawa, Y.

Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, " Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy," Phys. Rev. Lett. 94, 043001-(4) (2005).
[CrossRef] [PubMed]

Nazarkin, A.

A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, "Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals," Phys. Rev. A 67, 041804-(4) (2003).
[CrossRef]

Neutze, R.

R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature (London) 406,752-757 (2000).
[CrossRef] [PubMed]

Pfeifer, T.

T. Pfeifer, C. Spielmann, and G. Gerber, "Femtosecond x-ray science," Rep. Prog. Phys. 69,443-505 (2006).
[CrossRef]

Podorov, S.

A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, "Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals," Phys. Rev. A 67, 041804-(4) (2003).
[CrossRef]

Roberts, C. D.

R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, "Pair Creation and an X-Ray Free Electron Laser, " Phys. Rev. Lett. 87, 193902-(4) (2001).
[CrossRef] [PubMed]

Sauerbrey, R.

A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, "Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals," Phys. Rev. A 67, 041804-(4) (2003).
[CrossRef]

Schmidt, S. M.

R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, "Pair Creation and an X-Ray Free Electron Laser, " Phys. Rev. Lett. 87, 193902-(4) (2001).
[CrossRef] [PubMed]

Schulz, J.

H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005).
[CrossRef] [PubMed]

T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005).
[CrossRef]

Sekikawa, T.

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, "Nonlinear optics in the extreme ultraviolet," Nature (London) 432,605-608 (2004).
[CrossRef] [PubMed]

Spielmann, C.

T. Pfeifer, C. Spielmann, and G. Gerber, "Femtosecond x-ray science," Rep. Prog. Phys. 69,443-505 (2006).
[CrossRef]

Spoel, D.

R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature (London) 406,752-757 (2000).
[CrossRef] [PubMed]

Takahashi, E. J.

Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, " Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy," Phys. Rev. Lett. 94, 043001-(4) (2005).
[CrossRef] [PubMed]

Tsakiris, G. D.

E. P. Benis, D. Charalambidis, T. N. Kitsopoulos, G. D. Tsakiris, and P. Tzallas, " Two-photon double ionization of rare gases by a superposition of harmonics," Phys. Rev. A 74, 051402(R)-(4) (2006).
[CrossRef]

Tzallas, P.

E. P. Benis, D. Charalambidis, T. N. Kitsopoulos, G. D. Tsakiris, and P. Tzallas, " Two-photon double ionization of rare gases by a superposition of harmonics," Phys. Rev. A 74, 051402(R)-(4) (2006).
[CrossRef]

Uschmann, I.

A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, "Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals," Phys. Rev. A 67, 041804-(4) (2003).
[CrossRef]

Vaughan, D.

D. Vaughan, X-Ray Data Booklet (Lawrence Berkely Laboratory, Berkely, 1986).
[CrossRef]

Vinnik, D. V.

R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, "Pair Creation and an X-Ray Free Electron Laser, " Phys. Rev. Lett. 87, 193902-(4) (2001).
[CrossRef] [PubMed]

Wabnitz, H.

H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005).
[CrossRef] [PubMed]

T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005).
[CrossRef]

Warke, C. S.

S. S. Jha, and C. S. Warke, "Nonlinear electromagnetic response of Bloch electrons in a magnetic field," Nuovo Cimento 53B,120-135 (1968).

Watanabe, S.

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, "Nonlinear optics in the extreme ultraviolet," Nature (London) 432,605-608 (2004).
[CrossRef] [PubMed]

Weckert, E.

R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature (London) 406,752-757 (2000).
[CrossRef] [PubMed]

Woo, J. W. F.

S. S. Jha, and J. W. F. Woo, "Nonlinear Response of Bound Electrons to X Rays," Phys. Rev. B 5,4210-4212 (1972).
[CrossRef]

Wouts, R.

R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature (London) 406,752-757 (2000).
[CrossRef] [PubMed]

Nature (London) (2)

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, "Nonlinear optics in the extreme ultraviolet," Nature (London) 432,605-608 (2004).
[CrossRef] [PubMed]

R. Neutze, R. Wouts, D. Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature (London) 406,752-757 (2000).
[CrossRef] [PubMed]

Nuovo Cimento (1)

S. S. Jha, and C. S. Warke, "Nonlinear electromagnetic response of Bloch electrons in a magnetic field," Nuovo Cimento 53B,120-135 (1968).

Phys. Rev. (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, "Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry, " Phys. Rev. 174,813-822 (1968).
[CrossRef]

Phys. Rev. B (1)

S. S. Jha, and J. W. F. Woo, "Nonlinear Response of Bound Electrons to X Rays," Phys. Rev. B 5,4210-4212 (1972).
[CrossRef]

Proceedings of the IEEE (1)

N. Bloembergen, "Wave propagation in nonlinear electromagnetic media," Proceedings of the IEEE 51,124-131 (1963).
[CrossRef]

Rep. Prog. Phys. (1)

T. Pfeifer, C. Spielmann, and G. Gerber, "Femtosecond x-ray science," Rep. Prog. Phys. 69,443-505 (2006).
[CrossRef]

Other (16)

Techical Design Report of the European XFEL, DESY (2006-097). http://xfel.desy.de/tdr/tdr.

H. Hora, Physics of Laser Driven Plasmas (Wiley-Interscience, New York, 1981).

R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, "Pair Creation and an X-Ray Free Electron Laser, " Phys. Rev. Lett. 87, 193902-(4) (2001).
[CrossRef] [PubMed]

R. Moshammer Y. H. Jiang, L. Foucar, A. Rudenko, Th. Ergler, C. D. Schrter, S. Ldemann, K. Zrost, D. Fischer, J. Titze, T. Jahnke, M. Schffler, T. Weber, R. Drner, T. J. M. Zouros, A. Dorn, T. Ferger, K. U. Khnel, S. Dsterer, R. Treusch, P. Radcliffe, E. Plnjes, and J. Ullrich, " Few-Photon Multiple Ionization of Ne and Ar by Strong Free-Electron-Laser Pulses," Phys. Rev. Lett. 98, 203001-(4) (2007).
[CrossRef] [PubMed]

H. Wabnitz, A. R. B. de Castro, P. Grtler, T. Laarmann, W. Laasch, J. Schulz, and T. Mller, " Multiple Ionization of Rare Gas Atoms Irradiated with Intense VUV Radiation," Phys. Rev. Lett. 94, 023001-(4) (2005).
[CrossRef] [PubMed]

Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, " Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy," Phys. Rev. Lett. 94, 043001-(4) (2005).
[CrossRef] [PubMed]

T. Laarmann, A. R. B. de Castro, P. Grtler, W. Laasch, J. Schulz, H. Wabnitz, and T. Mller, " Photoionization of helium atoms irradiated with intense vacuum ultraviolet free-electron laser light. Part I. Experimental study of multiphoton and single-photon processes," Phys. Rev. A 72, 023409-(8) (2005).
[CrossRef]

E. P. Benis, D. Charalambidis, T. N. Kitsopoulos, G. D. Tsakiris, and P. Tzallas, " Two-photon double ionization of rare gases by a superposition of harmonics," Phys. Rev. A 74, 051402(R)-(4) (2006).
[CrossRef]

A. Nazarkin, S. Podorov, I. Uschmann, E. Frster, and R. Sauerbrey, "Nonlinear optics in the angstrom regime: Hard-x-ray frequency doubling in perfect crystals," Phys. Rev. A 67, 041804-(4) (2003).
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[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(Color online) X-ray electromagnetic band-structure for a FCC lattice (the inset shows the first Brillouin zone).

Fig. 2.
Fig. 2.

(Color online). (a) Solution of the nonlinear dispersion relation for the SPM Eq.(11), the continuous (dot-dashed) line is the forward (backward) propagating wave, the horizontal dashed line corresponds to the asymptotic value knl =-k; (b) divergence Vs input peak intensity I 0 in adimensional units.

Fig. 3.
Fig. 3.

(Color online). (a) Atomic- and (b) electronic-loss-limited intensity for nonlinear defocusing.

Tables (1)

Tables Icon

Table 1. Expression of Bloch modes for SC and FCC lattices.

Equations (25)

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× h = j + ε 0 t e ,
× e = μ 0 t h ,
t n + · ( n v ) = 0 ,
( t + v · ) v + q m ( e + μ 0 v × h ) = 0 ,
× H = i ω ε 0 ( 1 ω α 2 ω 2 ) E , × E = i ω μ 0 H ,
H H k = ( ω c ) 2 H k
· ( E × H * + E * × H ) + j Δ · E * = 0 .
n = n 0 + 2 [ n ( 1 ) e i ω t ] + 2 [ n ( 2 ) e 2 i ω t ] ,
v = 2 [ V ( 1 ) e i ω t ] + 2 [ V ( 2 ) e 2 i ω t ] ,
V ( 1 ) = i q ω m E , n ( 1 ) = q ω 2 m n 0 · E
V ( 2 ) = iq 2 4 ω 3 m 2 E 2
n ( 2 ) = q 2 2 ω 4 m 2 [ . ( E 2 n 0 ) + 1 4 . ( n 0 E 2 ) ] .
j ( 3 ) = iq 4 n 0 8 ω 5 m 3 E * 2 E 2 .
E = 2 η 0 A ( z ) y ̂ e ikz ,
H = 2 η 0 A ( z ) x ̂ e ikz ,
d A d z = iq 4 η 0 2 n 0 8 ω 5 m 3 A * e 2 ikz d 2 dz 2 ( A 2 e 2 ikz ) .
k nl + q 4 η 0 2 A 0 2 2 ω 5 m 3 ( k + k nl ) 2 = 0 .
2 γ A 0 2 d 2 a 1 dz 2 + i [ 1 + 8 γ A 0 2 ( k + k nl ) ] da 1 dz +
[ k nl + 8 A 0 2 ( k + k nl ) 2 γ ] a 1 + 4 γ ( k + k nl ) 2 a 1 * = 0 .
𝒩 n 2 I δ = λ 2 I 2 π P 0 ,
Φ nl ( r ) 4 k 2 γ LI 0 ( 1 2 r 2 w 0 2 )
f = w 0 2 16 k γ I 0 L .
tan ( θ ) = w 0 f = 16 k γ I 0 L w 0 .
f = 2 w 0 2 ξ L ( 1 + 8 ξ 1 + 16 ξ ) ,
tan ( θ ) = L ( 1 + 8 ξ 1 + 16 ξ ) 2 w 0 ξ .

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