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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  1. R. Dendy, Plasma physics: an in introductory course (Cambridge University Press, Cambridge, 1996).
  2. T. Pfeifer, C. Spielmann, and G. Gerber, “Femtosecond x-ray science,” Rep. Prog. Phys. 69, 443–505 (2006).
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  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]
  15. S. S. Jha and J. W. F. Woo, “Nonlinear Response of Bound Electrons to X Rays,” Phys. Rev. B 5, 4210–4212 (1972).
    [Crossref]
  16. N. Bloembergen, “Wave propagation in nonlinear electromagnetic media,” Proceedings of the IEEE 51, 124–131 (1963).
    [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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  20. T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990).
    [Crossref]
  21. C. A. Balanis, Advanced Engineering Electromagnetics (Jhon Wiley & Sons, Toronto, 1989).
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2007 (1)

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]

2006 (2)

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]

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

2005 (3)

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]

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]

2003 (1)

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]

2001 (1)

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]

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]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

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]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Jhon Wiley & Sons, Toronto, 1989).

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]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, New York, 2002).

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]

Dendy, R.

R. Dendy, Plasma physics: an in introductory course (Cambridge University Press, Cambridge, 1996).

Dorn, A.

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]

Drner, R.

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]

Dsterer, S.

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]

Ergler, Th.

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]

Ferger, T.

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]

Fischer, D.

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]

Foucar, L.

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]

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]

Hora, H.

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

Inui, T.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990).
[Crossref]

Jahnke, T.

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]

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]

Jiang, Y. H.

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]

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]

Khnel, K. U.

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]

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]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

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]

Ldemann, S.

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]

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.

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]

Moshammer, R.

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]

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]

Onodera, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990).
[Crossref]

Pfeifer, T.

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

Plnjes, E.

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]

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]

Radcliffe, P.

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]

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]

Rudenko, A.

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]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer & Verlag, Berlin, 2001).

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]

Schffler, M.

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]

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]

Schrter, C. D.

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]

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]

Tanabe, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990).
[Crossref]

Titze, J.

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]

Treusch, R.

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]

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]

Ullrich, J.

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]

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]

Weber, T.

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]

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]

Zouros, T. J. M.

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]

Zrost, K.

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]

Nature (2)

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]

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature (London)  432, 605–608 (2004).
[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. A (3)

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]

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]

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]

Phys. Rev. Lett. (4)

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]

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 (9)

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. W. Boyd, Nonlinear Optics (Academic Press, New York, 2002).

K. Sakoda, Optical Properties of Photonic Crystals (Springer & Verlag, Berlin, 2001).

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in Physics (Springer & Verlag, Berlin, 1990).
[Crossref]

C. A. Balanis, Advanced Engineering Electromagnetics (Jhon Wiley & Sons, Toronto, 1989).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

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

R. Dendy, Plasma physics: an in introductory course (Cambridge University Press, Cambridge, 1996).

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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)

Equations on this page are rendered with MathJax. Learn more.

× 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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