Abstract

The potential of hollow core photonic crystal fiber as a nonlinear gas cell for efficient high harmonic generation is discussed. The feasibility of phase-matching this process by modulating the phase of ionization electrons using a counter-propagating laser field is shown. In this way, harmonics with energies of several hundreds of eV can be produced using fs-laser pump pulses of µJ energy.

© 2008 Optical Society of America

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  1. H. C. Kapteyn, M. M. Murnane, and I. P. Christov, "Coherent x-rays from lasers: Applied attosecond science" Phys. Today 58, 39 (2005)
    [CrossRef]
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    [CrossRef] [PubMed]
  3. E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, "Soft x-ray-driven femtosecond molecular dynamics," Science 317, 1374 (2007)
    [CrossRef] [PubMed]
  4. P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994 (1993)
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
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  7. R. F. Cregan, B. J. Mangan, J. C. Knight,T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537 (1999)
    [CrossRef] [PubMed]
  8. P. J. Roberts, F. Couny, H. Sabert,B. J. Mangan, D. P. Williams, L. Farr, M.W. Mason, A. Tomlinson, T. A. Birks, and J. C. Knight, "Ultimate low loss of hollow-core photonic crystal fibres," Opt. Express 13, 236 (2005)
    [CrossRef] [PubMed]
  9. F. Benabid, J. Knight, G. Antonopoulos, and P. St.J. Russell, "Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber," Science 298, 399 (2002)
    [CrossRef] [PubMed]
  10. F. Benabid, G. Bouwmans, J. C. Knight, P. St.J. Russell, and F. Couny, "Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen," Phys. Rev. Lett. 93, 123903 (2004)
    [CrossRef] [PubMed]
  11. S. O. Konorov, A. B. Fedotov, and A. M. Zheltikov, "Enhanced four-wave mixing in a hollow-core photoniccrystal fiber," Opt. Lett. 28, 1448 (2003)
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  12. F. Benabid, P. S. Light, F. Couny, and P. St.J Russell, "Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF," Opt. Express 13, 5694 (2005)
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  19. B. Dromey, M. Zepf, M. Landreman, and S. M. Hooker, "Quasi-phasematching of harmonic generation via multimode beating in waveguides," Opt. Express 15, 7894 (2007)
    [CrossRef] [PubMed]
  20. O. Cohen, X. Zhang, A. L. Lytel, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, " Grating-assisted phase matching in extreme nonlinear optics," Phys. Rev. Lett. 99, 053902 (2007)
    [CrossRef] [PubMed]
  21. M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and of atomic ions in alternating electromagnetic field," Sov. Phys. JETP 64, 1191 (1986)
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    [CrossRef]
  23. M. Levenstein, P. Balcou, M. Y. Ivanov, A. L�??Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)
  24. D. G. Ouzounov et al., "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702 (2003)
    [CrossRef] [PubMed]
  25. G. Bouwmans, F. Luan, J. C. Knight, P. St.J. Russell, L. Farr, B. J. Mangan, and H. Sabert, "Properties of a hollow-core photonic bandgap fiber at 850 nm wavelength," Opt. Express 11, 1613 (2003)
    [CrossRef] [PubMed]
  26. E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov, "Phase-matching solutions for high-order harmonic generation in hollow-core photonic crystal fibers," Phys. Rev. E 70, 066619 (2004)
  27. O. Cohen, A. L. Litle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, "Optimizing quasi-phase matching using counterpropagating pulse trains," Opt. Lett. 32, 2975 (2007)
    [CrossRef] [PubMed]
  28. S. L. Voronov, I. Kohl, J. B. Madsen,  et al., "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001)
    [CrossRef] [PubMed]
  29. P. Salieres, B. Carre, L. Le Deroff,  et al., "Feynman�??s path-integral approach for intense-laser-atom interactions," Science,  292, 902 (2001)
    [CrossRef] [PubMed]
  30. E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov,"Broadband dynamic phase matching of high-order harmonic generation by a soliton pump field in a gas-filled holow photonic crystal fiber" Opt. Lett. 33, 977 (2008)
    [CrossRef] [PubMed]
  31. T. Hidaka,"Extremely low-loss waveguide for XUV light" Opt. Commun. 44, 90 (1982)
    [CrossRef]

2008 (1)

2007 (5)

O. Cohen, A. L. Litle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, "Optimizing quasi-phase matching using counterpropagating pulse trains," Opt. Lett. 32, 2975 (2007)
[CrossRef] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, "Quasi-phasematching and quantum path control of high-harmonic generation using counterpropagating light," Nature Physics 3, 270 (2007)
[CrossRef]

B. Dromey, M. Zepf, M. Landreman, and S. M. Hooker, "Quasi-phasematching of harmonic generation via multimode beating in waveguides," Opt. Express 15, 7894 (2007)
[CrossRef] [PubMed]

O. Cohen, X. Zhang, A. L. Lytel, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, " Grating-assisted phase matching in extreme nonlinear optics," Phys. Rev. Lett. 99, 053902 (2007)
[CrossRef] [PubMed]

E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, "Soft x-ray-driven femtosecond molecular dynamics," Science 317, 1374 (2007)
[CrossRef] [PubMed]

2006 (1)

A. Paul, E. A. Gibson, X. Zhang, A. Lytle, T. Popmintchev, X. Zhou, M. M. Murnane, I. P. Christov, and H. C. Kapteyn, "Phase-matching techniques for coherent soft x-ray generation," IEEE J. Quantum Electron. 42, 14 (2006)
[CrossRef]

2005 (4)

F. Benabid, P. S. Light, F. Couny, and P. St.J Russell, "Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF," Opt. Express 13, 5694 (2005)
[CrossRef] [PubMed]

H. C. Kapteyn, M. M. Murnane, and I. P. Christov, "Coherent x-rays from lasers: Applied attosecond science" Phys. Today 58, 39 (2005)
[CrossRef]

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

P. J. Roberts, F. Couny, H. Sabert,B. J. Mangan, D. P. Williams, L. Farr, M.W. Mason, A. Tomlinson, T. A. Birks, and J. C. Knight, "Ultimate low loss of hollow-core photonic crystal fibres," Opt. Express 13, 236 (2005)
[CrossRef] [PubMed]

2004 (2)

F. Benabid, G. Bouwmans, J. C. Knight, P. St.J. Russell, and F. Couny, "Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen," Phys. Rev. Lett. 93, 123903 (2004)
[CrossRef] [PubMed]

E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov, "Phase-matching solutions for high-order harmonic generation in hollow-core photonic crystal fibers," Phys. Rev. E 70, 066619 (2004)

2003 (4)

D. G. Ouzounov et al., "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702 (2003)
[CrossRef] [PubMed]

G. Bouwmans, F. Luan, J. C. Knight, P. St.J. Russell, L. Farr, B. J. Mangan, and H. Sabert, "Properties of a hollow-core photonic bandgap fiber at 850 nm wavelength," Opt. Express 11, 1613 (2003)
[CrossRef] [PubMed]

S. O. Konorov, A. B. Fedotov, and A. M. Zheltikov, "Enhanced four-wave mixing in a hollow-core photoniccrystal fiber," Opt. Lett. 28, 1448 (2003)
[CrossRef] [PubMed]

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

2002 (1)

F. Benabid, J. Knight, G. Antonopoulos, and P. St.J. Russell, "Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber," Science 298, 399 (2002)
[CrossRef] [PubMed]

2001 (2)

S. L. Voronov, I. Kohl, J. B. Madsen,  et al., "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001)
[CrossRef] [PubMed]

P. Salieres, B. Carre, L. Le Deroff,  et al., "Feynman�??s path-integral approach for intense-laser-atom interactions," Science,  292, 902 (2001)
[CrossRef] [PubMed]

1999 (2)

R. F. Cregan, B. J. Mangan, J. C. Knight,T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537 (1999)
[CrossRef] [PubMed]

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

1998 (1)

A. Rundquist, C. G. Durfee, Z. Chang et al., "Phase-matched generation of coherent soft x-rays," Science 280, 1412 (1998)
[CrossRef] [PubMed]

1994 (2)

P. L. Shkolnikov, A. Lago, and A. E. Kaplan, "Optimal quasi-phase-matching for high-order harmonic generation in gases and plasma," Phys.Rev. A 50, 4461 (1994)

M. Levenstein, P. Balcou, M. Y. Ivanov, A. L�??Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)

1993 (2)

M. J. Shaw, C. J. Hooker, and D. C. Wilson, "Measurement of the nonlinear refractive index of air and other gases at 248 nm,"Opt. Commun. 103, 153 (1993)
[CrossRef]

P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994 (1993)
[CrossRef] [PubMed]

1986 (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and of atomic ions in alternating electromagnetic field," Sov. Phys. JETP 64, 1191 (1986)

1982 (1)

T. Hidaka,"Extremely low-loss waveguide for XUV light" Opt. Commun. 44, 90 (1982)
[CrossRef]

1981 (1)

A. Bideau-mehu, Y. Guern, A. Abjean, and A. Johannin-gille, "Measurement of refractive indices of neon, argon, krypton and xenon in the 253.7 - 140.4 nm wavelength range. Dispersion relations and estimated oscillator strengths of the resonance lines," J. Quant. Spectrosc. Radiat. Transfer. 25, 395 (1981)
[CrossRef]

Abjean, A.

A. Bideau-mehu, Y. Guern, A. Abjean, and A. Johannin-gille, "Measurement of refractive indices of neon, argon, krypton and xenon in the 253.7 - 140.4 nm wavelength range. Dispersion relations and estimated oscillator strengths of the resonance lines," J. Quant. Spectrosc. Radiat. Transfer. 25, 395 (1981)
[CrossRef]

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight,T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537 (1999)
[CrossRef] [PubMed]

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and of atomic ions in alternating electromagnetic field," Sov. Phys. JETP 64, 1191 (1986)

Antonopoulos, G.

F. Benabid, J. Knight, G. Antonopoulos, and P. St.J. Russell, "Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber," Science 298, 399 (2002)
[CrossRef] [PubMed]

Aquila, A.

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Attwood, D. T.

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Audebert, P.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Backus, S.

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Balcou, P.

M. Levenstein, P. Balcou, M. Y. Ivanov, A. L�??Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)

Benabid, F.

F. Benabid, P. S. Light, F. Couny, and P. St.J Russell, "Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF," Opt. Express 13, 5694 (2005)
[CrossRef] [PubMed]

F. Benabid, G. Bouwmans, J. C. Knight, P. St.J. Russell, and F. Couny, "Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen," Phys. Rev. Lett. 93, 123903 (2004)
[CrossRef] [PubMed]

F. Benabid, J. Knight, G. Antonopoulos, and P. St.J. Russell, "Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber," Science 298, 399 (2002)
[CrossRef] [PubMed]

Bideau-mehu, A.

A. Bideau-mehu, Y. Guern, A. Abjean, and A. Johannin-gille, "Measurement of refractive indices of neon, argon, krypton and xenon in the 253.7 - 140.4 nm wavelength range. Dispersion relations and estimated oscillator strengths of the resonance lines," J. Quant. Spectrosc. Radiat. Transfer. 25, 395 (1981)
[CrossRef]

Birks, T. A.

P. J. Roberts, F. Couny, H. Sabert,B. J. Mangan, D. P. Williams, L. Farr, M.W. Mason, A. Tomlinson, T. A. Birks, and J. C. Knight, "Ultimate low loss of hollow-core photonic crystal fibres," Opt. Express 13, 236 (2005)
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight,T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537 (1999)
[CrossRef] [PubMed]

Bouwmans, G.

F. Benabid, G. Bouwmans, J. C. Knight, P. St.J. Russell, and F. Couny, "Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen," Phys. Rev. Lett. 93, 123903 (2004)
[CrossRef] [PubMed]

G. Bouwmans, F. Luan, J. C. Knight, P. St.J. Russell, L. Farr, B. J. Mangan, and H. Sabert, "Properties of a hollow-core photonic bandgap fiber at 850 nm wavelength," Opt. Express 11, 1613 (2003)
[CrossRef] [PubMed]

Carre, B.

P. Salieres, B. Carre, L. Le Deroff,  et al., "Feynman�??s path-integral approach for intense-laser-atom interactions," Science,  292, 902 (2001)
[CrossRef] [PubMed]

Chang, Z.

A. Rundquist, C. G. Durfee, Z. Chang et al., "Phase-matched generation of coherent soft x-rays," Science 280, 1412 (1998)
[CrossRef] [PubMed]

Christov, I. P.

A. Paul, E. A. Gibson, X. Zhang, A. Lytle, T. Popmintchev, X. Zhou, M. M. Murnane, I. P. Christov, and H. C. Kapteyn, "Phase-matching techniques for coherent soft x-ray generation," IEEE J. Quantum Electron. 42, 14 (2006)
[CrossRef]

H. C. Kapteyn, M. M. Murnane, and I. P. Christov, "Coherent x-rays from lasers: Applied attosecond science" Phys. Today 58, 39 (2005)
[CrossRef]

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Cocke, C. L.

E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, "Soft x-ray-driven femtosecond molecular dynamics," Science 317, 1374 (2007)
[CrossRef] [PubMed]

Cohen, O.

O. Cohen, A. L. Litle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, "Optimizing quasi-phase matching using counterpropagating pulse trains," Opt. Lett. 32, 2975 (2007)
[CrossRef] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, "Quasi-phasematching and quantum path control of high-harmonic generation using counterpropagating light," Nature Physics 3, 270 (2007)
[CrossRef]

O. Cohen, X. Zhang, A. L. Lytel, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, " Grating-assisted phase matching in extreme nonlinear optics," Phys. Rev. Lett. 99, 053902 (2007)
[CrossRef] [PubMed]

Corkum, P. B.

M. Levenstein, P. Balcou, M. Y. Ivanov, A. L�??Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)

P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994 (1993)
[CrossRef] [PubMed]

Couny, F.

Courtois, C.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight,T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537 (1999)
[CrossRef] [PubMed]

Cros, B.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Delone, N. B.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and of atomic ions in alternating electromagnetic field," Sov. Phys. JETP 64, 1191 (1986)

Dorchies, F.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Dromey, B.

Durfee, C. G.

A. Rundquist, C. G. Durfee, Z. Chang et al., "Phase-matched generation of coherent soft x-rays," Science 280, 1412 (1998)
[CrossRef] [PubMed]

Farr, L.

Fedotov, A. B.

Gagnon, E.

E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, "Soft x-ray-driven femtosecond molecular dynamics," Science 317, 1374 (2007)
[CrossRef] [PubMed]

Gaudiosi, D.

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Geindre, J. P.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Gibson, E. A.

A. Paul, E. A. Gibson, X. Zhang, A. Lytle, T. Popmintchev, X. Zhou, M. M. Murnane, I. P. Christov, and H. C. Kapteyn, "Phase-matching techniques for coherent soft x-ray generation," IEEE J. Quantum Electron. 42, 14 (2006)
[CrossRef]

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Guern, Y.

A. Bideau-mehu, Y. Guern, A. Abjean, and A. Johannin-gille, "Measurement of refractive indices of neon, argon, krypton and xenon in the 253.7 - 140.4 nm wavelength range. Dispersion relations and estimated oscillator strengths of the resonance lines," J. Quant. Spectrosc. Radiat. Transfer. 25, 395 (1981)
[CrossRef]

Gullikson, E. M.

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Hamoniaux, G.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Hidaka, T.

T. Hidaka,"Extremely low-loss waveguide for XUV light" Opt. Commun. 44, 90 (1982)
[CrossRef]

Hooker, C. J.

M. J. Shaw, C. J. Hooker, and D. C. Wilson, "Measurement of the nonlinear refractive index of air and other gases at 248 nm,"Opt. Commun. 103, 153 (1993)
[CrossRef]

Hooker, S. M.

Ivanov, M. Y.

M. Levenstein, P. Balcou, M. Y. Ivanov, A. L�??Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)

Johannin-gille, A.

A. Bideau-mehu, Y. Guern, A. Abjean, and A. Johannin-gille, "Measurement of refractive indices of neon, argon, krypton and xenon in the 253.7 - 140.4 nm wavelength range. Dispersion relations and estimated oscillator strengths of the resonance lines," J. Quant. Spectrosc. Radiat. Transfer. 25, 395 (1981)
[CrossRef]

Kaplan, A. E.

P. L. Shkolnikov, A. Lago, and A. E. Kaplan, "Optimal quasi-phase-matching for high-order harmonic generation in gases and plasma," Phys.Rev. A 50, 4461 (1994)

Kapteyn, H. C.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, "Quasi-phasematching and quantum path control of high-harmonic generation using counterpropagating light," Nature Physics 3, 270 (2007)
[CrossRef]

E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, "Soft x-ray-driven femtosecond molecular dynamics," Science 317, 1374 (2007)
[CrossRef] [PubMed]

O. Cohen, X. Zhang, A. L. Lytel, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, " Grating-assisted phase matching in extreme nonlinear optics," Phys. Rev. Lett. 99, 053902 (2007)
[CrossRef] [PubMed]

O. Cohen, A. L. Litle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, "Optimizing quasi-phase matching using counterpropagating pulse trains," Opt. Lett. 32, 2975 (2007)
[CrossRef] [PubMed]

A. Paul, E. A. Gibson, X. Zhang, A. Lytle, T. Popmintchev, X. Zhou, M. M. Murnane, I. P. Christov, and H. C. Kapteyn, "Phase-matching techniques for coherent soft x-ray generation," IEEE J. Quantum Electron. 42, 14 (2006)
[CrossRef]

H. C. Kapteyn, M. M. Murnane, and I. P. Christov, "Coherent x-rays from lasers: Applied attosecond science" Phys. Today 58, 39 (2005)
[CrossRef]

Knight, J.

F. Benabid, J. Knight, G. Antonopoulos, and P. St.J. Russell, "Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber," Science 298, 399 (2002)
[CrossRef] [PubMed]

Knight, J. C.

P. J. Roberts, F. Couny, H. Sabert,B. J. Mangan, D. P. Williams, L. Farr, M.W. Mason, A. Tomlinson, T. A. Birks, and J. C. Knight, "Ultimate low loss of hollow-core photonic crystal fibres," Opt. Express 13, 236 (2005)
[CrossRef] [PubMed]

F. Benabid, G. Bouwmans, J. C. Knight, P. St.J. Russell, and F. Couny, "Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen," Phys. Rev. Lett. 93, 123903 (2004)
[CrossRef] [PubMed]

G. Bouwmans, F. Luan, J. C. Knight, P. St.J. Russell, L. Farr, B. J. Mangan, and H. Sabert, "Properties of a hollow-core photonic bandgap fiber at 850 nm wavelength," Opt. Express 11, 1613 (2003)
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight,T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537 (1999)
[CrossRef] [PubMed]

Kohl, I.

S. L. Voronov, I. Kohl, J. B. Madsen,  et al., "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001)
[CrossRef] [PubMed]

Konorov, S. O.

Krainov, V. P.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and of atomic ions in alternating electromagnetic field," Sov. Phys. JETP 64, 1191 (1986)

Krausz, F.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

L???Huillier, A.

M. Levenstein, P. Balcou, M. Y. Ivanov, A. L�??Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)

Lago, A.

P. L. Shkolnikov, A. Lago, and A. E. Kaplan, "Optimal quasi-phase-matching for high-order harmonic generation in gases and plasma," Phys.Rev. A 50, 4461 (1994)

Landreman, M.

Le Deroff, L.

P. Salieres, B. Carre, L. Le Deroff,  et al., "Feynman�??s path-integral approach for intense-laser-atom interactions," Science,  292, 902 (2001)
[CrossRef] [PubMed]

Levenstein, M.

M. Levenstein, P. Balcou, M. Y. Ivanov, A. L�??Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117 (1994)

Light, P. S.

Litle, A. L.

Luan, F.

Lytel, A. L.

O. Cohen, X. Zhang, A. L. Lytel, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, " Grating-assisted phase matching in extreme nonlinear optics," Phys. Rev. Lett. 99, 053902 (2007)
[CrossRef] [PubMed]

Lytle, A.

A. Paul, E. A. Gibson, X. Zhang, A. Lytle, T. Popmintchev, X. Zhou, M. M. Murnane, I. P. Christov, and H. C. Kapteyn, "Phase-matching techniques for coherent soft x-ray generation," IEEE J. Quantum Electron. 42, 14 (2006)
[CrossRef]

Lytle, A. L.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, "Quasi-phasematching and quantum path control of high-harmonic generation using counterpropagating light," Nature Physics 3, 270 (2007)
[CrossRef]

Madsen, J. B.

S. L. Voronov, I. Kohl, J. B. Madsen,  et al., "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001)
[CrossRef] [PubMed]

Mangan, B. J.

Marques, J. R.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Mason, M.W.

Matthieussent, G.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Murnane, M. M.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, "Quasi-phasematching and quantum path control of high-harmonic generation using counterpropagating light," Nature Physics 3, 270 (2007)
[CrossRef]

E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, "Soft x-ray-driven femtosecond molecular dynamics," Science 317, 1374 (2007)
[CrossRef] [PubMed]

O. Cohen, X. Zhang, A. L. Lytel, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, " Grating-assisted phase matching in extreme nonlinear optics," Phys. Rev. Lett. 99, 053902 (2007)
[CrossRef] [PubMed]

O. Cohen, A. L. Litle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, "Optimizing quasi-phase matching using counterpropagating pulse trains," Opt. Lett. 32, 2975 (2007)
[CrossRef] [PubMed]

A. Paul, E. A. Gibson, X. Zhang, A. Lytle, T. Popmintchev, X. Zhou, M. M. Murnane, I. P. Christov, and H. C. Kapteyn, "Phase-matching techniques for coherent soft x-ray generation," IEEE J. Quantum Electron. 42, 14 (2006)
[CrossRef]

H. C. Kapteyn, M. M. Murnane, and I. P. Christov, "Coherent x-rays from lasers: Applied attosecond science" Phys. Today 58, 39 (2005)
[CrossRef]

Ouzounov, D. G.

D. G. Ouzounov et al., "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702 (2003)
[CrossRef] [PubMed]

Paul, A.

A. Paul, E. A. Gibson, X. Zhang, A. Lytle, T. Popmintchev, X. Zhou, M. M. Murnane, I. P. Christov, and H. C. Kapteyn, "Phase-matching techniques for coherent soft x-ray generation," IEEE J. Quantum Electron. 42, 14 (2006)
[CrossRef]

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Popmintchev, T.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, "Quasi-phasematching and quantum path control of high-harmonic generation using counterpropagating light," Nature Physics 3, 270 (2007)
[CrossRef]

O. Cohen, X. Zhang, A. L. Lytel, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, " Grating-assisted phase matching in extreme nonlinear optics," Phys. Rev. Lett. 99, 053902 (2007)
[CrossRef] [PubMed]

A. Paul, E. A. Gibson, X. Zhang, A. Lytle, T. Popmintchev, X. Zhou, M. M. Murnane, I. P. Christov, and H. C. Kapteyn, "Phase-matching techniques for coherent soft x-ray generation," IEEE J. Quantum Electron. 42, 14 (2006)
[CrossRef]

Ranitovic, P.

E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, "Soft x-ray-driven femtosecond molecular dynamics," Science 317, 1374 (2007)
[CrossRef] [PubMed]

Rebibo, S.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Roberts, P. J.

P. J. Roberts, F. Couny, H. Sabert,B. J. Mangan, D. P. Williams, L. Farr, M.W. Mason, A. Tomlinson, T. A. Birks, and J. C. Knight, "Ultimate low loss of hollow-core photonic crystal fibres," Opt. Express 13, 236 (2005)
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight,T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537 (1999)
[CrossRef] [PubMed]

Rundquist, A.

A. Rundquist, C. G. Durfee, Z. Chang et al., "Phase-matched generation of coherent soft x-rays," Science 280, 1412 (1998)
[CrossRef] [PubMed]

Russell, P. St.J

Russell, P. St.J.

F. Benabid, G. Bouwmans, J. C. Knight, P. St.J. Russell, and F. Couny, "Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen," Phys. Rev. Lett. 93, 123903 (2004)
[CrossRef] [PubMed]

G. Bouwmans, F. Luan, J. C. Knight, P. St.J. Russell, L. Farr, B. J. Mangan, and H. Sabert, "Properties of a hollow-core photonic bandgap fiber at 850 nm wavelength," Opt. Express 11, 1613 (2003)
[CrossRef] [PubMed]

F. Benabid, J. Knight, G. Antonopoulos, and P. St.J. Russell, "Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber," Science 298, 399 (2002)
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight,T. A. Birks, P. St.J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537 (1999)
[CrossRef] [PubMed]

Sabert, H.

Salieres, P.

P. Salieres, B. Carre, L. Le Deroff,  et al., "Feynman�??s path-integral approach for intense-laser-atom interactions," Science,  292, 902 (2001)
[CrossRef] [PubMed]

Sandhu, A. S.

E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, "Soft x-ray-driven femtosecond molecular dynamics," Science 317, 1374 (2007)
[CrossRef] [PubMed]

Scrinzi, A.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

Serebryannikov, E. E.

E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov,"Broadband dynamic phase matching of high-order harmonic generation by a soliton pump field in a gas-filled holow photonic crystal fiber" Opt. Lett. 33, 977 (2008)
[CrossRef] [PubMed]

E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov, "Phase-matching solutions for high-order harmonic generation in hollow-core photonic crystal fibers," Phys. Rev. E 70, 066619 (2004)

Seres, E.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

Seres, J.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

Shaw, M. J.

M. J. Shaw, C. J. Hooker, and D. C. Wilson, "Measurement of the nonlinear refractive index of air and other gases at 248 nm,"Opt. Commun. 103, 153 (1993)
[CrossRef]

Shkolnikov, P. L.

P. L. Shkolnikov, A. Lago, and A. E. Kaplan, "Optimal quasi-phase-matching for high-order harmonic generation in gases and plasma," Phys.Rev. A 50, 4461 (1994)

Spielmann, C.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

Streli, C.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

Tempea, G.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

Tobey, R.

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Tomlinson, A.

Tong, X. M.

E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, "Soft x-ray-driven femtosecond molecular dynamics," Science 317, 1374 (2007)
[CrossRef] [PubMed]

Velikoroussov, T.

F. Dorchies, J. R. Marques, B. Cros, G. Matthieussent, C. Courtois, T. Velikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux,  et al., "Monomode guiding of 1016W/cm2 laser pulses over 100 rayleigh lengths in hollow capillary dielectric tubes," Phys. Rev. Lett. 82, 4655 (1999)
[CrossRef]

Verhoef, A. J.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

von der Linde, D.

E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov,"Broadband dynamic phase matching of high-order harmonic generation by a soliton pump field in a gas-filled holow photonic crystal fiber" Opt. Lett. 33, 977 (2008)
[CrossRef] [PubMed]

E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov, "Phase-matching solutions for high-order harmonic generation in hollow-core photonic crystal fibers," Phys. Rev. E 70, 066619 (2004)

Voronov, S. L.

S. L. Voronov, I. Kohl, J. B. Madsen,  et al., "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001)
[CrossRef] [PubMed]

Wagner, N.

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, and D. T. Attwood, "Coherent soft x-ray generation in the water window with quasi-phase matching," Science 302, 95 (2003)
[CrossRef] [PubMed]

Williams, D. P.

Wilson, D. C.

M. J. Shaw, C. J. Hooker, and D. C. Wilson, "Measurement of the nonlinear refractive index of air and other gases at 248 nm,"Opt. Commun. 103, 153 (1993)
[CrossRef]

Worbrauschek, P.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Worbrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, "Source of coherent kiloelectronvolt x-rays," Nature 433, 596 (2005)
[CrossRef] [PubMed]

Yakovlev, V.

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Figures (3)

Fig. 1.
Fig. 1.

(Color online) The ionization fraction produced in Ne by a laser pulse with duration τ=50fs, central wavelength λ0=800nm, and a peak intensity of I 0=1×1015W/cm2. Also shown is the electric field of the ionizing laser pulse.

Fig. 2.
Fig. 2.

The coherence length Lc for HHG in HC-PCF with effective mode index n eff=0.99. The driving laser pulse has a Gaussian profile with the following parameters: τ=50fs, I 0=1×1015W/cm2, λ0=800nm. The nonlinear medium is Ne at p=15mbar, n 2=6×10-21 cm2/W. The insert shows the coherence length calculated for different gas pressures.

Fig. 3.
Fig. 3.

The field ratio parameter for the quasi-phase-matching of HHG in Ne: (a) short trajectory; (b) long trajectory. The parameters of the driving laser are I 0=1×1015W/cm2, λ0=800µm, τ=50fs. The wavelength of the counter-propagating laser light is 1.6µm.

Equations (11)

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Δ k q k 0 [ n ( λ q ) n eff ] + q N e r e λ 0 ( 1 1 q 2 )
q k 0 N a [ δ ( λ 0 ) δ ( λ q ) ] N 0 q k 0 ( 1 η ) n 2 I
η = 1 exp [ t w ( t ) dt ]
w ( t ) = ω p C n * 2 ( 4 ω p ω t ) 2 n * 1 exp ( 4 ω p 3 ω t ) ,
ω p = I p h ¯ , ω t = e | E 1 ( t ) | 2 m e I p , n * = Z I ph I p , | C n * | 2 = 2 2 n * n * Γ ( n * + 1 ) Γ ( n * ) ,
Λ = 2 L c = 2 π Δ k .
Λ = λ 2 2
q qpm = 2 λ 0 λ 2 [ n ( λ q ) n eff ] .
Δ Φ HHG ( z ) = Δ k · z + A cos ( 4 π z λ 2 )
E HHG = 0 L E HHG 0 exp ( i Δ Φ HHG ) d z
E HHG ( z = L ) = E HHG 0 L J 1 ( A )

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