Abstract

We investigate phase-matched harmonic generation of ultrashort (picosecond) laser pulses, tuned in the vicinity of a five-photon resonance in argon, confined in a hollow-core optical waveguide. This combines resonance enhancements and tight spatial confinement to increase the conversion efficiency toward vacuum-ultraviolet radiation pulses. We demonstrate that appropriate choice of the gas pressure maintains optimal phase-matching conditions also in the presence of inevitable dynamic level shifts at high intensities. Moreover, we reveal the considerable contribution of higher-order transversal waveguide modes to the total conversion efficiency and investigate the role of cascading frequency conversion processes. Finally, we study additional signal enhancements by buffer gas admixtures. The experimental data are compared with numerical simulations, taking higher transversal waveguide modes and cascade frequency conversion into account, identifying also the potential of quasi-phase matching by polarization mode beating. Our investigations show that proper choice of experimental parameters enables significant resonance enhancements in the conversion efficiency of harmonic generation.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
    [Crossref]
  2. M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10  fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996).
    [Crossref]
  3. C. G. Durfee, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
    [Crossref]
  4. S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express 17, 13050–13058 (2009).
    [Crossref]
  5. G. Hilber, A. Lago, and R. Wallenstein, “Broadly tunable vacuum-ultraviolet/extreme-ultraviolet radiation generated by resonant third-order frequency conversion in krypton,” J. Opt. Soc. Am. B 4, 1753–1764 (1987).
    [Crossref]
  6. M. N. R. Ashfold and J. D. Prince, “Third harmonic generation in molecular gases,” Mol. Phys. 73, 297–315 (1991).
    [Crossref]
  7. S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
    [Crossref]
  8. A. L. Huillier, P. Balcou, and L. A. Lompre, “Coherence and resonance effects in high-order harmonic generation,” Phys. Rev. Lett. 68, 166–169 (1992).
    [Crossref]
  9. E. S. Toma, P. Antoine, A. de Bohan, and H. G. Muller, “Resonance-enhanced high-harmonic generation,” J. Phys. B 32, 5843–5852 (1999).
    [Crossref]
  10. C. de Morisson Faria, R. Kopold, W. Becker, and J. Rost, “Resonant enhancements of high-order harmonic generation,” Phys. Rev. A 65, 023404 (2002).
    [Crossref]
  11. R. Taïeb, V. Véniard, J. Wassaf, and A. Maquet, “Roles of resonances and recollisions in strong-field atomic phenomena. II. High-order harmonic generation,” Phys. Rev. A 68, 033403 (2003).
    [Crossref]
  12. R. A. Ganeev, “Generation of high-order harmonics of high-power lasers in plasmas produced under irradiation of solid target surfaces by a prepulse,” Phys. Usp. 52, 55–77 (2009).
    [Crossref]
  13. P. Ackermann, H. Münch, and T. Halfmann, “Resonantly-enhanced harmonic generation in Argon,” Opt. Express 20, 13824–13832 (2012).
    [Crossref]
  14. L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001).
    [Crossref]
  15. H. O. Seo, T. Arion, F. Roth, D. Ramm, C. Lupulescu, and W. Eberhardt, “Improving the efficiency of high harmonic generation (HHG) by Ne-admixing into a pure Ar gas medium,” Appl. Phys. B 122, 1–6 (2016).
    [Crossref]
  16. M. Hermans, J. Gottmann, and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro Nanoeng. 9, 126–131 (2014).
    [Crossref]
  17. A. Agrawal, “A comprehensive review on gas flow in microchannels,” Int. J. Micro-Nano Scale Transp. 2, 1–40 (2011).
    [Crossref]
  18. Z. Yang and S. V. Garimella, “Rarefied gas flow in microtubes at different inlet-outlet pressure ratios,” Phys. Fluids 21, 052005 (2009).
    [Crossref]
  19. R. K. Nubling, “Launch conditions and mode coupling in hollow-glass waveguides,” Opt. Eng. 37, 2454 (1998).
    [Crossref]
  20. C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999).
    [Crossref]
  21. B. W. Shore, “Simple atoms and fields,” in The Theory of Coherent Atomic Excitation. Vol. 1. Simple Atoms and Fields, 1st ed. (Wiley, 1990).
  22. T. R. O’Brian, J.-B. Kim, G. Lan, T. J. McIlrath, and T. B. Lucatorto, “Verification of the ponderomotive approximation for the ac Stark shift in Xe Rydberg levels,” Phys. Rev. A 49, R649–R652 (1994).
    [Crossref]
  23. M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 1–4 (2007).
    [Crossref]
  24. H. Ren, A. Nazarkin, J. Nold, and P. St. J. Russell, “Quasi-phase-matched high harmonic generation in hollow core photonic crystal fibers,” Opt. Express 16, 17052–17059 (2008).
    [Crossref]
  25. A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
    [Crossref]
  26. T. Diskin, O. Kfir, A. Fleischer, and O. Cohen, “Phase modulation in polarization beating quasi-phase-matching of high-order-harmonic generation,” Phys. Rev. A 92, 1–6 (2015).
    [Crossref]
  27. W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulæ,” Opt. Express 18, 6613–6620 (2010).
    [Crossref]
  28. M. Tarazkar, D. A. Romanov, and R. J. Levis, “Higher-order nonlinearity of refractive index: The case of argon,” J. Chem. Phys. 140, 214316 (2014).
    [Crossref]
  29. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express 18, 3011–3012 (2010).
    [Crossref]
  30. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17, 13429–13434 (2009).
    [Crossref]
  31. J. K. Wahlstrand, Y. H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers–Kronig relations,” Phys. Rev. Lett. 109, 1–5 (2012).
    [Crossref]
  32. D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher-order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114, 093901 (2015).
    [Crossref]
  33. B. Dromey, M. Zepf, M. Landreman, and S. M. Hooker, “Quasi-phasematching of harmonic generation via multimode beating in waveguides,” Opt. Express 15, 7894–7900 (2007).
    [Crossref]
  34. E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
    [Crossref]
  35. F. Tani, J. C. Travers, and P. St. J. Russell, “Multimode ultrafast nonlinear optics in optical waveguides: numerical modeling and experiments in kagomé photonic-crystal fiber,” J. Opt. Soc. Am. B 31, 311–320 (2014).
    [Crossref]
  36. A. W. Snyder and J. D. Love, “Illumination, tilts and offsets,” in Optical Waveguide Theory (Springer, 1984), pp. 420–441
  37. A. Bideau-Mehu, Y. Guern, R. Abjean, and A. Johannin-Gilles, “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. Transf. 25, 395–402 (1981).
    [Crossref]
  38. R. W. Boyd, “Quantum-mechanical theory of the nonlinear optical susceptibility,” in Nonlinear Optics (Elsevier, 2008), pp. 135–206.
  39. W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149–171 (1992).
    [Crossref]
  40. X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
    [Crossref]
  41. R. W. Boyd, “The nonlinear optical susceptibility,” in Nonlinear Optics (Elsevier, 2008), pp. 1–67.
  42. P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104, 103903 (2010).
    [Crossref]
  43. D. Wang, Y. Leng, and Z. Xu, “Measurement of nonlinear refractive index coefficient of inert gases with hollow-core fiber,” Appl. Phys. B 111, 447–452 (2013).
    [Crossref]
  44. S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, “Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths,” Opt. Lett. 40, 5794–5797 (2015).
    [Crossref]

2016 (2)

H. O. Seo, T. Arion, F. Roth, D. Ramm, C. Lupulescu, and W. Eberhardt, “Improving the efficiency of high harmonic generation (HHG) by Ne-admixing into a pure Ar gas medium,” Appl. Phys. B 122, 1–6 (2016).
[Crossref]

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

2015 (3)

S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, “Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths,” Opt. Lett. 40, 5794–5797 (2015).
[Crossref]

T. Diskin, O. Kfir, A. Fleischer, and O. Cohen, “Phase modulation in polarization beating quasi-phase-matching of high-order-harmonic generation,” Phys. Rev. A 92, 1–6 (2015).
[Crossref]

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher-order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114, 093901 (2015).
[Crossref]

2014 (4)

F. Tani, J. C. Travers, and P. St. J. Russell, “Multimode ultrafast nonlinear optics in optical waveguides: numerical modeling and experiments in kagomé photonic-crystal fiber,” J. Opt. Soc. Am. B 31, 311–320 (2014).
[Crossref]

M. Tarazkar, D. A. Romanov, and R. J. Levis, “Higher-order nonlinearity of refractive index: The case of argon,” J. Chem. Phys. 140, 214316 (2014).
[Crossref]

M. Hermans, J. Gottmann, and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro Nanoeng. 9, 126–131 (2014).
[Crossref]

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

2013 (2)

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

D. Wang, Y. Leng, and Z. Xu, “Measurement of nonlinear refractive index coefficient of inert gases with hollow-core fiber,” Appl. Phys. B 111, 447–452 (2013).
[Crossref]

2012 (2)

J. K. Wahlstrand, Y. H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers–Kronig relations,” Phys. Rev. Lett. 109, 1–5 (2012).
[Crossref]

P. Ackermann, H. Münch, and T. Halfmann, “Resonantly-enhanced harmonic generation in Argon,” Opt. Express 20, 13824–13832 (2012).
[Crossref]

2011 (1)

A. Agrawal, “A comprehensive review on gas flow in microchannels,” Int. J. Micro-Nano Scale Transp. 2, 1–40 (2011).
[Crossref]

2010 (3)

W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulæ,” Opt. Express 18, 6613–6620 (2010).
[Crossref]

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express 18, 3011–3012 (2010).
[Crossref]

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104, 103903 (2010).
[Crossref]

2009 (4)

V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” Opt. Express 17, 13429–13434 (2009).
[Crossref]

Z. Yang and S. V. Garimella, “Rarefied gas flow in microtubes at different inlet-outlet pressure ratios,” Phys. Fluids 21, 052005 (2009).
[Crossref]

R. A. Ganeev, “Generation of high-order harmonics of high-power lasers in plasmas produced under irradiation of solid target surfaces by a prepulse,” Phys. Usp. 52, 55–77 (2009).
[Crossref]

S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express 17, 13050–13058 (2009).
[Crossref]

2008 (1)

2007 (2)

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 1–4 (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, 7894–7900 (2007).
[Crossref]

2003 (2)

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

R. Taïeb, V. Véniard, J. Wassaf, and A. Maquet, “Roles of resonances and recollisions in strong-field atomic phenomena. II. High-order harmonic generation,” Phys. Rev. A 68, 033403 (2003).
[Crossref]

2002 (1)

C. de Morisson Faria, R. Kopold, W. Becker, and J. Rost, “Resonant enhancements of high-order harmonic generation,” Phys. Rev. A 65, 023404 (2002).
[Crossref]

2001 (1)

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001).
[Crossref]

1999 (2)

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999).
[Crossref]

E. S. Toma, P. Antoine, A. de Bohan, and H. G. Muller, “Resonance-enhanced high-harmonic generation,” J. Phys. B 32, 5843–5852 (1999).
[Crossref]

1998 (1)

R. K. Nubling, “Launch conditions and mode coupling in hollow-glass waveguides,” Opt. Eng. 37, 2454 (1998).
[Crossref]

1997 (1)

1996 (1)

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10  fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996).
[Crossref]

1994 (1)

T. R. O’Brian, J.-B. Kim, G. Lan, T. J. McIlrath, and T. B. Lucatorto, “Verification of the ponderomotive approximation for the ac Stark shift in Xe Rydberg levels,” Phys. Rev. A 49, R649–R652 (1994).
[Crossref]

1992 (2)

A. L. Huillier, P. Balcou, and L. A. Lompre, “Coherence and resonance effects in high-order harmonic generation,” Phys. Rev. Lett. 68, 166–169 (1992).
[Crossref]

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149–171 (1992).
[Crossref]

1991 (1)

M. N. R. Ashfold and J. D. Prince, “Third harmonic generation in molecular gases,” Mol. Phys. 73, 297–315 (1991).
[Crossref]

1987 (1)

1981 (1)

A. Bideau-Mehu, Y. Guern, R. Abjean, and A. Johannin-Gilles, “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. Transf. 25, 395–402 (1981).
[Crossref]

1964 (1)

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Abdolvand, A.

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Abjean, R.

A. Bideau-Mehu, Y. Guern, R. Abjean, and A. Johannin-Gilles, “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. Transf. 25, 395–402 (1981).
[Crossref]

Ackermann, P.

Agrawal, A.

A. Agrawal, “A comprehensive review on gas flow in microchannels,” Int. J. Micro-Nano Scale Transp. 2, 1–40 (2011).
[Crossref]

Antoine, P.

E. S. Toma, P. Antoine, A. de Bohan, and H. G. Muller, “Resonance-enhanced high-harmonic generation,” J. Phys. B 32, 5843–5852 (1999).
[Crossref]

Arion, T.

H. O. Seo, T. Arion, F. Roth, D. Ramm, C. Lupulescu, and W. Eberhardt, “Improving the efficiency of high harmonic generation (HHG) by Ne-admixing into a pure Ar gas medium,” Appl. Phys. B 122, 1–6 (2016).
[Crossref]

Ashfold, M. N. R.

M. N. R. Ashfold and J. D. Prince, “Third harmonic generation in molecular gases,” Mol. Phys. 73, 297–315 (1991).
[Crossref]

Backus, S.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001).
[Crossref]

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999).
[Crossref]

C. G. Durfee, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
[Crossref]

Balcou, P.

A. L. Huillier, P. Balcou, and L. A. Lompre, “Coherence and resonance effects in high-order harmonic generation,” Phys. Rev. Lett. 68, 166–169 (1992).
[Crossref]

Bartels, R.

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001).
[Crossref]

Bartels, R. A.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

Becker, W.

C. de Morisson Faria, R. Kopold, W. Becker, and J. Rost, “Resonant enhancements of high-order harmonic generation,” Phys. Rev. A 65, 023404 (2002).
[Crossref]

Béjot, P.

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104, 103903 (2010).
[Crossref]

Bideau-Mehu, A.

A. Bideau-Mehu, Y. Guern, R. Abjean, and A. Johannin-Gilles, “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. Transf. 25, 395–402 (1981).
[Crossref]

Boyd, R. W.

R. W. Boyd, “Quantum-mechanical theory of the nonlinear optical susceptibility,” in Nonlinear Optics (Elsevier, 2008), pp. 135–206.

R. W. Boyd, “The nonlinear optical susceptibility,” in Nonlinear Optics (Elsevier, 2008), pp. 1–67.

Brion, C. E.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149–171 (1992).
[Crossref]

Burton, G. R.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149–171 (1992).
[Crossref]

Chan, W. F.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149–171 (1992).
[Crossref]

Chang, W.

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Cheng, Y. H.

J. K. Wahlstrand, Y. H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers–Kronig relations,” Phys. Rev. Lett. 109, 1–5 (2012).
[Crossref]

Christov, I. P.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

Cohen, O.

T. Diskin, O. Kfir, A. Fleischer, and O. Cohen, “Phase modulation in polarization beating quasi-phase-matching of high-order-harmonic generation,” Phys. Rev. A 92, 1–6 (2015).
[Crossref]

Cooper, G.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149–171 (1992).
[Crossref]

de Bohan, A.

E. S. Toma, P. Antoine, A. de Bohan, and H. G. Muller, “Resonance-enhanced high-harmonic generation,” J. Phys. B 32, 5843–5852 (1999).
[Crossref]

de Morisson Faria, C.

C. de Morisson Faria, R. Kopold, W. Becker, and J. Rost, “Resonant enhancements of high-order harmonic generation,” Phys. Rev. A 65, 023404 (2002).
[Crossref]

De Silvestri, S.

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10  fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996).
[Crossref]

Diskin, T.

T. Diskin, O. Kfir, A. Fleischer, and O. Cohen, “Phase modulation in polarization beating quasi-phase-matching of high-order-harmonic generation,” Phys. Rev. A 92, 1–6 (2015).
[Crossref]

Dromey, B.

B. Dromey, M. Zepf, M. Landreman, and S. M. Hooker, “Quasi-phasematching of harmonic generation via multimode beating in waveguides,” Opt. Express 15, 7894–7900 (2007).
[Crossref]

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 1–4 (2007).
[Crossref]

Durfee, C. G.

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001).
[Crossref]

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999).
[Crossref]

C. G. Durfee, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
[Crossref]

Eberhardt, W.

H. O. Seo, T. Arion, F. Roth, D. Ramm, C. Lupulescu, and W. Eberhardt, “Improving the efficiency of high harmonic generation (HHG) by Ne-admixing into a pure Ar gas medium,” Appl. Phys. B 122, 1–6 (2016).
[Crossref]

Elouga Bom, L. B.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Ettoumi, W.

Faucher, O.

Fleischer, A.

T. Diskin, O. Kfir, A. Fleischer, and O. Cohen, “Phase modulation in polarization beating quasi-phase-matching of high-order-harmonic generation,” Phys. Rev. A 92, 1–6 (2015).
[Crossref]

Foster, P.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 1–4 (2007).
[Crossref]

Gaeta, A. L.

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher-order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114, 093901 (2015).
[Crossref]

Ganeev, R. A.

R. A. Ganeev, “Generation of high-order harmonics of high-power lasers in plasmas produced under irradiation of solid target surfaces by a prepulse,” Phys. Usp. 52, 55–77 (2009).
[Crossref]

Gao, X.

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher-order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114, 093901 (2015).
[Crossref]

Garimella, S. V.

Z. Yang and S. V. Garimella, “Rarefied gas flow in microtubes at different inlet-outlet pressure ratios,” Phys. Fluids 21, 052005 (2009).
[Crossref]

Gobert, O.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Gottmann, J.

M. Hermans, J. Gottmann, and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro Nanoeng. 9, 126–131 (2014).
[Crossref]

Green, H.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

Guern, Y.

A. Bideau-Mehu, Y. Guern, R. Abjean, and A. Johannin-Gilles, “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. Transf. 25, 395–402 (1981).
[Crossref]

Guo, X.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149–171 (1992).
[Crossref]

Haessler, S.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Halfmann, T.

Henin, S.

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104, 103903 (2010).
[Crossref]

Hergott, J.-F.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Hermans, M.

M. Hermans, J. Gottmann, and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro Nanoeng. 9, 126–131 (2014).
[Crossref]

Herne, C.

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999).
[Crossref]

Herrmann, J.

Hertz, E.

Hilber, G.

Hiraoka, N.

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

Hölzer, P.

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Hooker, S. M.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 1–4 (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, 7894–7900 (2007).
[Crossref]

Huillier, A. L.

A. L. Huillier, P. Balcou, and L. A. Lompre, “Coherence and resonance effects in high-order harmonic generation,” Phys. Rev. Lett. 68, 166–169 (1992).
[Crossref]

Husakou, A.

Im, S.-J.

Johannin-Gilles, A.

A. Bideau-Mehu, Y. Guern, R. Abjean, and A. Johannin-Gilles, “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. Transf. 25, 395–402 (1981).
[Crossref]

Kang, X.

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

Kapteyn, H. C.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001).
[Crossref]

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999).
[Crossref]

C. G. Durfee, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
[Crossref]

Kasparian, J.

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104, 103903 (2010).
[Crossref]

W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulæ,” Opt. Express 18, 6613–6620 (2010).
[Crossref]

Kfir, O.

T. Diskin, O. Kfir, A. Fleischer, and O. Cohen, “Phase modulation in polarization beating quasi-phase-matching of high-order-harmonic generation,” Phys. Rev. A 92, 1–6 (2015).
[Crossref]

Khokhlova, M.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Kim, J.-B.

T. R. O’Brian, J.-B. Kim, G. Lan, T. J. McIlrath, and T. B. Lucatorto, “Verification of the ponderomotive approximation for the ac Stark shift in Xe Rydberg levels,” Phys. Rev. A 49, R649–R652 (1994).
[Crossref]

Kopold, R.

C. de Morisson Faria, R. Kopold, W. Becker, and J. Rost, “Resonant enhancements of high-order harmonic generation,” Phys. Rev. A 65, 023404 (2002).
[Crossref]

Lago, A.

Lan, G.

T. R. O’Brian, J.-B. Kim, G. Lan, T. J. McIlrath, and T. B. Lucatorto, “Verification of the ponderomotive approximation for the ac Stark shift in Xe Rydberg levels,” Phys. Rev. A 49, R649–R652 (1994).
[Crossref]

Landreman, M.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 1–4 (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, 7894–7900 (2007).
[Crossref]

Lavorel, B.

Leng, Y.

D. Wang, Y. Leng, and Z. Xu, “Measurement of nonlinear refractive index coefficient of inert gases with hollow-core fiber,” Appl. Phys. B 111, 447–452 (2013).
[Crossref]

Lepetit, F.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Levis, R. J.

M. Tarazkar, D. A. Romanov, and R. J. Levis, “Higher-order nonlinearity of refractive index: The case of argon,” J. Chem. Phys. 140, 214316 (2014).
[Crossref]

Liu, Y.

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

Lompre, L. A.

A. L. Huillier, P. Balcou, and L. A. Lompre, “Coherence and resonance effects in high-order harmonic generation,” Phys. Rev. Lett. 68, 166–169 (1992).
[Crossref]

Loriot, V.

Love, J. D.

A. W. Snyder and J. D. Love, “Illumination, tilts and offsets,” in Optical Waveguide Theory (Springer, 1984), pp. 420–441

Lucatorto, T. B.

T. R. O’Brian, J.-B. Kim, G. Lan, T. J. McIlrath, and T. B. Lucatorto, “Verification of the ponderomotive approximation for the ac Stark shift in Xe Rydberg levels,” Phys. Rev. A 49, R649–R652 (1994).
[Crossref]

Lupulescu, C.

H. O. Seo, T. Arion, F. Roth, D. Ramm, C. Lupulescu, and W. Eberhardt, “Improving the efficiency of high harmonic generation (HHG) by Ne-admixing into a pure Ar gas medium,” Appl. Phys. B 122, 1–6 (2016).
[Crossref]

Maquet, A.

R. Taïeb, V. Véniard, J. Wassaf, and A. Maquet, “Roles of resonances and recollisions in strong-field atomic phenomena. II. High-order harmonic generation,” Phys. Rev. A 68, 033403 (2003).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

McIlrath, T. J.

T. R. O’Brian, J.-B. Kim, G. Lan, T. J. McIlrath, and T. B. Lucatorto, “Verification of the ponderomotive approximation for the ac Stark shift in Xe Rydberg levels,” Phys. Rev. A 49, R649–R652 (1994).
[Crossref]

Milchberg, H. M.

S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, “Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths,” Opt. Lett. 40, 5794–5797 (2015).
[Crossref]

J. K. Wahlstrand, Y. H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers–Kronig relations,” Phys. Rev. Lett. 109, 1–5 (2012).
[Crossref]

Misoguti, L.

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001).
[Crossref]

Muller, H. G.

E. S. Toma, P. Antoine, A. de Bohan, and H. G. Muller, “Resonance-enhanced high-harmonic generation,” J. Phys. B 32, 5843–5852 (1999).
[Crossref]

Münch, H.

Murnane, M. M.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001).
[Crossref]

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999).
[Crossref]

C. G. Durfee, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
[Crossref]

Nazarkin, A.

Ni, D.

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

Nisoli, M.

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10  fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996).
[Crossref]

Nold, J.

Nubling, R. K.

R. K. Nubling, “Launch conditions and mode coupling in hollow-glass waveguides,” Opt. Eng. 37, 2454 (1998).
[Crossref]

O’Brian, T. R.

T. R. O’Brian, J.-B. Kim, G. Lan, T. J. McIlrath, and T. B. Lucatorto, “Verification of the ponderomotive approximation for the ac Stark shift in Xe Rydberg levels,” Phys. Rev. A 49, R649–R652 (1994).
[Crossref]

Ozaki, T.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Paul, A.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

Perdrix, M.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Petit, Y.

Prince, J. D.

M. N. R. Ashfold and J. D. Prince, “Third harmonic generation in molecular gases,” Mol. Phys. 73, 297–315 (1991).
[Crossref]

Ramm, D.

H. O. Seo, T. Arion, F. Roth, D. Ramm, C. Lupulescu, and W. Eberhardt, “Improving the efficiency of high harmonic generation (HHG) by Ne-admixing into a pure Ar gas medium,” Appl. Phys. B 122, 1–6 (2016).
[Crossref]

Ren, H.

Riedel, F.

M. Hermans, J. Gottmann, and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro Nanoeng. 9, 126–131 (2014).
[Crossref]

Romanov, D. A.

M. Tarazkar, D. A. Romanov, and R. J. Levis, “Higher-order nonlinearity of refractive index: The case of argon,” J. Chem. Phys. 140, 214316 (2014).
[Crossref]

Rost, J.

C. de Morisson Faria, R. Kopold, W. Becker, and J. Rost, “Resonant enhancements of high-order harmonic generation,” Phys. Rev. A 65, 023404 (2002).
[Crossref]

Roth, F.

H. O. Seo, T. Arion, F. Roth, D. Ramm, C. Lupulescu, and W. Eberhardt, “Improving the efficiency of high harmonic generation (HHG) by Ne-admixing into a pure Ar gas medium,” Appl. Phys. B 122, 1–6 (2016).
[Crossref]

Rundquist, A. R.

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999).
[Crossref]

Russell, P. St. J.

Salières, P.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Schmeltzer, R. A.

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Seo, H. O.

H. O. Seo, T. Arion, F. Roth, D. Ramm, C. Lupulescu, and W. Eberhardt, “Improving the efficiency of high harmonic generation (HHG) by Ne-admixing into a pure Ar gas medium,” Appl. Phys. B 122, 1–6 (2016).
[Crossref]

Shim, B.

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher-order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114, 093901 (2015).
[Crossref]

Shore, B. W.

B. W. Shore, “Simple atoms and fields,” in The Theory of Coherent Atomic Excitation. Vol. 1. Simple Atoms and Fields, 1st ed. (Wiley, 1990).

Snyder, A. W.

A. W. Snyder and J. D. Love, “Illumination, tilts and offsets,” in Optical Waveguide Theory (Springer, 1984), pp. 420–441

Strelkov, V.

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Svelto, O.

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10  fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996).
[Crossref]

Taïeb, R.

R. Taïeb, V. Véniard, J. Wassaf, and A. Maquet, “Roles of resonances and recollisions in strong-field atomic phenomena. II. High-order harmonic generation,” Phys. Rev. A 68, 033403 (2003).
[Crossref]

Tani, F.

Tarazkar, M.

M. Tarazkar, D. A. Romanov, and R. J. Levis, “Higher-order nonlinearity of refractive index: The case of argon,” J. Chem. Phys. 140, 214316 (2014).
[Crossref]

Tobey, R.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

Toma, E. S.

E. S. Toma, P. Antoine, A. de Bohan, and H. G. Muller, “Resonance-enhanced high-harmonic generation,” J. Phys. B 32, 5843–5852 (1999).
[Crossref]

Travers, J. C.

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

F. Tani, J. C. Travers, and P. St. J. Russell, “Multimode ultrafast nonlinear optics in optical waveguides: numerical modeling and experiments in kagomé photonic-crystal fiber,” J. Opt. Soc. Am. B 31, 311–320 (2014).
[Crossref]

Tsuei, K.-D.

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

Véniard, V.

R. Taïeb, V. Véniard, J. Wassaf, and A. Maquet, “Roles of resonances and recollisions in strong-field atomic phenomena. II. High-order harmonic generation,” Phys. Rev. A 68, 033403 (2003).
[Crossref]

Vieillard, T.

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104, 103903 (2010).
[Crossref]

Wahlstrand, J. K.

S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, “Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths,” Opt. Lett. 40, 5794–5797 (2015).
[Crossref]

J. K. Wahlstrand, Y. H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers–Kronig relations,” Phys. Rev. Lett. 109, 1–5 (2012).
[Crossref]

Wallenstein, R.

Wang, D.

D. Wang, Y. Leng, and Z. Xu, “Measurement of nonlinear refractive index coefficient of inert gases with hollow-core fiber,” Appl. Phys. B 111, 447–452 (2013).
[Crossref]

Wassaf, J.

R. Taïeb, V. Véniard, J. Wassaf, and A. Maquet, “Roles of resonances and recollisions in strong-field atomic phenomena. II. High-order harmonic generation,” Phys. Rev. A 68, 033403 (2003).
[Crossref]

Weerawarne, D. L.

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher-order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114, 093901 (2015).
[Crossref]

Weiman, S.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

Wolf, J.-P.

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104, 103903 (2010).
[Crossref]

W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulæ,” Opt. Express 18, 6613–6620 (2010).
[Crossref]

Xu, L.

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

Xu, X.

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

Xu, Z.

D. Wang, Y. Leng, and Z. Xu, “Measurement of nonlinear refractive index coefficient of inert gases with hollow-core fiber,” Appl. Phys. B 111, 447–452 (2013).
[Crossref]

Yang, K.

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

Yang, Z.

Z. Yang and S. V. Garimella, “Rarefied gas flow in microtubes at different inlet-outlet pressure ratios,” Phys. Fluids 21, 052005 (2009).
[Crossref]

Zahedpour, S.

Zepf, M.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 1–4 (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, 7894–7900 (2007).
[Crossref]

Zhu, L.-F.

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

Appl. Phys. B (2)

H. O. Seo, T. Arion, F. Roth, D. Ramm, C. Lupulescu, and W. Eberhardt, “Improving the efficiency of high harmonic generation (HHG) by Ne-admixing into a pure Ar gas medium,” Appl. Phys. B 122, 1–6 (2016).
[Crossref]

D. Wang, Y. Leng, and Z. Xu, “Measurement of nonlinear refractive index coefficient of inert gases with hollow-core fiber,” Appl. Phys. B 111, 447–452 (2013).
[Crossref]

Appl. Phys. Lett. (1)

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10  fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Int. J. Micro-Nano Scale Transp. (1)

A. Agrawal, “A comprehensive review on gas flow in microchannels,” Int. J. Micro-Nano Scale Transp. 2, 1–40 (2011).
[Crossref]

J. Chem. Phys. (1)

M. Tarazkar, D. A. Romanov, and R. J. Levis, “Higher-order nonlinearity of refractive index: The case of argon,” J. Chem. Phys. 140, 214316 (2014).
[Crossref]

J. Laser Micro Nanoeng. (1)

M. Hermans, J. Gottmann, and F. Riedel, “Selective, laser-induced etching of fused silica at high scan-speeds using KOH,” J. Laser Micro Nanoeng. 9, 126–131 (2014).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. B (2)

E. S. Toma, P. Antoine, A. de Bohan, and H. G. Muller, “Resonance-enhanced high-harmonic generation,” J. Phys. B 32, 5843–5852 (1999).
[Crossref]

X. Xu, D. Ni, X. Kang, Y. Liu, L. Xu, K. Yang, N. Hiraoka, K.-D. Tsuei, and L.-F. Zhu, “The absolute optical oscillator strengths of the 3p5 4  s and 3p5 4s’ excitations of argon measured by the dipole (γ, γ) method,” J. Phys. B 49, 64010 (2016).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (1)

A. Bideau-Mehu, Y. Guern, R. Abjean, and A. Johannin-Gilles, “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. Transf. 25, 395–402 (1981).
[Crossref]

Mol. Phys. (1)

M. N. R. Ashfold and J. D. Prince, “Third harmonic generation in molecular gases,” Mol. Phys. 73, 297–315 (1991).
[Crossref]

Nat. Photonics (1)

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Nature (1)

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421, 51–54 (2003).
[Crossref]

New J. Phys. (1)

S. Haessler, V. Strelkov, L. B. Elouga Bom, M. Khokhlova, O. Gobert, J.-F. Hergott, F. Lepetit, M. Perdrix, T. Ozaki, and P. Salières, “Phase distortions of attosecond pulses produced by resonance-enhanced high harmonic generation,” New J. Phys. 15, 013051 (2013).
[Crossref]

Opt. Eng. (1)

R. K. Nubling, “Launch conditions and mode coupling in hollow-glass waveguides,” Opt. Eng. 37, 2454 (1998).
[Crossref]

Opt. Express (7)

Opt. Lett. (2)

Phys. Fluids (1)

Z. Yang and S. V. Garimella, “Rarefied gas flow in microtubes at different inlet-outlet pressure ratios,” Phys. Fluids 21, 052005 (2009).
[Crossref]

Phys. Rev. A (5)

T. Diskin, O. Kfir, A. Fleischer, and O. Cohen, “Phase modulation in polarization beating quasi-phase-matching of high-order-harmonic generation,” Phys. Rev. A 92, 1–6 (2015).
[Crossref]

T. R. O’Brian, J.-B. Kim, G. Lan, T. J. McIlrath, and T. B. Lucatorto, “Verification of the ponderomotive approximation for the ac Stark shift in Xe Rydberg levels,” Phys. Rev. A 49, R649–R652 (1994).
[Crossref]

C. de Morisson Faria, R. Kopold, W. Becker, and J. Rost, “Resonant enhancements of high-order harmonic generation,” Phys. Rev. A 65, 023404 (2002).
[Crossref]

R. Taïeb, V. Véniard, J. Wassaf, and A. Maquet, “Roles of resonances and recollisions in strong-field atomic phenomena. II. High-order harmonic generation,” Phys. Rev. A 68, 033403 (2003).
[Crossref]

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton, and xenon,” Phys. Rev. A 46, 149–171 (1992).
[Crossref]

Phys. Rev. Lett. (7)

P. Béjot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, “Higher-order Kerr terms allow ionization-free filamentation in gases,” Phys. Rev. Lett. 104, 103903 (2010).
[Crossref]

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 87, 013601 (2001).
[Crossref]

A. L. Huillier, P. Balcou, and L. A. Lompre, “Coherence and resonance effects in high-order harmonic generation,” Phys. Rev. Lett. 68, 166–169 (1992).
[Crossref]

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99, 1–4 (2007).
[Crossref]

C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83, 2187–2190 (1999).
[Crossref]

J. K. Wahlstrand, Y. H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers–Kronig relations,” Phys. Rev. Lett. 109, 1–5 (2012).
[Crossref]

D. L. Weerawarne, X. Gao, A. L. Gaeta, and B. Shim, “Higher-order nonlinearities revisited and their effect on harmonic generation,” Phys. Rev. Lett. 114, 093901 (2015).
[Crossref]

Phys. Usp. (1)

R. A. Ganeev, “Generation of high-order harmonics of high-power lasers in plasmas produced under irradiation of solid target surfaces by a prepulse,” Phys. Usp. 52, 55–77 (2009).
[Crossref]

Other (4)

A. W. Snyder and J. D. Love, “Illumination, tilts and offsets,” in Optical Waveguide Theory (Springer, 1984), pp. 420–441

B. W. Shore, “Simple atoms and fields,” in The Theory of Coherent Atomic Excitation. Vol. 1. Simple Atoms and Fields, 1st ed. (Wiley, 1990).

R. W. Boyd, “Quantum-mechanical theory of the nonlinear optical susceptibility,” in Nonlinear Optics (Elsevier, 2008), pp. 135–206.

R. W. Boyd, “The nonlinear optical susceptibility,” in Nonlinear Optics (Elsevier, 2008), pp. 1–67.

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Figures (10)

Fig. 1.
Fig. 1. Experimental setup for phase-matched harmonic generation inside the 55 mm long waveguide (blue tube) with two holes for gas inlet each 1 cm from the waveguide end. The filter F1 as well as the mirrors M1 and M2 are inserted to image the laser focus on a camera chip (CMOS1) and the waveguide end onto CMOS2. The inserted plot shows the calculated static core gas pressure versus the chamber pressure p c .
Fig. 2.
Fig. 2. Level scheme of argon with the relevant atomic levels and the tuning range (gray area) of the fundamental with respect to the 4 s energy levels.
Fig. 3.
Fig. 3. Beam intensity profiles of the fundamental laser pulses at λ f = 512    nm , before (top left) and after the waveguide (top right). Red color indicates large intensity, purple, low intensity. Distances are normalized to the average bore radius a = 52    μm . The white dashed lines show the capillary aperture. The lower plots show cuts in horizontal (gray, solid dots) and vertical (black, hollow squares) direction through the 2D intensity profiles. The colored lines show simulations with the mode amplitudes as deduced from the experimental data (blue line), or with the slightly corrected mode amplitudes (green line); compare Table 1. Due to the slight ellipticity of the waveguide, the horizontal intensity distribution at the exit is shallower, which enhances the peak intensity compared to the simulation without taking the ellipticity into account.
Fig. 4.
Fig. 4. Fifth-harmonic pulse energy versus chamber gas pressure at a fundamental pump peak intensity of 4    TW / cm 2 (blue dots), 7 TW / cm 2 (red dots), and 8.3    TW / cm 2 (green dots). The fundamental wavelength is λ F = 512    nm . Solid lines show numerical simulations averaged over 1 nm spectral width. Data and simulations are normalized to the peak harmonic yield at 7    TW / cm 2 .
Fig. 5.
Fig. 5. Fifth-harmonic pulse energy versus pressure. Pump intensity I P = 7    TW / cm 2 , wavelength λ P = 512    nm , pulse bandwidth Δ λ P = 1    nm . Experimental data (black dots), full simulation with contributions from all waveguide modes (thick, solid red line), simulation assuming the full fundamental pulse energy to be in the lowest mode E H 11 (thick, dashed blue line). Thin colored lines show the distribution of the generated fifth harmonic (H5) into the waveguide modes.
Fig. 6.
Fig. 6. Model simulation of fifth-harmonic generation in a waveguide at constant argon pressure. (a) Solid lines show the fifth harmonic generated in mode E H 11 at p c = 8    mbar (orange) and E H 12 (cyan) at p c = 6    mbar . Dashed and dotted lines indicate the phase differences between the fundamental and the harmonic in E H 11 (orange, dotted) and E H 12 (cyan, dashed); (b) fundamental intensity variation by mode beating in the waveguide; (c) phase differences and fifth-harmonic power, generated in a QPM scheme for the E H 11 mode at p c = 16    mbar or the E H 12 mode at p c = 13    mb a r .
Fig. 7.
Fig. 7. Fifth-harmonic pulse energy (experimental data) versus pump fundamental wavelength and argon pressure. The pump intensity was I P = 7    TW / cm 2 . We note that the pulse duration from our laser system varied slightly with the wavelength. To calibrate for the effect, at each wavelength we deduced the pulse duration from a FROG trace. We compensate the variation in the pulse duration by slightly increased pulse power to acquire data at constant pump peak intensity, and normalize with respect to pulse area.
Fig. 8.
Fig. 8. Fifth-harmonic pulse energy (numerical simulation) versus pump fundamental wavelength and argon pressure. We assumed a fundamental peak intensity of 7.1    TW / cm 2 and averaged over a laser bandwidth of 1 nm. Black dots with error bars depict the pressures of maximal efficiency from Fig. 7. The red dashed line represents the pressure of maximal efficiency, neglecting the resonance shift. We note that due to the pronounced Stark shifts, at peak intensity the dispersion for the fifth harmonic increases. This results in a lower group velocity at the peak intensity and thus modifies the pulse shape of the fifth harmonic. The shaded, parabolic area indicates the region where parts of the fifth-harmonic pulse experience a group delay larger than half the pump pulse duration.
Fig. 9.
Fig. 9. (a) Dependence of χ ( 5 ) versus wavelength for a low, constant argon pressure (gray dashed line). Dependence of χ ( 5 ) versus wavelength at the corresponding QPM argon pressure (green solid line). (b) Fifth-harmonic signal maxima versus wavelength (black dots); experimental data taken from Fig. 8. The green thick line shows the results of the full numerical simulation, taken from Fig. 9. Thin lines show numerical simulations of normalized dependencies for negative values of χ ( 5 ) (dashed) as well as for a lower positive value (orange). The blue line shows the results without the contribution of cascading conversion processes, i.e., setting χ ( 3 ) = 0 . The shaded area indicates the wavelength region, where pulse propagation effects may occur, which are neglected in the simulations (compare Fig. 8).
Fig. 10.
Fig. 10. Fifth-harmonic pulse energy versus pressure in a mixture of argon with neon (blue data points), compared to pure argon (orange data points). Data taken at a fundamental peak intensity of 7.5    TW / cm 2 and a wavelength of 512 nm.

Tables (2)

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Table 1. Modal Power Decomposition of the Input Pump Beam Profile a

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Table 2. Nonlinear Susceptibilities Applied in Our Numerical Simulation

Equations (19)

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E j , m ( r , z , t ) = E ¯ j , m ( r , z ) · e i ( ω j    t ) · e T ,
E ¯ j , m = E j , m · J 0 ( u 1 m r a ) exp ( i 0 z γ j , m ( z ) d z ) .
γ j , m = k j 1 2 k j ( u 1 m a ) 2 i 1 2 k j ( u 1 m a ) 2 · 2 ν 1 , j k j a ,
ν 1 , j = ( n ˜ e 2 ( ω j ) + 1 ) / ( 2 n ˜ e 2 ( ω j ) 1 ) ,
Δ γ m m ( j ) = j γ 1 , m γ j , m ,
I in ( r , z = 0 ) = π Z | m E 1 , m ( r , 0 ) + c.c. | 2 ,
E 1 , m = 0 a J 0 ( u 1 m r · a 1 ) · F s ( r ) r d r 0 a J 0 2 ( u 1 m r · a 1 ) r d r .
E j , m z = i 2 γ j , m [ ( k j , m 2 γ j , m 2 ) E j , m + ω j 2 c 0 2 ε 0 P ˜ j , m N L ( z ) · e i 0 z γ j , m ( z ) d z ] ,
α b ( ω ) = e 2 2 ϵ 0 m n f n a ω n a [ ( ω n a ω i Γ 2 ) 1 + ( ω n a + ω + i Γ 2 ) 1 ] ,
Im [ α c ] = σ c c 0 ω ,
α ( ω ) = α b ( ω ) + α c ( ω ) ,
n ( ω , p ) = ( 1 + N ( p ) · α ( ω ) ) 1 / 2 ,
E ¯ ¯ j = m E ¯ j , m ( r , z )
P ¯ ¯ 1 N L ( r , z ) = ϵ 0 ( 3 χ ( ω , ω , ω ) ( 3 ) E ¯ ¯ 1 | E ¯ ¯ 1 | 2 + 10 χ ( ω , ω , ω , ω , ω ) ( 5 ) E ¯ ¯ 1 | E ¯ ¯ 1 | 4 )
P ¯ ¯ 3 N L ( r , z ) = ϵ 0 ( 3 χ ( ω , ω , ω ) ( 3 ) E ¯ ¯ 1 3 + 5 χ ( ω , ω , ω , ω , ω ) ( 5 ) E ¯ ¯ 1 3 | E ¯ ¯ 1 | 2 ) ,
P ¯ ¯ 5 N L ( r , z ) = ϵ 0 ( 3 χ ( ω , ω , ω ) ( 3 ) E ¯ ¯ 1 2 E ¯ ¯ 3 + χ ( ω , ω , ω , ω , ω ) ( 5 ) E ¯ ¯ 1 5 ) ,
P ˜ j , m N L ( z ) = 0 a J 0 ( u 1 m r · a 1 ) · P ¯ ¯ j N L ( r , z ) r d r 0 a J 0 2 ( u 1 m r · a 1 ) r d r ,
χ ( ω 1 , ω 2 , ω 3 , ω 4 ) ( 3 ) = A p ( 3 ) · N ( p ) · i = 1 4 α ( ω i ) ,
χ ( ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , ω 6 ) ( 5 ) = A p ( 5 ) · N ( p ) · i = 1 6 a ( ω i )

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