Abstract

High harmonic production can be dramatically increased by utilizing an interaction region much longer than a coherence length. Counter-propagating light pulses can be used to disrupt the out-of-phase harmonic emission from selected zones in the focus so that the remaining emission builds constructively. Counter-propagating light creates a standing field modulation repeating over a half laser wavelength in which phase cancellations for harmonic emission occur. A simple power-law model is used to demonstrate how such pulses can be designed to counteract geometrical phase mismatches and improve emission for individual harmonics by more than two orders of magnitude.

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References

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  1. LHuillier and Ph. Balcou, "High-Order harmonic Generation in Rare Gases with a 1-ps 1053nm Laser," Phys. Rev. Lett. 70, 774 (1993).
    [CrossRef]
  2. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Enhanced High-Harmonic Generation Using 25 Femtosecond Laser Pulses, Phys. Rev. Lett. 76, 752 (1996).
    [CrossRef] [PubMed]
  3. Ditmire, K. Kulander, J. K. Crane, H. Nguyen, M. D. Perry, "Calculation and Measurement of High-Order Harmonic Energy Yields in Helium," J. Opt. Soc. of Am. B 13, 406.
  4. Peatross, J. Zhou, I. Christov, A. Rundquist, M. M. Murnane, H. C. Kapteyn, High-Order Harmonic Generation with a 25 Femtosecond Laser Pulse, in Proceedings of the NATO Advanced Research Workshop on Super Intense Laser-Atom Physics IV (Moscow, Russia 1995) p. 455.
  5. V. T. Platonenko, V. V. Strelkov, G. Ferrante, V. Miceli, E. Fiordilino, "Control of the Spectral Width and Pulse Duration of a Single High-Order Harmonic," Laser Phys. 6, p. 1164-1167 (1996).
  6. Fiordilino and V. Miceli, "Laser Pulse Shape Effects in Harmonic Generation from a Two-Level Atom," J. Mod. Opt. 41, 1415-1426 (1994).
    [CrossRef]
  7. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, "Quantum Control of Wave Packet Evolution with Tailored Femtosecond Pulses," Phys. Rev. Lett. 74, 3360-3363 (1995).
    [CrossRef] [PubMed]
  8. LHuillier, K. J. Schafer, and K. C. Kulander, "Theoretical Aspects of Intense Field Harmonic Generation," J. Phys. B: At. Mol. Opt. Phys. 24, 3315 (1991).
    [CrossRef]
  9. L'Huillier, Ph. Balcou, S. Candel, K. J. Schafer, and K. C. Kulander, "Calculations of High-Order Harmonic-Generation Processes in Xenon at 1064 nm," Phys. Rev. A 46, 2778-2790 (1992).
    [CrossRef]
  10. Ph. Balcou and A. L'Huillier, "Phase-Matching Effects in Strong-Field Harmonic Generation," Phys. Rev. A 47, 1447-1459 (1993).
    [CrossRef] [PubMed]
  11. Peatross and D. D. Meyerhofer, "The Angular Distribution of High-Order Harmonics Emitted from Rare Gases at Low Density," Phys. Rev. A 51, R906 (1995).
    [CrossRef] [PubMed]
  12. Peatross and D. D. Meyerhofer, "Intensity-Dependent Phase Effects in High-Order Harmonic Generation," Phys. Rev. A 52, 3976-3987 (1995).
    [CrossRef] [PubMed]
  13. Peatross, M. V. Fedorov, K. C. Kulander, "Intensity-Dependent Phase-Matching Effects in Harmonic Generation," J. Opt. Soc. Am. B 12,863 (1995).
    [CrossRef]
  14. Salieres, A. L'Huillier, and M. Lewenstein, "Coherence Control of High-Order Harmonics," Phys. Rev. Lett. 74, 3776 (1995).
    [CrossRef] [PubMed]
  15. Wahlstrom, J. Larsson, A. Persson, T. Starczewski, and S. Svanberg, "High-Order Harmonic Generation in Rare Gases with an Intense Short-Pulse Laser," Phys. Rev. A. 48, 4709 (1993).
    [CrossRef] [PubMed]
  16. Altucci, T. Starczewski, E. Mevel, C.-G. Wahlstrom, B. Carre, A. L`Huillier, "Influence of Atomic Density in High-Order Harmonic Generation," J. Opt. Soc. Am. B 13, 148 (1996).
    [CrossRef]
  17. Lynga, A. L'Huillier, C.-G. Wahlstrom, "High-Order Harmonic Generation in Molecular Gases," J. Phys. B 29, 3293 (1996).
    [CrossRef]
  18. T. Ditmire, J. W. G. Tisch, D. J. Fraser, J. P. Marangos, N. Hay, M. H. R. Hutchinson, T. Donnelly, R. W. Falcone, M. D. Perry, "High-Order Harmonic Generation in Large Molecules and Atomic Clusters," in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, DC, 1996) p. 544.
  19. Wahlstrom, S. Borgstrom, J. Larsson, and S.-G. Pettersson, "High-Order Harmonic Generation in Laser-Produced Ions Using a Near-Infrared Laser," Phys. Rev. A 51, 585 (1995).
    [CrossRef] [PubMed]
  20. Norreys, M. Zepf, S. Moustaizis, A. P. Fews, J. Zhang, P. Lee, M. Bakarezos, C. N. Danson, A. Dyson, P. Gibbon, P. Loukakos, D. Neely, F. N. Walsh, J. S. Wark, A. E. Dangor, "Efficient Extreme UV Harmonics Generated from Picosecond Laser Pulse Interactions with Solid Targets," Phys. Rev. Lett. 76, 1832 (1996).
    [CrossRef] [PubMed]
  21. H. M. Milchberg, C. G. Durfee, T. J. McIlrath, High-Order Frequency Conversion in the Plasma Waveguide, Phys. Rev. Lett. 75, 2494 (1995).
    [CrossRef] [PubMed]
  22. Shkolnikov, A. E. Kaplan, and A. Lago, "Phase-Matching Optimization of Large-Scale Nonlinear Frequency Upconversion in Neutral and Ionized Gases," J. Opt. Soc. Am. B 13, 412 (1996).
    [CrossRef]
  23. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  24. Birulin, V. T. Platonenko, and V. V. Strelkov, "High-Harmonic Generation in Interfering Waves," JETP 83, 33 (1996).
  25. Peatross, J. Chaloupka, and D. D. Meyerhofer, "High-Order Harmonic Generation with an Annular Laser Beam," Opt. Lett. 19, 942 (1994).
    [CrossRef] [PubMed]

Other (25)

LHuillier and Ph. Balcou, "High-Order harmonic Generation in Rare Gases with a 1-ps 1053nm Laser," Phys. Rev. Lett. 70, 774 (1993).
[CrossRef]

Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Enhanced High-Harmonic Generation Using 25 Femtosecond Laser Pulses, Phys. Rev. Lett. 76, 752 (1996).
[CrossRef] [PubMed]

Ditmire, K. Kulander, J. K. Crane, H. Nguyen, M. D. Perry, "Calculation and Measurement of High-Order Harmonic Energy Yields in Helium," J. Opt. Soc. of Am. B 13, 406.

Peatross, J. Zhou, I. Christov, A. Rundquist, M. M. Murnane, H. C. Kapteyn, High-Order Harmonic Generation with a 25 Femtosecond Laser Pulse, in Proceedings of the NATO Advanced Research Workshop on Super Intense Laser-Atom Physics IV (Moscow, Russia 1995) p. 455.

V. T. Platonenko, V. V. Strelkov, G. Ferrante, V. Miceli, E. Fiordilino, "Control of the Spectral Width and Pulse Duration of a Single High-Order Harmonic," Laser Phys. 6, p. 1164-1167 (1996).

Fiordilino and V. Miceli, "Laser Pulse Shape Effects in Harmonic Generation from a Two-Level Atom," J. Mod. Opt. 41, 1415-1426 (1994).
[CrossRef]

Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, "Quantum Control of Wave Packet Evolution with Tailored Femtosecond Pulses," Phys. Rev. Lett. 74, 3360-3363 (1995).
[CrossRef] [PubMed]

LHuillier, K. J. Schafer, and K. C. Kulander, "Theoretical Aspects of Intense Field Harmonic Generation," J. Phys. B: At. Mol. Opt. Phys. 24, 3315 (1991).
[CrossRef]

L'Huillier, Ph. Balcou, S. Candel, K. J. Schafer, and K. C. Kulander, "Calculations of High-Order Harmonic-Generation Processes in Xenon at 1064 nm," Phys. Rev. A 46, 2778-2790 (1992).
[CrossRef]

Ph. Balcou and A. L'Huillier, "Phase-Matching Effects in Strong-Field Harmonic Generation," Phys. Rev. A 47, 1447-1459 (1993).
[CrossRef] [PubMed]

Peatross and D. D. Meyerhofer, "The Angular Distribution of High-Order Harmonics Emitted from Rare Gases at Low Density," Phys. Rev. A 51, R906 (1995).
[CrossRef] [PubMed]

Peatross and D. D. Meyerhofer, "Intensity-Dependent Phase Effects in High-Order Harmonic Generation," Phys. Rev. A 52, 3976-3987 (1995).
[CrossRef] [PubMed]

Peatross, M. V. Fedorov, K. C. Kulander, "Intensity-Dependent Phase-Matching Effects in Harmonic Generation," J. Opt. Soc. Am. B 12,863 (1995).
[CrossRef]

Salieres, A. L'Huillier, and M. Lewenstein, "Coherence Control of High-Order Harmonics," Phys. Rev. Lett. 74, 3776 (1995).
[CrossRef] [PubMed]

Wahlstrom, J. Larsson, A. Persson, T. Starczewski, and S. Svanberg, "High-Order Harmonic Generation in Rare Gases with an Intense Short-Pulse Laser," Phys. Rev. A. 48, 4709 (1993).
[CrossRef] [PubMed]

Altucci, T. Starczewski, E. Mevel, C.-G. Wahlstrom, B. Carre, A. L`Huillier, "Influence of Atomic Density in High-Order Harmonic Generation," J. Opt. Soc. Am. B 13, 148 (1996).
[CrossRef]

Lynga, A. L'Huillier, C.-G. Wahlstrom, "High-Order Harmonic Generation in Molecular Gases," J. Phys. B 29, 3293 (1996).
[CrossRef]

T. Ditmire, J. W. G. Tisch, D. J. Fraser, J. P. Marangos, N. Hay, M. H. R. Hutchinson, T. Donnelly, R. W. Falcone, M. D. Perry, "High-Order Harmonic Generation in Large Molecules and Atomic Clusters," in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, DC, 1996) p. 544.

Wahlstrom, S. Borgstrom, J. Larsson, and S.-G. Pettersson, "High-Order Harmonic Generation in Laser-Produced Ions Using a Near-Infrared Laser," Phys. Rev. A 51, 585 (1995).
[CrossRef] [PubMed]

Norreys, M. Zepf, S. Moustaizis, A. P. Fews, J. Zhang, P. Lee, M. Bakarezos, C. N. Danson, A. Dyson, P. Gibbon, P. Loukakos, D. Neely, F. N. Walsh, J. S. Wark, A. E. Dangor, "Efficient Extreme UV Harmonics Generated from Picosecond Laser Pulse Interactions with Solid Targets," Phys. Rev. Lett. 76, 1832 (1996).
[CrossRef] [PubMed]

H. M. Milchberg, C. G. Durfee, T. J. McIlrath, High-Order Frequency Conversion in the Plasma Waveguide, Phys. Rev. Lett. 75, 2494 (1995).
[CrossRef] [PubMed]

Shkolnikov, A. E. Kaplan, and A. Lago, "Phase-Matching Optimization of Large-Scale Nonlinear Frequency Upconversion in Neutral and Ionized Gases," J. Opt. Soc. Am. B 13, 412 (1996).
[CrossRef]

Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Birulin, V. T. Platonenko, and V. V. Strelkov, "High-Harmonic Generation in Interfering Waves," JETP 83, 33 (1996).

Peatross, J. Chaloupka, and D. D. Meyerhofer, "High-Order Harmonic Generation with an Annular Laser Beam," Opt. Lett. 19, 942 (1994).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1.
Fig. 1.

A mirror with a hole is used to extract high-order harmonics generated in counter-propagating laser beams.

Fig. 2.
Fig. 2.

Standing intensity and phase variations resulting when a plane wave is met by a counter-propagating plane wave one hundredth as intense.

Fig. 3.
Fig. 3.

The effective emission from a microscopic phase-matching interval for the 11th, 31st, and 51st harmonics as a function of relative counter-propagating field strength. The calculation assumes that harmonic emission follows a power law with p=5.

Fig. 4.
Fig. 4.

(a) Amplitude and phase of the 51st harmonic field arriving simultaneously in the center of a distant screen as a function of emission position (normalized by the Rayleigh range). The graph shows the field contributions associated with a particular point on the pulse temporal envelope as it travels through the focus. The net field for that instant is found by summing the contributions as in Eq. (8). (b) Real part of the field

Fig. 5.
Fig. 5.

(a) The relative intensity of counter-propagating pulses tailored to improve emission for the 51st harmonic. (b) The real part of the 51st harmonic field components at the center of a distant screen in the presence of counter-propagating pulses. The graph shows the field contributions associated with a particular point on the pulse temporal envelope as it travels through the focus (position normalized to the Rayleigh range). The net field for that instant is found by summing the contributions as in Eq. (8). Compare with Fig. 4(b).

Fig. 6.
Fig. 6.

(a) Relative intensity of the 51st harmonic at a distant screen as a function of angle, in the presence of the counter-propagating field (blue), in the absence of the counter-propagating field (green), in the absence of the counter-propagating field but with the gas restricted to a coherence length at the focal center (red). The laser profile is depicted for angular comparison (light blue). (b) The same figure shown with a linear scale.

Equations (10)

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E 1 e i ( kz ωt ) + E 2 e i ( kz + ωt ) = E t ( z ) e i ( kz ωt + ϕ ( z ) ) .
E t ( z ) = E 1 1 + ( E 2 E 1 ) 2 + 2 E 2 E 1 cos 2 kz ,
ϕ ( z ) = tan 1 E 2 E 1 sin 2 kz 1 + E 2 E 1 cos 2 kz .
V = I max I min I max + I min = 2 E 2 E 1 1 ( E 2 + E 1 ) 2 .
E q z t ~ E t p ( z ) e qi ( kz ωt + ϕ ( z ) ) .
λ 2 0 E q ( z , t + z c ) dz ~ e iqωt λ 2 0 E t p ( z ) e qiϕ ( z ) dz .
ξ = 2 λE 1 p λ 2 0 E t p ( z ) e qiϕ ( z ) dz .
E q θ T ~ E 1 p ( T ) dz N ( z ) ξ T z ηe q 2 η 2 p ( 2 f # θ ) 2 ( 1 + z 2 z o 2 ) ( p 1 ) 2 e iq tan 1 z z o + i tan 1 qz pz o + iq ( 2 f # θ ) 2 [ 1 q 2 η 2 p 2 ] z z o
z o z o = 1 + z 2 z o 2 .
Power ~ E q θ T 2 θdθ .

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