Abstract

We study semi-analytically and numerically the photon yield of high harmonic generation (HHG) on the level of the single atom response under ideal conditions: no initial depletion of the ground state prior to the main peak of the pump pulse. We show that the yield decreases exponentially as function of the cutoff energy starting at about 0.5keV in the case of a Ti:sapphire source and a helium target. We show that the yield in helium beyond the 1keV energy range can be increased by orders of magnitude when long wavelength driver sources in the range from 1.5µm to 3µm are used. This finding leads to the conclusion that significant HHG beyond 1keV is possible through long wavelength driver pulses.

© 2005 Optical Society of America

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References

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  1. Ch. Spielmann, N. H. Burnett, S. Santania, R. Koppitsch, M. Schn¨urer, C. Kan, M. Lenzner, P.Wobrauschek, and F. Krausz, �??Generation of Coherent X-rays in the Water Window Using 5-Femtosecond Laser Pulses,�?? Science 278, 661-664 (1997).
    [CrossRef]
  2. Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, �??Generation of Coherent Soft X Rays at 2.7 nm Using High Harmonics,�?? Phys. Rev. Lett. 79, 2967-2970 (1997).
    [CrossRef]
  3. P. B. Corkum, �??Plasma Perspective on Strong-Field Multiphoton Ionization,�?? Phys. Rev. Lett. 71, 1994-1997 (1993).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. T. Brabec and F. Krausz, �??Intense few-cycle laser fields: Frontiers of nonlinear optics,�?? Rev. Mod. Phys. 72, 545-591 (2000).
    [CrossRef]
  6. Ivan P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, �??Nonadiabatic Effects in High-Harmonic Generation with Ultrashort Pulses,�?? Phys. Rev. Lett. 77, 1743-1746 (1996).
    [CrossRef] [PubMed]
  7. M. Nisoli, S. De Silvestri, and O. Svelto, �??Generation of high energy 10 fs pulses by a new pulse compression technique,�?? Applied Phys. Lett. 68, 2793-2795 (1996).
    [CrossRef]
  8. E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, Science 302, �??Coherent Soft X-ray Generation in the Water Window with Quasi-Phase Matching,�?? 5642-5645 (2003).
    [CrossRef]
  9. G. Tempea, M. Geissler, M. Schnürer, and T. Brabec, �??Self-Phase-Matched High Harmonic Generation,�?? Phys. Rev. Lett. 84, 4329-4332 (2000).
    [CrossRef] [PubMed]
  10. E. Seres, J. Seres, F. Krausz, and Ch. Spielmann, �??Generation of Coherent Soft-X-Ray Radiation Extending Far Beyond the Titanium L Edge,�?? Phys. Rev. Lett 92, 163002 (2004).
    [CrossRef] [PubMed]
  11. J. Seres, E. Seres, A. J. Verhoff, G. Tempea, C. Streli, P. Wobrauschek, Y. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, �??Source of coherent kiloelectronvolt X-rays,�?? Nature 433, 596-596 (2005).
    [CrossRef] [PubMed]
  12. E. A. Gibson, A. Paul, N. Wagner, R. Tobey, S. Backus, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, �??High-Order Harmonic Generation up to 250 eV from Highly Ionized Argon,�?? Phys. Rev. Lett. 92, 033001 (2004).
    [CrossRef] [PubMed]
  13. E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, �??Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime,�?? Phys. Rev. A 61, 063801 (2000).
    [CrossRef]
  14. A. Scrinzi, M. Geissler, and T. Brabec, �??Ionization Above the Coulomb Barrier,�?? Phys. Rev. Lett 83, 706 (1999).
    [CrossRef]
  15. M. Yu. Ivanov, T. Brabec, and N. Burnett, �??Coulomb corrections and polarization effects in high-intensity high-harmonic emission,�?? Phys. Rev. A. 54, 742 (1996).
    [CrossRef] [PubMed]
  16. M. V. Ivanov, �??Complex rotation in two-dimensional mesh calculations for quantum systems in uniform electric fields,�?? J. Phys. B 34, 2447-2473 (2001).
    [CrossRef]
  17. B. Sheehy, J. D. D. Martin, L. F. DiMauro, P. Agostini, K. J. Schafer, M. B. Gaarde, and K. C. Kulander, �??High Harmonic Generation at Long Wavelengths,�?? Phys. Rev. Lett. 83, 5270-5273 (1999).
    [CrossRef]
  18. B. Shan and Z. Chang, �??Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,�?? Phys. Rev. A 65, 011804(R) (2002).

Applied Phys. Lett. (1)

M. Nisoli, S. De Silvestri, and O. Svelto, �??Generation of high energy 10 fs pulses by a new pulse compression technique,�?? Applied Phys. Lett. 68, 2793-2795 (1996).
[CrossRef]

J. Phys. B (1)

M. V. Ivanov, �??Complex rotation in two-dimensional mesh calculations for quantum systems in uniform electric fields,�?? J. Phys. B 34, 2447-2473 (2001).
[CrossRef]

Nature (1)

J. Seres, E. Seres, A. J. Verhoff, G. Tempea, C. Streli, P. Wobrauschek, Y. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, �??Source of coherent kiloelectronvolt X-rays,�?? Nature 433, 596-596 (2005).
[CrossRef] [PubMed]

Phys. Rev. (1)

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L�??Huillier, and P. B. Corkum, �??Theory of High Harmonic Generation by low-frequency laser fields,�?? Phys. Rev. A. 49, 2117-2132 (1994).
[CrossRef] [PubMed]

Phys. Rev. A (2)

E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, �??Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime,�?? Phys. Rev. A 61, 063801 (2000).
[CrossRef]

B. Shan and Z. Chang, �??Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,�?? Phys. Rev. A 65, 011804(R) (2002).

Phys. Rev. A. (1)

M. Yu. Ivanov, T. Brabec, and N. Burnett, �??Coulomb corrections and polarization effects in high-intensity high-harmonic emission,�?? Phys. Rev. A. 54, 742 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett (2)

A. Scrinzi, M. Geissler, and T. Brabec, �??Ionization Above the Coulomb Barrier,�?? Phys. Rev. Lett 83, 706 (1999).
[CrossRef]

E. Seres, J. Seres, F. Krausz, and Ch. Spielmann, �??Generation of Coherent Soft-X-Ray Radiation Extending Far Beyond the Titanium L Edge,�?? Phys. Rev. Lett 92, 163002 (2004).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

G. Tempea, M. Geissler, M. Schnürer, and T. Brabec, �??Self-Phase-Matched High Harmonic Generation,�?? Phys. Rev. Lett. 84, 4329-4332 (2000).
[CrossRef] [PubMed]

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, S. Backus, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, �??High-Order Harmonic Generation up to 250 eV from Highly Ionized Argon,�?? Phys. Rev. Lett. 92, 033001 (2004).
[CrossRef] [PubMed]

B. Sheehy, J. D. D. Martin, L. F. DiMauro, P. Agostini, K. J. Schafer, M. B. Gaarde, and K. C. Kulander, �??High Harmonic Generation at Long Wavelengths,�?? Phys. Rev. Lett. 83, 5270-5273 (1999).
[CrossRef]

Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, �??Generation of Coherent Soft X Rays at 2.7 nm Using High Harmonics,�?? Phys. Rev. Lett. 79, 2967-2970 (1997).
[CrossRef]

P. B. Corkum, �??Plasma Perspective on Strong-Field Multiphoton Ionization,�?? Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

Ivan P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, �??Nonadiabatic Effects in High-Harmonic Generation with Ultrashort Pulses,�?? Phys. Rev. Lett. 77, 1743-1746 (1996).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

T. Brabec and F. Krausz, �??Intense few-cycle laser fields: Frontiers of nonlinear optics,�?? Rev. Mod. Phys. 72, 545-591 (2000).
[CrossRef]

Science (2)

Ch. Spielmann, N. H. Burnett, S. Santania, R. Koppitsch, M. Schn¨urer, C. Kan, M. Lenzner, P.Wobrauschek, and F. Krausz, �??Generation of Coherent X-rays in the Water Window Using 5-Femtosecond Laser Pulses,�?? Science 278, 661-664 (1997).
[CrossRef]

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, Science 302, �??Coherent Soft X-ray Generation in the Water Window with Quasi-Phase Matching,�?? 5642-5645 (2003).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of the semiclassical model of HHG, that illustrates the calculation of the second depletion. The thick blue curve is a bunch of trajectories around the one with the largest kinetic energy upon return.

Fig. 2.
Fig. 2.

HHG normalized efficiency η0 as function of the cutoff energy for different pump wavelengths and a neutral helium target.

Fig. 3.
Fig. 3.

Simulated HHG spectra for hydrogen with the pump field given by Eq. (3), at the interval 0<ωt<2.04π, for ω=.057, corresponding to a Ti:sapphire source. The corresponding E 0 is denoted near each curve. The exponential decrease in the yield near the cutoff is evident.

Fig. 4.
Fig. 4.

HHG photon yield as function of the spectral cutoff for hydrogen, a 0.8µm and 1.5µm source, the driving field given by Eq. (3), and no initial depletion of the ground state. The points were obtained from simulations of the three dimensional Schrödinger equation. The solid curves were obtained from Eq. (6) and tabulated static ionization rates for hydrogen [16]. η0 was multiplied by a coefficient (the same for both curves) such that it would match the simulated yields at high cutoff energies.

Equations (7)

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ψ = e i I p t ( a ( t ) 0 + φ ) ,
x = a * ( t ) 0 x φ + c . c .
E ( t ) = E 0 sin ( ω t ) .
a ( t 1 ) = exp ( 1 2 0 1.884 ω w ( E 0 sin ( ω t ) ) d t ) = exp ( 1 2 ω 0 1.884 w ( E 0 sin ( θ ) ) d θ ) .
a ( t 2 ) = exp ( 1 2 0 5.97 ω w ( E 0 sin ( ω t ) ) d t ) = exp ( 1 2 ω 0 5.97 w ( E 0 sin ( θ ) ) d θ ) .
η 0 = ω 3 a ( t 1 ) a ( t 2 ) 2
N photons ( ω ) = 1 20 1 3 π c 3 0 2.04 π ω 1 x ̈ ( t ) e i ω t d t 2

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