## Scaling of keV HHG photon yield with drive wavelength

Optics Express, Vol. 13, Issue 8, pp. 2941-2947 (2005)

http://dx.doi.org/10.1364/OPEX.13.002941

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### 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 1.5*µ*m to 3*µ*m 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

## 1. Introduction

1. Ch. Spielmann, N. H. Burnett, S. Santania, R. Koppitsch, M. Schnürer, 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]

4. 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]

*I*+3.17

_{p}*U*, where

_{p}*I*is the ionization potential and

_{p}*U*is the ponderomotive energy [3

_{p}3. P. B. Corkum, “Plasma Perspective on Strong-Field Multiphoton Ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

4. 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]

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]

^{5}photons per second in 5% of bandwidth were obtained [10

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]

^{2}–10

^{3}photons per second in a 10% bandwidth. All these records were achieved using a Ti:sapphire source and a target of neutral helium. The decrease in yield as the cutoff is pushed from 500eV to 700eV and further to 1.3keV [11

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]

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]

*exponentially*. For a Ti:sapphire source and a neutral helium target, the exponential drop starts at about 0.5keV, and the yield decreases by about 5.5 orders of magnitude per keV as the cutoff energy is pushed upwards. Since helium is the best neutral atom candidate for achieving high HHG energy, having the highest ionization potential among them, we conclude that the potential of HHG in terms of photon energy with a Ti:sapphire source and a neutral atom target is nearly exhausted.

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]

## 2. The intracycle depletion of the ground state

4. 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]

*ψ*〉, which in atomic units (

*h*̄=

*m*=

*e*=1) has the form

*I*is the ionization potential of the atom, |0〉 is the ground state, and |

_{p}*φ*〉 is the part of the wavefunction describing the freed electron [4

**49**, 2117–2132 (1994). [CrossRef] [PubMed]

*a*(

*t*) is the amplitude of the ground state, whose modulus decreases in time due to ionization by the external field. The three step model associates HHG with the interference term

*ψ*|

*x*|

*ψ*〉, due to the recollision of the freed electron with the parent ion.

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

*a*(

*t*), the amplitude of the ground state, and the amplitude of

*ψ*. According to the three step model, the main contribution to harmonic emission near the spectral cutoff (the “plateau harmonics”) comes from electrons that were released around the time

*t*

_{1}≈(

*π*/2+0.313)ω

^{-1}, and return to the nucleus around the time

*t*

_{2}≈5.97ω

^{-1}[3

3. P. B. Corkum, “Plasma Perspective on Strong-Field Multiphoton Ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

*a*(

*t*)=1 at the beginning of the optical cycle, that is,

*even if no initial depletion of the ground state has occurred before the largest electric field in the pulse acts on the atom*, the amplitude of the dipole acceleration is proportional to |

*a*(

*t*

_{1})| and |

*a*(

*t*

_{2})|, the amplitude remaining in the ground state upon ionization, and upon the reencounter. These two factors are always smaller then one and may significantly suppress the HHG photon yield. It is crucial that the depletion of the ground state affects the HHG yield “twice”: Once at

*t*

_{1}and once at

*t*

_{2}, as pointed out in various publications [4

**49**, 2117–2132 (1994). [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]

*t*=0. For a more realistic pulse |

*a*(0)|<1, and the photon yield is obviously further suppressed by an additional factor of |

*a*(0)|

^{4}.

*w*(

*E*) be the static ionization rate as function of the electric field. keV HHG occurs mostly in the barrier suppression regime at the wavelengths we are considering here. Therefore the quasi-static approximation for the ionization rate is excellent [14

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

*a*(

*t*

_{1}) and

*a*(

*t*

_{2}) satisfy

*a*(

*t*

_{1})

*a*(

*t*

_{2}) enters as a prefactor to the dipole moment amplitude, and therefore the HHG photon yield will have |

*a*(

*t*

_{1})

*a*(

*t*

_{2})|

^{2}as a prefactor. While the HHG photon yield obviously has other factors in it [15

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]

*a*(

*t*

_{1})

*a*(

*t*

_{2})|

^{2}is the only factor that exponentially decreases with increasing

*E*

_{0}. Since we intend to compare the HHG efficiency for different drive wavelengths, we define the normalized efficiency

*ω*

^{3}factor comes from quantum diffusion, which degrades the yield [4

**49**, 2117–2132 (1994). [CrossRef] [PubMed]

_{0}enters as a prefactor to the HHG yield, and we wish to study its scaling as a function of the drive wavelength and intensity.

_{0}computed using the tabulated static ionization rates for helium [14

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

_{0}is calculated for a range of electric field amplitudes

*E*

_{0}, and for convenience we plot it as function of the HHG cutoff, obtained by the semiclassical formula, associated with each

*E*

_{0}. η

_{0}decays exponentially as function of the cutoff energy, which follows simply from the fact that ionization in the barrier suppression regime [5

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

*w*(

*E*) is nearly linearly growing with

*E*.

*µ*m pumps outperform a 0.8

*µ*m one, and as the cutoff increases, the ratio grows by orders of magnitude. Indeed the opposite is true at low energies due to quantum diffusion. However, the intracycle depletion becomes a much more dominant limitation for higher cutoff energies. Another interesting observation is that η

_{0}for a 0.8

*µ*m decreases by 3 orders of magnitude as the cutoff is pushed from 0.7 to 1.3keV, which agrees with recently reported experiments [10

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]

**433**, 596–596 (2005). [CrossRef] [PubMed]

_{0}we compare it with the overall HHG single-atom photon yield. We numerically solved the Schrödinger equation of a hydrogen atom in the field given by Eq. (3). In order to isolate the effect discussed here, we constructed the spectrum of the harmonics by Fourier-transforming the dipole acceleration in the time interval [0,2.04

*πω*

^{-1}]. At the end of this interval the most energetic trajectories, responsible for the highest harmonics, have reached the nucleus. Terminating the interval later, but not reaching the region

*ωt*≈2

*π*·1.5, makes the spectrum more complicated. It adds interference fringes, but the last peak before the cutoff varies very little, as expected from the three step model.

*πω*

^{-1}]. These spectra are obtained using the dipole radiation formula, which in atomic units reads

*c*is the speed of light, which equals the reciprocal of the fine structure constant in atomic units. The prefactor 1/20 is because the number of photons refers to a 5% bandwidth. This measure is in commonly used [10

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]

_{0}.

## 3. Discussion

*µ*m is even greater. For a given cutoff energy longer wavelength means a weaker field, and therefore weaker input intensity and weaker depletion of the ground state by the leading edge of the pulse, before its main peak hits the atom.

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]

## Acknowledgments

## References and links

1. | Ch. Spielmann, N. H. Burnett, S. Santania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of Coherent X-rays in the Water Window Using 5-Femtosecond Laser Pulses,” Science |

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. |

3. | P. B. Corkum, “Plasma Perspective on Strong-Field Multiphoton Ionization,” Phys. Rev. Lett. |

4. | 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. |

5. | T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. |

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. |

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. |

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 |

9. | G. Tempea, M. Geissler, M. Schnürer, and T. Brabec, “Self-Phase-Matched High Harmonic Generation,” Phys. Rev. Lett. |

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 |

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 |

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. |

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 |

14. | A. Scrinzi, M. Geissler, and T. Brabec, “Ionization Above the Coulomb Barrier,” Phys. Rev. Lett |

15. | M. Yu. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A. |

16. | M. V. Ivanov, “Complex rotation in two-dimensional mesh calculations for quantum systems in uniform electric fields,” J. Phys. B |

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. |

18. | B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(260.7200) Physical optics : Ultraviolet, extreme

(340.7480) X-ray optics : X-rays, soft x-rays, extreme ultraviolet (EUV)

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 8, 2005

Revised Manuscript: March 30, 2005

Published: April 18, 2005

**Citation**

Ariel Gordon and Franz Kärtner, "Scaling of keV HHG photon yield with drive wavelength," Opt. Express **13**, 2941-2947 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-2941

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### References

- 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]
- 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]
- 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]
- T. Brabec and F. Krausz, �??Intense few-cycle laser fields: Frontiers of nonlinear optics,�?? Rev. Mod. Phys. 72, 545-591 (2000). [CrossRef]
- 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]
- 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]
- 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]
- 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. 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]
- 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]
- 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]
- 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]
- A. Scrinzi, M. Geissler, and T. Brabec, �??Ionization Above the Coulomb Barrier,�?? Phys. Rev. Lett 83, 706 (1999). [CrossRef]
- 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]
- 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]
- 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]
- 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).

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