## The influence of plasma defocusing in high harmonic generation |

Optics Express, Vol. 19, Issue 23, pp. 22377-22387 (2011)

http://dx.doi.org/10.1364/OE.19.022377

Acrobat PDF (2635 KB)

### Abstract

We numerically investigate the influence of plasma defocusing in high harmonic generation (HHG) by solving the first-order wave equation in an ionized medium and defining an enhancement factor to quantitatively analyze the influence of plasma defocusing. While degrading the driver pulse intensity, plasma also has a strong impact on HHG phase-matching. Combined with the HHG wavelength scaling law, our results give an estimate of HHG efficiencies with different driver wavelengths and show a limited HHG efficiency in high density media.

© 2011 OSA

## 1. Introduction

1. Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnurer, 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**(5338), 661–664 (1997). [CrossRef]

3. M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft X-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. **105**(17), 173901 (2010). [CrossRef] [PubMed]

4. T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. **106**(26), 10516–10521 (2009). [CrossRef] [PubMed]

5. J. Moses, S.-W. Huang, K.-H. Hong, O. D. Mucke, E. L. Falcao-Filho, A. Benedick, F. O. Ilday, A. Dergachev, J. A. Bolger, B. J. Eggleton, and F. X. Kartner, “Highly stable ultrabroadband mid-infrared optical parametric chirped pulse amplifier optimized for superfluorescence suppression,” Opt. Lett. **34**(11), 1639–1641 (2009). [CrossRef] [PubMed]

3. M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft X-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. **105**(17), 173901 (2010). [CrossRef] [PubMed]

6. P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Dimauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. **4**(5), 386–389 (2008). [CrossRef]

7. E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Coherent water window x ray by phase-matched high-order harmonic generation in neutral media,” Phys. Rev. Lett. **101**(25), 253901 (2008). [CrossRef] [PubMed]

8. K.-H. Hong, J. T. Gopinath, D. Rand, A. M. Siddiqui, S.-W. Huang, E. Li, B. J. Eggleton, J. D. Hybl, T. Y. Fan, and F. X. Kartner, “High-energy, kHz-repetition-rate, ps cryogenic Yb:YAG chirped-pulse amplifier,” Opt. Lett. **35**(11), 1752–1754 (2010). [CrossRef] [PubMed]

_{1}

^{−(5~6)}, at a given HHG driver wavelength λ

_{1}[9

9. E. L. Falcao-Filho, M. Gkortsas, A. Gordon, and F. X. Kartner, “Analytic scaling analysis of high harmonic generation conversion efficiency,” Opt. Express **17**(13), 11217–11229 (2009). [CrossRef] [PubMed]

11. A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H.-C. Bandulet, D. Comtois, F. Legare, M. Giguere, J.-C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. **103**(7), 073902 (2009). [CrossRef] [PubMed]

9. E. L. Falcao-Filho, M. Gkortsas, A. Gordon, and F. X. Kartner, “Analytic scaling analysis of high harmonic generation conversion efficiency,” Opt. Express **17**(13), 11217–11229 (2009). [CrossRef] [PubMed]

12. C. Altucci, T. Starczewski, E. Mevel, C.-G. Wahlstrom, B. Carre, and A. L’Huillier, “Influence of atomic density in high-order harmonic generation,” J. Opt. Soc. Am. B **13**(1), 148–156 (1996). [CrossRef]

12. C. Altucci, T. Starczewski, E. Mevel, C.-G. Wahlstrom, B. Carre, and A. L’Huillier, “Influence of atomic density in high-order harmonic generation,” J. Opt. Soc. Am. B **13**(1), 148–156 (1996). [CrossRef]

12. C. Altucci, T. Starczewski, E. Mevel, C.-G. Wahlstrom, B. Carre, and A. L’Huillier, “Influence of atomic density in high-order harmonic generation,” J. Opt. Soc. Am. B **13**(1), 148–156 (1996). [CrossRef]

16. V. Tosa, K. T. Kim, and C. H. Nam, “Macroscopic generation of attosecond-pulse trains in strongly ionized media,” Phys. Rev. A **79**(4), 043828 (2009). [CrossRef]

17. E. Constant, D. Garzella, P. Breger, E. Mevel, Ch. Dorrer, C. Le Blanc, F. Salin, and P. Agostini, “Optimizing High Harmonic Generation in Absorbing Gases: Model and Experiment,” Phys. Rev. Lett. **82**(8), 1668–1671 (1999). [CrossRef]

## 2. Numerical model for plasma defocusing

18. M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light Propagation in Field-Ionizing Media: Extreme Nonlinear Optics,” Phys. Rev. Lett. **83**(15), 2930–2933 (1999). [CrossRef]

*E*is the complex representation of the electric field;

*z*and

*τ*are the propagation distance and the retarded time in the retarded time frame respectively;

*k*is the wave-vector at the carrier frequency;

*n*is the nonlinear index of refraction;

_{2}*ω*is the plasma frequency;

_{p}*I*is the ionization potential of the atom;

_{p}*ρ*is the number density of the ionized atoms. We assume that tunneling ionization is the only ionization mechanism, and therefore, the ADK formula gives an accurate description of the ionization rate

*E*is a function of

*z*,

*τ*, and the radial coordinate

*r*.

20. A. Behjat, G. J. Tallents, and D. Neely, “The characterization of a high-density gas jet,” J. Phys. D Appl. Phys. **30**(20), 2872–2879 (1997). [CrossRef]

21. J.-S. Wu, S.-Y. Chou, U.-M. Lee, Y.-L. Shao, and Y.-Y. Lian, “Parallel DSMC Simulation of a Single Under-Expanded Free Orifice Jet From Transition to Near-Continuum Regime,” J. Fluids Eng. **127**(6), 1161–1170 (2005). [CrossRef]

^{14}W/cm

^{2}and 4.4x10

^{14}W/cm

^{2}respectively. In this paper, the medium temperature is assumed to be 300K, and then 1bar pressure is equivalent to a density of 2.4x10

^{19}atoms/cm

^{3}if we assume the medium is an ideal gas. In a real gas jet where the temperature can be much lower than 300K due to free expansion, the pressure in our simulation should be referred as atomic density.

*z*to positive

*z*through a He jet with three different pressures. The laser pulse is focused to an intensity of 7x10

^{14}W/cm

^{2}at the center of the jets (

*z =*0). Figure 1 (a)–(c) show the peak intensity of the pulse during propagation. The jet is located at

*z =*0, and the Gaussian curve in each plot shows a relative He pressure (atomic density) profile along the z-dimension. Because the beam spot is much smaller than the characteristic length of the jet, we neglect the pressure variation in the transverse spatial dimensions. In Fig. 1(a), the gas density is small, so the plasma doesn’t have too much impact on the pulse, and the intensity evolution is similar to free space propagation. As the pressure becomes higher in Fig. 1 (b) and (c), plasma defocusing starts to play an important role. A drop of intensity near the center of the jet can be seen. Such a drop of intensity hinders the pulse from efficiently ionizing the gas and generating high harmonics. Figure 1 (d)–(f) show the distribution of the ionization level (i.e. the fraction of atoms that are ionized) after the entire pulse leaves the medium and before the plasma starts to diffuse or recombine with ions. The pressures in Fig. 1 (d)–(f) are the same as in Fig. 1 (a)–(c) respectively. Because ionization is a highly nonlinear process, the ionization level is more sensitive to plasma defocusing than the peak intensity of the laser field. While the ionization level in Fig. 1(d) distributes widely from negative to positive z domain, the ionized regions in Fig. 1 (e) and (f) are mostly restricted in the negative z domain. Furthermore, the strong defocusing effect in Fig. 1(f) even prevents the pulse from approaching the center of the gas jet with a high intensity, so the ionization and HHG can only occur in the low pressure wing before the gas jet. In such situation, the high density center of the gas jet doesn’t help HHG but reabsorbs the high harmonic photons instead. This effect decreases the advantage of using a high pressure gas jet.

^{14}W/cm

^{2}) at the center of the cell (

*z =*0). Compared with the jet case, the intensity and ionization level in the He cell are lower at the same pressure because the pulse has to propagate a longer distance in the cell and suffers more plasma defocusing before reaching the focus at the cell center. The ionization level again shows sensitive dependence on the laser pulse intensity. In Fig. 2(f), the maximum ionization is only 4x10

^{−4}, which is much less than the ionization level in the lower pressure case (Fig. 2(d)). An immediate effect of plasma defocusing can be seen from Fig. 1 and Fig. 2 is the drop of the pulse intensity that reduces the cutoff photon energy or even stops HHG. Besides, as will be shown in next section, the impact of plasma defocusing on the phase-matching condition is even more dramatic to HHG efficiency.

## 3. Phase-matching and HHG enhancement

17. E. Constant, D. Garzella, P. Breger, E. Mevel, Ch. Dorrer, C. Le Blanc, F. Salin, and P. Agostini, “Optimizing High Harmonic Generation in Absorbing Gases: Model and Experiment,” Phys. Rev. Lett. **82**(8), 1668–1671 (1999). [CrossRef]

22. E. J. Takahashi, Y. Nabekawa, H. Mashiko, H. Hasegawa, A. Suda, and K. Midorikawa, “Generation of Strong Optical Field in Soft X-ray Region by Using High-Order Harmonics,” IEEE J. Sel. Top. Quantum Electron. **10**(6), 1315–1328 (2004). [CrossRef]

23. S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A **83**(6), 063405 (2011). [CrossRef]

*q*harmonic is assumed to be in the following form:where

^{th}*A*is a proportional constant;

*η*(

*q,λ*), as a function of the harmonic order

_{1}*q*and the driver wavelength

*λ*, is the single atom efficiency (SAE);

_{1}*w = w*(

*r,z,τ*) is the ionization rate given by the ADK formula;

*P*(

*z*) is the medium pressure;

*Δk*is the mismatch between the fundamental and the

_{q}= qk_{1}-k_{q}*q*harmonic wave-vectors;

^{th}*β*(

*z*) represents the absorption of the harmonics. The integral with respect to

*z*is due to the coherent addition of the high harmonic field generated over the non-uniform pressure distribution

*P*(

*z*). The integral is over a range from

*–L/*2 to

*L/*2 that is long enough to cover the medium completely. Its magnitude square would be proportional to the harmonic intensity generated from the point (

*r,τ*) in the parameter space. Then, the integrals over

*r*and

*τ*consider the HHG contribution from every part of the driver pulse and result in the total amount of high-harmonic energy in a given harmonic. We neglect the depletion of the ground states because the ionization level considered here is always less than few percents. We define the integral part of Eq. (2) as the enhancement factor

*ξ*:because it shows how much the SAE is enhanced by the total HHG medium. Then the total high harmonic energy can be rewritten as

*ξ*comprises the propagation effects of HHG and is proportional to the high harmonic energy, so it is useful for the investigation of the plasma defocusing effect.

*Δk*is important in the calculation of the enhancement factor

_{q}*ξ*. For a many-cycle pulse, phase-matching mainly depends on the neutral atom dispersion

*δk*, the plasma dispersion

_{a}*δk*that is induced by the free electrons, and the mismatches

_{p}*δk*and

_{g}*δk*that result from the geometric phase and the dipole phase respectively [14

_{d}14. H. Dachraoui, T. Auguste, A. Helmstedt, P. Bartz, M. Michelswirth, N. Mueller, W. Pfeiffer, P. Salieres, and U. Heinzmann, “Interplay between absorption, dispersion and refraction in high-order harmonic generation,” J. Phys. At. Mol. Opt. Phys. **42**(17), 175402 (2009). [CrossRef]

*α*, where

_{s}I*I*is the laser intensity, and the coefficient

*α*can be calculated based on the harmonic order and the action of the semi-classical electron trajectory [25

_{s}25. M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A **52**(6), 4747–4754 (1995). [CrossRef] [PubMed]

*α*is 5x10

_{s}^{−14}rad-cm

^{2}/W for 0.8μm driver wavelength. For other driver wavelengths, the coefficients are scaled with the cube of the wavelength.

*r =*0 and

*τ =*0) under the same driver pulse condition and medium structure as Fig. 1 and Fig. 2 respectively. The harmonic energy considered in both figures is 500eV, which is the 807th harmonic of the 2μm driver wavelength. The refractive indices used here for the driver pulses and the harmonics are taken from Ref [26]. and Ref [27

27. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmision, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables **54**(2), 181–342 (1993). [CrossRef]

*P =*0.1bar), the geometric phase and the dipole phase are more dominant than the other two, and phase-matching (

*Δk*) can only be realized far behind the focus. As the pressure becomes higher (Fig. 3(b) and Fig. 4(b),

_{q}= 0*P =*1bar), all four contributions are comparable, and the total

*Δk*shows complicated behavior. Phase-matching is achieved at some point near the center of the medium. At even higher pressure (Fig. 3(c) and Fig. 4(c),

_{q}*P =*10bar), the total

*Δk*basically follows the profile of the neutral atom dispersion because the negative plasma dispersion is roughly cancelled by the positive dipole and geometric phases.

_{q}*Δk*0 for a long distance. Instead,

_{q}=*Δk*often crosses the

_{q}*Δk*0 line with some slope. To improve the HHG efficiency in such a situation, one should try to decrease the slope and move the crossing point to the high density part of the medium.

_{q}=*ξ*for a He jet and a He cell for six different driver wavelengths and varying pressure. In Fig. 5(a), there are sharp transitions at about 0.1bar, because at this pressure a transition from phase-mismatched HHG to phase-matched HHG occurs. At low pressure (<0.1bar), Guoy phase and dipole phase dominate, and the phase-matching point where

*Δk*0 is located far behind the gas jet. Therefore, phase-matching cannot be achieved with low density media density, and poor HHG efficiency results from it. When the pressure is low,

_{q}=*Δk*doesn’t change much with the increasing pressure. Therefore the HHG efficiency still increases quadratically with the medium pressure although it is not phase-matched at all. Once the pressure is high enough (>0.1bar), the neutral atom dispersion is able to compensate the other phase effects and results in a phase-matching point within the gas jet. Correspondingly the enhancement factor

_{q}*ξ*, and with it the HHG efficiency, increase drastically. This transition phenomenon is less sharp for shorter driver wavelengths due to the smaller Guoy and dipole phases and smaller harmonic orders that can be generated. With higher medium density, the strong plasma defocusing and reabsorption start to limit the HHG efficiency and result in a saturation feature with optimal medium pressure for

*ξ*. In Fig. 5(b), the basic features of a He cell are similar to those of a He jet except that the optimal pressure is lower due to a longer interaction length. Figure 6 shows the enhancement factor

*ξ*for Ne. The same sharp transition can be seen at lower medium pressures because Ne has larger neutral atom dispersion. Compared to He, the larger reabsorption of Ne limits the optimal pressure to lower values than those of He.

## 4. HHG efficiency

^{−8}[22

22. E. J. Takahashi, Y. Nabekawa, H. Mashiko, H. Hasegawa, A. Suda, and K. Midorikawa, “Generation of Strong Optical Field in Soft X-ray Region by Using High-Order Harmonics,” IEEE J. Sel. Top. Quantum Electron. **10**(6), 1315–1328 (2004). [CrossRef]

^{−7}[29

29. E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. **84**(1), 4–6 (2004). [CrossRef]

3. M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft X-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. **105**(17), 173901 (2010). [CrossRef] [PubMed]

7. E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Coherent water window x ray by phase-matched high-order harmonic generation in neutral media,” Phys. Rev. Lett. **101**(25), 253901 (2008). [CrossRef] [PubMed]

*λ*

_{1}

^{−5.5}−scaling starting from the 0.8μm reference points. For comparison, we also calculate the cases without any plasma defocusing and show the results by the green stars. Most points are 0~2 orders higher than the dashed curves, so it means that they have a few to hundreds times higher enhancements than the 0.8μm case. This only partially compensates the efficiency loss due to reduced SAE.

## 5. Summary

*ξ*that considers macroscopic characteristics including plasma defocusing, reabsorption of harmonics, and phase-matching. Geometric and dipole phases that are important in the phase-matching of HHG driven by mid-IR wavelengths are also included. Our numerical result shows good agreement with experiment and provides an easy way to calculate and explain HHG performance without referring to the complex microscopic behavior of strong field dynamics and the atomic parameters. Although increasing the medium pressure can partially make up the severe loss of SAE with longer driver wavelengths, the compensation is still limited by the plasma defocusing.

## Acknowledgments

## References and links

1. | Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnurer, 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. | J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Wobrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, “Laser technology: source of coherent kiloelectronvolt X-rays,” Nature |

3. | M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft X-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. |

4. | T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. |

5. | J. Moses, S.-W. Huang, K.-H. Hong, O. D. Mucke, E. L. Falcao-Filho, A. Benedick, F. O. Ilday, A. Dergachev, J. A. Bolger, B. J. Eggleton, and F. X. Kartner, “Highly stable ultrabroadband mid-infrared optical parametric chirped pulse amplifier optimized for superfluorescence suppression,” Opt. Lett. |

6. | P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Dimauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. |

7. | E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Coherent water window x ray by phase-matched high-order harmonic generation in neutral media,” Phys. Rev. Lett. |

8. | K.-H. Hong, J. T. Gopinath, D. Rand, A. M. Siddiqui, S.-W. Huang, E. Li, B. J. Eggleton, J. D. Hybl, T. Y. Fan, and F. X. Kartner, “High-energy, kHz-repetition-rate, ps cryogenic Yb:YAG chirped-pulse amplifier,” Opt. Lett. |

9. | E. L. Falcao-Filho, M. Gkortsas, A. Gordon, and F. X. Kartner, “Analytic scaling analysis of high harmonic generation conversion efficiency,” Opt. Express |

10. | J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. |

11. | A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H.-C. Bandulet, D. Comtois, F. Legare, M. Giguere, J.-C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. |

12. | C. Altucci, T. Starczewski, E. Mevel, C.-G. Wahlstrom, B. Carre, and A. L’Huillier, “Influence of atomic density in high-order harmonic generation,” J. Opt. Soc. Am. B |

13. | I. Jong Kim, G. H. Lee, S. B. Park, Y. S. Lee, T. K. Kim, C. H. Nam, T. Mocek, and K. Jakubczak, “Generation of submicrojoule high harmonics using a long gas jet in a two-color laser field,” Appl. Phys. Lett. |

14. | H. Dachraoui, T. Auguste, A. Helmstedt, P. Bartz, M. Michelswirth, N. Mueller, W. Pfeiffer, P. Salieres, and U. Heinzmann, “Interplay between absorption, dispersion and refraction in high-order harmonic generation,” J. Phys. At. Mol. Opt. Phys. |

15. | V. Tosa, E. Balogh, and K. Kovács, “Phase-matched generation of water-window x rays,” Phys. Rev. A |

16. | V. Tosa, K. T. Kim, and C. H. Nam, “Macroscopic generation of attosecond-pulse trains in strongly ionized media,” Phys. Rev. A |

17. | E. Constant, D. Garzella, P. Breger, E. Mevel, Ch. Dorrer, C. Le Blanc, F. Salin, and P. Agostini, “Optimizing High Harmonic Generation in Absorbing Gases: Model and Experiment,” Phys. Rev. Lett. |

18. | M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light Propagation in Field-Ionizing Media: Extreme Nonlinear Optics,” Phys. Rev. Lett. |

19. | M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP |

20. | A. Behjat, G. J. Tallents, and D. Neely, “The characterization of a high-density gas jet,” J. Phys. D Appl. Phys. |

21. | J.-S. Wu, S.-Y. Chou, U.-M. Lee, Y.-L. Shao, and Y.-Y. Lian, “Parallel DSMC Simulation of a Single Under-Expanded Free Orifice Jet From Transition to Near-Continuum Regime,” J. Fluids Eng. |

22. | E. J. Takahashi, Y. Nabekawa, H. Mashiko, H. Hasegawa, A. Suda, and K. Midorikawa, “Generation of Strong Optical Field in Soft X-ray Region by Using High-Order Harmonics,” IEEE J. Sel. Top. Quantum Electron. |

23. | S. Kazamias, S. Daboussi, O. Guilbaud, K. Cassou, D. Ros, B. Cros, and G. Maynard, “Pressure-induced phase matching in high-order harmonic generation,” Phys. Rev. A |

24. | Pascal Salières and Ivan Christov, “Macroscopic Effects in High-Order Harmonic Generation,” in Strong Field Laser Physics, Thomas Brabec, ed. (Springer, 2008). |

25. | M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A |

26. | M. J. Weber, |

27. | B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmision, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables |

28. | I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly Efficient High-Harmonic Generation in an Orthogonally Polarized Two-Color Laser Field,” Phys. Rev. Lett. |

29. | E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. |

**OCIS Codes**

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

(020.2649) Atomic and molecular physics : Strong field laser physics

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: August 19, 2011

Revised Manuscript: October 3, 2011

Manuscript Accepted: October 8, 2011

Published: October 24, 2011

**Citation**

Chien-Jen Lai and Franz X. Kärtner, "The influence of plasma defocusing in high harmonic generation," Opt. Express **19**, 22377-22387 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22377

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### References

- Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnurer, 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(5338), 661–664 (1997). [CrossRef]
- J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Wobrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, “Laser technology: source of coherent kiloelectronvolt X-rays,” Nature 433(7026), 596–596 (2005). [CrossRef] [PubMed]
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