## Space-time resolved simulation of femtosecond nonlinear light-matter interactions using a holistic quantum atomic model : Application to near-threshold harmonics |

Optics Express, Vol. 20, Issue 14, pp. 16113-16128 (2012)

http://dx.doi.org/10.1364/OE.20.016113

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### Abstract

We introduce a new computational approach for femtosecond pulse propagation in the transparency region of gases that permits full resolution in three space dimensions plus time while fully incorporating quantum coherent effects such as high-harmonic generation and strong-field ionization in a holistic fashion. This is achieved by utilizing a one-dimensional model atom with a delta-function potential which allows for a closed-form solution for the nonlinear optical response due to ground-state to continuum transitions. It side-steps evaluation of the wave function, and offers more than one hundred-fold reduction in computation time in comparison to direct solution of the atomic Schrödinger equation. To illustrate the capability of our new computational approach, we apply it to the example of near-threshold harmonic generation in Xenon, and we also present a qualitative comparison between our model and results from an in-house experiment on extreme ultraviolet generation in a femtosecond enhancement cavity.

© 2012 OSA

## 1. Introduction

26. J. Lee, D. R. Carlson, and R. J. Jones, “Optimizing intracavity high harmonic generation for xuv fs frequency combs,” Opt. Express **19**, 23315–23326 (2011). [CrossRef] [PubMed]

## 2. Computational approach and model

27. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E **70**, 036604 (2004). [CrossRef]

*E*

_{k⊥}(

*ω*,

*z*), where

*ω*is the frequency, and

*k*

_{⊥}denotes transverse wavenumbers, so that the space-time dependent electric field

*E*(

*r,z,t*) may always be reconstructed using a 3D Fourier (or Hankel) transform over

*k*

_{⊥},

*ω*. Note that the assumption of a linearly polarized field is consistent with our adoption of a 1D atomic model. Nonlinear effects enter the propagation equations through either the polarization (

*P*) or current-density (

*J*) terms. These are both evaluated as functionals of the electric field history

*E*(

*t*) at each spatial point, and then Fourier-transformed to the spectral representation, e.g.

*J*

_{k⊥}(

*ω*,

*z*, {

*E*}), that appears in the above propagation equation. The linear wavelength-dependent properties of a medium are represented through the susceptibility

*χ*(

*ω*).

1. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. **441**, 47–189 (2007). [CrossRef]

- A description of the linear dispersive properties of the gas via a complex-valued frequency dependent susceptibility
*χ*(*ω*). - The 1D
*δ*-potential quantum model for modeling the nonlinear current (*J*) due to ground state to continuum transitions. This will incorporate the effects of ionization, HHG, and ionization induced absorption and refraction in a holistic fashion. - A Kerr-like nonlinearity and associated nonlinear polarization (
*P*) to capture the nonlinear optical response due to the ground state to excited bound state transitions. This will contribute processes such as four-wave mixing, self-phase modulation, and self-focusing. (For a molecular gas we may also add a time-delayed nonlinear response to capture Raman effects)

*χ*,

*J*,

*P*are coupled into the field-propagation equation which encompasses all frequency components from the fundamental to high harmonics. While we use the Unidirectional Pulse Propagation Equation, we emphasize that this approach may be implemented with any pulse-propagation simulator that resolves the carrier wave of the optical field.

### Linear dispersive properties

*χ*(

*ω*), and the pulse propagation method utilizes this information in the propagation constant

*k*(

_{z}*ω*,

*k*

_{⊥}) in (2) at each frequency or wavelength resolved by the numerical simulation.

*http://henke.lbl.gov/optical_constants/*with the refractive-index parametrization obtained from Ref. [28

28. 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. Spect. Rad. Transfer **25**, 395–402 (1981). [CrossRef]

*harmonic. Note that with the real and imaginary parts of the susceptibility function coming from separate sources, and with no absorption data available for photon energies below 10 eV, our parametrization does not satisfy the Kramers-Kronig relations. To our best knowledge a more complete data set is currently not available. Nevertheless, while not a perfect model of the Xenon gas for all frequencies this data set employed serves a an illustration example in which all electromagnetic frequencies are treated on the same footing by the propagation solver, which operates on the full electric field. Generally the accuracy of the linear propagation will be limited by the quality of the available data for the linear susceptibility.*

^{th}### Quantum atomic model for the nonlinear response

*δ*-potential for the nonlinear current due to ground state to continuum transitions. More specifically the corresponding time-dependent Schrödinger equation in atomic units takes the form Here

*s*and

*τ*are the space and time variables in atomic units, the strength of the attractive potential is governed by the parameter

*B*, with the ionization potential of the ground state equal to

*B*

^{2}/2, and

*F*(

*τ*) denotes the time-dependent external field in atomic units applied to the atom.

*F*(

*t*) in atomic units. This drives the quantum system, and the induced current is computed as outlined in the Appendix. The nonlinear current is next converted from the atomic units, and multiplied by the number density atoms at the point in space. The resulting macroscopic current density is included in the right-hand-side of the UPPE equation. We remark that since we only incorporate the

*nonlinear*current from the quantum model into the UPPE, we do not double-count by erroneously including linear properties from the 1D atomic model. Also note that it is not an option to retain the linear part of the model’s response instead of

*χ*(

*ω*) introduced in the previous subsection. This is because in a real medium

*χ*(

*ω*) originates in virtual transitions among a large number of states, and it would be difficult to model this from first principles. The linear susceptibility arising from our 1D atomic model is far too simplistic to capture the chromatic properties of a gas to any realistic degree.

*δ*-potential atomic model displays features for HHG that are qualitatively similar to those obtained using the strong field approximation applied to the more exact 3D Hydrogen-like atomic model. To this end Fig. 2 shows an example of a harmonic spectrum of the nonlinear current induced in the 1D model atom for an ionization potential of 12 eV, characteristic of Xenon, and a ten-cycle pulse of center wavelength

*λ*= 800nm and peak intensity1.5 × 10

^{18}W/m

^{2}. This harmonic spectrum exhibits the characteristic high-harmonic generation plateau with a high frequency cut-off. The cut-off predicted by the formula from the standard HHG model [2

2. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high harmonics generation by low-frequency laser fields,” Phys. Rev. A **49**, 2117–2132 (1994). [CrossRef] [PubMed]

29. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett **71**, 1994–1997 (1993). [CrossRef] [PubMed]

30. K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. **70**, 1599–1602 (1993). [CrossRef] [PubMed]

*harmonic which occurs near the ionization potential up to around the 40*

^{th}*harmonic, and thus constitutes an example of near-threshold HHG where the generated harmonics straddle the ionization energy. What is shown in Fig. 2 is the spectrum of the nonlinear polarization*

^{th}*P*that appears in the Maxwell equations and not the spectrum of the radiation actually generated. Importantly the harmonic spectrum shown is exact, within the context of our 1D model, over the whole frequency range and in particular at low frequencies.

2. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high harmonics generation by low-frequency laser fields,” Phys. Rev. A **49**, 2117–2132 (1994). [CrossRef] [PubMed]

### The nonlinear Kerr effect

*δ*-potential atomic model only incorporates the nonlinear contribution of ground state to continuum transitions. To capture the nonlinear contribution of virtual ground state to bound state electronic transitions we include a term representing the nonlinear Kerr effect of the form where

*n*̄

_{2}is the nonlinear index and

*n*is the linear refractive index. This effect enters the propagation equation via the nonlinear polarization term, and gives rise to self-focusing and a cascade of lower-harmonic radiation.

_{b}## 3. Illustrations involving near-threshold high-harmonic generation

### HHG in a Xenon gas jet

*t*= 85 fs and center wavelength of 800 nm, and focused spot size

_{p}*w*

_{0}. We model the Xenon gas jet as a vertical column which is tapered along the propagation or z-axis according to the jet pressure profile indicated by the dotted line in Fig. 3. This pressure or density profile has a constant region of length

*L*= 200

*μ*m between

*z*= 100–300

*μ*m and tapers off on each side of this region. For the examples in this paper we used 20 Torr for the maximal pressure in the gas jet. Furthermore we consider the two distinct focusing cases where the input beam is focused at

*z*= 100

*μ*m close to the entrance to the gas jet, and also the case that the input beam is focused at

*z*= 300

*μ*m close to the exit of the gas jet (We use this descriptive nomenclature of exit and entrance to the gas jet with the caveat that HHG can arise before and after the exit and entrance due to the tapers in the gas density). The variation of the on-axis intensity along the z-axis is illustrated in Fig. 3 for the cases of a) a narrow beam with

*w*

_{0}= 7

*μ*m, and b) a wide beam with

*w*

_{0}= 15

*μ*m.

31. F. Schapper, M. Holler, T. Auguste, A. Zaïr, M. Weger, P. Salières, L. Gallmann, and U. Keller, “Spatial finger-print of quantum path interferences in high order harmonic generation,” Opt. Express **18**, 2987–2994 (2010). [CrossRef] [PubMed]

32. S. M. Teichmann, D. R. Austin, P. Bates, S. Cousin, A. Grün, M. Clerici, A. Lotti, D. Faccio, P. DiTrapani, A. Couairon, and J. Biegert, “Trajectory interferences in a semi-infinite gas cell,” Laser Phys. Lett. **9**, 207–211 (2012). [CrossRef]

### Tighter focus case

*w*

_{0}= 7

*μ*m giving a Rayleigh range of

*L*≃

*z*

_{0}and we expect that the focusing of the beam will impact phase-matching of the HHG via the Gouy phase-shift incurred by the Gaussian beam. The calculated angularly resolved spectra are shown in Fig. 4 for (a) focusing at the exit, and (b) focusing at the entrance of the gas jet, and we note that there is a marked difference between the angularly resolved spectra for the two cases. This is consistent with general expectations based on phase-matching [33

33. A. L’Huillier, K.J. Schafer, and K.C. Kulander, “High-order harmonic generation in Xenon at 1064 nm: The role of phase matching,” Phys. Rev. Lett. **66**, 2200–2203 (1991). [CrossRef]

35. E. Constant, D. Garzella, P. Breger, E. Mvel, C. Dorrer, C. Le Blanc, F. Sali, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: Model and experiment,” Phys. Rev. Lett. **82**, 1668–1671 (1999). [CrossRef]

34. P. Salières, A. L’Huillier, and M. Lewenstein, “Coherence control of high-order harmonics,” Phys. Rev. Lett. **74**3776–3779 (1995). [CrossRef] [PubMed]

*harmonic where case (a) is dominated by off-axis emission whereas case (b) is dominantly on-axis emission. So the results of our simulations are compatible with expectations based on on-axis phase-matching [33*

^{th}33. A. L’Huillier, K.J. Schafer, and K.C. Kulander, “High-order harmonic generation in Xenon at 1064 nm: The role of phase matching,” Phys. Rev. Lett. **66**, 2200–2203 (1991). [CrossRef]

34. P. Salières, A. L’Huillier, and M. Lewenstein, “Coherence control of high-order harmonics,” Phys. Rev. Lett. **74**3776–3779 (1995). [CrossRef] [PubMed]

4. M. Gaarde, J. Tate, and K. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys. **41**, 132001 (2008). [CrossRef]

34. P. Salières, A. L’Huillier, and M. Lewenstein, “Coherence control of high-order harmonics,” Phys. Rev. Lett. **74**3776–3779 (1995). [CrossRef] [PubMed]

*harmonic at three propagation distances. At the shortest and longest propagation distances note that the spectrum has its maximum on-axis, whereas at the on-axis and off-axis become comparable in the middle. It is interesting that the turnaround in the conversion efficiency for the 9th harmonic that occurs for a propagation distance of around 200*

^{th}*μ*m is therefore dominated by a reduction in the off-axis as opposed to the on-axis emission. This highlights the point that the spatial evolution of the various harmonics involves a complex interplay between on-axis and off-axis emissions, and it is not generally sufficient to assess the conversion efficiency based on the on-axis phase-matching considerations.

### Weaker focus case

*w*

_{0}= 15

*μ*m giving a Rayleigh range of

*L*<<

*z*

_{0}and we expect that the focusing of the beam will not impact phase-matching of the HHG in a major way. The calculated angularly resolved spectra for this case are shown in Fig. 7 for the cases of (a) focusing at the exit, and (b) focusing at the entrance of the gas jet, and we note that there is no major differences difference between the angularly resolved spectra for the two cases. There are small quantitative differences though, in particular in the visibility of the ring structure (cf. the 11th harmonic far-field patterns).

**74**3776–3779 (1995). [CrossRef] [PubMed]

31. F. Schapper, M. Holler, T. Auguste, A. Zaïr, M. Weger, P. Salières, L. Gallmann, and U. Keller, “Spatial finger-print of quantum path interferences in high order harmonic generation,” Opt. Express **18**, 2987–2994 (2010). [CrossRef] [PubMed]

**74**3776–3779 (1995). [CrossRef] [PubMed]

32. S. M. Teichmann, D. R. Austin, P. Bates, S. Cousin, A. Grün, M. Clerici, A. Lotti, D. Faccio, P. DiTrapani, A. Couairon, and J. Biegert, “Trajectory interferences in a semi-infinite gas cell,” Laser Phys. Lett. **9**, 207–211 (2012). [CrossRef]

32. S. M. Teichmann, D. R. Austin, P. Bates, S. Cousin, A. Grün, M. Clerici, A. Lotti, D. Faccio, P. DiTrapani, A. Couairon, and J. Biegert, “Trajectory interferences in a semi-infinite gas cell,” Laser Phys. Lett. **9**, 207–211 (2012). [CrossRef]

### HHG in a femtosecond enhancement cavity

36. R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett. **94**, 193201 (2005). [CrossRef] [PubMed]

37. C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, “A frequency comb in the extreme ultraviolet,” Nature **436**, 234–237 (2005). [CrossRef] [PubMed]

*w*

_{0}= 15

*μ*m within the jet. The fsEC has a 1% input coupler that allows pulse energy enhancement over 200 times relative to the incident pulse train leading to peak intensities in excess of 1 × 10

^{14}W/cm

^{2}when the gas jet is off. The harmonics are coupled out of the cavity using a thin sapphire plate, aligned at Brewster’s angle for the fundamental pulse, and resolved spatially by reflection from an XUV grating.

38. D. R. Carlson, J. Lee, J. Mongelli, E. M. Wright, and R. J. Jones, “Intracavity ionization and pulse formation in femtosecond enhancement cavities,” Opt. Lett. **36**, 2991–2993 (2011). [CrossRef] [PubMed]

38. D. R. Carlson, J. Lee, J. Mongelli, E. M. Wright, and R. J. Jones, “Intracavity ionization and pulse formation in femtosecond enhancement cavities,” Opt. Lett. **36**, 2991–2993 (2011). [CrossRef] [PubMed]

## 4. Conclusion

## 5. Appendix

*F*(

*t*) enters through the classical electron trajectory

*x*(

_{cl}*t*) (here we assume that

*x*is the direction of the optical field polarization) Assuming that the system was initially in its ground state, an auxiliary quantity

*A*(

*t*) is obtained first as a solution to the following integral equation: where the right-hand-side is Having calculated

*A*(

*t*), the nonlinear (in the strength of the external field

*F*) current is expressed as a sum with the components listed below:

## Acknowledgments

## References and links

1. | A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. |

2. | M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high harmonics generation by low-frequency laser fields,” Phys. Rev. A |

3. | L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. |

4. | M. Gaarde, J. Tate, and K. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys. |

5. | M. Nurhuda, A. Suda, and K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys. |

6. | E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett. |

7. | I. Christov, “Propagation of ultrashort pulses in gaseous medium:breakdown of the quasistatic approximation,” Opt. Express |

8. | E. Lorin, S. Chelkowski, and A. D. Bandrauk, “The WASP model: A Micro-Macro system of Wave-Schrödinger-Plasma equations for filamentation,” Commun. Comput. Phys. |

9. | Q. Su and J. Eberly, “Model atom for multiphoton physics,” Phys. Rev. A |

10. | G. P. Arrighini and M. Gavarini, “Ionization of a model atom by strong and superstrong electric fields,” Lett. Nuovo Cimento |

11. | W. Elberfeld and M. Kleber, “Tunneling from an ultrathin quantum well in a strong electrostatic field: A comparison of different methods,” Z. Phys. B: Cond. Matt. |

12. | R. M. Cavalcanti, P. Giacconi, and R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A: Math. Gen. |

13. | H. Uncu, H. Erkol, E. Demiralp, and H. Beker, “Solutions of the Schrödinger equation for dirac delta decorated linear potential,” C. Eur. J. Phys. |

14. | G. Álvarez and B. Sundaram, “Perturbation theory for the stark effect in a double quantum well,” J. Phys. A: Math. Gen. |

15. | V. M. Villalba and L. A. González-Díaz, “Particle resonance in the dirac equation in the presence of a delta interaction and a perturbative hyperbolic potential,” Eur. Phys. J. C |

16. | S. Geltman, “Ionisation dynamics of a model atom in an electrostatic field,” J. Phys. B: At. Mol. Phys. |

17. | G. V. Dunne and C. S. Gauthier, “Simple soluble molecular ionization model,” Phys. Rev. A |

18. | Q. Su, B. P. Irving, C. W. Johnson, and J. H. Eberly, “Stabilization of a one-dimensional short-range model atom in intense laser fields,” J. Phys. B: At. Mol. Opt. |

19. | T. Dziubak and J. Matulewski, “Stabilization of one-dimensional soft-core and singular model atoms,” Eur. Phys. J. D |

20. | A. Sanpera, Q. Su, and L. Roso-Franco, “Ionization suppression in a very-short-range potential,” Phys. Rev. A |

21. | S. Geltman, “Short-pulse model-atom studies of ionization in intense laser fields,” J. Phys. B - At. Mol. Opt. |

22. | Z. X. Zhao, B. D. Esry, and C. D. Lin, “Boundary-free scaling calculation of the time-dependent Schrödinger equation for laser-atom interactions,” Phys. Rev. A |

23. | A. Teleki, E. M. Wright, and M. Kolesik, “Microscopic model for the higher-order nonlinearity in optical filaments,” Phys. Rev. A |

24. | J. M. Brown, E. M. Wright, J. V. Moloney, and M. Kolesik, “On the relative roles of higher-order nonlinearity and ionization in ultrafast light-matter interactions,” Opt. Lett. |

25. | J. Brown, A. Lotti, A. Teleki, and M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A |

26. | J. Lee, D. R. Carlson, and R. J. Jones, “Optimizing intracavity high harmonic generation for xuv fs frequency combs,” Opt. Express |

27. | M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E |

28. | 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. Spect. Rad. Transfer |

29. | P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett |

30. | K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. |

31. | F. Schapper, M. Holler, T. Auguste, A. Zaïr, M. Weger, P. Salières, L. Gallmann, and U. Keller, “Spatial finger-print of quantum path interferences in high order harmonic generation,” Opt. Express |

32. | S. M. Teichmann, D. R. Austin, P. Bates, S. Cousin, A. Grün, M. Clerici, A. Lotti, D. Faccio, P. DiTrapani, A. Couairon, and J. Biegert, “Trajectory interferences in a semi-infinite gas cell,” Laser Phys. Lett. |

33. | A. L’Huillier, K.J. Schafer, and K.C. Kulander, “High-order harmonic generation in Xenon at 1064 nm: The role of phase matching,” Phys. Rev. Lett. |

34. | P. Salières, A. L’Huillier, and M. Lewenstein, “Coherence control of high-order harmonics,” Phys. Rev. Lett. |

35. | E. Constant, D. Garzella, P. Breger, E. Mvel, C. Dorrer, C. Le Blanc, F. Sali, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: Model and experiment,” Phys. Rev. Lett. |

36. | R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett. |

37. | C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, “A frequency comb in the extreme ultraviolet,” Nature |

38. | D. R. Carlson, J. Lee, J. Mongelli, E. M. Wright, and R. J. Jones, “Intracavity ionization and pulse formation in femtosecond enhancement cavities,” Opt. Lett. |

**OCIS Codes**

(190.4160) Nonlinear optics : Multiharmonic generation

(190.5940) Nonlinear optics : Self-action effects

(320.0320) Ultrafast optics : Ultrafast optics

(320.2250) Ultrafast optics : Femtosecond phenomena

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 6, 2012

Revised Manuscript: June 20, 2012

Manuscript Accepted: June 20, 2012

Published: June 29, 2012

**Citation**

M. Kolesik, E. M. Wright, J. Andreasen, J. M. Brown, D. R. Carlson, and R. J. Jones, "Space-time resolved simulation of femtosecond nonlinear light-matter interactions using a holistic quantum atomic model : Application to near-threshold harmonics," Opt. Express **20**, 16113-16128 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-16113

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### References

- A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep.441, 47–189 (2007). [CrossRef]
- M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high harmonics generation by low-frequency laser fields,” Phys. Rev. A49, 2117–2132 (1994). [CrossRef] [PubMed]
- L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys.70, 1633–1713 (2007). [CrossRef]
- M. Gaarde, J. Tate, and K. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys.41, 132001 (2008). [CrossRef]
- M. Nurhuda, A. Suda, and K. Midorikawa, “Generalization of the Kerr effect for high intensity, ultrashort laser pulses,” New J. Phys.10, 053006 (2008). [CrossRef]
- E. A. Volkova, A. M. Popov, and O. V. Tikhonova, “Nonlinear polarization response of an atomic gas medium in the field of a high-intensity femtosecond laser pulse,” JETP Lett.94, 519–524 (2011). [CrossRef]
- I. Christov, “Propagation of ultrashort pulses in gaseous medium:breakdown of the quasistatic approximation,” Opt. Express6, 34–39 (2000). [CrossRef] [PubMed]
- E. Lorin, S. Chelkowski, and A. D. Bandrauk, “The WASP model: A Micro-Macro system of Wave-Schrödinger-Plasma equations for filamentation,” Commun. Comput. Phys.9, 406–440 (2011).
- Q. Su and J. Eberly, “Model atom for multiphoton physics,” Phys. Rev. A44, 5997–6008 (1991). [CrossRef] [PubMed]
- G. P. Arrighini and M. Gavarini, “Ionization of a model atom by strong and superstrong electric fields,” Lett. Nuovo Cimento33, 353–358 (1982). [CrossRef]
- W. Elberfeld and M. Kleber, “Tunneling from an ultrathin quantum well in a strong electrostatic field: A comparison of different methods,” Z. Phys. B: Cond. Matt.73, 23–32 (1988). [CrossRef]
- R. M. Cavalcanti, P. Giacconi, and R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A: Math. Gen.36, 12065–12080 (2003). [CrossRef]
- H. Uncu, H. Erkol, E. Demiralp, and H. Beker, “Solutions of the Schrödinger equation for dirac delta decorated linear potential,” C. Eur. J. Phys.3, 303–323 (2005). [CrossRef]
- G. Álvarez and B. Sundaram, “Perturbation theory for the stark effect in a double quantum well,” J. Phys. A: Math. Gen.37, 9735–9748 (2004). [CrossRef]
- V. M. Villalba and L. A. González-Díaz, “Particle resonance in the dirac equation in the presence of a delta interaction and a perturbative hyperbolic potential,” Eur. Phys. J. C61, 519–525 (2009). [CrossRef]
- S. Geltman, “Ionisation dynamics of a model atom in an electrostatic field,” J. Phys. B: At. Mol. Phys.11, 3323–3337 (1978). [CrossRef]
- G. V. Dunne and C. S. Gauthier, “Simple soluble molecular ionization model,” Phys. Rev. A69, 053409 (2004). [CrossRef]
- Q. Su, B. P. Irving, C. W. Johnson, and J. H. Eberly, “Stabilization of a one-dimensional short-range model atom in intense laser fields,” J. Phys. B: At. Mol. Opt.29, 5755–5764 (1996). [CrossRef]
- T. Dziubak and J. Matulewski, “Stabilization of one-dimensional soft-core and singular model atoms,” Eur. Phys. J. D59, 321–327 (2010). [CrossRef]
- A. Sanpera, Q. Su, and L. Roso-Franco, “Ionization suppression in a very-short-range potential,” Phys. Rev. A47, 2312–2318 (1993). [CrossRef] [PubMed]
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