## Harmonic generation beyond the Strong-Field Approximation: the physics behind the short-wave-infrared scaling laws

Optics Express, Vol. 17, Issue 12, pp. 9891-9903 (2009)

http://dx.doi.org/10.1364/OE.17.009891

Acrobat PDF (232 KB)

### Abstract

The physics of laser-mater interactions beyond the perturbative limit configures the field of extreme non-linear optics. Although most experiments have been done in the near infrared (*λ*≤1*µ*m), the situation is changing nowadays with the development of sources at longer wavelengths (<5*µ*m), opening new perspectives in the synthesis of shorter XUV attosecond pulses and higher frequencies. The theory of intense-field interactions is based either on the exact numerical integration of the time-dependent Schrödinger equation or in the development of models, mostly based on the strong-field approximation. Recent studies in the short-wave infrared show a divergence between the predictions of these models and the exact results. In this paper we will show that this discrepancy reveals the incompleteness of our present understanding of high-order harmonic generation. We discuss the physical grounds, provide a theoretical framework beyond the standard approximations and develop a compact approach that accounts for the correct scaling of the harmonic yield.

© 2009 Optical Society of America

## 1. Introduction

2. F. H. M. Faisal, “Multiple Absorption of Laser Photons by Atoms,” J. Phys. B **6**, L89–L92 (1973).
[CrossRef]

3. H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A **22**, 1786–1813 (1980).
[CrossRef]

*standard*approach [4

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

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

6. W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, “A unified theory of high-harmonic generation: application to polarization properties of the harmonics,” Phys. Rev. A **56**, 645–656 (1997).
[CrossRef]

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

*I*+3.17

_{p}*U*,

_{p}*I*being the ionization energy, and

_{p}*U*=

_{p}*q*

^{2}

*E*

^{2}/4

*mω*

^{2}∝

*λ*

^{2}, the ponderomotive energy). The standard model also succeeds in predicting the mode locking of the highest order harmonics, chirps and modulation of the yields with the intensity [8

8. A. Zaïr, M. Holler, A. Guadalini, F. Shapper, J. Biegert, L. Gallmann, U. Keller, A. S. Wyatt, A. Monmayrant, I. A. Walmsley, E. Cornier, T. Auguste, J. P. Caumes, and P. Salières, “Quantum Path Interferences in High-Order Harmonic Generation,” Phys. Rev. Lett. **100**, 143902-1-4 (2008).
[CrossRef]

9. J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. Di Mauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. **98**, 013901-1-4 (2007).
[CrossRef]

10. K. Schiessl, K. L. Ishikawa, E. Person, and J. Burgdörfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. **99**, 253903-1-4 (2007).
[CrossRef]

12. G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated Single-Cycle Attosecond Pulses,” Science **314**, 443–446 (2006).
[CrossRef] [PubMed]

13. M. Janjusevic and Y. Hahn, “Testing of laser-dressed atomic states,” J. Opt. Soc. Am. B **7**, 592–597 (1990).
[CrossRef]

14. P. Krstić and M. H. Mittleman, “S-matrix theory of multiphoton ionization,” J. Opt. Soc. Am. B **7**, 587–591 (1990).
[CrossRef]

15. O. Smirnova, M. Spanner, and M. Ivanov, “Coulomb and polarization effects in sub-cycle dynamics of strong field ionization,” J. Phys. B: At. Mol. Opt. Phys. **39**, S307–321 (2006).
[CrossRef]

16. W. Becker, J. Chen, S. G. Chen, and D. B. Milosevic, “Dressed-state strong-field approximation for laser induced molecular ionization,” Phys. Rev. A **76**, 033403-1-7 (2007).
[CrossRef]

## 2. Theoretical approach

17. 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. Di Mauro, “Scaling strong-field interactions towards the classical limit,” Nature Phys. **4**, 386–389 (2008).
[CrossRef]

9. J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. Di Mauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. **98**, 013901-1-4 (2007).
[CrossRef]

10. K. Schiessl, K. L. Ishikawa, E. Person, and J. Burgdörfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. **99**, 253903-1-4 (2007).
[CrossRef]

11. M. V. Frolov, N. L. Manakov, and A. F. Starace, “Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. **100**, 173001-1-4 (2008).
[CrossRef]

11. M. V. Frolov, N. L. Manakov, and A. F. Starace, “Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. **100**, 173001-1-4 (2008).
[CrossRef]

9. J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. Di Mauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. **98**, 013901-1-4 (2007).
[CrossRef]

10. K. Schiessl, K. L. Ishikawa, E. Person, and J. Burgdörfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. **99**, 253903-1-4 (2007).
[CrossRef]

11. M. V. Frolov, N. L. Manakov, and A. F. Starace, “Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. **100**, 173001-1-4 (2008).
[CrossRef]

**E**·

**r**and

**p**·

**A**gauges (red and orange graphs), for the model beyond the SFA presented here (green) and the exact 3D TDSE result (blue). The failure of the three models to reproduce accurately the 3D TDSE spectrum in this window becomes apparent and suggest, as discussed above, that the process of harmonic radiation in the inner part of the plateau has a complex nature. On the other hand, the spectra computed at the window near the cut-off show a more accurate resemblance with the 3D TDSE (see figure 2), as will be discussed below.

*n*⋍3.9±0.4) than for the translating window near the cut-off (

*n*⋍5.6±0.1). This suggests that this latter case is less sensitive to the details of the electromagnetic pulse. Moreover, the difference between the scaling powers of the yields computed at each window suggests a different mechanism for HOHG. In any case, as a main conclusion, our results confirm the discrepancy between the scaling powers of the exact 3D TDSE results (

*n*⋍5.6) and the expectations of the S-Matrix SFA models (

*n*between 3 and 4, see Appendix B) also for the higher frequency part of the harmonic spectrum.

18. L. Plaja and J. A. Pérez-Hernández, “A quantitative S-Matrix approach to high-order harmonic generation from multiphoton to tunneling regimes,” Opt. Express **15**, 3629–3634 (2007).
[CrossRef] [PubMed]

19. J. A. Pérez-Hernández and L. Plaja, “Quantum description of the high-order harmonic generation in multiphoton and tunneling regimes,”Phys. Rev. A **76**, 023829-1-7 (2007).
[CrossRef]

*H*(

*t*)=

*H*+

_{a}*V*(

_{i}*t*),

*H*being the atomic Hamiltonian, and

_{a}*V*(

_{i}*t*) describing the interaction with a linearly polarized electromagnetic wave,

*V*(

_{i}*t*)=−(

*q*/

*mc*)

*A*(

*t*)

*p*+

_{z}*q*

^{2}/(2

*mc*

^{2})

*A*

^{2}(

*t*) (velocity gauge) or

*V*(

_{i}*t*)=−

*qE*(

*t*)

*z*(length gauge). Following the philosophy of Feshbach formalism, let us consider two general orthogonal projectors

*Q*̂ and

*P*̂. These operators are constructed accordingly to the different boundary conditions at infinity of bound states (

*Q*̂) and continuum states (

*P*̂). By definition, we find

*Q*̂+

*P*̂=1,

*Q*̂

*P*̂=

*P*̂

*Q*̂=0,

*Q*̂

^{2}=

*Q*̂,

*P*̂

^{2}=

*P*̂, and |

*ψ*(

*t*

_{0})〉=

*Q*̂|ψ(

*t*

_{0})〉 as the state is assumed initially to be the ground (note that this does not imply that the projector contains only one state). We further assume [

*H*,

_{a}*Q*̂]≃0 and [

*H*,

*P*̂]≃0. Imposing these definitions, Eq. (1) leads to two coupled equations (one for the bounded part of the wavefunction and other for the free part)

*rhs*of Eq. (2) describes the possibility of atomic excitation (ground-state dressing), while the

*rhs*of Eq. (3) describes ionization. The SFA consists in setting

*Q*̂

*V*

*(*

_{i}*t*′)

*Q*̂=0 in Eq. (2), considering that the field interaction leads invariably to ionization, together with the former condition [

*H*,

*P*̂]=0, which prevents from the recombination of an ionized state. Note that these expressions resemble partially to the truncated propagator in [15

15. O. Smirnova, M. Spanner, and M. Ivanov, “Coulomb and polarization effects in sub-cycle dynamics of strong field ionization,” J. Phys. B: At. Mol. Opt. Phys. **39**, S307–321 (2006).
[CrossRef]

*t*

_{0},

*t*′).

*a*̂=−(

*q*/

*m*)

*∂V*/∂

_{c}*z*(

*V*being the atomic Coulomb potential). Since the higher frequencies of the harmonic spectrum involve the most energetic transitions (i.e. continuum to bound states), the relevant part of the acceleration corresponds to

_{c}*a*and

_{b}*a*are two interfering contributions to the total acceleration, associated with transitions between the continuum and the bare atomic ground state and to its field dressing (not considered in the SFA). Respectively,

_{d}*P*̂

*V*(

_{i}*t*

_{1})

*Q*̂,

*P*̂

*G*

^{+}(

*t*,

*t*

_{1})

*P*̂,

*Q*̂

*V*(

_{i}*t*

_{2})

*Q*̂ and

*Q*̂

*G*

^{−}(

*t*,

*t*

_{2})

*Q*̂. In general this is not straightforward, but may be greatly simplified for the study of harmonic generation. High-order harmonics are generated through transitions from continuum to bound states that take place during rescattering. Therefore, to compute the higher frequency part of the spectrum is enough to describe the process during these recollision events. According to the semiclassical theory, the most energetic collisions take place near the zero of the electric field. The electric field being small, we shall assume that the transition lower state at the beginning of the recollision is the atomic ground state. During the collision, as a result of the field interaction, this state evolves perciptibly. Therefore, the bound state at time t results from the evolution of the atomic ground state during the time lapse

*δ t*during which the recollision takes place. Mathematically this is given by Eq. (2) but with the lower limit of the time integral set to

_{s}*t*−

*δ t*instead of

_{s}*t*

_{0}. Consequently, the same substitution is to be done in the lower limit of the outer integral in Eq. (6). For an estimation of the value of

*δ t*, see Appendix C.

_{s}*(given in Appendix D) resulting from the classical estimation of the field interaction energy*

_{s}*V*(

_{i}*τ*), averaged over the scattering time interval

*δt*. Approximating the ground-state by a free wave with negative kinetic energy, we have

_{s}*Q*̂

*H*(

*t*

_{2})

*Q*̂⋍

*p*̂

^{2}/2

*m*+Δ

_{s}*I*̂, and

*I*̂ being the identity operator), as well as

_{0}being the energy of the bare atomic ground state,

*H*|

_{a}*ψ*(

*t*

_{0})〉=ε

_{0}|

*ψ*(

*t*

_{0})〉.

*P*̂

*V*(

_{i}*t*

_{1})

*Q*̂ and

*P*̂

*G*+(

*t*,

*t*

_{1})

*P*̂ in Eqs. (3)–(6) can be evaluated accordingly to the standard procedure in SFA methods as follows. First we consider a planewave basis, {

**k**}, for the subspace defined by

*P*̂, therefore

**p**=

*h*̄

**k**under the influence of the electromagnetic field. Therefore,

*V*(

_{i}**k**,

*t*

_{1})=−(

*q*/

*mc*)

*A*(

*t*

_{1})

*k*+

_{z}*q*

^{2}/(2

*mc*

^{2})

*A*

^{2}(

*t*

_{1}). In addition we have

20. V. P. Krainov, “Ionization rates and energy and angular distributions at the barrier-suppression ionization of complex atoms and atomic ions,” J. Opt. Soc. Am. B **14**, 425431 (1997).
[CrossRef]

*C*/

_{F}*rn*=(2

*Z*

^{2}/

*n*

^{2}

*E*

_{0}

*r*) n(here n=1, Z=1), and with

**k**,τ)=

*h*̄

^{2}

*k*

^{2}/2

*m*−(

*q*/

*mc*)

*A*(

*τ*)

*k*+

_{z}*q*

^{2}/(2

*mc*

^{2})

*A*

^{2}(

*τ*). Introducing these definitions in Eqs. (5) and (6) (see Appendix E) we can compute the final acceleration as the sum of the transitions of each Volkov state to the bare atom and to the dressed part of the ground state

*ψ*(

*t*

_{0})〉=|

*ϕ*

_{0})〉, the ground state of the atom, and

*a*and

_{b}*a*) interfere destructively, leading to the total acceleration

_{d}*a*(

_{b}**k**) can be computed numerically very effectively without recurring to the saddle-point approximation and, thus, retaining the full quantum description of the process. This is done by integrating the set of (uncoupled) one dimensional differential equations, each associated with a particular Volkov wave k, that result from differentiating Eq. (14)

*a*(

_{d}*t*) to zero in Eq. (4) or (13).

*λ*

^{−n}in solid lines. Note that, linked to its better accuracy, our model reproduces also quite accurately the scaling of the yields with the wavelengths. Therefore, as a main result, we conclude that the physical reason for the divergence between the scaling laws predicted by S-Matrix SFA models and the exact 3D TDSE can be attributed to the influence of the electromagnetic field in the ground state.

*n*=3.61±0.16 and

*n*=3.06±0.06) are also consistent with the expectations of the standard model (

*n*between 3 and 4), as discussed above. On the other hand, it is also apparent the discrepancy of the S-matrix SFA models with the exact 3D TDSE results (

*n*=5.63±0.22), and with the prediction of our model (

*n*=5.41±0.21).

## 3. Conclusion

**98**, 013901-1-4 (2007).
[CrossRef]

*E*

_{0}(

*t*) being pulse envelope: a sin

^{2}shape for the 4 cycle pulse, a trapezoidal shape of 6 cycles (with 2 cycles of linear turn-on, 2 cycles of constant amplitude and 2 cycles of linear turn-off), and a trapezoidal shape of 9 cycles (with 1/2 cycle of linear turn-on, 8 cycles of constant amplitude and 1/2 cycle of linear turn-off). The later corresponds to the field used in the previous works [9

**98**, 013901-1-4 (2007).
[CrossRef]

**99**, 253903-1-4 (2007).
[CrossRef]

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

*τ*. The higher harmonics have associated typical rescattering times of the order of 3

*T*/4 (

*T*being the laser period). Therefore the first factor in the integral contributes approximately as

*T*

^{−3/2}∝

*λ*

^{−3/2}to the dipole amplitude. On the other hand, an additional

*ω*

^{−1}

_{0}∝

*λ*factor comes from the time integration of the remaining oscillating functions. Consequently, in the standard model, the dipole amplitude can be expected to be proportional to

*λ*

^{−1/2}.

*a*being the dipole acceleration (second derivative of the dipole amplitude, therefore scaling as

_{q}*λ*

^{−5/2}). Consequently, the estimation of the standard model for the harmonic power is a scaling

*λ*

^{−5}. As pointed out in [9

**98**, 013901-1-4 (2007).
[CrossRef]

**98**, 013901-1-4 (2007).
[CrossRef]

**99**, 253903-1-4 (2007).
[CrossRef]

*λ*, and the yield at a given harmonic frequency (product of the harmonic power times the interaction time) would scale as

*λ*

^{−4}, according to the above discussion. Note that the same scaling (

*λ*

^{−4}) is also expected if we consider the yield over a fixed energy interval. We must note, however, that previous works [9

**98**, 013901-1-4 (2007).
[CrossRef]

**99**, 253903-1-4 (2007).
[CrossRef]

*λ*

^{−3}. In any case, our numerical computations confirm a scaling exponent between −3 and −4, which is in conflict with the exact TDSE (≃−5.6), as is demonstrated in this paper.

*π*/

*ω*

_{0}, therefore the size of the wavepacket of the returning electron at the rescattering time is

*.*

_{s}*V*(

_{i}*τ*) over the collision time

*δt*,

_{s}*ω*

_{0}

*t*+

*ϕ*≃

*nπ*), we find 〈cos(

*ω*

_{0}

*t*+

*ϕ*)〉≃sin(

*ω*

_{0}

*δ t*)/

_{s}*ω*

_{0}

*δ t*and 〈cos

_{s}^{2}(

*ω*

_{0}t+

*ϕ*)〉≃1/2+sin(2

*ω*

_{0}

*δ t*)/4

_{s}*ω*

_{0}

*δ t*, therefore

_{s}*k*, we shall approximate the dressed state energy to the instantaneous energy of the bound state in the electromagnetic field 〈

_{z}*ϕ*

_{0}|

*H*(

*t*)|

*ϕ*

_{0}〉=

*ε*

_{0}+2

*U*cos

_{p}^{2}(

*ω*

_{0}

*t*+

*ϕ*) (|

*ϕ*

_{0}〉 being the ground state of the atom). Note from the virial theorem that 〈

*V*(

*r*)〉=2ε

_{0}. On the other hand, as the electric field during the collision is small, the departure from equilibrium is harmonic and, therefore, half of the interaction energy is invested into kinetic energy. Therefore, assuming energy equipartition, the kinetic energy along the field polarization direction is

*δt*, the solution of

_{s}*k*reads as

_{z}*P*̂

^{2}≡

*P*̂ and the definition (9) in Eq. (5), we have

*ψ*(

*t*

_{0})〉=|

*ϕ*

_{0}〉, therefore

**k**.

*p*̂

^{2}and

*a*̂ approximately commute and we may write

*t*−

*δ t*) does not contribute, as the wavefunction overlap before the rescattering is negligible. Therefore

_{s}## Acknowledgments

## References and links

1. | L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Zh. Eksp. Teor. Fiz. |

2. | F. H. M. Faisal, “Multiple Absorption of Laser Photons by Atoms,” J. Phys. B |

3. | H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A |

4. | M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A |

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

6. | W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, “A unified theory of high-harmonic generation: application to polarization properties of the harmonics,” Phys. Rev. A |

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

8. | A. Zaïr, M. Holler, A. Guadalini, F. Shapper, J. Biegert, L. Gallmann, U. Keller, A. S. Wyatt, A. Monmayrant, I. A. Walmsley, E. Cornier, T. Auguste, J. P. Caumes, and P. Salières, “Quantum Path Interferences in High-Order Harmonic Generation,” Phys. Rev. Lett. |

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

10. | K. Schiessl, K. L. Ishikawa, E. Person, and J. Burgdörfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. |

11. | M. V. Frolov, N. L. Manakov, and A. F. Starace, “Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. |

12. | G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated Single-Cycle Attosecond Pulses,” Science |

13. | M. Janjusevic and Y. Hahn, “Testing of laser-dressed atomic states,” J. Opt. Soc. Am. B |

14. | P. Krstić and M. H. Mittleman, “S-matrix theory of multiphoton ionization,” J. Opt. Soc. Am. B |

15. | O. Smirnova, M. Spanner, and M. Ivanov, “Coulomb and polarization effects in sub-cycle dynamics of strong field ionization,” J. Phys. B: At. Mol. Opt. Phys. |

16. | W. Becker, J. Chen, S. G. Chen, and D. B. Milosevic, “Dressed-state strong-field approximation for laser induced molecular ionization,” Phys. Rev. A |

17. | 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. Di Mauro, “Scaling strong-field interactions towards the classical limit,” Nature Phys. |

18. | L. Plaja and J. A. Pérez-Hernández, “A quantitative S-Matrix approach to high-order harmonic generation from multiphoton to tunneling regimes,” Opt. Express |

19. | J. A. Pérez-Hernández and L. Plaja, “Quantum description of the high-order harmonic generation in multiphoton and tunneling regimes,”Phys. Rev. A |

20. | V. P. Krainov, “Ionization rates and energy and angular distributions at the barrier-suppression ionization of complex atoms and atomic ions,” J. Opt. Soc. Am. B |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(190.4160) Nonlinear optics : Multiharmonic generation

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(260.5210) Physical optics : Photoionization

(270.6620) Quantum optics : Strong-field processes

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

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: February 17, 2009

Revised Manuscript: April 2, 2009

Manuscript Accepted: April 2, 2009

Published: May 28, 2009

**Citation**

J. A. Pérez-Hernández, L. Roso, and L. Plaja, "Harmonic generation beyond the Strong-Field Approximation: the physics behind the short-wave-infrared scaling laws," Opt. Express **17**, 9891-9903 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-12-9891

Sort: Year | Journal | Reset

### References

- L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Zh. Eksp. Teor. Fiz. 47, 1945-1956 (1964) [ Sov. Phys. JETP 20 1307-1314 (1965)].
- F. H. M. Faisal, "Multiple Absorption of Laser Photons by Atoms," J. Phys. B 6, L89-L92 (1973). [CrossRef]
- H. R. Reiss, "Effect of an intense electromagnetic field on a weakly bound system," Phys. Rev. A 22,1786-1813 (1980). [CrossRef]
- M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117-2132 (1994). [CrossRef] [PubMed]
- M. Lewenstein, P. Sali`eres, and A. L’Huiller, "Phase of the atomic polarization in high-order harmonic generation," Phys. Rev. A 52, 4747-4754 (1995). [CrossRef] [PubMed]
- W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, "A unified theory of high-harmonic generation: application to polarization properties of the harmonics," Phys. Rev. A 56, 645-656 (1997). [CrossRef]
- P. B. Corkum, "Plasma perspective on strong-field multiphoton ionization," Phys. Rev. Lett. 71, 1994-1997 (1993). [CrossRef] [PubMed]
- A. Za¨ır, M. Holler, A. Guadalini, F. Shapper, J. Biegert, L. Gallmann, U. Keller, A. S. Wyatt, A. Monmayrant, I. A. Walmsley, E. Cornier, T. Auguste, J. P. Caumes, and P. Sali`eres, "Quantum Path Interferences in High-Order Harmonic Generation," Phys. Rev. Lett. 100, 143902-1-4 (2008). [CrossRef]
- J. Tate, T. Auguste, H. G. Muller, P. Sali`eres, P. Agostini, and L. F. Di Mauro, "Scaling of wave-packet dynamics in an intense midinfrared field," Phys. Rev. Lett. 98,013901-1-4 (2007). [CrossRef]
- K. Schiessl, K. L. Ishikawa, E. Person, and J. Burgd¨orfer,"Quantum path interference in the wavelength dependence of high-harmonic generation," Phys. Rev. Lett. 99, 253903-1-4 (2007). [CrossRef]
- M. V. Frolov, N. L. Manakov, and A. F. Starace, "Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence," Phys. Rev. Lett. 100, 173001-1-4 (2008). [CrossRef]
- G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006). [CrossRef] [PubMed]
- M. Janjusevic and Y. Hahn, "Testing of laser-dressed atomic states," J. Opt. Soc. Am. B 7, 592-597 (1990). [CrossRef]
- P. Krsti’c and M. H. Mittleman, "S-matrix theory of multiphoton ionization," J. Opt. Soc. Am. B 7, 587-591 (1990). [CrossRef]
- O. Smirnova, M. Spanner, and M. Ivanov, "Coulomb and polarization effects in sub-cycle dynamics of strong field ionization," J. Phys. B: At. Mol. Opt. Phys. 39, S307-321 (2006). [CrossRef]
- W. Becker, J. Chen, S. G. Chen, and D. B. Milosevic, "Dressed-state strong-field approximation for laser induced molecular ionization," Phys. Rev. A 76, 033403-1-7 (2007). [CrossRef]
- 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. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008). [CrossRef]
- L. Plaja and J. A. P’erez-Hern’andez, "A quantitative S-Matrix approach to high-order harmonic generation from multiphoton to tunneling regimes," Opt. Express 15, 3629-3634 (2007). [CrossRef] [PubMed]
- J. A. P´erez-Hern´andez and L. Plaja, "Quantum description of the high-order harmonic generation in multiphoton and tunneling regimes," Phys. Rev. A 76, 023829-1-7 (2007). [CrossRef]
- V. P. Krainov, "Ionization rates and energy and angular distributions at the barrier-suppression ionization of complex atoms and atomic ions," J. Opt. Soc. Am. B 14, 425431 (1997). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.