## Semiclassical model for attosecond angular streaking |

Optics Express, Vol. 18, Issue 17, pp. 17640-17650 (2010)

http://dx.doi.org/10.1364/OE.18.017640

Acrobat PDF (1351 KB)

### Abstract

Attosecond angular streaking is a new technique to achieve unsurpassed time accuracy of only a few attoseconds. Recently this has been successfully used to set an upper limit on the electron tunneling delay time in strong laser field ionization. The measurement technique can be modeled with either the time-dependent Schrödinger equation (TDSE) or a more simple semiclassical approach that describes the process in two steps in analogy to the three-step model in high harmonic generation (HHG): step one is the tunnel ionization and step two is the classical motion in the strong laser field. Here we describe in detail a semiclassical model which is based on the ADK theory for the tunneling step, with subsequent classical propagation of the electron in the laser field. We take into account different ellipticities of the laser field and a possible wavelength-dependent ellipticity that is typically observed for pulses in the two-optical-cycle regime. This semiclassical model shows excellent agreement with the experimental result.

© 2010 OSA

## 1. Introduction and basic background

2. H. R. Reiss, “Limits on tunneling theories of strong-field ionization,” Phys. Rev. Lett. **101**(4), 043002 (2008). [CrossRef] [PubMed]

3. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**(13), 1994–1997 (1993). [CrossRef] [PubMed]

4. H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, “Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation,” Appl. Phys. B **69**(4), 327–332 (1999). [CrossRef]

6. P. Dietrich, F. Krausz, and P. B. Corkum, “Determining the absolute carrier phase of a few-cycle laser pulse,” Opt. Lett. **25**(1), 16–18 (2000). [CrossRef]

7. P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. **4**(7), 565–570 (2008). [CrossRef]

8. C. Smeenk, L. Arissian, A. Staudte, D. M. Villeneuve, and P. B. Corkum, “Momentum space tomographic imaging of photoelectrons,” J. Phys. B **42**(18), 185402 (2009). [CrossRef]

*ε*= 1) we observe a significant effect on the electron distribution in strong field ionization due to the high nonlinearity of the ionization process. Therefore it is essential to characterize carefully the ellipticity in the experiment. The idea of mapping the time of ionization to the final momentum is based on the fact that, in the case of strong laser fields, the wave packet of the ionized electron is moving essentially along its classical trajectory in the laser field. This can be seen most elegantly in the Feynman description of quantum mechanics [9

9. R. P. Feynman, “Space-Time Approach to Non-Relativistic Quantum Mechanics,” Rev. Mod. Phys. **20**(2), 367–387 (1948). [CrossRef]

10. P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science **292**(5518), 902–905 (2001). [CrossRef] [PubMed]

_{0},t

_{0}) to (x,t).

_{0}at t

_{0}and finishing in x at time t. These paths are weighted with the exponential factor

*iS*in units of ħ. In case of strong laser fields this is a strongly oscillating function, so only the neighborhood of those quantum paths contribute to the integral, where the phase, and so the action S, is stationary. This condition of stationary action defines the trajectories of particles in classical mechanics. Therefore we can approximate the momentum obtained by the particles in the field E(t) given by Eq. (2) by the classical momentum p where A(t) is the pulse envelope, ω

_{0}the center radial frequency of the laser pulse,

*x*→ and

*y*→ two orthogonal unit vectors in the plane of polarization, and ε describes the ellipticity and helicity of the pulse, where

_{i}is the instant of ionization. The approximation in Eq. (3) assumes a slowly varying pulse envelope that allows us to set the time derivative of the envelope to zero and neglect the last integral term in line 4 of Eq. (3). This approximation work extremely well even for few cycle laser pulses: A numerical comparison of this approximation with the exact result leads to a maximal error of the order of 7.5% of the maximum momentum for a 5.9-fs Gaussian, Fourier-limited laser pulse with a center wavelength of 780 nm. For a perfect circularly polarized pulse with

*ε*= 1, Eq. (3) implies that the final momentum shows a 90 - degree phase shift compared to the direction of the electric field of the pulse at the instant of ionization. This means that the maximum of the final momentum distribution is rotated by 90 degrees from the direction of the maximum electric field. The final momentum distribution depends on the ellipticity but can be calculated in a straightforward way as well. The temporal trajectory can be determined by numerical integration and is shown in Fig. 1 (Media 1).

3. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**(13), 1994–1997 (1993). [CrossRef] [PubMed]

## 2. Full characterization of elliptically polarized field

7. P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. **4**(7), 565–570 (2008). [CrossRef]

13. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. **68**(9), 3277–3295 (1997). [CrossRef]

14. C. Iaconis and I. A. Walmsley, “Self-Referencing Spectral Interferometry for Measuring Ultrashort Optical Pulses,” IEEE J. Quantum Electron. **35**(4), 501–509 (1999). [CrossRef]

7. P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. **4**(7), 565–570 (2008). [CrossRef]

15. M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, C. Spielmann, and F. Krausz, “A novel high-energy pulse compression system: generation of multigigawatt sub-5-fs pulses,” Appl. Phys. B **65**(2), 189–196 (1997). [CrossRef]

16. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B **79**(6), 673–677 (2004). [CrossRef]

^{2}pulse. Therefore the exact dispersion of the material of the retardation plate has to be taken into account. There are no single birefringent materials which show the desired retardation properties over a sufficient bandwidth. Therefore a combination of materials is being used for optimal dispersion compensation, which are referred to as achromatic retardation plates. In the visible and near-infrared regime they consist of a combination of MgF

_{2}and quartz. This also yields a small difference in group velocity and group delay dispersion (GDD) along the ordinary and extraordinary axes. The use of quartz complicates the propagation of the pulse through the retardation plate: G. Szivessy and Cl. Münster [17

17. G. Szivessy and C. Münster, “Über die Prüfung der Gitteroptik bei aktiven Kristallen,” Ann. Phys. **412**(7), 703–736 (1934). [CrossRef]

_{2}plate. The eigenmodes of the combination of both parts of the retardation plate are elliptical and the orientation of the major axes changes with wavelength. To calculate the electric field behind the retardation plate, the optical activity of quartz and the dispersion of quartz and MgF

_{2}have to be taken into account. The calculation of the propagation through the retardation plate is performed in two steps. First, the linear pulse is being propagated along one of the axis of the retardation plate. This step takes care of the dispersion of the pulse and ignores the effect of optical activity. Second, the stretched pulse is Fourier transformed, split into polarization eigenmodes (Eq. (4) and Eq. (5)) P1 and P2 and for each frequency component a frequency dependant phase shift

*δ*is acquired.

17. G. Szivessy and C. Münster, “Über die Prüfung der Gitteroptik bei aktiven Kristallen,” Ann. Phys. **412**(7), 703–736 (1934). [CrossRef]

19. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A **10**(11), 2383–2393 (1993). [CrossRef]

21. M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. **23**(12), 1980–1985 (1984). [CrossRef] [PubMed]

_{2}. The necessary data for the second propagation step through the retardation plate used by Eckle et al. [7

**4**(7), 565–570 (2008). [CrossRef]

**4**(7), 565–570 (2008). [CrossRef]

22. P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A **53**(3), 1725–1745 (1996). [CrossRef] [PubMed]

## 3. Final momentum distribution using ADK rate and Newtonian motion

**4**(7), 565–570 (2008). [CrossRef]

**4**(7), 565–570 (2008). [CrossRef]

23. N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B **8**(6), 1207–1211 (1991). [CrossRef]

_{s}for circular polarized, stochastic light [23

23. N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B **8**(6), 1207–1211 (1991). [CrossRef]

_{i}the ionization potential and ω the angular frequency of the laser light. Delone and Krainov used p

_{║}as the final momentum of the electron in the polarization plane and p

_{┴}as the final electron momentum perpendicular to it. This extension provides a momentum distribution along and perpendicular to the polarization directions for non-monochromatic circularly polarized light. Since we ignore the Coulomb potential and the spatial dependence of the electric field in our semiclassical model, the initial momentum distribution, centered at momentum 0 is transferred into the final momentum distribution by an offset equal to the momentum gained by the electron on its classical trajectory. To adapt the momentum distribution of Eq. (7) for an instantaneous ionization, we change the interpretation of p

_{┴}to the momentum perpendicular to the electric field vector at the instant of ionization. In tunneling theories the electron has zero velocity after tunneling through the potential barrier. We therefore shift the distribution of p

_{║}by E*/ω to zero. We use this adapted momentum distribution to provide initial conditions for the classical trajectories starting at t

_{i}with an electric field strength E = E(t

_{i}). Momentum frequency distributions are created for all distributions numerically using the acceptance–rejection method. Substituting the shifted p

_{║}in Eq. (7), this equation can be written as a product of three exponential functions, each of them depending either on the ionization potential, the shifted p

_{║}or p

_{┴}, showing the statistical independence of this quantities. Motivated by this independence we used Eq. (7) for the momentum distribution and the ionization rate was calculated by the ADK-formula for circularly polarized light [11].

^{14}W/cm

^{2}. The electron is then propagated in the electric field with no spatial dependency up to 200 fs after the pulse center. To match the experimental ion data, the temperature of the gas target of 2.8 K was taken into account by adding a velocity according to the Maxwell distribution. The final velocity is transformed to momentum and this value is then stored in a histogram weighted by the ADK Rate of the electric field at the instant of ionization, assuming that the ADK Rate for circularly polarized light is still a good approximation for the ionization rate of the elliptically polarized pulse with an ellipticity of about 0.9. For each starting time 400 trajectories have been computed, which can be done efficiently by calculating the propagation only once for each starting point and then calculating the initial and thermal velocities independently for each trajectory, since these velocities represent just an offset of the trajectory in velocity space. The intensity is used as a fitting parameter to the experimental data. The best fit was achieved at an intensity of 3.92 10

^{14}W/cm

^{2}resulting in a Keldysh parameter γ gamma of 1.14, which is well within the experimentally estimated values. The use of tunneling based theories for γ >1 has been successfully shown by Uiberacker et al. [25

25. M. Uiberacker, Th. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond real-time observation of electron tunnelling in atoms,” Nature **446**(7136), 627–632 (2007). [CrossRef] [PubMed]

**4**(7), 565–570 (2008). [CrossRef]

## 4. Limitations

26. C. M. Maharjan, A. S. Alnaser, X. M. Tong, B. Ulrich, P. Ranitovic, S. Ghimire, Z. Chang, I. V. Litvinyuk, and C. L. Cocke, “Momentum imaging of doubly charged ions of Ne and Ar in the sequential ionization region,” Phys. Rev. A **72**, 041403(R) (2005) [CrossRef]

^{16}W/cm

^{2}. This picture shows only the general behavior, since in this intensity region, where the peak intensity is far above the over-barrier intensity, the ADK rates are no longer valid. Since the ground state can be depleted almost completely within half a laser cycle, see Fig. 5, more accurate rates are needed for a quantitative calculation. Empirical formulas for the static field ionization rates are known for several elements [27

27. X. M. Tong and C. D. Lin, “Empirical formula for statistic field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B **38**(15), 2593–2600 (2005). [CrossRef]

27. X. M. Tong and C. D. Lin, “Empirical formula for statistic field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B **38**(15), 2593–2600 (2005). [CrossRef]

^{2}factor in Eq. (8) it follows that the angular momenta of ions and electrons have the same sign and the numerical analysis of this equation shows that L cannot be assumed to be zero. As a consequence the angular momentum has to be provided by the electric field. The time dependence of L and the kinetic energy of the system of ion and electron show two striking features in Fig. 6 . The amount of photons needed to provide the kinetic energy is different from the amount of photons needed to provide the angular momentum. The second feature is that the trend is not correct: In the region where the magnitude of the total angular momentum reaches its maximum between its second and third zero crossing, the kinetic energy is continuously increasing. There is a region where the magnitude of the total angular momentum decreases but the total kinetic energy increases. A decrease of angular momentum could be only due to emission of photons while an increase of kinetic energy indicates an absorption of photons. As long emission of low energy photon is excluded, this leads to a contradiction in terms of quantum mechanics. Its origin is the description of the laser field: One could theoretically produce the electric field used in Eq. (3) by a plate capacitor which rotates with frequency ω

_{0}while the electric field inside the capacitor has the amplitude A(t)∙(cos

^{2}(ω

_{0}t) + ε∙sin

^{2}(ω

_{0}t))

^{1/2}. This external plate capacitor will enforce the angular momentum conservation in a classical picture.

## 5. Summary

## Acknowledgments

## References and links

1. | L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Soviet Phys. JETP |

2. | H. R. Reiss, “Limits on tunneling theories of strong-field ionization,” Phys. Rev. Lett. |

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

4. | H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, “Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation,” Appl. Phys. B |

5. | G. G. Paulus, ““A Meter of the “Absolute” Phase of Few-Cycle Laser Pulses,” Laser Phys. |

6. | P. Dietrich, F. Krausz, and P. B. Corkum, “Determining the absolute carrier phase of a few-cycle laser pulse,” Opt. Lett. |

7. | P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. |

8. | C. Smeenk, L. Arissian, A. Staudte, D. M. Villeneuve, and P. B. Corkum, “Momentum space tomographic imaging of photoelectrons,” J. Phys. B |

9. | R. P. Feynman, “Space-Time Approach to Non-Relativistic Quantum Mechanics,” Rev. Mod. Phys. |

10. | P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science |

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

12. | T. Gabriel, “Dispersion-Free Reflective Phase Retarder for Few-Cycle Femtosecond Pulses”, Paper FB6, Optical Interference Coatings Topical Meeting, Tuscon (AZ), June 6–11 2010. |

13. | R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. |

14. | C. Iaconis and I. A. Walmsley, “Self-Referencing Spectral Interferometry for Measuring Ultrashort Optical Pulses,” IEEE J. Quantum Electron. |

15. | M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, C. Spielmann, and F. Krausz, “A novel high-energy pulse compression system: generation of multigigawatt sub-5-fs pulses,” Appl. Phys. B |

16. | C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B |

17. | G. Szivessy and C. Münster, “Über die Prüfung der Gitteroptik bei aktiven Kristallen,” Ann. Phys. |

18. | S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. |

19. | S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A |

20. | E. D. Palik, “Handbook of Optical Constants of Solids,” (Academic Press, 1985). |

21. | M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. |

22. | P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A |

23. | N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B |

24. | H. R. Schwarz, “Numerische Mathematik,” B. G. Teubner, ed. (Verlag, 1997). |

25. | M. Uiberacker, Th. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond real-time observation of electron tunnelling in atoms,” Nature |

26. | C. M. Maharjan, A. S. Alnaser, X. M. Tong, B. Ulrich, P. Ranitovic, S. Ghimire, Z. Chang, I. V. Litvinyuk, and C. L. Cocke, “Momentum imaging of doubly charged ions of Ne and Ar in the sequential ionization region,” Phys. Rev. A |

27. | X. M. Tong and C. D. Lin, “Empirical formula for statistic field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B |

**OCIS Codes**

(000.2700) General : General science

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: May 25, 2010

Revised Manuscript: July 23, 2010

Manuscript Accepted: July 23, 2010

Published: August 2, 2010

**Citation**

M. Smolarski, P. Eckle, U. Keller, and R. Dörner, "Semiclassical model for attosecond angular streaking," Opt. Express **18**, 17640-17650 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17640

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### References

- L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Soviet Phys. JETP 20, 1307 (1965).
- H. R. Reiss, “Limits on tunneling theories of strong-field ionization,” Phys. Rev. Lett. 101(4), 043002 (2008). [CrossRef] [PubMed]
- P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef] [PubMed]
- H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, “Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation,” Appl. Phys. B 69(4), 327–332 (1999). [CrossRef]
- G. G. Paulus, ““A Meter of the “Absolute” Phase of Few-Cycle Laser Pulses,” Laser Phys. 15, 843–854 (2005).
- P. Dietrich, F. Krausz, and P. B. Corkum, “Determining the absolute carrier phase of a few-cycle laser pulse,” Opt. Lett. 25(1), 16–18 (2000). [CrossRef]
- P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schöffler, H. G. Muller, R. Dörner, and U. Keller, “Attosecond angular streaking,” Nat. Phys. 4(7), 565–570 (2008). [CrossRef]
- C. Smeenk, L. Arissian, A. Staudte, D. M. Villeneuve, and P. B. Corkum, “Momentum space tomographic imaging of photoelectrons,” J. Phys. B 42(18), 185402 (2009). [CrossRef]
- R. P. Feynman, “Space-Time Approach to Non-Relativistic Quantum Mechanics,” Rev. Mod. Phys. 20(2), 367–387 (1948). [CrossRef]
- P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292(5518), 902–905 (2001). [CrossRef] [PubMed]
- M. V. Ammosov, N. B. Delone, and V. P. Kraĭnov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).
- T. Gabriel, “Dispersion-Free Reflective Phase Retarder for Few-Cycle Femtosecond Pulses”, Paper FB6, Optical Interference Coatings Topical Meeting, Tuscon (AZ), June 6–11 2010.
- R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68(9), 3277–3295 (1997). [CrossRef]
- C. Iaconis and I. A. Walmsley, “Self-Referencing Spectral Interferometry for Measuring Ultrashort Optical Pulses,” IEEE J. Quantum Electron. 35(4), 501–509 (1999). [CrossRef]
- M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, C. Spielmann, and F. Krausz, “A novel high-energy pulse compression system: generation of multigigawatt sub-5-fs pulses,” Appl. Phys. B 65(2), 189–196 (1997). [CrossRef]
- C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004). [CrossRef]
- G. Szivessy and C. Münster, “Über die Prüfung der Gitteroptik bei aktiven Kristallen,” Ann. Phys. 412(7), 703–736 (1934). [CrossRef]
- S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. 10(11), 2371–2382 (1993). [CrossRef]
- S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10(11), 2383–2393 (1993). [CrossRef]
- E. D. Palik, “Handbook of Optical Constants of Solids,” (Academic Press, 1985).
- M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. 23(12), 1980–1985 (1984). [CrossRef] [PubMed]
- P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A 53(3), 1725–1745 (1996). [CrossRef] [PubMed]
- N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B 8(6), 1207–1211 (1991). [CrossRef]
- H. R. Schwarz, “Numerische Mathematik,” B. G. Teubner, ed. (Verlag, 1997).
- M. Uiberacker, Th. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond real-time observation of electron tunnelling in atoms,” Nature 446(7136), 627–632 (2007). [CrossRef] [PubMed]
- C. M. Maharjan, A. S. Alnaser, X. M. Tong, B. Ulrich, P. Ranitovic, S. Ghimire, Z. Chang, I. V. Litvinyuk, and C. L. Cocke, “Momentum imaging of doubly charged ions of Ne and Ar in the sequential ionization region,” Phys. Rev. A 72, 041403(R) (2005) [CrossRef]
- X. M. Tong and C. D. Lin, “Empirical formula for statistic field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B 38(15), 2593–2600 (2005). [CrossRef]

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