## Asymmetries in the momentum distributions of electrons stripped by a XUV chirped pulse in the presence of a laser field |

Optics Express, Vol. 20, Issue 20, pp. 22475-22480 (2012)

http://dx.doi.org/10.1364/OE.20.022475

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### Abstract

The ionization of hydrogen by a chirped XUV pulse in the presence of a few cycle infrared laser pulse has been investigated. The electron momentum distribution has been obtained by treating the interaction of the atom with the XUV radiation at the first order of the time-dependent perturbation theory and describing the emitted electron through the Coulomb-Volkov wavefunction. The results of the calculations agree with the ones found by solving numerically the time-dependent Schrödinger equation. It has been found that depending on the delay between the pulses the combined effect of the XUV chirp and of the steering action on the infrared field brings about asymmetries in the electron momentum distribution. These asymmetries may give information on both the chirp and the XUV pulse duration.

© 2012 OSA

## 1. Introduction

1. P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Progr. Phys. **67**,, 813–855 (2004). [CrossRef]

2. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-Cycle Nonlinear Optics,” Science **320**, 1614–1617 (2008). [CrossRef] [PubMed]

3. A. Bouhal, R. Evans, G. Grillon, A. Mysyrowicz, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, and D. von der Linde, “Cross-correlation measurement of femtosecond noncollinear high-order harmonics,” J. Opt. Soc. Am. B **14**, 950–956 (1997). [CrossRef]

4. E. S. Toma, H. G. Muller, P. M. Paul, P. Breger, M. Cheret, P. Agostini, C. LeBlanc, G. Mullot, and G. Cheriaux, “Ponderomotive streaking of the ionization potential as a method for measuring pulse durations in the XUV domain with fs resolution,” Phys. Rev. A **62**, 061801(R) (2000). [CrossRef]

*E*Δ

*t*>

*h*̄. In fact, the sidebands disappear and, consequently, the ponderomotive shift is not longer observable, when the XUV duration is shorter than the period of the infrared radiation field. Different cross correlation methods have been proposed to measure attosecond pulse duration [5

5. M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature , **414**, 509 (2001). [CrossRef] [PubMed]

7. J. Itatani, F. Quéré, G. L. Yudin, M. Yu. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. **88**, 173903 (2002). [CrossRef] [PubMed]

5. M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature , **414**, 509 (2001). [CrossRef] [PubMed]

6. M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. Brabec, “Quantum Theory of Attosecond XUV Pulse Measurement by Laser Dressed Photoionization,” Phys. Rev. Lett. **88**, 173904 (2002). [CrossRef] [PubMed]

7. J. Itatani, F. Quéré, G. L. Yudin, M. Yu. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. **88**, 173903 (2002). [CrossRef] [PubMed]

8. E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science **305**, 1267–1269 (2004). [CrossRef] [PubMed]

9. Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A **71**, 011401 (2005). [CrossRef]

2. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-Cycle Nonlinear Optics,” Science **320**, 1614–1617 (2008). [CrossRef] [PubMed]

10. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. **81**, 163–234 (2009). [CrossRef]

11. L.-Y. Peng, E. A. Pronin, and A. Starace, “Attosecond pulse carrier-envelope phase effects on ionized electron momentum and energy distributions: roles of frequency, intensity and an additional IR pulse,” New J. Phys. **10**, 025030 (2008). [CrossRef]

## 2. Theory

12. D.B. Milošević and F. Ehlotzky, “Coulomb and rescattering effects in above-threshold ionization,” Phys. Rev. A **58**, 3124–3127 (1998); [CrossRef]

13. G. L. Yudin, S. Patchkovskii, and A. D. Bandrauk, “Chirp-dependent attosecond interference in the Coulomb-Volkov continuum,” J. Phys B: At. Mol Opt. Phys **41**, 045602 (2008). [CrossRef]

**q**≡ (

*q*,

_{x}*q*,

_{y}*q*) may be written, in atomic units, as Where

_{z}**q**(

*t*) =

**q**+

**A**

*(*

_{L}*t*)/

*c*is the instantaneous mechanical momentum,

**A**

*(*

_{L}*t*) the vector potential associated to the IR pulse,

**û**

*a unit vector directed along the z-axis,*

_{z}*E*(

_{H}*t*) the XUV electric field,

**d**

_{q}_{(t)}the field free dipole transition matrix element between the atomic ground state and the state describing the photoelectron emitted with mechanical momentum

**q**(

*t*),

*I*= −0.5

_{p}*a.u.*the ground state energy and

*τ*the total IR pulse duration. We remark that Eq. (1) does not account for the channel pertaining to the ionization due to the infrared action through the above threshold ionization (ATI). Therefore, the validity of

_{L}*P*(

**q**) is confined to photoelectron energy ranges well separated from the energies characterizing the photoelectrons generated by ATI. In our calculations an attosecond, linearly chirped Gaussian, XUV pulse will be assumed with the electric field given by where

*E*

_{0}

*is the field amplitude,*

_{H}*t*the instant at which the pulse reaches its maximum,

_{H}*τ*the pulse duration, taken as full width at half maximum (FWHM), for the transform-limited pulse, and

_{H}*ω*the central photon energy at

_{H}*t*=

*t*.

_{H}*β*stands for the dimensionless linear chirp rate: positive (negative) chirp corresponds to the instantaneous frequency increasing (decreasing) with time. The duration (FWHM) of the chirped pulse is

*ω*and field amplitude

_{L}*E*

_{0}

*, is taken as In Eq. (3)*

_{L}*f*(

*t*) = cos

^{2}

*πt*/

*τ*for −

_{L}*τ*/2 ≤

_{L}*t*≤

*τ*/2 and zero elsewhere. In order to have an integer number of cycles we assume

_{L}*τ*=

_{L}*n*, with

_{L}T_{L}*T*= 2

_{L}*π*/

*ω*the period of the carrier. The time lag between the maxima of the two pulses is given by

_{L}*t*. The vector potential associated to IR pulse, taken in Gaussian units as

_{H}*t*≤ −

*τ*/2 and

_{L}*t*≤

*τ*/2. We note that for

_{L}*t*>

*τ*/2,

_{L}**A**

*(*

_{L}*t*) = 0 and

**q**(

*t*) =

**q**.

*P*(

*q*,

_{x}*q*,

_{y}*q*) will be shown in the (

_{z}*q*,

_{x}*q*) plane, having put

_{z}*q*= 0 without loss of generality.

_{y}## 3. Results and discussion

*τ*= 150

_{H}*asec*FWHM,

*ω*= 90

_{H}*eV*and peak intensity

*I*= 1011

_{H}*W*/

*cm*

^{2}in the presence of a 6 cycle IR pulse with wavelength 750 nm and peak intensity

*I*= 2·10

_{L}^{13}

*W*/

*cm*

^{2}. The center of the attosecond pulse is assumed to be positioned at the peak of the laser field (

*t*= 0). The results of Fig. 1(a) show the breakdown of the photoelectron momentum distribution invariance under the reflection through the plane

_{H}*q*= 0. This invariance is commonly observed when the atomic ionization is caused by a sole very long XUV pulse. In particular, for the electron emitted along the z-direction (

_{z}*q*= 0), the two peaks centered at about the kinetic momenta

_{x}*q*> 0 is lower and broader than the other one centered at

_{z}*q*< 0. Calculations here not reported show that the peaks positions invert when

_{z}*β*changes sign, i.e. the photoelectron momentum distributions are invariant under both the transformations

*q*→ −

_{z}*q*and

_{z}*β*→ −

*β*. Moreover, it may be shown that by keeping fixed

*ω*,

_{L}*ω*and

_{H}*τ*,

_{H}*P*(

*q*,

_{x}*q*,

_{y}*q*) results to be very sensitive to the variations of both the linear chirp and the IR pulse intensity

_{z}*I*.

_{L}*t*=

_{H}*T*/4. In this case the main effect of the presence of IR pulse is the shifting of the EMD, in the momentum space, by Δ

_{L}*q*= −

_{z}*A*(

_{L}*t*)/

_{H}*c*, where

*A*(

_{L}*t*) is the value of the vector potential of the IR pulse at the instant of birth of the electron assumed to occur at the peak of the XUV pulse. This shift has been experimentally observed in the ionization of electrons ejected from the 4p state of krypton atoms under simultaneous irradiation of a 90

_{H}*eV*transform limited X-ray pulse and a femtosecond IR pulse (

*λ*= 750

*nm*) [6

6. M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. Brabec, “Quantum Theory of Attosecond XUV Pulse Measurement by Laser Dressed Photoionization,” Phys. Rev. Lett. **88**, 173904 (2002). [CrossRef] [PubMed]

*t*= 0, become vanishingly small for

_{H}*t*=

_{H}*T*/4. The main features of the EMD shown in the Fig. 1 may be conveniently illustrated by considering that, for a sufficiently short XUV pulse (

_{L}*τ*<

_{CH}*T*/4), Eq. (1) may be evaluated by expanding the integrand in power series of (

_{L}*t*−

*t*) and by keeping, in the exponent in curly brackets, terms up to the second order in (

_{H}*t*−

*t*). Then, the differential transition probability, taking the XUV radiation in the rotating wave approximation, assumes the simple form where with

_{H}14. D. Bauer and P. Koval, “Qprop: A Schrödinger-solver for intense laser-atom interaction,” Comp. Phys Commun. **174**, 396–421 (2006). [CrossRef]

*t*= 0 and

_{H}*t*=

_{H}*T*/4, already shown in Fig. 1. More generally, from Eq. (4), it turns out that the peaks of the photoelectron momentum distributions, for given values of

_{L}*q*, are located about at

_{x}*q*. For

_{z}*β*> 0 and fixed value of

*q*, the momentum distribution of the electron ejected with

_{x}**û**

*parallel to*

_{z}q_{z}**E**

*(*

_{L}*t*) are found to be broader than the ones pertaining to electron emission with

_{H}**û**

*opposite to*

_{z}q_{z}**E**

*(*

_{L}*t*), as already shown in Fig. 1(a) for the particular case

_{H}*t*= 0. These asymmetries originate from the combined effect of the XUV chirp and of the steering action of the laser pulse on the freed electron. They tend to disappear for electron ejection along the direction perpendicular to the laser electric field, or, as shown in Fig. 1(b), when the time delay is

_{H}*t*=

_{H}*T*/4, as the steering effect extinguishes being

_{L}*E*(

_{L}*T*/4) = 0. Moreover, we note that the EMD are invariant under the simultaneous transformations

_{L}**q →**−

**q**+ 2

**A**

*(*

_{L}*t*)/

*c*and

*β*→ −

*β*. From the above considerations it follows that the XUV chirp influence on the EMD characterized by opposite

*q*becomes more effective when

_{z}*t*= 0. This circumstance may be exploited to get information on both the chirp and XUV pulse duration. In fact, for electron emission along the z-direction and for time delay

_{H}*t*= 0, Eq. (4) predicts that the height of the peaks of the EMD for forward emission decreases monotonically by increasing

_{H}*I*, while for backward electron emission the height of the peaks first increases by increasing

_{L}*I*and, after reaching its maximum, decreases monotonically. In Fig. 3 the peaks height for, respectively, forward and backward electron emission, evaluated for two different values of

_{L}*β*by means of Eq. (4), is shown as a function of

*I*and compared with the results obtained by using Eq. (1) and with the ones found by performing the numerical integration of the TDSE. We note that (see Eq. (4)) for

_{L}*β*> 0 (

*β*< 0) the highest peak in the EMD occurs for backward (forward) electron emission. These results suggest a way for determining the value of

*β*and

*τ*. These parameters may be obtained by simultaneously detecting, as a function of

_{CH}*I*, the momentum distribution of the electrons emitted, respectively, in the forward and backward directions. This task can be accomplished by using the same stereodetector arrangement as the one used in Ref. [16

_{L}16. G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature **414**, 182–184 (2001). [CrossRef] [PubMed]

*A*(

_{L}*t*) = 0 when

_{H}*t*= 0, the highest peaks in EMD, by assuming

_{H}*β*> 0, occurs when

*b*given by Eq. (7) is zero, i.e. for such laser field strength

*Ē*

_{0}

*that*

_{L}*Ē*

_{0}

*̄ = −2*

_{L}q*βa*with

*R*the ratio between the values of the peaks of the EMD recorded respectively in the backward and forward direction at the field strength

*Ē*

_{0}

*, it is easily found that*

_{L}*β*,

*τ*) · ℱ, that is independent of the atomic system taken under consideration, the characteristics of the atom being incorporated into the dipole transition matrix element evaluated at the instantaneous mechanical electron momentum

_{H}**q**(

*t*).

_{H}## References and links

1. | P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Progr. Phys. |

2. | E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-Cycle Nonlinear Optics,” Science |

3. | A. Bouhal, R. Evans, G. Grillon, A. Mysyrowicz, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, and D. von der Linde, “Cross-correlation measurement of femtosecond noncollinear high-order harmonics,” J. Opt. Soc. Am. B |

4. | E. S. Toma, H. G. Muller, P. M. Paul, P. Breger, M. Cheret, P. Agostini, C. LeBlanc, G. Mullot, and G. Cheriaux, “Ponderomotive streaking of the ionization potential as a method for measuring pulse durations in the XUV domain with fs resolution,” Phys. Rev. A |

5. | M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature , |

6. | M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. Brabec, “Quantum Theory of Attosecond XUV Pulse Measurement by Laser Dressed Photoionization,” Phys. Rev. Lett. |

7. | J. Itatani, F. Quéré, G. L. Yudin, M. Yu. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. |

8. | E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science |

9. | Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A |

10. | F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. |

11. | L.-Y. Peng, E. A. Pronin, and A. Starace, “Attosecond pulse carrier-envelope phase effects on ionized electron momentum and energy distributions: roles of frequency, intensity and an additional IR pulse,” New J. Phys. |

12. | D.B. Milošević and F. Ehlotzky, “Coulomb and rescattering effects in above-threshold ionization,” Phys. Rev. A |

13. | G. L. Yudin, S. Patchkovskii, and A. D. Bandrauk, “Chirp-dependent attosecond interference in the Coulomb-Volkov continuum,” J. Phys B: At. Mol Opt. Phys |

14. | D. Bauer and P. Koval, “Qprop: A Schrödinger-solver for intense laser-atom interaction,” Comp. Phys Commun. |

15. | S. Bivona, G. Bonanno, R. Burlon, and C. Leone, “Radiation controlled energy of photoelectrons produced by two-color short pulses,” Eur. Phys. J ST |

16. | G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature |

**OCIS Codes**

(270.6620) Quantum optics : Strong-field processes

(320.0320) Ultrafast optics : Ultrafast optics

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: July 26, 2012

Revised Manuscript: September 4, 2012

Manuscript Accepted: September 6, 2012

Published: September 17, 2012

**Citation**

G. Bonanno, S. Bivona, R. Burlon, and C. Leone, "Asymmetries in the momentum distributions of electrons stripped by a XUV chirped pulse in the presence of a laser field," Opt. Express **20**, 22475-22480 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22475

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### References

- P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Progr. Phys.67,, 813–855 (2004). [CrossRef]
- E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-Cycle Nonlinear Optics,” Science320, 1614–1617 (2008). [CrossRef] [PubMed]
- A. Bouhal, R. Evans, G. Grillon, A. Mysyrowicz, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, and D. von der Linde, “Cross-correlation measurement of femtosecond noncollinear high-order harmonics,” J. Opt. Soc. Am. B14, 950–956 (1997). [CrossRef]
- E. S. Toma, H. G. Muller, P. M. Paul, P. Breger, M. Cheret, P. Agostini, C. LeBlanc, G. Mullot, and G. Cheriaux, “Ponderomotive streaking of the ionization potential as a method for measuring pulse durations in the XUV domain with fs resolution,” Phys. Rev. A62, 061801(R) (2000). [CrossRef]
- M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature, 414, 509 (2001). [CrossRef] [PubMed]
- M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. Brabec, “Quantum Theory of Attosecond XUV Pulse Measurement by Laser Dressed Photoionization,” Phys. Rev. Lett.88, 173904 (2002). [CrossRef] [PubMed]
- J. Itatani, F. Quéré, G. L. Yudin, M. Yu. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett.88, 173903 (2002). [CrossRef] [PubMed]
- E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science305, 1267–1269 (2004). [CrossRef] [PubMed]
- Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A71, 011401 (2005). [CrossRef]
- F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys.81, 163–234 (2009). [CrossRef]
- L.-Y. Peng, E. A. Pronin, and A. Starace, “Attosecond pulse carrier-envelope phase effects on ionized electron momentum and energy distributions: roles of frequency, intensity and an additional IR pulse,” New J. Phys.10, 025030 (2008). [CrossRef]
- D.B. Milošević and F. Ehlotzky, “Coulomb and rescattering effects in above-threshold ionization,” Phys. Rev. A58, 3124–3127 (1998); [CrossRef]
- G. L. Yudin, S. Patchkovskii, and A. D. Bandrauk, “Chirp-dependent attosecond interference in the Coulomb-Volkov continuum,” J. Phys B: At. Mol Opt. Phys41, 045602 (2008). [CrossRef]
- D. Bauer and P. Koval, “Qprop: A Schrödinger-solver for intense laser-atom interaction,” Comp. Phys Commun.174, 396–421 (2006). [CrossRef]
- S. Bivona, G. Bonanno, R. Burlon, and C. Leone, “Radiation controlled energy of photoelectrons produced by two-color short pulses,” Eur. Phys. J ST160, 23–31 (2008).
- G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature414, 182–184 (2001). [CrossRef] [PubMed]

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