## High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase

Optics Express, Vol. 11, Issue 12, pp. 1418-1429 (2003)

http://dx.doi.org/10.1364/OE.11.001418

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### Abstract

Angle-resolved energy spectra of high-order above-threshold ionization are calculated in the direction of the laser polarization for a linearly polarized four-cycle laser pulse (two cycles FWHM) as a function of the carrier-envelope relative phase (absolute phase). The spectra exhibit a characteristic left-right (backward-forward) asymmetry, which should allow one to determine the value of the absolute phase in a given experiment by comparison with the theoretical spectra. A classical analysis of the spectra calculated is presented. High-energy electron emission is found to occur in one or two ultrashort (≲0.7 fs) bursts. In the latter case, the spectra display a peak structure whose analysis reveals a time-domain image of electron emission.

© 2003 Optical Society of America

1. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. **72**, 545–591 (2000). [CrossRef]

2. D.J. Jones, S.A. Diddams, J.K. Ranka, A. Stentz, R.S. Windeler, J.L. Hall, and S.T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science **288**, 635–639 (2000). [CrossRef] [PubMed]

3. A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T.W. Hänsch, and F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. **85**, 740–743 (2000). [CrossRef] [PubMed]

4. M. Kakehata, H. Takada, Y. Kobayashi, K. Torizuka, Y. Fujihira, T. Homma, and H. Takahashi, “Single-shot measurement of carrier-envelope phase changes by spectral interferometry,” Opt. Lett. **26**, 1436–1438 (2001). [CrossRef]

5. A. Baltuška, T. Fuji, and T. Kobayashi, “Controlling the carrier-envelope phase of ultrashort light pulses with optical parametric amplifiers,” Phys. Rev. Lett. **88**, 133901 (1–4) (2002). [CrossRef]

6. S.T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D **35**, R43–R59 (2002). [CrossRef]

^{14}Wcm

^{-2}. While an infinitely extended monochromatic plane wave is, in principle, fully characterized by its frequency and intensity, at least two additional parameters are required to specify a few-cycle pulse: besides its duration, this is the carrier-envelope relative phase (or, briefly, the “absolute phase”), that is, the relative phase between the maximum of the pulse envelope and the nearest maximum of the carrier wave. The actual shape of a few-cycle pulse crucially depends on the value of the absolute phase, and so do the physical processes that such a pulse may induce. This is reminiscent of the relative phase between the two components of an infinitely extended two-color plane wave with commensurate frequencies [7

7. D.W. Schumacher and P.H. Bucksbaum, “Phase dependence of intense-field ionization,” Phys. Rev. A **54**, 4271–4278 (1996). [CrossRef] [PubMed]

8. G.G. Paulus, W. Becker, and H. Walther, “Classical rescattering effects in two-color above-threshold ionization,” Phys. Rev. A **52**, 4043–4053 (1995). [CrossRef] [PubMed]

9. F. Ehlotzky, “Atomic phenomena in bichromatic laser fields,” Phys. Rep. **345**, 175–264 (2001). [CrossRef]

10. A. de Bohan, Ph. Antoine, D.B. Milošević, and B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. **81**, 1837–1840 (1998). [CrossRef]

12. F. Krausz, T. Brabec, M. Schnürer, and C. Spielmann, “Extreme nonlinear optics: Exploring matter to a few periods of light,” Opt. Photon. News **9**, 46–51 (1998). [CrossRef]

13. G. Tempea, M. Geissler, and T. Brabec, “Phase sensitivity of high-order harmonic generation with few-cycle laser pulses,” J. Opt. Soc. Am. B **16**, 669–673 (1999). [CrossRef]

14. Z. Zeng, R. Li, W. Yu, and Z. Xu, “Effect of the carrier-envelope phase of the driving laser field on the high-order harmonic attosecond pulse,” Phys. Rev. A **67**, 013815 (1–6) (2003). [CrossRef]

15. A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. Yakovlev, A. Scrinzi, T.W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London) **421**, 611–615 (2003). [CrossRef]

*π*. The fact that atomic ionization depends on the absolute phase was emphasized in Refs. [16

16. E. Cormier and P. Lambropoulos, “Effect of the initial phase of the field in ionization by ultrashort laser pulses,” Eur. Phys. J. D **2**, 15–20 (1998). [CrossRef]

17. I.P. Christov, “Phase-dependent loss due to nonadiabatic ionization by sub-10-fs pulses,” Opt. Lett. **24**, 425–1427 (1999). [CrossRef]

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

19. S. Chelkowski and A.D. Bandrauk, “Sensitivity of spatial photoelectron distributions to the absolute phase of an ultrashort intense laser pulse,” Phys. Rev. A **65**, 061802(R) (1–4) (2002). [CrossRef]

20. M. Kakehata, Y. Kobayashi, H. Takada, and K. Torizuka, “Single-shot measurement of a carrier-envelope phase by use of a time-dependent polarization pulse,” Opt. Lett. **27**, 1247–1249 (2002). [CrossRef]

21. 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** (London), 182–185 (2001). [CrossRef] [PubMed]

24. A.D. Bandrauk, S. Chelkowski, and N.H. Shon, “Measuring the electric field of few-cycle laser pulses by attosecond cross correlation,” Phys. Rev. Lett. **89**, 283903 (1–4) (2002). [CrossRef]

25. A. Scrinzi, M. Geissler, and T. Brabec, “Attosecond cross correlation technique,” Phys. Rev. Lett. **86**, 412–415 (2001). [CrossRef] [PubMed]

26. P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H.G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science **292**, 1689–1692 (2001). [CrossRef] [PubMed]

27. M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, Ch. Spielmann, G.A. Reider, P.B. Corkum, and F. Krausz, “X-ray pulses approaching the attosecond frontier,” Science **291**, 1923–1927 (2001). [CrossRef] [PubMed]

28. 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 (London) **414**, 509–513 (2001). [CrossRef] [PubMed]

30. H. Niikura, F. Légaré, R. Hasbani, A.D. Bandrauk, M.Yu. Ivanov, D.M. Villeneuve, and P.B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature (London) **417**, 917–922 (2002). [CrossRef] [PubMed]

21. 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** (London), 182–185 (2001). [CrossRef] [PubMed]

^{13}Wcm

^{-2}and discuss their potential as a phasemeter. The results of the quantum-mechanical calculation are reconsidered from the point of view of a classical simple-man picture. We use atomic units (

*h̄*=

*e*=|

*e*|=

*m*=1) throughout the paper.

*T*

_{p}impinging during the interval 0≤

*t*≤

*T*

_{p}. An exact expression for the probability amplitude of detecting an ATI electron with momentum

**p**set free by this pulse from the atom is (see Ref. [31

31. W. Becker, F. Grasbon, R. Kopold, D.B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At., Mol., Opt. Phys. **48**, 35–98 (2002). [CrossRef]

*U*(

*t, t*′) is the time-evolution operator of the complete Hamiltonian

**r**·

**E**(

*t*) is the laser-field-electron interaction in the length gauge and the dipole approximation and

*V*(

**r**) is the atomic binding potential. The states

*ψ*

_{p}and

*ψ*

_{0}are a scattering state with asymptotic momentum

**p**and the ground state, respectively, of the atomic Hamiltonian

*H*

_{A}=-∇

^{2}/2+

*V*(

**r**). The time-evolution operator

*U*(

*t, t*′) satisfies the Dyson equation

*U*

_{F}(

*t, t*′) is the time-evolution operator that corresponds to the Hamiltonian

*H*

_{F}(

*t*)=-∇

^{2}/2+

**r**·

**E**(

*t*) of a free electron in the laser field. The eigenstates of the time-dependent Schrödinger equation with the Hamiltonian

*H*

_{F}(

*t*) are the Volkov states

**E**(

*t*) is denoted by

**A**(

*t*), and |

**q**〉 is a plane-wave state [〈

**r**|

**q**〉=(2

*π*)

^{-3/2}exp(

*i*

**q**·

**r**)]. The Volkov time-evolution operator is

**v**(

*t*)≡

**p**+

**A**(

*t*) where

**p**is a constant. If we choose a vector potential

**A**(

*t*) such that

**A**(

*t*)=

**0**for

*t*>

*T*

_{p}, then

**p**has the physical meaning of the kinetical momentum at the detector. Below, however, we will employ a vector potential [cf. Eq. (12)] such that

**A**(0)=

**A**(

*T*

_{p})≠

**0**. Then, in order that the Volkov state given by Eq. (4) really describe an electron with the final momentum

**p**at the position of the detector, we have to replace

**p**by

*ψ*

_{p}(

*t*)|

*U*(

*t, t*′) by 〈

*χ*

_{p̃}(

*t*)|

*U*(

*t, t*′) in Eq. (1) and

*U*by

*U*

_{F}on the right-hand side of Eq. (3). This yields

33. F.H.M. Faisal, “Multiple absorption of laser photons by atoms,” J. Phys. B **6**, L89–92 (1973). [CrossRef]

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

*t*) and rescattering (at time

*t*′), the electron does not feel the binding potential so that its propagation is governed by the Volkov propagator

*U*

_{F}(

*t*′,

*t*). In the rescattering amplitude

*t*′ may be arbitrarily late (the integral over

*t*′ extends from

*t*to infinity). However, if

*t*′≥

*T*

_{p}, then the laser pulse has passed through already at the time of rescattering. Rescattering itself is elastic, so the electron may only change its direction but not its energy in the process. The energy gain that does result from rescattering is due to

*subsequent*acceleration by the laser field. Since the maximal energy of the electron before rescattering is rather low1, rescattering in the absence of the laser field makes no contribution to the high-energy part of the spectrum. Hence, we will ignore it, by restricting in the amplitude given by Eq. (10) the integration over

*t*′ by the upper limit

*T*

_{p}in place of infinity.

*t*∈[0,

*T*

_{p}] and zero outside this interval. We assume an integer number of cycles so that

*T*

_{p}=

*n*

_{p}

*T*with

*T*=2

*π*/

*ω*. Moreover, we introduced the abbreviations

*ω*

_{p}=

*ω*/

*n*

_{p},

*ω*

_{0}=

*ω*,

*ω*

_{1,2}=

*ω*±

*ω*

_{p}, ℰ

_{0}=

*E*

_{0}/2, and ℰ

_{i}=-ℰ

_{0}/2(

*i*=1, 2). At the center of the pulse at

*t*=

*T*

_{p}/2, we have

*ϕ*is the absolute phase (carrier-envelope relative phase) as verbally defined above. Notice that specification of the absolute phase within the interval 0°≤

*ϕ*≤360° requires us to define one spatial direction as positive. HHG only allows us to read off the value of the absolute phase modulo 180°. The field given by Eq. (11) is trichromatic comprising the frequencies

*ω*and

*ω*±

*ω*

_{p}. This allows us to employ the formalism developed earlier in Refs. [36

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

37. D.B. Milošević and F. Ehlotzky, “S-matrix theory of above-threshold ionization in a bichromatic laser field,” J. Phys. B **32**, 1585–1596 (1999). [CrossRef]

*t*)

**A**(

*t*)=

**E**(

*t*), and we choose

**E**(0)=

**E**(

*T*

_{p})=

**0**. However, the vector potential given by Eq. (12) is, in general, different from zero for

*t*≤0 and

*t*≥

*T*

_{p}, so that we have to make the substitution given in Eq. (7). Still, we have

**A**(0)=

**A**(

*T*

_{p}) so that the integral over the electric field is zero,

*E*

_{p}=

**p**

^{2}/2 in the direction

*θ*(cos

*θ*=

**ê**·

**p̂**), which is

*M*is approximated by Eqs. (8)–(10). The explicit calculations will be done for a hydrogen-like model atom, for which

_{p}*V*(

*r*)=-(

*a*/

*r*)exp(-

*λr*) with

*a*=1 a.u. and λ=1 a.u. Notice that we take different potentials for binding and rescattering; for a justification see Refs. [36

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

37. D.B. Milošević and F. Ehlotzky, “S-matrix theory of above-threshold ionization in a bichromatic laser field,” J. Phys. B **32**, 1585–1596 (1999). [CrossRef]

*θ*=0° and for

*θ*=180°, for a four-cycle pulse. The absolute phase is

*ϕ*=0°. In this case, the field

**E**(

*t*) is symmetric and the vector potential

**A**(

*t*)-

**A**(

*T*

_{p}) is antisymmetric with respect to

*t*=

*T*

_{p}/2. This implies that the spectrum of the

*direct*electrons obeys left-right symmetry, as can easily be shown with the help of the matrix element

*rescattered*electrons, whose spectrum is governed by

*t*and the rescattering time

*t*′. This is borne out in Fig. 1: for low energies, the small difference between the spectra in the two directions that is visible is due to the contribution of the rescattered electrons. In contrast, in the rescattering regime where the direct electrons do not contribute, the two spectra are completely different.

*ϕ*=0° and

*ϕ*=180°, for otherwise the same parameters as in Fig. 1. Absolute phases -180°≤

*ϕ*≤0° yield the same spectra except that the left and the right direction are interchanged.

*ϕ*≲110° (and again for 310°≲

*ϕ*≤360°), emission into the direction

*θ*=0° is stronger than into

*θ*=180° for sufficiently high energies, while for 130°≲

*ϕ*≲290° more electrons go in the direction

*θ*=180°. Insomuch as there is an experimental lower detection threshold, detectors at

*θ*=0° and at

*θ*=180° will collect roughly the same number of high-energy electrons for absolute phases in the ranges 110°≲

*ϕ*≲130° and 290°≲

*ϕ*≲310°. (2) For all phases

*ϕ*outside the afore-mentioned two ranges, there exists a well-defined cross-over energy where left-hand emission becomes more prominent than right-hand emission (or vice versa). In the plots, this point moves from

*E*

_{p}=26 eV for

*ϕ*=135° via

*E*

_{p}=31 eV for

*ϕ*=180° and

*ϕ*=0° to

*E*

_{p}=43 eV for

*ϕ*=90°. These features could be employed to establish the value of the absolute phase in an experiment by comparison with a phase scan of calculated left-versus-right energy spectra such as Fig. 2.

35. G.G. Paulus, W. Becker, W. Nicklich, and H. Walther, “Rescattering effects in above-threshold ionization: a classical model,” J. Phys. B **27**, L703–L708 (1994). [CrossRef]

8. G.G. Paulus, W. Becker, and H. Walther, “Classical rescattering effects in two-color above-threshold ionization,” Phys. Rev. A **52**, 4043–4053 (1995). [CrossRef] [PubMed]

31. W. Becker, F. Grasbon, R. Kopold, D.B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At., Mol., Opt. Phys. **48**, 35–98 (2002). [CrossRef]

39. D.B. Milošević and F. Ehlotzky, “Coulomb corrections in above-threshold ionization in a bichromatic laser field,” J. Phys. B **31**, 4149–4161 (1998). [CrossRef]

*t*∈[0,

*T*

_{p}]. Solving Newton’s equation of motion

**v̇**(

*t*)=-

**E**(

*t*) in the presence of only the laser field, we obtain the electron momentum at some arbitrary later time

*t*

_{1}>

*t*:

*t*′ the electron returns to the origin, i.e., if

*t*)=∫

^{t}d

*t*′

**A**(

*t*′), then it may elastically rescatter off its parent ion, so that its kinetic energy immediately after the rescattering is

**v**

^{2}(

*t*′

_{+})/2=

**v**

^{2}(

*t*′_)/2 (where

*t*′

_{±}≡

*t*′±0

^{+}). Thereafter, the electron moves in the laser field only, up to the time

*T*

_{p}, when the laser pulse has passed through. For

*t*>

*T*

_{p}the electron does not change its momentum anymore. The final electron energy, registered at the detector, is

**v̂**(

*t*′

_{+}) is the unit vector in the direction of the electron momentum immediately after the rescattering. In this paper, we only consider the case when

**v̂**(

*t*′

_{+})·

**ê**=cos

*θ*=±1.

*ϕ*=0°, in the two directions

*θ*=0° and

*θ*=180°. It corresponds to Fig. 1 or to the left uppermost panel of Fig. 2. Figure 3 allows one to find the energy

*E*

_{p}at the detector of an electron that is born at the ionization time

*t*. Such an electron may return not at all, once, or several times to its parent ion. Each scenario leads to a different energy at the detector. This is why in the figure, in general, more than one energy corresponds to each ionization time.

*t*corresponds to a reasonably high electric field, say |

*E*(

*t*)|≳0.6

*E*

_{0}. Moreover, high rescattering energies

*E*

_{p}are only reached (i) if the electron rescatters close to a zero of the electric field and (ii) if the subsequent electric field still is sufficiently strong. This last condition marginalizes all solutions with ionization times much later than 2 optical cycles (o.c.). All of these conditions together have the consequence that high-energy electrons are only generated in one or two ultrashort bursts. In a very similar fashion, high-order harmonic generation by few-cycle pulses generates very few ultrashort bursts of uv radiation. In omparison, the number of high-energy-electron bursts in high-order ATI is smaller, since the above condition (ii) is absent for HHG.

*θ*=0° and

*θ*=180° separately.

*θ*=0°:

*E*

_{p}≳15 eV. Of these, the pair of solutions with ionization times below 0.5 o.c. plays absolutely no role, since for these times the electric field is still very weak. The second pair of solutions with ionization times

*t*≈1 o.c. has its cutoff at

*E*

_{p}=27 eV. No qualitative effect of this cutoff is visible in the energy spectrum of Fig. 1 (or the uppermost left panel of Fig. 2).We conclude that this is due to the still rather low value (0.45

*E*

_{0}) of the electric field at the ionization time. Hence, the energy spectrum is dominated by the remaining pair of solutions with cutoff

*E*

_{p}=35.5 eV, which start at

*t*≈1.5 o.c. The corresponding rescattering times

*t*′ can be read off from Fig. 4 and found to be in the interval 2.1 o.c.≤

*t*′≤2.35 o.c. for

*E*

_{p}>20 eV. Consequently, these electrons are emitted in one single burst with a temporal width of 0.68 fs.

31. W. Becker, F. Grasbon, R. Kopold, D.B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At., Mol., Opt. Phys. **48**, 35–98 (2002). [CrossRef]

*E*

_{p}=30 eV, the two spectra begin to separate. This is probably due to the increasing effect of the above-mentioned pair of solutions with their cutoff at 27 eV.

*θ*=180°:

*θ*=0°. In the lower panel of Fig. 3, we can identify four pairs of solutions that might be significant, which we characterize by their approximate ionization times and cutoffs: (

*t*(o.c.),

*E*

_{max}(eV)). They are (1, 31.1), (1, 15.7), (1.5, 12.6), and (2, 21.4). The last two have the highest |

**E**(

*t*)|, but the first has the highest cutoff. For 24 eV≲

*E*

_{p}≲32 eV, the solution (2, 21.4) strongly dominates the spectrum, but owing to its cutoff at 21.4 eV its magnitude continuously decreases for increasing energy. Therefore, with increasing energy, the quantitatively smaller contribution of (1, 31.1) becomes more and more competitive. Indeed, for

*E*

_{p}≳36 eV, the interference of the contributions of these two pairs of solutions, whose ionization and rescattering times lie one cycle apart, produces an ATI spectrum with well developed peaks. This is in contrast to the completely smooth spectrum for

*θ*=0°, which is generated by just one pair of solutions. For energies below about 25 eV, the spectrum again displays ATI peaks. These are probably due to interference of the dominant solution (2, 21.4) with (1.5, 12.6).

*T*. Also, the positions of the peaks depend on the value of the absolute phase.

*θ*=180° shows peak structures with an energy separation around

*h̄ω*/2. Better examples of the same effect can be found in Fig. 2, in particular for

*ϕ*=112.5° and

*θ*=0°. This spectrum exhibits high-contrast peaks with a precise separation of

*h̄ω*for

*E*

_{p}≲30 eV followed by a sequence of peaks with an approximate spacing of

*h̄ω*/2 and lesser but still good contrast for

*E*

_{p}≳30 eV. The

*h̄ω*/2 spacing requires contributions with rescattering times 2

*T*apart, in analogy to three-slit interference. A classical analysis analogous to Figs. 3 and 4 yields several contributions all of which have very low electric fields at the time of ionization. We have not been able to identify those that are responsible for the

*h̄ω*/2 peak structure. However, it seems worthwhile to notice that few-cycle pulses are capable of producing peaks with a separation of one half of the carrier frequency.

^{14}Wcm

^{-2}, except that its cutoff moves to higher and higher energy, roughly proportional to 10

*U*

_{p}, cf. Fig. 1. For higher intensity, the fixed-intensity long-pulse spectrum develops a lot of structure, mostly due to beating between the contributions of the shortest two orbits [31

**48**, 35–98 (2002). [CrossRef]

^{14}Wcm

^{-2}, we found that it will no longer be useful to measure the absolute phase when

*T*

_{p}≳7

*T*. We do not expect a pronounced effect of the detailed pulse shape on the energy spectrum and its asymmetry.

*π*.

## Footnotes

1 | For a linearly polarized plane-wave field, this maximal energy is 3.17U_{P}
, with U_{P}
the ponderomotive energy of the laser field, while the cutoff energy of the rescattered electrons is 10.01U_{P}
. |

## References and links

1. | T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. |

2. | D.J. Jones, S.A. Diddams, J.K. Ranka, A. Stentz, R.S. Windeler, J.L. Hall, and S.T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science |

3. | A. Apolonski, A. Poppe, G. Tempea, Ch. Spielmann, Th. Udem, R. Holzwarth, T.W. Hänsch, and F. Krausz, “Controlling the phase evolution of few-cycle light pulses,” Phys. Rev. Lett. |

4. | M. Kakehata, H. Takada, Y. Kobayashi, K. Torizuka, Y. Fujihira, T. Homma, and H. Takahashi, “Single-shot measurement of carrier-envelope phase changes by spectral interferometry,” Opt. Lett. |

5. | A. Baltuška, T. Fuji, and T. Kobayashi, “Controlling the carrier-envelope phase of ultrashort light pulses with optical parametric amplifiers,” Phys. Rev. Lett. |

6. | S.T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D |

7. | D.W. Schumacher and P.H. Bucksbaum, “Phase dependence of intense-field ionization,” Phys. Rev. A |

8. | G.G. Paulus, W. Becker, and H. Walther, “Classical rescattering effects in two-color above-threshold ionization,” Phys. Rev. A |

9. | F. Ehlotzky, “Atomic phenomena in bichromatic laser fields,” Phys. Rep. |

10. | A. de Bohan, Ph. Antoine, D.B. Milošević, and B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. |

11. | A. de Bohan, Ph. Antoine, D.B. Milošević, G.L. Kamta, and B. Piraux, “Phase sensitivity of harmonic emission with ultrashort laser pulses,” Laser Phys. |

12. | F. Krausz, T. Brabec, M. Schnürer, and C. Spielmann, “Extreme nonlinear optics: Exploring matter to a few periods of light,” Opt. Photon. News |

13. | G. Tempea, M. Geissler, and T. Brabec, “Phase sensitivity of high-order harmonic generation with few-cycle laser pulses,” J. Opt. Soc. Am. B |

14. | Z. Zeng, R. Li, W. Yu, and Z. Xu, “Effect of the carrier-envelope phase of the driving laser field on the high-order harmonic attosecond pulse,” Phys. Rev. A |

15. | A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. Yakovlev, A. Scrinzi, T.W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London) |

16. | E. Cormier and P. Lambropoulos, “Effect of the initial phase of the field in ionization by ultrashort laser pulses,” Eur. Phys. J. D |

17. | I.P. Christov, “Phase-dependent loss due to nonadiabatic ionization by sub-10-fs pulses,” Opt. Lett. |

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

19. | S. Chelkowski and A.D. Bandrauk, “Sensitivity of spatial photoelectron distributions to the absolute phase of an ultrashort intense laser pulse,” Phys. Rev. A |

20. | M. Kakehata, Y. Kobayashi, H. Takada, and K. Torizuka, “Single-shot measurement of a carrier-envelope phase by use of a time-dependent polarization pulse,” Opt. Lett. |

21. | 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 |

22. | D.B. Milošević, G.G. Paulus, and W. Becker, “Phase-dependent effects of a few-cycle laser pulse,” Phys. Rev. Lett. |

23. | D.B. Milošević, G.G. Paulus, and W. Becker, “Above-threshold ionization with few-cycle laser pulses and the relevance of the absolute phase,” Laser Phys. |

24. | A.D. Bandrauk, S. Chelkowski, and N.H. Shon, “Measuring the electric field of few-cycle laser pulses by attosecond cross correlation,” Phys. Rev. Lett. |

25. | A. Scrinzi, M. Geissler, and T. Brabec, “Attosecond cross correlation technique,” Phys. Rev. Lett. |

26. | P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H.G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science |

27. | M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, Ch. Spielmann, G.A. Reider, P.B. Corkum, and F. Krausz, “X-ray pulses approaching the attosecond frontier,” Science |

28. | 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 (London) |

29. | A.D. Bandrauk and S. Chelkowski, “Dynamic imaging of nuclear wave functions with ultrashort UV laser pulses,” Phys. Rev. Lett. |

30. | H. Niikura, F. Légaré, R. Hasbani, A.D. Bandrauk, M.Yu. Ivanov, D.M. Villeneuve, and P.B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature (London) |

31. | W. Becker, F. Grasbon, R. Kopold, D.B. Milošević, G.G. Paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effects,” Adv. At., Mol., Opt. Phys. |

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

33. | F.H.M. Faisal, “Multiple absorption of laser photons by atoms,” J. Phys. B |

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

35. | G.G. Paulus, W. Becker, W. Nicklich, and H. Walther, “Rescattering effects in above-threshold ionization: a classical model,” J. Phys. B |

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

37. | D.B. Milošević and F. Ehlotzky, “S-matrix theory of above-threshold ionization in a bichromatic laser field,” J. Phys. B |

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

39. | D.B. Milošević and F. Ehlotzky, “Coulomb corrections in above-threshold ionization in a bichromatic laser field,” J. Phys. B |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(190.4180) Nonlinear optics : Multiphoton processes

(270.4180) Quantum optics : Multiphoton processes

(270.6620) Quantum optics : Strong-field processes

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 5, 2003

Revised Manuscript: June 6, 2003

Published: June 16, 2003

**Citation**

D. B. Milošević, G. G. Paulus, and W. Becker, "High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase," Opt. Express **11**, 1418-1429 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-12-1418

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### References

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