Abstract

In this paper, a scaling law of photoionization of atoms irradiated by intense, few-cycle laser pulses is established. The scaling law sets a relation to the phase-dependent ionization with the kinetic energy of photoelectrons, the duration and peak intensity of short pulses, and the ionization potential of the target atoms. We find that it will be advantageous to manifest the phase-dependent photoionization by choosing the target atoms with larger ionization potential, using laser with smaller carrier-frequency, and increasing the pulse intensity.

©2007 Optical Society of America

1. Introduction

Recent developments in laser technology have made it possible to produce intense laser pluses with durations as short as only few optical cycles[1]. These ultrashort pulses have the advantage that higher intensity can be reached before the target being depleted, allowing much higher effective field strengths, thus provide a useful research tool widely used in various fields [2, 3]. For few-cycle pulses, the temporal shape of the electric field varies dramatically with the initial phase of the carrier wave with respect to the pulse envelope (the so-called carrier-envelope phase, CE phase), the physical process depending on the electric field is CE phase-dependent [4].

Measurements of the photoelectron angular distributions (PADs) manifested directly the CE phase of a short pulse [5, 6, 7, 8, 9]. The experiment performed by Paulus et al. detected the coincident photoelectrons emitted in opposite directions and showed the asymmetric ionization in few-cycle above-threshold ionization (ATI) [4]. From then on, the CE phase-dependence phenomena attract much more attentions in both theoretical and experimental studies [8, 10, 11, 12, 13].

Scaling is a powerful technique for analyzing many phenomena in intense laser fields [14, 15]. For relatively-long laser pulses, the scaling law established recently [16] reveals a remarkably simple physical mechanism: the main characters of PADs as well as of many strong-field phenomena are determined by three dimensionless numbers: (1) the ponderomotive number up = Up/ħω, the ponderomotive energy in the unit of laser-photon energy; (2) the binding number εb = Eb/ħω, the atomic binding energy in the unit of laser-photon energy; and (3) the absorbed-photon number q [16]. The physical essences of processes with the same three parameters are just the same, even though their dynamical parameters look quite different. While for few-cycle pluses, all related physical processes may strongly depend on the CE phase. Is the scaling law still applicable in the short-pulse case?

Very recently, the asymmetry in the ionization of Rydberg atoms by radio-frequency few-cycle pulses were demonstrated [17]. Similar results were also found in the photoionization of atoms with intense few-cycle laser pulses [18]. The similarity strongly suggests a scaling relation in various few-cycle pulses. In this paper, a scaling law of photoionization in the few-cycle case is given, which determines the similarities of photoionization in various short pulses. The scaling law is proved theoretically and verified numerically. A discussion with the present experimental observations is given in the last part of this paper.

2. Theoretical proof of the scaling law

The scaling law in few-cycle case can be stated as: For a given CE phase and the corresponding ATI peaks, the main characters of PADs for an atom of binding energy Eb irradiated by an ultrashort laser pulse (with fixed polarization) of duration τ, peak intensity I and carrier frequency ω are the same as that for an atom of binding energy rEb irradiated by an ultrashort laser pulse of duration τ/r, peak intensity r3I and carrier frequency rω, where the re-scaling parameter r is an arbitrarily positive number. The scaling law says that all PADs admit an arbitrary scaling transformation.

The scaling law is indicated by the following fact: the main characters of PADs in the few-cycle case are determined by five dimensionless numbers, i.e., the CE phase ϕ0, the cycle number n, the atomic binding number εb, the ponderomotive number up, and the number of absorbed photons q. The PADs are the same if the five parameters are the same. In contrast to the relatively-long-pulse case, the PADs in the few-cycle case are determined by two additional dimensionless numbers describing the shape of the electric field modulated by the pulse envelope, i.e. the CE phase ϕ0 and the cycle number n. Once the values of ϕ0 and n are given, the distribution of the electric field in the pulse envelope is fixed; thus the PADs of atoms irradiated by the short pulse satisfy the same scaling law as the one in the relatively-long-pulse case.

The theoretical proof of the scaling is based on our earlier works [19, 20]. We use a non-perturbative scattering theory of ATI [21], which treats the Volkov states as intermediate states while the final state of ATI is an electron-photon plane wave. The electric field of n-cycle pulses with carrier frequency ω is written as

E(t)=E0cos(ωt+ϕ0)sin2(ω2nt+π2),

where n is the cycle number defined by n =τω/(2π) through the pulse duration τ, and ϕ0 the CE phase. The defined CE phase is always zero when the maxima of the field and the envelope coincide [12]. A three-mode laser field is used to mimic an infinite sequence of n-cycle pulses:

ω1=ω;ω2=ω(1+1n);ω3=ω(11n)

with the intensity of the sideband modes being a quarter of that of the central mode. The advantage of this treatment is that the ionization rate can be obtained analytically [19]. Our treatment reproduces the asymmetric PADs, and also shows the reversal of the asymmetry with shifting the CE phase by π[19, 20]. The scaling law preserves these properties.

The differential ionization rate for a given ATI peak is (ħ = 1, c = 1) [20]

dWdΩpf=(2me3ω5)12(2π)2(qεb)12(qup)2Φi(Pfqk)2
×qi,jiχj1q1,j2q2,j3q3(zf)χj1,j2,j3(zf)*2,

where P f is the final momentum of the photoelectron, and k the central wave-vector of the laser pulse. The integer q relates the final kinetic energy of the photoelectron as

EkPf22me=qωEb.

Here q denotes the order of an ATI peak and gives the net absorbed-photon number. According to Eq. (2), the number of transferred photons in the ith mode in overall process, say qi(i = 1,2,3), satisfies the following integer equation

q=q1+q2+q3+(q2q3)n,

and the sum over qi is performed over all possible sets of {qi} satisfying above relation. The sum over ji is performed on the energy shell; up is the ponderomotive parameter defined as the ponderomotive energy in the unit of ω. The generalized phased Bessel (GPB) functions are given by

χj1,j2,j3(zf)=miXj1+2m1+m4+m5+m6+m7(ζ1f)Xj2+2m2+m4m5+m8+m9(ζ2f)
×Xj3+2m3+m6+m7+m8m9(ζ3f)Xm1(z1)Xm9(z9),

where the sum is performed over integer indices mi(i = 1,2,⋯9): -∞ < mi < ∞, and Xn(z) is phased Bessel function defined in terms of the ordinary Bessel function as

Xn(z)=Jn(z)exp(inarg(z)).

The arguments of the GPB function are defined as

ζ1f=2eΛ1meω1Pfε1,ζ2f=2eΛ2meω2Pfε2,ζ3f=2eΛ3meω3Pfε3,z1=e2Λ122meω1ε1ε1,z2=e2Λ222meω2ε2ε2,z3=e2Λ322meω3ε3ε3,z4=2e2Λ1Λ2ε1ε2me(ω1+ω2),z5=2e2Λ1Λ2ε1ε2*me(ω2ω1),z6=2e2Λ1Λ2ε1ε3me(ω1+ω3),z7=2e2Λ1Λ3ε1ε3*me(ω3ω1),z8=2e2Λ2Λ3ε2ε3me(ω2+ω3),z9=2e2Λ2Λ3ε2ε3*me(ω3ω2).

where εj(j = 1,2,3) is the polarization vector of the jth mode defined by

εj=[εxcos(ξ/2)+iεysin(ξ/2)]eiϕj,

where εx and εy are unit vectors vertical to each other; ξ determines the degree of polarization, such that ξ = π/2 denotes circular polarization and ξ = 0 and π linear polarization. The phase angle ϕj(j = 1,2,3) relate to the CE phase ϕ 0 as

ϕ1=0,ϕ2=ϕ0n,ϕ3=ϕ0n.

In Eq. (8), Λi(i = 1,2,3) is the classical-field amplitude of the ith mode and given by

Λ1=meωup2e;Λ2=nΛ12(n+1);Λ3=nΛ12(n1).

The photoelectron rate of a given ATI peak is obtained by integrating over the solid angle dΩPf = sin θfff of the final photoelectron, where θf is the scattering angle and ϕf is the azimuthal angle. The PAD denotes the ionization rate for different azimuths at a fixed scattering angle θf = π/2.

First, the PAD does not depend on the initial angular state of the atom. Even though the transition matrix element does contain the initial-state momentum wave function, the final-state momentum may still directly or indirectly incorporate with the initial-state momentum. This momentum wave function is squared and averaged over the initial orientations. Thus, the factor of the momentum wave function reduces to its radial part. Since the photon momentum is very small compared with the electron momentum, this factor does not affect the PADs [16].

Second, we show that the sum over the GPB functions is invariant under the scaling transformation by a constant r. First we show that the arguments of the GPB function are unchanged under this scaling change. By means of Eqs. (2), (4) and (9-11), we obtain the following relations

Pfωi=2me(qεb)ωωi,

which varies with q, n and εb;

eΛ1ω1=meup2,eΛ2ω2=meup4(nn+1)32,eΛ3ω3=meup4(nn1)32,

which depend only on up and n; and

εjεk=cosξei(ϕj+ϕk),εjεk*=ei(ϕj+ϕk),

which vary with ϕ 0 and n for a fixed polarization. Thus, the arguments given in Eq. (8) are completely determined by the five dimensionless parameters, that is, the CE phase ϕ 0, the cycle number n, the atomic binding number εb, the ponderomotive number up, and the absorbed-photon number q. This means that the arguments of the GPB function are kept unchanged under the scaling change. Here, the scaling change means that the five parameters mentioned above are kept unchanged but other quantities, such as the laser frequency and the binding energy, changed according to the scaling law. Next, we show the orders of the GPB functions, say ji and qi, do not change under the scaling change. In Eq. (3), the sum over ji(i = 1,2,3) is performed on the energy shell for a given value of q. If q doesn’t change,ji in the sum doesn’t change, too. The possible value of qi also determined by q, (see Eq. (5)). If q doesn’t change, all the possible value of qi are kept unchanged. Thus, for a fixed q, the orders of the GPB functions keep invariant under the scaling change. Finally, we show the number of transition channels is fixed in the scaling change. A set of qi satisfying Eq. (5) for a given integer q defines a transition channel. Since q and n are fixed numbers in the scaling change, all the possible value of qi as well as their combination are fixed.

Then, we conclude that the PADs in few-cycle case satisfy the scaling law. In the following we describe the tests to the scaling law using numerical methods.

3. Numerical verification of the scaling law

In few-cycle case, the photoelectron rates in a pair of opposite directions are not always equal to each other, which is termed as inversion asymmetry [19]. The inversion asymmetry varies with the CE phase, as shown in Fig. 1. The laser pulse is of five-cycle duration, and the pulse is linearly polarized and of peak intensity 4 × 1013 W/cm2. A direct comparison can be made between our calculation and the measurements of Linder et al. [8]. For a revealing calculation to verify the scaling law, we study the PADs of given ATI peaks of xenon in five-cycle pulses for various CE phases and compare them with the calculated PADs using doubled (r = 2) binding energy, doubled central frequency, and eight times of the laser intensity of the former one in five-cycle pulses. The PADs of the same ATI order q for the same CE phase show identical feature, thus the comparison verifies the scaling law.

 figure: Fig. 1.

Fig. 1. (Color online) The variation of the photoelectron rates in a pair of opposite directions along the CE phase (a) ϕf = 0; (b) ϕf = π. The linearly polarized laser pulse is of five-cycle duration and of peak intensity 4 × 1013 W/cm2 and the target atom is xenon.

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Direct experimental tests to the scaling law is somewhat difficult, since it requires the frequency, peak intensity and pulse duration of the short pulse as well as the atomic binding energy varying in a large scale at the same time. A favorite candidate will be the PADs of argon target irradiated by 616 nm laser pulses and that of xenon target irradiated by 800 nm laser pulses. Keeping the cycle number and the CE phase unchanged, choosing the peak intensity of the latter as 2.21 times that of the former, the PADs of the corresponding ATI peaks (with same q) will be the same, for linear polarization and for circular polarization, respectively. A numerical verification according to Eq. (3) is presented in Fig. 2.

Verifications from other sources can provide additional strong supports for the scaling law. The following are two. One is the calculation of the asymmetry degree, the ratio of the difference to the sum of the ionization rates in the two opposite directions, using the numerical solutions of the one-dimension time-dependent Schrodinger equation (TDSE). One result is shown in Fig. 3 (For detailed calculations, see Ref. [22]). The likeness of the asymmetry degree in (a) and (d), where the latter relates to the former with a re-scaling parameter r = 2, verifies the scaling law. The other lies in the data published by Chelkowski et al. [13], where the asymmetry degree in few-cycle laser pulses is calculated by three-dimension TDSE method. In Fig. 3 of Ref. [13], the asymmetry in photoionization of H atoms by a 788 nm pulse shows similar structure to that in a 800 nm pulse, but shifts to higher intensities. This likeness verifies the scaling law qualitatively.

4. Discussions

The scaling law in few-cycle regime sets a relation to the CE phase-dependent asymmetric ionization with the kinetic energy of photoelectrons, the duration and peak intensity of short pulses, and the ionization potential of the target atom. The CE phase is manifested by the inversion asymmetry that is caused by the interference among ionization channels. More channels can be formed in shorter pulses while less channels in longer pulses [19, 20]. If a pulse is not short enough that only one ionization channel may be formed, the PADs in such a pulse will be inversion symmetric then the CE phase plays no role. Thus a question is: How short should a pulse be, in which the CE phase as an important role comes into play? The answer to this question lies in the number of channels to form an ATI peak. The number of channels is counted as all possible combinations for different qi satisfying Eq. (5) for a given q and the channels with q 2 = q 3 are regarded as one because they are indistinguishable. The number of the ionization channels is determined by the kinetic energy and the pulse duration given in Eqs. (4) and (5). We find that when the cycle number is bigger than the order of an ATI peak, i.e.,

 figure: Fig. 2.

Fig. 2. (Color online) The polar plots of PADs for the second ATI peak (q = 9) in linearly polarized 5-cycle laser pulses: (a) for xenon atom with Eb, = 12.1 eV, λ = 800 nm, I = 5 × 1013 W/cm2; (b) for argon atom with Eb = 15.76 eV, λ = 616 nm, I = 11.5 × 1013 W/cm2. The two calculated PADs, each was normalized by its maximum, with same CE phases (ϕ = 0), but with different pulse intensities and frequencies for atoms with different binding energies show identical angular distributions, verifying the scaling law.

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n>q,

there is only one transition channel, thus the PAD is inversion symmetric and CE phase-independent. Since q relates with the kinetic energy of photoelectrons and the binding energy of the atom (see Eq. (4)), this relation indicates that the role of the CE phase varies with the kinetic energy of photoelectron and the binding energy of the target atom From Eq. (15) we find that more channels are available for the higher-energy electrons. Thus, it is reasonable that the PADs for low-energy electrons have inversion symmetry and that for high-energy electrons have inversion asymmetry in relatively longer few-cycle pulses [19]. The scaling law is also suggestive to the experimental observations of the CE phase-dependence. According to Eq. (15), it will be advantageous to choose the target atoms with larger binding energy or to use laser with smaller carrier-frequency, or to do the both. Moreover, increasing the laser-pulse intensity will help to manifest the CE phase in relative long few-cycle pulses, because the low ATI peaks (with small q) will be suppressed by the increased ponderomotive energy and the total ATI rate is contributed mainly by the higher ATI peaks, where the multi-channel ionization is easy to be formed in longer few-cycle pulses. This analysis agrees qualitatively with the observation of Gurtler et al. [ 17].

In the experiment performed by Paulus et al., the dependence of photoionization on the CE phase is very slight for 6-cycle laser pulses. For Kr atoms of binding energy 14eV irradiated by 800nm laser shot, 10 or more photons are to be absorbed for ionization. In energy spectra, photoelectrons distribute mostly to the 4th ∼ 6thorder ATI peaks (q ≃ 13 ∼ 15). Thus, it is expected that the CE phase makes an appreciable effect on those ATI peaks even for the pulses with more than 10 cycles.

 figure: Fig. 3.

Fig. 3. (Color online) The dependence on the CE phase of the asymmetric degree is calculated by the TDSE method: (a) Eb = 13.6 eV, λ = 800 nm, Ia = 3.5 × 1014 W/cm2; (b)-(e) are of double binding energy, double frequency to that in (a), but of different pulse intensities: (b) I = 2Ia; (c) I = 22 Ia;(d) I = 23 Ia; and (e) I = 24 Ia. In all the calculations, five-cycle pulses are used. The similarity in (a) and (d) verifies the scaling law.

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In our analytical treatment, the atomic structure was neglected which may lead to some uncertainties in the scaling relation. For example, if the carrier frequency is close to resonance with some bound states, the PADs will be different and the scaling relation will be poor. This implies that the scaling law becomes a better one when the resonant transition plays a negligible role. Generally, it is expected that the scaling law is more effective for rare gases, hydrogen and hydrogen-like atoms, as well as negative ions.

Acknowledgments

This work is supported by Shanghai Rising-Star Program, the China National NSF under Grant No. 60408008 and 973 Program of China.

References and links

01. M. Nisoli, S. De Silvestri, O. Svelto, R. Szipcs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997). [CrossRef]   [PubMed]  

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

03. A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London) 421, 611–615 (2003). [CrossRef]   [PubMed]  

04. 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 (London) 414, 182 (2001). [CrossRef]   [PubMed]  

05. 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]  

06. 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]  

07. S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Measurement of the carrier-envelope phase of few-cycle laser pulses by use of asymmetric photoionization,” Opt. Lett. 29, 1557–1559 (2004). [CrossRef]   [PubMed]  

08. F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004). [CrossRef]  

09. D. B. Milosevic, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–262 (2006). [CrossRef]  

10. D. B. Milosevic, G. G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Exp. 11, 1418–1429 (2003).

11. G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003). [CrossRef]  

12. S. Chelkowski and A.D. Bandrauk, “Asymmetries in strong-field photoionization by few-cycle laser pulses: Kinetic-energy spectra and semiclassical explanation of the asymmetries of fast and slow electrons,” Phys. Rev. A 71, 053815(9) (2005). [CrossRef]  

13. S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Phase-dependent asymmetries in strong-field photoioniza-tion by few-cycle laser pulses,” Phys. Rev A 70, 013815(9) (2004). [CrossRef]  

14. J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98. 013901 (2007) [CrossRef]   [PubMed]  

15. A. Gordon and F. Kartner, “Scaling of keV HHG photon yield with drive wavelength,” Opt. Exp. 13, 2941–2947 (2005). [CrossRef]  

16. D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003). [CrossRef]  

17. A. Gurtler, E. Robicheaux, W. J. Zande, and L. D. Noordam, “Asymmetry in the strong-field ionization of Rydberg atoms by few-cycle pulses,” Phys. Rev. Lett. 92, 033002(4) (2004). [CrossRef]   [PubMed]  

18. X. Zhang, J. Zhang, R. Li, Q. Gong, and Z. Xu, “Photoionization of H atoms in few-cycle laser pulses,” Eur. Phys. J. D 37, 457–462 (2006). [CrossRef]  

19. J. Zhang and Z. Xu, “Above-threshold ionization of Kr atoms in an infinite sequence of circularly polarized few-cycle pulses,” Phys. Rev. A 68, 013402(5) (2003). [CrossRef]  

20. J. Zhang, X. Feng, Z. Xu, and D.-S. Guo, “Phase-dependent angular distributions of photoelectrons in an infinite sequence of linearly polarized few-cycle pulses,” Phys. Rev. A 69, 043409(5) (2004). [CrossRef]  

21. D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A 40, 4997–5005(1989). [CrossRef]   [PubMed]  

22. X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Exp. 13, 8708–8716 (2005). [CrossRef]  

References

  • View by:

  1. M. Nisoli, S. De Silvestri, O. Svelto, R. Szipcs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997).
    [Crossref] [PubMed]
  2. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
    [Crossref]
  3. A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
    [Crossref] [PubMed]
  4. 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 (London)  414, 182 (2001).
    [Crossref] [PubMed]
  5. 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]
  6. 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]
  7. S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Measurement of the carrier-envelope phase of few-cycle laser pulses by use of asymmetric photoionization,” Opt. Lett. 29, 1557–1559 (2004).
    [Crossref] [PubMed]
  8. F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004).
    [Crossref]
  9. D. B. Milosevic, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–262 (2006).
    [Crossref]
  10. D. B. Milosevic, G. G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Exp. 11, 1418–1429 (2003).
  11. G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003).
    [Crossref]
  12. S. Chelkowski and A.D. Bandrauk, “Asymmetries in strong-field photoionization by few-cycle laser pulses: Kinetic-energy spectra and semiclassical explanation of the asymmetries of fast and slow electrons,” Phys. Rev. A 71, 053815(9) (2005).
    [Crossref]
  13. S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Phase-dependent asymmetries in strong-field photoioniza-tion by few-cycle laser pulses,” Phys. Rev A 70, 013815(9) (2004).
    [Crossref]
  14. J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98. 013901 (2007)
    [Crossref] [PubMed]
  15. A. Gordon and F. Kartner, “Scaling of keV HHG photon yield with drive wavelength,” Opt. Exp. 13, 2941–2947 (2005).
    [Crossref]
  16. D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003).
    [Crossref]
  17. A. Gurtler, E. Robicheaux, W. J. Zande, and L. D. Noordam, “Asymmetry in the strong-field ionization of Rydberg atoms by few-cycle pulses,” Phys. Rev. Lett. 92, 033002(4) (2004).
    [Crossref] [PubMed]
  18. X. Zhang, J. Zhang, R. Li, Q. Gong, and Z. Xu, “Photoionization of H atoms in few-cycle laser pulses,” Eur. Phys. J. D 37, 457–462 (2006).
    [Crossref]
  19. J. Zhang and Z. Xu, “Above-threshold ionization of Kr atoms in an infinite sequence of circularly polarized few-cycle pulses,” Phys. Rev. A 68, 013402(5) (2003).
    [Crossref]
  20. J. Zhang, X. Feng, Z. Xu, and D.-S. Guo, “Phase-dependent angular distributions of photoelectrons in an infinite sequence of linearly polarized few-cycle pulses,” Phys. Rev. A 69, 043409(5) (2004).
    [Crossref]
  21. D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A 40, 4997–5005(1989).
    [Crossref] [PubMed]
  22. X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Exp. 13, 8708–8716 (2005).
    [Crossref]

2007 (1)

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98. 013901 (2007)
[Crossref] [PubMed]

2006 (2)

X. Zhang, J. Zhang, R. Li, Q. Gong, and Z. Xu, “Photoionization of H atoms in few-cycle laser pulses,” Eur. Phys. J. D 37, 457–462 (2006).
[Crossref]

D. B. Milosevic, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–262 (2006).
[Crossref]

2005 (3)

S. Chelkowski and A.D. Bandrauk, “Asymmetries in strong-field photoionization by few-cycle laser pulses: Kinetic-energy spectra and semiclassical explanation of the asymmetries of fast and slow electrons,” Phys. Rev. A 71, 053815(9) (2005).
[Crossref]

A. Gordon and F. Kartner, “Scaling of keV HHG photon yield with drive wavelength,” Opt. Exp. 13, 2941–2947 (2005).
[Crossref]

X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Exp. 13, 8708–8716 (2005).
[Crossref]

2004 (5)

A. Gurtler, E. Robicheaux, W. J. Zande, and L. D. Noordam, “Asymmetry in the strong-field ionization of Rydberg atoms by few-cycle pulses,” Phys. Rev. Lett. 92, 033002(4) (2004).
[Crossref] [PubMed]

J. Zhang, X. Feng, Z. Xu, and D.-S. Guo, “Phase-dependent angular distributions of photoelectrons in an infinite sequence of linearly polarized few-cycle pulses,” Phys. Rev. A 69, 043409(5) (2004).
[Crossref]

S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Phase-dependent asymmetries in strong-field photoioniza-tion by few-cycle laser pulses,” Phys. Rev A 70, 013815(9) (2004).
[Crossref]

S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Measurement of the carrier-envelope phase of few-cycle laser pulses by use of asymmetric photoionization,” Opt. Lett. 29, 1557–1559 (2004).
[Crossref] [PubMed]

F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004).
[Crossref]

2003 (5)

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

D. B. Milosevic, G. G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Exp. 11, 1418–1429 (2003).

G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003).
[Crossref]

D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003).
[Crossref]

J. Zhang and Z. Xu, “Above-threshold ionization of Kr atoms in an infinite sequence of circularly polarized few-cycle pulses,” Phys. Rev. A 68, 013402(5) (2003).
[Crossref]

2002 (1)

2001 (1)

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 (London)  414, 182 (2001).
[Crossref] [PubMed]

2000 (2)

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

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]

1997 (1)

1989 (1)

D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A 40, 4997–5005(1989).
[Crossref] [PubMed]

Åberg, T.

D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A 40, 4997–5005(1989).
[Crossref] [PubMed]

Agostini, P.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98. 013901 (2007)
[Crossref] [PubMed]

Apolonski, A.

S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Phase-dependent asymmetries in strong-field photoioniza-tion by few-cycle laser pulses,” Phys. Rev A 70, 013815(9) (2004).
[Crossref]

S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Measurement of the carrier-envelope phase of few-cycle laser pulses by use of asymmetric photoionization,” Opt. Lett. 29, 1557–1559 (2004).
[Crossref] [PubMed]

Auguste, T.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98. 013901 (2007)
[Crossref] [PubMed]

Bai, L.

X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Exp. 13, 8708–8716 (2005).
[Crossref]

Baltuska, A.

F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004).
[Crossref]

G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003).
[Crossref]

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Bandrauk, A. D.

S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Measurement of the carrier-envelope phase of few-cycle laser pulses by use of asymmetric photoionization,” Opt. Lett. 29, 1557–1559 (2004).
[Crossref] [PubMed]

S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Phase-dependent asymmetries in strong-field photoioniza-tion by few-cycle laser pulses,” Phys. Rev A 70, 013815(9) (2004).
[Crossref]

Bandrauk, A.D.

S. Chelkowski and A.D. Bandrauk, “Asymmetries in strong-field photoionization by few-cycle laser pulses: Kinetic-energy spectra and semiclassical explanation of the asymmetries of fast and slow electrons,” Phys. Rev. A 71, 053815(9) (2005).
[Crossref]

Bauer, D.

D. B. Milosevic, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–262 (2006).
[Crossref]

Becker, W.

D. B. Milosevic, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–262 (2006).
[Crossref]

D. B. Milosevic, G. G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Exp. 11, 1418–1429 (2003).

Brabec, T.

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

Chelkowski, S.

S. Chelkowski and A.D. Bandrauk, “Asymmetries in strong-field photoionization by few-cycle laser pulses: Kinetic-energy spectra and semiclassical explanation of the asymmetries of fast and slow electrons,” Phys. Rev. A 71, 053815(9) (2005).
[Crossref]

S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Phase-dependent asymmetries in strong-field photoioniza-tion by few-cycle laser pulses,” Phys. Rev A 70, 013815(9) (2004).
[Crossref]

S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Measurement of the carrier-envelope phase of few-cycle laser pulses by use of asymmetric photoionization,” Opt. Lett. 29, 1557–1559 (2004).
[Crossref] [PubMed]

Corkum, P. B.

Crasemann, B.

D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A 40, 4997–5005(1989).
[Crossref] [PubMed]

De Silvestri, S.

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 (London)  414, 182 (2001).
[Crossref] [PubMed]

M. Nisoli, S. De Silvestri, O. Svelto, R. Szipcs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997).
[Crossref] [PubMed]

Dietrich, P.

DiMauro, L. F.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98. 013901 (2007)
[Crossref] [PubMed]

Feng, X.

J. Zhang, X. Feng, Z. Xu, and D.-S. Guo, “Phase-dependent angular distributions of photoelectrons in an infinite sequence of linearly polarized few-cycle pulses,” Phys. Rev. A 69, 043409(5) (2004).
[Crossref]

Ferencz, K.

Freeman, R. R.

D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003).
[Crossref]

Fu, P.

D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003).
[Crossref]

Gohle, Ch.

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Gong, Q.

X. Zhang, J. Zhang, R. Li, Q. Gong, and Z. Xu, “Photoionization of H atoms in few-cycle laser pulses,” Eur. Phys. J. D 37, 457–462 (2006).
[Crossref]

X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Exp. 13, 8708–8716 (2005).
[Crossref]

Gordon, A.

A. Gordon and F. Kartner, “Scaling of keV HHG photon yield with drive wavelength,” Opt. Exp. 13, 2941–2947 (2005).
[Crossref]

Goulielmakis, E.

F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004).
[Crossref]

G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003).
[Crossref]

Goullelmakis, E.

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Grasbon, F.

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 (London)  414, 182 (2001).
[Crossref] [PubMed]

Guo, D.-S.

J. Zhang, X. Feng, Z. Xu, and D.-S. Guo, “Phase-dependent angular distributions of photoelectrons in an infinite sequence of linearly polarized few-cycle pulses,” Phys. Rev. A 69, 043409(5) (2004).
[Crossref]

D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003).
[Crossref]

D.-S. Guo, T. Åberg, and B. Crasemann, “Scattering theory of multiphoton ionization in strong fields,” Phys. Rev. A 40, 4997–5005(1989).
[Crossref] [PubMed]

Gurtler, A.

A. Gurtler, E. Robicheaux, W. J. Zande, and L. D. Noordam, “Asymmetry in the strong-field ionization of Rydberg atoms by few-cycle pulses,” Phys. Rev. Lett. 92, 033002(4) (2004).
[Crossref] [PubMed]

Hansch, T. W.

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Henstchel, M.

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Holzwarth, R.

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Kakehata, M.

Kartner, F.

A. Gordon and F. Kartner, “Scaling of keV HHG photon yield with drive wavelength,” Opt. Exp. 13, 2941–2947 (2005).
[Crossref]

Kobayashi, Y.

Krausz, F.

F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004).
[Crossref]

G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003).
[Crossref]

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

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]

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

M. Nisoli, S. De Silvestri, O. Svelto, R. Szipcs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997).
[Crossref] [PubMed]

Lezius, M.

F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004).
[Crossref]

G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003).
[Crossref]

Li, R.

X. Zhang, J. Zhang, R. Li, Q. Gong, and Z. Xu, “Photoionization of H atoms in few-cycle laser pulses,” Eur. Phys. J. D 37, 457–462 (2006).
[Crossref]

Li, X.

D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003).
[Crossref]

Linder, F.

F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004).
[Crossref]

G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003).
[Crossref]

Milosevic, D. B.

D. B. Milosevic, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–262 (2006).
[Crossref]

D. B. Milosevic, G. G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Exp. 11, 1418–1429 (2003).

Muller, H. G.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98. 013901 (2007)
[Crossref] [PubMed]

Nisoli, M.

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 (London)  414, 182 (2001).
[Crossref] [PubMed]

M. Nisoli, S. De Silvestri, O. Svelto, R. Szipcs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997).
[Crossref] [PubMed]

Noordam, L. D.

A. Gurtler, E. Robicheaux, W. J. Zande, and L. D. Noordam, “Asymmetry in the strong-field ionization of Rydberg atoms by few-cycle pulses,” Phys. Rev. Lett. 92, 033002(4) (2004).
[Crossref] [PubMed]

Paulus, G. G.

D. B. Milosevic, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–262 (2006).
[Crossref]

F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004).
[Crossref]

D. B. Milosevic, G. G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Exp. 11, 1418–1429 (2003).

G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003).
[Crossref]

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 (London)  414, 182 (2001).
[Crossref] [PubMed]

Priori, E.

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 (London)  414, 182 (2001).
[Crossref] [PubMed]

Robicheaux, E.

A. Gurtler, E. Robicheaux, W. J. Zande, and L. D. Noordam, “Asymmetry in the strong-field ionization of Rydberg atoms by few-cycle pulses,” Phys. Rev. Lett. 92, 033002(4) (2004).
[Crossref] [PubMed]

Salieres, P.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98. 013901 (2007)
[Crossref] [PubMed]

Sartania, S.

Scrinzi, A.

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Spielmann, Ch.

Stagira, S.

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 (London)  414, 182 (2001).
[Crossref] [PubMed]

Svelto, O.

Szipcs, R.

Takada, H.

Tate, J.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98. 013901 (2007)
[Crossref] [PubMed]

Torizuka, K.

Udem, Th.

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Uibereacker, M.

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Villoresi, P.

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 (London)  414, 182 (2001).
[Crossref] [PubMed]

Walther, H.

F. Linder, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001(4) (2004).
[Crossref]

G. G. Paulus, F. Linder, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004(4) (2003).
[Crossref]

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 (London)  414, 182 (2001).
[Crossref] [PubMed]

Xu, Z.

X. Zhang, J. Zhang, R. Li, Q. Gong, and Z. Xu, “Photoionization of H atoms in few-cycle laser pulses,” Eur. Phys. J. D 37, 457–462 (2006).
[Crossref]

X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Exp. 13, 8708–8716 (2005).
[Crossref]

J. Zhang, X. Feng, Z. Xu, and D.-S. Guo, “Phase-dependent angular distributions of photoelectrons in an infinite sequence of linearly polarized few-cycle pulses,” Phys. Rev. A 69, 043409(5) (2004).
[Crossref]

J. Zhang and Z. Xu, “Above-threshold ionization of Kr atoms in an infinite sequence of circularly polarized few-cycle pulses,” Phys. Rev. A 68, 013402(5) (2003).
[Crossref]

D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003).
[Crossref]

Yakovlev, V.S.

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
[Crossref] [PubMed]

Zande, W. J.

A. Gurtler, E. Robicheaux, W. J. Zande, and L. D. Noordam, “Asymmetry in the strong-field ionization of Rydberg atoms by few-cycle pulses,” Phys. Rev. Lett. 92, 033002(4) (2004).
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Zhang, J.

X. Zhang, J. Zhang, R. Li, Q. Gong, and Z. Xu, “Photoionization of H atoms in few-cycle laser pulses,” Eur. Phys. J. D 37, 457–462 (2006).
[Crossref]

X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Exp. 13, 8708–8716 (2005).
[Crossref]

J. Zhang, X. Feng, Z. Xu, and D.-S. Guo, “Phase-dependent angular distributions of photoelectrons in an infinite sequence of linearly polarized few-cycle pulses,” Phys. Rev. A 69, 043409(5) (2004).
[Crossref]

J. Zhang and Z. Xu, “Above-threshold ionization of Kr atoms in an infinite sequence of circularly polarized few-cycle pulses,” Phys. Rev. A 68, 013402(5) (2003).
[Crossref]

D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003).
[Crossref]

Zhang, X.

X. Zhang, J. Zhang, R. Li, Q. Gong, and Z. Xu, “Photoionization of H atoms in few-cycle laser pulses,” Eur. Phys. J. D 37, 457–462 (2006).
[Crossref]

X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Exp. 13, 8708–8716 (2005).
[Crossref]

Eur. Phys. J. D (1)

X. Zhang, J. Zhang, R. Li, Q. Gong, and Z. Xu, “Photoionization of H atoms in few-cycle laser pulses,” Eur. Phys. J. D 37, 457–462 (2006).
[Crossref]

J. Phys. B (1)

D. B. Milosevic, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39, R203–262 (2006).
[Crossref]

Nature (2)

A. Baltuska, Th. Udem, M. Uibereacker, M. Henstchel, E. Goullelmakis, Ch. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature (London)  421, 611–615 (2003).
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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 (London)  414, 182 (2001).
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Opt. Exp. (3)

D. B. Milosevic, G. G. Paulus, and W. Becker, “High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase,” Opt. Exp. 11, 1418–1429 (2003).

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[Crossref]

X. Zhang, J. Zhang, L. Bai, Q. Gong, and Z. Xu, “Verification of a scaling law in few-cycle laser pulses,” Opt. Exp. 13, 8708–8716 (2005).
[Crossref]

Opt. Lett. (4)

Phys. Rev A (1)

S. Chelkowski, A. D. Bandrauk, and A. Apolonski, “Phase-dependent asymmetries in strong-field photoioniza-tion by few-cycle laser pulses,” Phys. Rev A 70, 013815(9) (2004).
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Phys. Rev. A (5)

D.-S. Guo, J. Zhang, Z. Xu, X. Li, P. Fu, and R. R. Freeman, “Practical scaling law for photoelectron angular distributions,” Phys. Rev. A 68, 043404(5) (2003).
[Crossref]

J. Zhang and Z. Xu, “Above-threshold ionization of Kr atoms in an infinite sequence of circularly polarized few-cycle pulses,” Phys. Rev. A 68, 013402(5) (2003).
[Crossref]

J. Zhang, X. Feng, Z. Xu, and D.-S. Guo, “Phase-dependent angular distributions of photoelectrons in an infinite sequence of linearly polarized few-cycle pulses,” Phys. Rev. A 69, 043409(5) (2004).
[Crossref]

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[Crossref]

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A. Gurtler, E. Robicheaux, W. J. Zande, and L. D. Noordam, “Asymmetry in the strong-field ionization of Rydberg atoms by few-cycle pulses,” Phys. Rev. Lett. 92, 033002(4) (2004).
[Crossref] [PubMed]

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Figures (3)

Fig. 1.
Fig. 1. (Color online) The variation of the photoelectron rates in a pair of opposite directions along the CE phase (a) ϕf = 0; (b) ϕf = π. The linearly polarized laser pulse is of five-cycle duration and of peak intensity 4 × 1013 W/cm2 and the target atom is xenon.
Fig. 2.
Fig. 2. (Color online) The polar plots of PADs for the second ATI peak (q = 9) in linearly polarized 5-cycle laser pulses: (a) for xenon atom with Eb , = 12.1 eV, λ = 800 nm, I = 5 × 1013 W/cm2; (b) for argon atom with Eb = 15.76 eV, λ = 616 nm, I = 11.5 × 1013 W/cm2. The two calculated PADs, each was normalized by its maximum, with same CE phases (ϕ = 0), but with different pulse intensities and frequencies for atoms with different binding energies show identical angular distributions, verifying the scaling law.
Fig. 3.
Fig. 3. (Color online) The dependence on the CE phase of the asymmetric degree is calculated by the TDSE method: (a) Eb = 13.6 eV, λ = 800 nm, Ia = 3.5 × 1014 W/cm2; (b)-(e) are of double binding energy, double frequency to that in (a), but of different pulse intensities: (b) I = 2Ia ; (c) I = 22 Ia ;(d) I = 23 Ia ; and (e) I = 24 Ia . In all the calculations, five-cycle pulses are used. The similarity in (a) and (d) verifies the scaling law.

Equations (17)

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E ( t ) = E 0 cos ( ω t + ϕ 0 ) sin 2 ( ω 2 n t + π 2 ) ,
ω 1 = ω ; ω 2 = ω ( 1 + 1 n ) ; ω 3 = ω ( 1 1 n )
d W d Ω p f = ( 2 m e 3 ω 5 ) 1 2 ( 2 π ) 2 ( q ε b ) 1 2 ( q u p ) 2 Φ i ( P f q k ) 2
× q i , j i χ j 1 q 1 , j 2 q 2 , j 3 q 3 ( z f ) χ j 1 , j 2 , j 3 ( z f ) * 2 ,
E k P f 2 2 m e = q ω E b .
q = q 1 + q 2 + q 3 + ( q 2 q 3 ) n ,
χ j 1 , j 2 , j 3 ( z f ) = m i X j 1 + 2 m 1 + m 4 + m 5 + m 6 + m 7 ( ζ 1 f ) X j 2 + 2 m 2 + m 4 m 5 + m 8 + m 9 ( ζ 2 f )
× X j 3 + 2 m 3 + m 6 + m 7 + m 8 m 9 ( ζ 3 f ) X m 1 ( z 1 ) X m 9 ( z 9 ) ,
X n ( z ) = J n ( z ) exp ( in arg ( z ) ) .
ζ 1 f = 2 e Λ 1 m e ω 1 P f ε 1 , ζ 2 f = 2 e Λ 2 m e ω 2 P f ε 2 , ζ 3 f = 2 e Λ 3 m e ω 3 P f ε 3 , z 1 = e 2 Λ 1 2 2 m e ω 1 ε 1 ε 1 , z 2 = e 2 Λ 2 2 2 m e ω 2 ε 2 ε 2 , z 3 = e 2 Λ 3 2 2 m e ω 3 ε 3 ε 3 , z 4 = 2 e 2 Λ 1 Λ 2 ε 1 ε 2 m e ( ω 1 + ω 2 ) , z 5 = 2 e 2 Λ 1 Λ 2 ε 1 ε 2 * m e ( ω 2 ω 1 ) , z 6 = 2 e 2 Λ 1 Λ 2 ε 1 ε 3 m e ( ω 1 + ω 3 ) , z 7 = 2 e 2 Λ 1 Λ 3 ε 1 ε 3 * m e ( ω 3 ω 1 ) , z 8 = 2 e 2 Λ 2 Λ 3 ε 2 ε 3 m e ( ω 2 + ω 3 ) , z 9 = 2 e 2 Λ 2 Λ 3 ε 2 ε 3 * m e ( ω 3 ω 2 ) .
ε j = [ ε x cos ( ξ / 2 ) + i ε y sin ( ξ / 2 ) ] e i ϕ j ,
ϕ 1 = 0 , ϕ 2 = ϕ 0 n , ϕ 3 = ϕ 0 n .
Λ 1 = m e ω u p 2 e ; Λ 2 = n Λ 1 2 ( n + 1 ) ; Λ 3 = n Λ 1 2 ( n 1 ) .
P f ω i = 2 m e ( q ε b ) ω ω i ,
e Λ 1 ω 1 = m e u p 2 , e Λ 2 ω 2 = m e u p 4 ( n n + 1 ) 3 2 , e Λ 3 ω 3 = m e u p 4 ( n n 1 ) 3 2 ,
ε j ε k = cos ξ e i ( ϕ j + ϕ k ) , ε j ε k * = e i ( ϕ j + ϕ k ) ,
n > q ,

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