Abstract

We present a new scheme to generate efficient isolated attosecond pulse by using the combination of a fundamental and a weak second harmonic (SH) field in the multi-cycle regime. Because of the symmetry breaking of the electric field, the ionization dynamics of the electron can be controlled by adjusting the relative phase of the two fields. Then attosecond (as) pulses will be generated each full optical cycle of the fundamental field. Our simulation shows that the intensity of the single attosecond pulse can be enhanced by an order of magnitude in our scheme compared with the scheme in [Opt. Lett. 31, 975 (2006)].

© 2007 Optical Society of America

1. Introduction

The appearance and development of attosecond-science have opened new fields of time-resolved studies with unprecedented resolution. Attosecond pulses can be used to temporally probe several fundamental atomic processes, such as inner-shell electronic relaxation or ionization by optical tunneling [2]. A typical attosecond pulse with a duration of several hundred attosecond covers a bandwidth of about 10 eV. The most well-known resource with the potential to produce ultrashort radiation over such bandwidths is High Harmonic Generation (HHG). The mechanism of HHG can be well explained by the rescattering model [3]. The electron is tunneling ionized by the driving laser, then combines with the core after acceleration in the laser field to emit a harmonic photon. It has been investigated theoretically [4] and experimentally [5] that attosecond pulse train with a periodicity of half optical cycle is generated from HHG. However, an isolated attosecond pulse is needed to operate a pump-probe experiment. Single attosecond pulse generation based on HHG has been realized by using a state-of-the-art 5-fs laser pulse with the wavelength of 800 nm [6, 7], which is a rather stringent requirement. Single attosecond pulse production using long driving field can be carried out with a time dependent polarization pulse [8, 9, 10]. Based on the use of phase-stabilized few-cycle driving pulse in combination with the polarization gating technique, isolated single-cycle attosecond pulse is generated in experiment [11], which opens the way to a new regime in ultrafast physics. Very recently the requirement for single attosecond production with single-cycle driving pulse has been released by using a multicycle-driver regime by adding a weak second-harmonic field [1]. It is shown that a minor addition of phase-locked second harmonic field to the fundamental driver pulse leads to a major difference in the maximum kinetic energies of the recombining electrons in adjacent half-cycles during HHG. Then attosecond pulses with a periodicity of one optical cycle can be produced. Therefore, the duration required for single attosecond pulse generation can be doubled and extended to the multi-cycle regime. However, the attosecond pulse is generated from the cut-off region with a rather low yield. The low intensity of the attosecond pulse will limit its further application.

In this paper, we propose a new scheme for relatively intense isolated attosecond pulse generation in the multi-cycle regime. The combination of a multi-cycle fundamental and a weak second-harmonic field is adopted. By adjusting the relative phase of the two fields, the electron ionization dynamics can be controlled to induce the change of harmonic efficiency in two successive half cycles. Then attosecond pulses can be generated each full optical cycle more efficiently from lower order harmonics compared to the scheme in reference [1]. An isolated attosecond pulse can be extracted with the intensity about an order of magnitude higher than that from the method in reference [1].

2. Results and discussion

We calculated the single atom-response to the driving laser field using Lewenstein’s analytic formula of atomic dipoles for HHG [12]. The neon atom is chosen in our calculation. The A-D-K tunnelling ionization model [13] is used to obtain the ionization rate. The driving pulse can be expressed in the form E(t) = f(t)[E 0 cos(ω0 t)+E 1 cos(2ω0 t + ϕ)] ,where f(t) presents the profile of the laser field, ω0 is the fundamental laser frequency, ϕ is the relative phase between the fundamental and the SH field, E 0 and E 1 denote the amplitudes of the fundamental (800 nm) and the SH field (400 nm), respectively. In our simulation, the sin2 pulse profile is adopted, E 0 = 0.12 and E 1 = 0.0168 in atomic units, which correspond to laser intensities of 5×1014W/cm2 and 1×1013W/cm2, respectively. Then the intensity of the SH field is 2% of the fundamental one. Figure 1 shows harmonic spectra and the corresponding spectrotemporal plot driven by a 21 fs bichromatic field at ϕ = 0.5π.

 

Fig. 1. Harmonic spectrum (a) and the spectrogram (b) of harmonic radiation from Neon exposed to a 21 fs, 800 nm, two-color field.

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It can be seen from Fig. 1(b) that there are two types of bows generated due to the breaking of the symmetry of the electric field in successive half cycles of the fundamental field. An intense bow with a relatively low peak is generated in the first half cycle of the fundamental field. While a relatively weak bow with a high peak is generated in the next half cycle. This leads to the occurrence of two separated cut-off regions in harmonic spectrum, which correspond to 68 – 78ω0 and 80 – 90ω0, respectively [see Fig. 1(a)]. Then a small amount of the SH field (2% of the fundamental intensity) results in the significant changes of both the maximum photon-energies and the harmonic intensities between two half fundamental cycles. The time dependence of the superpositions of harmonics in the two separated cut-off regions are shown in Fig. 2. Attosecond pulses with a periodicity of one optical cycle of the fundamental field are obtained in both regions. The intensity of the pulse train from the low frequency cut-off region is about an order of magnitude higher than that from the higher frequency cut-off region.

 

Fig. 2. Time profiles of the harmonic attosecond pulse trains from the first cut-off region (a) and the second cut-off region (b) in Fig. 1(a).

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In order to obtain a better comprehension and an intuitive picture of the process, the classical dynamics and the ionization rate of the electron in a continuous-wave (cw) bichromatic field are considered as shown in Fig. 3. Bows with low and high peaks are generated alternately due to the asymmetric laser field [see Fig. 3(a)]. An electron ionized when the electric field is positive will obtain more kinetic energy than the electron ionized when the electric field is negative. Then the recombination of the electron and the parent core will generate harmonics with different maximum frequencies between two adjacent half fundamental cycles. The intensity change of the successive bows can be explained by the ionization rate of the electron. It has been confirmed that the atomic ionization dynamics can be control by using a two-color field [14]. In our calculation, the depletion of the ground state is negligible because of the slight ionization possibility of the neutral atoms. Therefore, the harmonic efficiency is determined by the ionization rate of the electron based on the three-step model [3]. As shown in Fig. 3(b), the bows with low peaks are generated by electrons which are ionized near the peak of the laser field where the ionization rate is high. However, the bows with high peaks come from electrons which are ionized when the laser intensity is relatively low. Then there is a distinct contrast between the harmonic intensities in two successive half fundamental cycles. As a result, the addition of a SH field with a minor intensity can influence the electron evolution process as well as the ionization step of HHG seriously, which can ultimately lead to the two separated cut-off regions in the harmonic spectrum. Consequently, there are two mechanisms for attosecond pulse generation each full optical cycle of the fundamental field. The first one is due to the difference between the maximum kinetic energies of electrons from the successive half cycles of the laser field [Fig. 2(b)]. The second one is caused by the change of harmonic emission efficiency in two successive half cycles [Fig. 2(a)].

 

Fig. 3. (a) Classical ionization (black stars) and recombination times (red diamonds) as a function of the return kinetic energy in unit of the ponderomotive potential Up. The arrows demonstrate the evolution process of the electron. (b) Time-dependence of the cw field (blue solid line) amplitude and the ionization rate (black dashed line).

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Figure 4 demonstrates the relationships of the relative phase versus the maximum kinetic energy and the corresponding ionization rate of the electron in the cw case. The optimal phase for attosecond pulses generation based on the first mechanism is 1.6π, and the optimal phase for the second mechanism is ϕ = 0.9π. Then a direct application of the pulse trains is that isolated attosecond pulse can be extracted from such pulse trains driven by a bichromatic field with a duration twice as long as the traditional monochromatic case [6], i.e. a multi-cycle pulse can be used for single attosecond pulse production. Single as pulse based on the first mechanism, which is called the first scheme, has been investigated by T. Pfeifer et al [1]. Here, we present another scheme for single as pulse generation based on the second mechanism. It is more fascinating because of its potential to obtain more efficient isolated as pulse compared with the first scheme. With the help of the Lewenstein model and the fast fourier transforming technique, we calculated the harmonic attosecond pulse driven by a multi-cycle laser field with a duration of 10.5 fs. By applying the filtering method in the low frequency cut-off region of the harmonic spectrum, single attosecond pulse can be produced. Because of the use of a nonadiabatic short pulse instead of an adiabatic cw laser, the optimal phase for the second scheme is 0.8π. Figure 5 shows the comparison of single attosecond pulses generated from both schemes. It can be seen that the intensity of single as pulse from our scheme is about an order of magnitude higher than that from the scheme mentioned in reference [1]. This can be explained by the fact that in our scheme single attosecond pulse is produced from the electron ionized near the peak of the driving field corresponding to a higher ionization rate. Furthermore, the single as pulse is extracted from lower-order harmonics which are more efficiently produced. Our simulation also shows that the variation of ϕ as much as ±0.2π in the second scheme will maintain the isolated attosecond pulse production with the side peaks less than 10% of the main peak intensity. The fluctuation of 30% of the SH intensity will seldom influence the results in our scheme, the peak intensity of the single as pulse varies less than 30%. Furthermore, the sensitivity of single attosecond pulse generation on the carrier-envelope phase (CEP) of the fundamental field is also considered. It is shown that the value of the CEP can be varied as much as ±0.2π to keep the satellites below 15% of the main pulse intensity. These indicate that it is experimentally feasible to obtain efficient isolated attosecond pulse with our scheme.

 

Fig. 4. (a) Ratio of the maximum kinetic energies in the first versus the second half-cycle as a function of the relative phase between the fundamental and the SH fields. (b) Radio of the ionization rates of electrons with maximum kinetic energies in the first versus the second half-cycle as a function of the relative phase between the fundamental and the SH field.

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A fact must be addressed is that we choose the SH field with the intensity 2% rather than 1% of the fundamental intensity as is mentioned in reference [1]. This is because that a slightly more intense SH field is needed to induce a remarkable difference between the ionization rates in two half cycles. Then single attosecond pulse will be generated with satellites less than 2% of the main peak intensity. However it is rather valuable to obtain a much more intense attosecond pulse by increasing the SH field from 1% to 2% intensity of the fundamental field.

 

Fig. 5. Temporal profiles of the isolated attosecond pulses generated with the first scheme (dashed line) at ϕ = 1.5π and the second scheme (solid line) at ϕ = 0.8π.

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3. Conclusion

In conclusion, we present a method for efficient isolated attosecond pulse generation using the combination of a fundamental and a weak second harmonic field in the multi-cycle regime. It is achieved via controlling the ionization dynamics of the electron. This leads to more efficient attosecond pulses generation each full optical cycle of the fundamental field. In contrast to the method mentioned in reference [1], our scheme can enhance the intensity of attosecond pulse by about an order of magnitude. Furthermore, the reduced sensitivity of single attosecond pulse generation on the carrier-envelope phase of the fundamental field is also preserved in our scheme.

Acknowledgement

This work was supported by the National Natural Science Foundation of China under grant No. 10574050, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20040487023 and the National Basic Research Program of China under grant No. 2006CB806006.

† Author to whom correspondence should be addressed.

References and links

1. T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett 31,975–977 (2006). [CrossRef]   [PubMed]  

2. R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004). [CrossRef]  

3. P. B. Corkum, “Plasma Perspective on Strong-Field Multiphoton Ionization,” Phys. Rev. Lett 71,1994–1997 (1993). [CrossRef]   [PubMed]  

4. P. Antoine, A. L’’Huillier, and M. Lewenstein, “Attosecond Pulse Trains Using High-Order Harmonics,” Phys. Rev. Lett 77,1234–1237 (1996). [CrossRef]   [PubMed]  

5. P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosec-ond light bunching,” Nature (London) 426,267–271 (2003). [CrossRef]  

6. I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett 78,1251–1254 (1997). [CrossRef]  

7. M. Hentschel, R. Kienerger, C. 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]  

8. P. B. Corcum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett 19,1870–1872 (1994). [CrossRef]  

9. V. T. Platonenko and V. V. Strelkov, “Single attosecond soft-x-ray pulse generated with a limited laser beam,” J. Opt. Soc. Am. B 16,435–440 (1999). [CrossRef]  

10. V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005). [CrossRef]  

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

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

13. M. V. Ammesov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64,1191–1194 (1986).

14. S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004). [CrossRef]   [PubMed]  

References

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  1. T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett 31,975–977 (2006).
    [Crossref] [PubMed]
  2. R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
    [Crossref]
  3. P. B. Corkum, “Plasma Perspective on Strong-Field Multiphoton Ionization,” Phys. Rev. Lett 71,1994–1997 (1993).
    [Crossref] [PubMed]
  4. P. Antoine, A. L’’Huillier, and M. Lewenstein, “Attosecond Pulse Trains Using High-Order Harmonics,” Phys. Rev. Lett 77,1234–1237 (1996).
    [Crossref] [PubMed]
  5. P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosec-ond light bunching,” Nature (London) 426,267–271 (2003).
    [Crossref]
  6. I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett 78,1251–1254 (1997).
    [Crossref]
  7. M. Hentschel, R. Kienerger, C. 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]
  8. P. B. Corcum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett 19,1870–1872 (1994).
    [Crossref]
  9. V. T. Platonenko and V. V. Strelkov, “Single attosecond soft-x-ray pulse generated with a limited laser beam,” J. Opt. Soc. Am. B 16,435–440 (1999).
    [Crossref]
  10. V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
    [Crossref]
  11. G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S.De Silvestri, and M. Nisoli, “Isolated Single-Cycle Attosecond Pulses,” Science 314,443–446 (2006).
    [Crossref] [PubMed]
  12. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49,2117–2132 (1994).
    [Crossref] [PubMed]
  13. M. V. Ammesov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64,1191–1194 (1986).
  14. S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004).
    [Crossref] [PubMed]

2006 (2)

T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett 31,975–977 (2006).
[Crossref] [PubMed]

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

2005 (1)

V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
[Crossref]

2004 (2)

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004).
[Crossref] [PubMed]

2003 (1)

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosec-ond light bunching,” Nature (London) 426,267–271 (2003).
[Crossref]

2001 (1)

M. Hentschel, R. Kienerger, C. 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]

1999 (1)

1997 (1)

I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett 78,1251–1254 (1997).
[Crossref]

1996 (1)

P. Antoine, A. L’’Huillier, and M. Lewenstein, “Attosecond Pulse Trains Using High-Order Harmonics,” Phys. Rev. Lett 77,1234–1237 (1996).
[Crossref] [PubMed]

1994 (2)

P. B. Corcum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett 19,1870–1872 (1994).
[Crossref]

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

1993 (1)

P. B. Corkum, “Plasma Perspective on Strong-Field Multiphoton Ionization,” Phys. Rev. Lett 71,1994–1997 (1993).
[Crossref] [PubMed]

1986 (1)

M. V. Ammesov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64,1191–1194 (1986).

Abel, M. J.

T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett 31,975–977 (2006).
[Crossref] [PubMed]

Altucci, C.

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

Ammesov, M. V.

M. V. Ammesov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64,1191–1194 (1986).

Antoine, P.

P. Antoine, A. L’’Huillier, and M. Lewenstein, “Attosecond Pulse Trains Using High-Order Harmonics,” Phys. Rev. Lett 77,1234–1237 (1996).
[Crossref] [PubMed]

Avaldi, L.

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

Balcou, P.

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

Baltuska, A.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

Bammer, F.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

Benedetti, E.

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

Brabec, T.

M. Hentschel, R. Kienerger, C. 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]

Burnett, N. H.

P. B. Corcum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett 19,1870–1872 (1994).
[Crossref]

Calegari, F.

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

Charalambidis, D.

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosec-ond light bunching,” Nature (London) 426,267–271 (2003).
[Crossref]

Christov, I. P.

I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett 78,1251–1254 (1997).
[Crossref]

Constant, E.

V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
[Crossref]

Corcum, P. B.

P. B. Corcum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett 19,1870–1872 (1994).
[Crossref]

Corkum, P.

M. Hentschel, R. Kienerger, C. 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]

Corkum, P. B.

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

P. B. Corkum, “Plasma Perspective on Strong-Field Multiphoton Ionization,” Phys. Rev. Lett 71,1994–1997 (1993).
[Crossref] [PubMed]

Cormier, E.

V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
[Crossref]

Delone, N. B.

M. V. Ammesov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64,1191–1194 (1986).

Drescher, M.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

M. Hentschel, R. Kienerger, C. 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]

Flammini, R.

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

Gallmann, L.

T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett 31,975–977 (2006).
[Crossref] [PubMed]

Goulielmakis, E.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

Gurtler, A.

S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004).
[Crossref] [PubMed]

Heinzmann, U.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

M. Hentschel, R. Kienerger, C. 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]

Hentschel, M.

M. Hentschel, R. Kienerger, C. 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]

Ivanov, M. Y.

P. B. Corcum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett 19,1870–1872 (1994).
[Crossref]

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

Kapteyn, H. C.

I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett 78,1251–1254 (1997).
[Crossref]

Kienberger, R.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

Kienerger, R.

M. Hentschel, R. Kienerger, C. 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]

Kleineberg, U.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

Krainov, V. P.

M. V. Ammesov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64,1191–1194 (1986).

Krausz, F.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

M. Hentschel, R. Kienerger, C. 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]

L’’Huillier, A.

P. Antoine, A. L’’Huillier, and M. Lewenstein, “Attosecond Pulse Trains Using High-Order Harmonics,” Phys. Rev. Lett 77,1234–1237 (1996).
[Crossref] [PubMed]

L’Huillier, A.

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

Leone, S. R.

T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett 31,975–977 (2006).
[Crossref] [PubMed]

Lewenstein, M.

P. Antoine, A. L’’Huillier, and M. Lewenstein, “Attosecond Pulse Trains Using High-Order Harmonics,” Phys. Rev. Lett 77,1234–1237 (1996).
[Crossref] [PubMed]

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

Martens, R. L.

V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
[Crossref]

Mevel, E.

V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
[Crossref]

Milosevic, N.

M. Hentschel, R. Kienerger, C. 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]

Muller, H. G.

S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004).
[Crossref] [PubMed]

Murnane, M. M.

I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett 78,1251–1254 (1997).
[Crossref]

Neumark, D. M.

T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett 31,975–977 (2006).
[Crossref] [PubMed]

Ni, Y.

S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004).
[Crossref] [PubMed]

Nisoli, M.

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

Noordam, L. D.

S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004).
[Crossref] [PubMed]

Papadogiannis, N. A.

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosec-ond light bunching,” Nature (London) 426,267–271 (2003).
[Crossref]

Pfeifer, T.

T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett 31,975–977 (2006).
[Crossref] [PubMed]

Platonenko, V. T.

Poletto, L.

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

Reider, G. A.

M. Hentschel, R. Kienerger, C. 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]

Sansone, G.

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

Scrinzi, A.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

Silvestri, S.De

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

Spielmann, C.

M. Hentschel, R. Kienerger, C. 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]

Stagira, S.

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

Strelkov, V.

V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
[Crossref]

Strelkov, V. V.

Tcherbakoff, O.

V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
[Crossref]

Tsakiris, G. D.

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosec-ond light bunching,” Nature (London) 426,267–271 (2003).
[Crossref]

Tzallas, P.

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosec-ond light bunching,” Nature (London) 426,267–271 (2003).
[Crossref]

Uiberacker, M.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

Velotta, R.

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

Villoresi, P.

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

Vozzi, C.

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

Vrakking, M. J. J.

S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004).
[Crossref] [PubMed]

Wester-walbesloh, T.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

Witte, K.

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosec-ond light bunching,” Nature (London) 426,267–271 (2003).
[Crossref]

Yakovlev, V.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

Zair, A.

V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
[Crossref]

Zamith, S.

S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

V. Strelkov, A. Zair, O. Tcherbakoff, R. L. Martens, E. Cormier, E. Mevel, and E. Constant, “Single attosecond pulse production with an ellipticity-modulated driving IR pulse,” J. Phys. B 38,L161–L167 (2005).
[Crossref]

Nature (London) (3)

M. Hentschel, R. Kienerger, C. 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]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Wester-walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) 427,817–821 (2004).
[Crossref]

P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D. Tsakiris, “Direct observation of attosec-ond light bunching,” Nature (London) 426,267–271 (2003).
[Crossref]

Opt. Lett (3)

P. B. Corcum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett 19,1870–1872 (1994).
[Crossref]

T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett 31,975–977 (2006).
[Crossref] [PubMed]

S. Zamith, Y. Ni, A. Gurtler, L. D. Noordam, H. G. Muller, and M. J. J. Vrakking, “Control of atomic ionization by two-color few-cycle pulses,” Opt. Lett 29,2303–2305 (2004).
[Crossref] [PubMed]

Phys. Rev. A (1)

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

Phys. Rev. Lett (3)

I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett 78,1251–1254 (1997).
[Crossref]

P. B. Corkum, “Plasma Perspective on Strong-Field Multiphoton Ionization,” Phys. Rev. Lett 71,1994–1997 (1993).
[Crossref] [PubMed]

P. Antoine, A. L’’Huillier, and M. Lewenstein, “Attosecond Pulse Trains Using High-Order Harmonics,” Phys. Rev. Lett 77,1234–1237 (1996).
[Crossref] [PubMed]

Science (1)

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

Sov. Phys. JETP (1)

M. V. Ammesov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64,1191–1194 (1986).

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Figures (5)

Fig. 1.
Fig. 1.

Harmonic spectrum (a) and the spectrogram (b) of harmonic radiation from Neon exposed to a 21 fs, 800 nm, two-color field.

Fig. 2.
Fig. 2.

Time profiles of the harmonic attosecond pulse trains from the first cut-off region (a) and the second cut-off region (b) in Fig. 1(a).

Fig. 3.
Fig. 3.

(a) Classical ionization (black stars) and recombination times (red diamonds) as a function of the return kinetic energy in unit of the ponderomotive potential Up. The arrows demonstrate the evolution process of the electron. (b) Time-dependence of the cw field (blue solid line) amplitude and the ionization rate (black dashed line).

Fig. 4.
Fig. 4.

(a) Ratio of the maximum kinetic energies in the first versus the second half-cycle as a function of the relative phase between the fundamental and the SH fields. (b) Radio of the ionization rates of electrons with maximum kinetic energies in the first versus the second half-cycle as a function of the relative phase between the fundamental and the SH field.

Fig. 5.
Fig. 5.

Temporal profiles of the isolated attosecond pulses generated with the first scheme (dashed line) at ϕ = 1.5π and the second scheme (solid line) at ϕ = 0.8π.

Metrics