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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 17 — Aug. 13, 2012
  • pp: 19289–19296
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Coherent control of high-order harmonic generation by phase jump pulses

Yang Xiang, Yueping Niu, Hongmei Feng, Yihong Qi, and Shangqing Gong  »View Author Affiliations


Optics Express, Vol. 20, Issue 17, pp. 19289-19296 (2012)
http://dx.doi.org/10.1364/OE.20.019289


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Abstract

We theoretically investigate the high-order harmonic generation driven by laser pulses with a π -phase jump. The cutoff of high-order harmonic spectrum extends dramatically due to the phase jump which enlarges the asymmetry of the laser field. We find that the intensity and the coherence of the continuum can be controlled by the jump time. By selecting proper frequency of the continuum spectrum, a sub-100-attosecond pulse can be generated without any phase compensation.

© 2012 OSA

1. Introduction

In recent years, high-order harmonic generation (HHG) has become one of the hot topics of super intense laser physics. Its wide applications on the generation of attosecond (as) pulses make it attract great interests around the world [1

1. P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67(6), 813–855 (2004). [CrossRef]

, 2

2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009). [CrossRef]

]. The physical origin of the HHG process can be understood by the well-known three-step model (TSM) proposed by Corkum [3

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

]. Firstly a free electron is born by tunneling through the potential barrier formed by the electric field and the Coulomb potential, then oscillates in the laser field and recombines with the parent ion at last. During the recombination, a photon is emitted. Usually, the HHG spectrum contains only discrete odd harmonics since this process occurs in each half cycle of the laser field. In order to generate isolated attosecond pulse (IAP), a broad continuum spectrum is needed. Thanks to the development of the laser technology, ultra-short and ultra-strong laser pulses can be generated in experiment and the evolution of the electric field of the pulse can be well controlled [4

4. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]

, 5

5. D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39(14), R203–R262 (2006). [CrossRef]

], which provide powerful tools for the control of IAP. Till now, many control ways has been proposed for the generation of IAP, such as polarization gate [6

6. P. B. Corkum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett. 19(22), 1870–1872 (1994). [CrossRef] [PubMed]

8

8. Y. Yu, X. Song, Y. Fu, R. Li, Y. Cheng, and Z. Xu, “Theoretical investigation of single attosecond pulse generation in an orthogonally polarized two-color laser field,” Opt. Express 16(2), 686–694 (2008). [CrossRef] [PubMed]

], two-color (or multi-color) [9

9. Y. Xiang, Y. Niu, and S. Gong, “Proposal for isolated-attosecond-pulse generation in the multicycle regime through modulation of the carrier wave,” Phys. Rev. A 85(2), 023808 (2012). [CrossRef]

12

12. H.-C. Bandulet, D. Comtois, E. Bisson, A. Fleischer, H. Pepin, J.-C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Gating attosecond pulse train generation using multicolor laser fields,” Phys. Rev. A 81(1), 013803 (2010). [CrossRef]

], chirp [13

13. J. J. Carrera and S.-I. Chu, “Extension of high-order harmonic generation cutoff via coherent control of intense few-cycle chirped laser pulses,” Phys. Rev. A 75(3), 033807 (2007). [CrossRef]

] and static electric field control [14

14. S. Odžak and D. B. Milosevic, “Attosecond pulse generation by a coplanar circular and static field combination,” Phys. Lett. A 355(4-5), 368–372 (2006). [CrossRef]

, 15

15. W. Hong, P. Lu, W. Cao, P. Lan, and X. Wang, “Control of quantum paths of high-order harmonics and attosecond pulse generation in the presence of a static electric field,” J. Phys. B 40(12), 2321–2331 (2007). [CrossRef]

], and so on.

2. Principle and method

In our calculations, the target atom is the helium atom. The interaction between the helium atom and the laser field can be described by the following three-dimensional (3D) time-dependent Schrödinger equation (TDSE) with a single active electron [20

20. H. Xu, X. Tang, and P. Lambropoulos, “Nonperturbative theory of harmonic generation in helium under a high-intensity laser field: The role of intermediate resonance and of the ion,” Phys. Rev. A 46(5), R2225–R2228 (1992). [CrossRef] [PubMed]

] and dipole approximation [The atom units (a.u.) are used in all equations in this paper, unless otherwise mentioned]:
iψ(r,t)t=[122+Vc(r)r·E(t)]ψ(r,t),
(1)
where, ris the position vector, Vc(r) is the effective Coulomb potential, and E(t)=E(t)z^ is the electric field of the laser pulse with polarization direction along z axis. For helium atom, the effective Coulomb potential is expressed as Vc(r)=1r[1+(1+27r/16)e27r/8] with r=|r| [21

21. M. Gavrila, Atoms in Intense Laser Fields (Academic, New York, 1992).

] and the ionization potential Ip is 0.904a.u.. Equation (1) can be solved effectively by using a partial-wave decomposition of the wave function method, together with the Peaceman-Rachford scheme [22

22. D. R. Hartree, The Calculation of Atomic Structures (Wiley, New York, 1957).

]. When the time- dependent wave function ψ(r,t) is obtained, the mean acceleration can be calculated by means of the Ehrenfest’s theorem [23

23. K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, “Calculation of the background emitted during high-harmonic generation,” Phys. Rev. A 45(5), 3347–3349 (1992). [CrossRef] [PubMed]

]:
dA(t)z¨(t)=ψ(r,t)|V(r)zE(t)|ψ(r,t),
(2)
The HHG power spectrum P(ω) can be obtained by taking the Fourier transforms ofdA(t):

P(ω)=|dA(t)ejωtdt|2.
(3)

3. Numerical results and analysis

Usually, a laser pulse can be expressed as:
E(t)=Ff(t)cos(ωt+ϕi),
(4)
where, F, f(t), ω and ϕi are the amplitude, envelope, frequency and carrier-envelope phase of the laser pulse, respectively. As mentioned in [18

18. P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A 82(4), 045805 (2010). [CrossRef]

], the electric field of a pulse with a phase jump can be expressed as:
E(t)={Ff(t)cos(ωt+ϕi)ift<t0,Ff(t)cos(ωt+ϕi+ϕj)iftt0,
(5)
where, ϕj is the jump phase introduced into the electric field at t=t0. The phase jump laser field in Eq. (5) can be realized by femtosecond pulse shaping technology, which is described in [16

16. B. T. Torosov and N. V. Vitanov, “Coherent control of a quantum transition by a phase jump,” Phys. Rev. A 76(5), 053404 (2007). [CrossRef]

]. In this paper, F is 0.12 a.u. (corresponding to the intensity of 5.0×1014W/cm2) and ω is 0.057 a.u. (corresponding to the wavelength of 800nm). The envelope is f(t)=e2ln2(t/τ)2with the durationτ. In order to obtain a larger asymmetry of the electric filed, we consider the cases that the phase jump occurs at the zero points near the pulse center with a jump phase ofπ. Figure 1
Fig. 1 The HHG spectra driven by the pulses with different jump times.
displays the HHG spectra driven by the laser pulses with τ=10fs and ϕi=0 for jump time at t0=0.75T, 0.25T, 0.25T, and 0.75T, where T is the period of the laser field. As a reference, the HHG spectrum without phase jump is also shown. From this figure, we can see that the jump time t0 has significant effects on the HHG spectrum. For t0=0.75T, the cutoff of the HHG spectrum is about at 280th-order harmonic which is much larger than the well-known value Ip+3.17Up(about 77 harmonics, Up=F2/4ω2 is the ponderomotive energy of the electron) and a very broad continuum spectrum appears. Fort0=0.25T, the HHG cutoff extends to about 300 harmonics and the intensity of the continuum spectrum is higher than that for t0=0.75T, except that near the cutoff. Fort0=0.25T, the intensity of continuum spectrum increases further and the cutoff has not apparent variation. While fort0=0.75T, the intensity of the continuum spectrum is a little higher than that for t0=0.25T, while the cutoff decreases to about 270 harmonics.

In order to explore the underlying mechanism of the HHG spectrum variation with the jump time, we show in Fig. 2
Fig. 2 The RKE of the electron as a function of born and return time and the electric fields for different jump times. (a) t0=0.75T; (b) t0=0.25T; (c) t0=0.25T; (d) t0=0.75T.
the return kinetic energy (RKE) of the electron vs. born time and return time for different jump times, as well as the electric field. The RKE can be obtained by solving Newton equation. From Fig. 2(a), we can see that due to the enlargement of the asymmetry of the laser field by phase jump, the maximum RKE (MRKE) reaches about 13.8Up(corresponding to the cutoff of 280 harmonics). In Fig. 2(b), the MRKE increases to 14.8Up(corresponding to the cutoff of 299 harmonics), due to that the asymmetry of the laser field is enlarged compared with that in Fig. 2(a). In Fig. 2(c), the MRKE is about 15.0Up(corresponding to the cutoff of 303harmonics). This value is comparable with the MRKE in Fig. 2 (b), since the asymmetry of the electric field has little changes compared with that in Fig. 2(b). In Fig. 2(d), due to the decrease of the asymmetry of the laser field, the MRKE reduces to 13.5Up(corresponding to the cutoff of 274 harmonics). Comparing the classical results with the cutoffs of the HHG spectra in Fig. 1, it is easy to find that they agree well.

Now we begin to discuss the variation of the intensity of the continuum spectra with jump time. According to the TSM, the continuum spectrum comes from the electrons with RKE between the MRKE and the second MRKE, due to their few recollision times with the parent ion [24

24. M. Protopapas, D. G. Lappas, C. H. Keitel, and P. L. Knight, “Recollisions, bremsstrahlung, and attosecond pulses from intense laser fields,” Phys. Rev. A 53(5), R2933–R2936 (1996). [CrossRef] [PubMed]

]. From Fig. 2(a), we can see that the electrons with RKE between 3.17Up and 13.8Up contribute to the continuum spectrum, which are mainly born at tAi[the time corresponding to peak Ai(i=1,2,3,4), the same below]. In Fig. 2(b), the electrons with RKE between 3.0Up and 14.8Up mainly born at tBi(i=1,2,3,4) and return at tB, which contribute to the generation of continuum spectrum. According to the ADK theory [25

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

], the tunneling ionization rate of the electron increases nonlinearly with the electric field intensity, i.e., the stronger the electric field, the larger the birth rate of free electrons and the higher the HHG spectrum generated by them. Apparently, the electric field |E(tBi)| is stronger than |E(tAi)|(i=1,2,3. Though |E(tA4)| is a little higher than|E(tB4)|, both of them are very low and the electrons born at these peaks have not significant contributions to the HHG spectrum), so the intensity of the continuum spectrum for t0=0.25Tis higher than that for t0=0.75T. For the case of t0=0.25T, as shown in Fig. 2(c), the continuum spectrum mainly comes from the electrons born at tCi(i=1,2,3,4,5) and |E(tCi)| is stronger than |E(tBi)|(i=1,2,3,4). Therefore, the intensity of continuum spectrum for t0=0.25T enhances compared with that for t0=0.25T. As is displayed in Fig. 2(d), the electrons which contribute to the generation of continuum spectrum are mainly born at tDi(i=1,2,3,4,5). Compare Figs. 2(c) and 2(d), one can find |E(tDi)|>|E(tCi)|(i=1,2,3,4,5), so the continuum spectrum for t0=0.75 has a higher intensity.

As mentioned in the introduction, the generation of IAP requires continuum spectrum. On the other hand, the coherence of the continuum spectrum has a great influence on the temporal profile of the generated IAP. So it is necessary to discuss the coherence of the continuum spectra in Fig. 1. Usually, there are two dominant quantum paths with different burst times contribute to each harmonic, i.e., the long path and the short path. The long path means the electron is born at an earlier time but returns at a later time while the short path is just the contrary. The different burst times of the two paths may led to decoherence of the harmonics [26

26. P. Antoine, A. L’Huillier, and M. Lewenstein, “Attosecond pulse trains using high–order harmonics,” Phys. Rev. Lett. 77(7), 1234–1237 (1996). [CrossRef] [PubMed]

]. In the map of RKE vs. born time, the positive and negative slope sections are corresponding to the long and short paths respectively. While in the map of RKE vs. return time, the positive and negative slope sections are corresponding to the short and long paths respectively, as marked in Fig. 2 (a). From Fig. 2(a), we can see that only short paths exist for the electrons with RKE between 3.17Up and8.44Up. This means that the photons generated by these electrons burst with highly coherence, which is beneficial to the generation of IAP. Selecting the harmonics between the 120th and 160th order (These harmonics come from the electrons with RKE from 5.4Upto7.48Up. For comparison, we only consider the same harmonic orders in the following discussions), an isolated 76as pulse can be obtained, as show in Fig. 3(a)
Fig. 3 The generated IAPs from the spectra of Fig. 1 driven by different pulses. (a)t0=0.75T; (b)t0=0.25T; (c) t0=0.25T; (d) t0=0.75T. The harmonics used are from the 120th to 160th order.
. Fort0=0.25T, though both short and long paths exist for the selected continuum spectrum, a regular IAP can be obtained with duration about 65as, as Fig. 3(b) displays. As can be seen from Fig. 2(b), for the electrons with RKE between 5.4Up and7.48Up, there are four short and two long paths. Moreover, |E(tBil)|is much weaker than |E(tBis)|(i=1,2.tBil and tBis represent the born times of the long path and the short path of Bi in the selected RKE range, the same below respectively). Therefore, the birth rates of the electrons through long paths are much lower than those through short paths. So the harmonics come mainly from the short paths and an IAP can be obtained. For the same reason, an isolated 77as pulse can be generated fort0=0.25T, as shown in Fig. 3(c). However, as can be seen from Fig. 2(d), |E(tDil)| enhances significantly compared with|E(tBil)|(i=1,2). Thus, the long paths may have obvious effects on the HHG spectrum. So in Fig. 3(d), an apparent tail appears in the generated attosecond pulse. From the analysis above, considering the intensity and the duration of the generated IAP, we can see that the optimal jump times are at t0=0.25Tandt0=0.25T, i.e., the nearest two zero points to the pulse center.

To verify the classical analysis above, we perform the time-frequency analysis of the HHG spectra driven by the phase jump laser pulses with different jump times using the Morlet wavelet [27

27. X. M. Tong and S.-I. Chu, “Probing the spectral and temporal structures of high-order harmonic generation in intense laser pulses,” Phys. Rev. A 61(2), 021802 (2000). [CrossRef]

], as shown in Fig. 4
Fig. 4 The wavelet time-frequency profiles of the HHG spectra driven by the laser pulses with different phase jump times. (a)t0=0.75T; (b)t0=0.25T; (c) t0=0.25T; (d) t0=0.75T.
. In this figure, the peaks A’, B’, C’ and D’ are corresponding to the peaks A, B, C and D in Fig. 2, respectively. From this figure, we can clearly see that the intensity increases from peak A’ to D’. For peaks A’, B’, and C’, the intensity of the long path is much weaker than those of the short path, so the continuum spectra are highly coherent. While for peak D’, the intensity of the long path increases apparently compared with the other three peaks and are comparable with the short path, which leads to the phase mismatch of the continuum spectrum. These mean that the classical analysis agrees very well with the quantum theory.

For a further investigation, we consider the IAP generation for different CEPs. Figure 5
Fig. 5 The generated IAP from the HHG spectrum driven by phase jump laser pulses with different ϕi. The harmonic orders used are from 120 to 160.
shows the IAPs driven by phase jump laser pulses with different ϕi. The jump time t0=(0.5πϕi)/ω is the nearest zero point to the pulse center. As can be seen from Fig. 5, the duration of the IAP varies in a small range (65as~77as), when the CEP ϕi varies from π to 0. This means that for the pulse with any CEP, an isolated sub-100 as pulse can be obtained by controlling the π-phase jump at the nearest zero point to the pulse center, which confirms the conclusion we have drawn before.

4. Conclusions

In conclusion, we investigated the HHG spectrum driven by the laser pulses with a phase jump in details. Due to the enlargement of the electric field caused by the π-phase jump, the cutoff of the HHG spectrum extends dramatically. The intensity and the coherence of the continuum spectrum can be controlled by adjusting the jump time. If the jump time is at the zero point which is the nearest to the pulse center, a continuum spectrum with highly coherence and a high intensity can be obtained, from which a regular IAP with duration less than 100as can be generated without any phase compensation.

Acknowledgments

This work was supported by the National Nature Science Foundation of China (Grant Nos. 11074263, 60978013, and 60921004), the high-performance grid computing platform of and the Doctor Fund of Henan Polytechnic University (Grant No. B2011-076).

References and links

1.

P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67(6), 813–855 (2004). [CrossRef]

2.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009). [CrossRef]

3.

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

4.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]

5.

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39(14), R203–R262 (2006). [CrossRef]

6.

P. B. Corkum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett. 19(22), 1870–1872 (1994). [CrossRef] [PubMed]

7.

Q. Zhang, P. Lu, P. Lan, W. Hong, and Z. Yang, “Multi-cycle laser-driven broadband supercontinuum with a modulated polarization gating,” Opt. Express 16(13), 9795–9803 (2008). [CrossRef] [PubMed]

8.

Y. Yu, X. Song, Y. Fu, R. Li, Y. Cheng, and Z. Xu, “Theoretical investigation of single attosecond pulse generation in an orthogonally polarized two-color laser field,” Opt. Express 16(2), 686–694 (2008). [CrossRef] [PubMed]

9.

Y. Xiang, Y. Niu, and S. Gong, “Proposal for isolated-attosecond-pulse generation in the multicycle regime through modulation of the carrier wave,” Phys. Rev. A 85(2), 023808 (2012). [CrossRef]

10.

P. Lan, E. J. Takahashi, and K. Midorikawa, “Optimization of infrared two-color multicycle field synthesis for intense-isolated-attosecond-pulse generation,” Phys. Rev. A 82(5), 053413 (2010). [CrossRef]

11.

B. Kim, J. An, Y. Yu, Y. Cheng, Z. Xu, and D. E. Kim, “Optimization of multi-cycle two-color laser fields for the generation of an isolated attosecond pulse,” Opt. Express 16(14), 10331–10340 (2008). [CrossRef] [PubMed]

12.

H.-C. Bandulet, D. Comtois, E. Bisson, A. Fleischer, H. Pepin, J.-C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Gating attosecond pulse train generation using multicolor laser fields,” Phys. Rev. A 81(1), 013803 (2010). [CrossRef]

13.

J. J. Carrera and S.-I. Chu, “Extension of high-order harmonic generation cutoff via coherent control of intense few-cycle chirped laser pulses,” Phys. Rev. A 75(3), 033807 (2007). [CrossRef]

14.

S. Odžak and D. B. Milosevic, “Attosecond pulse generation by a coplanar circular and static field combination,” Phys. Lett. A 355(4-5), 368–372 (2006). [CrossRef]

15.

W. Hong, P. Lu, W. Cao, P. Lan, and X. Wang, “Control of quantum paths of high-order harmonics and attosecond pulse generation in the presence of a static electric field,” J. Phys. B 40(12), 2321–2331 (2007). [CrossRef]

16.

B. T. Torosov and N. V. Vitanov, “Coherent control of a quantum transition by a phase jump,” Phys. Rev. A 76(5), 053404 (2007). [CrossRef]

17.

J. Qian, Y. Qian, M. Ke, X. Feng, C. H. Oh, and Y. Wang, “Breakdown of the dipole blockade with a zero-area phase-jump pulse,” Phys. Rev. A 80(5), 053413 (2009). [CrossRef]

18.

P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A 82(4), 045805 (2010). [CrossRef]

19.

Y. Xiang, Y. Niu, and S. Gong, “Control of the high-order harmonics cutoff through the combination of a chirped laser and static electric field,” Phys. Rev. A 79(5), 053419 (2009). [CrossRef]

20.

H. Xu, X. Tang, and P. Lambropoulos, “Nonperturbative theory of harmonic generation in helium under a high-intensity laser field: The role of intermediate resonance and of the ion,” Phys. Rev. A 46(5), R2225–R2228 (1992). [CrossRef] [PubMed]

21.

M. Gavrila, Atoms in Intense Laser Fields (Academic, New York, 1992).

22.

D. R. Hartree, The Calculation of Atomic Structures (Wiley, New York, 1957).

23.

K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, “Calculation of the background emitted during high-harmonic generation,” Phys. Rev. A 45(5), 3347–3349 (1992). [CrossRef] [PubMed]

24.

M. Protopapas, D. G. Lappas, C. H. Keitel, and P. L. Knight, “Recollisions, bremsstrahlung, and attosecond pulses from intense laser fields,” Phys. Rev. A 53(5), R2933–R2936 (1996). [CrossRef] [PubMed]

25.

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

26.

P. Antoine, A. L’Huillier, and M. Lewenstein, “Attosecond pulse trains using high–order harmonics,” Phys. Rev. Lett. 77(7), 1234–1237 (1996). [CrossRef] [PubMed]

27.

X. M. Tong and S.-I. Chu, “Probing the spectral and temporal structures of high-order harmonic generation in intense laser pulses,” Phys. Rev. A 61(2), 021802 (2000). [CrossRef]

OCIS Codes
(190.4160) Nonlinear optics : Multiharmonic generation
(190.7110) Nonlinear optics : Ultrafast nonlinear optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 18, 2012
Revised Manuscript: July 25, 2012
Manuscript Accepted: July 26, 2012
Published: August 8, 2012

Citation
Yang Xiang, Yueping Niu, Hongmei Feng, Yihong Qi, and Shangqing Gong, "Coherent control of high-order harmonic generation by phase jump pulses," Opt. Express 20, 19289-19296 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-17-19289


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References

  1. P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys.67(6), 813–855 (2004). [CrossRef]
  2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys.81(1), 163–234 (2009). [CrossRef]
  3. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett.71(13), 1994–1997 (1993). [CrossRef] [PubMed]
  4. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys.72(2), 545–591 (2000). [CrossRef]
  5. D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B39(14), R203–R262 (2006). [CrossRef]
  6. P. B. Corkum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett.19(22), 1870–1872 (1994). [CrossRef] [PubMed]
  7. Q. Zhang, P. Lu, P. Lan, W. Hong, and Z. Yang, “Multi-cycle laser-driven broadband supercontinuum with a modulated polarization gating,” Opt. Express16(13), 9795–9803 (2008). [CrossRef] [PubMed]
  8. Y. Yu, X. Song, Y. Fu, R. Li, Y. Cheng, and Z. Xu, “Theoretical investigation of single attosecond pulse generation in an orthogonally polarized two-color laser field,” Opt. Express16(2), 686–694 (2008). [CrossRef] [PubMed]
  9. Y. Xiang, Y. Niu, and S. Gong, “Proposal for isolated-attosecond-pulse generation in the multicycle regime through modulation of the carrier wave,” Phys. Rev. A85(2), 023808 (2012). [CrossRef]
  10. P. Lan, E. J. Takahashi, and K. Midorikawa, “Optimization of infrared two-color multicycle field synthesis for intense-isolated-attosecond-pulse generation,” Phys. Rev. A82(5), 053413 (2010). [CrossRef]
  11. B. Kim, J. An, Y. Yu, Y. Cheng, Z. Xu, and D. E. Kim, “Optimization of multi-cycle two-color laser fields for the generation of an isolated attosecond pulse,” Opt. Express16(14), 10331–10340 (2008). [CrossRef] [PubMed]
  12. H.-C. Bandulet, D. Comtois, E. Bisson, A. Fleischer, H. Pepin, J.-C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Gating attosecond pulse train generation using multicolor laser fields,” Phys. Rev. A81(1), 013803 (2010). [CrossRef]
  13. J. J. Carrera and S.-I. Chu, “Extension of high-order harmonic generation cutoff via coherent control of intense few-cycle chirped laser pulses,” Phys. Rev. A75(3), 033807 (2007). [CrossRef]
  14. S. Odžak and D. B. Milosevic, “Attosecond pulse generation by a coplanar circular and static field combination,” Phys. Lett. A355(4-5), 368–372 (2006). [CrossRef]
  15. W. Hong, P. Lu, W. Cao, P. Lan, and X. Wang, “Control of quantum paths of high-order harmonics and attosecond pulse generation in the presence of a static electric field,” J. Phys. B40(12), 2321–2331 (2007). [CrossRef]
  16. B. T. Torosov and N. V. Vitanov, “Coherent control of a quantum transition by a phase jump,” Phys. Rev. A76(5), 053404 (2007). [CrossRef]
  17. J. Qian, Y. Qian, M. Ke, X. Feng, C. H. Oh, and Y. Wang, “Breakdown of the dipole blockade with a zero-area phase-jump pulse,” Phys. Rev. A80(5), 053413 (2009). [CrossRef]
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