Abstract

The detailed theoretical investigation into the spectral phase and intrinsic dispersion of high-order harmonics generated by monocycle pumping [a 1.93fs (T0), 580nm (λ0), 1.72.0×1015W/cm2 (I0), Gaussian-shaped pulse] showed that it is significantly important for further shortening of isolated attosecond pulses to select the spectral region that has the minimum intrinsic group-delay dispersion and higher-order dispersion in the middle region of the plateau, other than that close to the cutoff where the harmonic intensity drops rapidly. We demonstrated that the selection is carried out directly from the harmonic-order-dependent electron-recombination time, which is obtained from the simple semiclassical method and can be measured by experiment. Consequently, we found that monocycle pumping can generate efficiently 68as isolated pulses without chirp compensation as well as tunable (115 to 150eV), sub 70as pulses from He atoms by tuning the carrier-envelope phase. Particularly, nearly transform-limited 43as isolated pulses can be produced under the double-intensity (Ipump=2I0) and double- wavelength (λpump=2λ0, Tpump=2T0) pump. These results were also confirmed by the saddle-point analysis based on the Lewenstein model.

© 2011 Optical Society of America

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    [CrossRef]

2009 (5)

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

I. A. Ivanov and A. S. Kheifets, “Tailoring the waveforms to extend the high-order harmonic generation cutoff,” Phys. Rev. A 80, 023809 (2009).
[CrossRef]

Q. Li, P. Lu, W. Hong, Q. Zhang, and Z. Yang, “Propagation effects for attosecond ionization control of efficient broadband supercontinuum generation,” Phys. Rev. A 80, 043417 (2009).
[CrossRef]

L. E. Chipperfield, J. S. Robinson, J. W. G. Tisch, and J. P. Marangos, “Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period,” Phys. Rev. Lett. 102, 063003 (2009).
[CrossRef] [PubMed]

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102, 093002(2009).
[CrossRef] [PubMed]

2008 (6)

E. Matsubara, T. Sekikawa, and M. Yamashita, “Generation of ultrashort optical pulses using multiple coherent anti-Stokes Raman scattering in a crystal at room temperature,” Appl. Phys. Lett. 92, 071104 (2008).
[CrossRef]

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

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

A. Jullien, T. Pfeifer, M. J. Abel, P. M. Nagel, M. J. Bell, D. M. Neumark, and S. R. Leone, “Ionization phase-match gating for wavelength-tunable isolated attosecond pulse generation,” Appl. Phys. B 93, 433-442 (2008).
[CrossRef]

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4, 386-389(2008).
[CrossRef]

K. Yamane, T. Tanigawa, T. Sekikawa, and M. Yamashita, “Angularly-dispersed optical parametric amplification of optical pulses with one-octave bandwidth toward monocycle regime,” Opt. Express 16, 18345-18353 (2008).
[CrossRef] [PubMed]

2007 (4)

Z. N. Zeng, Y. Cheng, X. H. Song, R. X. Li, and Z. Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. 98, 203901 (2007).
[CrossRef] [PubMed]

E. Matsubara, K. Yamane, T. Sekikawa, and M. Yamashita, “Generation of 2.6fs optical pulses using induced-phase modulation in a gas-filled hollow fiber,” J. Opt. Soc. Am. B 24, 985-989(2007).
[CrossRef]

V. S. Yakovlev, M. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft x-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15, 15351-15364 (2007).
[CrossRef] [PubMed]

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, 2321-2331 (2007).
[CrossRef]

2006 (4)

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]

L. E. Chipperfield, P. L. Knight, J. W. G. Tisch, and J. P. Marangos, “Tracking individual electron trajectories in a high harmonic spectrum,” Opt. Commun. 264, 494-501(2006).
[CrossRef]

M. Yamashita, K. Yamane, and R. Morita, “Quasi-automatic phase-control technique for chirp compensation of pulses with over-one-octave bandwidth-generation of few- to mono-cycle optical pulses,” IEEE J. Sel. Top. Quantum Electron. 12, 213-222 (2006).
[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]

2005 (1)

L. E. Chipperfield, L. N. Gaier, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Conditions for the reliable production of attosecond pulses using ultra-short laser-generated high harmonics,” J. Mod. Opt. 52, 243-260 (2005).
[CrossRef]

2004 (5)

S. Kazamias and P. Balcou, “Intrinsic chirp of attosecond pulses: single-atom model versus experiment,” Phys. Rev. A 69, 063416 (2004).
[CrossRef]

T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature 432, 605-608 (2004).
[CrossRef] [PubMed]

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

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

G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli, “Nonadiabatic quantum path analysis of high-order harmonic generation: role of the carrier-envelope phase on short and long paths,” Phys. Rev. A 70, 013411 (2004).
[CrossRef]

2003 (2)

O. Tcherbakoff, E. Mével, D. Descamps, J. Plumridge, and E. Constant, “Time-gated high-order harmonic generation,” Phys. Rev. A 68, 043804 (2003).
[CrossRef]

A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

2001 (4)

D. Lee, H. Shin, Y. Cha, K. Hong, J. Kim, and C. Nam, “Selection of high-order harmonics from a single quantum path for the generation of an attosecond pulse train,” Phys. Rev. A 63, 021801 (2001).
[CrossRef]

M. Hentschel, R. Kienberger, C. Spielman, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heizmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509-513(2001).
[CrossRef] [PubMed]

G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: looking inside a laser cycle,” Phys. Rev. A 64, 013409(2001).
[CrossRef]

P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosevic, A. Sanpera, and M. Lewenstein, “Feynman's path-integral approach for intense-laser-atom interactions,” Science 292, 902-905(2001).
[CrossRef] [PubMed]

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 (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]

1993 (2)

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

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30,000eV, Z=1-92,” At. Data Nucl. Data Tables 54, 181-342 (1993).
[CrossRef]

1991 (1)

H. Kinoshita, K. Kurihara, and H. Takenaka, “Soft x-ray reduction lithography using multilayer mirrors,” Jpn. J. Appl. Phys. 30, 3048-3052 (1991).
[CrossRef]

1986 (2)

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

M. Yamashita, M. Ishikawa, K. Torizuka, and T. Sato, “Femtosecond-pulse laser chirp compensated by cavity-mirror dispersion,” Opt. Lett. 11, 504-506 (1986).
[CrossRef] [PubMed]

Abel, M. J.

A. Jullien, T. Pfeifer, M. J. Abel, P. M. Nagel, M. J. Bell, D. M. Neumark, and S. R. Leone, “Ionization phase-match gating for wavelength-tunable isolated attosecond pulse generation,” Appl. Phys. B 93, 433-442 (2008).
[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]

Agostini, P.

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102, 093002(2009).
[CrossRef] [PubMed]

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4, 386-389(2008).
[CrossRef]

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

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]

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and 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]

Aquila, A. L.

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

Attwood, D. T.

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

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.

S. Kazamias and P. Balcou, “Intrinsic chirp of attosecond pulses: single-atom model versus experiment,” Phys. Rev. A 69, 063416 (2004).
[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]

Baltuska, A.

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

A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

Bammer, F.

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

Becker, W.

P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosevic, A. Sanpera, and M. Lewenstein, “Feynman's path-integral approach for intense-laser-atom interactions,” Science 292, 902-905(2001).
[CrossRef] [PubMed]

Bell, M. J.

A. Jullien, T. Pfeifer, M. J. Abel, P. M. Nagel, M. J. Bell, D. M. Neumark, and S. R. Leone, “Ionization phase-match gating for wavelength-tunable isolated attosecond pulse generation,” Appl. Phys. B 93, 433-442 (2008).
[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]

Blaga, C. I.

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D. Lee, H. Shin, Y. Cha, K. Hong, J. Kim, and C. Nam, “Selection of high-order harmonics from a single quantum path for the generation of an attosecond pulse train,” Phys. Rev. A 63, 021801 (2001).
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H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
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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).
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M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191-1194(1986).

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G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102, 093002(2009).
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H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
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A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611-615 (2003).
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P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4, 386-389(2008).
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R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817-821 (2004).
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M. Hentschel, R. Kienberger, C. Spielman, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heizmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509-513(2001).
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M. Hentschel, R. Kienberger, C. Spielman, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heizmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509-513(2001).
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E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614-1617 (2008).
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A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611-615 (2003).
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D. Lee, H. Shin, Y. Cha, K. Hong, J. Kim, and C. Nam, “Selection of high-order harmonics from a single quantum path for the generation of an attosecond pulse train,” Phys. Rev. A 63, 021801 (2001).
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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, 2321-2331 (2007).
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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).
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A. Jullien, T. Pfeifer, M. J. Abel, P. M. Nagel, M. J. Bell, D. M. Neumark, and S. R. Leone, “Ionization phase-match gating for wavelength-tunable isolated attosecond pulse generation,” Appl. Phys. B 93, 433-442 (2008).
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T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature 432, 605-608 (2004).
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M. Yamashita, R. Morita, and N. Karasawa, “Active chirp compensation for ultra broadband optical pulses,” in Mono-Cycle Photonics and Optical Scanning Tunneling Microscopy, M.Yamashita, H.Shigekawa, and R.Morita, eds. (Springer, 2005), pp. 103-151.
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H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
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I. A. Ivanov and A. S. Kheifets, “Tailoring the waveforms to extend the high-order harmonic generation cutoff,” Phys. Rev. A 80, 023809 (2009).
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Kienberger, R.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614-1617 (2008).
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R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817-821 (2004).
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M. Hentschel, R. Kienberger, C. Spielman, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heizmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509-513(2001).
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D. Lee, H. Shin, Y. Cha, K. Hong, J. Kim, and C. Nam, “Selection of high-order harmonics from a single quantum path for the generation of an attosecond pulse train,” Phys. Rev. A 63, 021801 (2001).
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E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614-1617 (2008).
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R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817-821 (2004).
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L. E. Chipperfield, P. L. Knight, J. W. G. Tisch, and J. P. Marangos, “Tracking individual electron trajectories in a high harmonic spectrum,” Opt. Commun. 264, 494-501(2006).
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L. E. Chipperfield, L. N. Gaier, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Conditions for the reliable production of attosecond pulses using ultra-short laser-generated high harmonics,” J. Mod. Opt. 52, 243-260 (2005).
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P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosevic, A. Sanpera, and M. Lewenstein, “Feynman's path-integral approach for intense-laser-atom interactions,” Science 292, 902-905(2001).
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T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature 432, 605-608 (2004).
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M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191-1194(1986).

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E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614-1617 (2008).
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A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

M. Hentschel, R. Kienberger, C. Spielman, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heizmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509-513(2001).
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K. C. Kulander, K. J. Schafer, and J. L. Krause, “Time-dependent studies of multiphoton processes,” in Atoms in Intense Laser Fields, M.Gavrila, ed. (Academic, 1992), pp. 247-300.

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H. Kinoshita, K. Kurihara, and H. Takenaka, “Soft x-ray reduction lithography using multilayer mirrors,” Jpn. J. Appl. Phys. 30, 3048-3052 (1991).
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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, 2321-2331 (2007).
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P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosevic, A. Sanpera, and M. Lewenstein, “Feynman's path-integral approach for intense-laser-atom interactions,” Science 292, 902-905(2001).
[CrossRef] [PubMed]

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D. Lee, H. Shin, Y. Cha, K. Hong, J. Kim, and C. Nam, “Selection of high-order harmonics from a single quantum path for the generation of an attosecond pulse train,” Phys. Rev. A 63, 021801 (2001).
[CrossRef]

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A. Jullien, T. Pfeifer, M. J. Abel, P. M. Nagel, M. J. Bell, D. M. Neumark, and S. R. Leone, “Ionization phase-match gating for wavelength-tunable isolated attosecond pulse generation,” Appl. Phys. B 93, 433-442 (2008).
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Q. Li, P. Lu, W. Hong, Q. Zhang, and Z. Yang, “Propagation effects for attosecond ionization control of efficient broadband supercontinuum generation,” Phys. Rev. A 80, 043417 (2009).
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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, 2321-2331 (2007).
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L. E. Chipperfield, J. S. Robinson, J. W. G. Tisch, and J. P. Marangos, “Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period,” Phys. Rev. Lett. 102, 063003 (2009).
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L. E. Chipperfield, L. N. Gaier, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Conditions for the reliable production of attosecond pulses using ultra-short laser-generated high harmonics,” J. Mod. Opt. 52, 243-260 (2005).
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E. Matsubara, T. Sekikawa, and M. Yamashita, “Generation of ultrashort optical pulses using multiple coherent anti-Stokes Raman scattering in a crystal at room temperature,” Appl. Phys. Lett. 92, 071104 (2008).
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H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
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M. Yamashita, R. Morita, and N. Karasawa, “Active chirp compensation for ultra broadband optical pulses,” in Mono-Cycle Photonics and Optical Scanning Tunneling Microscopy, M.Yamashita, H.Shigekawa, and R.Morita, eds. (Springer, 2005), pp. 103-151.
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P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4, 386-389(2008).
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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).
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G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli, “Nonadiabatic quantum path analysis of high-order harmonic generation: role of the carrier-envelope phase on short and long paths,” Phys. Rev. A 70, 013411 (2004).
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P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4, 386-389(2008).
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P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosevic, A. Sanpera, and M. Lewenstein, “Feynman's path-integral approach for intense-laser-atom interactions,” Science 292, 902-905(2001).
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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).
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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).
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P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosevic, A. Sanpera, and M. Lewenstein, “Feynman's path-integral approach for intense-laser-atom interactions,” Science 292, 902-905(2001).
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G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli, “Nonadiabatic quantum path analysis of high-order harmonic generation: role of the carrier-envelope phase on short and long paths,” Phys. Rev. A 70, 013411 (2004).
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E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614-1617 (2008).
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R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817-821 (2004).
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E. Matsubara, T. Sekikawa, and M. Yamashita, “Generation of ultrashort optical pulses using multiple coherent anti-Stokes Raman scattering in a crystal at room temperature,” Appl. Phys. Lett. 92, 071104 (2008).
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K. Yamane, T. Tanigawa, T. Sekikawa, and M. Yamashita, “Angularly-dispersed optical parametric amplification of optical pulses with one-octave bandwidth toward monocycle regime,” Opt. Express 16, 18345-18353 (2008).
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T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature 432, 605-608 (2004).
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H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
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D. Lee, H. Shin, Y. Cha, K. Hong, J. Kim, and C. Nam, “Selection of high-order harmonics from a single quantum path for the generation of an attosecond pulse train,” Phys. Rev. A 63, 021801 (2001).
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Z. N. Zeng, Y. Cheng, X. H. Song, R. X. Li, and Z. Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. 98, 203901 (2007).
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M. Hentschel, R. Kienberger, C. Spielman, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heizmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509-513(2001).
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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]

G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli, “Nonadiabatic quantum path analysis of high-order harmonic generation: role of the carrier-envelope phase on short and long paths,” Phys. Rev. A 70, 013411 (2004).
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L. E. Chipperfield, J. S. Robinson, J. W. G. Tisch, and J. P. Marangos, “Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period,” Phys. Rev. Lett. 102, 063003 (2009).
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L. E. Chipperfield, L. N. Gaier, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Conditions for the reliable production of attosecond pulses using ultra-short laser-generated high harmonics,” J. Mod. Opt. 52, 243-260 (2005).
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Udem, T.

A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611-615 (2003).
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E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614-1617 (2008).
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R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817-821 (2004).
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A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611-615 (2003).
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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]

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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]

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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).
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G. Sansone, C. Vozzi, S. Stagira, and M. Nisoli, “Nonadiabatic quantum path analysis of high-order harmonic generation: role of the carrier-envelope phase on short and long paths,” Phys. Rev. A 70, 013411 (2004).
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P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milosevic, A. Sanpera, and M. Lewenstein, “Feynman's path-integral approach for intense-laser-atom interactions,” Science 292, 902-905(2001).
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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, 2321-2331 (2007).
[CrossRef]

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T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, “Nonlinear optics in the extreme ultraviolet,” Nature 432, 605-608 (2004).
[CrossRef] [PubMed]

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R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817-821 (2004).
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G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102, 093002(2009).
[CrossRef] [PubMed]

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Z. N. Zeng, Y. Cheng, X. H. Song, R. X. Li, and Z. Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. 98, 203901 (2007).
[CrossRef] [PubMed]

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R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817-821 (2004).
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E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614-1617 (2008).
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V. S. Yakovlev, M. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft x-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15, 15351-15364 (2007).
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A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

Yamane, K.

Yamashita, M.

K. Yamane, T. Tanigawa, T. Sekikawa, and M. Yamashita, “Angularly-dispersed optical parametric amplification of optical pulses with one-octave bandwidth toward monocycle regime,” Opt. Express 16, 18345-18353 (2008).
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E. Matsubara, T. Sekikawa, and M. Yamashita, “Generation of ultrashort optical pulses using multiple coherent anti-Stokes Raman scattering in a crystal at room temperature,” Appl. Phys. Lett. 92, 071104 (2008).
[CrossRef]

E. Matsubara, K. Yamane, T. Sekikawa, and M. Yamashita, “Generation of 2.6fs optical pulses using induced-phase modulation in a gas-filled hollow fiber,” J. Opt. Soc. Am. B 24, 985-989(2007).
[CrossRef]

M. Yamashita, K. Yamane, and R. Morita, “Quasi-automatic phase-control technique for chirp compensation of pulses with over-one-octave bandwidth-generation of few- to mono-cycle optical pulses,” IEEE J. Sel. Top. Quantum Electron. 12, 213-222 (2006).
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Q. Li, P. Lu, W. Hong, Q. Zhang, and Z. Yang, “Propagation effects for attosecond ionization control of efficient broadband supercontinuum generation,” Phys. Rev. A 80, 043417 (2009).
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Figures (10)

Fig. 1
Fig. 1

CEP dependence of the spectral intensity (thick black curves) and the spectral phase (thin blue curves and thick red curves for enlarged ones) by monocycle pumping. Here thin dashed red arrows denote the eye-guided shifts employed for the enlargement; the labels (I and II) for the upper branch I and II, respectively; green dotted curves denote the N b I , blue dotted curves denote the N peak I , and orange dotted curves denote the N peak I I .

Fig. 2
Fig. 2

(a) Electron kinetic energy as functions of emission (thick black dotted curves) and recombination (thick red dotted curves) times with the electrical field (thin blue curve) by monocycle-sine pumping; (b) spectral intensity of each short (dashed curves) and long (solid curves) trajectories of three main pairs of quantum orbits (B, C and D) by monocycle-sine pumping.

Fig. 3
Fig. 3

Spectral intensity and phase of short [ I S C ( ω ) and ϕ S C ( ω ) ] and long [ I L C ( ω ) and ϕ L C ( ω ) ] trajectories of the quantum orbit C from SPA by monocycle-sine pumping. The spectral phase ϕ S C ( ω ) (thick red solid curve) and the intensity spectrum I S C ( ω ) (thick black solid curve) from the short trajectory of the quantum orbit C as well as the corresponding spectral phase ϕ L C ( ω ) (thick red dotted curve) and intensity spectrum I L C ( ω ) (thick black dotted curve) from the long trajectory; the calculated spectral phase ϕ S , L ( ω ) [thin blue solid curve; from Eq. (3)] and intensity spectrum I S , L ( ω ) | E S , L ( ω ) | 2 [thin green solid curve from Eq. (2)] show an excellent agreement with the steplike spectral phase of the high-order harmonics ϕ ( ω ) (thin red solid curve) and the modulated intensity spectrum I ( ω ) (thin black solid curve).

Fig. 4
Fig. 4

Temporal intensity profile [thick black curve, I S C ( t ) and I L C ( t ) components] obtained by the IFT at the same spectral bandwidth (from upper-branch thirty-second to eighty-eighth orders) with the electrical field curve E ( t ) (blue curve) by monocycle- sine pumping.

Fig. 5
Fig. 5

The intensity spectra I S C ( ω ) in the upper branch I from CEP = 0.0 π to 0.7 π and those I S B ( ω ) in the upper branch II from CEP = 0.8 π to 0.9 π (for easy comparison, we increased the intensity to four and two times at CEP = 0.8 π and 0.9 π , respectively), which were calculated from the SPA.

Fig. 6
Fig. 6

(a) Harmonic-order-dependent recombination time t ( N ) from the S–C or S–B trajectory from CEP = 0.0 π to 0.9 π , which was calculated from the simple semiclassical method; (b) the corresponding N-dependent positive GDD ϕ S i ( N ) . For both (a) and (b), dotted lines, thin solid lines, and thick solid lines show the total trajectories, the upper branch and the optimum region, respectively, and the open square (□), open circle (○), and solid circle (●) show the ends of upper branch, the ends of optimum region, and the center of optimum region, respectively.

Fig. 7
Fig. 7

(a) Pulse profile and duration [inset Fig. 7a’] on as a function of the bandwidth Δ N at a constant central harmonic order N C = 54.5 for CEP = 0.5 π ; (b) pulse profile and duration [inset Fig. 7b’] on as a function of the central harmonic order N C at a constant bandwidth Δ N = 29 for CEP = 0.5 π .

Fig. 8
Fig. 8

(a) Temporal profile of the shortest IAP for different CEPs (thin black solid curves are from SPA and thick red solid curves are from t ( N ) ; they are exactly overlapped, so are not easily distinguished and then only the red solid curves are shown); (b) CEP dependence of the pulse duration (black curve is from SPA and red curve is from t ( N ) ); (c) enlarged temporal profile of the shortest IAP for CEP = 0.9 π [here we can see the difference between the SPA and t ( N ) from the inset figure].

Fig. 9
Fig. 9

(a) Shortest pulse profile after the optimum selection in three different cases; (b) the corresponding GDD ϕ ( N ) curve. Those were obtained by N H N L = 90 63 ( N H and N L are the highest and lowest harmonic-order edges, respectively) and CEP = 0.3 π , where the spectrum contains the cutoff region (thin green curves); by N H N L = 92 50 and CEP = 0.1 π , where the spectrum contains the entire upper branch (thick blue curves); and by N H N L = 71 41 and CEP = 0.3 π , where the spectrum contains the plateau region only (thick red curves), respectively. The open circle (○) and solid circle (●) show the ends of optimum region and the center of optimum region, respectively.

Fig. 10
Fig. 10

(a) Normalized pulse profile I S C opt ( t ) after the optimum selection; (b) N-dependent GDD ϕ S C ( N ) curves at CEP = 0.3 π : black curve for the intensity ( I 0 ) and wavelength ( λ 0 ) pump case, red curve for the double-intensity ( 2 I 0 ) and wavelength ( λ 0 ) pump case, and blue curve for the double-intensity ( 2 I 0 ), double-wavelength ( 2 λ 0 ) pump case; ( N eff = N / 2 ). The open square (□) and open circle (○) show the ends of upper branch and the ends of optimum region, respectively.

Tables (1)

Tables Icon

Table 1 Shortest Pulse Duration t p and the Corresponding GDD, TOD, and Spectral Range ( N H N L ) Obtained by Two Methods [ t ( N ) and SPA] for Each CEP

Equations (5)

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x ( t ) = i 0 t d t d 3 p d x * ( p A ( t ) ) × E ( t ) d x ( p A ( t ) ) exp [ i t t d t ( [ p A ( t ) ] 2 2 + I p ) ] + c . c .
E ( t ) = E 0 exp ( ( 2 ln 2 ) t 2 T 1 2 ) cos [ ω 0 t + φ 0 ] ,
x s ( ω ) = 2 π i [ det S ( t , t ) ] 1 / 2 | t s ( ω ) , t s ( ω ) [ π / ( ε + i 2 ( t s ( ω ) t s ( ω ) ) ) ] 3 / 2 × d x * [ p s ( ω ) A ( t s ( ω ) ) ] × E ( t s ( ω ) ) × d x [ p s ( ω ) A ( t s ( ω ) ) ] × exp [ i ( S ( t s ( ω ) , t s ( ω ) ) + ω t s ( ω ) ) ] .
E S , L ( ω ) = I S C ( ω ) 1 / 2 exp ( i ϕ S C ( ω ) ) + I L C ( ω ) 1 / 2 exp ( i ϕ L C ( ω ) ) = [ I S C ( ω ) + I L C ( ω ) + 2 ( I S C ( ω ) × I L C ( ω ) ) 1 / 2 × cos ϕ 0 ( ω ) ] 1 / 2 × exp ( i ϕ S , L ( ω ) ) ,
ϕ S , L ( ω ) = tan 1 { [ I S C ( ω ) 1 / 2 sin ϕ S C ( ω ) + I L C ( ω ) 1 / 2 sin ϕ L C ( ω ) ] [ I S C ( ω ) 1 / 2 cos ϕ S C ( ω ) + I L C ( ω ) 1 / 2 cos ϕ L C ( ω ) ] } ,

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