Abstract

We perform a quantum-orbit analysis for the dependence of high-order-harmonic yield on the driving field ellipticity and the polarization properties of the generated high harmonics. The electron trajectories responsible for the emission of particular harmonics are identified. It is found that, in elliptically polarized driving field, the electrons have ellipticity-dependent initial velocities, which lead to the decrease of the ionization rate. Thus the harmonic yield steeply decreases with laser ellipticity. Besides, we show that the polarization properties of the harmonics are related to the complex momenta of the electron. The physical origin of the harmonic ellipticity is interpreted as the consequence of quantum-mechanical uncertainty of the electron momentum. Our results are verified with the experimental results as well as the numerical solutions of the time dependent Schrödinger equation from the literature.

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  3. 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,” Science320, 1614–1617 (2008).
    [CrossRef] [PubMed]
  4. R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London)427, 817–821 (2004).
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  5. M. I. Stockman, M. F. Kling, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photon.1, 539–544 (2007).
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  6. Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovac̆ev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft x-rays,” Science302, 1540–1543 (2003).
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  19. L. Arissian, C. Smeenk, F. Turner, C. Trallero, A.V. Sokolov, D. M. Villeneuve, A. Staudte, and P. B. Corkum, “Direct test of laser tunneling with electron momentum imaging,” Phys. Rev. Lett.105, 133002 (2010).
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  21. D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan, D. Descamps, E. Mvel, S. Petit, H. J. Wörner, B. Pons, N. Dudovich, and Y. Mairesse, “Role of the ionic potential in high harmonic generation,” Phys. Rev. Lett.108, 203001 (2012).
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    [CrossRef] [PubMed]
  24. W. Becker, A. Lohr, and M. Kleber, “A unified theory of high-harmonic generation: Application to polarization properties of the harmonics,” Phys. Rev. A56, 645–656 (1997).
    [CrossRef]
  25. K. Yuan and A. D. Bandrauk, “Circularly polarized attosecond pulses from molecular high-order harmonic generation by ultrashort intense bichromatic circularly and linearly polarized laser pulses,” J. Phys. B: At. Mol. Opt. Phys.45, 074001 (2012).
    [CrossRef]
  26. Max Möller, Y. Cheng, S. D. Khan, B. Zhao, K. Zhao, M. Chini, G. G. Paulus, and Z. Chang, “Dependence of high-order-harmonic-generation yield on driving-laser ellipticity,” Phys. Rev. A86, 011401 (2012).
    [CrossRef]
  27. M. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A54, 742–745 (1996).
    [CrossRef] [PubMed]
  28. N. Dudovich, J. Levesque, O. Smirnova, D. Zeidler, D. Comtois, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Attosecond temporal gating with elliptically polarized light,” Phys. Rev. Lett.97, 253903 (2006).
    [CrossRef]
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    [CrossRef]
  30. V. V. Strelkov, A. A. Gonoskov, I. A. Gonoskov, and M. Yu. Ryabikin, “Origin for ellipticity of high-order harmonics generated in atomic gases and the sublaser-cycle evolution of harmonic polarization,” Phys. Rev. Lett.107, 043902 (2011).
    [CrossRef] [PubMed]
  31. V. V. Strelkov, M. A. Khokhlova, A. A. Gonoskov, I. A. Gonoskov, and M. Yu. Ryabikin, “High-order harmonic generation by atoms in an elliptically polarized laser field: Harmonic polarization properties and laser threshold ellipticity,” Phys. Rev. A86, 013404 (2012).
    [CrossRef]
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    [CrossRef]
  33. P. Balcou, P Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: The role of field-gradient forces,” Phys. Rev. A55, 3204–3210 (1997).
    [CrossRef]
  34. 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]
  35. M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys.41, 132001 (2008).
    [CrossRef]
  36. D. B. Milos̆ević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A66, 063417 (2002).
    [CrossRef]
  37. R. Kopold, D. B. Milos̆ević, and W. Becker, “Rescattering processes for elliptical polarization: A quantum trajectory analysis,” Phys. Rev. Lett.84, 3831–3834 (2000).
    [CrossRef] [PubMed]
  38. D. B. Milos̆ević, W. Becker, and R. Kopold, “Generation of circularly polarized high-order harmonics by two-color coplanar field mixing,” Phys. Rev. A61, 063403 (2000).
    [CrossRef]
  39. D. I. Bondar, “Instantaneous multiphoton ionization rate and initial distribution of electron momentum,” Phys. Rev. A78, 015405 (2008).
    [CrossRef]
  40. M. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt.52, 165–184 (2005).
    [CrossRef]
  41. Some clarification about the transverse velocity. Here, “transverse” means the direction perpendicular to the instantaneous polarization direction of the driving field at the moment of ionization, not the direction perpendicular to the major polarization axis.
  42. S. K. Son, D. A. Telnov, and S. I. Chu, “Probing the origin of elliptical high-order harmonic generation from aligned molecules in linearly polarized laser fields,” Phys. Rev. A82, 043829 (2010).
    [CrossRef]
  43. O. Smirnova, M. Spanner, and M. Ivanov, “Analytical solutions for strong field-driven atomic and molecular one-and two-electron continua and applications to strong-field problems,” Phys. Rev. A77, 033407 (2008).
    [CrossRef]
  44. X. Zhu, M. Qin, Q. Zhang, W. Hong, Z. Xu, and P. Lu, “Role of the Coulomb potential on the ellipticity in atomic high-order harmonics generation,” Opt. Express20, 16275–16284 (2012).
    [CrossRef]

2012

Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical simulations including electron correlations for sequential double ionization,” Phys. Rev. Lett.109, 053004 (2012).
[CrossRef] [PubMed]

D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan, D. Descamps, E. Mvel, S. Petit, H. J. Wörner, B. Pons, N. Dudovich, and Y. Mairesse, “Role of the ionic potential in high harmonic generation,” Phys. Rev. Lett.108, 203001 (2012).
[CrossRef] [PubMed]

K. Yuan and A. D. Bandrauk, “Circularly polarized attosecond pulses from molecular high-order harmonic generation by ultrashort intense bichromatic circularly and linearly polarized laser pulses,” J. Phys. B: At. Mol. Opt. Phys.45, 074001 (2012).
[CrossRef]

Max Möller, Y. Cheng, S. D. Khan, B. Zhao, K. Zhao, M. Chini, G. G. Paulus, and Z. Chang, “Dependence of high-order-harmonic-generation yield on driving-laser ellipticity,” Phys. Rev. A86, 011401 (2012).
[CrossRef]

V. V. Strelkov, M. A. Khokhlova, A. A. Gonoskov, I. A. Gonoskov, and M. Yu. Ryabikin, “High-order harmonic generation by atoms in an elliptically polarized laser field: Harmonic polarization properties and laser threshold ellipticity,” Phys. Rev. A86, 013404 (2012).
[CrossRef]

M. V. Frolov, N. L. Manakov, T. S. Sarantseva, and A. F. Starace, “High-order-harmonic-generation spectroscopy with an elliptically polarized laser field,” Phys. Rev. A86, 063406 (2012).
[CrossRef]

X. Zhu, M. Qin, Q. Zhang, W. Hong, Z. Xu, and P. Lu, “Role of the Coulomb potential on the ellipticity in atomic high-order harmonics generation,” Opt. Express20, 16275–16284 (2012).
[CrossRef]

2011

V. V. Strelkov, A. A. Gonoskov, I. A. Gonoskov, and M. Yu. Ryabikin, “Origin for ellipticity of high-order harmonics generated in atomic gases and the sublaser-cycle evolution of harmonic polarization,” Phys. Rev. Lett.107, 043902 (2011).
[CrossRef] [PubMed]

2010

L. Arissian, C. Smeenk, F. Turner, C. Trallero, A.V. Sokolov, D. M. Villeneuve, A. Staudte, and P. B. Corkum, “Direct test of laser tunneling with electron momentum imaging,” Phys. Rev. Lett.105, 133002 (2010).
[CrossRef]

S. K. Son, D. A. Telnov, and S. I. Chu, “Probing the origin of elliptical high-order harmonic generation from aligned molecules in linearly polarized laser fields,” Phys. Rev. A82, 043829 (2010).
[CrossRef]

2009

Q. Zhang, P. Lu, W. Hong, Q. Liao, and S. Wang, “Control of high-order harmonic generation from molecules lacking inversion symmetry with a polarization gating method,” Phys. Rev. A79, 053406 (2009).
[CrossRef]

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

2008

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,” Science320, 1614–1617 (2008).
[CrossRef] [PubMed]

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]

D. I. Bondar, “Instantaneous multiphoton ionization rate and initial distribution of electron momentum,” Phys. Rev. A78, 015405 (2008).
[CrossRef]

M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys.41, 132001 (2008).
[CrossRef]

O. Smirnova, M. Spanner, and M. Ivanov, “Analytical solutions for strong field-driven atomic and molecular one-and two-electron continua and applications to strong-field problems,” Phys. Rev. A77, 033407 (2008).
[CrossRef]

2007

M. I. Stockman, M. F. Kling, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photon.1, 539–544 (2007).
[CrossRef]

P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A76, 011402(R) (2007).
[CrossRef]

P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys.3, 381–387 (2007).
[CrossRef]

2006

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,” Science20, 443–446 (2006).
[CrossRef]

N. Dudovich, J. Levesque, O. Smirnova, D. Zeidler, D. Comtois, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Attosecond temporal gating with elliptically polarized light,” Phys. Rev. Lett.97, 253903 (2006).
[CrossRef]

V. V. Strelkov, “Theory of high-order harmonic generation and attosecond pulse emission by a low-frequency elliptically polarized laser field,” Phys. Rev. A74, 013405 (2006).
[CrossRef]

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]

2005

M. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt.52, 165–184 (2005).
[CrossRef]

A. Becker and F. H. M. Faisal, “Intense-field many-body S-matrix theory,” J. Phys. B: At. Mol. Opt. Phys.38, R1–R56 (2005).
[CrossRef]

2004

Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A70, 043802 (2004).
[CrossRef]

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

2003

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovac̆ev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft x-rays,” Science302, 1540–1543 (2003).
[CrossRef] [PubMed]

2002

C. Figueira de Morisson Faria, H. Schomerus, and W. Becker, “High-order above-threshold ionization: The uniform approximation and the effect of the binding potential,” Phys. Rev. A66, 043413 (2002).
[CrossRef]

D. B. Milos̆ević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A66, 063417 (2002).
[CrossRef]

2000

R. Kopold, D. B. Milos̆ević, and W. Becker, “Rescattering processes for elliptical polarization: A quantum trajectory analysis,” Phys. Rev. Lett.84, 3831–3834 (2000).
[CrossRef] [PubMed]

D. B. Milos̆ević, W. Becker, and R. Kopold, “Generation of circularly polarized high-order harmonics by two-color coplanar field mixing,” Phys. Rev. A61, 063403 (2000).
[CrossRef]

1997

P. Balcou, P Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: The role of field-gradient forces,” Phys. Rev. A55, 3204–3210 (1997).
[CrossRef]

W. Becker, A. Lohr, and M. Kleber, “A unified theory of high-harmonic generation: Application to polarization properties of the harmonics,” Phys. Rev. A56, 645–656 (1997).
[CrossRef]

1996

Ph. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, Theory of high-order harmonic generation by an elliptically polarized laser field, Phys. Rev. A53, 1725–1745 (1996).
[CrossRef] [PubMed]

M. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A54, 742–745 (1996).
[CrossRef] [PubMed]

1995

F. A. Weihe, S. K. Dutta, G. Korn, D. Du, P. H. Bucksbaum, and P. L. Shkolnikov, “Polarization of high-intensity high-harmonic generation,” Phys. Rev. A51, R3433–R3436 (1995).
[CrossRef] [PubMed]

1994

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A49, 2117–2132 (1994).
[CrossRef] [PubMed]

1993

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett.70, 1599–1602 (1993).
[CrossRef] [PubMed]

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

Agostini, P.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovac̆ev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft x-rays,” Science302, 1540–1543 (2003).
[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,” Science20, 443–446 (2006).
[CrossRef]

Antoine, Ph.

Ph. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, Theory of high-order harmonic generation by an elliptically polarized laser field, Phys. Rev. A53, 1725–1745 (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,” Science320, 1614–1617 (2008).
[CrossRef] [PubMed]

Arissian, L.

L. Arissian, C. Smeenk, F. Turner, C. Trallero, A.V. Sokolov, D. M. Villeneuve, A. Staudte, and P. B. Corkum, “Direct test of laser tunneling with electron momentum imaging,” Phys. Rev. Lett.105, 133002 (2010).
[CrossRef]

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,” Science320, 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,” Science20, 443–446 (2006).
[CrossRef]

Balcou, P.

P. Balcou, P Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: The role of field-gradient forces,” Phys. Rev. A55, 3204–3210 (1997).
[CrossRef]

Balcou, Ph.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A49, 2117–2132 (1994).
[CrossRef] [PubMed]

Baltuska, A.

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

Bandrauk, A. D.

K. Yuan and A. D. Bandrauk, “Circularly polarized attosecond pulses from molecular high-order harmonic generation by ultrashort intense bichromatic circularly and linearly polarized laser pulses,” J. Phys. B: At. Mol. Opt. Phys.45, 074001 (2012).
[CrossRef]

Becker, A.

A. Becker and F. H. M. Faisal, “Intense-field many-body S-matrix theory,” J. Phys. B: At. Mol. Opt. Phys.38, R1–R56 (2005).
[CrossRef]

Becker, W.

C. Figueira de Morisson Faria, H. Schomerus, and W. Becker, “High-order above-threshold ionization: The uniform approximation and the effect of the binding potential,” Phys. Rev. A66, 043413 (2002).
[CrossRef]

D. B. Milos̆ević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A66, 063417 (2002).
[CrossRef]

R. Kopold, D. B. Milos̆ević, and W. Becker, “Rescattering processes for elliptical polarization: A quantum trajectory analysis,” Phys. Rev. Lett.84, 3831–3834 (2000).
[CrossRef] [PubMed]

D. B. Milos̆ević, W. Becker, and R. Kopold, “Generation of circularly polarized high-order harmonics by two-color coplanar field mixing,” Phys. Rev. A61, 063403 (2000).
[CrossRef]

W. Becker, A. Lohr, and M. Kleber, “A unified theory of high-harmonic generation: Application to polarization properties of the harmonics,” Phys. Rev. A56, 645–656 (1997).
[CrossRef]

P. Salières, B. Carré, L. L. Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milos̆ević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science292, 902–905.

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,” Science20, 443–446 (2006).
[CrossRef]

Bondar, D. I.

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F. A. Weihe, S. K. Dutta, G. Korn, D. Du, P. H. Bucksbaum, and P. L. Shkolnikov, “Polarization of high-intensity high-harmonic generation,” Phys. Rev. A51, R3433–R3436 (1995).
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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,” Science320, 1614–1617 (2008).
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N. Dudovich, J. Levesque, O. Smirnova, D. Zeidler, D. Comtois, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Attosecond temporal gating with elliptically polarized light,” Phys. Rev. Lett.97, 253903 (2006).
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Li, C.

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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P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A76, 011402(R) (2007).
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D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan, D. Descamps, E. Mvel, S. Petit, H. J. Wörner, B. Pons, N. Dudovich, and Y. Mairesse, “Role of the ionic potential in high harmonic generation,” Phys. Rev. Lett.108, 203001 (2012).
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M. V. Frolov, N. L. Manakov, T. S. Sarantseva, and A. F. Starace, “High-order-harmonic-generation spectroscopy with an elliptically polarized laser field,” Phys. Rev. A86, 063406 (2012).
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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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Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovac̆ev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft x-rays,” Science302, 1540–1543 (2003).
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D. B. Milos̆ević, W. Becker, and R. Kopold, “Generation of circularly polarized high-order harmonics by two-color coplanar field mixing,” Phys. Rev. A61, 063403 (2000).
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P. Salières, B. Carré, L. L. Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milos̆ević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science292, 902–905.

Möller, Max

Max Möller, Y. Cheng, S. D. Khan, B. Zhao, K. Zhao, M. Chini, G. G. Paulus, and Z. Chang, “Dependence of high-order-harmonic-generation yield on driving-laser ellipticity,” Phys. Rev. A86, 011401 (2012).
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Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovac̆ev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft x-rays,” Science302, 1540–1543 (2003).
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Moon, E.

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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Muller, H. G.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovac̆ev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft x-rays,” Science302, 1540–1543 (2003).
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D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan, D. Descamps, E. Mvel, S. Petit, H. J. Wörner, B. Pons, N. Dudovich, and Y. Mairesse, “Role of the ionic potential in high harmonic generation,” Phys. Rev. Lett.108, 203001 (2012).
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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,” Science20, 443–446 (2006).
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Paulus, G. G.

Max Möller, Y. Cheng, S. D. Khan, B. Zhao, K. Zhao, M. Chini, G. G. Paulus, and Z. Chang, “Dependence of high-order-harmonic-generation yield on driving-laser ellipticity,” Phys. Rev. A86, 011401 (2012).
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P. Salières, B. Carré, L. L. Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milos̆ević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science292, 902–905.

Petit, S.

D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan, D. Descamps, E. Mvel, S. Petit, H. J. Wörner, B. Pons, N. Dudovich, and Y. Mairesse, “Role of the ionic potential in high harmonic generation,” Phys. Rev. Lett.108, 203001 (2012).
[CrossRef] [PubMed]

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,” Science20, 443–446 (2006).
[CrossRef]

Pons, B.

D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan, D. Descamps, E. Mvel, S. Petit, H. J. Wörner, B. Pons, N. Dudovich, and Y. Mairesse, “Role of the ionic potential in high harmonic generation,” Phys. Rev. Lett.108, 203001 (2012).
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Qin, M.

Ryabikin, M. Yu.

V. V. Strelkov, M. A. Khokhlova, A. A. Gonoskov, I. A. Gonoskov, and M. Yu. Ryabikin, “High-order harmonic generation by atoms in an elliptically polarized laser field: Harmonic polarization properties and laser threshold ellipticity,” Phys. Rev. A86, 013404 (2012).
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V. V. Strelkov, A. A. Gonoskov, I. A. Gonoskov, and M. Yu. Ryabikin, “Origin for ellipticity of high-order harmonics generated in atomic gases and the sublaser-cycle evolution of harmonic polarization,” Phys. Rev. Lett.107, 043902 (2011).
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Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovac̆ev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft x-rays,” Science302, 1540–1543 (2003).
[CrossRef] [PubMed]

Salières, P

P. Balcou, P Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: The role of field-gradient forces,” Phys. Rev. A55, 3204–3210 (1997).
[CrossRef]

Salières, P.

Ph. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, Theory of high-order harmonic generation by an elliptically polarized laser field, Phys. Rev. A53, 1725–1745 (1996).
[CrossRef] [PubMed]

P. Salières, B. Carré, L. L. Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milos̆ević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science292, 902–905.

Sanpera, A.

P. Salières, B. Carré, L. L. Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milos̆ević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science292, 902–905.

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,” Science20, 443–446 (2006).
[CrossRef]

Sarantseva, T. S.

M. V. Frolov, N. L. Manakov, T. S. Sarantseva, and A. F. Starace, “High-order-harmonic-generation spectroscopy with an elliptically polarized laser field,” Phys. Rev. A86, 063406 (2012).
[CrossRef]

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M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys.41, 132001 (2008).
[CrossRef]

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett.70, 1599–1602 (1993).
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R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London)427, 817–821 (2004).
[CrossRef]

Shafir, D.

D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan, D. Descamps, E. Mvel, S. Petit, H. J. Wörner, B. Pons, N. Dudovich, and Y. Mairesse, “Role of the ionic potential in high harmonic generation,” Phys. Rev. Lett.108, 203001 (2012).
[CrossRef] [PubMed]

Shakya, M. M.

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]

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F. A. Weihe, S. K. Dutta, G. Korn, D. Du, P. H. Bucksbaum, and P. L. Shkolnikov, “Polarization of high-intensity high-harmonic generation,” Phys. Rev. A51, R3433–R3436 (1995).
[CrossRef] [PubMed]

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L. Arissian, C. Smeenk, F. Turner, C. Trallero, A.V. Sokolov, D. M. Villeneuve, A. Staudte, and P. B. Corkum, “Direct test of laser tunneling with electron momentum imaging,” Phys. Rev. Lett.105, 133002 (2010).
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O. Smirnova, M. Spanner, and M. Ivanov, “Analytical solutions for strong field-driven atomic and molecular one-and two-electron continua and applications to strong-field problems,” Phys. Rev. A77, 033407 (2008).
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N. Dudovich, J. Levesque, O. Smirnova, D. Zeidler, D. Comtois, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Attosecond temporal gating with elliptically polarized light,” Phys. Rev. Lett.97, 253903 (2006).
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M. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt.52, 165–184 (2005).
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D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan, D. Descamps, E. Mvel, S. Petit, H. J. Wörner, B. Pons, N. Dudovich, and Y. Mairesse, “Role of the ionic potential in high harmonic generation,” Phys. Rev. Lett.108, 203001 (2012).
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Sokolov, A.V.

L. Arissian, C. Smeenk, F. Turner, C. Trallero, A.V. Sokolov, D. M. Villeneuve, A. Staudte, and P. B. Corkum, “Direct test of laser tunneling with electron momentum imaging,” Phys. Rev. Lett.105, 133002 (2010).
[CrossRef]

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S. K. Son, D. A. Telnov, and S. I. Chu, “Probing the origin of elliptical high-order harmonic generation from aligned molecules in linearly polarized laser fields,” Phys. Rev. A82, 043829 (2010).
[CrossRef]

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O. Smirnova, M. Spanner, and M. Ivanov, “Analytical solutions for strong field-driven atomic and molecular one-and two-electron continua and applications to strong-field problems,” Phys. Rev. A77, 033407 (2008).
[CrossRef]

M. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt.52, 165–184 (2005).
[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,” Science20, 443–446 (2006).
[CrossRef]

Starace, A. F.

M. V. Frolov, N. L. Manakov, T. S. Sarantseva, and A. F. Starace, “High-order-harmonic-generation spectroscopy with an elliptically polarized laser field,” Phys. Rev. A86, 063406 (2012).
[CrossRef]

Staudte, A.

L. Arissian, C. Smeenk, F. Turner, C. Trallero, A.V. Sokolov, D. M. Villeneuve, A. Staudte, and P. B. Corkum, “Direct test of laser tunneling with electron momentum imaging,” Phys. Rev. Lett.105, 133002 (2010).
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M. I. Stockman, M. F. Kling, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photon.1, 539–544 (2007).
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V. V. Strelkov, M. A. Khokhlova, A. A. Gonoskov, I. A. Gonoskov, and M. Yu. Ryabikin, “High-order harmonic generation by atoms in an elliptically polarized laser field: Harmonic polarization properties and laser threshold ellipticity,” Phys. Rev. A86, 013404 (2012).
[CrossRef]

V. V. Strelkov, A. A. Gonoskov, I. A. Gonoskov, and M. Yu. Ryabikin, “Origin for ellipticity of high-order harmonics generated in atomic gases and the sublaser-cycle evolution of harmonic polarization,” Phys. Rev. Lett.107, 043902 (2011).
[CrossRef] [PubMed]

V. V. Strelkov, “Theory of high-order harmonic generation and attosecond pulse emission by a low-frequency elliptically polarized laser field,” Phys. Rev. A74, 013405 (2006).
[CrossRef]

Taïzeb, R.

Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovac̆ev, R. Taïzeb, B. Carré, H. G. Muller, P. Agostini, and P. Salères, “Attosecond synchronization of high-harmonic soft x-rays,” Science302, 1540–1543 (2003).
[CrossRef] [PubMed]

Tate, J. L.

M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys.41, 132001 (2008).
[CrossRef]

Telnov, D. A.

S. K. Son, D. A. Telnov, and S. I. Chu, “Probing the origin of elliptical high-order harmonic generation from aligned molecules in linearly polarized laser fields,” Phys. Rev. A82, 043829 (2010).
[CrossRef]

Tisch, J. W. G.

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]

Trallero, C.

L. Arissian, C. Smeenk, F. Turner, C. Trallero, A.V. Sokolov, D. M. Villeneuve, A. Staudte, and P. B. Corkum, “Direct test of laser tunneling with electron momentum imaging,” Phys. Rev. Lett.105, 133002 (2010).
[CrossRef]

Turner, F.

L. Arissian, C. Smeenk, F. Turner, C. Trallero, A.V. Sokolov, D. M. Villeneuve, A. Staudte, and P. B. Corkum, “Direct test of laser tunneling with electron momentum imaging,” Phys. Rev. Lett.105, 133002 (2010).
[CrossRef]

Uiberacker, M.

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,” Science320, 1614–1617 (2008).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, 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,” Science20, 443–446 (2006).
[CrossRef]

Villeneuve, D. M.

L. Arissian, C. Smeenk, F. Turner, C. Trallero, A.V. Sokolov, D. M. Villeneuve, A. Staudte, and P. B. Corkum, “Direct test of laser tunneling with electron momentum imaging,” Phys. Rev. Lett.105, 133002 (2010).
[CrossRef]

N. Dudovich, J. Levesque, O. Smirnova, D. Zeidler, D. Comtois, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Attosecond temporal gating with elliptically polarized light,” Phys. Rev. Lett.97, 253903 (2006).
[CrossRef]

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,” Science20, 443–446 (2006).
[CrossRef]

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,” Science20, 443–446 (2006).
[CrossRef]

Walther, H.

P. Salières, B. Carré, L. L. Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milos̆ević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science292, 902–905.

Wang, S.

Q. Zhang, P. Lu, W. Hong, Q. Liao, and S. Wang, “Control of high-order harmonic generation from molecules lacking inversion symmetry with a polarization gating method,” Phys. Rev. A79, 053406 (2009).
[CrossRef]

Wang, X.

P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A76, 011402(R) (2007).
[CrossRef]

Weihe, F. A.

F. A. Weihe, S. K. Dutta, G. Korn, D. Du, P. H. Bucksbaum, and P. L. Shkolnikov, “Polarization of high-intensity high-harmonic generation,” Phys. Rev. A51, R3433–R3436 (1995).
[CrossRef] [PubMed]

Westerwalbesloh, Th.

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

Wörner, H. J.

D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan, D. Descamps, E. Mvel, S. Petit, H. J. Wörner, B. Pons, N. Dudovich, and Y. Mairesse, “Role of the ionic potential in high harmonic generation,” Phys. Rev. Lett.108, 203001 (2012).
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Xu, Z.

Yakovlev, V.

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

Yakovlev, V. S.

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,” Science320, 1614–1617 (2008).
[CrossRef] [PubMed]

Yang, B.

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett.70, 1599–1602 (1993).
[CrossRef] [PubMed]

Yuan, K.

K. Yuan and A. D. Bandrauk, “Circularly polarized attosecond pulses from molecular high-order harmonic generation by ultrashort intense bichromatic circularly and linearly polarized laser pulses,” J. Phys. B: At. Mol. Opt. Phys.45, 074001 (2012).
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Some clarification about the transverse velocity. Here, “transverse” means the direction perpendicular to the instantaneous polarization direction of the driving field at the moment of ionization, not the direction perpendicular to the major polarization axis.

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Figures (6)

Fig. 1
Fig. 1

Example of ionization and recombination times for HHG by a helium atom (Ip = 0.9035) with an 810nm elliptically laser field having ellipticity of 0.2 and intensity of 7.7 × 1014W/cm2. (a) and (b) are real and imaginary parts of ionization (blue line) and recombination (red line) times of the long (dashed line) and short (solid line) trajectories, respectively.

Fig. 2
Fig. 2

Schematic sketch of the quantum orbits in elliptically polarized laser fields driven at different wavelengths. The arrows indicate the direction of the electron traveling. The ellipticity of the laser field is 0.2.

Fig. 3
Fig. 3

(a) Parallel and (b) perpendicular components of the electron initial velocity as a function of harmonic ellipticity. The 19th harmonic is selected. Laser parameters are the same as Fig. 1.

Fig. 4
Fig. 4

Harmonic yield as a function of laser ellipticity. Solid lines represent the numerical results calculated by Eq. (9). Filled hexagrams in (a) and (c) and filled diamonds in (b) and (d) are the experiment data taken from Ref. [26]. Dashed lines represent the results calculated by Eq. (11). Target atom is neon in (a) and (b) and helium in (c) and (d). Details of the laser parameters are in main text.

Fig. 5
Fig. 5

(a) Parallel and (b) perpendicular components of the electron momentum at the moment of recombination by a 1300nm elliptically laser field having ellipticity of 0.1 and intensity of 2.2×1014W/cm2 in argon (Ip = 0.5790). Red solid lines and blue dashed lines represent real and imaginary parts of momentum respectively.

Fig. 6
Fig. 6

Harmonic ellipticity as a function of harmonic order. Blue solid line represents the result using QO, and red filled square represents the numerical result of TDSE in Ref. [30]. The laser parameters are the same as Fig. 5.

Equations (15)

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d ( Ω ) = i d t d t d pC ( p , t , t ) exp [ i S ( p , t , t ) + i Ω t ] ,
S ( p , t , t ) = t t d t ( [ p + A ( t ) ] 2 2 + I p ) ,
p s = 1 t i t r t i t r d t A ( t )
[ p s + A ( t i ) ] 2 2 + I p = 0
[ p s + A ( t r ) ] 2 2 + I p = n ω .
F ( t ) = F 1 + ε 2 [ cos ( ω t ) e | | + ε sin ( ω t ) e ] ,
A ( t ) = F 1 + ε 2 ω [ sin ( ω t ) e | | + ε cos ( ω t ) e ] .
R ( t ) = p s ( t t i ) + t i t d t A ( t ) ,
Γ ~ exp [ 2 Im S ( p s , t i , t r ) ] .
Φ ( v ) = Φ ( 0 ) exp ( v 2 2 τ ) .
Γ ~ exp ( [ Re v ( t i ) ] 2 2 τ ) .
ζ = 1 + R 2 1 + 2 R 2 cos 2 δ + R 4 1 + R 2 + 1 + 2 R 2 cos 2 δ + R 4 ,
d d | | = M ( v ) M | | ( v ) ,
M ( v ) = v f ( v ) .
d d | | = v v | | .

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