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.

© 2013 OSA

1. P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys.3, 381–387 (2007). [CrossRef]
2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys.81, 163–234 (2009). [CrossRef]
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). [CrossRef]
5. M. I. Stockman, M. F. Kling, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photon.1, 539–544 (2007). [CrossRef]
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). [CrossRef] [PubMed]
7. 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]
8. 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]
9. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett.71, 1994–1997 (1993). [CrossRef] [PubMed]
10. 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]
11. 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]
12. 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]
13. C. C. Chirilă, “Analysis of the strong field approximation for harmonic generation and multiphoton ionization in intense ultrashort laser pulses,” PhD Thesis http://massey.dur.ac.uk/resources/cpchirila/chirilathesis.pdf .
14. 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.
15. 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]
16. Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A70, 043802 (2004). [CrossRef]
17. 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]
18. 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]
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). [CrossRef]
20. 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]
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). [CrossRef] [PubMed]
22. 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]
23. 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]
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]
29. 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]
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]
32. 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]
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]

J. Mod. Opt.

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

J. Phys. B: At. Mol. Opt. Phys.

• 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]
• 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]
• 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]

Nat. Photon.

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

Nat. Phys.

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

Nature (London)

• 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]

Opt. Commun.

• 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]

Opt. Express

• 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]

Phys. Rev. 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]
• 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. 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]
• 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]
• 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]
• 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]
• D. B. Milos̆ević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A66, 063417 (2002). [CrossRef]
• 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]
• D. I. Bondar, “Instantaneous multiphoton ionization rate and initial distribution of electron momentum,” Phys. Rev. A78, 015405 (2008). [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]
• 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]
• 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]
• 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]
• 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]
• Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A70, 043802 (2004). [CrossRef]
• 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]
• 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]
• 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]

Phys. Rev. Lett.

• 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]
• 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, 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]
• 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]
• 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]
• 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. 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]

Rev. Mod. Phys.

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

Science

• 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]
• 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]
• 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.
• 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]

Other

• C. C. Chirilă, “Analysis of the strong field approximation for harmonic generation and multiphoton ionization in intense ultrashort laser pulses,” PhD Thesis http://massey.dur.ac.uk/resources/cpchirila/chirilathesis.pdf .
• 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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