## Quantum-orbit analysis for yield and ellipticity of high order harmonic generation with elliptically polarized laser field |

Optics Express, Vol. 21, Issue 4, pp. 4896-4907 (2013)

http://dx.doi.org/10.1364/OE.21.004896

Acrobat PDF (886 KB)

### 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.

© 2013 OSA

## 1. Introduction

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]

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]

7. P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A **76**, 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. A **49**, 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. A **66**, 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.

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,” Science **20**, 443–446 (2006). [CrossRef]

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. A **79**, 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. A **51**, 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. A **53**, 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. A **56**, 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]

16. Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A **70**, 043802 (2004). [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. A **86**, 011401 (2012). [CrossRef]

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]

## 2. Dependence of harmonic yield on laser ellipticity

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. A **49**, 2117–2132 (1994). [CrossRef] [PubMed]

*t*is associated with the recombination time,

*t*′ is the moment of ionization,

**p**is the canonical momentum of the electron and the pre-factor

**C**(

**p**,

*t*,

*t*′) takes care of the amplitudes of ionization at

*t*′ and recombination at

*t*. The contribution of this pre-factor to the harmonic yield is less than 10% [19

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]

*S*(

**p**,

*t*,

*t*′) is the quasi-classical action of the form: with

**A**(

*t*) the vector potential of the laser field and

*I*the ionization potential.

_{p}**p**,

*t*′ and

*t*, yielding correspondingly: Here

**p**

*,*

_{s}*t*and

_{i}*t*denote the solutions to these saddle-point equations

_{r}**p**,

*t*′ and

*t*, respectively. Equations (3)–(5) have a transparent physical interpretation. Equation (3) expresses the return condition for the electron. Equations (4) and (5) represent the energy conservation at the moment of ionization and recombination, respectively.

*F*, ellipticity

*ε*and frequency

*ω*.

**e**

_{‖}and

**e**

_{⊥}denote the unit vectors parallel and perpendicular to the major polarization axis of the laser field. The corresponding vector potential is: Substituting Eq. (7) into the saddle-point Eqs. (3)–(5) and introducing

**p**

*=*

_{s}**p**

*(*

_{s}*t*,

_{i}*t*) from Eq. (3) into Eqs. (4) and (5), we obtain a system of two equations, which can be solved numerically for the complex variables

_{r}*t*and

_{i}*t*. In Fig. 1, we present examples of the solutions. Real and imaginary parts of the ionization and recombination times are plotted in Figs. 1(a) and 1(b), respectively. As one can see, there are two quantum orbits with different travel times within one optical cycle, which are well known as short and long trajectories. It is also known that when taking propagation effects into account, long trajectories have an undesirable phase-matching conditions and thus only short trajectories contribute to the harmonics [33

_{r}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. A **55**, 3204–3210 (1997). [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]

**R**(

*t*) occupied by the electron is given by [15

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,” Science **20**, 443–446 (2006). [CrossRef]

36. D. B. Milos̆ević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A **66**, 063417 (2002). [CrossRef]

*t*∈ [Re(

*t*), Re(

_{i}*t*)]. In general, the orbit described by Eq. (8) is complex and satisfies the conditions

_{r}**R**(

*t*) =

_{r}**R**(

*t*) = 0. The real part of

_{i}*R*(

*t*) is displayed in Fig. 2, with elliptically polarized fields of different wavelengths. Some interesting features can be observed. First, the starting points of the trajectories are not at the origin (the position of the parent ion) but the vicinity of the origin. Mathematically speaking, since

*t*is a complex variable,

_{i}**R**(Re(

*t*)) ≠

_{i}**R**(

*t*). Physically, this is due to the fact that the electron must tunnel out and it can’t appear at the origin. Second, The return positions are almost at the origin. Note that we do not add this constraint a priori. It is a natural property of the quantum orbit calculation. This is very different from classical trajectories in elliptically polarized fields. As we pointed out above, using CTM, the electrons can not return to the ion core, for the reason that the transverse component of the driving field would cause transverse displacements of the electron trajectories. Third, we can deduce from the second feature that the electron has non-zero initial velocity at the tunnel exit. The initial velocity compensates the transverse displacement caused by electric field. The same conclusion can also be drawn from the saddle-point Eqs. (3)–(5) mathematically. There is critical difference between the saddle-point equations in linearly and elliptically polarized field [37

_{i}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. A **61**, 063403 (2000). [CrossRef]

**p**

*is along the laser field. Thus, the initial velocity*

_{s}**p**

*+*

_{s}**A**(

*t*) of the electron is purely imaginary considering Eq. (4). However, for elliptical polarization, the initial velocity has a none-zero real part. The imaginary part emerges from tunneling, and the real part compensates the transverse displacement and forces the electron to return to the origin. The initial velocities are displayed in Fig. 3. For non-zero ellipticity, the real part of the two velocity components, along and perpendicular the major polarization axis, are not zero, and both of them increase with the laser elipticity. Below we will show that this phenomenon would greatly influence the harmonic yield.

_{i}**p**

*,*

_{s}*t*,

_{i}*t*) with laser ellipticity varying from 0 to 1. Inserting the saddle-points into Eq. (2), the stationary points of the action

_{r}*S*(

**p**

*,*

_{s}*t*,

_{i}*t*) are obtained. They are all complex too. The modulation of the harmonic yield Γ as a function of the laser ellipticity is given by [39

_{r}39. D. I. Bondar, “Instantaneous multiphoton ionization rate and initial distribution of electron momentum,” Phys. Rev. A **78**, 015405 (2008). [CrossRef]

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

^{14}W/cm

^{2}with the wavelength of 810nm. The 27th harmonic is selected. Laser intensity in panel (b) is 5.4×10

^{14}W/cm

^{2}with the wavelength of 405nm. The 11th harmonic is selected. In panels (c) and (d), the target atom is helium. The intensity of the laser pulse used in panel (c) is 7.7×10

^{14}W/cm

^{2}with the wavelength of 810nm. The 19th harmonic is selected. Laser intensity in panel (d) is 7.7×10

^{14}W/cm

^{2}with the wavelength of 405nm. The 11th harmonic is selected. As one can see, when increasing the ellipticity, the harmonic yields steeply decrease. The dependence of the harmonic yields on laser ellipticity are approximately Gaussian. For laser ellipticity lager than 0.5, there are almost no harmonics generated. Another characteristic we can find from Fig. 4 is that the threshold ellipticity (the fundamental ellipticity for which the harmonic yield is two times lower than for the linearly polarized fundamental field) is smaller for long wavelength (810nm) than short wavelength (405nm). As a result, we find that the HHG sensitivity to the driving laser ellipticity increases with the drive wavelength. The theoretical results are compared with a recent experiment [26

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. A **86**, 011401 (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. A **86**, 011401 (2012). [CrossRef]

## 3. Harmonic polarization properties in elliptically polarized laser field

24. W. Becker, A. Lohr, and M. Kleber, “A unified theory of high-harmonic generation: Application to polarization properties of the harmonics,” Phys. Rev. A **56**, 645–656 (1997). [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]

29. V. V. Strelkov, “Theory of high-order harmonic generation and attosecond pulse emission by a low-frequency elliptically polarized laser field,” Phys. Rev. A **74**, 013405 (2006). [CrossRef]

*ζ*is calculated by [42

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. A **82**, 043829 (2010). [CrossRef]

*R*is the amplitude ratio of the two orthogonal dipole calculated by

*R*= |

*d*

_{⊥}|/|

*d*

_{‖}| and

*δ*is the phase difference by

*δ*= arg(

*d*

_{⊥}) − arg(

*d*

_{‖}). The ratio of

*d*

_{⊥}and

*d*

_{‖}satisfies: where

**M**(

**v**) is the recombination dipole matrix element and

**v**is the drift momentum at the instant of recombination. For a spherically symmetrical atom ground state,

**M**(

**v**) can be expressed as a product of

**v**and a scalar that only depends on the absolute value of

**v**[27

27. M. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A **54**, 742–745 (1996). [CrossRef] [PubMed]

*v*

_{⊥}and

*v*

_{‖}should be complex in general. We present

*v*

_{⊥}and

*v*

_{‖}as a function of harmonic order in Fig. 5. It is shown that the imaginary part of

*v*

_{‖}is negligible, but the imaginary part of

*v*

_{⊥}is comparable with the real part. Thus the phase difference is not 0 or

*π*. In comparison, CTM would give real valued

*v*

_{‖}and

*v*

_{⊥}. This is why harmonic ellipticity can not be explained by CTM.

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. A **77**, 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. Express **20**, 16275–16284 (2012). [CrossRef]

## 4. Conclusion

## Acknowledgment

## References and links

1. | P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. |

2. | F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. |

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,” Science |

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) |

5. | M. I. Stockman, M. F. Kling, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photon. |

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,” Science |

7. | P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A |

8. | K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. |

9. | P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. |

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. A |

11. | A. Becker and F. H. M. Faisal, “Intense-field many-body S-matrix theory,” J. Phys. B: At. Mol. Opt. Phys. |

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. A |

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,” Science |

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,” Science |

16. | Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A |

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. |

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. A |

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. |

20. | Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical simulations including electron correlations for sequential double ionization,” Phys. Rev. Lett. |

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. |

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. A |

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. A |

24. | W. Becker, A. Lohr, and M. Kleber, “A unified theory of high-harmonic generation: Application to polarization properties of the harmonics,” Phys. Rev. A |

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. |

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. A |

27. | M. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A |

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. |

29. | V. V. Strelkov, “Theory of high-order harmonic generation and attosecond pulse emission by a low-frequency elliptically polarized laser field,” Phys. Rev. A |

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. |

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. A |

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. A |

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. A |

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. |

35. | M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At. Mol. Opt. Phys. |

36. | D. B. Milos̆ević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A |

37. | R. Kopold, D. B. Milos̆ević, and W. Becker, “Rescattering processes for elliptical polarization: A quantum trajectory analysis,” Phys. Rev. Lett. |

38. | D. B. Milos̆ević, W. Becker, and R. Kopold, “Generation of circularly polarized high-order harmonics by two-color coplanar field mixing,” Phys. Rev. A |

39. | D. I. Bondar, “Instantaneous multiphoton ionization rate and initial distribution of electron momentum,” Phys. Rev. A |

40. | M. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt. |

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. A |

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. A |

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. Express |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(260.3230) Physical optics : Ionization

(270.6620) Quantum optics : Strong-field processes

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: January 4, 2013

Revised Manuscript: February 11, 2013

Manuscript Accepted: February 13, 2013

Published: February 20, 2013

**Citation**

Yang Li, Xiaosong Zhu, Qingbin Zhang, Meiyan Qin, and Peixiang Lu, "Quantum-orbit analysis for yield and ellipticity of high order harmonic generation with elliptically polarized laser field," Opt. Express **21**, 4896-4907 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4896

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### References

- P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys.3, 381–387 (2007). [CrossRef]
- F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys.81, 163–234 (2009). [CrossRef]
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