## Role of the Coulomb potential on the ellipticity in atomic high-order harmonics generation |

Optics Express, Vol. 20, Issue 15, pp. 16275-16284 (2012)

http://dx.doi.org/10.1364/OE.20.016275

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### Abstract

The role of the Coulomb potential on the generation of elliptically polarized high-order harmonics from atoms driven by elliptically polarized laser is investigated analytically. It is found that the Coulomb effect makes a contribution to the harmonic ellipticity for low harmonic orders and short quantum path. By using the strong-field eikonal-Volkov approximation, we analyze the influence of the Coulomb potential on the dynamics of the continuum state and find that the obtained harmonic ellipticity in our simulation originates from the break of symmetry of the continuum state.

© 2012 OSA

## 1. Introduction

1. Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft X-rays,” Science **302**, 1540–1543 (2003). [CrossRef] [PubMed]

7. A. Etches and L. B. Madsen, “Extending the strong-field approximation of high-order harmonic generation to polar molecules: gating mechanisms and extension of the harmonic cutoff,” J. Phys. B: At. Mol. Opt. Phys. **43**, 155602 (2010). [CrossRef]

8. M. Lein, “Molecular imaging using recolliding electrons,” J. Phys. B: At. Mol. Opt. Phys. **40**, R135–R173 (2007). [CrossRef]

12. Y. Chen and B. Zhang, “Tracing the structure of asymmetric molecules from high-order harmonic generation,” Phys. Rev. A **84**, 053402 (2011). [CrossRef]

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

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

15. Ph. Antoine, B. Carré, A. L’Huillier, and M. Lewenstein, “Polarization of high-order harmonics,” Phys. Rev. A **55**, 1314–1324 (1997). [CrossRef]

16. 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]

15. Ph. Antoine, B. Carré, A. L’Huillier, and M. Lewenstein, “Polarization of high-order harmonics,” Phys. Rev. A **55**, 1314–1324 (1997). [CrossRef]

17. 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]

18. 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]

19. X. Zhou, R. Lock, N. Wagner, W. Li, H. C. Kapteyn, and M. M. Murnane, “Elliptically polarized high-order harmonic emission from molecules in linearly polarized laser fields,” Phys. Rev. Lett. **102**, 073902 (2009). [CrossRef] [PubMed]

24. K. J. 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]

## 2. Theoretical model

^{14}W/cm

^{2}, elliptically polarized driving lasers with ellipticity of 0.1. The major axis of the polarization ellipse is parallel to the

*x*axis of the laboratory frame and the minor axis is parallel to the

*y*axis. To obtain the ellipticity of the HH, the time-dependent transition dipoles of the system are calculated by [28

28. M. Gühr, B. K. McFarland, J. P. Farrell, and P. H. Bucksbaum, “High harmonic generation for N2 and CO2 beyond the two-point model,” J. Phys. B: At. Mol. Opt. Phys. **40**, 3745–3755 (2007). [CrossRef]

29. J. Levesque, Y. Mairesse, N. Dudovich, H. Pépin, J. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Polarization state of high-order harmonic emission from aligned molecules,” Phys. Rev. Lett. **99**, 243001 (2007). [CrossRef]

*ψ*

_{0}is the ground state of the atom and

*ψ*refers to the continuum state.

_{k}**k**is the recombination momentum, which is associated with the recombination time

*t*.

*ψ*

_{0}is obtained by solving the time-dependent Schrödinger equation (TDSE) with imaginary-time propagation. The Hamiltonian is (in atomic units) with where the soft-core parameter

*a*is set to be 0.388 to fit the ground-state energy of Ar

*E*= −0.5794 a.u. Through this paper, the target atom is Ar.

_{g}*e*

^{ik·r}= cos(

**k**·

**r**) +

*i*sin(

**k**·

**r**), the real and imaginary parts of

*ψ*are symmetric and antisymmetric on the spatial coordinates respectively. As the ground state

_{k}*ψ*

_{0}is also symmetric, when its space-independent term

*e*

^{−iEgt}is also omitted, the result of the integral

*d*(

_{i}*t*;

*k*) = ∫

*ψ*

_{0}

*r*must be a pure imaginary number and the phase difference between the two components

_{i}ψ_{k}dr*d*and

_{y}*d*is either 0 or

_{x}*π*[23

23. M. Y. Qin, X. S. Zhu, Q. B. Zhang, W. Y. Hong, and P. X. Lu, “Broadband large-ellipticity harmonic generation with polar molecules,” Opt. Express **19**, 25084–25092 (2011). [CrossRef]

29. J. Levesque, Y. Mairesse, N. Dudovich, H. Pépin, J. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Polarization state of high-order harmonic emission from aligned molecules,” Phys. Rev. Lett. **99**, 243001 (2007). [CrossRef]

30. O. Smirnova, M. Spanner, and M. Ivanov, “Coulomb and polarization effects in sub-cycle dynamics of strong-field ionization,” J. Phys. B: At. Mol. Opt. Phys. **39**, S307–S321 (2006). [CrossRef]

32. D. I. Bondar, W. K. Liu, and M. Yu. Ivanov, “Two-electron ionization in strong laser fields below intensity threshold: Signatures of attosecond timing in correlated spectra,” Phys. Rev. A **79**, 023417 (2009). [CrossRef]

*σ*(

_{k}**r**,

*t*) is the correction to the plane wave considering the effect of the Coulomb potential and is given by where describes the motion of the continuum electron in the laser field.

18. 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]

**k**at time

*t*, SF-EVA describes the Coulomb correction to the continuum state by tracing the motion of the electron in the process from

*t*to

_{b}*t*. Therefore, with this approach, it is possible to in detail investigate influence of the Coulomb potential on the dynamics of the continuum electrons with different trajectories and to study the role of the Coulomb potential on the harmonic ellipticity. In ref. [31

31. 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]

25. 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]

*t*,

*t*and

_{b}**k**can be obtained by finding the approximate solutions to the saddle-point equations [17

17. 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]

33. 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]

35. 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.

**k**

*(*

_{L}*t*) and the classical trajectory of the continuum electron can be calculated by substituting the solutions into the equations of the electron motion for both x and y components.

*R*= |

*d*|/|

_{y}*d*| and

_{x}*δ*=

*arg*[

*d*] −

_{y}*arg*[

*d*] respectively, and the ellipticity

_{x}*ε*is finally obtained by [36

36. 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*= 0 or the phase difference

*δ*= 0 or

*π*, the ellipticity

*ε*= 0 and the HH is linearly polarized, otherwise it will be elliptically or circularly polarized. In this work, in order to study the role of Coulomb effect on the harmonic ellipticity, we only take into account the influence of the Coulomb potential in the model and the other effects such as the quantum mechanical uncertainty of the electron motion are not included.

## 3. Result and discussion

*U*, 2

_{p}*U*and 3

_{p}*U*are presented, where

_{p}*F*

_{0}and

*ω*being the amplitude and angular frequency of laser. Panels (a) and (b) correspond to the short and long quantum paths respectively. The continuum electron starts from the origin with zero velocity and recombines to the core at

*x*= 0, which is the same as in ref. [25

25. 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]

33. 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]

*U*corresponds to HH radiation at the cutoff in the harmonic spectrum, and the recombination kinetic energy of 2

_{p}*U*, 1

_{p}*U*correspond to HH of lower orders in the plateau region (the 89th and 53rd orders respectively). As shown in Fig. 1(a) for short quantum path, when the recombination kinetic energy grows higher, the trajectory grows longer and the motion of the continuum electron is farther from the core. The maximum distance increases from no more than 20 a.u. to nearly 60 a.u. As shown in Fig. 1(b) for long quantum path, the trajectories are even longer for HH from the cutoff to lower orders. At the same time, with the trajectory extending longer, the transverse shift grows up to 40 a.u. Figs. 1(c) and (d) are the snap shots of the evolution of the probability density distribution of the wavepacket at the instant of recombination. The evolution is obtained by solving the TDSE numerically. In the quantum view of the dynamic of the continuum electron, one can see that although the center of the continuum wavepacket shifts about 30 a.u. transversely, it is still wide enough to interfere with the bound state to generate the HH. For the numerical evolution, the laser parameters are the same as those given in Section 2 and the initial state is also obtained by solving TDSE with imaginary-time propagation with the Hamiltonian given by Eq. (2).

_{p}*σ*(

_{k}**r**,

*t*) in Eq. (5), which is mainly determined by the integral for the Coulomb potential along the trajectories,

*i.e. V*[

**r**

*(*

_{L}*t*′)]. After considering this modification in the phase, the continuum state will be no longer a plane wave. As a result, the symmetry of the wave function

*ψ*is broken and the obtained

_{k}*d*,

_{x}*d*are not pure imaginary numbers any more. Then, the value of the phase difference

_{y}*δ*varies in the interval of [0,2

*π*) rather than fixes to 0 or

*π*[23

23. M. Y. Qin, X. S. Zhu, Q. B. Zhang, W. Y. Hong, and P. X. Lu, “Broadband large-ellipticity harmonic generation with polar molecules,” Opt. Express **19**, 25084–25092 (2011). [CrossRef]

*σ*(

_{k}**r**,

*t*) at the instant of recombination with kinetic energy of 1

*U*and 3

_{p}*U*and for short and long quantum paths respectively. All the four panels are plotted in the same color scale. Comparing the second column to the first column in Fig. 2, the Coulomb effect is weaker for long quantum path. This is because for the long quantum path, the electron leaves directly far away from the core, drifts in the faraway region where the Coulomb potential

_{p}*V*(

**r**) is weak and enters the vicinity of the core shortly before recombination. While for short quantum path, the electron only drifts in the vicinity of the core, where the effect of the Coulomb potential is much stronger.

*V*[

**r**

*(*

_{L}*t*)] is much bigger. Thus it is obvious that the Coulomb effect is strongest for the low order harmonics from short quantum path as shown in panel (a). For the long quantum path, although the trajectory for 1

*U*is longer and farther from the core than that for 3

_{p}*U*, the Coulomb effect is already very weak for both faraway electrons. The determining factor is that, the accumulated Coulomb phase is greater for the slow electron especially in the processes when the electron leaves and comes back to the vicinity of the core. Mathematically speaking, the integral between

_{p}*t*and

_{b}*t*in Eq. (5) is bigger for HH of low harmonic order.

*t*are presented in Fig 3. The recombination at about

*t*= 0.7 optical cycle is associated with the HH at the cutoff region in the harmonic spectrum, and recombination at earlier or later time corresponds to HH of lower harmonic orders in the plateau region. The rotation angle is defined by the direction of the electron momentum at the instant of return [25

25. 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]

**107**, 043902 (2011). [CrossRef] [PubMed]

*t*into curves as a function of the harmonic orders, it will be more obvious for the result to be analyzed. This is shown in Fig 4(a) and the phase differences between the two orthogonal components

*δ*are also plotted in Fig 4(b). As the relationship between

*t*and

**k**has been obtained by solving the saddle-point equations in the x component, the recombination momentum is then transformed into the harmonic order

*q*according to the relation

*qω*=

*k*

^{2}/2 −

*E*.

_{g}**107**, 043902 (2011). [CrossRef] [PubMed]

*π*due to the symmetry of the ground state and the continuum state

*ψ*≈

_{k}*e*

^{ik·r}. For low harmonic orders, where the Coulomb effect is strong enough to modify the continuum state,

*ψ*is no longer a plane wave and the symmetry is broken. As a result, the phase different deviate from

_{k}*π*and the nonzero ellipticity appears.

**107**, 043902 (2011). [CrossRef] [PubMed]

**107**, 043902 (2011). [CrossRef] [PubMed]

**107**, 043902 (2011). [CrossRef] [PubMed]

**107**, 043902 (2011). [CrossRef] [PubMed]

**107**, 043902 (2011). [CrossRef] [PubMed]

30. O. Smirnova, M. Spanner, and M. Ivanov, “Coulomb and polarization effects in sub-cycle dynamics of strong-field ionization,” J. Phys. B: At. Mol. Opt. Phys. **39**, S307–S321 (2006). [CrossRef]

*V*(

_{s}**r**). In this relatively faraway area, the Coulomb effect is too weak to lead to the elliptically polarized HHG. However, for the HH of the lowest orders of short quantum path, the motions of the continuum electrons are confined in a very small range. Take the 21st harmonic order for example, the maximum distance of the electron trajectory from the core is only about 2.5 a.u. In this area, the difference of the Coulomb effects on these trajectories between the short- and long-range potentials is small. Therefore, the harmonic ellipticity of these orders is still high and become close to that obtained with soft-core potential. Similarly, as shown by the dashed curves in Fig. 4(b), the phase differences also converge to the value of

*π*much more rapidly due to the short range of the Coulomb potential.

## 4. Conclusion

## Acknowledgment

## References and links

1. | Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft X-rays,” Science |

2. | 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 |

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

4. | W. Y. Hong, P. X. Lu, P. F. Lan, Q. G. Li, Q. B. Zhang, Z. Y. Yang, and X. B. Wang, “Method to generate directly a broadband isolated attosecond pulse with stable pulse duration and high signal-to-noise ratio,” Phys. Rev. A |

5. | W. Y. Hong, P. X. Lu, P. F. Lan, Q. B. Zhang, and X. B. Wang, “Few-cycle attosecond pulses with stabilized-carrier-envelope phase in the presence of a strong terahertz field,” Opt. Express |

6. | V. Strelkov, “Role of autoionizing state in resonant high-order harmonic generation and attosecond pulse production,” Phys. Rev. Lett. |

7. | A. Etches and L. B. Madsen, “Extending the strong-field approximation of high-order harmonic generation to polar molecules: gating mechanisms and extension of the harmonic cutoff,” J. Phys. B: At. Mol. Opt. Phys. |

8. | M. Lein, “Molecular imaging using recolliding electrons,” J. Phys. B: At. Mol. Opt. Phys. |

9. | S. Haessler, J. Caillat, and P. Salières, “Self-probing of molecules with high harmonic generation,” J. Phys. B: At. Mol. Opt. Phys. |

10. | J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature |

11. | E. Hijano, C. Serrat, G. N. Gibson, and J. Biegert, “Orbital geometry determined by orthogonal high-order harmonic polarization components,” Phys. Rev. A |

12. | Y. Chen and B. Zhang, “Tracing the structure of asymmetric molecules from high-order harmonic generation,” Phys. Rev. A |

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

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

15. | Ph. Antoine, B. Carré, A. L’Huillier, and M. Lewenstein, “Polarization of high-order harmonics,” Phys. Rev. A |

16. | 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 |

17. | 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 |

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

19. | X. Zhou, R. Lock, N. Wagner, W. Li, H. C. Kapteyn, and M. M. Murnane, “Elliptically polarized high-order harmonic emission from molecules in linearly polarized laser fields,” Phys. Rev. Lett. |

20. | O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, and D. Villeneuve, “Attosecond circular dichroism spectroscopy of polyatomic molecules,” Phys. Rev. Lett. |

21. | A. Etches, C. B. Madsen, and L. B. Madsen, “Inducing elliptically polarized high-order harmonics from aligned molecules with linearly polarized femtosecond pulses,” Phys. Rev. A |

22. | S. Ramakrishna, P. A. J. Sherratt, A. D. Dutoi, and T. Seideman, “Origin and implication of ellipticity in high-order harmonic generation from aligned molecules,” Phys. Rev. A |

23. | M. Y. Qin, X. S. Zhu, Q. B. Zhang, W. Y. Hong, and P. X. Lu, “Broadband large-ellipticity harmonic generation with polar molecules,” Opt. Express |

24. | K. J. 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. |

25. | 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. |

26. | P. A. J. Sherratt, S. Ramakrishna, and T. Seideman, “Signatures of the molecular potential in the ellipticity of high-order harmonics from aligned molecules,” Phys. Rev. A |

27. | E. V. van der Zwan and M. Lein, “Two-center interference and ellipticity in high-order harmonic generation from |

28. | M. Gühr, B. K. McFarland, J. P. Farrell, and P. H. Bucksbaum, “High harmonic generation for N2 and CO2 beyond the two-point model,” J. Phys. B: At. Mol. Opt. Phys. |

29. | J. Levesque, Y. Mairesse, N. Dudovich, H. Pépin, J. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Polarization state of high-order harmonic emission from aligned molecules,” Phys. Rev. Lett. |

30. | O. Smirnova, M. Spanner, and M. Ivanov, “Coulomb and polarization effects in sub-cycle dynamics of strong-field ionization,” J. Phys. B: At. Mol. Opt. Phys. |

31. | 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 |

32. | D. I. Bondar, W. K. Liu, and M. Yu. Ivanov, “Two-electron ionization in strong laser fields below intensity threshold: Signatures of attosecond timing in correlated spectra,” Phys. Rev. A |

33. | 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 |

34. | W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G. G. Paulus, and H. Walther, “Above-threshold ionization: From classical features to quantum effects,” Adv. At. Mol. Opt. Phys. |

35. | 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. |

36. | 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 |

**OCIS Codes**

(190.4160) Nonlinear optics : Multiharmonic generation

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 22, 2012

Revised Manuscript: June 12, 2012

Manuscript Accepted: June 26, 2012

Published: July 2, 2012

**Citation**

Xiaosong Zhu, Meiyan Qin, Qingbin Zhang, Weiyi Hong, Zhizhan Xu, and Peixiang Lu, "Role of the Coulomb potential on the ellipticity in atomic high-order harmonics generation," Opt. Express **20**, 16275-16284 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16275

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### References

- Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft X-rays,” Science302, 1540–1543 (2003). [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,” Science320, 1614–1617 (2008). [CrossRef] [PubMed]
- P. F. Lan, P. X. Lu, W. Cao, Y. H. Li, and X. L. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A76, 011402(R) (2007). [CrossRef]
- W. Y. Hong, P. X. Lu, P. F. Lan, Q. G. Li, Q. B. Zhang, Z. Y. Yang, and X. B. Wang, “Method to generate directly a broadband isolated attosecond pulse with stable pulse duration and high signal-to-noise ratio,” Phys. Rev. A78, 063407 (2008). [CrossRef]
- W. Y. Hong, P. X. Lu, P. F. Lan, Q. B. Zhang, and X. B. Wang, “Few-cycle attosecond pulses with stabilized-carrier-envelope phase in the presence of a strong terahertz field,” Opt. Express17, 5139–5146 (2009). [CrossRef] [PubMed]
- V. Strelkov, “Role of autoionizing state in resonant high-order harmonic generation and attosecond pulse production,” Phys. Rev. Lett.104, 123901 (2010). [CrossRef] [PubMed]
- A. Etches and L. B. Madsen, “Extending the strong-field approximation of high-order harmonic generation to polar molecules: gating mechanisms and extension of the harmonic cutoff,” J. Phys. B: At. Mol. Opt. Phys.43, 155602 (2010). [CrossRef]
- M. Lein, “Molecular imaging using recolliding electrons,” J. Phys. B: At. Mol. Opt. Phys.40, R135–R173 (2007). [CrossRef]
- S. Haessler, J. Caillat, and P. Salières, “Self-probing of molecules with high harmonic generation,” J. Phys. B: At. Mol. Opt. Phys.44, 203001 (2011). [CrossRef]
- J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature432, 867–871 (2004). [CrossRef] [PubMed]
- E. Hijano, C. Serrat, G. N. Gibson, and J. Biegert, “Orbital geometry determined by orthogonal high-order harmonic polarization components,” Phys. Rev. A81, 041401(R) (2010). [CrossRef]
- Y. Chen and B. Zhang, “Tracing the structure of asymmetric molecules from high-order harmonic generation,” Phys. Rev. A84, 053402 (2011). [CrossRef]
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