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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 4 — Feb. 25, 2013
  • pp: 4896–4907
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Quantum-orbit analysis for yield and ellipticity of high order harmonic generation with elliptically polarized laser field

Yang Li, Xiaosong Zhu, Qingbin Zhang, Meiyan Qin, and Peixiang Lu  »View Author Affiliations


Optics Express, Vol. 21, Issue 4, pp. 4896-4907 (2013)
http://dx.doi.org/10.1364/OE.21.004896


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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

High-order harmonic generation (HHG) from atoms and molecules exposed to intense laser fields has been intensively investigated in the past two decades [1

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

, 2

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

]. As a highly nonlinear phenomenon, it provides versatile source of applications such as generating coherent attosecond pulses in the extreme ultraviolet (XUV) regime as well as monitoring and controlling electron dynamics with attosecond and Ångstörm resolutions [4

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

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]

]. There is a general agreement that the physical origin of HHG process can be understood within the classical three-step model (CTM) [8

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

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

]. First, a electron tunnels into the continuum through the potential barrier formed by the Coulomb potential and the laser field. Then it is accelerated in the laser field treated as a free particle. Finally, it may recombine with the parent ion and a high energy XUV photon is emitted. The quantum-mechanical version of CTM based on strong-field approximation (SFA), known as Lewenstein model, is able to give a quantitative treatment of HHG [10

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

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]

]. With the saddle-point approximation applied to SFA [12

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]

], the harmonic amplitude can be expressed as a coherent sum of a few complex electron trajectories (see [13

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.

] for details). These complex trajectories, involving complex times and momenta, can be interpreted using the concept of Feynman’s path integrals [14

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 292, 902–905.

]. We refer to this model as quantum-orbit model (QO).

HHG driven by elliptically polarized laser pulse attracts more attention motivated by developing new techniques for generating isolated attosecond pulse [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]

18

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]

] and studying basic physical processes such as tunnel ionization in strong laser field [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]

]. The ellipticity of the laser field provides an additional control parameter for laser-matter interactions and introduces some new features in strong field laser-matter interaction process [20

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

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]

]. It is known that if the laser pulse is elliptically polarized, elliptically polarized high harmonics would be obtained [22

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]

]. However, CTM can not explain HHG for elliptical polarization, because the electrons following classical trajectories can not recombine with the parent ion. Some properties of the generated elliptically polarized high harmonics, e.g., the rotation angle of the harmonics, can be explained by CTM as the angle between the direction of the electron momentum at the instant of the recombination and the major polarization axis of the laser field. But the harmonic ellipticity and the dependence of the harmonic yield on the driving field ellipticity can’t be explained using CTM. Lewenstein model based on SFA can provide a theoretical treatment for the harmonic ellipticity and the dependence of the harmonic yield on the driving field ellipticity [23

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

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]

]. However, a qualitative explanation for the physical origin is missed. This question is essential for researchers to gain a better understanding of HHG process and further optimize techniques to generate circularly polarized attosecond pulses [25

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]

] or isolated linearly polarized attosecond pulses [16

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

].

In this paper, we perform a systematical quantum-orbit description of HHG with intense elliptically polarized laser field. The dependence of the harmonic yield on the laser ellipticity and polarization properties of the high harmonics are investigated. Quantum electron trajectories responsible for the generation of harmonics are identified. We show that for quantum trajectories contributing to the generation of harmonics, the electron has an nonzero initial velocity at the tunnel exit which facilitate the recombination of the electron with the parent ion. This initial velocity naturally emerges from the quantum mechanical phenomenon of tunneling and we do not require a priori assumption. The initial velocity changes the ionization probability of the electron trajectory thus has a significant influence on the harmonic yield. This physical picture has been well documented in the literatures [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]

28

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]

]. In this work, a simple formula is given to calculate the ellipticity dependence of the harmonic yield. Our calculation is based on the same physical picture, but we applied less additional simplifications and provided more accurate results. Another point of our results is about the polarization properties of HHG. We show that two components of the electron velocity, along and perpendicular to the major polarization axis of the lase field, dictate the vectorial properties of the HHG and are mapped into the HHG polarization state. Since we deal with quantum trajectories, momenta of the electrons are complex. The non-zero phase difference leads the harmonics to be ellipticity polarized. A qualitative understanding of the physical origin is offered as well.

This paper is organized as follows. In Sec. 2 we briefly discuss QO used here to describe HHG in elliptically polarized field. Quantum trajectories responsible for the generation of particular harmonics are identified. The results of QO are utilized to calculate the dependence of the harmonic yield on the laser ellipticity. In Sec. 3 we derive formulas to calculate HHG ellipticity of different harmonic orders, using complex electron momentum at the instant of recombination obtained by QO. A physical interpretation is provided. Finally, we conclude in Sec. 4.

2. Dependence of harmonic yield on laser ellipticity

We will begin with the standard analysis relied on SFA [10

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]

]. The expression of the induced dipole moment at the frequency Ω can be written as (atomic units are used throughout):
d(Ω)=idtdtdpC(p,t,t)exp[iS(p,t,t)+iΩt],
(1)
where 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]

], so we neglect this part below. The phase S(p, t, t′) is the quasi-classical action of the form:
S(p,t,t)=ttdt([p+A(t)]22+Ip),
(2)
with A(t) the vector potential of the laser field and Ip the ionization potential.

The quantum orbits are the trajectories along which the phase of the multidimensional integral Eq. (1) is stationary. They are obtained by finding the stationary points of the action, i.e., by differentiating the action with respect to the integration variables p, t′ and t, yielding correspondingly:
ps=1titrtitrdtA(t)
(3)
[ps+A(ti)]22+Ip=0
(4)
[ps+A(tr)]22+Ip=nω.
(5)
Here ps, ti and tr denote the solutions to these saddle-point equations 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.

We consider the HHG process in a monochromatic elliptically polarized laser field with electric vector:
F(t)=F1+ε2[cos(ωt)e||+εsin(ωt)e],
(6)
with amplitude 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:
A(t)=F1+ε2ω[sin(ωt)e||+εcos(ωt)e].
(7)
Substituting Eq. (7) into the saddle-point Eqs. (3)(5) and introducing ps = ps(ti, tr) from Eq. (3) into Eqs. (4) and (5), we obtain a system of two equations, which can be solved numerically for the complex variables ti and tr. 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

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

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]

]. Below we will concentrate on the shot trajectories only.

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.

To gain an intuitive picture about the quantum electron orbit, we give a visualized presentation by plotting the real parts of the trajectories. The position 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

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

]
R(t)=ps(tti)+titdtA(t),
(8)
with t ∈ [Re(ti), Re(tr)]. In general, the orbit described by Eq. (8) is complex and satisfies the conditions R(tr) = R(ti) = 0. The real part of 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 ti is a complex variable, R(Re(ti)) ≠ R(ti). 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

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

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]

]. For the former, owing to Eq. (3), ps is along the laser field. Thus, the initial velocity ps + A(ti) 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.

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 (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 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.

3. Harmonic polarization properties in elliptically polarized laser field

We now discuss the polarization properties of HHG in elliptically polarized field. A unified theory of HHG for an elliptically polarized driving field based on SFA provides theoretical treatment for polarization properties of harmonics [24

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]

]. However, it doesn’t gives a clear explanation of the physical origin of the non-zero harmonic ellipticity. Very recently, Strelkov et al. propose the explanation of the origin of the harmonic ellipticity with their analytical model [30

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]

]. They show that this ellipticity originates from quantum-mechanical uncertainty of the electron motion. Their analytical model is based on semiclassical electron trajectories and includes tunneling coherently [29

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]

]. Below, we will analyze the harmonic ellipticity fully quantum mechanically based on QO and a coincident conclusion can be made.

The harmonic ellipticity 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]

]
ζ=1+R21+2R2cos2δ+R41+R2+1+2R2cos2δ+R4,
(12)
where 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:
dd||=M(v)M||(v),
(13)
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]

], i.e.
M(v)=vf(v).
(14)
Substituting Eq. (14) into Eq. (13), we find
dd||=vv||.
(15)
This expression indicates that the two orthogonal momentum components at the moment of recombination are mapped onto the harmonic ellipticity. Since we deal with quantum trajectories, 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.

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.

Next, we calculate harmonic ellipticity using Eq. (15), shown in Fig. 6. One can see that the QO result provides very good agreement with the numerical result for almost all harmonics. Small deviations appear in low harmonic orders. This may due to that we don’t consider the influence of Coulomb potential. The Coulomb effects can be added coherently using strong-field eikonal approximation [43

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]

]. It is shown in Ref. [44

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]

] that the Coulomb modification for low harmonic orders is stronger than that for high harmonic orders. The effect of the Coulomb potential on high energy electrons is much weaker. Thereby for higher harmonics orders, our results agree quite well with the numerical results.

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.

4. Conclusion

In summary, we have performed a systematical quantum-orbit analysis of high harmonic generation in intense elliptically polarized laser field. Both the dependence of the harmonic yield on laser ellipticity and the harmonic polarization properties have been investigated. For the former, we have shown that when increasing the laser ellipticity, the initial velocity of the electron increases, causing the reduce of the ionization rate, thereby diminish the generation of harmonics. A physically transparent formula has been presented to calculated the ellipticity dependence of harmonic yield and the results show very good agreement with a recently done experiment. For the latter, we have found that for a spherically symmetrical atom ground state, the parallel and perpendicular components of the electron’s return momentum are mapped on the polarization state of the harmonics. Because the two two orthogonal components are complex quantity, the phase difference of them are nonzero, leading the harmonics to be elliptically polarized. Our calculations match well with the numerical solutions of TDSE. Furthermore, we have provided an explanation of the physical origin of the harmonic ellipticity. The harmonic ellipticity originates from quantum uncertainty of the electron momentum of the relevant quantum electron trajectory. Our work can help researchers gain more insight into HHG process in elliptically polarized laser field and develop new techniques to generated elliptically polarized attosecond pulse.

Acknowledgment

This work was supported by the NNSF of China under grants 11234004 and 60925021, the 973 Program of China under grant 2011CB808103, and the Doctoral fund of Ministry of Education of China under grant 20100142110047.

References and links

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,” Science 320, 1614–1617 (2008). [CrossRef] [PubMed]

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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]

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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.

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

16.

Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A 70, 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. 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]

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]

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]

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. A 74, 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. A 86, 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. A 86, 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. A 55, 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. A 66, 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. A 61, 063403 (2000). [CrossRef]

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]

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 82, 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. 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]

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

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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). [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]

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