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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 28676–28684
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Ellipticity dependence of the near-threshold harmonics of H2 in an elliptical strong laser field

Hua Yang, Peng Liu, Ruxin Li, and Zhizhan Xu  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 28676-28684 (2013)
http://dx.doi.org/10.1364/OE.21.028676


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Abstract

We study the ellipticity dependence of the near-threshold (NT) harmonics of pre-aligned H2 molecules using the time-dependent density functional theory. The anomalous maximum appearing at a non-zero ellipticity for the generated NT harmonics can be attributed to multiphoton effects of the orthogonally polarized component of the elliptical driving laser field. Our calculation also shows that the structure of the bound-state, such as molecular alignment and bond length, can be sensitively reflected on the ellipticity dependence of the near-threshold harmonics.

© 2013 Optical Society of America

1. Introduction

High-order harmonic generation (HHG) occurs when atoms or molecules are exposed to strong laser fields, emitting a spectrum from the extreme ultraviolet to the keV region [1

1. T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012). [CrossRef] [PubMed]

]. HHG can be understood by the three-step model [2

2. P. B. Corkum, “Plasma Perspective on Strong Field Multiphoton Ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef] [PubMed]

] which includes tunneling ionization, acceleration of electrons, and recombination. A quantum presentation of this model under strong-field approximation (SFA) [3

3. M. Lewenstein, P. 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(3), 2117–2132 (1994). [CrossRef] [PubMed]

] has also been developed. When the intense laser field is elliptically polarized, the orthogonal polarized component of the electric field can push the free electrons away from the parent ions, thus suppress the recombination and HHG yield. It has been demonstrated that the ellipticity dependence of HHG can be used to probe the wavepacket of the recombining free electrons both in experiments [4

4. A. Flettner, J. König, M. B. Mason, T. Pfeifer, U. Weichmann, R. Düren, and G. Gerber, “Ellipticity dependence of atomic and molecular high harmonic generation,” Eur. Phys. J. D 21(1), 115–119 (2002). [CrossRef]

6

6. Y. Mairesse, N. Dudovich, J. Levesque, M. Y. Ivanov, P. B. Corkum, and D. M. Villeneuve, “Electron wavepacket control with elliptically polarized laser light in high harmonic generation from aligned molecules,” New J. Phys. 10(2), 025015 (2008). [CrossRef]

] and by calculations [7

7. P. 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(3), 1725–1745 (1996). [CrossRef] [PubMed]

, 8

8. A. S. Landsman, A. N. Pfeiffer, C. Hofmann, M. Smolarski, C. Cirelli, and U. Keller, “Rydberg state creation by tunnel ionization,” New J. Phys. 15(1), 013001 (2013). [CrossRef]

].

However, for the harmonics with the photon energy near the ionization potential, i.e., the near-threshold (NT) harmonics, anomalous ellipticity dependence has been observed in atoms [9

9. N. H. Burnett, C. Kan, and P. B. Corkum, “Ellipticity and polarization effects in harmonic generation in ionizing neon,” Phys. Rev. A 51(5), R3418–R3421 (1995). [CrossRef] [PubMed]

11

11. M. Kakehata, H. Takada, H. Yumoto, and K. Miyazaki, “Anomalous ellipticity dependence of high-order harmonic generation,” Phys. Rev. A 55(2), R861–R864 (1997). [CrossRef]

] as well as aligned molecules [12

12. H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, “Near-Threshold High-Order Harmonic Spectroscopy with Aligned Molecules,” Phys. Rev. Lett. 105(14), 143904 (2010). [CrossRef] [PubMed]

]: As the ellipticity increases, the intensity of some of the NT harmonics shows an anomalous maximum (AM) at a non-zero ellipticity. Comparing with the harmonic emission driven by the linearly polarized intense laser field, an enhancement of nearly one order in magnitude for the AM has been observed [9

9. N. H. Burnett, C. Kan, and P. B. Corkum, “Ellipticity and polarization effects in harmonic generation in ionizing neon,” Phys. Rev. A 51(5), R3418–R3421 (1995). [CrossRef] [PubMed]

, 12

12. H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, “Near-Threshold High-Order Harmonic Spectroscopy with Aligned Molecules,” Phys. Rev. Lett. 105(14), 143904 (2010). [CrossRef] [PubMed]

]. More properties of the AMs such as alignment dependence have also been investigated and reported in the experiment of O2 [12

12. H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, “Near-Threshold High-Order Harmonic Spectroscopy with Aligned Molecules,” Phys. Rev. Lett. 105(14), 143904 (2010). [CrossRef] [PubMed]

].

In this paper we perform an ab initio calculation on the HHG of aligned H2 molecules by means of time-dependent density functional theory (TDDFT), where the ionic potential and the electron-electron effects are attentively treated. AMs exist in the NT harmonics, which is consistent with the reported experimental observations on atoms [9

9. N. H. Burnett, C. Kan, and P. B. Corkum, “Ellipticity and polarization effects in harmonic generation in ionizing neon,” Phys. Rev. A 51(5), R3418–R3421 (1995). [CrossRef] [PubMed]

11

11. M. Kakehata, H. Takada, H. Yumoto, and K. Miyazaki, “Anomalous ellipticity dependence of high-order harmonic generation,” Phys. Rev. A 55(2), R861–R864 (1997). [CrossRef]

] and O2 molecules [12

12. H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, “Near-Threshold High-Order Harmonic Spectroscopy with Aligned Molecules,” Phys. Rev. Lett. 105(14), 143904 (2010). [CrossRef] [PubMed]

]. The AMs can be attributed to the system-dependent multiphoton effects of the orthogonally polarized component of the elliptical driving laser field. Our calculation shows that the structure of the bound-state, such as molecular alignment and bond length, can be sensitively reflected on the ellipticity dependence of the near-threshold harmonics. Thus information of the multiphoton process as well as the dynamical features of the bound states in strong laser fields can be got from study of the near-threshold harmonics, which can also offer better understanding of the HHG mechanisms and knowledge of extending the HHG spectroscopy to the threshold region.

2. Theoretical method

The three-dimensional TDDFT calculation is performed by propagating the molecular wavefunctions under the time-dependent Kohn-Sham (KS) equation,
itψi(r,t)=[122+veff(r,t)]ψi(r,t),i=1,2,...,N.
(1)
where N is the total number of the KS orbitals. As the KS orbital of H2 is double-occupied, we do not specify the spin-orbitals throughout the propagation. Since N = 1 for H2, no unoccupied orbital is introduced, and no radiation from the transition of excited states is included. veff is the time-dependent effective potential, which can be written in the general form:

veff(r,t)=ρ(r',t)|rr'|dr'+vxc(r,t)+vps+E(t)r,
(2)

The first term is the classical Hartree potential that describes the electron-electron interaction and ρ is the electron density. The second term is the exchange-correlation potential including all non-trivial many body effects. The Leeuwen-Baerends functional [17

17. P. R. T. Schipper, O. V. Gritsenko, S. J. A. van Gisbergen, and E. J. Baerends, “Molecular calculations of excitation energies and (hyper)polarizabilities with a statistical average of orbital model exchange-correlation potentials,” J. Chem. Phys. 112(3), 1344–1352 (2000). [CrossRef]

], whose accuracy has been extensively benchmarked in many calculations of HHG [18

18. J. Heslar, D. Telnov, and S. I. Chu, “High-order-harmonic generation in homonuclear and heteronuclear diatomic molecules: Exploration of multiple orbital contributions,” Phys. Rev. A 83(4), 043414 (2011). [CrossRef]

21

21. X. Chu and G. C. Groenenboom, “Time-dependent density-functional-theory calculation of high-order-harmonic generation of H2,” Phys. Rev. A 85(5), 053402 (2012). [CrossRef]

], is employed in the second term. The last two terms account for the potential due to the interaction of electrons with nuclei and external laser field, respectively. The nuclear distance of the molecule is fixed during the propagation. vps gives the ionic potential modeled by the norm-conserving, nonlocal Troullier-Martines pseudopotential [22

22. N. Troullier and J. L. Martins, “Efficient pseudopotentials for plane-wave calculations,” Phys. Rev. B Condens. Matter 43(3), 1993–2006 (1991). [CrossRef] [PubMed]

]. The potential of the laser field is expressed using the dipole approximation and the length gauge. The calculation of the ground state and the propagation of the KS orbital are carried out using the real-space code Octopus [23

23. X. Andrade, J. Alberdi-Rodriguez, D. A. Strubbe, M. J. T. Oliveira, F. Nogueira, A. Castro, J. Muguerza, A. Arruabarrena, S. G. Louie, A. Aspuru-Guzik, A. Rubio, and M. A. L. Marques, “Time-dependent density-functional theory in massively parallel computer architectures: the OCTOPUS project,” J. Phys. Condens. Matter 24(23), 233202 (2012). [CrossRef] [PubMed]

].

The harmonic spectrum in each polarization direction is obtained from Fourier transformation of the dipole acceleration calculated using Ehrenfest’s theorem [24

24. K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, “Calculation of the background emitted during high-harmonic generation,” Phys. Rev. A 45(5), 3347–3349 (1992). [CrossRef] [PubMed]

], which has been suggested as the most accurate method for calculating HHG when dealing with molecular systems [25

25. G. L. Kamta and A. D. Bandrauk, “Three-dimensional time-profile analysis of high-order harmonic generation in molecules: Nuclear interferences in H2+,” Phys. Rev. A 71(5), 053407 (2005). [CrossRef]

]:
Hp(ω)|ap(ω)|2=|d¨p(t)eiωtdt|2,
(3)
where p presents the polarization direction along x or z in our calculation.

The elliptically polarized laser field is described as
E(t)=E0f(t)(11+ε2)1/2[ezsin(ωt)+exεsin(ωtπ/2)],
(4)
where E0 stands for the amplitude of the electric field, f (t) is the laser envelope, and ex and ez are unit vectors. We choose the peak intensity of 1.5 × 1014 W/cm2, the wavelength of 800 nm, the time-step of dt = 0.02 a.u. (atomic units) and the ellipticity of 0 ≤ ε ≤ 0.5. A 9-cycle, trapezoidal envelope, which stays constant for 6 cycles between a 1.5-cycle linear ramp and a 1.5-cycle linear decay, as shown in Fig. 1
Fig. 1 Amplitude of the z- and x- polarization component of the elliptical laser field at ε = 0.2.
, is used. The middle cycles correspond to most of the HHG yields, and prevent the amplitude of the orthogonal laser component from cycle-to-cycle change.

The molecule is placed in a three-dimensional rectangular grid cell of |xmax| = 20 a.u. (1 a.u. = 0.0529 nm) and |ymax| = 15 a.u.. The size of the |zmax| is chosen according to the electron excursion distance. Classically the shortest excursion distance of the electrons for the cut-off harmonic is zcut = 1.13 Ez /ω2 ~23 a.u., while the maximum excursion distance of the long trajectory is zlong = 2Ez /ω2 ~40 a.u.. Since the long trajectory can affect the ellipticity dependence of the near-threshold harmonics and no AM was observed in experiments for the long path [12

12. H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, “Near-Threshold High-Order Harmonic Spectroscopy with Aligned Molecules,” Phys. Rev. Lett. 105(14), 143904 (2010). [CrossRef] [PubMed]

], we take |zmax| = 29 a.u., which is larger than zcut but shorter than zlong, thus can roughly block out the long trajectory [15

15. V. V. Strelkov, M. A. Khokhlova, A. A. Gonoskov, I. A. Gonoskov, and M. Y. Ryabikin, “High-order harmonic generation by atoms in an elliptically polarized laser field: Harmonic polarization properties and laser threshold ellipticity,” Phys. Rev. A 86(1), 013404 (2012). [CrossRef]

]. The electron wavepackets originally in long trajectories may return and reach the nucleus area, by which the molecules are prematurely ionized. However, the influence to the bound state is still small, because of the lateral diffusion of electron wavepackets during the long excursion. Thus the premature ionization is not important for the bound state. The grid space of dx = dy = dz = 0.4 a.u. is used in the ground state calculation and the time-propagation. For H2 whose equilibrium bond length is ~1.4 a.u., the calculated ionization energy is 15.01 eV, which gives a small difference compared with the experimental result of 15.37 eV [26

26. W. Bleakney, “The Ionization Potential of Molecular Hydrogen,” Phys. Rev. 40(4), 496–501 (1932). [CrossRef]

]. A bonus imaginary potential of Δr = 6 a.u. and V0 = 0.6 a.u. outside the grid cell in each dimension is appended to absorb the reflecting wavefunction in the boundary region [27

27. D. Neuhasuer and M. Baer, “The time-dependent Schrödinger equation: Application of absorbing boundary conditions,” J. Chem. Phys. 90(8), 4351 (1989). [CrossRef]

]:

V(r)={0for0<r<R,iV0sin1/8[π(rR)2Δr]forR<r<R+Δr.
(5)

3. Results and discussions

In our calculation the nuclei of H2 are fixed with the equilibrium distance of 1.4 a.u. as well as the elongated distance of 2.2 and 3.0 a.u.. Different interatomic distances are considered thus different bound-state structures and ionization potenticals are employed. For the case of such elongated H2, the Leeuwen-Baerends functional used in our calculation has been demonstrated to give accurate ionization energies and HHG yields [21

21. X. Chu and G. C. Groenenboom, “Time-dependent density-functional-theory calculation of high-order-harmonic generation of H2,” Phys. Rev. A 85(5), 053402 (2012). [CrossRef]

]. We first discuss the results for the bond length of 2.2 a.u., which is close to the bond length of O2 (~2.28 a.u.) and gives representative results. The harmonics are also distinguished under perpendicular and parallel alignment of molecules, where the polarization direction of the major laser component (Ez) is perpendicular or parallel to the molecular axis, respectively. The ionization energy of the KS orbital calculated at the bond length of 2.2 a.u. is ~12.9 eV, corresponding to the harmonic order of 8. Harmonic spectrum at zero ellipticity for H2 at the bond length of 2.2 a.u. under both alignments is given in Fig. 2
Fig. 2 Harmonic spectrum at zero ellipticity for H2 at the bond length of 2.2 a.u. under both alignments.
. It is noteworthy that the harmonics beyond cutoff (~H33) may be generated from the imperfect absorption of the absorbing boundaries [28

28. F. He, C. Ruiz, and A. Becker, “Absorbing boundaries in numerical solutions of the time-dependent Schrödinger equation on a grid using exterior complex scaling,” Phys. Rev. A 75(5), 053407 (2007). [CrossRef]

], which gives little influence to the lower harmonics in our calculations.

Figure 3
Fig. 3 Normalized harmonic intensity as a function of ellipticity for perpendicularly (left) and parallelly (right) aligned H2 at the bond length of 2.2 a.u.. Each harmonic is normalized to its intensity at zero ellipticity.
shows the calculated intensity of the generated harmonics (Hx + Hz) from the perpendicularly and parallelly aligned H2 molecules, where the polarization direction of Ez is perpendicular or parallel to the molecular axis, respectively. Each harmonic is normalized to its intensity at ε = 0. The AMs can be clearly distinguished in the NT harmonics. For the perpendicular alignment [Fig. 3(a)], H5 as well as H7 reaches to the maximum intensity at ε ~0.2. For the parallel alignment [Fig. 3(b)], an AM appears at ε ~0.1 for H11. No AM is found in the harmonics well above the threshold. Instead, they exhibit a near-Gaussian ellipticity dependence with a constant width due to quantum effects of the tunnelling process, as reported previously [29

29. M. Y. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt. 52(2-3), 165–184 (2005). [CrossRef]

]

Figure 4
Fig. 4 z-component of the harmonics (Hz) as a function of ellipticity for perpendicularly (left) and parallelly (right) aligned H2 at the bond length of 2.2 a.u.. Each harmonic is normalized to its intensity at zero ellipticity.
plots the z-component of the harmonics (Hz) under perpendicular and parallel alignments, respectively. Similar ellipticity dependence for all harmonics for both alignment conditions is observed and no AM can be found. As one can see, the lower harmonics are more insensitive to the ellipticity than the near-cutoff harmonics. This phenomenon has been observed in experiments using the laser intensities not only in tunnelling regime [4

4. A. Flettner, J. König, M. B. Mason, T. Pfeifer, U. Weichmann, R. Düren, and G. Gerber, “Ellipticity dependence of atomic and molecular high harmonic generation,” Eur. Phys. J. D 21(1), 115–119 (2002). [CrossRef]

, 9

9. N. H. Burnett, C. Kan, and P. B. Corkum, “Ellipticity and polarization effects in harmonic generation in ionizing neon,” Phys. Rev. A 51(5), R3418–R3421 (1995). [CrossRef] [PubMed]

, 30

30. A. Flettner, J. König, M. B. Mason, T. Pfeifer, U. Weichmann, and G. Gerber, “Atomic and molecular high-harmonic generation: A comparison of ellipticity dependence based on the three-step model,” J. Mod. Opt. 50, 529–537 (2003).

] but also in multiphoton-ionization regime [31

31. Y. Liang, M. V. Ammosov, and S. L. Chin, “High-order harmonic generation in argon by elliptically polarized picosecond dye laser pulses,” J. Phys. At. Mol. Opt. Phys. 27(6), 1269–1276 (1994). [CrossRef]

, 32

32. K. S. Budil, P. Salières, A. L’Huillier, and M. D. Perry, “Influence of ellipticity on harmonic generation,” Phys. Rev. A 48(5), R3437–R3440 (1993). [CrossRef] [PubMed]

]. In the latter case, the deviation of the ellipticity dependence was found to be small from the prediction of perturbation theory, especially for the lowest harmonics. The ellipticity sensitivity is predicted to increase with the harmonic order, which is the case in our calculation result for these near-threshold harmonics.

Since the Hz of the NT harmonics indicates a smooth variation, the x-component of the harmonics must be essential to the AMs observed in the total harmonics shown in Fig. 3. In Fig. 5
Fig. 5 Spectra of the x-component of the harmonics as a function of ellipticity under perpendicular (left) and parallel (right) alignments at the bond length of 2.2 a.u.. The maximum of each harmonic is marked by the square.
the spectra of the x-component of the harmonics (Hx) as a function of ellipticity under both alignments are given. In contrary to Hz, the maximum intensity appears at a non-zero ellipticity for each Hx. Meanwhile, the intensity of each Hx is zero at ε = 0, regardless of the alignment. The non-monotonic ellipticity dependence of Hx has been experimentally observed in the near-threshold harmonics of Ne [9

9. N. H. Burnett, C. Kan, and P. B. Corkum, “Ellipticity and polarization effects in harmonic generation in ionizing neon,” Phys. Rev. A 51(5), R3418–R3421 (1995). [CrossRef] [PubMed]

] by respective measurements of Hx and Hz. At ε = 0 Hx was observed to have a minimum while Hz exhibited a maximum, which is consistent with our calculation. When it comes to the above-threshold harmonics, the three-step model regime begins to take effect, and the intensity of Hx shows a rapid decay as the harmonic order increases. The decay is more significant under perpendicular alignment.

For the near-threshold harmonics, when the x-component is comparable with or larger than the z-component, the AM can appear in the total harmonic emission. The x- and z- components of the NT harmonics with AMs are retrieved and plotted in Fig. 6
Fig. 6 Intensities of different polarization components of H5 (a) and H7 (b) under perpendicular alignment and H11 (c) under parallel alignment at the bond length of 2.2 a.u..
. For H5 under perpendicular alignment [Fig. 6(a)], the x-component (H5x) is significant compared with the z-component (H5z) thus an AM can appear. While H5x maximizes at the ellipticity of ~0.35, the total intensity of H5 peaks at the ellipticity of ~0.2. The adjacent H7 [Fig. 6(b)] shows similar curves but a stronger AM, because of a larger ratio of H7x over H7z. H11 under the parallel alignment [Fig. 6(c)], whose energy is above-threshold, obeys the same rule. As H11x is relatively weak compared with H11z, totally H11 has an AM but less pronounced than H5 and H7. For other harmonics, Hx is weak compared with Hz, therefore no AM exists.

To gain a better understanding of the HHG mechanisms in different energy regimes, we perform the time-frequency analysis by wavelet transformation of the dipole acceleration [33

33. X. M. Tong and S. I. Chu, “Probing the spectral and temporal structures of high-order harmonic generation in intense laser pulses,” Phys. Rev. A 61(2), 021802 (2000). [CrossRef]

], where a Gaussian filter centred at a chosen frequency is interposed:

aω(t)=a(t')exp[ω2(tt')22τ2]exp[iω(tt')]dt'
(6)

As shown in our calculation, the AMs of near-threshold harmonics can mainly be attributed to the non-monotonous ellipticity dependence of the Hx, while the Hz shows the monotonous one. This suggests that the Hz and the Hx are generated from different mechanisms. It is notable that the intensity of the orthogonal laser component is always much smaller than that of the z-component in our calculation. They may have different performance on the generation of the near-threshold harmonics, although can be both regarded as playing a part in the multiphoton mechanism in board terms.

As no unoccupied orbital is introduced in the calculation, the multiphoton mechanism may derive from a multiphoton resonance of the electrons interacting with the laser fields. A trajectorial description without involving the tunnelling process was proposed to give a qualitative interpretation for the generation of the below-threshold harmonics [12

12. H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, “Near-Threshold High-Order Harmonic Spectroscopy with Aligned Molecules,” Phys. Rev. Lett. 105(14), 143904 (2010). [CrossRef] [PubMed]

]. It was suggested that the intermediate resonance of the bound charge could contribute to the generation of these low harmonics. Intuitively speaking, unlike the semi-classical process which dominates the generation of the near-cutoff harmonics, the orthogonal laser field with a low intensity may not prohibit but contribute to Hx. However, if the ellipticity is large and the x-component of the laser field is strong enough to incur an over-excitation, the intensity of Hx may be suppressed.

Our calculation suggests that the resonance depends on the alignment, especially for Hx of the lowest harmonics. It can be seen from Fig. 5 that the absolute intensity of Hx is stronger for the case of perpendicular alignment, where the orthogonal laser component is in fact parallel to the elongating direction of the electron orbital. Thus the resonance amplitude may be enhanced due to a large electron polarization, and consequently the AMs tend to be more dominant. However, for the aligned O2 [12

12. H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, “Near-Threshold High-Order Harmonic Spectroscopy with Aligned Molecules,” Phys. Rev. Lett. 105(14), 143904 (2010). [CrossRef] [PubMed]

], the AMs are more pronounced for the parallel alignment. Besides the unknown detecting efficiency for each polarization component in experiments, the reason may also lies in the fact that the outmost occupied orbital of O2 has a πg symmetry where the near-core electron tends to spread outwards because of the anti-bonding nature.

Ionization potential of the bound state may play a similar role. The electron with a lower ionization potential is more loosely bounded, thus is expected to exhibit stronger AMs. We perform the calculations at different bond lengths of 3.0 and 1.4 a.u., using the same calculating parameters. The calculated ionization potentials are 11.8 and 15.01 eV, respectively. Generally the near-threshold harmonics exhibit similar features as in Figs. 35, and the AMs are also found. The normalized near-threshold harmonics under different alignments are plotted in Fig. 8
Fig. 8 Calculated harmonics under perpendicular (left) and parallel (right) alignments at the bond length of 3.0 a.u. (top) and 1.4 a.u. (bottom). Each harmonic is normalized to its intensity at zero ellipticity.
. For the large bond length of 3.0 a.u., the enhancement of the AM by a factor of ~8 can be found. In contrary, for the bond length of 1.4 a.u., only a weak AM can be found under perpendicular alignment.

4. Conclusion

Ellipticity dependence of the near-threshold harmonics are calculated by means of TDDFT. For the simple molecular system of H2, anomalous maximums can be found, as those observed in experiments of atoms and aligned molecules. Hx is found to dominant the appearance of the AM, as long as its intensity is significant compared with Hz. For these low harmonics, multiphoton effects play a dominate role rather than the three-step process, but may still be different between Hx and Hz. Our calculation suggests that the ellipticity dependence of the low harmonics can be sensitive to both alignment and bond length, especially for Hx. Thus the near-threshold harmonic can offer an effective tool for the study of the multiphoton process as well as the dynamical features of the bound states in strong laser fields. The calculation also has the potential to be applied to more complex systems, due to the multi-electron features of TDDFT. Our calculation indicates that the polarization direction of the NT harmonics can be strongly deviated. If there were conditions that the radiation was highly elliptical, it would be very appealing for future HHG studies since it would provide a way to create elliptical harmonics without losing efficiency. However, we still lack knowledge of their phase relationship. The phase difference of the two components can’t be easily determined from the time-dependent dipole or its frequency spectrum via Fourier transformation in our calculation, which still needs further investigation.

References and links

1.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336(6086), 1287–1291 (2012). [CrossRef] [PubMed]

2.

P. B. Corkum, “Plasma Perspective on Strong Field Multiphoton Ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef] [PubMed]

3.

M. Lewenstein, P. 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(3), 2117–2132 (1994). [CrossRef] [PubMed]

4.

A. Flettner, J. König, M. B. Mason, T. Pfeifer, U. Weichmann, R. Düren, and G. Gerber, “Ellipticity dependence of atomic and molecular high harmonic generation,” Eur. Phys. J. D 21(1), 115–119 (2002). [CrossRef]

5.

T. Kanai, S. Minemoto, and H. Sakai, “Ellipticity Dependence of High-Order Harmonic Generation from Aligned Molecules,” Phys. Rev. Lett. 98(5), 053002 (2007). [CrossRef] [PubMed]

6.

Y. Mairesse, N. Dudovich, J. Levesque, M. Y. Ivanov, P. B. Corkum, and D. M. Villeneuve, “Electron wavepacket control with elliptically polarized laser light in high harmonic generation from aligned molecules,” New J. Phys. 10(2), 025015 (2008). [CrossRef]

7.

P. 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(3), 1725–1745 (1996). [CrossRef] [PubMed]

8.

A. S. Landsman, A. N. Pfeiffer, C. Hofmann, M. Smolarski, C. Cirelli, and U. Keller, “Rydberg state creation by tunnel ionization,” New J. Phys. 15(1), 013001 (2013). [CrossRef]

9.

N. H. Burnett, C. Kan, and P. B. Corkum, “Ellipticity and polarization effects in harmonic generation in ionizing neon,” Phys. Rev. A 51(5), R3418–R3421 (1995). [CrossRef] [PubMed]

10.

M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A 54(1), 742–745 (1996). [CrossRef] [PubMed]

11.

M. Kakehata, H. Takada, H. Yumoto, and K. Miyazaki, “Anomalous ellipticity dependence of high-order harmonic generation,” Phys. Rev. A 55(2), R861–R864 (1997). [CrossRef]

12.

H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, “Near-Threshold High-Order Harmonic Spectroscopy with Aligned Molecules,” Phys. Rev. Lett. 105(14), 143904 (2010). [CrossRef] [PubMed]

13.

A. T. Le, R. R. Lucchese, S. Tonzani, T. Morishita, and C. D. Lin, “Quantitative rescattering theory for high-order harmonic generation from molecules,” Phys. Rev. A 80(1), 013401 (2009). [CrossRef]

14.

V. V. Strelkov, A. A. Gonoskov, I. A. Gonoskov, and M. Y. Ryabikin, “Origin for Ellipticity of High-Order Harmonics Generated in Atomic Gases and the Sublaser-Cycle Evolution of Harmonic Polarization,” Phys. Rev. Lett. 107(4), 043902 (2011). [CrossRef] [PubMed]

15.

V. V. Strelkov, M. A. Khokhlova, A. A. Gonoskov, I. A. Gonoskov, and M. Y. Ryabikin, “High-order harmonic generation by atoms in an elliptically polarized laser field: Harmonic polarization properties and laser threshold ellipticity,” Phys. Rev. A 86(1), 013404 (2012). [CrossRef]

16.

Y. Li, X. S. Zhu, Q. B. Zhang, M. Qin, and P. X. Lu, “Quantum-orbit analysis for yield and ellipticity of high order harmonic generation with elliptically polarized laser field,” Opt. Express 21(4), 4896–4907 (2013). [CrossRef] [PubMed]

17.

P. R. T. Schipper, O. V. Gritsenko, S. J. A. van Gisbergen, and E. J. Baerends, “Molecular calculations of excitation energies and (hyper)polarizabilities with a statistical average of orbital model exchange-correlation potentials,” J. Chem. Phys. 112(3), 1344–1352 (2000). [CrossRef]

18.

J. Heslar, D. Telnov, and S. I. Chu, “High-order-harmonic generation in homonuclear and heteronuclear diatomic molecules: Exploration of multiple orbital contributions,” Phys. Rev. A 83(4), 043414 (2011). [CrossRef]

19.

D. A. Telnov, K. E. Sosnova, E. Rozenbaum, and S. I. Chu, “Exterior complex scaling method in time-dependent density-functional theory: Multiphoton ionization and high-order-harmonic generation of Ar atoms,” Phys. Rev. A 87(5), 053406 (2013). [CrossRef]

20.

X. Chu and G. C. Groenenboom, “Role of resonance-enhanced multiphoton excitation in high-harmonic generation of N2: A time-dependent density-functional-theory study,” Phys. Rev. A 87(1), 013434 (2013). [CrossRef]

21.

X. Chu and G. C. Groenenboom, “Time-dependent density-functional-theory calculation of high-order-harmonic generation of H2,” Phys. Rev. A 85(5), 053402 (2012). [CrossRef]

22.

N. Troullier and J. L. Martins, “Efficient pseudopotentials for plane-wave calculations,” Phys. Rev. B Condens. Matter 43(3), 1993–2006 (1991). [CrossRef] [PubMed]

23.

X. Andrade, J. Alberdi-Rodriguez, D. A. Strubbe, M. J. T. Oliveira, F. Nogueira, A. Castro, J. Muguerza, A. Arruabarrena, S. G. Louie, A. Aspuru-Guzik, A. Rubio, and M. A. L. Marques, “Time-dependent density-functional theory in massively parallel computer architectures: the OCTOPUS project,” J. Phys. Condens. Matter 24(23), 233202 (2012). [CrossRef] [PubMed]

24.

K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, “Calculation of the background emitted during high-harmonic generation,” Phys. Rev. A 45(5), 3347–3349 (1992). [CrossRef] [PubMed]

25.

G. L. Kamta and A. D. Bandrauk, “Three-dimensional time-profile analysis of high-order harmonic generation in molecules: Nuclear interferences in H2+,” Phys. Rev. A 71(5), 053407 (2005). [CrossRef]

26.

W. Bleakney, “The Ionization Potential of Molecular Hydrogen,” Phys. Rev. 40(4), 496–501 (1932). [CrossRef]

27.

D. Neuhasuer and M. Baer, “The time-dependent Schrödinger equation: Application of absorbing boundary conditions,” J. Chem. Phys. 90(8), 4351 (1989). [CrossRef]

28.

F. He, C. Ruiz, and A. Becker, “Absorbing boundaries in numerical solutions of the time-dependent Schrödinger equation on a grid using exterior complex scaling,” Phys. Rev. A 75(5), 053407 (2007). [CrossRef]

29.

M. Y. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt. 52(2-3), 165–184 (2005). [CrossRef]

30.

A. Flettner, J. König, M. B. Mason, T. Pfeifer, U. Weichmann, and G. Gerber, “Atomic and molecular high-harmonic generation: A comparison of ellipticity dependence based on the three-step model,” J. Mod. Opt. 50, 529–537 (2003).

31.

Y. Liang, M. V. Ammosov, and S. L. Chin, “High-order harmonic generation in argon by elliptically polarized picosecond dye laser pulses,” J. Phys. At. Mol. Opt. Phys. 27(6), 1269–1276 (1994). [CrossRef]

32.

K. S. Budil, P. Salières, A. L’Huillier, and M. D. Perry, “Influence of ellipticity on harmonic generation,” Phys. Rev. A 48(5), R3437–R3440 (1993). [CrossRef] [PubMed]

33.

X. M. Tong and S. I. Chu, “Probing the spectral and temporal structures of high-order harmonic generation in intense laser pulses,” Phys. Rev. A 61(2), 021802 (2000). [CrossRef]

34.

P. Antoine, B. Piraux, and A. Maquet, “Time profile of harmonics generated by a single atom in a strong electromagnetic field,” Phys. Rev. A 51(3), R1750–R1753 (1995). [CrossRef] [PubMed]

35.

S. De. Luca and E. Fiordilino, “Wavelet analysis of short harmonics generated in presence of a two colour laser field,” J. Mod. Opt. 45(9), 1775–1783 (1998).

OCIS Codes
(020.4180) Atomic and molecular physics : Multiphoton processes
(020.2649) Atomic and molecular physics : Strong field laser physics

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: October 7, 2013
Revised Manuscript: November 5, 2013
Manuscript Accepted: November 5, 2013
Published: November 14, 2013

Citation
Hua Yang, Peng Liu, Ruxin Li, and Zhizhan Xu, "Ellipticity dependence of the near-threshold harmonics of H2 in an elliptical strong laser field," Opt. Express 21, 28676-28684 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28676


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References

  1. T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science336(6086), 1287–1291 (2012). [CrossRef] [PubMed]
  2. P. B. Corkum, “Plasma Perspective on Strong Field Multiphoton Ionization,” Phys. Rev. Lett.71(13), 1994–1997 (1993). [CrossRef] [PubMed]
  3. M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A49(3), 2117–2132 (1994). [CrossRef] [PubMed]
  4. A. Flettner, J. König, M. B. Mason, T. Pfeifer, U. Weichmann, R. Düren, and G. Gerber, “Ellipticity dependence of atomic and molecular high harmonic generation,” Eur. Phys. J. D21(1), 115–119 (2002). [CrossRef]
  5. T. Kanai, S. Minemoto, and H. Sakai, “Ellipticity Dependence of High-Order Harmonic Generation from Aligned Molecules,” Phys. Rev. Lett.98(5), 053002 (2007). [CrossRef] [PubMed]
  6. Y. Mairesse, N. Dudovich, J. Levesque, M. Y. Ivanov, P. B. Corkum, and D. M. Villeneuve, “Electron wavepacket control with elliptically polarized laser light in high harmonic generation from aligned molecules,” New J. Phys.10(2), 025015 (2008). [CrossRef]
  7. P. 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(3), 1725–1745 (1996). [CrossRef] [PubMed]
  8. A. S. Landsman, A. N. Pfeiffer, C. Hofmann, M. Smolarski, C. Cirelli, and U. Keller, “Rydberg state creation by tunnel ionization,” New J. Phys.15(1), 013001 (2013). [CrossRef]
  9. N. H. Burnett, C. Kan, and P. B. Corkum, “Ellipticity and polarization effects in harmonic generation in ionizing neon,” Phys. Rev. A51(5), R3418–R3421 (1995). [CrossRef] [PubMed]
  10. M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A54(1), 742–745 (1996). [CrossRef] [PubMed]
  11. M. Kakehata, H. Takada, H. Yumoto, and K. Miyazaki, “Anomalous ellipticity dependence of high-order harmonic generation,” Phys. Rev. A55(2), R861–R864 (1997). [CrossRef]
  12. H. Soifer, P. Botheron, D. Shafir, A. Diner, O. Raz, B. D. Bruner, Y. Mairesse, B. Pons, and N. Dudovich, “Near-Threshold High-Order Harmonic Spectroscopy with Aligned Molecules,” Phys. Rev. Lett.105(14), 143904 (2010). [CrossRef] [PubMed]
  13. A. T. Le, R. R. Lucchese, S. Tonzani, T. Morishita, and C. D. Lin, “Quantitative rescattering theory for high-order harmonic generation from molecules,” Phys. Rev. A80(1), 013401 (2009). [CrossRef]
  14. V. V. Strelkov, A. A. Gonoskov, I. A. Gonoskov, and M. Y. Ryabikin, “Origin for Ellipticity of High-Order Harmonics Generated in Atomic Gases and the Sublaser-Cycle Evolution of Harmonic Polarization,” Phys. Rev. Lett.107(4), 043902 (2011). [CrossRef] [PubMed]
  15. V. V. Strelkov, M. A. Khokhlova, A. A. Gonoskov, I. A. Gonoskov, and M. Y. Ryabikin, “High-order harmonic generation by atoms in an elliptically polarized laser field: Harmonic polarization properties and laser threshold ellipticity,” Phys. Rev. A86(1), 013404 (2012). [CrossRef]
  16. Y. Li, X. S. Zhu, Q. B. Zhang, M. Qin, and P. X. Lu, “Quantum-orbit analysis for yield and ellipticity of high order harmonic generation with elliptically polarized laser field,” Opt. Express21(4), 4896–4907 (2013). [CrossRef] [PubMed]
  17. P. R. T. Schipper, O. V. Gritsenko, S. J. A. van Gisbergen, and E. J. Baerends, “Molecular calculations of excitation energies and (hyper)polarizabilities with a statistical average of orbital model exchange-correlation potentials,” J. Chem. Phys.112(3), 1344–1352 (2000). [CrossRef]
  18. J. Heslar, D. Telnov, and S. I. Chu, “High-order-harmonic generation in homonuclear and heteronuclear diatomic molecules: Exploration of multiple orbital contributions,” Phys. Rev. A83(4), 043414 (2011). [CrossRef]
  19. D. A. Telnov, K. E. Sosnova, E. Rozenbaum, and S. I. Chu, “Exterior complex scaling method in time-dependent density-functional theory: Multiphoton ionization and high-order-harmonic generation of Ar atoms,” Phys. Rev. A87(5), 053406 (2013). [CrossRef]
  20. X. Chu and G. C. Groenenboom, “Role of resonance-enhanced multiphoton excitation in high-harmonic generation of N2: A time-dependent density-functional-theory study,” Phys. Rev. A87(1), 013434 (2013). [CrossRef]
  21. X. Chu and G. C. Groenenboom, “Time-dependent density-functional-theory calculation of high-order-harmonic generation of H2,” Phys. Rev. A85(5), 053402 (2012). [CrossRef]
  22. N. Troullier and J. L. Martins, “Efficient pseudopotentials for plane-wave calculations,” Phys. Rev. B Condens. Matter43(3), 1993–2006 (1991). [CrossRef] [PubMed]
  23. X. Andrade, J. Alberdi-Rodriguez, D. A. Strubbe, M. J. T. Oliveira, F. Nogueira, A. Castro, J. Muguerza, A. Arruabarrena, S. G. Louie, A. Aspuru-Guzik, A. Rubio, and M. A. L. Marques, “Time-dependent density-functional theory in massively parallel computer architectures: the OCTOPUS project,” J. Phys. Condens. Matter24(23), 233202 (2012). [CrossRef] [PubMed]
  24. K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, “Calculation of the background emitted during high-harmonic generation,” Phys. Rev. A45(5), 3347–3349 (1992). [CrossRef] [PubMed]
  25. G. L. Kamta and A. D. Bandrauk, “Three-dimensional time-profile analysis of high-order harmonic generation in molecules: Nuclear interferences in H2+,” Phys. Rev. A71(5), 053407 (2005). [CrossRef]
  26. W. Bleakney, “The Ionization Potential of Molecular Hydrogen,” Phys. Rev.40(4), 496–501 (1932). [CrossRef]
  27. D. Neuhasuer and M. Baer, “The time-dependent Schrödinger equation: Application of absorbing boundary conditions,” J. Chem. Phys.90(8), 4351 (1989). [CrossRef]
  28. F. He, C. Ruiz, and A. Becker, “Absorbing boundaries in numerical solutions of the time-dependent Schrödinger equation on a grid using exterior complex scaling,” Phys. Rev. A75(5), 053407 (2007). [CrossRef]
  29. M. Y. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt.52(2-3), 165–184 (2005). [CrossRef]
  30. A. Flettner, J. König, M. B. Mason, T. Pfeifer, U. Weichmann, and G. Gerber, “Atomic and molecular high-harmonic generation: A comparison of ellipticity dependence based on the three-step model,” J. Mod. Opt.50, 529–537 (2003).
  31. Y. Liang, M. V. Ammosov, and S. L. Chin, “High-order harmonic generation in argon by elliptically polarized picosecond dye laser pulses,” J. Phys. At. Mol. Opt. Phys.27(6), 1269–1276 (1994). [CrossRef]
  32. K. S. Budil, P. Salières, A. L’Huillier, and M. D. Perry, “Influence of ellipticity on harmonic generation,” Phys. Rev. A48(5), R3437–R3440 (1993). [CrossRef] [PubMed]
  33. X. M. Tong and S. I. Chu, “Probing the spectral and temporal structures of high-order harmonic generation in intense laser pulses,” Phys. Rev. A61(2), 021802 (2000). [CrossRef]
  34. P. Antoine, B. Piraux, and A. Maquet, “Time profile of harmonics generated by a single atom in a strong electromagnetic field,” Phys. Rev. A51(3), R1750–R1753 (1995). [CrossRef] [PubMed]
  35. S. De. Luca and E. Fiordilino, “Wavelet analysis of short harmonics generated in presence of a two colour laser field,” J. Mod. Opt.45(9), 1775–1783 (1998).

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