Abstract

By solving the time-dependent Schrödinger equation both in simplified one-dimensional coordinate and three-dimensional cylindrical coordinate systems, the high-order harmonic generation from H2+ in spatially symmetric and asymmetric nonhomogeneous laser fields was studied. At large internuclear distances, minima were clearly observed in high energy part of harmonic spectra, which can be attributed to two-center interference in diatomic molecule. Compared with previous studies, the minima in nonhomogeneous laser field are more distinct. Remarkably, the positions of the minima are different in these two types of fields, which demonstrate that interference effects are greatly influenced by laser parameters. Besides, the asymmetric nonhomogeneous field leads to an asymmetric recollision of the ionized electron, and both odd and even order harmonics could be emitted, which is explained in detail based on quantum dynamics calculations.

© 2016 Optical Society of America

1. Introduction

As a promising source of coherent light in the extreme ultraviolet, soft X-ray regions, and isolated attosecond pulses, high-order harmonic generation (HHG) has attracted much attention in recent decades [1–4]. It has been applied to image the structure of an atom or a molecule, and to probe multi-electron dynamics [5,6]. The generation of high-order harmonics can be understood by a semiclassical three-step model [7]: the electron’s tunneling ionization, its acceleration in the laser electric field, and its recombination with the ionic core. It is well known that the emitted HHG spectrum generally exhibits a plateau with a cutoff at the energy 3.17Up + Ip, where Up is the ponderomotive potential, Ip is the ionization potential of the target. For the use of HHG, there are two important aspects should be considered: the cutoff energy and conversion efficiency [8].

Recently, Kim et al. [9] experimentally showed that due to surface plasmon resonance, the laser intensity can be enhanced by several orders which can be used to increase the cutoff energy of harmonic spectrum. Later, many theoretical researches were concentrated on the HHG from atoms in spatially nonhomogeneous laser field [10–22]. Pérez-Hernández et al. successfully extended the cutoff far beyond the usual semiclassical limit, and their scheme has been proven capable of generating coherent ultraviolet photons beyond the carbon K edge [10]. Moreover, Yavuz et al. [13] obtained a 130 as pulse by employing a single four-optical-cycle plasmon-enhanced field. Depending on the shape of the metal nanostructure, the nonhomogeneous laser field can be divided into spatially symmetric nonhomogeneous laser field (SNLF) and asymmetric nonhomogeneous laser field (ANLF), as presented in Fig. 1. The spatial symmetry of the nonhomogeneous field significantly influences the movement of electronic wave packets, and subsequently affects harmonic emission. For example, in the ANLF, both odd and even harmonics will be emitted, which is different from the case in SNLF [22].

 figure: Fig. 1

Fig. 1 The nonhomogeneous fields for the input laser intensify of 1.0 × 1014 W/cm2 and β = 0.004. (a) Asymmetric case (ANLF), and (b) symmetric case (SNLF).

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As far as we know, most previous researches focused on the interaction of atom with the spatially nonhomogeneous laser fields. For more complex systems, such as polyatomic molecules and crystals [23,24], there exist much more nonlinear effects. In this work, we pay attention to HHG and electronic dynamics of the prototype molecular system, H2+, in spatially nonhomogeneous laser fields. The interaction of molecules with intense laser pulses is much more complex because of the multi-center effect and nuclear motion. The effect of charge resonance and new cutoff laws have been demonstrated [25–32]. Lein proposed theoretically that the harmonic conversion efficiency in a molecule is approximately proportional to the squared modulus of the nuclear autocorrelation function [28]. By performing full-dimensional non-Born-Oppenheimer calculations, Vafaee and associates [29] have well explained the isotopic effects on HHG of H2+ and D2+ when nuclear motion is taken into account. To our knowledge, Yavuz et al. [30] investigated the HHG from H2+ near plasmon-enhanced laser fields for the first time, and they found that the efficiency of the harmonics is enhanced relative to that of static nuclei when nuclear vibrations are enabled. Afterward, they utilized the plasmon-enhanced inhomogeneous fields to further control electron localization in H2+ [31].

Another important phenomenon is the “minimum” in the harmonic spectrum which results from two-center interference. In theory, Lein et al. [32,33] first predicted the minimum in harmonic spectrum from H2+: Destructive interference occurs when the de Broglie wavelength (λ) of the returning electron, the internuclear distance (R), and alignment angle (θ) satisfy the relationship

Rcosθ=(2m+1)λ/2,m=0,1,2,....

From Eq. (1), we can see that the location of the minimum reflects the internuclear distance. For an isolated molecule, to efficiently measure its full dimensional structure is not easy, especially for complex molecules. The ultimate goal scientists are struggling for is to read the structure of molecules in real time. Real-time reading of the molecular structure is the most direct method to study the photochemical reactions and molecular dynamics. Although electron diffraction and ultrafast X-ray diffraction are the most used methods for tracing the structure of molecules, the interference minimum in harmonic spectrum mentioned here is a potential method. The use of the recollision of ionized electrons as a probe has been proved to be an effective scheme to measure the quantum mechanical nuclear dynamics [34–36].

For the potential value of the minimum, the underlying mechanism has attracted a lot of attention. For example, Kamta and Bandrauk [37] indicated that the interferences of the harmonic spectra from H2+ are shown to be maximum at certain harmonic orders as a function of molecular orientation, while Rost et al. [38] claim that for N2, the interference presents a different pattern because the valence orbital has admixture of both atomic s and p orbitals. For heteronuclear diatomic molecules CO, the location of the minimum is shifted to lower harmonic orders compared with that in a homonuclear case, such as H2, N2, O2 [39], which is attributed to additional phase shifts [38]. By calculating HHG including orbital distortion for N2, Madsen et al. [40] explained why the minima were not observed experimentally in other works [41,42]. Also, the interference between excited state and ground state of H2+ will affect the location of the minimum [43]. Recently, Hu et al. [44] investigated the dependence of the spectral minima positions of H2+ on the carrier envelope phase. In brief, the previous discussions were mainly toward the molecules with small internuclear distance, and detailed investigations for the case at large internuclear distance are still lacking. To our knowledge, Chen’s work [45] is the only study on the minimum of HHG from molecule with large internuclear distance. They revealed that the interference between different recombination electron trajectories plays an important role in the minimum location of HHG from H2+, however, the illustrated minima are not so obvious. As we expect, the clearer the position of the minimum, the better for us to obtain the internuclear distance information.

In this paper, we theoretically investigate the HHG from the interaction of H2+ with nonhomogeneous laser field by solving one-dimensional (1D) and three-dimensional (3D) time-dependent Schrödinger equation (TDSE). Under the condition of nonhomogeneous laser field, a series of clear “minima” in high energy region of the harmonic spectra can be observed. The harmonic spectra between two spatial nonhomogeneous fields (SNLF and ANLF) are also compared. We find that the location of the minimum in the case of SNLF is slightly higher in energy than the case of ANLF. We demonstrate that the external field will have an observable influence on the interference minimum. Besides, odd and even harmonics are observed in ANLF, but not in SNLF.

The paper is organized as follows. We will briefly introduce the theoretical model and numerical method in section 2. The results and discussion are presented in section 3. The conclusion of our paper is in section 4.

2. Theoretical methods

All the calculations have been performed using the attosecond resolution quantum dynamics program LZH-DICP [46,47]. In the following, atomic units are used throughout unless otherwise stated. In the dipole approximation, the 1D TDSE for H2+ is given by

itψ(z;t)=[122z2+V(z)+zE(z;t)]ψ(z;t),
where V(z)=1/R1/(zR/2)2+11/(z+R/2)2+1, R is internuclear distance, z is the electronic coordinate. The 3D TDSE for H2+ in cylindrical coordinates can be described as [48–50]:
itψ(z,ρ;t)=[TR+Tz+Tρ+Vc(R,z,ρ)+kzE(z;t)]ψ(z,ρ;t),
whereV(R,z,ρ)=1/R1/(zR/2)2+ρ21/(z+R/2)2+ρ2,TR=1mp2R2,Tz=2mp+me4mpme2z2,Tρ=2mp+me4mpme(2ρ2+1ρρ),k=1+me/(2mp+me),me and mp are the mass of electron and proton. The harmonic spectrum is calculated by Fourier transforming the time-dependent acceleration which is obtained from Ehrenfest’s theorem [51]. The TDSE is solved by standard second-order split-operator approach. We use a 10-cycle trapezoidal laser field with a linear turn-on and turn-off of 3 cycles as input laser. Due to the spatial dependence, the laser field can be expressed as E(z;t)=E0f(t)[(1+βz)βR2z]cos(ωt)(for ANLF) or E(z;t)=E0f(t)[1+β|z|]cos(ωt)(for SNLF), where E0 is the peak amplitude, f(t) is pulse envelope, β is a small parameter that characterizes the inhomogeneity region and ω is the frequency of the electromagnetic radiation [22,52,53].

The 11-point finite-difference method and the sine discrete variable representation were respectively applied in ρ direction (0~30 a.u. with 75 adaptive grids) and the z direction (−235 a.u.< z <235 a.u. with 2350 grids). The dipole acceleration is

a(t)=ψ(z,ρ;t)|V(R,z,ρ)z+E(t)|ψ(z,ρ;t),
then, the harmonic spectra are calculated by Fourier transforming the time-dependent dipole acceleration a(t):

P(ω)=|12π0Ta(t)eiwtdt|2.

To better investigate the temporal structures of HHG, we also perform time-frequency analyses by using the wavelet transformation of the dipole acceleration [54,55],

A(t,ω)=a(t)ωW(ω(tt))dt,
where W(ω(tt)) is the mother wavelet with the formula W(x)=(1τ)eixex2/2τ2, and τ = 50 in our calculations.

3. Results and discussions

In our calculations, the polarization of the laser is set to be parallel to the molecular axis of H2+. Also, the molecular axis is in line with nanostructures’ main axis. The laser intensity, λ and β are chosen as 1.0 × 1014 W/cm2, 1064 nm and 0.004, respectively. Here, the quoted intensity is the plasmonic-enhanced value, not the input laser intensity that could be much smaller. For simplicity, the internuclear distance was fixed during the propagation of the electronic wave packet. Typical 3D numerical results are shown in Fig. 2, where harmonic spectra for H2+ are plotted under the condition of ANLF [Fig. 2(a)] and SNLF [Fig. 2(b)]. We focus on the large internuclear distances because the harmonic minima at small internuclear distances close to the equilibirum bond length have been previously addressed. With the advances in modern laser technologies to prepare vibrationally excited state, the large internuclear distances up to 20 a.u. are accessible. The interesting feature of the spectra is the pronounced minimum indicated by arrows. For comparison, we also show the harmonic spectra of homogeneous (β = 0) case for R = 12 a.u. in Fig. 2. In the homogeneous field, we can hardly recognize the minimum structure from electronic interferences. To gain more clear estimation of the minimum position, convolution with a Gaussian of appropriate width was used to smooth the spectra,

Psmooth(ω)=P(ω')exp((ω'ω)2/σ2)dω'.
As presented in Figs. 2(a) and 2(b), the smoothed spectra in solid lines were obtained with σ = 5ωL, here ωL is the frequency of the driving laser.

 figure: Fig. 2

Fig. 2 Harmonic spectra from 3D calculations for H2+ in (a) ANLF and (b) SNLF, where the solid lines give the smoothed spectra, and the yellow and dark green arrows indicate the location of the minima for m = 5 and 6 respectively according to Eq. (1). (c) Harmonic spectra from |G1(ω)|2+|G2(ω)|2 of H2+ without interferences at R = 13 a.u. (red line for SNLF, black line for ANLF). (d) The time-frequency distribution for HHG in SNLF at R = 13 a.u., where the white dashed line corresponds to the minima. The harmonic spectra for R = 12 a.u. in (a), (b), and the harmonic spectra for ANLF in (c) are the raw data, whereas the harmonic spectra for R = 13, 14, 15, and 16 a.u. in (a), (b), and the harmonic spectra for SNLF in (c) are shifted up vertically for clarity.

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The minimums could result from two-center interference or from absorbing of the ionized wavepacket by the boundary (plasmonic nanostructures). In our simulations, the box we used is large enough for the moving of ionized wavepacket under the condition of present laser parameters. Similar to previous work [37], the spectrum can be approximately written as

P(ω)=|G1(ω)|2+|G2(ω)|2+2Re[G1(ω)G2*(ω)],
where Gj(ω)=eiωtψ(t)|Vj(r)|ψ(t)dt, with j = 1, 2. |Gj(ω)|2 can be interpreted as the harmonic spectrum originating from the nucleus j, in the presence of the other nucleus. Figure 2(c) shows the harmonic spectra |G1(ω)|2+|G2(ω)|2 of H2+ without interferences for the internuclear distance R = 13 (red line for SNLF, black line for ANLF). It is clear that the minimum originates from the last interference term in Eq. (7). To get more insightful information, we use a Gabor analysis [56,57] that provides the time profiles of the harmonic spectra. The time-frequency distribution for HHG of H2+ at R = 13 a.u. in SNLF is presented in Fig. 2(d). There is an obvious minimum in the time-frequency distribution indicated by white dashed line. By contrast, the spectral minima in other work are concealed to some extent for large internuclear distance [45] due to the interference of the long and short trajectories. What we should notice is that the minimum we concentrate on is in the high energy region of the HHG spectrum. As studied previously [58], in nonhomogeneous laser field short trajectories dominate in high-energy spectrum and no clear harmonic cutoff is visible. That is why only the minima located in high energy region could be observed clearly. So, there is enough evidence in present calculations that the two-center interference instead of the normal trajectory interference contributes to the minima in harmonic spectra.

To further address more characteristics of the two-center interference minima in nonhomogeneous laser fields, we plot a collection of data in Fig. 3 regarding the relation between the internuclear distance and the location of the minimums in HHG spectrums. According to Eq. (1), for a given m, the larger is the internuclear distance, the lower is the kinetic energy of the recolliding electron, which leads to a lower position of the minimum in the HHG spectrum. The data in Fig. 3 match Eq. (1) qualitatively but not quantitatively. There exist many factors that affect the interference minimum. Wu et al. [59] and Chen et al. [45] pointed out that the intensity of the laser could influence the two-center interference by changing the recombination process and ionization process. The effects of the difference of additional phase shifts [38], excited states [43] and Coulomb continuum wave functions [60] on the interference minima have also been studied. Therefore, the interference pattern cannot be completely described by Eq. (1). Here, we pay attention to another feature observed in Fig. 3, that is, the location of the minima under the condition of SNLF are always higher than those in ANLF. We can see that the difference could be distinguished clearly. That is to say, different space shape of the laser field could influence the molecular orbitals, and then leads to the different interference minimums. Madsen group [40] once investigated the influence of field-induced orbital distortion on HHG by an extended strong-field approximation theory. In this article, we demonstrate again that the external field will have an observable influence on HHG from hydrogen molecular ion by solving the TDSE.

 figure: Fig. 3

Fig. 3 Relationship between the location of interference minimnum and internuclear distance.

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The spatial symmetry of the nonhomogeneous field influences the movement of electronic wave packets, and subsequently affects harmonic emission. Time-dependent electronic density which is defined as p(z;t)=|ψ(z,ρ;t)|2ρdρ are provided in Fig. 4 for R = 15 a.u. in the ANLF and SNLF. Under the condition of ANLF, most of the wave packet ionized along the negative direction can’t be drived back which can be proved by comparing Fig. 4(a) with Fig. 4(b). In order to check the phenomenon of asymmetric recollision directly, “virtual detector” [46] was used to detect the electronic probability flux for recolliding to the nuclei along different direction. The probability flux can be defined as

F(t)=Im[ψ*δ(zz0)zψ]ρdρ,
where z0 is the position of flux analysis. Figures 4(c) and 4(d) show the results with z0 = ± 15 a.u. in ANLF and SNLF respectively. In the case of ANLF, the wave packet ionized along the negative direction will hardly come back.

 figure: Fig. 4

Fig. 4 Time-dependent electronic density for R = 15 a.u. in (a) ANLF and (b) SNLF. Recollision flux of the electronic wavepacket in the case of (c) ANLF and (d) SNLF.

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In [50] and [57], odd and even harmonics were observed for atom system in ANLF, and this characteristic was interpreted as the break of the symmetry of the total potential. Here, we also get the characteristic as presented in Fig. 5(b), and we attribute this phenomenon to the asymmetric recollision. From strong field approximation theory, under the condition of symmetrical potential and laser field, for each trajectory x1(t) starting at the moment t0 during the first quarter cycle and returning at some moment tf, there exists a mirror symmetric trajectory x2(t) which starts at t0′ = t0 + π/ω and returns at tf′ = tf + π/ω: x1(t) = –x2(t + π/ω) [61]. The contribution from these two trajectories to the field-induced dipole moment has a form

ΔNexp[iΘ(tf,t0)]exp[iΘ(tf',t0')],
where Θ(t,t0)=12t0t[A(t0)A(t")]2dt"+Ip(tt0)Nωt, Ip is ionization potential, A(t) is the field vector potential, and N indicates the harmonic order. For symmetrical potential and laser field, Θ(tf,t0)=Θ(tf',t0')Nπ, as a result, ΔN1exp(iNπ). So in the case of SNLF, the harmonic spectra will have constructive interference for odd N, but destructive for even N. Whereas in the case of ANLF, for the asymmetric recollision, the trajectory x2(t) would disappear or almost be ignored which can be found by comparing Fig. 5(a) and Fig. 2(d). Therefore, even harmonic would not be weakened coherently which leads to both odd and even harmonics in the spectra.

 figure: Fig. 5

Fig. 5 (a) The time-frequency distribution for HHG in ANLF at R = 13 a.u.. (b) Harmonic spectra of H2+ at R = 13 a.u. in ANLF and SNLF.

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4. Conclusion

In summary, a detailed quantum dynamics study of HHG from H2+ in both symmetric and asymmetric nonhomogeneous laser fields is presented. Due to that in nonhomogeneous laser field only short trajectory dominates in the high energy region, the two-nuclear-center interference leads to the minimum in HHG by minimizing the contribution from short-long trajectory interference, therefore, the minima in harmonic spectra are more distinctly observed in high energy region compared to previous work. We note that the location of the interference minimum could be influenced by the laser field profile. In spatially symmetric nonhomogeneous field, the energy of the minimum is slightly larger than the case in asymmetric nonhomogeneous field. Moreover, in asymmetric nonhomogeneous field, both odd and even order harmonic could be emitted because of constructive interference, which is explained in more detail based on asymmetric recollision. For more potential applications, asymmetric recollision by the plasmon-enhanced asymmetric nonhomogeneous field can also localize the charge on the desired site of the biomacromolecule, and then predetermine the molecular reactivity [62]. Thus, this work will be of broad interest in strong field physics and related research field.

Funding

National Natural Science Foundation of China (NSFC) (21373113, 61275103), the Fundamental Research Funds for the Central Universities (30920140111008, 30916011105).

Acknowledgments

SC Jiang gratefully acknowledges the support of Scientific Research Innovation Projects of Jiangsu Province for University Graduate Students with Grant No. KYLX15_0407.

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42. Y. Mairesse, J. Levesque, N. Dudovich, P. B. Corkum, and D. M. Villeneuve, “High harmonic generation from aligned molecules–amplitude and polarization,” J. Mod. Opt. 55(16), 2591–2602 (2008). [CrossRef]  

43. Y. C. Han and L. B. Madsen, “Minimum in the high-order harmonic generation spectrum from molecules: role of excited states,” J. Phys. At. Mol. Opt. Phys. 43(10), 225601 (2010). [CrossRef]  

44. P. Hu, Y. Niu, Y. Xiang, S. Gong, and C. Liu, “Carrier-envelope phase dependence of molecular harmonic spectral minima induced by mid-infrared laser pulses,” Opt. Express 23(18), 23834–23844 (2015). [CrossRef]   [PubMed]  

45. Y. J. Chen and J. Liu, “High-order harmonic generation from diatomic molecules with large internuclear distance: The effect of two-center interference,” Phys. Rev. A 77(1), 013410 (2008). [CrossRef]  

46. R. F. Lu, P. Y. Zhang, and K. L. Han, “Attosecond-resolution quantum dynamics calculations for atoms and molecules in strong laser fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(6), 066701 (2008). [CrossRef]   [PubMed]  

47. J. Hu, K. L. Han, and G. Z. He, “Correlation quantum dynamics between an electron and D2+ molecule with attosecond resolution,” Phys. Rev. Lett. 95(12), 123001 (2005). [CrossRef]   [PubMed]  

48. V. Roudnev, B. D. Esry, and I. Ben-Itzhak, “Controlling HD+ and H2+ dissociation with the carrier-envelope phase difference of an intense ultrashort laser pulse,” Phys. Rev. Lett. 93(16), 163601 (2004). [CrossRef]   [PubMed]  

49. H. X. He, R. F. Lu, P. Y. Zhang, Y. H. Guo, K. L. Han, and G. Z. He, “Theoretical investigation of the origin of the multipeak structure of kinetic-energy-release spectra from charge-resonance-enhanced ionization of H2+ in intense laser fields,” Phys. Rev. A 84(3), 033418 (2011). [CrossRef]  

50. H. X. He, R. F. Lu, P. Y. Zhang, K. L. Han, and G. Z. He, “Direct multi-photon ionizations of H2+ in intense laser fields,” J. Phys. At. Mol. Opt. Phys. 45(12), 085103 (2012). [CrossRef]  

51. 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]  

52. J. R. Hiskes, “Dissociation of molecular ions by electric and magnetic fields,” Phys. Rev. 122(4), 1207–1217 (1961). [CrossRef]  

53. M. Yamaguchi and K. Nobusada, “Photodissociation path in H2+ induced by nonuniform optical near fields: Two-step excitation via vibrationally excited states,” Phys. Rev. A 93(2), 023416 (2016). [CrossRef]  

54. 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(R) (2000).

55. X. Chu and S. I. Chu, “Optimization of high-order harmonic generation by genetic algorithm and wavelet time-frequency analysis of quantum dipole emission,” Phys. Rev. A 64(2), 021403(R) (2001).

56. 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]  

57. L. V. Vela-Arevalo and J. E. Marsden, “Time-frequency analysis of the restricted three-body problem: transport and resonance transitions,” Class. Quantum Gravity 21(3), S351–S375 (2004). [CrossRef]  

58. M. F. Ciappina, J. Biegert, R. Quidant, and M. Lewenstein, “High-order-harmonic generation from inhomogeneous fields,” Phys. Rev. A 85(3), 033828 (2012). [CrossRef]  

59. Y. Wu, J. T. Zhang, H. L. Ye, and Z. Z. Xu, “Intensity-dependent interference effect in high-order harmonic generation from aligned H2+ molecules,” Phys. Rev. A 83(2), 023417 (2011). [CrossRef]  

60. M. F. Ciappina, C. C. Chirilă, and M. Lein, “Influence of Coulomb continuum wave functions in the description of high-order harmonic generation with H2+,” Phys. Rev. A 75(4), 043405 (2007). [CrossRef]  

61. M. Ivanov, P. B. Corkum, T. Zuo, and A. Bandrauk, “Routes to control of intense-field atomic polarizability,” Phys. Rev. Lett. 74(15), 2933–2936 (1995). [CrossRef]   [PubMed]  

62. N. V. Golubev and A. I. Kuleff, “Control of charge migration in molecules by ultrashort laser pulses,” Phys. Rev. A 91(5), 051401(R) (2015).

References

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    [Crossref] [PubMed]
  47. J. Hu, K. L. Han, and G. Z. He, “Correlation quantum dynamics between an electron and D2+ molecule with attosecond resolution,” Phys. Rev. Lett. 95(12), 123001 (2005).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  50. H. X. He, R. F. Lu, P. Y. Zhang, K. L. Han, and G. Z. He, “Direct multi-photon ionizations of H2+ in intense laser fields,” J. Phys. At. Mol. Opt. Phys. 45(12), 085103 (2012).
    [Crossref]
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    [Crossref] [PubMed]
  52. J. R. Hiskes, “Dissociation of molecular ions by electric and magnetic fields,” Phys. Rev. 122(4), 1207–1217 (1961).
    [Crossref]
  53. M. Yamaguchi and K. Nobusada, “Photodissociation path in H2+ induced by nonuniform optical near fields: Two-step excitation via vibrationally excited states,” Phys. Rev. A 93(2), 023416 (2016).
    [Crossref]
  54. 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(R) (2000).
  55. X. Chu and S. I. Chu, “Optimization of high-order harmonic generation by genetic algorithm and wavelet time-frequency analysis of quantum dipole emission,” Phys. Rev. A 64(2), 021403(R) (2001).
  56. 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]
  57. L. V. Vela-Arevalo and J. E. Marsden, “Time-frequency analysis of the restricted three-body problem: transport and resonance transitions,” Class. Quantum Gravity 21(3), S351–S375 (2004).
    [Crossref]
  58. M. F. Ciappina, J. Biegert, R. Quidant, and M. Lewenstein, “High-order-harmonic generation from inhomogeneous fields,” Phys. Rev. A 85(3), 033828 (2012).
    [Crossref]
  59. Y. Wu, J. T. Zhang, H. L. Ye, and Z. Z. Xu, “Intensity-dependent interference effect in high-order harmonic generation from aligned H2+ molecules,” Phys. Rev. A 83(2), 023417 (2011).
    [Crossref]
  60. M. F. Ciappina, C. C. Chirilă, and M. Lein, “Influence of Coulomb continuum wave functions in the description of high-order harmonic generation with H2+,” Phys. Rev. A 75(4), 043405 (2007).
    [Crossref]
  61. M. Ivanov, P. B. Corkum, T. Zuo, and A. Bandrauk, “Routes to control of intense-field atomic polarizability,” Phys. Rev. Lett. 74(15), 2933–2936 (1995).
    [Crossref] [PubMed]
  62. N. V. Golubev and A. I. Kuleff, “Control of charge migration in molecules by ultrashort laser pulses,” Phys. Rev. A 91(5), 051401(R) (2015).

2016 (3)

H. Y. Zhong, J. Guo, W. Feng, P. C. Li, and X. S. Liu, “Comparison of high harmonic generation and attosecond pulse from 3D hydrogen atom in three kinds of inhomogeneous fields,” Phys. Lett. A 380(1–2), 188–193 (2016).
[Crossref]

I. Yavuz, M. F. Ciappina, A. Chacón, Z. Altun, M. F. Kling, and M. Lewenstein, “Controlling electron localization in H2+ by intense plasmon-enhanced laser fields,” Phys. Rev. A 93(3), 033404 (2016).
[Crossref]

M. Yamaguchi and K. Nobusada, “Photodissociation path in H2+ induced by nonuniform optical near fields: Two-step excitation via vibrationally excited states,” Phys. Rev. A 93(2), 023416 (2016).
[Crossref]

2015 (7)

P. Hu, Y. Niu, Y. Xiang, S. Gong, and C. Liu, “Carrier-envelope phase dependence of molecular harmonic spectral minima induced by mid-infrared laser pulses,” Opt. Express 23(18), 23834–23844 (2015).
[Crossref] [PubMed]

N. V. Golubev and A. I. Kuleff, “Control of charge migration in molecules by ultrashort laser pulses,” Phys. Rev. A 91(5), 051401(R) (2015).

I. Yavuz, Y. Tikman, and Z. Altun, “High-order-harmonic generation from H2+ molecular ions near plasmon-enhanced laser fields,” Phys. Rev. A 92(2), 023413 (2015).
[Crossref]

X. L. Ge, H. Du, Q. Wang, J. Guo, and X. S. Liu, “Selection of quantum path in high-order harmonics and isolated sub-100 attosecond generation in few-cycle spatially inhomogeneous laser fields,” Chin. Phys. B 24(2), 023201 (2015).
[Crossref]

L. Q. Feng, “Molecular harmonic extension and enhancement from H2+ ions in the presence of spatially inhomogeneous fields,” Phys. Rev. A 92(5), 053832 (2015).
[Crossref]

C. Yu, H. Wei, X. Wang, A. T. Le, R. Lu, and C. D. Lin, “Reconstruction of two-dimensional molecular structure with laser-induced electron diffraction from laser-aligned polyatomic molecules,” Sci. Rep. 5, 15753 (2015).
[Crossref] [PubMed]

Y. Chou, P. C. Li, T. S. Ho, and S. I. Chu, “Generation of an isolated few-attosecond pulse in optimized inhomogeneous two-color fields,” Phys. Rev. A 92(2), 023423 (2015).
[Crossref]

2014 (5)

X. Cao, S. Jiang, C. Yu, Y. Wang, L. Bai, and R. Lu, “Generation of isolated sub-10-attosecond pulses in spatially inhomogenous two-color fields,” Opt. Express 22(21), 26153–26161 (2014).
[Crossref] [PubMed]

Q. Liao and U. Thumm, “Attosecond time-resolved photoelectron dispersion and photoemission time delays,” Phys. Rev. Lett. 112(2), 023602 (2014).
[Crossref] [PubMed]

C. Yu, Y. H. Wang, X. Cao, S. C. Jiang, and R. F. Lu, “Isolated few-attosecond emission in a multicycle asymmetrically nonhomogeneous twocolor laser field,” J. Phys. At. Mol. Opt. Phys. 47(22), 225602 (2014).
[Crossref]

H. Ahmadi, A. Maghari, H. Sabzyan, A. R. Niknam, and M. Vafaee, “Effect of nuclear motion on high-order-harmonic generation of H2+ in intense ultrashort laser pulses,” Phys. Rev. A 90(4), 043411 (2014).
[Crossref]

M. D. Śpiewanowski, A. Etches, and L. B. Madsen, “High-order-harmonic generation from field-distorted orbitals,” Phys. Rev. A 89(4), 043407 (2014).
[Crossref]

2013 (4)

I. Yavuz, “Gas population effects in harmonic emission by plasmonic fields,” Phys. Rev. A 87(5), 053815 (2013).
[Crossref]

J. H. Luo, Y. Li, Z. Wang, Q. B. Zhang, and P. X. Lu, “Ultra-short isolated attosecond emission in mid-infrared inhomogeneous fields without CEP stabilization,” J. Phys. At. Mol. Opt. Phys. 46(14), 145602 (2013).
[Crossref]

J. A. Pérez-Hernández, M. F. Ciappina, M. Lewenstein, L. Roso, and A. Zaïr, “Beyond carbon K-edge harmonic emission using a spatial and temporal synthesized laser field,” Phys. Rev. Lett. 110(5), 053001 (2013).
[Crossref] [PubMed]

Y. Y. Yang, A. Scrinzi, A. Husakou, Q. G. Li, S. L. Stebbings, F. Süßmann, H. J. Yu, S. Kim, E. Rühl, J. Herrmann, X. C. Lin, and M. F. Kling, “High-harmonic and single attosecond pulse generation using plasmonic field enhancement in ordered arrays of gold nanoparticles with chirped laser pulses,” Opt. Express 21(2), 2195–2205 (2013).
[Crossref] [PubMed]

2012 (4)

I. Yavuz, E. A. Bleda, Z. Altun, and T. Topcu, “Generation of a broadband xuv continuum in high-order-harmonic generation by spatially inhomogeneous fields,” Phys. Rev. A 85(1), 013416 (2012).
[Crossref]

M. F. Ciappina, S. S. Aćimović, T. Shaaran, J. Biegert, R. Quidant, and M. Lewenstein, “Enhancement of high harmonic generation by confining electron motion in plasmonic nanostrutures,” Opt. Express 20(24), 26261–26274 (2012).
[Crossref] [PubMed]

H. X. He, R. F. Lu, P. Y. Zhang, K. L. Han, and G. Z. He, “Direct multi-photon ionizations of H2+ in intense laser fields,” J. Phys. At. Mol. Opt. Phys. 45(12), 085103 (2012).
[Crossref]

M. F. Ciappina, J. Biegert, R. Quidant, and M. Lewenstein, “High-order-harmonic generation from inhomogeneous fields,” Phys. Rev. A 85(3), 033828 (2012).
[Crossref]

2011 (4)

Y. Wu, J. T. Zhang, H. L. Ye, and Z. Z. Xu, “Intensity-dependent interference effect in high-order harmonic generation from aligned H2+ molecules,” Phys. Rev. A 83(2), 023417 (2011).
[Crossref]

X. Zhu, Q. Zhang, W. Hong, P. Lan, and P. Lu, “Two-center interference in high-order harmonic generation from heteronuclear diatomic molecules,” Opt. Express 19(2), 436–447 (2011).
[Crossref] [PubMed]

H. X. He, R. F. Lu, P. Y. Zhang, Y. H. Guo, K. L. Han, and G. Z. He, “Theoretical investigation of the origin of the multipeak structure of kinetic-energy-release spectra from charge-resonance-enhanced ionization of H2+ in intense laser fields,” Phys. Rev. A 84(3), 033418 (2011).
[Crossref]

S. L. Stebbings, F. Süßmann, Y.-Y. Yang, A. Scrinzi, M. Durach, A. Rusina, M. I. Stockman, and M. F. Kling, “Generation of isolated attosecond extreme ultraviolet pulses employing nanoplasmonic field enhancement: optimization of coupled ellipsoids,” New J. Phys. 13(11), 073010 (2011).
[Crossref]

2010 (1)

Y. C. Han and L. B. Madsen, “Minimum in the high-order harmonic generation spectrum from molecules: role of excited states,” J. Phys. At. Mol. Opt. Phys. 43(10), 225601 (2010).
[Crossref]

2009 (1)

A. T. Le, R. R. Lucchese, M. T. Lee, and C. D. Lin, “Probing molecular frame photoionization via laser generated high-order harmonics from aligned molecules,” Phys. Rev. Lett. 102(20), 203001 (2009).
[Crossref] [PubMed]

2008 (5)

G. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80(1), 117–140 (2008).
[Crossref]

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

Y. Mairesse, J. Levesque, N. Dudovich, P. B. Corkum, and D. M. Villeneuve, “High harmonic generation from aligned molecules–amplitude and polarization,” J. Mod. Opt. 55(16), 2591–2602 (2008).
[Crossref]

Y. J. Chen and J. Liu, “High-order harmonic generation from diatomic molecules with large internuclear distance: The effect of two-center interference,” Phys. Rev. A 77(1), 013410 (2008).
[Crossref]

R. F. Lu, P. Y. Zhang, and K. L. Han, “Attosecond-resolution quantum dynamics calculations for atoms and molecules in strong laser fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(6), 066701 (2008).
[Crossref] [PubMed]

2007 (1)

M. F. Ciappina, C. C. Chirilă, and M. Lein, “Influence of Coulomb continuum wave functions in the description of high-order harmonic generation with H2+,” Phys. Rev. A 75(4), 043405 (2007).
[Crossref]

2006 (1)

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirilă, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312(5772), 424–427 (2006).
[Crossref] [PubMed]

2005 (4)

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]

B. Zimmermann, M. Lein, and J. M. Rost, “Analysis of recombination in high-order harmonic generation in molecules,” Phys. Rev. A 71(3), 033401 (2005).
[Crossref]

J. Hu, K. L. Han, and G. Z. He, “Correlation quantum dynamics between an electron and D2+ molecule with attosecond resolution,” Phys. Rev. Lett. 95(12), 123001 (2005).
[Crossref] [PubMed]

M. Lein, “Attosecond probing of vibrational dynamics with high-harmonic generation,” Phys. Rev. Lett. 94(5), 053004 (2005).
[Crossref] [PubMed]

2004 (4)

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 432(7019), 867–871 (2004).
[Crossref] [PubMed]

V. Roudnev, B. D. Esry, and I. Ben-Itzhak, “Controlling HD+ and H2+ dissociation with the carrier-envelope phase difference of an intense ultrashort laser pulse,” Phys. Rev. Lett. 93(16), 163601 (2004).
[Crossref] [PubMed]

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 432(7019), 867–871 (2004).
[Crossref] [PubMed]

L. V. Vela-Arevalo and J. E. Marsden, “Time-frequency analysis of the restricted three-body problem: transport and resonance transitions,” Class. Quantum Gravity 21(3), S351–S375 (2004).
[Crossref]

2003 (1)

H. Niikura, F. Légaré, R. Hasbani, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421(6925), 826–829 (2003).
[Crossref] [PubMed]

2002 (3)

M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Role of the intramolecular phase in high-harmonic generation,” Phys. Rev. Lett. 88(18), 183903 (2002).
[Crossref] [PubMed]

M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Interference effects in high-order harmonic generation with molecules,” Phys. Rev. A 66(2), 023805 (2002).
[Crossref]

H. Niikura, F. Légaré, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417(6892), 917–922 (2002).
[Crossref] [PubMed]

2001 (4)

M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, C. Spielmann, G. A. Reider, P. B. Corkum, and F. Krausz, “X-ray pulses approaching the attosecond frontier,” Science 291(5510), 1923–1927 (2001).
[Crossref] [PubMed]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

X. Chu and S. I. Chu, “Optimization of high-order harmonic generation by genetic algorithm and wavelet time-frequency analysis of quantum dipole emission,” Phys. Rev. A 64(2), 021403(R) (2001).

2000 (1)

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(R) (2000).

1999 (2)

N. A. Papadogiannis, B. Witzel, C. Kalpouzos, and D. Charalambidis, “Observation of attosecond light localization in higher order harmonic generation,” Phys. Rev. Lett. 83(21), 4289–4292 (1999).
[Crossref]

A. D. Bandrauk and H. Yu, “High-order harmonic generation by one- and two-electron molecular ions with intense laser pulses,” Phys. Rev. A 59(1), 539–548 (1999).
[Crossref]

1998 (1)

R. Kopold, W. Becker, and M. Kleber, “Model calculations of high-harmonic generation in molecular ions,” Phys. Rev. A 58(5), 4022–4038 (1998).
[Crossref]

1995 (2)

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]

M. Ivanov, P. B. Corkum, T. Zuo, and A. Bandrauk, “Routes to control of intense-field atomic polarizability,” Phys. Rev. Lett. 74(15), 2933–2936 (1995).
[Crossref] [PubMed]

1993 (2)

M. Y. Ivanov and P. B. Corkum, “Generation of high-order harmonics from inertially confined molecular ions,” Phys. Rev. A 48(1), 580–590 (1993).
[Crossref] [PubMed]

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

1992 (1)

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]

1961 (1)

J. R. Hiskes, “Dissociation of molecular ions by electric and magnetic fields,” Phys. Rev. 122(4), 1207–1217 (1961).
[Crossref]

Acimovic, S. S.

Agostini, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Ahmadi, H.

H. Ahmadi, A. Maghari, H. Sabzyan, A. R. Niknam, and M. Vafaee, “Effect of nuclear motion on high-order-harmonic generation of H2+ in intense ultrashort laser pulses,” Phys. Rev. A 90(4), 043411 (2014).
[Crossref]

Altun, Z.

I. Yavuz, M. F. Ciappina, A. Chacón, Z. Altun, M. F. Kling, and M. Lewenstein, “Controlling electron localization in H2+ by intense plasmon-enhanced laser fields,” Phys. Rev. A 93(3), 033404 (2016).
[Crossref]

I. Yavuz, Y. Tikman, and Z. Altun, “High-order-harmonic generation from H2+ molecular ions near plasmon-enhanced laser fields,” Phys. Rev. A 92(2), 023413 (2015).
[Crossref]

I. Yavuz, E. A. Bleda, Z. Altun, and T. Topcu, “Generation of a broadband xuv continuum in high-order-harmonic generation by spatially inhomogeneous fields,” Phys. Rev. A 85(1), 013416 (2012).
[Crossref]

Antoine, P.

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]

Auge, F.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Bai, L.

Baker, S.

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirilă, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312(5772), 424–427 (2006).
[Crossref] [PubMed]

Balcou, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Bandrauk, A.

M. Ivanov, P. B. Corkum, T. Zuo, and A. Bandrauk, “Routes to control of intense-field atomic polarizability,” Phys. Rev. Lett. 74(15), 2933–2936 (1995).
[Crossref] [PubMed]

Bandrauk, A. D.

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]

H. Niikura, F. Légaré, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417(6892), 917–922 (2002).
[Crossref] [PubMed]

A. D. Bandrauk and H. Yu, “High-order harmonic generation by one- and two-electron molecular ions with intense laser pulses,” Phys. Rev. A 59(1), 539–548 (1999).
[Crossref]

Becker, W.

R. Kopold, W. Becker, and M. Kleber, “Model calculations of high-harmonic generation in molecular ions,” Phys. Rev. A 58(5), 4022–4038 (1998).
[Crossref]

Ben-Itzhak, I.

V. Roudnev, B. D. Esry, and I. Ben-Itzhak, “Controlling HD+ and H2+ dissociation with the carrier-envelope phase difference of an intense ultrashort laser pulse,” Phys. Rev. Lett. 93(16), 163601 (2004).
[Crossref] [PubMed]

Biegert, J.

Bleda, E. A.

I. Yavuz, E. A. Bleda, Z. Altun, and T. Topcu, “Generation of a broadband xuv continuum in high-order-harmonic generation by spatially inhomogeneous fields,” Phys. Rev. A 85(1), 013416 (2012).
[Crossref]

Brabec, T.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Breger, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Burnett, K.

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]

Cao, X.

X. Cao, S. Jiang, C. Yu, Y. Wang, L. Bai, and R. Lu, “Generation of isolated sub-10-attosecond pulses in spatially inhomogenous two-color fields,” Opt. Express 22(21), 26153–26161 (2014).
[Crossref] [PubMed]

C. Yu, Y. H. Wang, X. Cao, S. C. Jiang, and R. F. Lu, “Isolated few-attosecond emission in a multicycle asymmetrically nonhomogeneous twocolor laser field,” J. Phys. At. Mol. Opt. Phys. 47(22), 225602 (2014).
[Crossref]

Chacón, A.

I. Yavuz, M. F. Ciappina, A. Chacón, Z. Altun, M. F. Kling, and M. Lewenstein, “Controlling electron localization in H2+ by intense plasmon-enhanced laser fields,” Phys. Rev. A 93(3), 033404 (2016).
[Crossref]

Charalambidis, D.

N. A. Papadogiannis, B. Witzel, C. Kalpouzos, and D. Charalambidis, “Observation of attosecond light localization in higher order harmonic generation,” Phys. Rev. Lett. 83(21), 4289–4292 (1999).
[Crossref]

Chen, Y. J.

Y. J. Chen and J. Liu, “High-order harmonic generation from diatomic molecules with large internuclear distance: The effect of two-center interference,” Phys. Rev. A 77(1), 013410 (2008).
[Crossref]

Chirila, C. C.

M. F. Ciappina, C. C. Chirilă, and M. Lein, “Influence of Coulomb continuum wave functions in the description of high-order harmonic generation with H2+,” Phys. Rev. A 75(4), 043405 (2007).
[Crossref]

S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirilă, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing proton dynamics in molecules on an attosecond time scale,” Science 312(5772), 424–427 (2006).
[Crossref] [PubMed]

Chou, Y.

Y. Chou, P. C. Li, T. S. Ho, and S. I. Chu, “Generation of an isolated few-attosecond pulse in optimized inhomogeneous two-color fields,” Phys. Rev. A 92(2), 023423 (2015).
[Crossref]

Chu, S. I.

Y. Chou, P. C. Li, T. S. Ho, and S. I. Chu, “Generation of an isolated few-attosecond pulse in optimized inhomogeneous two-color fields,” Phys. Rev. A 92(2), 023423 (2015).
[Crossref]

X. Chu and S. I. Chu, “Optimization of high-order harmonic generation by genetic algorithm and wavelet time-frequency analysis of quantum dipole emission,” Phys. Rev. A 64(2), 021403(R) (2001).

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(R) (2000).

Chu, X.

X. Chu and S. I. Chu, “Optimization of high-order harmonic generation by genetic algorithm and wavelet time-frequency analysis of quantum dipole emission,” Phys. Rev. A 64(2), 021403(R) (2001).

Ciappina, M. F.

I. Yavuz, M. F. Ciappina, A. Chacón, Z. Altun, M. F. Kling, and M. Lewenstein, “Controlling electron localization in H2+ by intense plasmon-enhanced laser fields,” Phys. Rev. A 93(3), 033404 (2016).
[Crossref]

J. A. Pérez-Hernández, M. F. Ciappina, M. Lewenstein, L. Roso, and A. Zaïr, “Beyond carbon K-edge harmonic emission using a spatial and temporal synthesized laser field,” Phys. Rev. Lett. 110(5), 053001 (2013).
[Crossref] [PubMed]

M. F. Ciappina, S. S. Aćimović, T. Shaaran, J. Biegert, R. Quidant, and M. Lewenstein, “Enhancement of high harmonic generation by confining electron motion in plasmonic nanostrutures,” Opt. Express 20(24), 26261–26274 (2012).
[Crossref] [PubMed]

M. F. Ciappina, J. Biegert, R. Quidant, and M. Lewenstein, “High-order-harmonic generation from inhomogeneous fields,” Phys. Rev. A 85(3), 033828 (2012).
[Crossref]

M. F. Ciappina, C. C. Chirilă, and M. Lein, “Influence of Coulomb continuum wave functions in the description of high-order harmonic generation with H2+,” Phys. Rev. A 75(4), 043405 (2007).
[Crossref]

Cooper, J.

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]

Corkum, P.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Corkum, P. B.

Y. Mairesse, J. Levesque, N. Dudovich, P. B. Corkum, and D. M. Villeneuve, “High harmonic generation from aligned molecules–amplitude and polarization,” J. Mod. Opt. 55(16), 2591–2602 (2008).
[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,” Nature 432(7019), 867–871 (2004).
[Crossref] [PubMed]

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 432(7019), 867–871 (2004).
[Crossref] [PubMed]

H. Niikura, F. Légaré, R. Hasbani, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421(6925), 826–829 (2003).
[Crossref] [PubMed]

H. Niikura, F. Légaré, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417(6892), 917–922 (2002).
[Crossref] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, C. Spielmann, G. A. Reider, P. B. Corkum, and F. Krausz, “X-ray pulses approaching the attosecond frontier,” Science 291(5510), 1923–1927 (2001).
[Crossref] [PubMed]

M. Ivanov, P. B. Corkum, T. Zuo, and A. Bandrauk, “Routes to control of intense-field atomic polarizability,” Phys. Rev. Lett. 74(15), 2933–2936 (1995).
[Crossref] [PubMed]

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

M. Y. Ivanov and P. B. Corkum, “Generation of high-order harmonics from inertially confined molecular ions,” Phys. Rev. A 48(1), 580–590 (1993).
[Crossref] [PubMed]

Drescher, M.

M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, C. Spielmann, G. A. Reider, P. B. Corkum, and F. Krausz, “X-ray pulses approaching the attosecond frontier,” Science 291(5510), 1923–1927 (2001).
[Crossref] [PubMed]

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Du, H.

X. L. Ge, H. Du, Q. Wang, J. Guo, and X. S. Liu, “Selection of quantum path in high-order harmonics and isolated sub-100 attosecond generation in few-cycle spatially inhomogeneous laser fields,” Chin. Phys. B 24(2), 023201 (2015).
[Crossref]

Dudovich, N.

Y. Mairesse, J. Levesque, N. Dudovich, P. B. Corkum, and D. M. Villeneuve, “High harmonic generation from aligned molecules–amplitude and polarization,” J. Mod. Opt. 55(16), 2591–2602 (2008).
[Crossref]

Durach, M.

S. L. Stebbings, F. Süßmann, Y.-Y. Yang, A. Scrinzi, M. Durach, A. Rusina, M. I. Stockman, and M. F. Kling, “Generation of isolated attosecond extreme ultraviolet pulses employing nanoplasmonic field enhancement: optimization of coupled ellipsoids,” New J. Phys. 13(11), 073010 (2011).
[Crossref]

Esry, B. D.

V. Roudnev, B. D. Esry, and I. Ben-Itzhak, “Controlling HD+ and H2+ dissociation with the carrier-envelope phase difference of an intense ultrashort laser pulse,” Phys. Rev. Lett. 93(16), 163601 (2004).
[Crossref] [PubMed]

Etches, A.

M. D. Śpiewanowski, A. Etches, and L. B. Madsen, “High-order-harmonic generation from field-distorted orbitals,” Phys. Rev. A 89(4), 043407 (2014).
[Crossref]

Feng, L. Q.

L. Q. Feng, “Molecular harmonic extension and enhancement from H2+ ions in the presence of spatially inhomogeneous fields,” Phys. Rev. A 92(5), 053832 (2015).
[Crossref]

Feng, W.

H. Y. Zhong, J. Guo, W. Feng, P. C. Li, and X. S. Liu, “Comparison of high harmonic generation and attosecond pulse from 3D hydrogen atom in three kinds of inhomogeneous fields,” Phys. Lett. A 380(1–2), 188–193 (2016).
[Crossref]

Ge, X. L.

X. L. Ge, H. Du, Q. Wang, J. Guo, and X. S. Liu, “Selection of quantum path in high-order harmonics and isolated sub-100 attosecond generation in few-cycle spatially inhomogeneous laser fields,” Chin. Phys. B 24(2), 023201 (2015).
[Crossref]

Gerber, G.

G. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80(1), 117–140 (2008).
[Crossref]

Golubev, N. V.

N. V. Golubev and A. I. Kuleff, “Control of charge migration in molecules by ultrashort laser pulses,” Phys. Rev. A 91(5), 051401(R) (2015).

Gong, S.

Guo, J.

H. Y. Zhong, J. Guo, W. Feng, P. C. Li, and X. S. Liu, “Comparison of high harmonic generation and attosecond pulse from 3D hydrogen atom in three kinds of inhomogeneous fields,” Phys. Lett. A 380(1–2), 188–193 (2016).
[Crossref]

X. L. Ge, H. Du, Q. Wang, J. Guo, and X. S. Liu, “Selection of quantum path in high-order harmonics and isolated sub-100 attosecond generation in few-cycle spatially inhomogeneous laser fields,” Chin. Phys. B 24(2), 023201 (2015).
[Crossref]

Guo, Y. H.

H. X. He, R. F. Lu, P. Y. Zhang, Y. H. Guo, K. L. Han, and G. Z. He, “Theoretical investigation of the origin of the multipeak structure of kinetic-energy-release spectra from charge-resonance-enhanced ionization of H2+ in intense laser fields,” Phys. Rev. A 84(3), 033418 (2011).
[Crossref]

Han, K. L.

H. X. He, R. F. Lu, P. Y. Zhang, K. L. Han, and G. Z. He, “Direct multi-photon ionizations of H2+ in intense laser fields,” J. Phys. At. Mol. Opt. Phys. 45(12), 085103 (2012).
[Crossref]

H. X. He, R. F. Lu, P. Y. Zhang, Y. H. Guo, K. L. Han, and G. Z. He, “Theoretical investigation of the origin of the multipeak structure of kinetic-energy-release spectra from charge-resonance-enhanced ionization of H2+ in intense laser fields,” Phys. Rev. A 84(3), 033418 (2011).
[Crossref]

R. F. Lu, P. Y. Zhang, and K. L. Han, “Attosecond-resolution quantum dynamics calculations for atoms and molecules in strong laser fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(6), 066701 (2008).
[Crossref] [PubMed]

J. Hu, K. L. Han, and G. Z. He, “Correlation quantum dynamics between an electron and D2+ molecule with attosecond resolution,” Phys. Rev. Lett. 95(12), 123001 (2005).
[Crossref] [PubMed]

Han, Y. C.

Y. C. Han and L. B. Madsen, “Minimum in the high-order harmonic generation spectrum from molecules: role of excited states,” J. Phys. At. Mol. Opt. Phys. 43(10), 225601 (2010).
[Crossref]

Hasbani, R.

H. Niikura, F. Légaré, R. Hasbani, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421(6925), 826–829 (2003).
[Crossref] [PubMed]

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H. Y. Zhong, J. Guo, W. Feng, P. C. Li, and X. S. Liu, “Comparison of high harmonic generation and attosecond pulse from 3D hydrogen atom in three kinds of inhomogeneous fields,” Phys. Lett. A 380(1–2), 188–193 (2016).
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Class. Quantum Gravity (1)

L. V. Vela-Arevalo and J. E. Marsden, “Time-frequency analysis of the restricted three-body problem: transport and resonance transitions,” Class. Quantum Gravity 21(3), S351–S375 (2004).
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J. Mod. Opt. (1)

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J. Phys. At. Mol. Opt. Phys. (4)

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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 432(7019), 867–871 (2004).
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New J. Phys. (1)

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Opt. Express (5)

Phys. Lett. A (1)

H. Y. Zhong, J. Guo, W. Feng, P. C. Li, and X. S. Liu, “Comparison of high harmonic generation and attosecond pulse from 3D hydrogen atom in three kinds of inhomogeneous fields,” Phys. Lett. A 380(1–2), 188–193 (2016).
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Figures (5)

Fig. 1
Fig. 1 The nonhomogeneous fields for the input laser intensify of 1.0 × 1014 W/cm2 and β = 0.004. (a) Asymmetric case (ANLF), and (b) symmetric case (SNLF).
Fig. 2
Fig. 2 Harmonic spectra from 3D calculations for H2+ in (a) ANLF and (b) SNLF, where the solid lines give the smoothed spectra, and the yellow and dark green arrows indicate the location of the minima for m = 5 and 6 respectively according to Eq. (1). (c) Harmonic spectra from | G 1 ( ω ) | 2 + | G 2 ( ω ) | 2 of H2+ without interferences at R = 13 a.u. (red line for SNLF, black line for ANLF). (d) The time-frequency distribution for HHG in SNLF at R = 13 a.u., where the white dashed line corresponds to the minima. The harmonic spectra for R = 12 a.u. in (a), (b), and the harmonic spectra for ANLF in (c) are the raw data, whereas the harmonic spectra for R = 13, 14, 15, and 16 a.u. in (a), (b), and the harmonic spectra for SNLF in (c) are shifted up vertically for clarity.
Fig. 3
Fig. 3 Relationship between the location of interference minimnum and internuclear distance.
Fig. 4
Fig. 4 Time-dependent electronic density for R = 15 a.u. in (a) ANLF and (b) SNLF. Recollision flux of the electronic wavepacket in the case of (c) ANLF and (d) SNLF.
Fig. 5
Fig. 5 (a) The time-frequency distribution for HHG in ANLF at R = 13 a.u.. (b) Harmonic spectra of H2+ at R = 13 a.u. in ANLF and SNLF.

Equations (10)

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R cos θ = ( 2 m + 1 ) λ / 2 , m = 0 , 1 , 2 , . . . .
i t ψ ( z ; t ) = [ 1 2 2 z 2 + V ( z ) + z E ( z ; t ) ] ψ ( z ; t ) ,
i t ψ ( z , ρ ; t ) = [ T R + T z + T ρ + V c ( R , z , ρ ) + k z E ( z ; t ) ] ψ ( z , ρ ; t ) ,
a ( t ) = ψ ( z , ρ ; t ) | V ( R , z , ρ ) z + E ( t ) | ψ ( z , ρ ; t ) ,
P ( ω ) = | 1 2 π 0 T a ( t ) e i w t d t | 2 .
A ( t , ω ) = a ( t ) ω W ( ω ( t t ) ) d t ,
P s m o o t h ( ω ) = P ( ω ' ) exp ( ( ω ' ω ) 2 / σ 2 ) d ω ' .
P ( ω ) = | G 1 ( ω ) | 2 + | G 2 ( ω ) | 2 + 2 Re [ G 1 ( ω ) G 2 * ( ω ) ] ,
F ( t ) = Im [ ψ * δ ( z z 0 ) z ψ ] ρ d ρ ,
Δ N exp [ i Θ ( t f , t 0 ) ] exp [ i Θ ( t f ' , t 0 ' ) ] ,

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