## 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 H_{2}^{+} 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 3.17*U*_{p} + *I*_{p}, where *U*_{p} is the ponderomotive potential, *I*_{p} 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].

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, H_{2}^{+}, 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 H_{2}^{+} and D_{2}^{+} when nuclear motion is taken into account. To our knowledge, Yavuz *et al.* [30] investigated the HHG from H_{2}^{+} 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 H_{2}^{+} [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 H_{2}^{+}: Destructive interference occurs when the de Broglie wavelength (*λ*) of the returning electron, the internuclear distance (*R*), and alignment angle (*θ*) satisfy the relationship

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 H_{2}^{+} are shown to be maximum at certain harmonic orders as a function of molecular orientation, while Rost et al. [38] claim that for N_{2}, 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 H_{2}, N_{2}, O_{2} [39], which is attributed to additional phase shifts [38]. By calculating HHG including orbital distortion for N_{2}, 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 H_{2}^{+} will affect the location of the minimum [43]. Recently, Hu *et al.* [44] investigated the dependence of the spectral minima positions of H_{2}^{+} 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 H_{2}^{+}, 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 H_{2}^{+} 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 H_{2}^{+} is given by

*R*is internuclear distance,

*z*is the electronic coordinate. The 3D TDSE for H

_{2}

^{+}in cylindrical coordinates can be described as [48–50]:

*m*

_{e}and

*m*

_{p}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 $E\left(z;t\right)={E}_{0}f\left(t\right)[(1+\beta z)-\frac{\beta {R}^{2}}{z}]\mathrm{cos}\left(\omega t\right)$(for ANLF) or $E\left(z;t\right)={E}_{0}f\left(t\right)[1+\beta \left|z\right|]\mathrm{cos}\left(\omega t\right)$(for SNLF), where

*E*

_{0}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*):

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],

*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 H_{2}^{+}. Also, the molecular axis is in line with nanostructures’ main axis. The laser intensity, *λ* and *β* are chosen as 1.0 × 10^{14} W/cm^{2}, 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 H_{2}^{+} 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,

*σ*= 5

*ω*

_{L}, here

*ω*

_{L}is the frequency of the driving laser.

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

*j*= 1, 2. |

*G*

_{j}(

*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 ${\left|{G}_{1}(\omega )\right|}^{2}\text{+}{\left|{G}_{2}(\omega )\right|}^{2}$ of H

_{2}

^{+}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 H

_{2}

^{+}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.

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)={{\displaystyle \int \left|\psi (z,\rho ;t)\right|}}^{2}\rho d\rho $ 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

*z*

_{0}is the position of flux analysis. Figures 4(c) and 4(d) show the results with

*z*

_{0}= ± 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.

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 *x*_{1}(*t*) starting at the moment *t*_{0} during the first quarter cycle and returning at some moment *t*_{f}, there exists a mirror symmetric trajectory *x*_{2}(*t*) which starts at *t*_{0}′ = *t*_{0} + *π*/*ω* and returns at *t*_{f}′ = *t*_{f} + *π*/*ω*: *x*_{1}(*t*) = –*x*_{2}(*t* + *π*/*ω*) [61]. The contribution from these two trajectories to the field-induced dipole moment has a form

*I*

_{p}is ionization potential,

*A*(t) is the field vector potential, and

*N*indicates the harmonic order. For symmetrical potential and laser field, $\Theta ({t}_{f},{t}_{0})=\Theta ({t}_{f}\text{'},{t}_{0}\text{'})-N\pi ,$ as a result, ${\Delta}_{N}\propto 1-\mathrm{exp}(-iN\pi ).$ 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

*x*

_{2}(

*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.

## 4. Conclusion

In summary, a detailed quantum dynamics study of HHG from H_{2}^{+} 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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