Multiple rescattering processes in high-order harmonic generation from molecular system

Cai-Ping Zhang; Chang-Long Xia; Xiang-Fu Jia; Xiang-Yang Miao

doi:10.1364/OE.24.020297

1. Introduction

The high-order harmonic generation (HHG) is a hot topic in intense laser field with the potential application tracing the ultrafast electron motion [1, 2] and detecting the molecular structure [3]. Moreover, the related physical process has been explained by semiclassical three-step model [4]: ionization, acceleration and recombination. By focusing on the first rescattering in HHG process, numerous useful results have been obtained. For example, the generation of an isolated attosecond pulse both in homogeneous fields [5–7] and inhomogeneous fields [8–12] or disentangling the HHG features of different quantum paths by spectrally and spatially resolved the atomic harmonic spectra [13]. Moreover, the two-center interference [14], the nuclear signature effects [15] and the electron localization [16] have been further learned from the molecular HHG. Especially the asymmetric molecule, with longer lifetime of the first excited state [17], can provide more novel physical images in harmonic emission. Considering the electron transition between the ground state and the first excited state, Bian and Bandrauk have clearly elucidated HHG process via four-step model [18, 19]. The odd and even harmonic emissions from oriented asymmetric molecules have also been deeply studied to detect the molecular structure [20, 21]. Furthermore, multiple recombination channels contributing to the asymmetric molecular harmonics have been distinguished [22] and efficiently manipulated [23] in our previous works.

In fact, the ionized electrons may revisit the parent ion several times before emitting the harmonic photon, in other words, the multiple rescattering processes (MRP) may contribute to the harmonic emission. Recently, the related research has attracted more researchers to further discuss the harmonic emission [24–30]. Tate et al. [24] have found that the MRP can contribute to HHG in examining the wavelength scaling by solving the time-dependent Schrödinger equation (TDSE). Hickstein et al. [25] have proposed that the variation of the photoelectron angular distribution is associated with the MRP. He et al. [26] have shown that based on the strong-field approximation, the contributions from high-order rescattering events with respect to the first rescattering increase with wavelength up to about 1600nm and become stable beyond that wavelength. This result has been confirmed within the TDSE approach [27] as well as within the quantum orbits theory [28] by Le et al.. Furthermore, they have also discovered that the high-order rescattering would contribute insignificantly to the total macroscopic HHG yields for the gas jet near the laser focus; moreover, the related contribution would not survive in typical phase-matching condition especially for harmonics in higher plateau [27]. Miller et al. [29] have provided an efficient scheme to control the MRP by adjusting the time delay between vacuum ultraviolet pulse and infrared pulse. Moreover, Hernández-García et al. [30] have obtained the zeptosecond waveform with the interference of x-ray emissions from two rescattering events in midinfrared laser field. These results are mainly about the atomic MRP, however, the ionized electron for molecular system has two choices in the recombination process (i.e. the recombination with parent nucleus and with neighboring nucleus). So the corresponding molecular MRP would be more complex and contain more underlying physical information.

In this paper, we research the molecular HHG at different internuclear distances considering the MRP by solving the one-dimensional (1D) and three-dimensional (3D) TDSE and classical motion equation. Both quantum results are almost identical with the classical results that the molecular MRP are similar with those of atomic system at smaller internuclear distance, but significant differences are displayed at larger internuclear distance. By establishing the physical model of the molecular MRP, both rescatterings originating from recombination with parent nucleus and with neighboring nucleus are distinguished. Two channels recombining with neigh-boring nucleus which are sensitive to the internuclear distance contribute to the molecular HHG. One is attributed to the recombination with neighbor nucleus after reversing the direction of external laser field, while the other one before changing the direction. Moreover, the harmonics are almost emitted at the same time for the two kinds of first rescatterings originating from recombination with neighboring nucleus at larger internuclear distance. This model may provide the theoretical guide for further study about the harmonics from the complicated molecules or atom clusters with large enough distance between atoms, especially for the recombination with neighboring nucleus. Atomic units (a.u.) are used in this paper unless otherwise stated.

2. Theoretical method

We theoretically simulate the molecular MRP in an intense laser field by solving the 1D and 3D TDSE. For simplicity, $H_{2}^{+}$ is taken as the simple model. Providing the linear laser field along with the molecular axis, in the dipole approximation and the length gauge, the 1D TDSE [31–33] is written as

i \frac{\partial}{\partial t} ψ (z, t) = H (z, t) ψ (z, t),

H (z, t) = - \frac{1}{2} \frac{\partial^{2}}{\partial z^{2}} + V_{c} (z) + k z E (t),

V_{c} (z) = \frac{1}{R} - \frac{1}{\sqrt{{(z - R / 2)}^{2} + 1}} - \frac{1}{\sqrt{{(z - R / 2)}^{2} - 1}},

where R is the internuclear distance, z is the electron coordinate (with respect to the nuclear center of mass). V_c is the soft Coulomb potential. In calculation, the converged numerical parameters are as follows: the grid ranges from −1000a.u. to 1000a.u. in z direction with 10000 points. And the corresponding time step is set to be 0.02a.u..

For the laser field linearly polarized along the z-axis, the 3D TDSE can be expressed as

i \frac{\partial ψ (z, ρ; t)}{\partial t} = [T_{z} + T_{ρ} + V_{c} (z, ρ) + k z E (t)],

T_{z} = - \frac{2 m_{p} + m_{e}}{4 m_{p} m_{e}} \frac{\partial^{2}}{\partial z^{2}}, T_{p} = - \frac{2 m_{p} + m_{e}}{4 m_{p} m_{e}} (\frac{\partial^{2}}{\partial ρ^{2}} + \frac{1}{ρ} \frac{\partial}{\partial ρ}),

V_{c} (z, ρ) = \frac{1}{R} - \frac{1}{\sqrt{{(z - R / 2)}^{2} + ρ^{2}}} - \frac{1}{\sqrt{{(z - R / 2)}^{2} - ρ^{2}}},

where k=1+m_e/(2m_p+m_e), additionally, m_e and m_p are the masses of the electron and the nucleus, respectively. The related converged numerical parameters are as follows: the grid ranges from −1000a.u. to 1000a.u. in z direction and from 0a.u. to 15a.u. in ρ direction, with 10000 and 75 points, respectively. E(t) is the adopted laser field, and the time step is also 0.02a.u.. The time-dependent wave function is advanced using the standard second-order split-operator method [34–36].

To make detail information of harmonic emission, the time-frequency distribution by means of the wavelet transform has been shown

d_{ω} (t) = \int a (t^{'}) \sqrt{ω_{0}} W (ω_{0} (t^{'} - t)) d t^{'},

where a(t′) and W(ω₀(t′ − t)) are dipole acceleration and Morlet wavelet, respectively.

3. Results and discussion

For the diatomic molecular harmonic emission, not only can the electron recombine with the parent nucleus where it is ionized but also the neighboring nucleus [37–40]. Moreover, the ionized electron may revisit the nucleus more than once after the acceleration in external field. It means that the MRP would contribute to the molecular HHG process [26]. With complicated recombination process, the molecular MRP will contain more intriguing physics. To get a clear insight, we establish the physical model of the molecular MRP originating from recombination with parent nucleus and with neighboring nucleus in Figs. 1(a) and 1(b), respectively. In the model, two nuclei are set as N₁ and N₂ fixed at −R/2 and R/2, respectively. In order to observe the phenomenon of molecular multiple rescatterings intuitively, the intense laser pulse E(t)=E₀cosωt which lasts for 5 optical cycles (o.c.) with the peak intensity of 2×10¹⁴W/cm² is selected in our work. 1o.c.=5.34fs corresponds to the wavelength of 1600nm. Due to the symmetry, the electron motion around N₁ nucleus is set as an example in the following discussion. In Fig. 1(a), the magenta filled circle represents the ionized electron around N₁ nucleus. The opened circle represents the unscattered electron. The blue, red and green filled circles represent the first, second and third returning electrons, respectively. In intense laser field, the electron around N₁ nucleus is firstly ionized around A and accelerated in the external field, finally, it recombines with the parent nucleus around B. This process is the first rescattering [4]. Nevertheless, the electron may be unscattered at its first visit to N₁ nucleus around B but rather reaccelerated. When it revisits N₁ nucleus around C, the second rescattering takes place. With similar mechanism, much higher-order rescattering may contribute to the harmonic emission [26]. How about the recombination with neighboring nucleus and the related MRP? As depicted in Fig. 1(b), the electron ionized around A₁ will experience the complete ionization-acceleration-recombination process to emit higher-energy harmonic photons around C₁, namely the channel 1 [37–40]. In fact, the real acceleration process based on the three-step mode [4] can be viewed as an acceleration-deceleration-reacceleration process. During the deceleration period, the laser field is negative. Under this condition, the potential well of N₁ nucleus is dressed-up while that of N₂ nucleus is dressed-down taking the laser-dressed double-well model into account [22]. So the electron can directly transit from N₁ nucleus to N₂ nucleus around B₁ (i.e. before laser field reversing its direction) with the emission of lower-energy harmonics, namely the channel 2. These are two kinds of first rescattering originating from recombination with neighboring nucleus. For channel 1, the ionized electron may be reaccelerated around C₁ rather, and then recombine with N₂ nucleus for the second time around C₂. So does the third rescattering. It’s worth noting that the electron around N₁ nucleus can recombine with N₂ nucleus via channel 2 only in the negative laser field. So the corresponding second and third rescatterings take place around B₂ and B₃, respectively.

Fig. 1 (a) and (b) are the schematic of electron motion in MRP originating from recombination with parent nucleus and with neighboring nucleus, respectively. The magenta filled circle represents the ionized electron around N₁ nucleus. The opened circle represents unscattered electron. The blue, red and green filled circles represent the first, second and third returning electrons, respectively.

Download Full Size | PDF

Based on the electron motion in molecular MRP above mentioned, we try to distinguish these rescattering events in molecular HHG. Figures 2(a) and 2(b) provide the time-frequency maps based on the 1D and 3D quantum simulations for smaller internuclear distance R=2a.u., respectively. The corresponding ionization potential is about 1.25a.u.. Both numerical models exhibit similar time-frequency structures. As the laser field is linearly polarized along the z-axis in 3D model, the electron motion along ρ-axis may be less affected in harmonic emission. To economize on computation time, the 1D model is adopted in the following investigations. Some additional quantum trajectories besides the first rescattering can be clearly observed from Fig. 2(a), which is similar with the results in References [24] and [26]. It shows that the harmonic emission from the molecule at smaller internuclear distance exhibits similar characters with that from the atom. To gain better understanding, the classical electron motion is described as follows:

v (t) = - \int_{t_{i}}^{t_{r}} E (t) d t = - \frac{E_{0}}{ω} [sin (ω t_{r}) - sin (ω t_{i})],

z (t) = - \int_{t_{i}}^{t_{r}} v (t) d t = \frac{E_{0}}{ω} [cos (ω t_{r}) - cos (ω t_{i}) + sin (ω t_{i}) (t_{r} - t_{i})] .

For the ionization from N₁ nucleus, the recombination with parent nucleus (i.e. N₁ nucleus), z(t)=−R/2, namely,

\frac{E_{0}}{ω} [cos (ω t_{r}) - cos (ω t_{i}) + sin (ω t_{i}) (t_{r} - t_{i})] = 0 .

The corresponding classical returning-kinetic-energy map of recombination with parent nucleus is presented in Fig. 2(c). The classical results prove that multiple rescatterings do exist in the molecular harmonic emission. As described in Fig. 1(a), when the difference between ionization and recombination time (t_ir) is about 0.5o.c., the harmonic emission originates from the first rescattering (black line). Other high-order rescattering events in Fig. 2(c) are second (megaton line) (t_ir is about 1.0o.c.), third (green line) (t_ir is about 1.5o.c.) and forth rescatterings (blue line) (t_ir is about 2.0o.c.), respectively. Furthermore, the electron can also recombine with the neighboring nucleus (i.e. N₂ nucleus), z(t)=R/2, namely,

\frac{E_{0}}{ω} [cos (ω t_{r}) - cos (ω t_{i}) + sin (ω t_{i}) (t_{r} - t_{i})] = R .

Since R is far less than E₀/ω², the Eq. (11) is approximately equal to the Eq. (10). The cutoff energy of recombination with neighboring nucleus is close to that with parent nucleus at smaller internuclear distance. Take the first rescattering from neighboring nucleus as an example, the corresponding cutoff energy is also about 3.17U_p as demonstrated in Fig. 2(d). Here, U_p is the ponderomotive energy. However, some differences still exit between the recombination with parent nucleus and with neighboring nucleus. On the one hand, there is an additional channel for the recombination with neighboring nucleus in lower-energy region. On the other hand, for N₁ nucleus, the recombination with parent nucleus occurs every half optical cycle, while with neighboring nucleus every one optical cycle.

Fig. 2 (a) and (b) are 1D and 3D time-frequency maps for $H_{2}^{+}$ with R=2a.u. with same color-coding scale, respectively. (c) and (d) are classical returning-kinetic-energy maps of recombination with N₁ nucleus and with N₂ nucleus for ionization from N₁ nucleus, respectively. The laser parameters are same as those in Fig. 1.

Download Full Size | PDF

To discover the related physical mechanism, Figs. 3(a)–3(c) provide the classical returning-kinetic-energy maps at different wavelengths with the internuclear distance of 2a.u., 7a.u. and 10a.u., respectively. The ionization potential is 0.81a.u. (0.77a.u.) for R=7a.u.(R=10a.u.). The corresponding wavelengths are 800nm (green line) with 1o.c=2.67fs, 1200nm (black line) with 1o.c.=4.01fs and 1600 nm (red line) with 1o.c.=5.34fs. The peak intensity is 2×10¹⁴W/cm² identical with that in Fig. 1. Obviously, the cutoff energy of the additional channel in lower-energy region (shown as the arrow) keeps almost constant with the variation of wavelength. In other words, this channel is independent on the wavelength. As we know that the electron motion is greatly affected by the Coulomb potential at smaller internuclear distance for the recombination with neighboring nucleus. The electron around N₁ nucleus may directly recombine with N₂ nucleus (i.e. the intermolecular transition). Chao et al. [41] have shown that the corresponding cutoff energy is just dependent on the internuclear distance and the peak intensity, moreover, the energy satisfies with E_cutoff = E₀R. So the harmonic energies can be calculated, namely 4.10eV, 14.36eV and 20.54eV for the internuclear distances of 2a.u., 7a.u. and 10a.u., respectively. Based on the three-step model [4], the related energies as indicated in Figs. 3(a)–3(c) are 3.72eV, 13.74eV and 19.81eV, which are close to the calculated values. So the additional lower-energy channel may be viewed as the intermolecular transition at smaller internuclear distance. To verify the above analysis, we further discuss the characters of the lower-energy channel combined with the laser field (depicted by gray line) in Fig. 3(d). The channel appears periodically at the negative laser field (the blue arrow indicated). Recall the description about the electron motion in recombination with neighboring nucleus, the electrons around N₁ nucleus can easily transit to N₂ nucleus in this condition. Therefore the recombination with neighboring nucleus for different nuclei may dominate in different periods compared with the recombination with parent nucleus as shown in Fig. 2(d). Meanwhile, it’s further proved that the lower-energy channel from recombination with neighboring nucleus is mainly attributed to the direct transition between two nuclei.

Fig. 3 (a)–(c) are the classical returning-kinetic-energy maps for $H_{2}^{+}$ with the wavelengths of 800nm (green line), 1200nm (black line) and 1600nm (red line). The internuclear distances are 2a.u., 7a.u. and 10a.u., respectively. (d) The classical returning-kinetic-energy map for $H_{2}^{+}$ with R=2a.u. in the intense laser field (gray line) with the wavelength of 1600nm.

Download Full Size | PDF

For the recombination with parent nucleus, the related physical mechanism is similar with the atomic harmonic emission. Additionally, it is independent on the internuclear distance. Nevertheless, the cutoff energy obtained from the recombination with neighboring nucleus is closely associated with the internuclear distance [37–39]. Moreno et al. [37, 38] have shown that the kinetic energy can be extended to 8U_p when the internuclear distance reaches (2n+1)πE₀/ω² (n is the integer). Based on this mechanism, Lan et al. [39] have theoretically obtained the sub-100 attosecond pulses with the internuclear distance ranging 70a.u. to 100a.u. in proper laser field. M. Lein and J. M. Rost have also proposed that the ultrahigh harmonics can be obtained from ion-atom collisions with the internuclear distances about hundreds of a.u. [40]. Next, we will focus on the MRP originating from recombination with neighboring nucleus at larger internuclear distances.

Figures 4(a)–4(c) offer the quantum calculations for different internuclear distances of 0.5α, 1.0α and 1.5α, respectively. α=E₀/ω² is the quivering amplitude of the electron. The value is about 93a.u. for the adopted laser field with the wavelength of 1600nm and the peak intensity of 2×10¹⁴W/cm², which is far beyond the equilibrium distance. The ionization potential is 0.69a.u. for 0.5α, moreover, it is 0.68a.u.for the internuclear distance larger than α. Under this condition, the diatomic molecular ion model can be viewed as a simple atom-atom model. Here we mainly pay attention to the variation of the first rescattering from neighboring nucleus with the increase of the internuclear distance. The black thick line represents the corresponding returning energy for electron around N₁ nucleus. Notably, the classical results agree well with the quantum ones. On the one side, the cutoff energy obtained via the recombination with neighboring nucleus tends to be higher with the increase of the internuclear distance. It’s understandable that the larger internuclear distance leads to the longer acceleration time for the electron ionized from N₁ nucleus [38–40]. On the other hand, the difference of emission time between two channels for recombination with neighboring nucleus decreases with the increase of the internuclear distance. For the stretched molecule, the interaction between two nuclei is weaker and the direct intermolecular transition is transformed to the ionization-acceleration-recombination process between two atoms (IAR). When the laser field is negative, the potential well of N₁ nucleus is dressed-up while that of N₂ nucleus is dressed-down [22]. So the electron is easy to overcome the constraint of N₁ nucleus (i.e. ionization), and then the electron moves to N₂ nucleus driven by the laser field (i.e. acceleration). Finally, the electron enters into the potential well of N₂ nucleus before the laser field reverses its direction (i.e. recombination). The whole process experiences about 0.5o.c.. With the periodical ionization process in harmonic emission, Fig. 4(d) illustrates the ionization-kinetic-energy maps for 0.5α, 1.0α and 1.5α, from 0.00o.c. to 1.00o.c.. The trajectory with later ionization time before 0.25o.c. corresponds to the short trajectory of the higher-energy (STH) channel (i.e. the short trajectory of the channel 1), which are marked as S₁, S₂ and S₃ for 0.5α, 1.0α and 1.5α, respectively. The corresponding ionization trajectories (IT) of IAR channel after 0.25o.c. are marked as IT₁, IT₂ and IT₃, respectively. With the increase of the internuclear distance, two trajectories above mentioned own close ionization times. For larger internuclear distance, the electron will experience longer time from N₁ nucleus to N₂ nucleus via IAR channel. As a result, the emission time of the IAR channel is close to that of the STH channel. Moreover, the difference of emission times between long and short trajectories is smaller at larger internuclear distance as reported in the Reference [39]. In short, two channels for recombination with neighboring nucleus tend to emit the harmonics at same moment for stretched molecule. What’s more, as shown in the time-frequency maps, much more additional quantum trajectories can also contribute to the harmonic emission at larger internuclear distance. What we wonder is how the multiple rescatterings from neighboring nucleus contribute to the harmonic emission. The following work will provide further discussion.

Fig. 4 (a)–(c) The time-frequency maps for different internuclear distances of 0.5α, 1.0α and 1.5α with same color-coding scale. α is the quiver amplitude of the electron in the laser field. The related wavelength is 1600nm and the peak intensity is 2×10¹⁴W/cm². The black thick line represents the classical returning-kinetic-energy of the first rescattering from neighboring nucleus for electron around N₁ nucleus. (d) The corresponding ionization-kinetic-energy maps.

Download Full Size | PDF

To distinguish these additional quantum trajectories, we present the molecular HHG at 2.0α in Fig. 5. Since the variation of the internuclear distance has less effect on the MRP from recombination with parent nucleus, the corresponding higher-order rescattering events are identical with the case of 2a.u. shown in Fig. 2(b). Apart from the second rescattering from parent nucleus around 1.25o.c., an additional quantum channel with lower energy is noticeable. Furthermore, more additional quantum channels tend to contribute to the harmonic emission after 1.25o.c.. Based on the different emission times of the MRP, the related classical simulations of recombination with neighboring nucleus both for the ionization from N₁ nucleus and N₂ nucleus are provided in Fig. 5(a) based on Eq. (11). The black thin line, black dash-dotted line and green solid line represent the first (t_ir <0.5o.c.), second (0.5o.c.<t_ir <1.0o.c.) and third (1.0o.c.<t_ir <1.5o.c.) rescatterings from recombination with neighboring nucleus for ionization from N₁ nucleus, respectively. While the black dash line, magenta solid line and blue solid line represent the first, second and third rescatterings from recombination with neighboring nucleus for ionization from N₂ nucleus, respectively. The classical results still well correspond to those quantum results. It means that the MRP from recombination with neighboring nucleus can also contribute to the harmonic emission, at the same time, the related high-order rescatterings are also well distinguished. Additionally, when the emission time is larger than 1.5o.c., the quantum trajectories are less smooth as illustrated in Fig. 5(a). To get clear understanding, the MRP both from recombination with parent nucleus and with neighboring nucleus are taken into consideration in Fig. 5(b)), where the red solid line, black dotted line and black thick line represent the first, second and third rescatterings from parent nucleus. It can be seen that the cutoff energy of the first rescattering from parent nucleus is close to that of the third rescattering from neighboring nucleus. As a result, these two rescattering events interfere easily with each other (as indicated by the shadow circle) leading to the less smooth structure after 1.5o.c. in the time-frequency analysis. Moreover, it’s worth noting that the identical high-order rescattering from neighboring nucleus exhibits different cutoff energies periodically, which is significantly different from those of the multiple rescatterings from parent nucleus.

Fig. 5 (a) The 1D time-frequency maps for internuclear distance of 2.0α. (b) The corresponding classical returning-kinetic-energy map. The laser parameters are same as those in Fig. 4. The red solid line, black dotted line and black thick line represent the first, second and third rescatterings from parent nucleus, respectively. The black thin line, black dash-dotted line and green solid line (black dash line, magenta solid line and blue solid line) represent the first, second and third rescatterings from neighboring nucleus for the ionization from N₁ nucleus (N₂ nucleus), respectively.

Download Full Size | PDF

Next, the difference between MRP from recombination with parent nucleus and with neighboring nucleus at larger internuclear distance will be investigated in detail. Figure 6 plots the classical ionization-recombination kinetic-energy maps of multiple rescatterings from recombination with parent nucleus in (a) and with neighboring nucleus in (c) for ionization from N₁ nucleus at 2.0α. For clear observation, the same order rescattering event is just depicted once. The related laser parameters shown in (b) are same with those in Fig. 2. The red dotted line, green dash line and black thin line (magenta solid line, blue dash-dotted line and black dotted line) represent the ionization processes (recombination processes) of the first, second and third rescatterings from parent nucleus. The blue solid line, red solid line and green solid line (magenta dash line, red dash line and black thick line) represent the ionization processes (recombination processes) of the first, second and third rescatterings from neighboring nucleus. It can be observed that the ionization processes of the MRP from recombination with parent nucleus almost take place around the peak intensity. However, it covers about 0.5o.c. for the first rescattering from neighboring nucleus with maximum cutoff energy at R =2.0α. In this case, the short trajectory of the channel 1 overlaps with the IAR channel. And then the second and the third rescatterings exhibit different characters with those rescatterings from parent nucleus. As shown in Fig. 6(c), there are two kinds of second and third rescatterings from neighboring nucleus, and the cutoff energies are different for the same order rescattering event. For example, the cutoff energy of the second rescattering from neighboring nucleus around 1.25o.c. is different from that around 1.75o.c.. With the classical ionization kinetic-energy maps, we can know that the first kind of the second rescattering (FSR) event around 1.25o.c. originates from the long trajectory of the channel 1, while the second kind around 1.75o.c. (SSR) from the IAR channel. Moreover, the cutoff energy of the first one is lower than that of the second one, and both energies are lower than that of the first rescattering. To obtain clear physical mechanism, the electron motion is demonstrated in Fig. 6(b). The electron around N₁ nucleus is first ionized around A₁, and then experiences acceleration-deceleration process. The velocity (v) is decreased to 0 around A₂, additionally, tA₁=tA₂ considering the symmetry of the laser field. Finally, the electron reverses its direction and is reaccelerated from A₂ to recombine with N₂ nucleus around A₃. Thus the so-called acceleration process actually takes place from A₂ to A₃ in HHG process. Considering the kinetic energy (E_k) obtained in the external field satisfying with $E_{k} = \frac{1}{2} v^{2}$ and v = − ∫ E(t)dt, we can adopt the shaded area S_A to estimate the cutoff energy of the first rescattering indirectly. With the same mechanism, the electron is ionized around B₁ (C₁) for the FSR event (SSR event), and v is decreased to 0 around B₂ (C₂) with t_B1=t_B2 (t_C1=t_C2). Around B₃ (C₃), the electron recombines with N₂ nucleus. And then S_B and S_C can be employed to estimate the corresponding cutoff energies. Notably, S_A>S_C >S_B in Fig. 6(b), it means that the cutoff energies of the first rescattering, SSR and FSR decrease in turn. Furthermore, this method can also be used to explain the variations about the cutoff energies both of the third rescattering from neighboring nucleus and those multiple rescatterings from parent nucleus.

Fig. 6 (a) Multiple rescattering events from recombination with parent nucleus for ionization from N₁ nucleus. The red dotted line, green dash line and black thin line (magenta solid line, blue dash-dotted line and black dotted line) represent the ionization processes (recombination processes) of the first, second and third rescatterings. (b) The sketch of laser field with same parameters in Fig. 1. (c) Multiple rescattering events from recombination with neighboring nucleus for ionization from N₁ nucleus. The blue solid line, red solid line and green solid line (magenta dash line, red dash line and black thick line) represent the ionization processes (recombination processes) of the first, second and third rescatterings.

Download Full Size | PDF

The harmonic emission at smaller internuclear distance exhibits similar characters both in 1D simulation and 3D simulation as shown in Fig. 2, how about the case of larger internuclear distance? Figure 7 demonstrates the harmonic emission for the internuclear distance of 2.0α based on the 3D simulation. Obviously, the time-frequency in Fig. 7(a) shows similar structure with that in Fig. 5(a) based on 1D simulation. Moreover, two plateau regions can be observed in the harmonic spectra both in 1D case (blue line) and 3D case (red line) displayed in Fig. 7(b). One is from recombination with parent nucleus while the other is from recombination neighboring nucleus. The corresponding harmonic intensity from recombination with neighboring nucleus is almost one order of magnitude lower than that from recombination with parent nucleus. It means that the recombination from neighboring nucleus still plays an important role in harmonic emission for larger internuclear distance in 3D case.

Fig. 7 (a) The 3D time-frequency maps for internuclear distance of 2.0α. (b) The related harmonic spectra for 1D (blue line) and 3D (red line).

Download Full Size | PDF

4. Conclusion

In summary, the multiple rescattering processes of stretched molecule have been investigated by numerically solving the one-dimensional and three-dimensional time-dependent Schrödinger equation in intense laser field. The multiple rescattering events originating from recombination with parent nucleus exhibit similar characters with those of the atomic system. However, those originating from recombination with neighboring nucleus are more sensitive to variation of the internuclear distance. Moreover, there are two channels recombining with neighboring nucleus to contribute to the harmonic emission before and after reversing the direction of the laser field, respectively. Additionally, the boundary of the corresponding first rescatterings becomes weaker with increasing the internuclear distance. They can both induce higher-order rescattering events with different cutoff energies periodically, which is greatly different from those recombination with parent nucleus. Looking ahead, the multiple rescatterings originating from recombination with neighboring nucleus may play an important role in the harmonic generation from the complicated molecules or atom clusters with large internuclear distance. The simple model in this work would provide the foundation to further investigate the dynamics of more complicated systems in strong-filed physics.

Funding

National Natural Science Foundation of China (NSFC) (Grant No.11404204, 11447208, 11504221), Natural Science Foundation for Young Scientists of Shanxi Province, China (Grant No. 2015021023), and Program for the Top Young Academic Leaders of Higher learning Institutions of Shanxi Province, China.

Acknowledgments

The authors sincerely thank Prof. Keli Han and Dr. Ruifeng Lu for providing the LZH-DICP code.

References and links

1. T. Brabec and F. Krausz, “Intense few-cycle laser fields:frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]

2. M. R. Miller, A. Jaroń-Becker, and A. Becker, “High-harminc spectroscopy of laser-driven nonadiabatic electron dynamics in the hydrogeen molcecular ion,” Phys. Rev. A 93(1), 013406 (2016). [CrossRef]

3. M. V. Frolov, N. L. Manakov, T. S. Sarantseva, M. Yu Emelin, M. Yu Ryabikin, and A. F. Starace, “Analytic description of the high-energy plateau in harmonic generation by atoms:can the harmonic power increase with increasing laser wavelenghs?” Phys. Rev. Lett. 102(24), 243901 (2009). [CrossRef] [PubMed]

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

5. Z. H. Jiao, G. L. Wang, P. C. Li, and X. X. Zhou, “Polarization control of ultrabroadband supercontinuum generation from middinfrared laser-induced harmonic emission,” Phys. Rev. A 90(2), 025401 (2014). [CrossRef]

6. Y. H. Wang, C. Yu, Q. Shi, Y. D. Zhang, X. Cao, S. C. Jiang, and R. F. Lu, “Reexamination of wavelength scaling of harmonic yield in intense midinfrared fields,” Phys. Rev. A 89(2), 023825 (2014). [CrossRef]

7. Y. H. Wang, H. P. Wu, Y. Qian, Q. Shi, C. Yu, Y. Chen, K. M. Deng, and R. F. Lu, “Laser-parameter effects on the generation of ultrabroad harmonic and ultrashort attosecond pulse in a long-pulse-short scheme,” J. Mod. Opt. 59(19), 1640–1649 (2012). [CrossRef]

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

9. Z. Wang, L. X. He, J. H. Luo, P. F. Lan, and P. X. Lu, “High-order harmonic generation from Rydberg atoms in inhomogeneous fields,” Opt. Express 22(21), 25909–25922 (2014). [CrossRef] [PubMed]

10. J. Wang, G. Chen, S. Y. Li, D. J. Ding, J. G. Chen, F. M. Guo, and Y. J. Yang, “Ultrashort-attosecond-pulse generation by reducing harmonic chirp with a spatially inhomogenous electric field,” Phys. Rev. A 92(3), 033848 (2015). [CrossRef]

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

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

13. L. X. He, P. F. Lan, Q. B. Zhang, C. Y. Zhai, F. Wang, W. J. Shi, and P. X. Lu, “Spectrally resolved spatiotemporal features of quantum paths in high-order-harmonic generation,” Phys. Rev. A 92(4), 043403 (2015). [CrossRef]

14. A. Picón, A. Bahabad, H. C. Kapteyn, M. M. Murnane, and A. Backer, “Two-center interferences in photoionization of a dissociating $H_{2}^{+}$ molecule,” Phys. Rev. A 83(1), 013414 (2011). [CrossRef]

15. N. T. Nguyen, V. H. Hoang, and V. H. Le, “Probing nuclear vibration using high-order harmonic generation,” Phys. Rev. A 88(2), 023824 (2013). [CrossRef]

16. Z. Wang, K. L. Liu, P. F. Lan, and P. X. Lu, “Steering the electron in dissociating $H_{2}^{+}$ via manipulating two-sate population dynamics by a weak low-frequency field,” Phys. Rev. A 91(4), 043419 (2015). [CrossRef]

17. I. Ben-Itzhak, I. Gertner, O. Heber, and B. Rosner, “Experimental evidence for the existence of the 2pσ bound state of HeH²⁺ and its decay mechanism,” Phys. Rev. Lett. 71(9), 1347–1350 (1993). [CrossRef] [PubMed]

18. X. B. Bian and A. D. Bandrauk, “Nonadiabatic molecular high-order harmonic generation from polar molecules: spectral redshift,” Phys. Rev. A 83(4), 041403 (2011). [CrossRef]

19. X. B. Bian and A. D. Bandrauk, “Orientation dependence of nonadiabatic molecular high-order-harmonic generation from resonant polar molecules,” Phys. Rev. A 86(5), 053417 (2012). [CrossRef]

20. Y. J. Chen, L. B. Fu, and J. Liu, “Asymmetric molecular imaging through decoding odd-even high-order harmonics,” Phys. Rev. Lett. 111(7), 073902 (2013). [CrossRef] [PubMed]

21. B. Zhang, S. J. Yu, Y. J. Chen, X. Q. Jiang, and X. D. Sun, “Time-resolved dynamics of odd and even harmonic emission from orited asymmetric molecules,” Phys. Rev. A 92(5), 053833 (2015). [CrossRef]

22. X. Y. Miao and C. P. Zhang, “Multichannel recombination in high-order-harmonic generation from asymmetric molecular ions,” Phys. Rev. A 89(3), 033410 (2014). [CrossRef]

23. X. Y. Miao and C. P. Zhang, “Manipulation of the recombination channels and isolated attosecond pulse generation from HeH²⁺ with multicycle combined field,” Laser Phys. Lett. 11(11), 115301 (2014). [CrossRef]

24. J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98(1), 013901 (2007). [CrossRef] [PubMed]

25. D. D. Hickstein, P. Ranitovic, S. Witte, X. M. Tong, Y. Huismans, P. Arpin, X. B. Zhou, K. E. Keister, C. W. Hogle, B. Zhang, C. Y. Ding, P. Johnsson, N. Toshima, M. J. J. Vrakking, M. M. Murnane, and H. C. Kapteyn, “Direct visualizaiotn of laser-driven electron multiple scattering and tunneling disstance in strong-field ionization,” Phys. Rev. Lett. 109(7), 073004 (2012). [CrossRef] [PubMed]

26. L. X. He, Y. Li, Z. Wang, Q. B. Zhang, P. F. Lan, and P. X. Lu, “Quantum trajectories for high-order-harmonic generation from multiple rescatteing events in the long-wavelength regime,” Phys. Rev. A 89(5), 053417 (2014). [CrossRef]

27. A. T. Le, H. Wei, C. Jin, V. N. Tuoc, T. Morishita, and C. D. Lin, “Universality of returning electron wave packet in high-order harmonic generation with midinfrared laser pulses,” Phys. Rev. Lett. 113(3), 033001 (2014). [CrossRef] [PubMed]

28. A. T. Le, H. Wei, C. Jin, and C. D. Lin, “Strong–field approximation and its extension for high-order harmonic generation with mid-infrared lasers,” J. Phys. B: At. Mol. Opt. Phys. 49(5), 053001 (2016). [CrossRef]

29. M. R. Miller, C. Hernández-García, A. Jaroń-Becker, and A. Becker, “Targeting multiple rescattering through VUV-controlled high-order-harmonic generation,” Phys. Rev. A 90(5), 053409 (2014). [CrossRef]

30. C. Hernández-García, J. A. Pérez-Hernández, T. Popmintchev, M. M. Murnane, H. C. Kapteyn, A. Jaroń-Becker, A. Becker, and L. Plaja, “Zeptosecond high harmonic KeV X-Ray waveforms driven by midinfrared laser pulses,” Phys. Rev. Lett. 111(3), 033002 (2013). [CrossRef] [PubMed]

31. 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 77(6), 066701 (2008). [CrossRef]

32. J. Hu, K. L. Han, and G. Z. He, “Correlation quantum dynamics between an electron and $D_{2}^{+}$ molecule with attosecond resolution,” Phys. Rev. Lett. 95(12), 123001 (2005). [CrossRef] [PubMed]

33. L. Q. Feng, “Molecular harmonic extension and enhancement from $H_{2}^{+}$ ions in the presence of spatially inhomogeneous fields,” Phys. Rev. A 92(5), 053832 (2015). [CrossRef]

34. T. S. Chu, Y. Zhang, and K. L. Han, “The time-dependent quantum wave packet approach to the electronically nonadiabatic processes in chemical reaction,” Int. Rev. Phys. Chem. 25(1–2), 201–235 (2006). [CrossRef]

35. X. Y. Miao and H. N. Du, “Theoretical study of high-order-harmonic generation from asymmetric diatomic molecules,” Phys. Rev. A 87(5), 053403 (2013). [CrossRef]

36. R. F. Lu, Y. Chao, Y. H. Wang, Q. Shi, and Y. D. Zhang, “Control of electron localization to isolate and enhance molecular harmonic plateau in asymmetric HeH²⁺ system,” Phys. Lett. A 378(1–2), 90–94 (2014). [CrossRef]

37. P. Moreno, L. Plaja, and L. Roso, “High-order harmonic generation by electron-proton recombination,” Europhys. Lett. 28(9), 629–633 (1994). [CrossRef]

38. P. Moreno, L. Plaja, and L. Roso, “Ultrahigh harmonic generation from diatomic molecular ions in highly excited vibrational states,” Phys. Rev. A 55(3), R1593–R1596 (1997). [CrossRef]

39. P. F. Lan, P. X. Lu, W. Cao, X. L. Wang, and G. Yang, “Phase-locked high-order-harmonic and sub-100-as pulse generation from stretched molecules,” Phys. Rev. A 74(6), 063411 (2006). [CrossRef]

40. M. Lein and J. M. Rost, “Ultrahigh harmonics from laser-assisted ion-atom collisions,” Phys. Rev. Lett. 91(24), 243901 (2003). [CrossRef] [PubMed]

41. Y. Chao, H. X. He, Y. H. Wang, Q. Shi, Y. D. Zhang, and R. F. Lu, “Isolated few-attosecond emission in a multi-cycle asymmetrically nonhomogeneous two-color laser field,” J. Phys. B: At. Mol. Opt. Phys. 47(22), 055601 (2014).

Multiple rescattering processes in high-order harmonic generation from molecular system

Abstract

1. Introduction

2. Theoretical method

3. Results and discussion

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (7)

Equations (11)

Optics Express