Chinese Journal of Chemical Physics  2018, Vol. 31 Issue (4): 471-476

The article information

Hong-jing Liang, Qiao-xia Wang, Xin Fan, Li-yu Shan, Shuang Feng, Bing Yan, Ri Ma, Hai-feng Xu
梁红静, 王巧霞, 范鑫, 单立宇, 冯爽, 闫冰, 马日, 徐海峰
High-order Harmonic Generation of Aligned Acetylene in Elliptically Polarized Strong Laser Fields
椭圆偏振强激光场中准直的乙炔分子的高次谐波发射
Chinese Journal of Chemical Physics, 2018, 31(4): 471-476
化学物理学报, 2018, 31(4): 471-476
http://dx.doi.org/10.1063/1674-0068/31/cjcp1805117

Article history

Received on: May 25, 2018
Accepted on: July 20, 2018
High-order Harmonic Generation of Aligned Acetylene in Elliptically Polarized Strong Laser Fields
Hong-jing Liang, Qiao-xia Wang, Xin Fan, Li-yu Shan, Shuang Feng, Bing Yan, Ri Ma, Hai-feng Xu     
Dated: Received on May 25, 2018; Accepted on July 20, 2018
Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China
Author: Hai-feng Xu is now a professor in Institute of Atomic and Molecular Physics in Jilin University, China. He obtained his B.S. degree from Department of Chemical Physics in University of Science and Technology of China in 1997, and the Ph.D. degree from the same department in 2002 under supervision of Prof. Xin-xiao Ma and Prof. Shi-lin Liu. After that, Dr. Xu worked as a postdoctor successively in UC-Davis, StonyBrook University and University of Maryland at College Park. In 2008, Dr. Xu joined Jilin University and was promoted to professor in 2010. His research interest is focused on the dynamics of molecular electronic excited states as well as the atomic/molecular physics in ultrafast strong laser fields. To date, Dr. Hai-feng Xu has coauthored with more than 70 publications, and is PI of several projects granted from National Natural Science Foundation of China, Ministry of Education of China, Department of Science and Technology of Jilin Province, etc.
*Author to whom correspondence should be addressed. Ri Ma, E-mail:rma@jlu.edu.cn; Hai-feng Xu, E-mail:xuhf@jlu.edu.cn
Part of the special issue for celebration of "the 60th Anniversary of University of Science and Technology of China and the 30th Anniversary of Chinese Journal of Chemical Physics"
Abstract: We perform an experimental study on high-order harmonic generation (HHG) of aligned acetylene molecules induced by a 35-fs 800-nm strong laser field, by using a home-built HHG spectrometer. It is observed that the molecular HHG probability declines with increasing the laser ellipticity, which is in consistence with the deduction from the well-known tunneling-plus-rescattering scenario. By introducing a weak femtosecond laser pulse to nonadiabatically align the molecules, we investigated the molecular orbital effect on the HHG in both linearly and elliptically polarized driving laser fields. The results show that the harmonic intensity is maximum for the molecular axis aligned perpendicularly to the laser electric field. It indicates that both the highest occupied molecular orbitals (HOMO) and HOMO-1 contribute to the strong-field HHG of acetylene molecules. Our study should pave the way for understanding the interaction of molecules with ultrafast strong laser fields.
Key words: High-order harmonic generation    Aligned molecule    Elliptically polarized strong laser    
Ⅰ. INTRODUCTION

When atoms or molecules are subjected to a laser field with pulse duration of tens of femtoseconds and peak intensity over 10$^{13}$ W/cm$^{2}$, various non-perturbative and highly non-linear physical processes could be induced, such as high-order above threshold ionization (HATI), non-sequential double ionization (NSDI), high-order harmonic generation (HHG), neutral high Rydberg state excitation (RSE), and so on [1-8]. Among them, HHG process has attracted intense research interest during the past decades, in which an atom or a molecule absorbs energy from a strong-laser pulse to emit radiation with photon-energy of many multiples of that of the driving laser pulse. HHG emerges as a versatile spectroscopic method for monitoring chemical reactions [9], imaging molecular orbitals [10], and developing novel table-top ultrafast extreme ultraviolet light sources and advanced attosecond lasers [11].

It is now well-known that the underlying mechanism of gas-phase atomic/molecular HHG in ultrafast strong laser fields can be understood in the frame of the so-called three-step re-scattering process [12]: an electron in an atom or a molecule first tunnels out through the barrier formed by combination of the laser field and the Columb potential, then it is accelerated and can be driven back to the parent ion by the oscillating laser field. HHG photons with high energy could be emitted if the tunneling electron recombines to the ground state of the atom/molecule. Based on this scenario, one can expect that the HHG probability is declined in an elliptically polarized laser field, since the recombination with the parent ion diminishes due to the drift momentum spread of the returning electron wave-packet. On the other hand, the investigation on the behavior of atoms or molecules in elliptically polarized laser fields could be important to reveal and control the dynamics of the strong laser-driven electron re-scattering process, for example, effect of "long orbit" of the returning electron (or multiple-return collision trajectory) in tunneling-plus-rescattering processes [13, 14]. Comparing to atoms, molecules have much more complicated geometric and electronic-state structure, as well as additional nuclear vibrational and rotational motions. The interaction of molecules with strong laser fields thus exhibits a large variety of peculiar behaviors, of which the underlying physics is far from completely understood [15-17]. For molecular HHG in strong laser fields, several experimental and theoretical studies have shown that the molecular properties, such as geometric structure, molecular orbital and molecular vibration, play a significant role in respect to the yield, the plateau range, and the cut-off energy of HHG [18-20]. Moreover, understanding the molecular structure effect in strong-field HHG is also of vital importance in application of tomographic imaging of individual molecular orbitals [10] and attosecond pulse generation [11].

Here, we experimentally investigate the HHG spectra of a small linear molecule, acetylene (C$_{2}$H$_{2}$), in femtosecond strong laser fields. There are several studies reported in the literature on the HHG of C$_{2}$H$_{2}$. In 2010, Vozzi et al. investigated HHG of various hydrocarbons (including C$_{2}$H$_{2}$) with ionization potentials ($I_{\text{p}}$) in the range 9.07-11.52 eV using few-cycle mid-infrared strong laser pulses [21]. The results show signatures of the molecular structure in HHG and extend the high harmonic spectroscopy to complex molecules. It has also been demonstrated that a broad HHG energy range up to 60 eV can be experimentally achieved using longer wavelengths of the strong laser (i.e., 1000-1500 nm), especially for molecules with relatively low $I_{\text{p}}$ (such as C$_{2}$H$_{2}$) [22]. Later, Negro et al. reported the molecular orbital tomography of nitrous oxide (N$_{2}$O) as well as C$_{2}$H$_{2}$ using high-order harmonic spectroscopy [23]. Very recently, Mulholland and Dundas performed a theoretical study on HHG of C$_{2}$H$_{2}$ using a mixed quantum-classical approach, the results of which imply the significant effect of different structure of the molecular orbitals on molecular HHG [24]. Nevertheless, further experimental and theoretical studies on the HHG of C$_{2}$H$_{2}$ are still in demand, especially for the aligned molecules in elliptically polarized strong laser fields which has been sparsely investigated in the literature.

In our work, we measure the HHG of aligned C$_{2}$H$_{2}$ molecules induced by a 35-fs 800-nm strong laser field, by using a home-built HHG spectrometer. The harmonic yield is investigated for different laser ellipticity as well as the angle between the molecular axis and the laser field. Based on our experimental results and the previous investigations in the literature, we discuss the molecular orbital effect on the HHG of C$_{2}$H$_{2}$ in both linearly and elliptically polarized driving laser fields.

Ⅱ. EXPERIMENTS

The home-built spectrometer for generation and detection of HHG induced by strong laser fields have been described in detail in our previous studies [25, 26]. Briefly, an output from a Ti: sapphire laser system (Spectra-Physics, ACE 35FIK) with a center 35 fs and a repetition rate of 1 kHz, is split into two beams by a 30:70 beam splitter. The weaker laser beam (namely, the aligning laser), which was linearly polarized, was used to induce nonadiabatic alignment of acetylene molecules. The stronger laser beam (namely, the driving laser), which was either linearly or elliptically polarized, was used to interact with the aligned molecules to drive HHG. The time delay between the two laser beams was manipulated by a computer-controlled translation stage. The diffraction efficiency of spherical grating (Shimadzu 30 002) depends on the angle between the groove of the grating and the harmonics polarization direction. In order to minimize this influence, the ellipticity of the driving laser was controlled by varying an 800 nm zero-order half-wave plate before it passed an 800 nm zero-order quarter-wave plate which was fixed. In this way the major axis of the elliptical polarization plane was kept parallel to the groove of the spherical grating. The intensity of each laser beam was adjusted by a half-wave plate and a Glan-laser polarizer. An adjustable iris was used to vary the beam size of the aligning laser bigger than that of the driving laser in the focusing region of supersonic gas jet. The two laser beams were collinearly focused into the interaction vacuum chamber by a lens (f=300 mm). The peak intensity of the driving laser was about 5$\times10^{13}$-7$\times10^{13}$ W/cm$^{2}$, and that of the aligning laser was kept less than 10$^{13}$ W/cm$^{2}$ to avoid any strong-field processes. Acetylene molecules were introduced into the interaction chamber through a nozzle with a diameter of 100 μm. The density of gas in the interaction region was about $10^{17}$ cm$^{-3}$ with estimated rotational temperature less than 100 K. After irradiated by the driving laser, HHG in the gas jet was reflected by an Au-coated mirror into home-made flat field XUV spectrometer, then was spectrally resolved by a spherical grating and finally was imaged on a microchannel plate detector equipped with a phosphor screen. A CCD camera (Hamamatsu ORCA R2) with high dynamic range was used to record HHG signals.

Ⅲ. RESULTS AND DISCUSSION

FIG. 1 presents part of the HHG spectrum of acetylene (C$_{2}$H$_{2}$) molecules irradiated by a 35-fs, 800-nm linearly polarized driving laser with peak intensity of 6$\times10^{13}$ W/cm$^{2}$. We use notation $H_{x}$ to represent the xth order harmonic. Because of the symmetry restriction, only the odd-order harmonics can be observed in the HHG spectrum. In our study we investigate the HHG intensity under different ellipticity of the driving laser as well as the delay between the aligning and the driving lasers. It should be mentioned that the main results obtained in the study are similar for different $H_{x}$, thus only the results for $H_{15}$ are presented in the following discussion.

FIG. 1 Upper panel: part of the HHG spectrum of C$_{2}$H$_{2}$ molecules irradiated by a 35-fs 800-nm linearly polarized driving laser with peak intensity of 6$\times$10$^{13}$ W/cm$^{2}$. (Lower panel): the corresponding integrated HHG intensity. The order of the harmonics are labeled as $H_{x}$ in the figure.

We first measured the intensity of $H_{15}$ as a function of the ellipticity of the driving laser. As shown in FIG. 2, the probability of HHG depends strongly on the driving laser polarization: the intensity has a maximal value in linearly polarized laser fields ($\varepsilon$=0) and completely disappears at the laser ellipticity $\varepsilon$$\approx$0.5. This observation is in consistence with the deduction from a tunneling-plus-rescattering process in strong laser fields, which can be attributed to the greater drift momentum spread of the electron wave packet in the elliptically polarized laser fields. According to the quasistatic strong field approximation (SFA), the electron at the tunnel exit has zero longitudinal and nonzero initial transverse velocity $v_{\text{0}}$. For maximal HHG probability, the tunneling electron needs to return to the tunnel exit ($ x=y=z=0, \ z$ axis refers to the direction of the laser electric field), to recombine with the parent ion. In an elliptically polarized laser field, additional drift motion of the tunneling electron is induced by the laser field. This indicates that the electrons released with $v_{0}$=0 will never return to $ x=y=0$. And only those electrons with non-zero $v_{\text{0}}$ that can compensate the drift motion allow the return. In other words, the required $v_{0}$ is higher for larger ellipticity. On the other hand, the distribution of $v_{0}$ for the tunneling ionization is a Gaussian distribution with maximum centered at $v_{0}$=0. As a result, the HHG yield is suppressed with increasing the laser ellipticity.

FIG. 2 Yield of C$_{2}$H$_{2}$ HHG vs. ellipticity of the strong 800 nm laser field. Black circles are the present experimental results and the blue solid curve is the analytical results based on the SFA model. The experimental results are normalized to the maximal value at linearly polarization. See text for the detail.

According to this SFA model, an analytical result is derived for describing the dependence of HHG yield ($I_{\text{HHG}}$) on the laser ellipticity [27]:

$ \begin{eqnarray}\frac{{{I_{{\rm{HHG}}}}(\varepsilon )}}{{{I_{{\rm{HHG}}}}(\varepsilon = 0)}} \approx \exp \left( { - \frac{{1.25\sqrt {2{I_{\rm{p}}}} E}}{{{\omega ^2}}}{\varepsilon ^2}} \right) \end{eqnarray} $ (1)

where $I_{\text{p}}$, $E$, and $\omega$ (in atomic units, a.u.) refer to the ionization potential of the target, the electric field amplitude and the frequency of the driving laser respectively. For C$_{2}$H$_{2}$ $(I_{\text{p}}$=11.4 eV=0.42 a.u.), the dependence of the HHG signal on ellipticity obtained using the above equation based on SFA is also presented in FIG. 2 (solid line). It can be seen that the experimental data are in good agreement with the result from the SFA model, which further supports the fact that strong-dependence of HHG signal on laser ellipticity can be understood in the tunneling-plus-rescattering frame.

In FIG. 3, we present the intensity of $H_{15}$ as a function of time delay between the aligning and the driving laser pulses. Both lasers are linearly polarized and the directions of their polarizations are kept parallel. The intensity is normalized to that without alignment. The weak aligning laser pulse first interacts with the target and gives the molecule a "kick" force. This introduces a coherent rotational-state population that revives periodically after the aligning laser pulse is turned off. The effect of such "field-free" non-adiabatic impulsive alignment on the molecules can be observed as the intensity modulation of the signal vs. the time delay. As shown in FIG. 3, the maximum modulation of the harmonic intensity occurs at the time delay of ~7 and ~14 ps, corresponding to a half revival at $T_{\text{rot}}/2$ and a full revival at $T_{\text{rot}}$ respectively ($T_{\text{rot}}$ is the rotational period of C$_{2}$H$_{2}$ which is 14.2 ps). At these delays between the driving and aligning pulses, the molecules are maximally aligned around the aligning electric field. The time delay of 7.0 ps, which was used in the following measurements, corresponds to the molecular axis aligned along the polarization of the aligning pulse.

FIG. 3 The harmonic intensity as a function of time delay between the aligning and the driving laser pulses. The HHG recorded from the aligned molecules are normalized to that of the unaligned ones. The delays of half revival and full revival are labeled in the figure. See text for the detail.

In order to further understand the molecular structure effect on the strong-field HHG, we investigated the harmonic intensity of the aligned C$_{2}$H$_{2}$ in elliptically polarized driving laser fields. In FIG. 4, we show the harmonic intensity as a function of the angel $\theta$ between the major axis of the ellipse of the driving field and the polarization direction of the aligning field. The HHG recorded from the aligned molecules is normalized to that of the unaligned ones. The harmonic signals recorded by linearly polarized ($\varepsilon$=0) and elliptically polarized ($\varepsilon$=0.15 and 0.3) driving laser fields are presented in the figure. We can see from FIG. 4 that for either of the ellipticities of the driving laser, the harmonic intensity increases monotonically as the $\theta$ angle increases. The harmonic intensity is maximum (minimum) for the molecular axis is aligned perpendicular (parallel) to the laser electric field. Another characteristic is that for a larger laser ellipticity, less difference is observed between the harmonic intensity at the perpendicular alignment ($\theta$=90$^{\circ}$) and at the parallel alignment ($\theta$=0$^{\circ}$).

FIG. 4 Harmonic intensity as a function of the angel between the major axis of the ellipse of the driving field and the polarization direction of the aligning field, with linearly polarization ($\varepsilon$=0) and elliptically polarization ($\varepsilon$=0.15 and 0.3). The HHG recorded from the aligned molecules are normalized to that of the unaligned ones.

As we have aforementioned, HHG induced by strong laser fields is directly related to the tunneling ionization probability. The HHG signal from aligned molecules reflects the structure of the molecular orbital from which the electron tunnels out. C$_{2}$H$_{2}$ is a linear molecule with the ground-state electron configuration of $(1\sigma_{\text{g}})^{2}$$(1\sigma_{\text{u}})^{2}$$(2\sigma_{\text{g}})^{2}$$(2\sigma_{\text{u}})^{2}$$(3\sigma_{\text{g}})^{2}$$(1\pi_{\text{u}})^{4}$. FIG. 5 shows the structure of the highest occupied molecular orbitals, HOMO and HOMO-1 of C$_{2}$H$_{2}$ calculated by the density functional theory (DFT) method at the $\text{B3LYP/6-311++G}^{**}$ level using the Gaussian 03 [28] program and the molecular orbital plotted with Multiwfn package [29]. The HOMO of C$_{2}$H$_{2}$ is a two-fold degenerate $\pi_{\text{u}}$ orbital (FIG. 5 (a) and (b)), while the HOMO-1 is a $\sigma_{\text{g}}$ orbital (FIG. 5(c)). According to Koopmans' theorem, ionization from these orbitals results in the ground and the first excited electronic states of the cations respectively. Our calculated energies for these cationic states are 11.40 and 16.80 eV above the neutral ground electronic state respectively, which are in good agreement with the experimental results [30].

FIG. 5 Structures of HOMO ((a) and (b)) and HOMO-1 (c) of C$_{2}$H$_{2}$ calculated by the DFT method at the $\text{B3LYP/6-311++G}^{**}$ level.

It can be seen that the electron density of the two-fold degenerate $\pi_{\text{u}}$ HOMO of C$_{2}$H$_{2}$ lies perpendicular to the molecular axis, and there is a nodal plane along the axis (see FIG. 5 (a) and (b)). This indicates that the ionization probability (and the corresponding HHG intensity) from the HOMO would be greatly suppressed if C$_{2}$H$_{2}$ is aligned parallel to the driving laser ($\theta$=0$^{\circ}$) and would be maximal at the perpendicular alignment ($\theta$=90$^{\circ}$). On the other hand, the electron density of the $\sigma_{\text{g}}$ HOMO-1 of C$_{2}$H$_{2}$ is along the molecular axis (see FIG. 5 (c)). Thus one can expect an opposite effect on the HHG, i.e., the HHG signal is mainly from the HOMO-1 if the molecule is aligned parallel to the driving laser. It is worth mentioning that in a strong laser field, an electron in the HOMO of a molecule has the largest ionization probability. Thus the HHG signal would be increased from the parallel alignment to the perpendicular alignment, as we have observed in the experiments (FIG. 4). Our study clearly shows the effect of molecular orbitals on the HHG of C$_{2}$H$_{2}$ induced by strong 800 nm femtosecond laser fields.

The contribution of different molecular orbitals on the HHG of C$_{2}$H$_{2}$ was theoretically investigated very recently, in which a longer wavelength of 1450 nm was employed as the driving laser [24]. Our experimental results are qualitatively consistent with this theoretical prediction. We should also mention that the difference in the measured HHG signal between the parallel and the perpendicular alignments is not so significant, comparing to the theoretical results [24]. A possible reason is that in experiments, one can never achieve completely pure alignment of the molecule axis parallel or perpendicular to the laser, indicating that both HOMO and HOMO-1 could be attributed to the observed HHG for either $\theta$=0$^{\circ}$ or $\theta$=90$^{\circ}$ thus may smear out the difference. More importantly, the difference between the parallel and the perpendicular alignments becomes more significant for higher orders of the harmonics, especially close to the cut-off energy at $I_{\text{p}}$+3.17$U_{\text{p}}$, where $U_{\text{p}}$ is the ponderomotive energy of the electron obtained from the oscillating laser field (${U_{\rm{p}}}$=${e}/{m_{\rm{e}}}{(E\lambda /4\pi {\rm{c}})^2}$, where e, $m_{\text{e}}$, $\text{c}$, $E$, and $\lambda$ are the electron charge, the electron mass, the speed of light, the electric field amplitude, and the laser wavelength, respectively) [24]. Thus one expects a more significant effect of the molecular orbital effect if measuring the HHG with higher photon energy.

We now turn to discuss the different behavior of HHG of the aligned C$_{2}$H$_{2}$ molecules in the elliptically polarized driving laser fields. Generally, the HHG probability decreases as the laser ellipticity is increased (see FIG. 2), due to the additional drift motion of the tunneling electron induced by the x-component of the electric force of the elliptically polarized laser field. On the other hand, the HHG signal is enhanced at the parallel alignment but is more suppressed at the perpendicular alignment in the elliptically polarized driving laser fields comparing to that in the linearly polarized driving fields (FIG. 4). This fact indicates that the HOMO-1 of C$_{2}$H$_{2}$ may have a more significant contribution to HHG driving by an elliptically polarized laser. To the best of our knowledge, this is the first report to show the evidence for molecular orbital effect on HHG in elliptically polarized strong laser fields. However, its underlying physics couldn't be revealed without insight theoretical studies, which is very challenging considering complicated molecular structure and large calculating box required for elliptically polarized laser fields, which is beyond the study in this paper. In addition, the molecular orbital effect on HHG may also lead to multi-center interference, which has been identified in several linear molecules in the harmonic spectrum near the cut-off energy [21-23]. While this effect is not observed in the present study, it is expected to play a role in the case of C$_{2}$H$_{2}$ if a driving laser with longer wavelength (for example, mid-IR) is employed and the HHG signal with higher energy is detected. We leave these interesting open questions to further experimental and theoretical studies, which would be important to shed more light on the interaction of molecules with strong laser fields.

Ⅳ. CONCLUSION

In summary, the harmonic signal of aligned C$_{2}$H$_{2}$ molecules irradiated by a 35-fs 800-nm strong laser field was investigated experimentally by using a home-built HHG spectrometer. Strong dependence of the molecular HHG probability on the laser ellipticity was observed, which agrees well with the prediction of the SFA model. It was observed that the harmonic yield increases monotonically as the angle between the molecular axis and the laser field increases, in both linearly and elliptically polarized driving laser fields. Our results strongly indicate the effect of molecular orbitals in strong laser HHG of C$_{2}$H$_{2}$ molecules. Further studies on this interesting subject is still in demand, both experimentally and theoretically, particular for high energy HHG of aligned molecules driving by an elliptically polarized laser field with a long wavelength.

Ⅴ. ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (No.U1532138, No.11474130, and No.91750104) and the Natural Science Foundation of Jilin Province, China (No.20180101289JC and No.20160101332JC).

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椭圆偏振强激光场中准直的乙炔分子的高次谐波发射
梁红静, 王巧霞, 范鑫, 单立宇, 冯爽, 闫冰, 马日, 徐海峰     
吉林大学原子与分子物理研究所, 长春 130012
摘要: 本文利用自制的高次谐波光谱仪,实验研究了35-fs 800-nm强激光场诱导的准直的乙炔分子的高次谐波发射(HHG).实验观测到分子HHG几率随着激光椭偏率的增加而降低,这与隧穿后重散射的理论框架的预言是一致的.通过引入一束弱飞秒激光作用对分子非绝热准直,研究了线偏振和椭圆偏振光场下分子结构效应对HHG的影响.结果表明,分子轴垂直于激光电场方向时谐波产率最大.分析表明,最高占据分子轨道(HOMO)以及内层轨道(HOMO-1)对乙炔分子的强场HHG都有贡献.本文将有助于深入理解分子与超快强激光场的相互作用.
关键词: 高次谐波发射    准直分子    椭圆偏振强激光