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Abstract
The spectral features of high-order harmonic spectra can provide rich information for probing the structure and dynamics of molecules in intense laser fields. We theoretically study the high harmonic spectrum with the laser polarization direction perpendicular to the N2O molecule and find a minimum structure in the plateau region of the harmonic spectrum. Through analyzing the time-dependent survival probability of different electronic orbitals and the time-dependent wave packet evolution, it is found that this minimum position is caused by the harmonic interference of HOMO a, HOMO-1, and HOMO-3 a orbitals. Moreover, this interference minimum is discovered over a wide frequency range of 0.087 a.u. to 0.093 a.u., as well as a range of driving laser intensities with peak amplitudes between 0.056 a.u. and 0.059 a.u.. This study sheds light on the multi-electron effects and ultrafast dynamics of inner-shell electrons in intense laser pulses, which are crucial for understanding and controlling chemical reactions in molecules.
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1. Introduction
High-order harmonics are generated [1–9] when atoms, molecules, and solid materials are irradiated by intense laser pulses. As the harmonic energy increases, the intensity of the harmonic spectrum presents a plateau structure, and the cut-off energy of this platform can provide information about the ionization potential of the material [10–12]. Since high-order harmonic generation (HHG) originated from the recombination of ionized electrons with their ions, the harmonic spectrum can be used to detect processes such as chemical reactions, molecular vibrations, and electron transitions on a femtosecond time scale [13–17], as well as to investigate properties of molecules in strong fields, such as molecular rotational wave packets and transition dipole moments [18].
The "three-step" model provides a framework for understanding the HHG mechanism [19–22]. According to this model, the process begins with the ionization of bound electrons in the system by the laser electric field [23–25]; these electrons then gain energy from the laser field and some of them can recombine with ions, which results in emitting harmonic photons. During this process, the released photons can reflect the instantaneous state of the molecule at the moment of the recombination. Therefore, molecular harmonic spectra have been applied to the tomographic imaging of molecular electronic orbitals [26,27]. Owing to the presence of multiple emission centers of molecular systems [28,29], a characteristic feature of the harmonic emission spectrum is the presence of spectral minima, which has been observed in many experiments about the molecular HHG.
Due to the complexity of the molecular harmonic generation process, there are many mechanisms to produce the minima of the harmonic spectrum. The cooper minima from the nodal structure in the bound-free transition dipole moment, which is not affected by laser parameters, can be applied to probe the structural characteristics of molecules [30–34]. The most commonly studied harmonic minimum is generated by the interference of the harmonics from two centers of the molecule. This spectral energy is largely insensitive to laser parameters and only related to the nuclear spacing and the angle of the laser polarization. As a result, it is often used to detect the bond length of the molecule. In recent years, similar minimum characteristics have been observed in the study of high-order harmonics from crystals [35] and liquids [36]. This characteristic can be employed to extract band information in solid materials and investigate the physical and chemical properties of liquid systems. Another mechanism for the harmonic minima is the interference between two orbitals in the molecule. Because the ionization energies of different orbitals in the molecule are close to each other, when the molecule is driven by a laser pulse, the ionization caused by multiple orbitals leads to coherent harmonic emission. Smirnova et al. studied the case where the polarization of the laser pulse is along the molecular axis, and found that HOMO-1 and HOMO orbitals of the $\mathrm {{N}_{2}}$ molecule produce the smallest coherent harmonic emission. Moreover, the position of the harmonic minima was significantly affected by the driving laser parameters [37–40]. Recently, Liang et al. discovered that when a laser pulse acts along the molecular axis, the dynamic interference between HOMO a and HOMO-1 orbitals in the $\mathrm {{N}_{2}{O}}$ molecule can generate the spectral minimum in the harmonic spectrum [41,42].
In the strong field ionization process, the ionization rate is exponentially dependent on the ionization energy of the system, so the ionization energy has a significant influence on the ionization. The impact of the multi-orbital effect on the harmonic generation is mainly focused on HOMO and HOMO-1 orbitals, which have been typically observed. It should be noted that the shapes of the electron density distributions from different electronic orbitals in the molecule may vary greatly and exhibit orientation characteristics. Therefore, under the action of a linearly polarized driving laser, it is also possible that the inner shell orbitals may produce a large ionization [43]. It is proved that, with the enhancement of the peak intensity of the driving laser pulse, not only the electrons on the highest occupied molecular orbital (HOMO) but also the inner-shell electrons may be ionized [41,44–47]. Guo et al. utilized the intensity ratio of parallel and perpendicular components of the $\mathrm {{N}_{2}}$ molecule HHG to reveal the contribution of inner-shell orbitals to the harmonic spectrum. More importantly, the harmonic emission is a dynamic process of the ionized electron recombination, which depends not only on the ionization probability but also on the magnitude of the dipole moments of the free electron jumping to different electronic orbitals [48–50].
In this paper, we theoretically investigate the harmonic emission spectrum of the $\mathrm {{N}_{2}{O}}$ molecule under the pulse laser with the polarization perpendicular to the molecular axis and find the existence of the minimum structure in the harmonic spectrum. By analyzing the harmonics from different orbitals, it is demonstrated that this minimum is due to the interference of the harmonics from the inner shell and valence electronic orbitals of the molecule. This paper is organized as follows: Section 2 introduces the theoretical method for the molecular HHG. In Section 3, the HHG spectra of $\mathrm {{N}_{2}{O}}$ molecules and the analysis process of the results are presented. In Section 4, we summarize the work. (Atomic units are used throughout the paper unless otherwise specified.)
2. Theoretical model and numerical method
We chose the time-dependent density functional theory (TDDFT) to investigate molecular HHG [51]. The Kohn-Sham method, proposed by Walter Kohn and Pierre Hohenberg, is a ground-state density functional theory (DFT) approach [52–54]. The time-dependent Kohn-Sham (TDKS) method extends the Kohn-Sham method to describe the time-dependent electronic systems under the influence of the laser pulses. TDKS orbital equation in the molecule is:
Here, ${V}_{{H}}[\rho ]({r}, t)=\int d^3 {r}^{\prime } \frac {\rho \left ({r}^{\prime }, t\right )}{\left |{r}-{r}^{\prime }\right |}$ represents the Hartree potential, $V_{ne}({r})$ is the potential energy that the valence electron feels from the real ion, which is described by the norm-conserved Troullier-Martins pseudopotential [55], and ${V}_{xc}[\rho ]({r}, t)$ is the exchange-correlation potential that accounts for non-trivial many-body effects through the exchange-correlation function. The last term $V_{laser}({r, t})$ is the interaction between the laser field and electrons. In this work, the exchange-correlation function used is the self-interaction corrected local density approximation (SIC-LDA) proposed by Perdew and Zung [56]. The obtained ground-state energy ($-E_{ks}$ =12.81 eV) is in good agreement with the experimental value (${I_p}$=12.89 eV) [57].
TDKS is numerically solved using octopus [58–60]. In the simulation, we use a time step of $\Delta t=0.04 \text { a.u. }$ and a spatial step of $\Delta r=0.4 \text { a.u. }$. The size of the simulation box is chosen as $30 a.u. \times 70 a.u. \times 120 a.u.$. To avoid non-physical reflections of ionized electrons at the boundaries, we use the complex absorbing potential [61] in the calculation.
HHG is obtained by a time-dependent dipole acceleration : $H(\omega ) \propto {\left | {\int {\frac {{{d^2}}}{{d{t^2}}}} d(t){e^{ - i\omega t}}dt} \right |^2}$.
To investigate the difference in the harmonic efficiency of different orbitals, we calculate the transition dipole moment matrix for the ground and free states [50,67]:
To understand the process of HHG, by using wavelet transform, the physical mechanism of high-order harmonic emission can be analyzed from time-domain and frequency-domain information. The specific formula is:
${\left | {A({t_0},{\omega })} \right |^2}$ represents the change information of HHG intensity with time and frequency, ${w_{{t_0},\omega }}(t) = \sqrt \omega W\left [ {\omega (t - {t_0})} \right ]$ is the kernel of the wavelet transform , ${W(z)}$ is the Morlet wavelet [68]:
3. Simulation results
The 800 nm laser pulse with the z-direction polarization is used to interact with the $\mathrm {{N}_{2}{O}}$ molecule along the y-axis of the molecular axis, and the specific schematic is shown in Fig. 1. The laser field is ${{E}}({{t}}) = {{E_0}}f({{t}})\sin (\omega t)$, where $f(t) = {\sin ^2}(\omega t/2N)$, ${N=2}$. The field intensity amplitude ${E_0}$ is 0.09 ($2.84 \times {10^{14}}\;{W/c{m^2}}$).
Fig. 1. The harmonic spectrum of the laser polarization orthogonal to the molecular axis. The laser parameters adopted are: a wavelength of 800 nm with a peak intensity $2.84 \times {10^{14}}\;{W/c{m^2}}$ (black solid line) and a wavelength of 1100 nm with a peak intensity $1.26 \times {10^{14}}\;{W/c{m^2}}$ (red dashed line).
The harmonic spectrum exhibits a cutoff around the 45th order and a minimum value at the 31st order, as apparent from the black solid line in Fig. 1. The direction of the driving laser polarization is perpendicular to the molecular axis, it is unlikely for the two emission centers to generate this harmonic minimum. Additionally, we also provide the harmonic emission spectrum when the driving laser wavelength is 1100 nm and the field intensity amplitude is $1.26 \times {10^{14}}\;{W/c{m^2}}$, as exhibited by the red dashed line in Fig. 1. It can be noticed that the generation of the harmonic minimum cannot be observed under the driving laser condition mentioned. Therefore, it can be inferred that the mechanism behind this minimum does not pertain to the Cooper minimum, which is caused by the molecular structure itself.
In order to clarify the origin of the harmonic minimum in the emission spectrum, we conducted individual calculations of the HHG for each orbital component of the $\mathrm {{N}_{2}{O}}$ molecule, as depicted in Fig. 2. This method has been utilized in many studies [69–71]. As presented in Fig. 2(a), the spectrum of each split orbital exhibits distinct harmonic features, with the HOMO-3 a orbital displaying the highest harmonic efficiency among them (indicated by the blue dotted line). The other two more efficient harmonic emissions arise from the HOMO a (represented by the black solid line) and HOMO-1 (denoted by the red dotted line) orbitals. In Fig. 2(b), the total HHG spectrum by the laser pulse is shown as the brown-red solid line, exhibiting the clear minima. The pink dotted line corresponds to the synthesized total harmonic spectrum (Component) through the HOMO a, HOMO-1, and HOMO-3 an orbital. It is noticeable that three orbitals with higher harmonic yields are present in Fig. 2(a). Although these individual harmonic spectra do not have minimum values, there is a minimum structure in the harmonic spectrum generated coherently by these three orbitals, which matches well with the total harmonic spectrum (Total). Hence, the origin of this harmonic minimum can be attributed to the coherent emission of multiple orbits that include inner-shell orbitals. At the same time, the Ip of HOMO a, HOMO-1, and HOMO-3 orbitals were observed to be 12.81 eV, 15.51 eV, and 19.59 eV (Fig. 2(c)), respectively. The corresponding cut-off positions predicted by $3.17U_{p} + I_{p}$ [20] are 52 eV (black solid line arrow in Fig. 2(b)), 54 eV (red dotted line arrow in Fig. 2(b)), and 60 eV (blue dot arrow in Fig. 2(b)), respectively. The harmonic cutoff position of the numerical simulation is also extended as ${I_p}$ increases due to the different ${I_p}$ of the three orbits. However, it is not entirely consistent with the predicted value. This is because, for molecular harmonics, many factors will affect the cut-off position, such as the angle between the molecular axis and the laser polarization direction, the geometric structure of the molecule [72], etc. Furthermore, a schematic diagram illustrating this mechanism is shown in Fig. 2(d). To better understand the cause of this interference of the harmonics from these orbitals, we first investigate the time-dependent survival probability of different electron orbitals.
Fig. 2. (a) HHG of each molecular orbital. (b) HHG of the three orbitals, the harmonic spectrum synthesized by the three orbitals (Component), and the total harmonic spectrum(Total). (c) Schematic representation of the HOMO a, HOMO-1, HOMO-3 a orbitals of the ground state, their symmetries and energies. (d) Schematic diagram of the three orbitals combined to produce the harmonic minima.
In Fig. 3, we present the time-dependent survival probability of the $\mathrm {{N}_{2}{O}}$ molecule’s individual orbital. As shown by Fig. 3(a), the time-dependent survival probability oscillate with the electric field of the driving laser, and the HOMO a (black solid line), HOMO-1 (red dotted line), and HOMO-3 a (blue dot line) orbitals show significant variations. After the laser irradiation, the time-dependent survival probability of the HOMO a orbital decreases the most, indicating the highest chance of ionization, while the time-dependent survival probability of HOMO-1 and HOMO-3 a orbital is less than 0.8${\% }$. To understand the ionization behavior of different electronic states, we display the symmetry of each molecular orbital and the orbital energy calculated by DFT in Fig. 3(b). As depicted in the figure, the HOMO a ($\pi$ orbital) electron cloud has zero probability along the direction of the laser polarization in the y-z plane at y=0. The HOMO b ($\pi$ orbital) electron cloud has a very weak probability distribution in the y-z plane, while the HOMO-1 ($\sigma$ orbital) electron cloud has zero probability distribution at the three nuclei. On the other hand, the HOMO-3 a ($\pi$ orbital) electron cloud has a significant probability distribution along the y-z plane. The higher ionization probabilities of the HOMO a and HOMO-1 orbitals can be attributed to their relatively smaller ionization energies, as ionization rate coefficients are directly proportional to the ionization energy. In addition, the HOMO-3 a orbital has a high ionization energy, making the ionization relatively difficult. The observed higher ionization probability can be attributed to the electron orbital shape, which has a greater distribution in the direction of the laser polarization, thus leading to a higher ionization probability than the HOMO-2 orbital. In general, high ionization probability orbitals tend to produce higher harmonic efficiency in the case of low ground-state loss [73]. However, as we have analyzed above, the HOMO-3 a orbital does not have the highest ionization probability but has the highest harmonic efficiency. Therefore, it is necessary to further investigate the reasons for this phenomenon.
Fig. 3. (a) The time-dependent survival probability of individual orbitals in $\mathrm {{N}_{2}{O}}$. (b) Schematic representation of the orbital electron density distribution and energy of each molecular orbital by DFT
As the harmonic generation process involves ionized electrons recombining with ions, the harmonic efficiency is influenced by both the probability of ionized electrons and the transition probability of ionized electrons returning to different electronic orbitals. The inset of Fig. 4 shows transition dipole moments obtained using the plane wave approximation [74] for transitions from the continuum state to HOMO a, HOMO-1, and HOMO-3 a orbitals. The inset in the figure also gives the ground state electron density distribution of the three orbitals. For photon energies from 44 eV to 54 eV, which is near the harmonic spectral minimum in Fig. 2(b), Fig. 4 presents transition dipole moments of these orbitals in the energy region. From the figure, it is evident that the transition dipole moment of the HOMO-3 a orbital(shown by the black solid line) is the largest, significantly higher than those of HOMO a (shown by the blue dot line) and HOMO-1 (shown by the red dotted line). This indicates that, despite HOMO-3 a having a low ionization probability, its transition dipole moment is considerably high, which leads to the highest orbital harmonic yield of HOMO-3 a.
Fig. 4. The transition dipole moment of HOMO a, HOMO-1, HOMO-3 a. Inset is the transition dipole moment of the photon energy at 30-75 eV.
To verify our speculation of this mechanism, we observe the dynamical process of the time-dependent evolution of various orbital wave packets, as illustrated in Fig. 5. Figures 5(a)-(d) illustrate the distribution of wave packets of the HOMO a orbital at different times t=0.8 o.c., t=1.0 o.c., t=1.2 o.c., and t=1.4 o.c., respectively. Figures 5(e)-(h) show the same for the HOMO-1 orbital, and Figs. 5(i)-(l) present the corresponding results for the HOMO-3 a orbital. In Fig. 5, red lines represent the laser field and red dots indicate the laser field at the corresponding moment of the wave packet motion. As can be seen from the figure, at the instantaneous moment t=0.8 o.c., the driving laser field is relatively weak, which results in the majority of electrons being located near the nucleus. When t=1.0 o.c., the bound electrons begin to move away from the nucleus due to the continuous influence of the laser field, leading to the gradual spreading of the electron wave packet. At t=1.2 o.c., the laser field changes direction for a certain moment, which causes the portion of ionized electrons to return and collide with the nucleus. Finally, at t=1.4 o.c., the electron wave packet re-enters the nucleus and continues to move leftward, gradually moving away from the nuclear region. The figure clearly demonstrates that probability distributions of the wave packets for the HOMO a orbital and HOMO-1 orbital are negligible near z=0, whereas the HOMO-3 a orbital has a significantly higher probability density in this region. That is, in the evolution process, the wave packet density distribution still maintains the shape of the ground state electron wave packet, which further indicates that HOMO-3 a orbital electrons have more chance to collide with the parent core, and the transition dipole moment between the HOMO-3 a orbital and the continuum state is the largest.
Fig. 5. HOMO a, HOMO-1, and HOMO-3 an orbital electron density distribution at times from 0.8 o.c. to 1.4 o.c.
After clarifying the reason for the high harmonic yield from the HOMO-3 a orbital, further analysis was conducted to understand the cause of the harmonic minima. As the harmonic efficiency from HOMO a and HOMO-1 orbitals differed significantly from that of the HOMO-3 a orbital, we coherently overlapped the harmonics from HOMO a and HOMO-1 orbitals to understand the harmonic interference process. It is found that the harmonic efficiency was similar to that of the HOMO-3 a orbital, and no minimum structure was observed. Furthermore, the combined harmonic spectrum between HOMO a and HOMO-1 is used to study the coherent emission between these orbitals and HOMO-3 a. In Fig. 6(a), the combined harmonic spectrum (red dashed line) of HOMO a and HOMO-1 orbitals, the harmonic spectrum component (blue dot-dashed line) of the three orbitals, and the total harmonic spectrum (black solid line) are shown. The blue dot line in Fig. 6 represents the harmonic spectrum generated by the HOMO-3 a orbital. As can be observed from the figure, the harmonic emission spectrum generated by the joint contribution of HOMO a and HOMO-1 is comparable to that of HOMO-3 a, which may produce a minimum structure of the harmonic spectrum. Fig. 6(b) presents the phase difference (${\Delta Phase = Phase(\mathrm {HOMO} - 3{{ }}a) - Phase(\mathrm {HOMO}{{ }}a + \mathrm {HOMO} - 1)}$) between the harmonics from the HOMO-3 a orbital and the HOMO a + HOMO-1 orbitals in the energy region from 44 eV to 54 eV, which is near the minima of the total harmonic spectrum. As can be seen from the figure, at the minimum value of 47.8 eV, the phase difference is $\pi$, as well as the harmonic intensities are almost equal, resulting in destructive interference. To observe the harmonic emission process more carefully, the temporal frequency behavior of the process was simulated. Figures 6(c)–6(e) display the time-frequency analysis of the HOMO a + HOMO-1, HOMO-3 a, and total harmonic spectra, respectively. It can be noticed that the emission times of the HOMO a + HOMO-1 and HOMO-3 a harmonics are almost identical, and the lower order harmonics are enhanced after the harmonic coherence between them. Furthermore, it is observed that the destructive interference between the harmonics from HOMO a + HOMO-1 and HOMO-3 a orbitals occurs near 1.3 o.c., which leads to the minimum of the wavelet analysis spectrum in Fig. 6(e).
Fig. 6. (a) Total (black line), HOMO a + HOMO-1 (red dotted line), HOMO-3 a (blue dot line) harmonic spectra, and Component (blue dot dotted line) HHG spectrum from three orbitals. (b) Harmonic spectrum of the laser polarization direction (black solid line), and the harmonic phase difference $\Delta Phase$ between the HOMO-3 a orbital and the HOMO a + HOMO-1 orbitals (red dot line). (c-e) Time-frequency analysis of HOMO a + HOMO-1, HOMO-3 a, and total harmonic spectra, respectively.
Due to the fact that the perfect alignment and orientation of molecules cannot be achieved in typical experiments, the effect of the polarization direction of the laser electric field on the harmonic generation needs to be taken into account. Fig. 7(a) shows the dependence of the harmonic spectra with the angle between the laser polarization direction and the molecular axis. Here, the angle range is changed from 85 to 95 degrees, and the laser pulse with a wavelength of 800 nm and peak intensity $2.84 \times {10^{14}}\;{W/c{m^2}}$ is adopted. It is observed that the minimum value of the harmonic spectra shifts to a higher energy region as the angle increases. When the angle between the molecular axis and the laser polarization direction is 90 degrees, the energy region with low harmonic yield becomes wider. Meanwhile, by averaging the harmonic efficiency within the angle range from 85 to 95 degrees, as depicted in Fig. 7(b), the interference minimum of the harmonic spectrum is still observable. Therefore, this interference minimum can be experimentally detected within a certain range of the molecular alignment. Based on this, the researchers further analyzed the impact of the driving laser parameters on the minimum structure of the harmonic spectrum. In Fig. 7(c), the shift of the harmonic spectrum minimum position with the variation of the laser field amplitude is illustrated. The figure indicates that the interference minimum structure can still be observed within the chosen range of the laser intensity, but as the driving laser intensity increases, the minimum position moves towards the high-energy region of the harmonic spectrum. Fig. 7(d) presents the variation of the harmonic spectrum minimum position with respect to the central frequency of the driving laser pulse. Here, the peak intensity of the laser pulse is set as $2.84 \times {10^{14}}\;{W/c{m^2}}$. As shown in the figure, the minimum is still observable as the frequency changes and its position shifts towards the low-energy region of the harmonic spectrum with the increasing frequency.
Fig. 7. (a) Dependence of the harmonic spectra with the angle between the laser polarization direction and the molecular axis. The laser pulse parameters are a wavelength of 800 nm, and a peak intensity of $2.84 \times {10^{14}}\;{W/c{m^2}}$. (b) The average harmonic spectra from angles between 85 and 95. (c) The minimum position of the harmonic spectra with the amplitude of the field intensity, the central wavelength of the driving laser pulse is 800 nm. (d) The change of the minimum position of the harmonic spectra with the central frequency, and the peak intensity of the laser pulse is $2.84 \times {10^{14}}\;{W/c{m^2}}$
4. Conclusion
To summarize, we employed the TDDFT method to numerically simulate the high-order harmonic emission produced by a vertically incident linearly polarized laser on ${\mathrm {{N}_{2}{O}}}$ molecules. A minimum structure is found in the harmonic emission spectrum. Through the analysis of the high-order harmonics generated by different orbitals of the system, as well as the time-dependent survival probability variation and transition dipole moments of these orbitals, it is discovered that the interference minimum structure observed in the harmonic spectrum is caused by the combined harmonics from three specific orbitals: HOMO-3 a, HOMO a, and HOMO-1. Additionally, further investigation was conducted to explore the impact of driving laser frequency, field strength amplitude, and molecular orientation on this harmonic minimum structure. Our study sheds light on a minimum structure resulting from complex multi-orbital effects induced by molecular inner shell orbitals. By uncovering the underlying physics of these effects, our research has the potential to enhance our understanding of the mechanisms that provide detailed information about molecular structure, thereby paving the way for molecular imaging.
Funding
National Key Research and Development Program of China (No. 2019YFA0307700, No. 2022YFE0134200); National Natural Science Foundation of China (11774129, 11975012, 12074145, 12204214); Jilin Provincial Research Foundation for Basic Research, China (20220101003JC); Outstanding Youth Project of Taizhou University (2019JQ002); the Zhejiang Provincial Natural Science Foundation of China (Y23A040001); the Foundation of Education Department of Liaoning Province, China (LJKMZ20221435).
Acknowledgments
The authors acknowledge the High-Performance Computing Center of Jilin University for supercomputer time.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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