The study of high-order harmonic generation (HHG) in confined quantum systems is essential for developing a comprehensive physical description of harmonic generation from atoms to bulk solids. Using the time-dependent density-functional theory, we demonstrate how the symmetry of the system modulates the generation of high-order harmonic in fullerene C_{60} molecules along different orthogonal directions, as well as the effects of amplitude and photon energy of a linearly polarized laser on high-order harmonics generation. We found that the generation of high-order harmonics perpendicular to the laser polarization direction (LPD) is related to the symmetry of molecules along the LPD and the symmetry of molecules perpendicular to the LPD. Within a certain parameter range, the cut-off energy is linearly proportional to the laser amplitude and the laser photon energy.

## I. INTRODUCTION

High-order harmonics are a crucial aspect of strong-field physics, playing a vital role in various fields, including the generation of coherent x rays,^{1} chemical reaction tracking attosecond science,^{2–4} and others. When a strong laser pulse interacts with substances in different states such as gases and solids,^{5–10} it emits coherent radiation with a frequency that is an integer multiple of the original driving frequency. In order to maximize the utilization of high-order harmonic light sources, people mainly focus on optimizing harmonics from the perspectives of achieving better control of harmonic polarization and increasing the cut-off energy of harmonics.^{11,12}

The selection rules of high-order harmonic generation (HHG) are determined by the joint spatiotemporal symmetry formed by the spatial symmetry of the target material and the symmetry of the laser electric field.^{13–18} Orientation dependence has also been observed in many molecular ionizations.^{19,20} Symmetry can enable higher order harmonics to have controllable polarization states and unique spectral characteristics.^{12,21} When excited by linearly polarized laser, a series of selection rules will affect the harmonic components parallel and perpendicular to the laser polarization direction (LPD). In both experiments and theory, people often study the higher order harmonic spectra in these two orthogonal directions. Their modulation behavior deeply reflects the symmetry of the system, which helps deepen the understanding of high-order harmonic physical processes. For example, the study of CO molecules shows that even harmonics are generated with the polarization perpendicular to the laser polarization when the molecular axis of CO is perpendicular to the laser polarization.^{22} Our previous study shows that multi-directional pure even harmonics are simultaneously generated from planar molecules in linearly polarized laser fields.^{23}

For gas-phase atoms and molecules,^{24,25} according to the semi-classical three-step model, the cut-off energy of the high-order harmonic spectrum is E_{cutoff} = I_{p} + 3.17U_{p}, where I_{P} is the ionization potential and U_{p} is the ponderomotive energy. E_{cutoff} is proportional to the square of the laser amplitude and wavelength. The mechanism of generating high order harmonics in solids is different from that of gas phase atoms and molecules. For instance, in solids, the band structure and Berry curvature play a crucial role in the generation of high-order harmonics.^{26} The results of both experiments^{6} and theories^{27–30} show that the cut-off energy of high-order harmonics in solid is proportional to the laser amplitude, not the square of the laser amplitude. At present, the relationship between the cut-off energy of solid-state harmonics and laser wavelength is still under debate.^{31–33} Another typical feature of solid-state high-order harmonic spectrum is the emergence of a multi-platform structure, which indicates strong inter-band couplings involving multiple single-particle bands.^{34}

Studying HHG in a confined quantum system is crucial for establishing a complete physical image of harmonic generation from atoms and molecules to bulk solids.^{35,36} At the nanoscale, HHG will have a significant impact on many fields as it will enable people to achieve more compact devices.^{37} Recently, HHG in fullerene attracted much attention. Experimentally, a strong harmonic signal from C_{60} plasma has been reported.^{38,39} In the single-particle picture, researchers found strong HHG from a C_{60} molecule^{40–43} and solid C_{60}.^{39} In general, HHG requires a strong laser field. More interestingly, previous studies have found that in C_{60} molecules, relatively weak laser fields are sufficient to generate harmonics.^{40}

In this paper, we first investigated how symmetry modulates HHG in fullerene C_{60} molecules in different orthogonal directions under the action of a linear laser field. Next, more importantly, we found the relationship between the high-order harmonic cut-off energy and the amplitude and photon energy of the laser. It should be pointed out that we did not consider the medium propagation effects in harmonic generation, that is, we did not consider the influence of phase matching.

## II. NUMERICAL METHOD

_{60}molecules is studied using time-dependent density-functional theory (TDDFT).

^{44}In TDDFT, the essential variable is the electron density, which significantly reduces the computational expenses. Therefore, it can handle multi-electron systems and is a promising tool for studying HHG.

^{14,44,45}The time-dependent Kohn–Sham equation is

*v*

_{eff}(

**r**,

*t*) is the time-dependent Kohn–Shan potential, and it is defined as

*n*(

**r**,

*t*) is calculated as

*ψ*

_{i}(

**r**,

*t*) represents the time-dependent Kohn–Sham orbitals. In Eq. (1),

*v*

_{ion}(

**r**) is the potential formed by atomic nuclei

^{46}and is independent of the density of electrons. Since the mass of electrons is much smaller than that of atomic nuclei, it can be assumed that the motion of atomic nuclei is slow. In our calculations, we disregarded the motion of the atomic nuclei. The geometry of C

_{60}is fixed. Therefore,

*v*

_{ion}(

**r**) does not depend on time. Although

*v*

_{ion}(

**r**) has a clear expression, pseudo-potential is generally used instead of Coulomb potential in TDDFT calculations. The Hartwigsen–Goedecker–Hutter pseudo-potential is used to describe the interaction. The time-dependent Hartree potential $vh(n;r,t)=\u222bd3r\u2032n(r,t)r\u2212r\u2032$ describes the repulsive potential between electrons.

*v*

_{xc}(

*n*;

**r**,

*t*) is the exchange-correlation potential

^{47}and contains the many-body interaction. In principle, it depends on the initial wave function

*ψ*

_{i}(

**r**, 0) and the history of changing electron density. Unfortunately, its specific expressions are unknown. We use the adiabatic local-density approximation $vxc(n;r,t)=vxcALDA(n0(t);r,t)$, which is the simplest approximation for the exchange correlation potential. The exchange correlation potential at each moment only depends on the electron density at that moment, rather than the entire history of electron density changes. $vxcALDA=\delta Exc\delta n0$, where

*Exc*is the exchange correlation energy. Under the local-density approximation, it only depends on the density of the uniform electron gas and has no specific expression.

^{47}It is obtained through approximate calculation. $vlaserr,t=r\u22c5E$ describes the relationship between the electrons and the laser field. The linearly polarized laser field is defined by

**E**(

*t*) =

**E**

_{0}

*f*(

*t*)cos(

*ω*

_{0}

*t*), where

**E**

_{0}represents the maximum-amplitude of the laser field. The trapezoidal envelope

*f*(

*t*) linearly increases during the first two cycles, remains constant for the four cycles, and decreases to zero again in the last two cycles.

*ω*

_{0}is the angular frequency. The harmonic spectra can be derived from the time-dependent dipole moment

**D**(t) as

**D**(t) is given by

The formula for the harmonic spectrum component is $Hj\u221d\u222b(d2dt2Dj(t))ei\omega tdt2$, *j* = *x*, *y*, *z*.^{48} In this case, ω represents the frequency of the high harmonics.

^{49}We conducted our simulations using Cartesian coordinates. The simulation domain was defined by assigning a sphere around the center of molecules with a radius of 45 Å and a uniform mesh grid with a spacing of 0.3 Å. The time step was set to 0.002 fs. To improve the quality of the spectra by avoiding the formation of standing density waves, the absorbing regions are considered in the calculation, which follow the edges of the simulation box. The absorbing boundaries are treated with a complex absorbing potential,

^{50}

*η*(=−1 a.u.) represents the height of the imaginary potential and

*L*(=13.2 a.u.) represents the absorbing length.

## III. RESULTS

Figures 1(a)–1(c) show the HHG spectra from the C_{60} molecule when the linear laser polarization direction is along different directions. Figures 1(az)–1(cz) show the schematic diagrams of HHG shown in Figs. 1(a)–1(c), respectively. Figures 1(ai)–1(ci) show the view of C_{60} along the i(i = x, y, z) axis. As shown by the red arrow in Fig. 1, the LPD is along the z-axis direction. In the schematic diagram, the direction of the blue arrow represents the polarization direction of high-order harmonics along the z-axis. The direction of the green arrow indicates the polarization direction of higher order harmonics parallel to the xy plane. The laser wavelength is 800 nm, and the amplitude of the laser electric field is 2.6 V/Å. The laser peak intensity is 9.0 × 10^{13} W/cm^{2}. Along the z-axis, as shown in Fig. 1(az), C_{60} is a central axis symmetric pattern. When the LPD is along this direction, due to the high symmetry, only odd harmonics are generated. No high-order harmonics will be generated in any direction perpendicular to the polarization of the laser. In Fig. 1(bz), the z-axis is the 5° rotational symmetry axis. In this situation, the C_{60} molecule is symmetric about the yz plane along the x-axis direction and asymmetric about the xz plane along the y-axis direction. When the LPD is along the z-axis, as shown in Fig. 1(bz), odd harmonics are generated along both the z-axis and y-axis direction. There is no HHG along the x-axis direction. Because of the symmetry, odd harmonics are generated along all green arrow directions, as shown in Fig. 1(bz). This means that odd harmonics in multiple directions are simultaneously generated from the C_{60} molecule in a linearly polarized laser field. This result is different from that of our previous research on planar molecules.^{23} In our previous study,^{23} when the LPD is along the z-axis and is perpendicular to the plane where the planar molecules are locates, the molecule is also symmetric about the yz plane and asymmetric about the xz plane. In this situation, even harmonics are generated along the y-axis direction.^{23} Through comparison, we found that this difference stems from the symmetry of molecules along the z-axis direction. Along the z-axis direction, the planar molecule is symmetric, while the C_{60} molecule is asymmetric. In this case, for the three-dimensional molecule, whether odd or even harmonics are generated along the direction perpendicular to laser polarization also depends on the symmetry of the three-dimensional molecule along the LPD.

When the linear LPD is along the z-axis, as shown in Fig. 1(cz), the C_{60} molecule has no reflection symmetry plane in the direction perpendicular to the LPD. Under this condition, the results of Fig. 1(c) show that harmonics are generated along all axes. Due to the symmetry, high-order harmonics are generated along the direction, as shown in Fig. 1(cz). Moreover, the results in Fig. 1 also show that the harmonic spectrum exhibits a plateau characterized by modulations in peak intensities. This phenomenon may be due to multi-center interference effects.^{17} The most visible minima are all located in the regions between the 19th and 21st harmonic orders. For a fixed laser-amplitude and laser-frequency, when the linear LPD is along different directions of C_{60}, the minimal harmonic orders are basically unchanged.

In order to further explain the HHG, we give dipole accelerations, as shown in Fig. 2. Figures 2(a)–2(c), respectively, show the dipole accelerations in the x, y, and z axes. The LPD for Fig. 2(a) is shown in Fig. 1(az), and Fourier transformations of the dipole accelerations give the harmonic spectra, as shown in Fig. 1(a). The LPDs for Figs. 2(b) and 2(c) are shown in Figs. 1(bz) and 1(cz), respectively. At adjacent half periods, the dipole acceleration along the z axis (blue curve) has opposite phases due to the change in the driving laser field sign. Therefore, along the z-axis, odd higher harmonics are generated.^{23} As shown in Fig. 2(a), along the y-axis direction, because C_{60} is symmetric about the xz plane as shown in Fig. 1(az), there is no dipole acceleration. Figures 2(b) and 2(c) show that compared with the z-axis direction, the dipole acceleration phase along the y-axis direction is about delayed by π. Thus, along the y-axis, odd higher harmonics are also generated, as shown in Figs. 1(b) and 1(c). Along the x-axis direction, as shown in Figs. 2(a) and 2(b), because C_{60} is symmetric about the yz plane as shown in Figs. 1(az) and 1(bz), there is no dipole acceleration. As shown in Fig. 2(c), along the x-axis direction, there is dipole acceleration, and odd higher harmonics are also generated because of the same reason. Interestingly, the results of Fig. 2 show that if there is dipole acceleration along the direction perpendicular to the LPD, these waveforms of dipole acceleration are basically the same and all are about delayed by π compared with the dipole acceleration along the z-axis direction.

In this section, we study the relationship between the high-order harmonic cut-off energy and the amplitude and frequency of the linearly polarized laser. First, we investigate the change law of the cut-off energy with the increase in the laser amplitude when the laser photon energy is constant. Figures 3(a)–3(c) show the HHG spectra from the C_{60} molecule along the LPD. The polarization direction of linear laser is shown in Fig. 1(cz). The laser amplitudes shown in Figs. 3(a)–3(c) are 2.71, 2.86, and 2.96 V/Å, respectively. The laser photon energy is 1.553 eV. In other words, the laser wavelength is 800 nm. It should be noted that the maximum harmonic order is the harmonic order at the sharp cut-off in the typical harmonic spectra.^{51} The maximum harmonic order shown in Figs. 3(a)–3(c) is 27th, 29^{th}, and 31^{st}, respectively . As shown in Fig. 1(c), the maximum harmonic order is 25th. The results show that the high-order harmonic cut-off energy increases linearly with the increase in laser amplitude, which is the same as that of solid and is different from gas-phase atoms and small molecules.

Second, we study the change law of the cut-off energy with the increase in the laser photon energy when the laser amplitude is constant. Figures 4(a)–4(d) show the HHG spectra from the C_{60} molecule along the LPD. The polarization direction of linear laser is shown in Fig. 1(cz). The laser photon energy shown in Figs. 4(a)–4(d) is 1.29, 1.38, 1.81, and 2.07 eV, respectively. In other words, the wavelengths of these lasers are 960, 900, 686, and 600 nm, respectively. The laser amplitude is 2.6 V/Å. The maximum harmonic orders shown in Figs. 4(a)–4(d) are 35th, 31st, 19th, and 13^{th}, respectively. The results show that the high-order harmonic cut-off energy decreases linearly with the increase in the laser photon energy, which is also different from that of gas-phase atoms and small molecules.

The results of Figs. 3 and 4 show that the cut-off energy is proportional to the laser amplitude and is also proportional to the laser photon energy. In order to find out the specific dependence between the cut-off energy and laser parameters, we determined the cut-off energy calculated by TDDFT by separately changing the laser amplitude and the laser photon energy, as shown in Figs. 5(a-A) and 5(b-C). Meanwhile, the fitting straight line formula is given, as shown in Figs. 5(a-B) and 5(b-D). The cut-off energy is E_{cutoff} = I_{P} + 24E′ − 19.67hv. The unit of energy is electron-volts. Here, I_{P} is 7.54 eV, which is approximately equal to the ionization potential of C_{60}. E′ = E_{0}/(V/Å) ^{∗} eV. E_{0} is the amplitude of the laser, and hv is the laser photon energy. Figure 5(a) shows the change in the cut-off energy with the increase in the laser amplitude when the laser photon energy is 1.553 eV. Figure 5(b) shows that when the laser amplitude E_{0} is 2.6 V/Å, the cut-off energy changes with the increase in the laser photon energy. The results shown in Fig. 5 indicate that when the laser amplitude is less than 3 V/Å, the cut-off energy is linearly proportional to the laser amplitude. In addition, when the laser photon energy is greater than 1.3 eV, the cut-off energy is linearly proportional to the laser photon energy. In brief, within a certain range of parameters, the cut-off energy is linearly proportional to the laser amplitude and the laser photon energy.

The results shown in Fig. 5(a) also indicate that a relatively weak laser field can generate harmonics in the C_{60} molecule. This is in line with a previous study.^{40} In atoms and small molecules, because the density of states is low, a strong laser is needed to generate HHG. In C_{60}, the density of states is higher, and there are many multiple excitation channels.^{40} Consequently, HHG in C_{60} can be generated easily. Moreover, we think this phenomenon is also related to the ionization potential of the molecule. According to the semi-classical three-step model, in the first-step, the outer electron escapes from the parent ion when the laser field suppresses the potential barrier. The outer electrons transition from a bound state to a continuous state. The smaller the ionization potential of a molecule, the easier it is for outer electrons to escape from the parent ion. Compared with other gas-phase atoms and molecules, the ionization potential of the C_{60} molecule is relatively smaller. Therefore, the relatively smaller ionization potential is another reason why a relatively weaker laser field can generate harmonics in C_{60}.

Figures 3 and 4 also exhibit the intense modulation law of the harmonic spectrum along the plateau with the change in the laser amplitude and the laser photon energy. First, we found that when the laser photon energy is 1.553 eV, with the change in the laser amplitude, the most visible minima are located in the regions between the 21st and 23rd harmonic orders. This result shows that the minimal harmonic orders are basically unchanged with the increase in the laser amplitude when the laser photon energy is constant. Second, we found that when the laser amplitude is 2.6 eV/Å, with the increase in the photon energy, the most visible minima are approximately located in 29th, 21st, and 17th harmonic orders, as shown in Figs. 4(a)–4(c). When the laser photon energy is 2.07 eV, the high-order harmonic spectrum has no other minimum except the maximum harmonic order, as shown in Fig. 4(d). The results show that with the increase in the laser photon energy, the minimal harmonic orders decrease gradually and then disappear. This also indicates that the position of the minimal harmonic orders mainly depends on the laser photon energy.

## IV. CONCLUSION

In conclusion, we found that symmetry modulates the generation of high-order harmonics in C_{60} along different orthogonal directions. Along the linear LPD, when C_{60} has a central axis symmetric pattern, there is no harmonic generation in any direction perpendicular to the laser polarization. When C_{60} has no reflection symmetry plane in the LPD and in the direction perpendicular to the LPD, harmonics are generated in any direction perpendicular to the laser polarization. In addition, by changing the parameters of the laser, we also studied the relationship between the high-order harmonic cut-off energy from the C_{60} molecule and the amplitude and photon energy of the laser. We found that within a certain range of parameters, the cut-off energy of high-order harmonics is proportional to the laser amplitude and the laser photon energy. This study will contribute to establish a complete physical picture of harmonic generation from atoms to bulk solids.

## ACKNOWLEDGMENTS

We acknowledge the financial support from the Projects of Kaili University (Grant No. BSFZ202201), the Innovative Talents at the “Thousand” Level in Guizhou Province of China (Grant No. [2021]201504), the Research Projects in the Education Department of Guizhou Province of China (Qian Jiao He KY Zi [2016] Grant No. 308), the National Natural Science Foundation of China (Grant No. 11464023), and the Guizhou High Level Innovative Talents Training Plan (Grant No. [2021]201504).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Haifeng Yin**: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Funding acquisition (equal); Writing – original draft (lead). **Dandan Liu**: Investigation (equal); Software (equal). **Fanju Zeng**: Formal analysis (equal). **Wenjing Chen**: Formal analysis (equal); Software (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.