Molecular High-order Harmonic Generation Research Articles

Coulomb-induced emission time shifts in high-order harmonic generation from H2+

Accurate emission times of high-order harmonic generation (HHG) are vital for high-precision ultrafast detection in attosecond science, but a quantitative analysis of Coulomb effects on this time is absent in the molecular HHG. Here, we investigate the Coulomb-induced emission-time shift in HHG of H2+ with two different internuclear distances R, where the times obtained via the Gabor transform of numerical data from solving the time-dependent Schrödinger equation are used as simulation experiment results. Based on the molecular strong-field approximation, we develop a trajectory-resolved classical model that takes into account the molecular two-center structure. By selecting appropriate electron trajectories and including Coulomb interactions, the classical trajectory method can reproduce Gabor emission times well. This consistence reveals that Coulomb tails cause an emission-time shift of ∼35 as at the R = 2.0 a.u. case and of ∼40-60 as at the R = 2.6 a.u. case under the present laser parameters when compared to the Coulomb-free quantum-orbit model. Our results are of significance to probe the attosecond dynamics via two-center interference.

Double-blind decoupling of molecular rotation and high-order harmonic generation with a neural network

Double-blind decoupling of molecular rotation and high-order harmonic generation with a neural network

High-order harmonic generation from aligned HCN molecules under orthogonally and linearly polarized two-color laser fields

Molecular high-order harmonic generation and molecular orbital ionization probabilities are calculated under orthogonally and linearly polarized two-color laser fields. When a second-harmonic field is applied, the high-order harmonics generated under the linearly polarized two-color laser fields in the antiparallel case are stronger than those generated in the orthogonal polarization case and even stronger than those of the parallel polarization case. The results show that ionization probabilities of various orbitals and harmonic orders are dependent on spatial symmetry of molecular orbitals. It is found that the ionization of low-lying Kohn-Sham molecular orbitals contributes significantly to the ionization and molecular high-order harmonic generation processes. The ionization probability maximum occurs when molecular orbital densities are maximum in the direction of laser field polarization. Furthermore, we show that the degeneracy of π orbitals is broken when the laser-molecule alignment angle deviates from the field axis. Accordingly, we indicated one component of the π orbital is effectively contributed to the ionization and high-order harmonic generation processes. Finally, to confirm the recollision model in the high-order harmonic generation, the quantum time-frequency analysis is used to extract electron paths information on subcycle time scales.

Molecular high-order harmonic generation by counter-rotating elliptically polarized intense pulses

We present a scheme to produce the selected left or right elliptically polarized high-order harmonic generation. By adopting bichromatic counter-rotating colinear elliptically polarized pulses, their interaction with oxygen molecule are calculated under the framework of time-dependent density functional theory. Here we show that the helicity-selective high harmonics with tunable ellipticity are generated by changing the ellipticity of one of the pump pulses. Because of the interference of the degenerate highest occupied molecular orbitals, tunable ellipticity in a wide spectral range is obtained. This scheme can be used to synthesize attosecond pulses with controlled ellipticity and helicity.

Plasmon-enhanced high order harmonic generation of open-ended finite-sized carbon nanotubes: The effects of incident field's intensity and frequency and the interference between the incident and scattered fields.

The nonlinear optical properties of hybrid systems composed of a silver nanosphere and an open-ended finite-sized armchair single-walled carbon nanotube (SWCNT) are systematically investigated by the hybrid time-dependent Hartree-Fock (TDHF)/finite difference time domain (FDTD) approach, which combines the real-time TDHF approach for the molecular electronic dynamics with the classical computational electrodynamics approach, the FDTD, for solving Maxwell's equations. The high order harmonic generation (HHG) spectra of SWCNTs are studied as a function of the intensity (I0) and frequency (ω0) of the incident field, and SWCNTs length as well. It is found that the near field generated by a Ag nanoparticle has an overall enhancement to the molecular HHG in all the energy range, and it extends the HHG spectra to high energy. The inhomogeneity of the near field results in the appearance of even-order harmonics, and their corresponding spectral intensities are sensitive to ω0, therefore the near field's gradient. When ω0 is far away from the frequency of plasmon resonance of the silver nanosphere (ωc), the interference between the incident and scattering light beams extends the spectral range and makes the HHG spectra more sensitive to I0, while at ω0 = ωc, the impact of the interference on the spectra is negligible.

Polarization of high-order harmonic generation in oriented molecules with intense ultrashort laser pulses

Using numerical solutions of the time-dependent Schr\odinger equation we show that in oriented molecules one can control the polarization of molecular high-order harmonic generation (MHOHG) by intense linearly polarized pulses. For oriented triangular molecules, ${\mathrm{H}}_{3}{}^{+}/{\mathrm{H}}_{3}{}^{2+}$, one obtains even-order harmonics perpendicular to the laser polarization, whereas all odd-order harmonics have the same polarization as the laser polarization. For ${\mathrm{H}}_{2}{}^{+}/{\mathrm{H}}_{2}$ we show that only odd-order harmonics are generated for any orientation and their polarization is nearly parallel to the molecular axis. We describe these phenomena based on perturbation theory. Redshifts of the MHOHG spectra occur due to laser induced electron localization in non-Born-Oppenheimer dissociating molecules, reflecting the effect of nuclear dynamics. The numerical results illustrate the sensitivity of harmonic polarization to molecular geometry and nuclear dynamics.

Quantum interference of multi-orbital effects in high-harmonic spectra from aligned carbon dioxide and nitrous oxide**Project supported by the National Natural Science Foundation of China (Grant Nos. 91750104, 11534004, 11874179, 11127403, 11474130) and the Natural Science Foundation of Jilin Province, China (Grant No. 20180101289JC).

We investigate experimentally multi-orbital effects in high-order harmonic generation (HHG) from aligned CO2 and N2O molecules by intense femtosecond laser fields with linear and elliptical polarizations. For either of the aligned molecules, a minimum in the harmonic spectrum is observed, the position of which shifts to lower-order harmonics when decreasing the intensity or increasing the ellipticity of the driving laser. This indicates that the minimum originates from the dynamic interference of different channels, of which the tunneling ionization and recombination are contributed via different molecular orbitals. The results show that both the highest occupied molecular orbital (HOMO) and low-lying HOMO-2 in CO2 (or HOMO-1 in N2O) contribute to the molecular HHG in both linearly and elliptically polarized strong laser fields. Our study would pave a way for understanding multi-electron dynamics from polyatomic molecules irradiated by strong laser fields.

Carrier-envelope phase-dependent molecular high-order harmonic generation from H2+ in a multi-cycle regime.

Carrier-envelope phase (CEP) dependence of high-order harmonic generation (HHG) from H2+ in a multi-cycle laser pulse is investigated by solving the non-Born-Oppenheimer time-dependent Schrödinger equation (TDSE). It is found that high harmonics in the plateau exhibit counterintuitive frequency modulation (FM) as the CEP of the multi-cycle laser varies. Based on the classical electron trajectories and time-frequency analysis, this multi-cycle CEP-dependent FM is demonstrated to result from the interference of half-cycle HHG radiations, which is modulated by laser-driven nuclear motion. The mechanism of the CEP-dependent FM is further confirmed by simulations based on a simple algorithm in the time domain, which satisfactorily reproduces the TDSE results. The CEP-dependent FM encodes rich information on the correlated electron and nuclear dynamics, which paves the way for probing nuclear motion with attosecond resolution.

High-Order harmonic generation of aligned acetylene in elliptically polarized strong laser fields

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 non-adiabatically 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.

Nonadiabatic effects in the molecular high-order harmonic generation from two-center molecular ions

The nonadiabatic effects in the molecular high-order harmonic generation (MHHG) from H2+ and its isotope have been theoretically investigated beyond the Born-Oppenheimer approximations. It is found that (i) due to the asymmetric contributions of the harmonic emission on the rising and the falling parts of the laser field, the frequency red-shifts in the MHHG can be found. As the initial vibrational state or the pulse intensity increases, the red-shifts of the harmonics are decreased and even the blue-shifts of the harmonics can be found in the higher pulse intensity with the higher vibrational state. With the increase of the pulse duration, the frequency red-shifts and the blue-shifts in the MHHG are decreased and enhanced, respectively. The shifts of the harmonics from the heavy nuclei are decreased compared with those from the light nuclei. (ii) Due to the coupling of the electron-nuclear dynamics, the non-odd harmonics can be produced at the larger internuclear distance. With the increase of the initial vibrational state or the pulse duration, the generations of the non-odd harmonics can be enhanced. As the pulse intensity increases, the intensities of the non-odd harmonics are first increased and then decreased at the much larger pulse intensity. Due to the slower nuclear motion, the generations of the non-odd harmonics are decreased as the nuclear mass increases. (iii) With the increase of the alignment angle of the molecule ions, the nonadiabatic effects in MHHG (including the shifts of the harmonics and the generations of the non-odd harmonics) are decreased.

Symmetry in circularly polarized molecular high-order harmonic generation with intense bicircular laser pulses

We present symmetry effects of laser fields and molecular geometries in circularly polarized high-order harmonic generation by bichromatic counter-rotating circularly polarized laser pulses. Simulations are performed on oriented molecules by numerically solving time-dependent Schr\odinger equations. We discuss how electron recollision trajectories by the orthogonal laser field polarizations influence the harmonic polarization by using a time-frequency analysis of harmonics. It is found that orientation-dependent asymmetric ionization in linear molecules due to Coulomb potentials gives rise to a dependence of the polarization on the harmonic frequency. Effects of Coriolis forces are also presented on harmonic generation. Electron recollision trajectories illustrate the effects of the relative symmetry of the field and the molecule, thus paving a method for circularly polarized attosecond pulse generation and molecular orbital imaging in more complex systems.

Intensity distributions and isolated attosecond pulse generation from molecular high-order harmonic generation in H2+ driven by nonhomogeneous field

Intensity distributions and isolated attosecond pulse generation from the molecular high-order harmonic generation (MHHG) in H2+ and T2+ driven by the nonhomogeneous field have been theoretically investigated. (i) Generally speaking, the intensities of the harmonics driven by the homogeneous field can be enhanced as the initial vibrational state increases and much more intense harmonics can be obtained from the light nuclei. However, with the introduction of the nonhomogeneous effect, the enhanced ratios of the harmonic yields are decreased as the initial vibrational state increases. Moreover, the intensities of the harmonics from H2+ and T2+ are very sensitive to the nonhomogeneous effect of the laser field. (ii) The contributions of the MHHG from the two-H nuclei present the periodic variation as a function of the laser phase for the case of the symmetric nonhomogeneous field. However, for the case of the positive and the negative asymmetric nonhomogeneous fields, the left-H and the right-H play the dom...

Electron-Nuclear Dynamics on Amplitude and Frequency Modulation of Molecular High-Order Harmonic Generation from H2 + and its Isotopes

Abstract Electron-nuclear dynamics of molecular high-order harmonic generation from H2 + and its isotopes has been theoretically investigated beyond the Born-Oppenheimer approximations. The results show that (i) due to the different ionisation probabilities and the harmonic emission times, the intensities of the harmonics from H2 + and its isotopes are very sensitive to the initial vibrational state, the pulse duration, and the pulse intensity. (ii) Due to the nonadiabatic effects in molecular high-order harmonic generation, the red-shifts of the harmonics can be found in the lower pulse intensity. With the increase of the pulse intensity, the harmonics are from the red-shifts to the blue-shifts. Moreover, as the pulse duration increases, the blue-shifts of the harmonics can be enhanced. As the initial vibrational state increases, the red-shifts of the harmonics can be decreased, whereas the blue-shifts of the harmonics can be enhanced. However, the shifts of the harmonics are decreased as the nuclear mass increases. (iii) Due to the coupled electron-nuclear dynamics in molecules, the spatial symmetry of the system is broken. As a result, non-odd harmonics can be generated at the larger internuclear distance. With the increase of the initial vibrational state or the nuclear mass, the generation of the non-odd harmonics can be enhanced and reduced, respectively. As the pulse duration or the pulse intensity increase, the generation of the non-odd harmonics can be enhanced. However, the intensities of the non-odd harmonics are decreased when using the longer pulse duration with the much higher laser intensity.

Electron‐Nuclear Dynamics in Molecular Harmonic Generation Driven by a Plasmonic Nonhomogeneous Field

Electron (z)‐nuclear (R) dynamics in the molecular high‐order harmonic generation (MHHG) from H2+ driven by the plasmonic nonhomogeneous field, generated by the surface plasmon polaritons in the bowtie‐shaped nanostructure, have been theoretically investigated through solving the two dimensional time‐dependent Schrödinger equation with the Non‐Bohn‐Oppenheimer approximation. It is found that (i) due to the plasmonic enhancement of the laser intensity, the harmonic cutoff can be extended when the spatial position of H2+ is away from the gap center of the nanostructure. However, due to the limit of the gap size, the threshold value of the harmonic cutoff can be obtained at a given position of H2+. (ii) Due to the asymmetric enhancement of the laser intensity in space, the extended higher harmonics are respectively from E(t) > 0 a.u. or E(t) < 0 a.u. for the cases of the positive and the negative spatial position of H2+. As a result, the intensities of the extended higher harmonics are different and can be controlled by changing the carrier‐envelope phase and the pulse duration of the laser field. (iii) In the few‐cycle pulse duration, the MHHG mainly comes from the multi‐photon resonance ionization (MPRI), while as the pulse duration increases, the MPRI, the charge‐resonance enhanced ionization (CREI) and even the dissociative ionization (DI) are contributed to the MHHG. Moreover, as the spatial position of H2+ moves, the contributions of the MHHG from the MPRI, the CERI and the DI can be controlled. (iv) The contributions of the MHHG from the two‐H nuclei have been investigated and found that when E(t) > 0 a.u., the intensities of the harmonics from the negative‐H is higher than those from the positive‐H; while when E(t) < 0 a.u., the intensities of the harmonics from the positive‐H plays the main role in the MHHG. Moreover, the multi‐minima, caused by the two‐center interference can also be found. (v) Finally, by superposing a properly selected harmonics, a single isolated attosecond pulse (SIAP) with the full width at half maximum (FWHM) of 34 as can be obtained.

Molecular harmonic generation in bowtie-shaped nanostructure

Molecular high-order harmonic generation (MHHG) from in the bowtie-shaped nanostructure has been investigated. The results show that (i) due to the surface plasmon polaritons in the bowtie-shaped nanostructure, the laser intensity can be enhanced and present the nonhomogeneous effect in space. As a result, the extension of the harmonic cutoff can be achieved when the position of is injected away from the gap center of the nanostructure. However, due to the limited gap size of the nanostructure, the inherent threshold value of the harmonic cutoff can be found. (ii) Due to the asymmetric enhancement of the laser intensity in space, the extended higher harmonics come from E(t) > 0 or E(t) < 0 for the cases of the positive or the negative injected positions of , respectively. (iii) In the few-cycle pulse, the extended higher harmonics mainly comes from the multi-photon resonance ionization (MPRI) around R = 4.0 and R = 4.5 but from only one harmonic emission event (HEE), thus leading to the intensity difference of the extended higher harmonics from the positive and the negative positions of . In the multi-cycle pulse, the extended higher harmonics are also attributed to the MPRI but from R = 3.5 to R = 6.0 and there are three HEEs contributed to the extended higher harmonics, thus leading to the similar intensity of the extended higher harmonics from the positive and the negative positions of . (iv) Due to the slower nuclear motion of the heavy nuclei, the intensities of the harmonics from are lower than those from . (v) The contributions of the MHHG from the two-H nuclei show that when E(t) > 0 or E(t) < 0, the left-H or the right-H plays the dominating role in the MHHG. Moreover, as moves away from the gap center, the intensity difference of the higher harmonics from the two-H nuclei can be enhanced. But, it can be decreased as the pulse duration increases.

Chirp control of multi-photon resonance ionization and charge-resonance enhanced ionization on molecular harmonic generation

The effects of the multi-photon resonance ionization (MPRI) and the charge-resonance enhanced ionization (CREI) on the molecular high-order harmonic generation (MHHG) from H2+ have been investigated by using the chirped pulses. It is found that the MHHG only comes from the MPRI in the shorter pulse duration. As the pulse duration increases, both the MPRI and the CREI contribute to the MHHG. But the MPRI plays the main role in the generations of the above-threshold harmonics and the CREI mainly contributes to the below-threshold harmonics. With the introductions of the up-chirped and the down-chirped pulses, the contributions of the MHHG from the CREI and the MPRI can be enhanced, respectively. Finally, the isotopic investigation (e.g. T2+) shows that due to the slower nuclear motion of the heavy nuclei, the contributions of MHHG from the CERI can be suppressed in the heavy nuclei.

The shift of harmonics with different initial vibrational states in the H molecular ion

Molecular high-order harmonic generation of H and its isotopes is investigated by numerical simulations of the non-Born–Oppenheimer time-dependent Schrödinger equations. The general characteristic of the typical high-order harmonic generation (HHG) spectra for the H molecule indicates that only the odd harmonics can be generated. Here we show that how the initial vibrational states and nuclear dynamics break down this standard characteristic, i.e. a redshift or blueshift of the harmonics appears. We investigate the effect of the initial vibrational states on the redshift or blueshift of the HHG spectrum under trapezoidal laser pulses. The ionization probability and time-frequency analysis are used to illustrate the physical mechanism of the shift of the harmonics. We also show the HHG spectra from the different isotopes of H molecule with different initial vibrational states.

Laser control on molecular harmonic emission and isolated attosecond pulse generation

Laser control on the generations of the molecular high-order harmonic generation (MHHG) and the isolated attosecond pulse (IAP) from H2+ have been theoretically investigated through solving the non-Bohn–Oppenheimer time-dependent Schrödinger equation. It is found that (i) in the middle laser intensity (i.e., I = 4.0 × 1014 W/cm2), the contribution from negative-H to the MHHG is higher than that from the positive-H. With the decrease (i.e., I = 2.0 × 1014 W/cm2) or the increase (i.e., I = 7.0 × 1014 W/cm2) in laser intensity, the asymmetric contributions from the two H nuclei to the MHHG are decreased. (ii) Pulse duration investigation shows that the distributions of the MHHG in two H nuclei present similar contributions as the pulse duration is enhanced. (iii) Laser phase investigation shows that when the laser phase is chosen from 0.0π to 0.6π and from 1.7π to 2.0π, the contribution from the negative-H plays the main role in MHHG. When the laser phase is chosen from 0.7π to 1.6π, the contribution from the positive-H to the MHHG is remarkably enhanced and becomes greater than that from the negative-H. The time–frequency analyses of the MHHG and the time-dependent wave function are shown to explain the harmonic emission process and the electron motion in H2+. (iv) With the introduction of one or two half-cycle controlling pulses, the cutoff and the intensity of the MHHG spectra can be controlled. Finally, by selecting the harmonics generated during a single harmonic emission event, an IAP with the full width at half maximum of 48 as can be produced.

Alignment dependent ultrafast electron-nuclear dynamics in molecular high-order harmonic generation.

We investigated the high-order harmonic generation (HHG) process of diatomic molecular ion H2+ in non-Born-Oppenheimer approximations (NBOA). The corresponding three-dimensional time-dependent Schrödinger equation is solved with arbitrary alignment angles. It is found that the nuclear motion can lead to spectral modulation of HHG in both the tunneling and multiphoton ionization regimes. The universal redshifts of the whole spectrum are unique in molecular HHG. The spectral width of HHG increases in NBOA. We calculated possible influences on redshifts of HHG in real experimental conditions and found that redshifts decrease with the increase of alignment angles of the molecules and are sensitive to the initial vibrational states. It can be used to extract the ultrafast electron-nuclear dynamics and image molecular structure. It will be instructive to related experiments.

Laser phase effect on asymmetric harmonic distribution in H * *Project supported by the National Natural Science Foundation of China (Grant No. 11504151), the Doctoral Scientific Research Foundation of Liaoning Province, China (Grant No. 201501123), and the Scientific Research Foundation of Liaoning Provincial Education Department, China (Grant No. L2014242).

The laser phase effect on the spatial distribution of the molecular high-order harmonic generation (MHHG) spectrum from H is theoretically investigated through solving the Non-Bohn–Oppenheimer (NBO) time-dependent Schrödinger equation (TDSE). The results are shown as follows. (i) The generated harmonics from the two nuclei each present an asymmetric distribution. Particularly, when the laser phases are chosen from 0.0π to 0.6π and from 1.7π to 2.0π, the contribution from the negative-H plays a main role in harmonic generation. When the laser phases are chosen from 0.7π to 1.6π, the contribution from the positive-H to the harmonic generation is remarkably enhanced and becomes greater than that from the negative-H. The electron localization, the time-frequency analyses of the harmonic spectrum and the time-dependent wave function are shown to explain the asymmetric harmonic distribution in H, which provides us with a method to control the electron motion in molecules. (ii) As the pulse duration increases, the asymmetric distributions of the MHHG in two H nuclei decrease. (iii) Isotope investigation shows that the asymmetric harmonic distribution can be reduced by introducing the heavy nucleus (i.e., D.