High-order harmonic generation of ZnO crystals in chirped and static electric fields

High harmonic generation in ZnO crystals under chirped single-color field and static electric field are investigated by solving the semiconductor Bloch equation(SBE). It is found that when the chirp pulse is introduced, the interference structure becomes obvious while the harmonic cutoff is not extended. Furthermore, the harmonic efficiency is improved when the static electric field is included. These phenomena are demonstrated by the classical recollision model in real space affected by the waveform of laser field and inversion symmetry. Specifically, the electron motion in k-space shows that the change of waveform and the destruction of the symmetry of the laser field lead to the incomplete X-structure of the crystal-momentum-resolved(k-resolved) inter-band harmonic spectrum. Furthermore, a pre-acceleration process in the solid four-step model is confirmed.

Elliptically polarized high-order harmonic generation of Ar atom in an intense laser field

High-order harmonic generation(HHG) of Ar atom in an elliptically polarized intense laser field is experimentally investigated in this work.Interestingly,the anomalous ellipticity dependence on the laser ellipticity(ε) in the lower-order harmonics is observed,specifically in the 13rd-order,which displays a maximal harmonic intensity at ε ≈ 0.1,rather than at ε = 0 as expected.This contradicts the general trend of harmonic yield,which typically decreases with the increase of laser ellipticity.In this study,we attribute this phenomenon to the disruption of the symmetry of the wave function by the Coulomb effect,leading to the generation of a harmonic with high ellipticity.This finding provides valuable insights into the behavior of elliptically polarized harmonics and opens up a potential way for exploring new applications in ultrafast spectroscopy and light–matter interactions.

Elliptically polarized high-order harmonic generation in nitrogen molecules with cross-linearly polarized two-color laser fields

Circularly and elliptically polarized high-order harmonics have unique advantages when used in studying the chiral and magnetic features of matter.Here,we studied the polarization properties of high-order harmonics generated from alignment nitrogen molecules driven by cross-linearly polarized two-color laser fields.Through adjusting various laser parameters and targets,such as the relative phase,the crossing angle,the intensity ratio of the driving fields,and the molecular alignment angle,we obtained highly elliptically polarized high-order harmonics with the same helicity in a wide spectral range.This provides a possible effective way to generate elliptically polarized attosecond pulses.Finally,we showed the probability of controlling the spectral range of elliptically polarized harmonics.

Role of excited states in helium-like ions on high-order harmonic generation

We theoretically investigate high-order harmonic generation(HHG) of helium(He), lithium cation(Li+), and beryllium dication(Be2+) using the time-dependent Hartree–Fock method to solve the three-dimensional time-dependent Schr ¨odinger equation. It is found that the intensity of the HHG increases significantly from a certain harmonic order below the ionization threshold, and the initial position of the enhancement does not depend on the intensity or the wavelength of the driving laser field. Further analysis shows that excited states play an important role on this enhancement,consistent with the excited-state tunneling mechanism [Phys. Rev. Lett. 116 123901(2016)]. Our results unambiguously show that excited-state tunneling is essential for understanding the enhancement of HHG. Accordingly, a four-step model is herein proposed to illustrate the multiphoton excitation effect in helium-like ions, which enriches the physics of HHG enhancement.

Phase-coherence dynamics of frequency-comb emission via high-order harmonic generation in few-cycle pulse trains

Frequency-comb emission via high-order harmonic generation(HHG)provides an alternative method for the coherent vacuum ultraviolet(VUV)and extreme ultraviolet(XUV)radiation at ultrahigh repetition rates.In particular,the temporal and spectral features of the HHG were shown to carry profound insight into frequency-comb emission dynamics.Here we present an ab initio investigation of the temporal and spectral coherence of the frequency comb emitted in HHG of He atom driven by few-cycle pulse trains.We find that the emission of frequency combs features a destructive and constructive coherences caused by the phase interference of HHG,leading to suppression and enhancement of frequency-comb emission.The results reveal intriguing and substantially different nonlinear optical response behaviors for frequency-comb emission via HHG.The dynamical origin of frequency-comb emission is clarified by analyzing the phase coherence in HHG processes in detail.Our results provide fresh insight into the experimental realization of selective enhancement of frequency comb in the VUV–XUV regimes.

Tailoring OAM spectrum of high-order harmonic generation driven by two mixed Laguerre–Gaussian beams with nonzero radial nodes

The extreme ultraviolet(XUV)light beam carrying orbital angular momentum(OAM)can be produced via high-order harmonic generation(HHG)due to the interaction of an intense vortex infrared laser and a gas medium.Here we show that the OAM spectrum of vortex HHG can be readily tailored by varying the radial node(from 0 to 2)in the driving laser consisting of two mixed Laguerre-Gaussian(LG)beams.We find that due to the change in spatial profile of HHG,the distribution range of the OAM spectrum can be broadened and its shape can be modified by increasing the radial node.We also show that the OAM mode range becomes much wider and its distribution shape becomes more symmetric when the harmonic order is increased from the plateau to the cutoff when the driving laser has the nonzero radial nodes.Through the map of coherence length and the evolution of harmonic field in the medium,we reveal that the favorable off-axis phase-matching conditions are greatly modified due to the change of intensity and phase distributions of driving laser with the radial node.We anticipate this work to stimulate some interests in generating the XUV vortex beam with tunable OAM spectrum through the gaseous HHG process achieved by manipulating the mode properties of the driving laser beam.

High-order harmonic generation of the cyclo[18]carbon molecule irradiated by circularly polarized laser pulse

High-order harmonic generation of the cyclo[18]carbon(C_(18) ) molecule under few-cycle circularly polarized laser pulse is studied by time-dependent density functional theory. Compared with the harmonic emission of the ring molecule C_(6)H_(6) having similar ionization potential, the C_(18) molecule has higher efficiency and cutoff energy than C_(6)H_(6) with the same laser field parameters. Further researches indicate that the harmonic efficiency and cutoff energy of the C_(18) molecule increase gradually with the increase of the laser intensity of the driving laser or decrease of the wavelength, both are larger than those of the C_(6)H_(6) molecule. Through the analysis of the time-dependent evolution of the electronic wave packets, it is also found that the higher efficiency of harmonic generation can be attributed to the larger spatial scale of the C_(18) molecule,which leads to a greater chance for the ionized electrons from one atom to recombine with others of the parent molecule.Selecting the suitable driving laser pulse, it is demonstrated that high-order harmonic generation in the C_(18) molecule has a wide range of applications in producing circularly polarized isolated attosecond pulse.

Calibration of quantitative rescattering model for simulating vortex high-order harmonic generation driven by Laguerre–Gaussian beam with nonzero orbital angular momentum

We calibrate the macroscopic vortex high-order harmonic generation(HHG)obtained by the quantitative rescattering(QRS)model to compute single-atom induced dipoles against that by solving the time-dependent Schr?dinger equation(TDSE).We show that the QRS perfectly agrees with the TDSE under the favorable phase-matching condition,and the QRS can accurately predict the main features in the spatial profiles of vortex HHG if the phase-matching condition is not good.We uncover that harmonic emissions from short and long trajectories are adjusted by the phase-matching condition through the time-frequency analysis and the QRS can simulate the vortex HHG accurately only when the interference between two trajectories is absent.This work confirms that it is an efficient way to employ the QRS model in the single-atom response for precisely simulating the macroscopic vortex HHG.

Extension of high-order harmonics and generation of an isolated attosecond pulse in the chirped laser field

This paper theoretically investigates the high-order harmonic generation cutoff extension using intense few-cycle linearly chirped laser pulses. It shows that the cutoff of the harmonic can be extended remarkably by optimising the chirping parameters. The time-frequency characteristics of high-order harmonics with different chirping parameters are analysed by means of wavelet transform of the dipole acceleration. It also gives out the classical three-step model pictures of electron. By superposing a properly selected range of the harmonic spectrum, it obtains an isolated 65as pulse.

Effect of laser intensity on quantum trajectories in the macroscopic high-order harmonic generation

We macroscopically investigate the effect of the laser intensity and gas density on quantum trajectories in the highorder harmonic generation of Ne atoms irradiated by few-cycle, 800-nm laser pulses. The time–frequency profile of the harmonics shows that the long quantum trajectory is dominant at both lower and higher gas densities for a low laser intensity. At high laser intensities, the long quantum trajectory plays an important role for lower gas densities, while the short quantum trajectory is dominant at higher gas densities. An analysis of the phase mismatch for high-order harmonic generation shows that the primary emission of the quantum trajectories is determined by dynamic changes in the laser electric field during the propagation process.

Selection of quantum path in high-order harmonics and isolated sub-100 attosecond generation in few-cycle spatially inhomogeneous laser fields

We theoretically study the selection of the quantum path in high-order harmonics（HHG） and isolated attosecond pulse generation from a one-dimensional（1D） model of a H_2~＋ molecule in few-cycle inhomogeneous laser fields.We show that the inhomogeneity of the laser fields play an important role in the HHG process.The cutoff of the harmonics can be extended remarkably,and the harmonic spectrum becomes smooth and has fewer modulations.We investigate the time-frequency profile of the time-dependent dipole,which shows that the short quantum path is enhanced and the long quantum path disappears in spatially inhomogeneous fields.The semi-classical three-step model is also applied to illustrate the physical mechanism of HHG.The influence of driving field carrier-envelop phase（CEP） on HHG is also discussed.By superposing a series of properly selected harmonics,an isolated attosecond pulse（IAP） with duration 53 as can be obtained by a 15-fs,1600-nm laser pulse with the parameter ε = 0.0013（e is the parameter that determines the order of inhomogeneity of the laser field）.

Restraint of spatial distribution in high-order harmonic generation from a model of hydrogen molecular ion

The spatial distribution in high-order harmonic generation（HHG） is theoretically investigated by using a few-cycle laser pulse from a two-dimensional model of a hydrogen molecular ion. The spatial distribution in HHG demonstrates that the harmonic spectra are sensitive to the carrier envelope phase and the duration of the laser pulse. The HHG can be restrained by a pulse with the duration of 5 fs in the region from the 90 th to 320th order. This characteristic is illustrated by the probability density of electron wave packet distribution. The electron is mainly located near the nucleus along the positive-x direction from 3.0 o.c. to 3.2 o.c., which is an important time to generate the HHG in the plateau area. We also demonstrate the time-frequency distribution in the region of the positive-and negative-x direction to explain the physical mechanism.

A scaling law of high-order harmonic generation

Using a nonperturbative quantum electrodynamics theory of high-order harmonic generation （HHG）, a scaling law of HHG is established. The scaling law states that when the atomic binding energy Eb, the wavelength ）, and the intensity I of the laser field change simultaneously to kEb, λ/k, and k3I, respectively. The characteristics of the HHG spectrum remain unchanged, while the harmonic yield is enhanced k3 times. That HHG obeys the same scaling law with above-threshold ionization is a solid proof of the fact that the two physical processes have similar physical mechanisms. The variation of integrated harmonic yields is also discussed.

Carrier envelope phase effect on the spatial distribution of high-order harmonic generation in asymmetric molecule

The spatial distribution in high-order harmonic generation（HHG） from the asymmetric diatomic molecule He H^（2＋） is investigated by numerically solving the non-Born–Oppenheimer time-dependent Schr？dinger equation（TDSE）. The spatial distribution of the HHG spectra shows that there is little contribution in HHG around the geometric center of two nuclei（z = 1.17 a.u.） and the equilibrium internuclear position of the H nucleus（z = 3.11 a.u.）. We demonstrate the carrier envelope phase（CEP） effect on the spatial distribution of HHG in a few-cycle laser pulse. The HHG process is investigated by the time evolution of the electronic density distribution. The time–frequency analysis of HHG from two nuclei in HeH^（2＋） is presented to further explain the underlying physical mechanism.

Role of Bloch oscillation in high-order harmonic generation from periodic structure

The high-order harmonic generation from a model solid structure driven by an intense laser pulse is investigated using the semiconductor Bloch equations(SBEs). The main features of harmonic spectrum from SBEs agree well with the result of the time-dependent Schroodinger equation(TDSE), and the cut-off energy can be precisely estimated by the recollision model. With increasing the field strength, the harmonic spectrum shows an extra plateau. Based on the temporal population of electron and the time–frequency analysis, the harmonics in the extra plateau are generated by the Bloch oscillation. Due to the ultrafast time response of the Bloch electron, the generated harmonics provide a potential source of shorter isolated attosecond pulse.

High-order harmonic generation in a two-color strong laser field with Bohmian trajectory theory

We theoretically study the high-order harmonic generation （HHG） in a two-color laser field using the Bohmian mechanics. Our results show that, for tile case of a weak second-color laser field, the simulation of the HHG with only one central Bohmian trajectory is in a good agreement with the ab initio time-dependent Schrodinger equation （TDSE） results. In contrast, with the increase of the amplitude of the second-color laser field, the HHG spectra from the single central Bohmian trajectory deviate from the TDSE results more and more significantly. By analyzing the Bohmian trajectories, we find that the significant deviation is due to the fact that the central Bohmian trajectory leaves the core quickly in the two-color laser field with the breaking of inversion symmetry. Interestingly, we find that another Bohmian trajectory with different initial position, which keeps oscillating around the core, could qualitatively well reproduce the TDSE results. Furthermore, we study the HHG spectrum in a two-color laser field with inversion symmetry and find that the HHG spectrum in TDSE can be still well simulated with the central Bohmian trajectory. These results indicate that, similar to the case of one color laser field, the HHG spectra in a two-color laser field can be also reproduced with a single Bohmian trajectory, although the initial position of the trajectory is dependent on the symmetry of the laser field. Our work thus demonstrates that Bohmian trajectory theory can be used as a promising tool in investigating the HHG process in a two-color laser field.

High-order harmonic generation of N2molecule in two-color circularly polarized laser fields

The generation of high-order harmonics and the attosecond pulse of the N2 molecule in two-color circularly polarized laser fields are investigated by the strong-field Lewenstein model. We show that the plateau of spectra is dramatically extended and a continuous harmonic spectrum with the bandwidth of 113 eV is obtained. When a static field is added to the x direction, the quantum path control is realized and a supercontinuum spectrum can be obtained, which is beneficial to obtain a shorter attosecond pulse. The underlying physical mechanism is well explained by the time-frequency analysis and the semi-classical three-step model with a finite initial transverse velocity. By superposing several orders of harmonics in the combination of two-color circularly polarized laser fields and a static field, an isolated attosecond pulse with a duration of 30 as can be generated.

Controlling the contributions to high-order harmonic generation from different nuclei of N_2 with an orthogonally polarized two-color laser field

The generation of high-order harmonic and the attosecond pulse of the N2 molecule with an orthogonally polarized two-color laser field are investigated by the strong-field Lewenstein model.We show that the control of contributions to high-order harmonic generation（HHG） from different nuclei is realized by properly selecting the relative phase.When the relative phase is chosen to be φ= 0.4π,the contribution to HHG from one nucleus is much more than that from another.Interference between two nuclei can be suppressed greatly; a supercontinuum spectrum of HHG appears from 40 e V to125 e V.The underlying physical mechanism is well explained by the time–frequency analysis and the semi-classical threestep model with a finite initial transverse velocity.By superposing several orders of harmonics,an isolated attosecond pulse with a duration of 80 as can be generated.

Generation of non-integer high-order harmonics and significant enhancement of harmonic intensity

High-order harmonics from helium atom in the orthogonally two-color(OTC) laser field are investigated by solving the two-dimensional time-dependent Schrodinger equation.Non-integer high-order harmonics are obtained in some ratio of frequencies of two components.Pure odd and even harmonics from atoms could be separated in two components by adjusting the ratio of frequencies in OTC scheme,and the resolution of harmonics is improved at the same time.The physical mechanism is explained by the periodicity of dipole.With the same intensity of the incident laser,the intensity of the high-order harmonics from the OTC field scheme is improved by three orders of magnitude compared to the monochromatic laser field scheme.A theoretical scheme is provided for experimentally achieving improving energy resolution and separation of pure odd and even harmonics in atoms.Also,we provide a means for improving harmonic intensity.

High-order harmonic generation with a two-color laser pulse

We theoretically investigate the electron dynamics of the high-order harmonics generation process by combining a near-infrared 800 nm driving pulse with a mid-infrared 2000 nm control field. We also investigate the emission time of harmonics using time-frequency analysis to illustrate the physical mechanisms of high-order harmonic generation. We calculate the ionization rate using the Ammoso Delone-Krainov model and interpret the variations in harmonic intensity for different control field strengths and delays. We find that the width of the harmonic plateau can be extended when the control electric field is added, and a supercontinuum from 198 to 435 eV is generated, from which an isolated 61-as pulse can be directly obtained.