Effect of Types and Orders of Electromagnetic Field Finite Element Meshes on Power Communication Harmonic Parameters Calculation Results of Tubular Hydrogenerators

In generator design field,waveform total harmonic distortion(THD)and telephone harmonic factor(THF)are parameters commonly used to measure the impact of generator no-load voltage harmonics on the power communication quality.Tubular hydrogenerators are considered the optimal generator for exploiting low-head,high-flow hydro resources,and they have seen increasingly widespread application in China's power systems recent years.However,owing to the compact and constrained internal space of such generators,their internal magnetic-field harmonics are pronounced.Therefore,accurate calculation of their THD and THF is crucial during the analysis and design stages to ensure the quality of power communication.Especially in the electromagnetic field finite element modeling analysis of such generators,the type and order of the finite element meshes may have a significant impact on the THD and THF calculation results,which warrants in-depth research.To address this,this study takes a real 34 MW large tubular hydrogenerator as an example,and establishes its electromagnetic field finite element model under no-load conditions.Two types of meshes,five mesh densities,and two mesh orders are analyzed to reveal the effect of electromagnetic field finite element mesh types and orders on the calculation results of THD and THF for such generators.

Low-order harmonic generation of hydrogen molecular ion in laser field studied by the two-state model

The low-order harmonic generation of hydrogen molecular ion interacting with a linearly polarized laser field has been investigated theoretically by using a simple two-state model. The validity of the two-state model is carefully examined by comparing the harmonic spectra of hydrogen molecular ion obtained from this model with those from the three-dimensional time-dependent Schr¨odinger equation. When combined with the Morlet transform of quantum time-frequency spectrum,the two-state model can be used to study the dynamical origin of the low-order harmonic generation of hydrogen molecular ion driven by low-frequency pulses. In addition, some interesting structures of the time profiles for low order harmonics are obtained.

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.

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.

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.

Quantitative Measurement of the Proportions of High-Order Harmonics for the 4B7B Soft-X-Ray Source at Beijing Synchrotron Radiation Facility

A transmission grating coupled with an X-ray charge coupled device （CCD） is used to quantitatively measure the proportion of high-order harmonics of the soft-X-ray source of beam line 4B7B. The results show that the monochromatic X-ray has third-order and second-order harmonics. The proportion of second-order harmonic of 4B7B is less than 9.0% and the third- order harmonic is below 0.7% when no suppressing method is applied. When suppression methods are used, the proportion of second-order harmonic is less than 1.7% and the third-order harmonic is ignorable.

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.

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）.

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.

Primary resonance of fractional-order Duffing–van der Pol oscillator by harmonic balance method

The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.

Atomic high-order harmonic generation from a circularly polarized laser pulse

We demonstrate theoretically that the high-order harmonic of an atom can be generated by a circularly polarized laser pulse.The harmonic spectrum shows a clear cutoff with an energy Ip ＋ 2Up.In particular,the high-order harmonic generation comes from the multiple recombination of the ionized electron with non-zero initial velocity.These results are verified by the classical model theory and the time-frequency analysis of a harmonic spectrum.

Extension of high-order harmonic cutoff frequency by synthesizing the waveform of a laser field via the optimization of classical electron trajectory in the laser field

We theoretically investigate high-order harmonic generation by employing strong-field approximation （SFA） and present a new approach to the extension of the high-order harmonic cutoff frequency via an exploration of the dependence of high-order harmonic generation on the waveform of laser fields. The dependence is investigated via detailed analysis of the classical trajectories of the ionized electron moving in the continuum in the velocity-position plane. The classical trajectory consists of three sections （Acceleration Away, Deceleration Away, and Acceleration Back）, and their relationship with the electron recollision energy is investigated. The analysis of classical trajectories indicates that, besides the final （Acceleration Back） section, the electron recollision energy also relies on the previous two sections. We simultaneously optimize the waveform in all three sections to increase the electron recollision energy, and an extension of the cutoff frequency up to Ip ＋ 20.26Up is presented with a theoretically synthesized waveform of the laser field.

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.

Theoretical study of the odd-even-order harmonic generation for asymmetric ions in non-Born-Oppenheimer approximation

We calculated the harmonic spectra generated from the asymmetric molecules of HD^＋ and HeH^2＋. It is found that HD＋produces only odd harmonics, while HeH^2＋produces both odd and even harmonics. Further analysis reveals that for both HD^＋ and HeH^2＋, the nuclear dipole acceleration can generate even harmonics, but it is three orders of magnitude lower than that of the electron. Hence, the electronic dipole acceleration dominates the harmonic generation. For HD^＋,the electronic dipole acceleration only contributes to the generation of odd harmonics, but for HeH^2＋it contributes to the generation of both odd and even harmonics. Besides, one concept of the broken degree of system-symmetry is proposed to explain the different odd-even property between the harmonic spectra of HD^＋ and HeH^2＋.

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.

High-order harmonic generation spectrum of an excited one-dimensional Coulomb atom in an intense laser field

Response of the wave packet of a one-dimensional Coulomb atom to an intense laser field is calculated using the symmetrized split operator fast Fourier method. The high-order harmonic generation （HHG） of the initial state separately being the ground and excited states is presented. When the hardness parameter a in the soft Coulomb potential V（x） =-1√x^2＋α is chosen to be small enough, the so-called hard Coulomb potential V（x）=1/｜x｜ can be obtained. It is well known that the hard one-dimensional Coulomb atom has an unstable ground state with an energy eigenvalue of - 0.5 and it has no states corresponding to physical states in the true atoms, and has the first and second excited states being degenerate. The parity effects on the HHG can be seen from the first and second excited states of the hard one-dimensional Coulomb atom. The HHG spectra of the excited states from both the soft and hard Coulomb atom models are shown to have more complex structures and to be much stronger than the corresponding HHG spectrum of the ground state of the soft Coulomb model with a = 2 in the same laser field. Laser-induced non-resonant one-photon emission is also observed.

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.

Harmonic Retrieval Embedded in Hybrid Colored Noise: a Novel Pisarenko Method Based on Cross high order Cumulant

Spectral estimation method is an effective method in the case of parameter estimation of sinusoid signal in noises.It always applies cross spectral estimation theory and high order self spectral estimation methods up to now.Yule Walker equation based on cross fourth order cumulant is developed and Pisarenko method is provided in this paper.Simulation results show that the method is effective to retrieve sinusoid signal frequencies from hybrid colored noises and has good performance.

High-order harmonic generations in tilted Weyl semimetals

We investigate high-order harmonic generations(HHGs)under comparison of Weyl cones in two types.Due to the hyperboloidal electron pocket structure,strong noncentrosymmetrical generations in high orders are observed around a single type-ⅡWeyl point,especially at zero frequency.Such a remarkable DC signal is proved to have attributions from the intraband transition after spectral decomposition.Under weak pulse electric field,the linear optical response of a nontilted Weyl cone is consistent with the Kubo theory.With extensive numerical simulations,we conclude that the non-zero chemical potential can enhance the even-order generations,from the slightly tilted system to the over-tilted systems.In consideration of dynamical symmetries,type-Ⅰand type-ⅡWeyl cones also show different selective responses under the circularly polarized light.Finally,using a more realistic model containing two pairs of Weyl points,we demonstrate that paired Weyl points with opposite chirality can suppress the overall even-order generations.