Odd Harmonics Research Articles

Efficient non-perturbative high-harmonic generation from nonlinear metasurfaces with low pump intensity

High-harmonic generation (HHG) from solids in non-perturbative regime has recently attracted great interest due to the high efficiency compared with atomic gaseous media, compact experimental configurations, and promising applications for spectroscopy and coherent extreme ultraviolet sources. However, it needs still high pump intensity in the order of several hundreds GW/cm2. In this work, we present comprehensive results of efficient HHG with low pump threshold by employing high field enhancements in dielectric metasurfaces and hybrid metasurfaces comprising plasmonic nanoantennae. The numerical results show that ZnO nanodisk arrays supported by gold substrate enable us to generate high-order harmonics with efficiency in the order of 10−5–10−6% in plateau region and cut-off order of up to 21 by pumping at 1550 nm with a peak intensity below 70 GW/cm2. When using metasurfaces of plasmonic nanoantennae containing ZnO nanoparticles in their gaps and supported by dielectric substrate, HHG can be realized by using pump with much lower peak intensity and the spatial inhomogeneity of plasmon-induced local field plays an important role, leading to the appearance of even order harmonics. For peak pump intensity of about 0.4 GW/cm2, the resultant HHG efficiency in plateau region is in the orders of 10−8% for odd order harmonics and 10−9% for even order ones. In particular, when the plasmonic metasurfaces are supported by dielectric-coated metallic substrate, the local field in the gap region is further enhanced, resulting in HHG efficiency 1–2 orders-of-magnitude higher than the case of using dielectric substrate.

Revisiting frequency domain analysis method for water hammer effects on pipeline systems

This paper revisits frequency domain analysis method for water hammer study in a hydraulic system. Detailed investigation of the effect of the flow parameters is conducted. Weaknesses on previous research works were pointed out. A brief presentation of the frequency response function determined from motions equations of transient water flow is firstly presented. A special emphasis was devoted to the effect of frictional flow on resonance pulsations. It was obtained that, unlike previous findings, resonance frequencies in a frictional flow are decayed from odd harmonics. This decay is more or less significant depending on the friction coefficient. The observations related to various effects are also illustrated through a set of test cases that were numerically analyzed. Additionally relative uncertainties highlighting the frictional flow effect on the accuracy of resonant pulsations detection are evaluated.

Theoretical spectral analysis of FEL radiation from multi-harmonic undulators.

A theoretical study of the spontaneous and stimulated undulator radiation (UR) from electrons in undulators with multiple periods in both transversal directions is presented. Exact expressions are derived for the UR intensities in terms of the generalized Bessel and Airy functions, accounting for undulator field harmonics of arbitrary strength and for real parameters of the beams and installations. Theoretical results are verified with numerical and experimental data for SWISS-XFEL, PAL-XFEL, LEUTL, LCLS etc. The spectrum, UR line shape and width, and the harmonic evolution along the undulators are analyzed and compared with the available data for these experiments. Moreover, the effect of the field harmonics is elucidated. It is demonstrated that the third field harmonic can cause distinct odd UR harmonics. The asymmetric undulator field configuration is identified, which allows intense radiation of these harmonics. The power evolution in a free-electron laser (FEL) with such an undulator is studied by means of an analytical FEL model. The latter is enhanced by a true description of the gradual power saturation of harmonics. A FEL with elliptic undulator and electron-photon phase-shifting is proposed and modeled. It is shown that the resulting harmonic power from the phase-shifted elliptic undulator can be significantly higher than from a planar undulator with the same phase-shifting.

A Tandem Axial-Piston Unit Based Strategy for the Reduction of Noise Sources in Hydraulic Systems

This article presents a novel passive fluid borne noise source reduction strategy, based on tandem axial-piston unit indexing with the usage of symmetric lines. The strategy consists of setting the phase between the two synchronous units to accomplish destructive interference in targeted unit harmonics. A strategy capable of achieving destructive interference in all odd harmonics is investigated first analytically and then confirmed by a simulation study. Experiments on the proposed strategy confirmed its effectiveness at the first and third pump fundamental harmonics, and pressure ripple reduction was accomplished. The fluid borne noise source reduction in the first and third harmonic is verified to be propagated to pipe vibration and sound power. Regarding the first harmonic, pressure ripple was reduced by up to 18 dB; while for third harmonic, pressure ripple was reduced by up to 11 dB. In the experiment, however, noise cancellation is not achieved for the higher odd harmonics, as is instead found in the simulation. Conversely, transfer functions form pressure ripple to pipe wall acceleration are obtained experimentally, and a critical vibration band from 2000 Hz to 3000 Hz is identified as being crucial for effective overall sound power reduction.

A Design Technique for Shaping of Back-EMF Using Heuristic Optimization and Winding Factors in Brushless AC and DC Machines

One of the requirements for the electric machine designer is to provide an optimal back-EMF waveform in a high-performance application such as electric vehicles (EV) or hybrid EVs (HEV) systems. This paper presents a design technique for the shaping of back-EMF waveforms in brushless permanent magnet (PM) AC and DC machines. At first, back-EMF waveforms are calculated by an analytical model based on the subdomain field model and the complex permeance function. The finite element and experimental results are used for the verification of the analytical model. Secondarily, the heuristic optimization algorithm is applied to the analytical model in order to search for optimal solutions where the objective function is composed of harmonics in target back-EMF. As a result, the optimal back-EMF waveform can be traced by matching the odd harmonics of the design candidate with those of the target waveform. Also, a detailed investigation of the winding factor is carried out to provide the key design techniques regarding the shape of back-EMF.

Non-dipole effect in vortex high-order harmonic generation

We discuss the non-dipole effect in high-order harmonic generation from atoms driving by a linearly polarized vortex beam. Different from the selection rule under the dipole approximation, where only odd high harmonics are allowed, both even and odd harmonics are generated in the vortex beam. Moreover, the polarization states of the harmonics are modulated. Elliptically polarized high harmonics can be directly generated with a linearly polarized vortex beam. We show that these non-dipole effects arise from the orbital angular momentum and spin angular momentum of the vortex beam.

Wave propagation in a one-dimensional bar with rate-independent hysteresis

A single frequency ultrasonic wave propagation in nonlinear materials generates higher harmonics. The generation of higher harmonics depends on the type of material nonlinearity considered, like odd and even harmonics in quadratically nonlinear materials, and only odd harmonics in cubically nonlinear material. The objective of this study is to present numerical study of one dimensional wave propagation in rate-independent hysteretic media by considering one dimensional chain of spring masses. Preisach-Mayergoyz (1994) and Hodgdon hysteretic models (1988) are commonly used in the theoretical and numerical study of nonlinear wave propagation studies. Here we implement the famous scalar Bouc-Wen model (1976) and a recently developed two-state hysteresis model by Biswas and Chatterjee (2015) to consider hysteresis in the large spring mass system. Number of spring masses are decided based on the spatial resolution needed for the nonlinear wave propagation to capture higher harmonics. The study presents interesting comparison between the wave propagation through an aluminum bar damaged under low cycle fatigue modeled as two different rate-independent hysteretic models. Both models show higher harmonics in responses. However, being a scalar model, the Bouc-Wen model cannot capture small minor loops due to partial reversals, whereas the two-state model captures the minor loops well.

Research of a Flexible Space-Vector-Based Hybrid PWM Transition Algorithm between SHEPWM and SHMPWM for Three-Level NPC Inverters

In this paper, one model of flexible space-vector-based hybrid pulse width modulation (HPWM) transition algorithm consisting of selective harmonic elimination pulse width modulation (SHEPWM) and selective harmonic mitigation pulse width modulation (SHMPWM) is applied and examined in a 10kV grid with a three-level neutral point clamped (3L-NPC) grid-connected inverter. These two modulation techniques are used to produce the appropriate firing pulses for 3L-NPC grid-connected inverters in different cases. SHMPWM is adopted to the grid-connected inverters to mitigate the required odd non-triplen harmonics according to the requirements of grid codes EN 50160 and CIGRE WG 36-05, while the firing pulses generated using SHEPWM is used to eliminate the primary low-order odd non-triplen harmonics completely. Meanwhile, one smooth and fast transition scheme is proposed by providing a suitable switching angles set at the transition point. Finally, it is demonstrated and validated by the MATLAB/SIMULINK model that smooth and quick transition is realized and there is no sudden change of current during the transition, as expected. Furthermore, this hybrid PWM technique is universal for different PWM methods based on the specific operating conditions.

A Digital Control Strategy With Simple Transfer Matrix for Three-Phase Buck Rectifier Under Unbalanced AC Input Conditions

In this letter, a digital control strategy with a simple transfer matrix between three-phase ac input voltages and dc-link voltage is proposed for a three-phase buck rectifier (3ph-BR) under unbalanced ac input conditions. The proposed digital control strategy can eliminate both the odd harmonics of the ac input currents and the even harmonics of the output voltage ripple under unbalanced ac input conditions. Compared with the state-of-the-art control strategies, the proposed control strategy does not require the extraction of the positive sequence voltage and the negative sequence voltage, thus a three-phase lock loop and complex calculation are not required. The proposed control strategy is essentially an input voltage feedforward control, which is independent of the loop bandwidth of the feedback system. Therefore, the processing speed of the digital processor is no longer the critical factor for the 3ph-BR under the unbalanced ac input conditions. The proposed control strategy makes it possible to utilize low-cost and small-sized digital processors. An experimental prototype of a 1.5-kW 3ph-BR with the proposed control strategy is built to verify the analysis results.

Vibration Reduction Characteristics of Permanent Magnet DC Motors With Sawtooth Edge Poles

It is known that the slot frequency vibration is the one of the main components in electromagnetic vibration spectrum of permanent magnet DC motors (PMDCM). After finding that the vibration reduction scheme of the piecewise varying pole width (PVPW) is only effective for the odd order harmonic of slot frequency vibrations by analytical method, the PM pole with sawtooth edge is put forward to reduce all harmonic components of slot frequency vibrations of PMDCM. In this article, after the air gap flux density and radial force density are analyzed, the two vibration sources, radial force and bending moment, are derived for the motor with sawtooth edge poles (SEP). Then the optimized pole shape is obtained to reduce all harmonic components of both the radial force and bending moment at the same time. Three dimensional finite element method of electromagnetic field and structure are used to calculate the magnetic flux density, radial force and vibration for the motors of PVPW and SEP. Three permanent magnet motors with fixed pole width (FPW), PVPW and SEP are manufactured and tested. The comparison and verification show that SEP is better than the other two schemes in general.

Forecasting the Reliability of Components Subjected to Harmonics Generated by Power Electronic Converters

This paper aims at refining an experimentally based reliability model for the insulation of power components subjected to the randomly varying harmonics generated by power electronic converters. Compared to previous papers of the same authors and to the existing literature, here the model is re-formulated from the theoretical viewpoint focusing on the foremost role played by low percentiles of time to failure—in particular by the 1st percentile—selected as the rated life in the framework of modern probabilistic design of components. This is not only more correct from the viewpoint of component design, but also on the safe side as for the reliability of devices. Moreover, the application of the model is broadened to treat the whole sequence of odd voltage harmonics from the 5th to the 25th, i.e., those taken as the most significant in power systems according to international standards. The limits to voltage distortion set in Standard EN50160 are the reference for establishing parametrically a series of typical distorted voltage waveshape analyzed in the applicative part, which account for the possible phase-shift angles between voltage harmonics. The effect of current harmonics is also considered, from both the theoretical and applicative viewpoint. As a last, but not least novelty, the reliability model is used here for life and reliability estimates not only of Medium Voltage (MV)/Low Voltage (LV) capacitors and cables—already studied in the previous stages of this investigation—but also of induction motors and transformers in the presence of harmonics from power converters.

The Range of Jupiter's Flow Structures That Fit the Juno Asymmetric Gravity Measurements

AbstractThe asymmetric gravity field measured by the Juno spacecraft has allowed the estimation of the depth of Jupiter's zonal jets, showing that the winds extend approximately 3,000 km beneath the cloud level. This estimate was based on an analysis using a combination of all measured odd gravity harmonics, J3, J5, J7, and J9, but the wind profile's dependence on each of them separately has yet to be investigated. Furthermore, these calculations assumed the meridional profile of the cloud‐level wind extends to depth. However, it is possible that the interior jet profile varies somewhat from that of the cloud level. Here we analyze in detail the possible meridional and vertical structure of Jupiter's deep jet streams that can match the gravity measurements. We find that each odd gravity harmonic constrains the flow at a different depth, with J3 the most dominant at depths below 3,000 km, J5 the most restrictive overall, whereas J9 does not add any constraint on the flow if the other odd harmonics are considered. Interior flow profiles constructed from perturbations to the cloud‐level winds allow a more extensive range of vertical wind profiles, yet when the meridional profiles differ substantially from the cloud level, the ability to match the gravity data significantly diminishes. Overall, we find that while interior wind profiles that do not resemble the cloud level are possible, they are statistically unlikely. Finally, inspired by the Juno microwave radiometer measurements, assuming the brightness temperature is dominated by the ammonia abundance, we find that depth‐dependent flow profiles are still compatible with the gravity measurements.

Updated Equipotential Shapes of Jupiter and Saturn Using Juno and Cassini Grand Finale Gravity Science Measurements

AbstractA commonly used shape model for the giant plants of Jupiter and Saturn is an oblate ellipsoid, a simplified model of the equipotential shape. The ellipsoidal shape models were originally derived from radio occultation data and gravity data after the Voyager flybys in 1979. Through precise Doppler tracking of NASA's Juno and Cassini spacecraft telecommunications links, zonal coefficients in a spherical harmonic expansion of the gravity field of Jupiter and Saturn have been resolved to degree 10, including the detection of nonzero odd zonal harmonics, which have been interpreted as differential rotation in the atmosphere for both planets. In this work, we construct the equipotential surfaces of Jupiter and Saturn using the recently measured gravity fields determined by Juno through perijove 8 and Cassini through end of mission. For both planets, even zonal harmonics dominate the equipotential shape, differing from the reference ellipsoid up to ~32 km at Jupiter and ~125 km at Saturn in the midlatitude regions. Saturn's internal rotation period estimated with ring seismology produces a shape that is fully consistent with the Pioneer and Voyager radio occultation measurements, and Jupiter's shape is fully consistent with the Pioneer and Voyager radio occultation measurements. With current equipotential theory, the recent analysis of the depth of the deep zonal flow on Jupiter and Saturn cannot fully be explained by the shape from current radio occultation measurements; additional occultation measurements and reanalysis of the Pioneer and Voyager radio occultations and will be useful for further constraining the shape of the planets.

3/2-Order Dual-Mode Fractional Repetitive Control for Harmonic Vibration Suppression in Magnetically Suspended Rotor

Harmonic currents in magnetically suspended rotor systems are usually caused by mass unbalance and displacement sensor errors, which will produce harmonic vibrations. Repetitive control (RC) is used to eliminate harmonic currents in the control system, but for the magnetic bearing control system, the odd harmonics often dominates, and the characteristics of the even harmonic frequency are not expected to change greatly. At the same time, the ratio of the sampling frequency to the rotor frequency is required to be an integer in the conventional repetitive controller (CRC), which greatly limits its range of application. Therefore, a 3/2-order dual-mode fractional-order repetitive control scheme with a three-parallel structure is proposed. The three branches in the scheme can not only eliminate odd and even harmonics independently,but also improve the robustness of frequency fluctuations. In addition, this method converts the fraction into an integer polynomial by Lagrange interpolation, which can accurately suppress the harmonic current at any frequency. The system delay is reduced to 0.5N sampling periods, and the system dynamic response rate is accelerated. Simulation results show that the proposed algorithm can effectively achieve harmonic vibration suppression at any frequency.

Effect of Molecular Charge Asymmetry on Even-to-odd Ratio of High-order Harmonic Generation

Recently, asymmetric molecules, such as HeH\(_2^+\), CO, OCS, HCl, have been evolved much attention since its rich information in the high-order harmonic generation (HHG), whose ratio of adjacent even and odd harmonics characterizes the asymmetry of molecules. In this paper, we study the dependence of even-to-odd ratio on the asymmetric parameters, in particular, the nuclear-charge ratio, and the permanent dipole, by exploiting a simple but general model of asymmetric molecules \(Z_1Z_2\) subjected to an intense laser pulse. The HHG is simulated by the numerical method of solving the time-dependent Schrödinger equation. We find out that this even-to-odd ratio strongly depends on the nuclear-charge ratio. In particular, the even-to-odd ratio reaches its maximum when the nuclear-charge ratio is about from 0.5 to 0.7. Besides, the dependence on the permanent dipole of the even-to-odd ratio has a non-trivial law. To explain, we calculate the analytical ratio of the transition dipole according to the emission of even and odd harmonics, and this ratio is well consistent with the even-to-odd ratio of the HHG.

A high-current 1.5-stage LED driver with current-mode BCM PFC and low-voltage film capacitors

In this paper, current-mode boundary conduction mode (BCM) of a high-efficiency LED driver with high power factor, utilizing low-voltage film capacitors, is presented, and its theoretical design and considerations are discussed. The 1.5-stage driver consists of a forward-flyback and buck converter. Higher efficiency was reached using the reduced redundant power processing approach. Most of the processed energy is transferred to the output through flyback, while the rest removes the low harmonic ripple in the output current through the forward-buck path. An acceptable amount of odd harmonics is injected into the input current by using current-mode BCM control method, further reducing bus capacitance. To validate these principles, a laboratory prototype has been built and experimental results of a high-current 85 W (29.5 V/2.9 A) lamp with 50 Hz, 185–255 V power supply are provided. The results show that output current ripple is low, and the input current harmonics are within the standard regulations.

Generation of Terahertz Radiation by Multicolor Ionizing Pulses

Analytical formulas have been obtained for the optimal waveform of an ionizing pulse and corresponding maximum density of residual terahertz currents. The found analytical solutions can be implemented with multicolor femtosecond pulses containing the field at the fundamental frequency and several of its lowest harmonics. The amplitudes of odd harmonics in the field optimal for the generation of terahertz radiation are larger than those for neighboring even harmonics. The maximum residual current density increases with an increase in the number of harmonics, in the wavelength of the fundamental harmonic, and in the ionization potential of gas particles and with a decrease in the duration of the ionizing pulse. This maximum residual current density approaches limiting values for saturated ionization when few-cycle pulses are used.

Polarization-resolved analysis of high-order harmonic generation in monolayer MoS2

We employ a theoretical model based on the density-matrix equation in the velocity gauge to calculate high harmonic generation from monolayer MoS2. This approach incorporates the tight-binding model, enabling the full consideration of both crystal symmetry and multiple band effects. In additional to the usual odd harmonics, even harmonics are also presented in the case of observing two different polarization components, which are parallel and perpendicular to the polarization of linearly driving pulses. We detailedly analyze the crystal orientation dependence for the parallel and perpendicular components of both odd- and even-harmonics. It is found that they exhibits different modulation behavior with rotating the crystal orientation. The simulation results capture all important orientation-dependent features observed in the recent experiment, thus demonstrating that the Berry curvature of MoS2 has been appropriately considered in our proposed model. In order to facilitate analysis of the underlying mechanism, we examine the channel current in terms of the contribution from different density-matrix elements, and identify their role in the orientation modulation of high harmonics. We further use simplified one-dimensional integral model to explain the appearance of perpendicular components of even harmonics. Our analysis shows that the multi-band coupling effect is the origin of the parallel even harmonics, while the broken inversion symmetry of phase difference of momentum matrix elements along two orthogonal directions determines the perpendicular even harmonic generation. Additionally, the relationship between the concept of Berry curvature and our theoretical framework is discussed. These demonstrations show that polarization-resolved high harmonics might provide an all-optical way for imaging material’s Berry curvature.

Current-assisted Raman activation of the Higgs mode in superconductors

The Higgs mode in superconductors is a scalar mode without electric or magnetic dipole moment. Thus, it is commonly believed that its excitation is restricted to a nonlinear two-photon Raman process. However, recent efforts have shown that a linear excitation in the presence of a supercurrent is possible, resulting in a new resonant enhancement at $\Omega=2\Delta$ with the driving light frequency $\Omega$ and the energy of the Higgs mode $2\Delta$. This is in contrast to the usual $2\Omega = 2\Delta$ resonance condition found in nonlinear third-harmonic generation experiments. In this communication, we show that such a linear excitation can still be described as an effective Raman two-photon process, with one photon at $\omega=2\Delta$ and one virtual photon at $\omega=0$ which represents the dc supercurrent. At the same time we demonstrate that a straightforward infrared activation with a single photon excitation is negligible. Moreover, we give a general context to our theory, providing an explanation for how the excitation of the Higgs mode in both THz quench and drive experiments can be understood within a conventional difference-frequency generation or sum-frequency generation process, respectively. In such a picture, the observed new resonance condition $\Omega = 2\Delta$ is just a special case. With the same approach, we further discuss another recent experiment, where we find a suppression of odd order higher harmonics in the presence of a dc supercurrent.

Advances on odd-even high-harmonic spectroscopy

High harmonic generation (HHG) has important applications in attosecond science, such as generating attosecond pulses and probing the structure and the electron dynamics of atoms and molecules with attosecond resolution. Different from symmetric cases, because of symmetry breaking, asymmetric molecules can emit both odd-order and even-order harmonics in strong single-color laser fields. Ultrafast probing of polar molecules with HHG is a hot issue in recent years, as polar molecules are generally very active in chemical reactions. With dividing the HHG signal into odd and even parts, recent studies have shown that odd and even harmonics from asymmetric molecules have different properties in frequency domain. To explain this typical difference, a theoretical model has been developed to describe the generation mechanism of odd versus even HHG from asymmetric molecules. It is shown that during the process of tunneling, the gerade component of the asymmetric orbital plays a dominating role and the contributions of the ungerade component can be neglected. Thereafter, the recombination of the rescattering electron with the gerade (ungerade) component of the asymmetric orbital mainly contributes to the emission of odd (even) harmonics. As a result, odd and even harmonics carry different information of the asymmetric system. Recently, it has been shown that the contributions of the ungerade component of asymmetric orbital to ionization are strongly suppressed due to the destructive intramolecular interference. This suppressing effect is general for strong-field ionization of polar molecules, implying that the theoretical model of odd-even HHG holds for a wide parameter region. These results open a new perspective for probing polar molecules with odd-even HHG. Specifically, when it is relatively difficult to read the information of the molecular structure or the electron dynamics through the whole harmonic spectrum of asymmetric molecules, one can extract the characteristic information of odd or even harmonic separately, then this characteristic information can be integrated and inverted to get the information of the whole asymmetric system. These discussions suggest ultrafast probing of asymmetric molecules with odd-even high-harmonic spectroscopy (HHS), where odd and even harmonics are treated independently. This idea of odd-even HHS and the theoretical model of odd-even HHG mentioned above have been widely used in ultrafast probing of asymmetric molecules in theoretical studies. For example, it has been shown that odd-even HHS and this theoretical model can be used to reconstruct the asymmetric molecular orbital, probe the nuclear dynamics of polar molecules, calibrate the degree of orientation of asymmetric molecules, and probe the structure of multi-center molecules, etc. Experimentally, the idea of odd-even HHS has also been widely used in probing the structure and the electron dynamics of polar molecules. In particular, this theoretical model of odd-even HHG has also been used in probing the shape resonance and charge transport of asymmetric molecules in experiments. In addition, theoretical studies based on the first-principle calculations have also shown the applicability of this theoretical model of odd-even HHG, which provides the theoretical tool for the present odd-even HHS. The research and development of odd-even HHS plays an important role in promoting the application of HHG in attosecond science and chemical reactions.