Generation Of Odd Harmonics Research Articles

Saddle point analysis on selective enhancement of high-order harmonic generation

We theoretically demonstrate control of selective enhancement of the near-cutoff odd order harmonic generation by dual-color chirped laser field. The sharp enhancements of a single harmonic order ranging from H37 to H59 (λ h =21.6nm-13.5nm) are achieved by adjusting the intensity of the fundamental frequency field. The saddle point method is applied to the analysis of high-order harmonic generation using the distorted-waveform laser field. A series of orbits of principal contribution to enhancement are extracted using the saddle point method, and the enhancement of the key orbits’ interference is reconstructed by quantum orbital theory. Through the analysis of the interference mechanism of enhancement, we give a full explanation of the dependence of enhanced harmonic order on laser intensity.

Low projectile density contributions in the dilute-dense CGC framework for two-particle correlations

At leading-order, the standard dilute-dense Color Glass Condensate formula used for two-particle correlations in proton-nucleus collisions, whose symmetries prevent the generation of odd azimuthal anisotropy harmonics, is the dilute projectile limit of the dense-dense formalism. However, when the projectile is genuinely dilute, the complete formulation contains additional contributions at the same leading order in the strong coupling constant. In this work we investigate those low projectile density contributions that are relevant when the particles are produced at forward rapidities. We find that they are responsible for non-zero odd harmonics which are negative, in qualitative agreement with recent experimental measurements at the Relativistic Heavy-Ion Collider.

Multiple conical odd harmonics from filament-inscribed nanogratings

We report on the observation of conical third, fifth, seventh, and ninth harmonics that gradually emerge during the supercontinuum generation by filamentation of femtosecond midinfrared pulses in lithium strontium hexafluoroaluminate crystal. We show that the generation of conical odd harmonics is an optical signature of light-driven material reorganization in the form of volume nanogratings at the site irradiated by repetitive femtosecond filaments. The angle-resolved spectral measurements demonstrate remarkably broad spectra of individual odd harmonics, benefiting from a spectrally broadened pump pulse (supercontinuum), and reveal that filament-inscribed nanogratings represent photonic structures that are able to provide ultrabroad phase-matching bandwidths covering the wavelength range from the ultraviolet to the near infrared. We propose a scenario that interprets the generation of conical fifth, seventh, and ninth harmonics as nanograting phase-matched cascaded noncollinear four-wave mixing processes.

On the physics of generation of even and odd harmonics in the undulators of free electron lasers

We theoretically explore the physics of even and odd harmonics generation and radiation in undulators of single pass free electron lasers (FELs). Planar undulators are studied as a most common design. The results are obtained analytically and some of them also numerically. In the analytical form the contributions of various factors to the harmonic generation are distinguished and their effect is analyzed. The results are verified with numerical simulations and compared with reported experimental data; the data for best documented FELs in terms of harmonic generation, LEUTL and LCLS, is analyzed. The reasons for even harmonic generation are identified; we show that they differ for the spontaneous undulator radiation and for the stimulated FEL radiation. Comparative analysis is performed with various theoretical models for the FEL harmonic powers. The advantages and possibilities of amplification and radiation the second FEL harmonic in the LCLS-II FEL with variable deflection parameter is demonstrated for the harmonic lasing self-seed (HLSS) FEL design. The possibility to suppress the radiated second harmonic in a FEL by a guiding field of a weak second harmonic of the main undulator field with negative phase is shown. This can be used to reduce the reflected signal of the second harmonic generation (SHG) when the nonlinear harmonic response of the mater is investigated.

Harmonic Generation in Biased Semiconductor Superlattices.

Semiconductor superlattices are proven nanomaterials for THz nonlinear optics by means of high order harmonic generation. Seminal approaches leading to a perfectly antisymmetric current-voltage (I–V.) curve predict the generation of odd harmonics only in the absence of a bias. However, even harmonics at high orders have been detected in several experiments. Their generation has been explained by considering deviations from the current flow symmetry that break the exact antisymmetry of the I–V. curve. In this paper, we focus on another issue found experimentally that has also not been explained, namely the harmonic power output asymmetry from negative to positive applied bias. Once more, breaking the I–V. flow symmetry explains the experiments and leads to a further tool to design the power output of these materials. Furthermore, a new approach for the Boltzmann Equation under relaxation-rate approximation eliminates numerical difficulties generated by a previous theory. This leads to very efficient analytical expressions that can be used for both fundamental physics/optics/material sciences and realistic device development and simulations.

High-harmonic spectroscopy of light-driven nonlinear anisotropic anomalous Hall effect in a Weyl semimetal

Weyl semimetals are promising quantum materials that offer unique topological properties. Lately, it has been shown that laser-driven electron dynamics have characteristic signatures in two-dimensional and three-dimensional Dirac semimetals. The transition from Dirac to Weyl semimetal requires the breaking of either inversion or time-reversal symmetry. The present work shows that the laser-driven electron dynamics in a Weyl semimetal with broken time-reversal symmetry has intriguing features in its high-harmonic spectrum. It is found that the parity and magnitude of the non-zero Berry curvature's components control the direction and strength of the anomalous current, which leads to the generation of the anomalous odd harmonics. We demonstrate that the non-trivial topology of the Berry curvature in time-reversal symmetry broken quantum materials can be probed by measuring the polarisation of the emitted anomalous odd harmonics. Our findings unequivocally illustrate that laser-driven electron dynamics leads to the generation of nonlinear anisotropic anomalous Hall effect in time-reversal symmetry broken quantum materials on an ultrafast timescale.

A hyperbolic framework for shear sound beams in nonlinear solids

In soft elastic solids, directional shear waves are in general governed by coupled nonlinear KZK-type equations for the two transverse velocity components, when both quadratic nonlinearity and cubic nonlinearity are taken into account. Here we consider spatially two-dimensional wave fields. We propose a change of variables to transform the equations into a quasi-linear first-order system of partial differential equations. Its numerical resolution is then tackled by using a path-conservative MUSCL-Osher finite volume scheme, which is well-suited to the computation of shock waves. We validate the method against analytical solutions (Green’s function, plane waves). The results highlight the generation of odd harmonics and of second-order harmonics in a Gaussian shear-wave beam.

Modeling Ultrasound Propagation in the Moving Brain: Applications to Shear Shock Waves and Traumatic Brain Injury.

Traumatic brain injury (TBI) studies on the living human brain are experimentally infeasible due to ethical reasons and the elastic properties of the brain degrade rapidly postmortem. We present a simulation approach that models ultrasound propagation in the human brain, while it is moving due to the complex shear shock wave deformation from a traumatic impact. Finite difference simulations can model ultrasound propagation in complex media such as human tissue. Recently, we have shown that the fullwave finite difference approach can also be used to represent displacements that are much smaller than the grid size, such as the motion encountered in shear wave propagation from ultrasound elastography. However, this subresolution displacement model, called impedance flow, was only implemented and validated for acoustical media composed of randomly distributed scatterers. Herein, we propose a generalization of the impedance flow method that describes the continuous subresolution motion of structured acoustical maps, and in particular of acoustical maps of the human brain. It is shown that the average error in simulating subresolution displacements using impedance flow is small when compared to the acoustical wavelength ( λ /1702). The method is then applied to acoustical maps of the human brain with a motion that is imposed by the propagation of a shear shock wave. This motion is determined numerically with a custom piecewise parabolic method that is calibrated to ex vivo observations of shear shocks in the porcine brain. Then the fullwave simulation tool is used to model transmit-receive imaging sequences based on an L7-4 imaging transducer. The simulated radio frequency data are beamformed using a conventional delay-and-sum method and a normalized cross-correlation method designed for shock wave tracking is used to determine the tissue motion. This overall process is an in silico reproduction of the experiments that were previously performed to observe shear shock waves in fresh porcine brain. It is shown that the proposed generalized impedance flow method accurately captures the shear wave motion in terms of the wave profile, shock front characteristics, odd harmonic spectrum generation, and acceleration at the shear shock front. We expect that this approach will lead to improvements in image sequence design that takes into account the aberration and multiple reflections from the brain and in the design of tracking algorithms that can more accurately capture the complex brain motion that occurs during a traumatic impact. These methods of modeling ultrasound propagation in moving media can also be applied to other displacements, such as those generated by shear wave elastography or blood flow.

The frequency multiplication in electrooptical modulator of the Mach–Zehnder

The analysis of parameters of work of the electrooptical modulator of the Mach–Zehnder circuits for frequency multiplication of the radio signal. The possibilities of frequency multiplication are revealed and the optimal parameters of signals are determined. It is determined that for a given geometry of the modulator electrodes for longer laser waves, the half-wave voltage is higher than for shorter wavelengths. The maximum permissible modulation index is higher in Mach–Zehnder electrooptical modulators with a shorter wavelength of laser radiation. The number and amplitude of the spectral components at the output of the modulator depends on the amplitude of the modulating microwave signal, as well as on the location of the working point on the transfer function of the modulator. Without shifting the operating point, the Mach–Zehnder electrooptical modulator multiplies the signal frequency by 2, 4 and 6 times (to obtain even harmonics). If the modulator allows for a larger modulation index, even harmonics of a larger order can be obtained. The shift of the working point of the Mach–Zehnder modulator to the linear region of the transfer function (by half the half-wave voltage) allows to obtain odd harmonics. When the displacement voltage of the working point is equal to a quarter of the half-wave voltage, the Mach–Zehnder electrooptical modulator allows to obtain both even and odd harmonics. The maximum frequency of harmonics is limited by the allowable voltage amplitude for RF electrodes (the permissible amplitude of the modulating voltage). A further significant increase in the frequency multiplication factor is possible in circuits with two or more Mach- Zehnder modulators. Diagram with two modulators of the Mach–Zehnder allows you to get a grid of frequencies. The Mach–Zehnder modulator in the mode of odd harmonic generation can be used as a converter of sinusoidal voltage to a periodic voltage close to the pulse voltage. Radio-photon frequency multipliers have a wide band of operating frequencies. It is determined by the frequency properties of the modulator and photodetector. Frequency multipliers signal current in radiophonic systems to transfer sinusoidal frequencies in the microwave range, to synthesize a grid of frequencies.

Atomic even-harmonic generation due to symmetry-breaking effects induced by spatially inhomogeneous field**Project supported by the National Natural Science Foundation of China (Grant Nos. 91850114, 11774131, 11604119, and 11627807).

We ab initio investigate the interaction between the hydrogen atom and the inhomogeneous field which is induced by resonant plasmons within a metal nanostructure. Same as normal laser pulse (homogeneous field), only odd-harmonic generation occurs when the bow-tie nanostructure is utilized. For the single nanotip case, the even-harmonic generation can be distinctly found in the harmonic emission spectrum. By investigating the symmetry and trajectories of different inhomogeneous fields, we demonstrate that the breaking symmetry of system can enable even high harmonic generations.

Modeling and simulations of two dimensional propagation of shear shock waves in relaxing soft solids

Soft solids, such as gelatin or soft tissue, have a shear wave speed that is smaller than the compressional wave speed. Recent observations of shear shock wave generation in the brain, which can easily reach a large Mach number regime, suggest that the cubic nonlinear behavior of soft solids could be responsible for traumatic brain injury. However, currently there are no two-dimensional (2D) models describing the propagation of linearly-polarized shear shock waves in relaxing soft solids. These models are required to model the fundamental wave propagation physics like diffraction, focusing etc., and are related to traumatic brain injuries as it can be used to model the skull/brain morphology. In this work, we present a two-dimensional system of first-order equations modeling the propagation of shear shock waves in a relaxing soft solid, and then this system is solved numerically using the piecewise parabolic method, a high-order finite volume method. The numerical solutions are validated in two parts. First, the nonlinear component, which is designed for large Mach numbers, is validated with a step-shock Riemann problem. Then relaxation mechanisms based on a generalized Maxwell body, which model the non-classical attenuation that occurs in soft tissue, are compared to analytical solutions with an error of 5%-10%. The validation of attenuation also includes dispersion due to causality which is determined by the Kramers-Kronig relations. Finally, the full numerical method, which includes nonlinearity, attenuation, and dispersion, is compared to ultrasonic measurements of shear shock wave propagation in tissue-mimicking phantoms. Two experiments were performed based on high frame-rate ultrasound imaging and tracking in a gelatin phantom 1) planar shear shock waves, and 2) focused shear shock wave propagation. The experimental and numerical waveforms closely match, e.g. the RMS amplitude error is between 12.05% and 12.27%. Moreover, the frequency-content of the temporal signal was compared for third and fifth multiples of fundamental harmonic validating the generation of odd-harmonics due to cubic nonlinearity. Furthermore, the numerical tool was able to estimate the nonlinear parameter in the phantom (β=4.4) using a grid-parameter-search. In context of traumatic brain injury, the current method can be used to study the shear shock formation in 2D-sections of human skull, and can also be used for nonlinear parameter estimation in brain.

Pure even and odd harmonics produced in modeled hydrogen molecular ions in single-color strong laser fields

The even and odd high-order harmonic generation in a modeled hydrogen molecular ion in strong laser fields is studied by numerically simulating the time-dependent Schrodinger equation. By applying a linearly polarized laser pulse whose polarization axis is perpendicular to the molecular axis, the pure odd harmonics polarized along the laser polarization direction are produced. Meanwhile, the pure even harmonics polarized along the molecular axis may be produced simultaneously either by adding an extra weak direct-current electric field along the molecular axis, or by artificially setting asymmetric charges for the two nuclei. The Bohmian trajectories reveal that the pure even harmonics are contributed by the dipole formed by the asymmetric expansion of the ionized wave packet along the molecular axis, instead of the asymmetric Coulomb attraction for the rescattering electron as expected. Moreover, the relationship between the symmetry of molecular orbital and the parity of high-order harmonics is also explored.

Light bullet supercontinuum

Some new results of experimental and numerical investigations on light bullet spectra are depicted. The scenario of supercontinuum and light field evolution from odd harmonics generation to the appearance of broadband spectrum and pulse compression up to a single-cycle light bullet accompanied by the onset of isolated an anti-Stokes wing in spectra and high-frequency modulation of the light field is shown. An extension of the dispersion equation for the anti-Stokes band spectral maximum to all known experimental results obtained up to now for light bullet spectra in different transparent dielectrics is presented. The decisive role of multiphoton ionization in the short-wavelength cutoff of a supercontinuum is revealed experimentally, numerically, and analytically.

Polarization-Resolved Study of High Harmonics from Bulk Semiconductors.

The polarization property of high harmonics from gallium selenide is investigated using linearly polarized midinfrared laser pulses. With a high electric field, the perpendicular polarization component of the odd harmonics emerges, which is not present with a low electric field and cannot be explained by the perturbative nonlinear optics. A two-dimensional single-band model is developed to show that the anisotropic curvature of an energy band of solids, which is pronounced in an outer part of the Brillouin zone, induces the generation of the perpendicular odd harmonics. This model is validated by three-dimensional quantum mechanical simulations, which reproduce the orientation dependence of the odd-order harmonics. The quantum mechanical simulations also reveal that the odd- and even-order harmonics are produced predominantly by the intraband current and interband polarization, respectively. These experimental and theoretical demonstrations clearly show a strong link between the band structure of a solid and the polarization property of the odd-order harmonics.

Nonlinear optical effects of opening a gap in graphene

Graphene possesses remarkable electronic, optical and mechanical properties that have taken the research of two-dimensional relativistic condensed matter systems to prolific levels. However, the understanding of how its nonlinear optical properties are affected by relativistic-like effects has been broadly uncharted. It has been recently shown that highly-nontrivial currents can be generated in free-standing samples, notably leading to the generation of even harmonics. Since graphene monolayers are centrosymmetric media, for which such harmonic generation is deemed inaccessible, this light-driven phenomenon is both startling and promising. More realistically, graphene samples are often deposited on a dielectric substrate, leading to additional intricate interactions. Here, we present a treatment to study this instance by gapping the spectrum and we show this leads to the appearance of a Berry phase in the carrier dynamics. We analyse the role of such a phase in the generated nonlinear current and conclude that it suppresses odd-harmonic generation. The pump energy can be tuned to the energy gap to yield interference among odd harmonics mediated by interband transitions, allowing even harmonics to be generated. Our results and general methodology pave the way for understanding the role of gap-opening physical factors in the nonlinear optics of hexagonal two-dimensional lattices.

Optical Rectification and Generation of Harmonics Under Condition of Propagation of Few-Cycle Pulses in the Birefringent Medium with Asymmetric Molecules

Without using the approximation of slowly varying envelopes, we investigate spectral transformations of vector few-cycle and quasi-monochromatic solitons under propagation in the birefringent medium of asymmetric molecules possessing permanent dipole moments. We show that when an ordinary pulse enters the medium, the self-modulation effect arises. As a result, the spectrum of the ordinary component undergoes significant changes from the violet shift and the generation of odd harmonics with close frequency satellites to the supercontinuum including zero frequency. The spectrum of the extra-ordinary few-cycle pulse also has the properties of a supercontinuum. When a quasi-monochromatic ordinary pulse enters the medium in the spectrum of an extra-ordinary component, the effect of optical rectification with simultaneous generation of the second harmonics is clearly manifested.

Thermally and electrically controllable multiple high harmonics generation by harmonically driven quasi-two-dimensional electron gas

In this paper, we consider the activation processes in nonlinear meta-stable system based on a lateral (quasi-two-dimensional) superlattice and study the dynamics of such a system externally driven by a harmonic force. The internal control parameters are the longitudinal applied electric field and the sample temperature. The spontaneous transverse electric field is considered as an order parameter. The forced violations of order parameter are considered as a response of a system to periodic driving. We investigate the cooperative effects of self-organization and high harmonic forcing from the viewpoint of catastrophe theory and show the possibility of generation of third and higher odd harmonics in output signal that lead to distortion of its wave front. A higher harmonics detection strategy is further proposed and explained in detail by exploring the influences of system parameters on the response output of the system that are discussed through numerical simulations.

FDTD Modeling of Nonlinear Phenomena in Wave Transmission Through Graphene

A novel finite-difference time-domain modeling method is proposed to simulate the nonlinear electrodynamic responses of graphene at terahertz frequencies. The relation between currents in graphene and electromagnetic waves is governed by a $J\text{--}E$ characteristic formula at the frequencies where the conductivity of graphene exhibits resonant behavior. Simulation results demonstrate nonlinear phenomena in wave transmission through graphene including odd-harmonic generation and frequency-mixing effect. The proposed modeling method can be used as a full-wave solution to the design of nonlinear graphene-based structures.

Simulation of nonlinear transient elastography: finite element model for the propagation of shear waves in homogeneous soft tissues.

In this study, visco-hyperelastic Landau's model, which is widely used in acoustical physic field, is introduced into a finite element formulation. It is designed to model the nonlinear behaviour of finite amplitude shear waves in soft solids, typically, in biological tissues. This law is used in finite element models based on elastography, experiments reported in Jacob et al, the simulations results show a good agreement with the experimental study: It is observed in both that a plane shear wave generates only odd harmonics and a nonplane wave generates both odd and even harmonics in the spectral domain. In the second part, a parametric study is performed to analyse the influence of different factors on the generation of odd harmonics of plane wave. A quantitative relation is fitted between the odd harmonic amplitudes and the non-linear elastic parameter of Landau's model, which provides a practical guideline to identify the non-linearity of homogeneous tissues using elastography experiment.

Theoretical study of the odd–even-order harmonic generation for asymmetric ions in non-Born–Oppenheimer approximation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11404153, 11135002, 11475076, and 11405077) and the Fundamental Research Funds for the Central Universities of China (Grants Nos. lzujbky-2016-29, lzujbky-2016-31, and lzujbky-2016-209).

We calculated the harmonic spectra generated from the asymmetric molecules of HD and HeH. It is found that HDproduces only odd harmonics, while HeH produces both odd and even harmonics. Further analysis reveals that for both HD and HeH, 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 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.