Nonlinear Three-wave Interaction Research Articles

CUDA-based focused Gaussian beams second-harmonic generation efficiency calculator

We present an object-oriented programming (OOP) CUDA-based package for fast and accurate simulation of second-harmonic generation (SHG) efficiency using focused Gaussian beams. The model includes linear as well as two-photon absorption that can ultimately lead to thermal lensing due to self-heating effects. Our approach speeds up calculations by nearly 40x (11x) without (with) temperature profiles with respect to an equivalent implementation using CPU. The package offers a valuable tool for experimental design and study of 3D field propagation in nonlinear three-wave interactions. It is useful for optimization of SHG-based experiments and mitigates undesired thermal effects, enabling improved oven designs and advanced device architectures, leading to stable, efficient high-power SHG. Program summaryProgram Title:cuSHGCPC Library link to program files:https://doi.org/10.17632/hn76s7x848.1Developer's repository link:https://github.com/alfredos84/cuSHGLicensing provisions: MITProgramming language:▪, CUDANature of problem: The problem which is solved in this work is that of second-harmonic generation (SHG) performance degradation in a nonlinear crystal with focused Gaussian beams due to thermal effects. By placing the nonlinear crystal in an oven that controls temperature, the package computes the involved electric fields along the medium. The implemented model includes the linear and nonlinear absorption which occasionally lead to self-heating effect, degrading the performance of the SHG.Solution method: The coupled differential equations for three-wave interactions, which describe the field evolution along the crystal, are solved using the well-known Split-Step Fourier method. The temperature profiles are estimated using the finite-elements method. The field evolution and thermal effects are embedded in a self-consistent algorithm that sequentially and separately solves the electromagnetic and thermal problems until the system reaches the steady state. Due to the eventual computational demand that some problems may have, we chose to implement the coupled equations in the ▪/CUDA programming language. This allows us to significantly speed up simulations, thanks to the computing power provided by a graphics processing unit (GPU) card. The output files obtained are the interacting electric fields and the temperature profile, which have to be analyzed during post-processing.

Measurement of nonlinear three-wave interaction using shear-vertical-wave point-focusing electromagnetic acoustic transducers

Abstract We developed a method of nonlinear three-wave interaction using shear-vertical-wave point-focusing electromagnetic acoustic transducers (SV-wave PF-EMATs). EMATs can excite and detect ultrasonic waves without couplants through interaction between electromagnetic fields and solid mechanics and are generally unsuitable for the nonlinear ultrasound measurement due to their low signal conversion efficiency. However, our originally developed PF-EMAT can focus SV waves at a focal point in phase, achieving a large local displacement. Here, two PF-EMATs with different driving frequencies are placed facing each other so as to match their focal points, where the focused SV waves intersect to induce the third waves with the frequencies of the sum and difference of the incident frequencies through the nonlinear three-wave interaction. We demonstrate that the local nonlinearity can be measured by this method with EMATs and demonstrate its applicability to evaluation of fatigue damage in a fatigue crack initiation phase with aluminum alloy specimens.

Impact of longitudinal phase-matching variations on three-wave nonlinear interactions

A general study of three-wave nonlinear mixing in the presence of longitudinal variations in phase-matching conditions is presented. The efficiency of second-harmonic generation and optical parametric amplification is quantified using a normalized set of equations and a polynomial description of the wave-vector mismatch as a function of the longitudinal coordinate. These modeling results are used to estimate the impact of spatial variations in wave-vector mismatch experimentally obtained for five partially deuterated potassium dihydrogen phosphate crystals. The longitudinal inhomogeneities in the properties of crystals of similar quality are not expected to have a significant impact on their use for second-harmonic generation and optical parametric amplification, but the efficiency of nonlinear processes in crystals with larger variations could decrease.

Nickel-Based Superalloy Inconel 718, Relation between Non-Linear Ultrasonic and Microstructure Changes under the Creep Damages.

Nickel-based superalloy, Inconel 718, has high temperature strength effected by γ"(Ni3Nb) phase at approximately 973 K. It has been found that the creep damage of Inconel 718 was induced by γ"phase disappearance. In the study, we applied the nonlinear ultrasonic method, nonlinear three-wave interaction, for evaluation of the creep damages and the microstructural degradation in γ" phase for Inconel 718 at 973 K, 310MPa. In the nonlinear three-wave interaction method, two intersecting ultrasonic waves produced a scattered wave when the resonance condition was satisfied. The amplitude in resonant scattering wave was measured. We analyzed the relation between the microstructural change and nonlinear acoustic characterizations by the electromagnetic acoustic resonance (EMAR) throughout the creep life, and the nonlinearity corresponded to change of the dislocation density. This was supported with X-ray diffraction, EBSD and SEM observation.

The simulation of ELMs mitigation by pedestal coherent mode in EAST using BOUT++

A general phenomenon that the edge localized modes (ELMs) can be effectively mitigated with the enhanced coherent modes (CMs) has been observed on EAST. For this phenomenon, the experimental statistical analysis and electromagnetic (EM) simulations have been performed. There is a threshold value of the CM intensity in the experiments, which plays a key role in ELMs mitigation. Through the ELITE and conventional BOUT++ analysis, we found that when the insignificant ELM and enhanced CM co-exist, the pedestal is located in unstable P–B region and the ELM is relatively large. The simulation results only using the experimental profiles without considering other factors cannot reproduce the no significant ELM experiment. The CM enhances the edge turbulence, which can control ELMs. Therefore, the effects of CM are considered to explain the ELM mitigation. Modifying the three-field reduced model in BOUT++, an imposed perturbation is added as the CM. The simulation results indicate that: without the CM, the ELM size belongs to the relative large ELM region; after considering the CM, the ELM is mitigated and the energy loss is reduced by about 44.5%. Analysis shows that the CM enhances the three-wave nonlinear interactions in the pedestal and reduces the phase coherence time (PCT) between the pressure and potential, which lead the perturbation to tend to be ‘multiple-mode’ coupling. The competition of free energy between the multiple modes leads to the lack of obvious filament structures and the decreased energy loss. The above reveals that there is a competitive relationship between turbulence and ELMs, and the CM-enhanced turbulence can effectively reduce ELM energy loss. In addition, through the parameter scanning, there is a threshold of the amplitude A, which is consistent with the statistical results in the experiments.

Explosive instability of dust settling in a protoplanetary disc

ABSTRACT It is shown that gas-dust perturbations in a disc with dust settling to the disc mid-plane exhibit the non-linear three-wave resonant interactions between streaming dust wave (SDW) and two inertial waves (IW). In the particular case considered in this paper, SDW at the wavenumber k• = 2κ/(g$z$ts), where κ, g$z$, and ts are, respectively, epicyclic frequency, vertical gravitational acceleration, and particle’s stopping time, interacts with two IW at the lower wavenumbers k′ and k″ such that k′ < kDSI < k″ < k•, where kDSI = κ/(g$z$ts) is the wavenumber of the linear resonance between SDW and IW associated with the previously discovered linear dust settling instability. The problem is solved analytically in the limit of the small dust fraction. As soon as the dynamical dust back reaction on gas is taken into account, k•, k′, and k″ become slightly non-collinear and the emerging interaction of waves leads to simultaneous explosive growth of their amplitudes. This growth is explained by the conservative exchange with energy between the waves. The amplitudes of all three waves grow because the negative energy SDW transfers its energy to the positive energy IW. The product of the dimension-less amplitude of initially dominant wave and the time of explosion can be less than Keplerian time in a disc. It is shown that, generally, the three-wave resonance of an explosive type exists in a wide range of wavenumbers 0 < k• ≤ 2κ/(g$z$ts). An explosive instability of gas-dust mixture may facilitate the dust clumping and the subsequent formation of planetesimals in young protoplanetary discs.

The simulation of ELMs mitigation by pedestal coherent mode in EAST using BOUT++

Abstract A general phenomenon that the edge localized modes (ELMs) can be effectively mitigated with the enhanced coherent modes (CMs) has been observed on EAST. For this phenomenon, the experimental statistical analysis and electromagnetic (EM) simulations have been performed. There is a threshold value of the CM intensity in the experiments, which plays a key role in ELMs mitigation. Through the ELITE and conventional BOUT++ analysis, we found that when the insignificant ELM and enhanced CM co-exist, the pedestal is located in unstable P-B region and the ELM is relatively large. The simulation results only using the experimental profiles without considering other factors cannot reproduce the no significant ELM experiment. The CM enhances the edge turbulence, which can control ELMs. Therefore, the effects of CM are considered to explain the ELM mitigation. Modifying the three-field reduced model in BOUT++, an imposed perturbation is added as the CM. The simulation results indicate that: without the CM, the ELM size belongs to the relative large ELM region; after considering the CM, the ELM is mitigated and the energy loss is reduced by about 44.5%. Analysis shows that the CM enhances the three-wave nonlinear interactions in the pedestal and reduces the phase coherence time (PCT) between the pressure and potential, which lead the perturbation to tend to be ‘multiple-mode’ coupling. The competition of free energy between the multiple modes leads to the lack of obvious filament structures and the decreased energy loss. The above reveals that there is a competitive relationship between turbulence and ELMs, and the CM-enhanced turbulence can effectively reduce ELM energy loss. In addition, through the parameter scanning, there is a threshold of the amplitude A, which is consistent with the statistical results in the experiments.

Nonlinear interaction of EGAM with DW turbulence in the Dimits shift region

Based on the nonlinear three-wave interaction model including driving and dissipation, we construct a system to describe the nonlinear interaction between energetic particle induced geodesic acoustic mode (EGAM) and the drift waves turbulence in the Dimits shift region, and study both analytically and numerically the linear growth and nonlinear oscillation phases of the system, respectively. Further numerical results show that, without the contribution of EGAM, the system goes through limited cycle oscillation to period doubling, and finally route to chaos with the change of the linear drive/dissipation rate. On this basis, the nonlinear saturated “Dimits region” of the system is constructed, which is then used to study the influence of EGAM on the drift wave in the Dimits region. The results show that for EGAM with different amplitude and frequency, the modulated drift wave can be either excited or suppressed, partly reproduces the results from large scale simulation. Finally, We use the method of phase space analysis to give the corresponding explanation.

Energy Spectra of Ensemble of Nonlinear Capillary Waves on a Fluid

The problem of spectral description of the nonlinear capillary waves on the fluid surface is discussed. Usually, three-wave nonlinear interactions are considered as a major factor determined by the energy spectrum of these waves in the kinetic wave turbulent regime. We demonstrate that four-wave interactions should be taken into account. In this case, there are two possible scenarios for the transfer of energy over the wave spectrum: kinetic and dynamic. The first is described by the averaged stochastic interaction of waves using the kinetic equation, while the second is described by dynamic equations written for discrete modes. In this article, we compare the time scales, spectral shapes, and other properties of both energy cascades, allowing them to be identified in an experiment.

Damping of Oscillations in Red Giants by Resonant Mode Coupling

Abstract Asteroseismic studies of red giants generally assume that the oscillation modes can be treated as linear perturbations to the background star. However, observations by the Kepler mission show that the oscillation amplitudes increase dramatically as stars ascend the red giant branch. The importance of nonlinear effects should therefore be assessed. In previous work, we found that mixed modes in red giants are unstable to nonlinear three-wave interactions over a broad range of stellar mass and evolutionary state. Here we solve the amplitude equations that describe the mode dynamics for large networks of nonlinearly coupled modes. The networks consist of stochastically driven parent modes coupled to resonant secondary modes (daughters, granddaughters, etc.). We find that nonlinear interactions can lower the energy of gravity-dominated mixed modes by ≳80% compared to linear theory. However, they have only a mild influence on the energy of pressure-dominated mixed modes. Expressed in terms of the dipole-mode visibility V 2, i.e., the summed amplitudes of dipole modes relative to radial modes, we find that V 2 can be suppressed by 50%–80% relative to the linear value for highly evolved red giants whose frequency of maximum power ν max ≲ 100 μ Hz . However, for less evolved red giants with 150 ≲ ν max ≲ 200 μ Hz , V 2 is suppressed by only 10%–20%. We conclude that resonant mode coupling can have a potentially detectable effect on oscillations at ν max ≲ 100 μ Hz but it cannot account for the population of red giants that exhibit dipole modes with unusually small amplitudes at high ν max .

Neoclassical Tearing Mode Seeding by Nonlinear Three-Wave Interactions in Tokamaks.

We report the experimental observation of seed magnetic island formation by nonlinear three-wave coupling of magnetic island triplets. In this experiment, disruptive 2,1 islands are seeded by the coupling of 4,3 and 3,2 tearing modes to a central 1,1 sawtooth precursor. Three-wave interactions between these modes are conclusively identified by bispectral analysis, indicating fixed phase relationships in agreement with theory. This new observation of this seeding mechanism has important implications for future reactors that must operate in stable plasma equilibria, free of disruptive 2,1 islands.

Eigenmodes of the Boundary of a Magnetic Barrier Flowed Around by Plasma: the Boundary Membrane Model, Linear and Nonlinear Resonances, and Couplings with Internal Modes

The linear and nonlinear resonances and cascades affecting significantly the interaction of a collisionless plasma flow with a magnetic barrier and the plasma penetration into the barrier are studied using the near-Earth magnetopause as an example. Our linear theory of the membrane instability of the boundary of a magnetic barrier, the magnetopause, explains the structure of the spectra and bi-spectra of fluctuations on different segments of the boundary layers near the magnetopause and the resonances at frequencies 0.05–0.5 mHz. The recorded compressibility of waves under the magnetopause implies that their excitation by Alfv’en resonances alone is impossible. We assume, as a general principle, an approximate coincidence of the frequencies of different harmonics of different resonances as a condition for the amplification of resonances to nonlinear amplitudes and their propagation into the ionosphere. We confirm the presence of resonant waves near the magnetopause and demonstrate a nonlinear three-wave cascade interaction of compressible (0.05–5 mHz) and incompressible waves under the magnetopause that provides the excitation of resonances of magnetic field lines and waveguide resonances at frequencies 1–25 mHz without a close coincidence of the frequencies with linear resonances at the magnetopause and the bow shock. This can also provide a nonlinear coupling of linear and nonlinear resonances in other multilayered media (for example, in the plasma and neutral sheets of the geomagnetic tail and between it and the magnetopause). An example of the excitation of a waveguide mode inside the magnetopause by nonlinear membrane waves is given. We have detected a nonlinear excitation and interaction of most “magic” harmonics under the magnetopause.

Experimental observation of resonance manifold shrinking under zonal flow shear.

In two-dimensional turbulent systems the redistribution of energy can be described by quadratic nonlinear three-wave interactions, which are limited by resonance conditions. The set of coupling modes can be understood as resonant manifold. It has been predicted by theory that, in the presence of a shear flow, the resonant manifold in wave-number space shrinks in time favoring large-scale structures. The phenomenon of manifold shrinking in the presence of shear flows is studied the first time experimentally in drift wave turbulence at the stellarator TJ-K by bicoherence analysis. By estimating effective mode numbers characterizing the width of the manifold, it is demonstrated that increasing shear leads to a shrinking of the resonance manifold.

Nonlinear energy transfer from low frequency electromagnetic fluctuations to broadband turbulence during edge localized mode crashes

Long ELMy H-mode KSTAR discharges are used to show that low frequencies () of the ion-scale density fluctuations contain electromagnetic components that transfer energy from to higher frequency () fluctuations during ELM crashes via three-wave non-linear interactions. Such non-linear interactions are not observed during the inter-ELM phases. These observations suggest that investigations on the enhanced fluctuation and transport levels during ELM crashes require a careful analysis of the non-linear couplings between low frequency electromagnetic and broadband fluctuations in addition to other causes such as open (stochastic) magnetic fields and advective transport. Our results are based on conditionally ensemble-averaged second-order and third-order spectral analyses applied to 2, 660 ELMs obtained from three different KSTAR discharges to ensure statistical validity.

The Shallow-Water Magnetohydrodynamic Theory of Stratified Rotating Astrophysical Plasma Flows: Beta-Plane Approximation and Magnetic Rossby Waves

Rotating magnetohydrodynamic flows of a thin stratified plasma layer in a gravitational field with a free boundary in an external vertical magnetic field were investigated. Magnetohydrodynamic equations were obtained in the two-layer shallow-water approximation in an external magnetic field when plasma is divided into two layers of different densities. A system of shallow-water equations for rotating stratified plasma in an external magnetic field was obtained in the beta-plane approximation. A linear theory was developed, and solutions in the form of magnetic Rossby waves, along with corrections that describe the effects of stratification, were obtained for stationary solutions in the form of vertical or horizontal magnetic field. Qualitative analysis of dispersion curves reveals the presence of three-wave nonlinear interactions of magnetic Rossby waves for each of the stationary states. The existence of parametric instabilities was demonstrated and their increments were found.

Low-frequency hiss-like whistler-mode waves generated by nonlinear three-wave interactions outside the plasmasphere

Whistler-mode waves play an important role in the inner magnetospheric particle dynamics. The direction and efficiency of energy transfer between waves and particles strongly depend on the wave frequency distribution characteristics. Recent observations have drawn attention to the low-frequency whistler-mode chorus waves below 0.1 times the equatorial gyrofrequency fce of electrons outside the plasmasphere. Here, on the basis of the analysis of the Van Allen Probes wave data, we describe a new type of low-frequency hiss-like whistler-mode wave around or below 0.1 fce. We show that two normal chorus wave bands split by the 0.5 fce gap can produce the low-frequency hiss-like whistler-mode waves through the nonlinear three-wave interaction.

Machine Learning Analysis for the Soliton Formation in Resonant Nonlinear Three-Wave Interactions

For the prediction of nonlinear phenomena in a three-wave Raman backscattering for laser amplification, a machine learning technology is applied to predict the generation of solitons in complicated multi-dimensional parameter spaces. The generation of the soliton in the resonant three-wave system is simulated with one-dimensional fluid equations. The solitons are generated in the early phase of the three-wave interaction, and the slow propagation speeds play an important role. Using a pattern matching method comparing the simulation data with the analytic solution, the generation of solitons are automatically detected. After collecting enough data sets by autonomous parameter scanning in the numerical simulation, nonlinear regression and k-nearest neighbor algorithms are utilized for the prediction of the existence of solitons.

Dragon-kings death in nonlinear wave interactions

Extreme events are by definition rare and exhibit unusual values of relevant observables. In literature, it is possible to find many studies about the predictability and suppression of extreme events. In this work, we show the existence of dragon-kings extreme events in nonlinear three-wave interactions. Dragon-king extreme events, identified by phase transitions, tipping points and catastrophes, affects fluctuating systems. We show that these events can be avoided by adding a perturbing small amplitude wave to the system.

Rossby waves in the magnetic fluid dynamics of a rotating plasma in the shallow-water approximation

We have studied rotating magnetohydrodynamic flows of a thin layer of astrophysical plasma with a free boundary in the β-plane. Nonlinear interactions of the Rossby waves have been analyzed in the shallow-water approximation based on the averaging of the initial equations of the magnetic fluid dynamics of the plasma over the depth. The shallow-water magnetohydrodynamic equations have been generalized to the case of a plasma layer in an external vertical magnetic field. We have considered two types of the flow, viz., the flow in an external vertical magnetic field and the flow in the presence of a horizontal magnetic field. Qualitative analysis of the dispersion curves shows the presence of three-wave nonlinear interactions of the magnetic Rossby waves in both cases. In the particular case of zero external magnetic field, the wave dynamics in the layer of a plasma is analogous to the wave dynamics in a neutral fluid. The asymptotic method of multiscale expansions has been used for deriving the nonlinear equations of interaction in an external vertical magnetic field for slowly varying amplitudes, which describe three-wave interactions in a vertical external magnetic field as well as three-wave interactions of waves in a horizontal magnetic field. It is shown that decay instabilities and parametric wave amplification mechanisms exist in each case under investigation. The instability increments and the parametric gain coefficients have been determined for the relevant processes.

Evaluation of fatigue damage in a pure copper with nonlinear three-wave interaction

We investigated nonlinear ultrasonic characterization, nonlinear three-wave interaction, of the pure copper during fatigue with EMAR (Electromagnetic Acoustic Resonance), which was the combination with ultrasonic resonance and non-contacting transducer, EMAT (Electromagnetic Acoustic Transducer). In nonlinear three-wave interaction method, two intersecting ultrasonic waves produced a scattered wave when the resonance condition was satisfied. The amplitude in resonant scattering wave was measured. Nonlinear three-wave interaction method exhibited high sensitivity to micro-structural change of the damaged material. It rapidly increased from 50% of fatigue life to the fracture. TEM (Transmission Electron Microscope) and EBSD (Electron Backscatter Diffraction) observations supported this phenomenon caused by dislocation movement. The sensitivity in three-wave interaction method was higher than that in linear methods. The noncontact resonance-EMAT measurement can monitor the evolution of nonlinearity throughout the fatigue life and has a potential to assess the damage advance and to predict the fatigue life of metals.