Effect of parametric oscillatory instability in axially symmetric modes of test masses in gravitational antennas LIGO

We demonstrate the importance of using the method of superposition for the precise calculation of frequencies and forms of axially symmetric elastic modes of test masses in gravitational antennas LIGO for detailed analysis of the effect of parametric oscillatory instability. The method allows one to construct analytical expressions for particular solutions to the medium equations of motion in cylindrical coordinates, which allow one to satisfy all zero boundary conditions. The obtained results allow considerably more reliable methods (as compared to meshing methods) to be used for predicting the number of combinations of optical and elastic modes that lead to the undesirable effect of the parametric oscillatory instability.     

Elastic modes frequencies in the mirrors with “ears” for parametric oscillatory instability in advanced LIGO interferometer

We discuss the importance of accurate numerical calculation of elastic modes in the mirrors with suspension “ears” in advanced LIGO interferometer to enable precise predictions of parametric oscillatory instability problem. We show that “ears” of test masses produce additional shift of elastic modes frequencies which is larger than relaxation rate of optical modes. These shifts may increase the possibility of parametric oscillatory instability.     

Numerical calculations of elastic modes frequencies for parametric oscillatory instability in Advanced LIGO interferometer

We discuss the importance of accurate numerical calculation of elastic modes in the mirrors of Advanced LIGO to enable precise predictions of the problem of parametric oscillatory instability. We propose accuracy estimations through use of analytical solutions based on Chree-Lamb modes. Small deviations from cylindrical test mass shape may produce splitting of non-axial symmetric elastic modes into doublets. This splitting may increase the possibility of parametric oscillatory instability.     

The effect of a laser beam displacement on parametric oscillatory instabilities for Advanced LIGO

The arm cavities of real gravitational wave detectors can show small deviations like a tilt or a spatial shift between the cavity mirrors. These deviations lead to a separation of the optical mode centres with respect to the mirror?s centre. In this Letter we perform the computation of parametric instable modes considering the described displacement. We further analyse the possibility of parametric oscillatory instability in the Advanced LIGO interferometer for the case of a displaced arm cavity. Our results reveal an additional number of optical and elastic mode combinations due to a displacement that can give rise to the undesirable effect of parametric oscillatory instability.     

The effect of parametric oscillatory instability in a Fabry-Perot cavity of the Einstein telescope

The nonlinear effect of a parametric oscillatory instability in a Fabry-Perot cavity of the Einstein Telescope is investigated. Unstable combinations of elastic and optical modes are calculated for two possible configurations of the third-generation gravitational-wave detector. The results are compared with those for the LIGO gravitational-wave interferometer.     

Parametric oscillatory instability in a Fabry–Perot cavity of the Einstein Telescope with different mirror's materials

We discuss the parametric oscillatory instability in a Fabry–Perot cavity of the Einstein Telescope. Unstable combinations of elastic and optical modes for two possible configurations of gravitational wave third-generation detector are deduced. The results are compared with the results for gravitational wave interferometers LIGO and LIGO Voyager.     

Lagrangian approach for analysis of radiation pressure induced parametric instability in micro-resonators

We introduce a new scheme for determining radiation pressure coupling to microresonator devices. It is shown that for the input pump powers there exists a threshold value for instability behavior. Radiation pressure can couple mechanical modes of a cavity to it's optical modes, leading to parametric oscillation instability. Here we present an approximate analysis of such nonlinear effect with Hamilton's least action principle. Our Lagrangian includes no interaction term, but interaction between optical and mechanical modes has been taken into account through integration limit.     

Electromechanical models of nanoresonators

The goal of this study is to construct simple electromechanical models of nanoresonators as mass detectors. A major obstacle in the achievement of sufficient measurement accuracy for the resonant frequency associated with the adsorption of additional mass onto the graphene layer is a low quality factor of the oscillatory system containing the graphene layer. A graphene resonator can be considered as an elastic system with distributed parameters. The application of the Galerkin method to study nearly resonant vibrational modes reduces the problem to considering an oscillatory system with a few degrees of freedom with pronounced nonlinear properties. These properties are, first of all, due to the nonlinear dependence of the forces produced by the electric field on the graphene deflection and, second, due to the nonlinear dependence of the graphene layer tension on its deflection. Taking into account the nonlinear properties leads to the appearance of characteristic drops in the resonance curve which allow for a more accurate resonant frequency measurement. Resonance curves with such characteristic drops can be obtained using a demonstration experimental macromodel of the resonator. Two absolutely new layouts are proposed, such as a differential resonator and resonator with parametric excitation. The oscillations excited in the differential resonator that contains two graphene layers resemble beats. In this case, small changes in the mass of the main layer correspond to significant changes in the frequency of the envelope. This effect is illustrated by oscillograms obtained for an experimental macromodel of the differential resonator. The parametric resonator has one graphene layer between two conducting surfaces. Parametric excitation of steady-state high amplitude oscillations is possible in this resonator only in a narrow frequency band close to the eigenfrequency. The band width reduces with a decrease in the quality factor of the oscillatory system. The latter fact can be useful for the improvement of eigenfrequency measurement accuracy at a low quality factor of the oscillatory system.     

Thermosolutal instability in porous medium in a partially ionized plasma

The thermosolutal instability in porous medium in a partially ionized plasma in the presence of a uniform vertical magnetic field is considered. The presence of each, magnetic field and stable solute gradient, brings oscillatory modes which were nonexistent in their absence. The collisional effects may also bring in oscillatory modes. The stable solute gradient and magnetic field are found to have stabilizing effect whereas the medium permeability and collisional frequency have destabilizing effect on the thermosolutal instability in porous medium.     

On the ponderomotive force and the effect of loss reaction on parametric instability

In this paper, the growth rate, ponderomotive force and the exciting condition for parametric instability are derived by considering the loss reaction using a new method. On the basis of the hydrodynamic equations, we take the production and loss reactions in plasma into account to derive the coupling equations for the electron plasma oscillation and ion acoustic oscillation, and obtain the growth rate for the parametric instability, the ponderomotive force and the exciting condition. The result shows that (a) the production reaction has no effect on the parametric instability, and the effect of loss reaction on the parametric instability is a damping one, (b) the more intensive the external field or pump is, the larger the growth rate is, (c) there exist two modes of the ponderomotive force, i.e.\ the high frequency mode and the low frequency mode, and (d) when ponderomotive force counteracts the damping force, the oscillations become non-damping and non-driving. The ratio of the electron plasma oscillation to ion acoustic oscillation is independent of the loss reaction and the external field.     

Modeling and visualization of quantum bifurcations in phase space

We investigate quantum-mechanical counterpart of a classical instability in a phase space by the numerical method of quantum trajectories with moving basis. As an application the model of coupled two oscillators driven by a monochromatic force in the presence of dissipation (intracavity second harmonic generation) is analyzed. The system of interest is characterized by two bifurcations leading to ranges of instability: the Hopf bifurcation which connects a steady state dynamics of the oscillatory modes to a self-pulsing temporal dynamics and the bifurcation of the period-doubling. The both two regimes are analyzed on the framework of the semiclassical phase trajectories and the Wigner functions of the oscillatory modes in phase space.     

各向异性表面张力对深胞晶界面形态稳定性的影响

应用匹配渐近展开法和多变量展开法研究各向异性表面张力对定向凝固中深胞晶界面形态稳定性的影响, 通过寻找定向凝固系统的模式解获得了深胞晶界面形态满足的量子化条件. 结果表明, 与各向同性的定向凝固系统中深胞晶界面形态稳定性比较, 考虑各向异性表面张力的定向凝固中深胞晶生长界面形态也有两种整体不稳定性机制: 整体波动不稳定性和低频不稳定性. 随着各向异性表面张力的增加, 中性模式产生强振荡的枝晶结构的整体波动不稳定性的不稳定区域减小, 中性模式产生弱振荡的胞晶结构的低频不稳定性的不稳定区域增加.     

PLANETARY GEAR PARAMETRIC INSTABILITY CAUSED BY MESH STIFFNESS VARIATION

Parametric instability is investigated for planetary gears where fluctuating stiffness results from the changing contact conditions at the multiple tooth meshes. The time-varying mesh stiffnesses of the sun-planet and ring-planet meshes are modelled as rectangular waveforms with different contact ratios and mesh phasing. The operating conditions leading to parametric instability are analytically identified. Using the well-defined properties of planetary gear vibration modes, the boundaries separating stable and unstable conditions are obtained as simple expressions in terms of mesh parameters. These expressions allow one to suppress particular instabilities by adjusting the contact ratios and mesh phasing. Tooth separation from parametric instability is numerically simulated to show the strong impact of this non-linearity on the response.     

Test mass ring dampers with minimum thermal noise

Advanced laser interferometers gravitational wave detectors may need to substantially reduce the Q-factor of test mass normal modes to eliminate parametric instability. In this Letter we investigate various ring damper configurations for two different laser beam geometries. We show that there is a well-defined location near the mid point of a test mass where the thermal noise degradation from the ring damper is minimised. A Q-factor reduction by a factor of 5 can be obtained for at least 30% of the investigated normal modes at the cost of a 1% increase in thermal noise as seen by a 5 cm diameter incident laser beam. Ring dampers can be up to about 10 mm wide while maintaining minimum thermal noise effect contribution. Of the remaining modes, 30% are very weakly damped.     

Effect of multiplicative noise on parametric instabilities

We report a study on the effect of external multiplicative noise on parametric instabilities using two different experimental systems: an electronic RLC circuit, parametrically pumped with a voltage-variable capacitor, and surface waves generated by vertically vibrating a layer of fluid (the Faraday instability). Both systems are forced by the superposition of a sinusoidal and a noisy component. We study the statistical properties of the response of both systems to noisy parametric forcing and compare them with theoretical predictions. When the detuning from parametric resonance is such that the bifurcation in the absence of noise is supercritical, both systems behave in the same way under the influence of noise. We find that the effect of noise is twofold: on one hand, it triggers the instability before its deterministic onset under the form of oscillatory bursts; on the other hand, it inhibits the nonlinearly saturated oscillatory response above the deterministic onset. When the detuning is such that the bifurcation is subcritical, we find that the two systems behave differently. In the case of the electronic oscillator, noise mostly triggers random transitions between the two states of the bistable region that exists in the absence of noise, whereas in the surface wave experiment new states are created by noise and the bistable region is strongly enlarged.     

Parallel velocity shear driven electrostatic waves in a very dense nonuniform magnetoplasma

It is shown that the parallel (magnetic field-aligned) velocity shear can drive the low-frequency (in comparison with the ion gyrofrequency) electrostatic (LF-ES) waves in an ultracold super-dense nonuniform magnetoplasma. By using an electron density response arising from the balance between the electrostatic and quantum Bohm forces, as well as the ion density response deduced from the continuity and momentum equations, a wave equation for the LF-ES waves is derived. In the local approximation, a new dispersion relation is obtained by Fourier transforming the wave equation. The dispersion relation reveals an oscillatory instability of dispersive drift-like modes in super-dense quantum magnetoplasmas.     

Parametric Instability of Surface Waves on the Second Harmonic of Electron Cyclotron Frequency in a Plasma Layer

The parametric instability of surface waves on the second harmonic of electron cyclotron frequency (SWCF) in a plasma filled dielectric wave guide is examined in a kinetic approximation. The studied surface waves are extraordinary polarized modes and propagate across the external steady magnetic field. The amplitude of the electrical pump wave is assumed to be small. Simple expressions for increments of the parametric instability of the SWCF are calculated. The otained results can be used in controlled fusion researches in order to avoid undesirable regimes of plasma periphery heating in that fusion devices which use the resonance electron cyclotron heating method.     

Modulation instabilities in a system of four coupled, nonlinear Schrödinger equations

The modulation instability of continuous waves for a system of four coupled nonlinear Schrödinger equations, two of which are in the unstable regime, is studied. In earlier studies, plane or continuous waves for a system of two coupled, nonlinear Schrödinger equations is shown to exhibit modulation instability (MI), even if both modes are in the normal dispersion regime, provided that the coefficient of cross phase modulation (XPM) is larger than that of self phase modulation (SPM). Requirements for MI in this system of four coupled, nonlinear Schrödinger equations can be relaxed. MI can occur even if the magnitude of XPM is less than that of SPM, and the magnitude of instability is generally larger than that of each mode alone. The implications for parametric process and wavelength exchange in optical physics with two pump waves are discussed.     

Mechanism for chaos in the duffing equation

It is shown that period-doubling bifurcation in the Duffing equation arise from parametric instability. Our mechanism requires this to be preceded by generation of zero- and second-harmonic modes by parametric instability. Numerical calculations are presented which confirm the analytic findings.     

Finite Larmor radius effect on thermosolutal instability of a plasma in porous medium

The thermosolutal instability of a plasma in porous medium is considered in the presence of finite Larmor radius effect. The finite Larmor radius, stable solute gradient and magnetic field introduce oscillatory modes in the systems which were nonexistent in their absence. For stationary convection, the finite Larmor radius and stable solute gradient have stabilizing effects on the thermosolutal instability in porous medium. In presence of finite Larmor radius effect, the medium permeability has a destabilizing (or stabilizing) effect and the magnetic field has a stabilizing (or destabilizing) effect under certain condition whereas in the absence of finite Larmor radius effect, the medium permeability and the magnetic field have destabilizing and stabilizing effects, respectively, on thermosolutal instability of a plasma in porous medium. The sufficient conditions for nonexistence of overstability are obtained.The financial assistance to Mr. Sunil in the form of Senior Research Fellowship of the Council of Scientific and Industrial Research (CSIR), New Delhi is gratefully acknowledged.