Three-mode Coupling Research Articles

Characterization of Three-Mode Combination Internal Resonances in Electrostatically Actuated Flexible–Flexible Microbeams

Abstract Nonlinear intermodal coupling based on internal resonances in MEMS resonators has advanced significantly over the past two decades for various real-world applications. In this study, we demonstrate the existence of various three-mode combination internal resonances between the first five flexural modes of electrostatically actuated flexible–flexible beams and dynamic modal interaction between three modes via internal resonance. We first calculate the natural frequencies of the beam as a function of the stiffnesses of the transverse and rotational springs of the flexible supports, utilizing both analytical formulation and finite element analysis (FEA). Following this, we identify six combination internal resonances among the first five modes and use applied DC voltage to validate the exactness of one commensurable internal resonance condition (ω2=ω5−ω4). Subsequently, we studied a detailed forced vibration analysis corresponding to this resonance condition by solving the five-mode coupled governing equations through numerical time integration and the method of multiple scales. The results compellingly exhibit three-mode intermodal coupling among the second, fourth, and fifth modes as a function of excitation amplitude and frequency. Alongside this, intriguing nonlinear phenomena such as threshold behavior, saturation phenomena, and autoparametric instability are observed. Finally, this paper provides a systematic methodology for investigating three-mode combination internal resonances and related nonlinear dynamics, offering significant insights that could be used in observing phonon or mechanical lasing phenomena in MEMS resonators.

Non-linear three-mode coupling of gravity modes in rotating slowly pulsating B stars

Context. Slowly pulsating B (SPB) stars display multi-periodic variability in the gravito-inertial mode regime with indications of non-linear resonances between modes. Several have undergone asteroseismic modeling in the past few years to infer their internal properties, but only in a linear setting. These stars rotate fast, so that rotation is typically included in the modeling by means of the traditional approximation of rotation (TAR). Aims. We aim to extend the set of tools available for asteroseismology, by describing time-independent (stationary) resonant non-linear coupling among three gravito-inertial modes within the TAR. Such coupling offers the opportunity to use mode amplitude ratios in the asteroseismic modeling process, instead of only relying on frequencies of linear eigenmodes, as has been done so far. Methods. Following observational detections, we derive expressions for the resonant stationary non-linear coupling between three gravito-inertial modes in rotating stars. We assess selection rules and stability domains for stationary solutions. We also predict non-linear frequencies and amplitude ratio observables that can be compared with their observed counterparts. Results. The non-linear frequency shifts of stationary couplings are negligible compared to typical frequency errors derived from observations. The theoretically predicted amplitude ratios of combination frequencies match with some of their observational counterparts in the SPB targets. Other, unexplained observed ratios could be linked to other saturation mechanisms, to interactions between different modes, or to different opacity gradients in the driving zone. Conclusions. For the purpose of asteroseismic modeling, our non-linear mode coupling formalism can explain some of the stationary amplitude ratios of observed resonant mode couplings in single SPB stars monitored during 4 years by the Kepler space telescope.

Triple electromagnetically induced transparency generated slow light for multiple carbon nanotube resonators

Hybrid spin-mechanical systems offer a promising platform for advancing quantum science and technology. However, practical implementation of applications within these hybrid quantum systems demands the seamless integration of supplementary physical components. In this context, we present a proposal for a multi-mode spin-mechanical setup, featuring the utilization of three-mode coupling nanomechanical carbon nanotube (CNT) resonators. These resonators interact with each other via a phase-dependent phonon-exchange mechanism, which is coupled to the same nitrogen vacancy (NV) centers in diamond. Based on the modulation of the phonon–phonon coupling phase and leveraging the triple Fano-like resonance phenomenon, a tripling of electromagnetically induced transparency (EIT) becomes achievable within the system. This tripling is accompanied by swift dispersion, leading to a subtle advancement or delay in outcomes. The phenomenon of triple Fano-like resonance, alongside the resulting triple EIT, engenders noteworthy slow-to-fast and fast-to-slow light effects, which is theoretically demonstrated in CNT resonators, with both identical and distinct frequencies. The findings underscore that CNT resonators with varying frequencies can evoke a more pronounced transition in the slow–fast–slow and fast–slow–fast light effects. This study lays the foundation for the application of phonon-mediated optical information storage and processing.

Revealing the quantum property of suppressed middle plexciton state in three-mode coupling system

The three-mode coupling among excitons–surface plasmon(SP)–excitons has attracted extensive attention owing to its richer modulation capability compared to traditional two-mode coupling. However, the coupling mechanism of middle hybrid state in three-mode systems has not been clear. Here, we develop a full quantum theory based on the Jaynes–Cummings model to investigate the properties of middle plexciton branch (MPB) in excitons–SP–excitons system. The calculated scattering spectrum includes three plexciton branches, which shows that the MPB has the narrowest linewidth and lowest peak amplitude compared to the other two branches. The derived Hopfield coefficients demonstrate that the narrowest linewidth of MPB is due to its low SP fraction compared to that of excitons, which is different from the fractions of SP and excitons for the other two branches. The calculated results indicate that MPB peak amplitude is lowest because the detuning between two excitons is smaller than the coupling coefficient g, and MPB has the lowest population compared with those of the other two branches. Additionally, to verify our theory, we design an Au nanorod coated with two layers of J-aggregate nanoshells by the finite-difference time-domain (FDTD) simulation which results accord well with our theory. Our work has wide applications in quantum optical devices, quantum information, and quantum computing.

Plexcitonic optical chirality in the chiral plasmonic structure-microcavity-exciton strong coupling system.

Chiral plexcitonic systems exhibit a novel chiroptical phenomenon, which can provide a new route to design chiroptical devices. Reported works focused on the two-mode strong coupling between chiral molecules and nanoparticles, while multiple-mode coupling can provide richer modulation. In this paper, we proposed a three-mode coupling system consisting of a chiral Au helices array, a Fabry-Pérot cavity, and monolayer WSe2, which can provide an extra chiral channel, a more widely tunable region, and more tunable methods compared to two-mode coupled systems. The optical response of this hybrid system was investigated based on the finite element method. Mode splitting observed in the circular dichroism (CD) spectrum demonstrated that the chiroptical response successfully shifted from the resonant position of the chiral structure to three plexcitons through strong coupling, which provided a new route for chiral transfer. Furthermore, we used the coupled oscillator model to obtain the energy and Hopfield coefficients of the plexciton branches to explain the chiroptical phenomenon of the hybrid system. Moreover, the tunability of the hybrid system can be achieved by tuning the temperature and period of the helices array. Our work provides a feasible strategy for chiral sensing and modulation devices.

Resonant Mode Coupling in δ Scuti Stars

Delta Scuti (δ Sct) variables are intermediate-mass stars that lie at the intersection of the main sequence and the instability strip on the Hertzsprung–Russell diagram. Various lines of evidence indicate that nonlinear mode interactions shape their oscillation spectra, including the particularly compelling detection of resonantly interacting mode triplets in the δ Sct star KIC 8054146. Motivated by these observations, we use the theory of three-mode coupling to study the strength and prevalence of nonlinear mode interactions in 14 δ Sct models that span the instability strip. For each model, we calculate the frequency detunings and nonlinear coupling strengths of ∼104 unique combinations of mode triplets. We find that all the models contain at least ∼100 well-coupled triplets whose detunings and coupling strengths are consistent with the triplets identified in KIC 8054146. Our results suggest that resonant mode interactions can be significant in δ Sct stars and may explain why many exhibit rapid changes in amplitude and oscillation period.

Parametric instability in warped astrophysical discs: growth, saturation, and feedback

ABSTRACT Attempts to understand the dynamics of warped astrophysical discs have garnered significant attention, largely motivated by the growing catalogue of observed distorted systems. Previous studies have shown that the evolution of the warp is crucially regulated by the internal flow fields established by the undulating geometry. These are typically modelled as laminar horizontal, shearing flows which oscillate back and forth at approximately the orbital frequency. However this shearing motion is known to be susceptible to a hydrodynamic, parametric instability of inertial waves which might modify the warped dynamics. Whilst the linear growth phase is well understood, the subsequent non-linear saturation combined with the self-consistent feedback onto the warp has not been studied. In this work, we implement a novel numerical setup using the recent ring model framework of Fairbairn and Ogilvie, within the Lagrangian code gizmo. We formally identify several locally growing modes in the simulation, as predicted by a three-mode coupling analysis of the instability, and find decent agreement with the theoretical growth rates. We understand the saturation mechanism as a wave breaking process which suppresses the growth of shorter wavelength parametric couplings first, whilst allowing the longest mode to dominate the final quasi-steady, wave-like turbulence. The Reynolds stresses, transporting energy from the warp to the small scales, can be effectively modelled using a time-dependent, anisotropic viscous alpha model which closely captures the amplitude and phase evolution of the warp. Consequently, this model might help inform future global studies which are commonplace but typically do not resolve the parametric instability.

Strong frequency-dependent beam steering dynamics, Zitterbewegung effect, and Klein tunneling in a ternary plasmonic-dielectric superlattice

Band structures are intrinsically influenced by the mode interaction in waveguide array, which can support many intriguing phenomena such as negative refraction, on-chip lensing, and quantum-analog dynamics like Bloch oscillation and topological interface transportation. Multiple branches of band may emerge due to the comprehensive mode interactions associated with high-order photonic modes supported in high refractive index dielectric waveguide. Furthermore, the number of photonic modes confined in a high-index waveguide is determined by both the width and working frequency. Hence, the mode interaction and resultant band structure can be effectively tuned by the frequency. As such, a diversity of equifrequency contours could be generated and modulated among a wide frequency range, which, however, remains largely unexplored. Here, we present several frequency-determined beam dynamics in a composite ternary plasmonic-dielectric waveguide array, including collimation effect, angle-dependent beam branching/multibranching, conicallike beam diffraction. Based on the evolution of the band structure, photonic Zitterbewegung effect and Klein tunneling can also be observed. In contrast to the coherent coupling between symmetric and antisymmetric modes that gives rise to the photonic Zitterbewegung effect, we show that three-mode coherent coupling in the ternary system can yield a superimposed extreme oscillation of the beam. This represents an extension of the general Zitterbewegung effect in photonic system. Additionally, the Klein tunneling sandwiched in two types of artificial waveguide arrays with different Dirac points is demonstrated, and approximately unimpeded penetration is exhibited. The configurations discussed here can be readily implemented in on-chip systems for a variety of potential applications including wave routing, selectively directional coupling, and multiplexing.

Engineering the Spectral and Spatial Dispersion of Thermal Emission via Polariton-Phonon Strong Coupling.

Strong coupling between optical modes can be implemented into nanophotonic design to modify the energy-momentum dispersion relation. This approach offers potential avenues for tuning the thermal emission frequency, line width, polarization, and spatial coherence. Here, we employ three-mode strong coupling between propagating and localized surface phonon polaritons, with zone-folded longitudinal optic phonons within periodic arrays of 4H-SiC nanopillars. Energy exchange, mode evolution, and coupling strength between the three polariton branches are explored experimentally and theoretically. The influence of strong coupling upon the angle-dependent thermal emission was directly measured, providing excellent agreement with theory. We demonstrate a 5-fold improvement in the spatial coherence and 3-fold enhancement of the quality factor of the polaritonic modes, with these hybrid modes also exhibiting a mixed character that could enable opportunities to realize electrically driven emission. Our results show that polariton-phonon strong coupling enables thermal emitters, which meet the requirements for a host of IR applications in a simple, lightweight, narrow-band, and yet bright emitter.

Dynamic In-Line Routing Between Distant Cores of a Multi-Core Fiber

Dynamic and coherent coupling between distant cores in a commercially available optical four-core fiber is demonstrated. Wavelength-selective excitation of cladding modes is experimentally realized by an acousto-optic mode coupler by which higher-order cladding modes are visualized and compared with simulations. By comparing the coupling conditions to a global cladding mode, the relative propagation constants of the local core modes are precisely measured at an accuracy level down to 10−6. Core-cladding-core three-mode coupling operation dynamically switches optical signals between two arbitrary distant cores in the four-core fiber and achieves an inter-core switching efficiency of 55.4%.

Tidal Asteroseismology: Possible Evidence of Nonlinear Mode Coupling in an Equilibrium State in Kepler Eclipsing Binary KIC 3230227

Abstract Previously, a series of tidally-excited oscillations were discovered in the eccentric eclipsing binary KIC 3230227. The pulsation amplitudes and phases suggest the observed oscillations are prograde quadruple modes. In this paper, we refine the analysis and extract more oscillation frequencies. We also study the temporal variations of amplitudes and phases and show that almost all modes have stable phases and amplitudes. We then focus on the non-orbital-harmonic oscillations. We consider two formation mechanisms: (1) nonlinear response of the surface convective layer, and (2) nonlinear three/multi-mode coupling. Although the former can explain some of the observed features, we find the latter mechanism is more probable. Assuming that these are coupled modes, the constant amplitude/phase over four years can be explained by either an equilibrium state in the mode coupling or modes undergoing limit cycles with very long periods. The observed frequency detuning and the calculated damping rates of the daughter modes favor the equilibrium-state interpretation. This is verified by integrating the amplitude equations of three-mode coupling. We find that the steady-state relation derived in Weinberg et al., which relates the observed frequency detuning, phase detuning, and mode damping rates, is approximately satisfied for one mode triplet. We also try to identify the azimuthal number of the modes based on the observed mode amplitude ratios and the selection rules in nonlinear three-mode coupling. We discuss further implications of these observations on nonlinear tidal asteroseismology.

The dynamics of three-mode coupling in a trapped Bose gas

ABSTRACTMultiple mode couplings in topological coherent modes of Bose–Einstein condensate are considered, by introducing an external alternating (resonating) field in the system. This analysis is based on the analytical solutions of nonlinear Gross–Pitaevskii equation for a trapped Bose gas at nearly absolute zero temperature. The dynamics of fractional populations of the generated coherent modes are analysed, particularly for a three-level system in the limit of small to large detuning of the intermediate state. These coupled topological modes, though nonlinear, are analogous to a resonant atom and exhibit a variety of significant non-trivial phenomena (effects), like: dynamic phase transitions, interference patterns, critical phenomena, mode-locking and chaotic motion.

The effect of background flow shear on gyrokinetic turbulence in the cold ion limit

The cold ion limit of the local gyrokinetic model is rigorously taken to produce a nonlinear system of fluid equations that includes background flow shear. No fluid closure is required. By considering a simple slab geometry with magnetic drifts, but no magnetic shear, these fluid equations reduce to the Charney–Hasegawa–Mima model in the presence of flow shear. Analytic solutions to this model are found to study the impact of E × B flow shear on the stability of a single parallel velocity gradient driven mode. Additionally, the model is used to investigate the effect of background E × B flow shear on the basic three-mode nonlinear coupling, which reveals differences between zonal and non-zonal modes. These analytic results agree with gyrokinetic simulations and can serve to benchmark the numerical implementation of flow shear and nonlinear coupling.

Interference-modulated photon statistics in whispering-gallery-mode microresonator optomechanics

Whispering-gallery-mode (WGM) microresonator optomechanical systems that can attain high quality factors, exhibit small optical mode volume, and can be excited through their evanescent field are versatile platforms for both theoretical and experimental studies in quantum and nonlinear optics. Investigating photon statistical properties in WGM microresonator optomechanical systems is an important avenue to understand their inner interaction mechanism. Here, the interference-modulated photon statistics in a three-mode coupling, i.e., a pair of counterpropagating optical cavity modes and a mechanical mode, in WGM microresonator optomechanical systems is studied. In the case that one optical mode is driven by an external field, strong antibunching photon statistics can be observed in the presence of mode coupling. When the two cavity modes are driven simultaneously, it is found that the photon statistical properties can be well steered by modulating the interference between different transition paths with the help of the amplitudes of the two input fields and their relative phase. Especially, we show in detail that the antibunching photon statistics can be optimized within the weak optomechanical coupling regime by properly adjusting the relative phase. We also find that it is necessary to prepare the mechanical resonator near to the ground state to eliminate the detrimental effect of thermal phonon number on the photon statistical properties. This investigation can deepen our understanding of the interaction between clockwise or counterclockwise light and mechanical motion as well as be useful for the construction of integrated on-chip single-photon sources.

Landau damping and frequency-shift of (0, 0, 2) scissors mode in a disc-shaped Bose-Einstein condensate

应用哈特里-福克-博戈留波夫平均场理论近似和基于托马斯-费米近似的解析方法, 研究碟形玻色-爱因斯坦凝聚体中(0, 0, 2)剪刀模的朗道阻尼和频移, 计算阻尼系数和频移大小以及它们的温度依赖. 计算中, 在集体激发本征频移微扰关系中考虑元激发弛豫及其弛豫之间的正交关系以获得阻尼和频移的计算公式, 把凝聚体基态波函数取为高斯分布函数的一级近似以消除托马斯-费米近似中三模耦合矩阵元的发散. 采用与相关实验研究相同的粒子数、囚禁频率和各向异性参量, 理论计算结果与相关实验测量结果相符合. 由于理论的复杂性和计算的困难性, 在大多数基于平均场理论的单分量和两分量玻色-爱因斯坦凝聚集体激发阻尼和频移的研究中采用半经典近似, 把准粒子激发能谱看成是连续的来积分计算各个准粒子跃迁对阻尼和频移的贡献, 而本文和本文前期工作按分立的准粒子激发频谱计算阻尼或频移, 并在研究过程中提出了考虑元激发弛豫及弛豫之间正交关系的改进方法, 希望这种方法对今后的工作有一定参考价值.

Full-Vectorial Mode Coupling in Optical Fibers

Optical full vectorial modes have closely attracted much attention because of their unique optical features and thus may develop the fundamental researches and new functionalities of light. Beyond the conventional linear polarization modal coupling, we thoroughly investigate the inherent polarizationdependence and polarization-isolation features for full vectorial mode coupling in optical fibers, as well as the coupling crosstalk in view of the modal degeneracy. We present the general full vectorial coupling mechanism and numerically simulate the full vectorial mode coupling for both fiber directional and grating assisted coupling cases. Furthermore, we give the detailed analyses of coupling crosstalk based on the unified threemode coupling model. Apart from lifting the modal degeneracy, significantly, the crosstalk can also be reduced by weakening the coupling coefficient and meanwhile increasing the coupling length. This extended full vectorial mode coupling mechanismmay provide theoretical guidance on exploiting new all fiberbased polarization-sensitive passive elements to control fullvectorial fields and modal polarization states.

Three-mode coupling interference patterns in the dynamic structure factor of a relaxor ferroelectric

A longstanding controversy for relaxor ferroelectrics has been the origin of the ``waterfall'' effect in the phonon dispersion curves, in which low-energy transverse phonons cascade into vertical columns. Originally interpreted as phonons interacting with polar nanoregions (PNRs), it was later explained as an interference effect of coupling damped optic and acoustic phonons. In light of a recently discovered PNR vibrational mode near the ``waterfall'' wave vector [M. E. Manley, J. W. Lynn, D. L. Abernathy, E. D. Specht, O. Delaire, A. R. Bishop, R. Sahul, and J. D. Budai, Nat. Commun. 5, 3683 (2014)], we have reexamined this feature using neutron scattering on [100]-poled PMN-30%PT $[0.6\mathrm{Pb}(\mathrm{M}{\mathrm{g}}_{1/3}\mathrm{N}{\mathrm{b}}_{2/3}){\mathrm{O}}_{3}\phantom{\rule{0.16em}{0ex}}\ensuremath{-}0.3\mathrm{PbTi}{\mathrm{O}}_{3}]$. We find that the PNR mode couples to both optic and acoustic phonons and that this results in complex patterns in the dynamic structure factor, including intensity pockets and peaks localized in momentum-energy space. These features are fully explained by extending the mode-coupling model to include three coupled damped harmonic oscillators representing the transverse optic, acoustic, and PNR modes.

Enhancing the Surface Sensitivity of Metallic Nanostructures Using Oblique-Angle-Induced Fano Resonances.

Surface sensitivity is an important factor that determines the minimum amount of biomolecules detected by surface plasmon resonance (SPR) sensors. We propose the use of oblique-angle-induced Fano resonances caused by two-mode coupling or three-mode coupling between the localized SPR mode and long-range surface plasmon polariton modes to increase the surface sensitivities of silver capped nanoslits. The results indicate that the coupled resonance between the split SPR (−kSPR) and cavity modes (two-mode coupling) has a high wavelength sensitivity for small-angle incidence (2°) due to its short decay length. Additionally, three-mode coupling between the split SPR (−kSPR), substrate (+kSub) and cavity modes has a high intensity sensitivity for large-angle incidence due to its short decay length, large resonance slope and enhanced transmission intensity. Compared to the wavelength measurement, the intensity measurement has a lower detectable (surface) concentration below 1 ng/ml (0.14 pg/mm2) and is reduced by at least 3 orders of magnitude. In addition, based on the calibration curve and current system noise, a theoretical detection limit of 2.73 pg/ml (0.38 fg/mm2) can be achieved. Such a surface concentration is close to that of prism-based SPR with phase measurement (0.1–0.2 fg/mm2 under a phase shift of 5 mdeg).

A strongly coupled Λ-type micromechanical system

We study a classical Λ-type three-level system based on three high-Q micromechanical beam resonators embedded in a gradient electric field. By modulating the strength of the field at the difference frequency between adjacent beam modes, we realize strong dynamic two-mode coupling, via the dielectric force. Driving adjacent pairs simultaneously, we observe the formation of a purely mechanical “dark” state and an all-phononic analog of coherent population trapping—signatures of strong three-mode coupling. The Λ-type micromechanical system is a natural extension of previously demonstrated “two-level” micromechanical systems and adds to the toolbox for engineering of all-phononic micromechanical circuits and arrays.

Rayleigh-Taylor vortices in a pair-ion plasma

The Rayleigh-Taylor (RT) vortices and the analytical solution of three-mode coupling in pair-ion plasmas are investigated. It is shown that the E×B convection of polarization drift is responsible for the saturation of growing RT instability and as a result the localized dipole vortex structures are formed. The shear flow generation due to the destruction of vortex structures is discussed by the Fourier mode analysis.