Trapped Modes Research Articles

Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbation

We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide Πlε formed by the union of an infinite strip and a narrow box-shaped perturbation of size 2l×ε, where ε>0 is a small parameter. We prove the existence of the length parameter lkε=πk+O(ε) with any k=1,2,3,... such that the waveguide Πlkεε supports a trapped mode with an eigenvalue λkε=π2−4π4l2ε2+O(ε3) embedded into the continuous spectrum. This eigenvalue is unique in the segment [0,π2], and it is absent in the case l≠lkε. The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main difficulty is caused by the rather specific shape of the perturbed wall ∂Πlε, namely a narrow rectangular bulge with corner points, and we discuss available generalizations for other piecewise smooth boundaries.

Trapping a moving droplet train by bubble guidance in microfluidic networks.

Trapping a train of moving droplets into preset positions within a microfluidic device facilitates the long-term observation of biochemical reactions inside the droplets. In this paper, a new bubble-guided trapping method, which can remarkably improve the limited narrow two-phase flow rate range of uniform trapping, was proposed by taking advantage of the unique physical property that bubbles do not coalescence with two-phase fluids and the hydrodynamic characteristic of large flow resistance of bubbles. The flow behaviors of bubble-free and bubble-guided droplet trains were compared and analyzed under the same two-phase flow rates. The experimental results show that the droplets trapped by bubble-free guided trapping exhibit the four trapping modes of sequentially uniform trapping, non-uniform trapping induced by break-up and collision, and failed trapping due to squeezing through, and the droplets exhibit the desired uniform trapping in a relatively small two-phase flow rate range. Compared with bubble-free guided droplets, bubble-guided droplets also show four trapping modes. However, the two-phase flow rate range in which uniform trapping occurs is increased significantly and the uniformity of the trapped droplet array is improved. This investigation is beneficial to enhance the applicability of microfluidic chips for storing droplets in a passive way.

Tuning the terahertz trapped modes of conductively coupled Fano-resonators in reflectional and rotational symmetry

We experimentally investigate the terahertz (THz) response of conductively coupled asymmetric split ring resonator-based meta-molecules in the layout of reflection and rotational symmetry. In the reflectional symmetry case, the horizontally polarized THz excites a couple of trapped modes: the low-order one is a coupled Fano-resonance, and the high-order one is a decoupled dipole oscillator. The vertically polarized THz excites an inductive-capacitor resonance as a low-order trapped mode below the frequency of a high-order intrinsic mode. The quality factors (Q factors) of trapped modes decrease with the displacement of top-and-bottom gap increasing. In the rotational symmetry case, the horizontally polarized THz excites a single trapped mode owing to coupled Fano-resonance. The vertically polarized THz excites a high-order trapped mode of coupled multiple dipole oscillations beyond the frequency of intrinsic low-order dipole oscillation. The Q factors of trapped modes increase with the displacement of the top-and-bottom gap increase. For the first time, our results reveal the trapped modes’ evolution owing to the interaction of Fano-resonators conductively coupled under different symmetry.

The investigation of trapped thickness shear modes in a contoured AT-cut quartz plate using the power series expansion technique

The dynamic model about the anti-plane vibration of a contoured quartz plate with thickness changing continuously is established by ignoring the effect of small elastic constant c56. The governing equation is solved using the power series expansion technique, and the trapped thickness shear modes caused by bulge thickness are revealed. Theoretically, the proposed method is more general, which can be capable of handling various thickness profiles defined mathematically. After the convergence of the series is demonstrated and the correctness is numerically validated with the aid of finite element method results, systematic parametric studies are subsequently carried out to quantify the effects of the geometry parameter upon the trapped modes, including resonant frequency and mode shape. After that, the band structures of thickness shear waves propagation in a periodically contoured quartz plate, as well as the power transmission spectra, are obtained based on the power series expansion technique. It is revealed that broad stop bands below cut-off frequency exist owing to the trapped modes excited by the geometry inhomogeneity, which has little relationship with the structural periodicity, and its physical mechanism is different from the Bragg scattering effect. The outcome is widely applicable, and can be utilized to provide theoretical and practical guidance for the design and manufacturing of quartz resonators and wave filters.

Unavoidable trapped mode in the interaction region of colliding beams

We discuss the nature of the electromagnetic fields excited by the beams in the beam pipe of an interaction region. In trying to find an optimum geometry for this region with a minimum of electromagnetic wave excitation, we have discovered one mode, which remains even in a very smooth geometry. This mode has a longitudinal electrical component and can be easily excited by the beam. By analyzing the structure of this mode we have found a way to absorb this mode. The work was done in connection with a proposal for a future electron-positron collider.

Role of turbulence regime on determining the local density gradient

In this paper we show that the local density gradient in the plasma core depends on the calculated mode-frequency of the most unstable linear mode and reaches a maximum when this frequency is close to zero. Previous theoretical and experimental work on AUG has shown that the ratio of electron to ion temperature, and as such the frequency of the dominant linear gyrokinetic mode, affects the local density gradient close to (Fable et al 2010 Plasma Phys. Control. Fusion 52 015007, Angioni et al 2011 Nucl. Fusion 51 023006). On DIII-D we find that by adding electron cyclotron heating, we modify the dominant unstable linear gyrokinetic mode from an ion temperature gradient (ITG) mode to a trapped electron mode (TEM), which means that the frequency of the dominant mode changes sign (from the ion to the electron direction). Local density peaking around mid-radius increases by 50% right around the cross-over between the ITG and TEM regimes. By comparing how the particle flux changes, through the derivative of the electron density, ne, with respect to time, , we find that the particle flux also exhibits the same trend versus mode frequency. As a result, we find that the changes in local particle transport are inversely proportional to the changes in electron density, indicating that the changes are driven by a change in thermo-diffusive pinch.

Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies

Elastic bodies with cuspidal singularities at the surface are known to support wave processes in a finite volume which lead to absorption of elastic and acoustic oscillations (this effect is recognized in the engineering literature as Vibration Black Holes). With a simple argument we will give the complete description of the phenomenon of wave propagation towards the tip of a three-dimensional cusp and provide a formulation of the Mandelstam (energy) radiation conditions based on the calculation of the Umov-Poynting vector of energy transfer. Outside thresholds, in particular, above the lower bound of the continuous spectrum, these conditions coincide with ones due to the traditional Sommerfeld principle but also work at the threshold frequencies where the latter principle cannot indicate the direction of wave propagation. The energy radiation conditions supply the problem with a Fredholm operator of index zero so that a solution is determined up to a trapped mode with a finite elastic energy and exists provided the external loading is orthogonal to these modes. An example of a symmetric cuspidal body is presented which supports an infinite sequence of eigenfrequencies embedded into the continuous spectrum and the corresponding trapped modes with the exponential decay at the cusp tip. We also determine (unitary and symmetric) scattering matrices of two types and derive a criterion for the existence of a trapped mode with the power-low behavior near the tip.

A new look at bubbles during biofilm inoculation reveals pronounced effects on growth and patterning.

Specially designed microfluidic bioflow cells were used to temporarily trap microbubbles during different inoculation stages of Pseudomonas sp. biofilms. Despite being eliminated many hours before biofilm appearance, templated growth could occur at former bubble positions. Bubble-templated growth was either continuous or in ring patterns, depending on the stage of inoculation when the bubbles were introduced. Templated biofilms were strongly enhanced in terms of their growth kinetics and structural homogeneity. High resolution confocal imaging showed two separate bubble-induced bacterial trapping modes, which were responsible for the altered biofilm development. It is concluded that static bubbles can be exploited for fundamental improvements to bioreactor performance, as well as open new avenues to study isolated bacteria and small colonies.

Dispersion Energy Analysis of Rayleigh and Love Waves in the Presence of Low-Velocity Layers in Near-Surface Seismic Surveys

High-frequency surface-wave analysis methods have been effectively and widely used to determine near-surface shear (S) wave velocity. To image the dispersion energy and identify different dispersive modes of surface waves accurately is one of key steps of using surface-wave methods. We analyzed the dispersion energy characteristics of Rayleigh and Love waves in near-surface layered models based on numerical simulations. It has been found that if there is a low-velocity layer (LVL) in the half-space, the dispersion energy of Rayleigh or Love waves is discontinuous and ‘‘jumping’’ appears from the fundamental mode to higher modes on dispersive images. We introduce the guided waves generated in an LVL (LVL-guided waves, a trapped wave mode) to clarify the complexity of the dispersion energy. We confirm the LVL-guided waves by analyzing the snapshots of SH and P–SV wavefield and comparing the dispersive energy with theoretical values of phase velocities. Results demonstrate that LVL-guided waves possess energy on dispersive images, which can interfere with the normal dispersion energy of Rayleigh or Love waves. Each mode of LVL-guided waves having lack of energy at the free surface in some high frequency range causes the discontinuity of dispersive energy on dispersive images, which is because shorter wavelengths (generally with lower phase velocities and higher frequencies) of LVL-guided waves cannot penetrate to the free surface. If the S wave velocity of the LVL is higher than that of the surface layer, the energy of LVL-guided waves only contaminates higher mode energy of surface waves and there is no interlacement with the fundamental mode of surface waves, while if the S wave velocity of the LVL is lower than that of the surface layer, the energy of LVL-guided waves may interlace with the fundamental mode of surface waves. Both of the interlacements with the fundamental mode or higher mode energy may cause misidentification for the dispersion curves of surface waves.

Asymmetric trapped modes in a tube waveguide with a bulge

Asymmetric trapped modes in a waveguide of a cylindrical hollow tube with a local bulge are studied. The problem is converted to solving a nonaxisymmetric boundary value problem associated with the three-dimensional Helmholtz equation subject to Dirichlet boundary condition. The domain decomposition method and matching technique are invoked for an infinitely long tube with a bulge and a semi-infinitely long tube with an end bulge. Asymmetric trapped modes along with the frequencies are determined and can be decomposed into a linear combination of those with the n-fold periodic symmetry. For each n-fold periodic trapped mode, whether azimuthal trapped modes exist depends on the radius and width of the bulge. The influence of the bulge’s size on the frequencies and intensity location of localized vibration is analyzed. The obtained results can be extended to analyze bound states in quantum wires.

On the substrate contribution to the back action trapping of plasmonic nanoparticles on resonant near-field traps in plasmonic films.

Nanoparticles trapped on resonant near-field apertures/engravings carved in plasmonic films experience optical forces due to the steep intensity gradient field of the aperture/engraving as well as the image like interaction with the substrate. For non-resonant nanoparticles the contribution of the substrate interaction to the trapping force in the vicinity of the trap (aperture/engraving) mode is negligible. But, in the case of plasmonic nanoparticles, the contribution of the substrate interaction to the low frequency stable trapping mode of the coupled particle-trap system increases as their resonance is tuned to the trap resonance. The strength of the substrate interaction depends on the height of the nanoparticle above the substrate. As a result, a difference in back action mechanism arises for nanoparticle displacements perpendicular to the substrate and along it. For nanoparticle displacements perpendicular to the substrate, the self induced back action component of the trap force arises due to changing interaction with the substrate as well as the trap. On the other hand, for displacements along the substrate, it arises solely due to the changing interaction with the trap. This additional contribution of the substrate leads to more pronounced back action. Numerical simulation results are presented to illustrate these effects using a bowtie engraving as the near-field trap and a nanorod as the trapped plasmonic nanoparticle. The substrate's role may be important in manipulation of plasmonic nanoparticles between successive traps of on-chip optical conveyor belts, because they have to traverse over regions of bare substrate while being handed off between these traps.

A trapped mode by higher-order Fano-like interference in a symmetric plasmonic structure

In this work, we have proposed a trapped mode arising from higher-order Fano-like interference in a symmetric plasmonic structure, which results in an electromagnetically induced transparency-like spectrum. The current oscillation of this trapped mode is rigorously anti-symmetric about the mirror plane parallel to the current oscillation and symmetric about the mirror plane perpendicular to the current oscillation, so the radiation loss is extremely weak. A modified model of two-driven coupled oscillators with frequency-dependent dissipation coefficients has been employed to describe the Fano-like interference. The Fano-like interference between two localized modes will evolve into the interference between a surface lattice mode and a localized mode on condition that the lattice mode forms through diffractive coupling. The resulting trapped mode simultaneously possesses both characteristics of localized and collective oscillation. Slow light with a group index as large as 90 could be realized by this interference.

Trapped modes in a wave guide with a circular cylinder

ABSTRACTTrapped mode in three-dimensional waveguide with parallel vertical walls a distance 2d apart, when there is a circular cylinder of radius placed symmetrically between them, is investigated. The potential is the solution of the three-dimensional Laplace equation in the fluid region between the cylinder and the parallel walls and the normal derivatives of potential are zero on both the walls and the cylinder. It is shown that for a cylinder of sufficiently small radius there exists a trapped mode, having a frequency close to the cutoff frequency, which is antisymmetric about the centerline of the guide and symmetric about a line through the center of the cylinder perpendicular to the centerline. The method used in this paper is to utilize the Fourier transform of complex form to the formula derived in the Appendix which can easily transform the solution in polar coordinates into the form useful for the rectangular coordinates. Thus, it is possible to express the unknown function in the solution in rectangular coordinates in terms of the coefficients of the solution in polar coordinates.

Single-shot energy measurement of a single atom and the direct reconstruction of its energy distribution

An ensemble of atoms in steady-state, whether in thermal equilibrium or not, has a well defined energy distribution. Since the energy of single atoms within the ensemble cannot be individually measured, energy distributions are typically inferred from statistical averages. Here, we show how to measure the energy of a single atom in a single experimental realization (single-shot). The energy distribution of the atom over many experimental realizations can thus be readily and directly obtained. We apply this method to a single-ion trapped in a linear Paul trap for which energy measurement in a single-shot is applicable from 10 K and above. Our energy measurement agrees within 5% to a different thermometry method which requires extensive averaging. Apart from the total energy, we also show that the motion of the ion in different trap modes can be distinguished. We believe that this method will have profound implications on single particle chemistry and collision experiments.

Three-Dimensional Numerical Modeling of Acoustic Trapping in Glass Capillaries

Acoustic traps are used to capture and handle suspended microparticles and cells in microfluidic applications. A particular simple and much-used acoustic trap consists of a commercially available, millimeter-sized, liquid-filled glass capillary actuated by a piezoelectric transducer. Here, we present a three-dimensional numerical model of the acoustic pressure field in the liquid coupled to the displacement field of the glass wall, taking into account mixed standing and traveling waves as well as absorption. The model predicts resonance modes well suited for acoustic trapping, their frequencies and quality factors, the magnitude of the acoustic radiation force on a single test particle as a function of position, and the resulting acoustic retention force of the trap. We show that the model predictions are in agreement with published experimental results, and we discuss how improved and more stable acoustic trapping modes might be obtained using the model as a design tool.

Understanding ECH density pump-out in DIII-D H-mode plasmas

In this paper, we show that the often observed ‘density pump-out’ with electron cyclotron heating (ECH) (Angioni et al 2004 Nucl. Fusion 44 827, 2009 Plasma Phys. Control. Fusion 51 124017), at low density and/or collisionality is the result of an increase in turbulence drive at the plasma edge (Angioni et al 2004 Nucl. Fusion 44 827, 2005 Phys. Plasmas 12 040701, Mordijck et al 2015 Nucl. Fusion 43 113025). Prior results were limited to comparison of steady-state conditions, before and after the ECH was applied, and thus failed to capture the dynamics of the density pump-out. In this paper, we find, similar to prior results, that when the plasma reaches a new equilibrium after ECH is applied, gyro-kinetic simulations indicate that the plasma has transitioned from the ion temperature gradient (ITG) to a trapped electron mode (TEM) regime around mid-radius. However, this transition from ITG to TEM only occures in the core after 100 ms. The pump-out on the other hand, starts immediately and is strongest around . Linear gyrokinetic simulations with TGLF show that there is an increase in turbulence drive simultaneously with the density pump-out and the doppler backscattering (DBS) measures an instant increase in density fluctuations at the same radial location. On the other hand, around mid-radius the DBS measures no increase in density fluctuations. All these calculations along with experimental measurements show that the density pump-out is not the result of a change in turbulence type (i.e. not caused by a change from ITG to TEM), but the result of a change in turbulence drive (an increase in linear growth rates), which is later followed by the ITG to TEM transition. This highlights the need for studying not just the equilibrium conditions after a transition, but also the time-dependent changes.

KIC 10001893: a pulsating sdB star with multiple trapped modes

KIC 10001893 is a V1093 Her type pulsating subdwarf-B star, which was observed extensively by the Kepler spacecraft. It was a part of the survey searching for compact pulsators in the Kepler field of view. An amplitude spectrum of the star demonstrates a rich content of g-modes between 102 and 496 {\mu}Hz as well as a few p-modes above 2000 {\mu}Hz. In total, we found 110 frequencies. The g-mode region contains 104 frequencies, while the p-mode region contains just six, altogether indicating the hybrid nature of KIC 10001893. The main goal of our analysis was to identify the detected modes and to find some features, which will significantly help modeling of the star. We found no multiplets, which points at a pole-on orientation, however, we defined modal degrees and relative radial orders using asymptotic period spacing. Eventually, we assigned 32 dipole l = 1 and 18 quadrupole l = 2 modes. The most remarkable feature we found are trapped modes, which are clearly seen in a reduce period diagram. It is the first time that three trapped modes are detected in one pulsating sdB star. Since the more trapped modes we find, the better sounding of the stellar interior we can do, this feature provides important constraints on the physical structure of the star. Mode trapping is likely caused by the He-H transition region and therefore it provides crucial constraints for making realistic theoretical models of hot subdwarfs.

Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability

Coherent features of a turbulent Mach 0.9, Reynolds number$10^{6}$jet are educed from a high-fidelity large eddy simulation. Besides the well-known Kelvin–Helmholtz instabilities of the shear layer, a new class of trapped acoustic waves is identified in the potential core. A global linear stability analysis based on the turbulent mean flow is conducted. The trapped acoustic waves form branches of discrete eigenvalues in the global spectrum, and the corresponding global modes accurately match the educed structures. Discrete trapped acoustic modes occur in a hierarchy determined by their radial and axial order. A local dispersion relation is constructed from the global modes and found to agree favourably with an empirical dispersion relation educed from the simulation data. The product between direct and adjoint modes is then used to isolate the trapped waves. Under certain conditions, resonance in the form of a beating occurs between trapped acoustic waves of positive and negative group velocities. This resonance explains why the trapped modes are prominently observed in the simulation and as tones in previous experimental studies. In the past, these tones were attributed to external factors. Here, we show that they are an intrinsic feature of high-subsonic jets that can be unambiguously identified by a global linear stability analysis.

Trapping mechanism for long waves over circular islands with power function profiles

Long waves such as tsunamis can be trapped by islands due to wave refraction, and these trapped waves will cause huge damage even in the sheltered shoreline of the island. That all waves propagating into the topography and finally reaching the coastline are called perfect trapped modes, while any waves escaping from the topography are called leaky modes. Whether these long waves can be trapped is dependent on the depth profile of the island. This paper presents analytic solutions of the ray path for waves propagating into the circular island with power function profiles. Wave height distributions over the island are further investigated based on the principia that crowded rays correspond to large wave height and sparse rays correspond to small wave height. The trapped mechanism for water waves over the island is revealed based on their ray paths. Furthermore, the perfectly trapped criterion is derived, that is, when the slope gradient at the topography toe is greater than twice the ratio of the water depth to the radial distances, all wave rays propagating on the island will finally reach the coastline, and the waves are perfectly trapped.

Plasmonic Light Scattering in Textured Silicon Solar Cells with Indium Nanoparticles from Normal to Non-Normal Light Incidence.

In this study, we sought to improve the light trapping of textured silicon solar cells using the plasmonic light scattering of indium nanoparticles (In NPs) of various dimensions. The light trapping modes of textured-silicon surfaces with and without In NPs were investigated at an angle of incidence (AOI) ranging from 0° to 75°. The optical reflectance, external quantum efficiency (EQE), and photovoltaic performance were first characterized under an AOI of 0°. We then compared the EQE and photovoltaic current density-voltage (J-V) as a function of AOI in textured silicon solar cells with and without In NPs. We observed a reduction in optical reflectance and an increase in EQE when the cells textured with pyramidal structures were coated with In NPs. We also observed an impressive increase in the average weighted external quantum efficiency (∆EQEw) and short-circuit current-density (∆Jsc) in cells with In NPs when illuminated under a higher AOI. The ∆EQEw values of cells with In NPs were 0.37% higher than those without In NPs under an AOI of 0°, and 3.48% higher under an AOI of 75°. The ∆Jsc values of cells with In NPs were 0.50% higher than those without In NPs under an AOI of 0°, and 4.57% higher under an AOI of 75°. The application of In NPs clearly improved the light trapping effects. This can be attributed to the effects of plasmonic light-scattering over the entire wavelength range as well as an expanded angle of incident light.