Modes Of Different Orders Research Articles

Landau damping due to octupoles of non-rigid head–tail modes

Octupole magnets are often used as a passive mitigation for the collective instabilities in ring accelerators, also for the unstable head–tail modes in bunches. An important advantage of the octupole mitigation is the ability to suppress the intra-bunch oscillations. We investigate the non-rigid oscillations of the head–tail modes using particle tracking simulations. Stability thresholds are determined and compared with the analytic predictions. The stabilization of the modes above the thresholds, and the related emittance blowup, are demonstrated and quantified for modes of different order.

Excitation and resonant enhancement of axisymmetric internal wave modes

To date, axisymmetric internal wave fields, which have relevance to atmospheric internal wave fields generated by storm cells and oceanic near-inertial wave fields generated by surface storms, have been experimentally realized using an oscillating sphere or torus as the source. Here, we use a novel wave generator configuration capable of exciting axisymmetric internal wave fields of arbitrary radial form to generate axisymmetric internal wave modes. After establishing the theoretical background for axisymmetric mode propagation, taking into account lateral and vertical confinement, and also accounting for the effects of weak viscosity, we experimentally generate and study modes of different order. We characterize the efficiency of the wave generator through careful measurement of the wave amplitude based upon group velocity arguments. This established, we investigate the ability of vertical confinement to induce resonance, identifying a series of experimental resonant peaks that agree well with theoretical predictions. In the vicinity of resonance, the wave fields undergo a transition to non-linear behaviour that is initiated on the central axis of the domain and proceeds to erode the wave field throughout the domain.

Low frequency band gaps of Helmholtz resonator coupled with membrane

<sec> In this paper, a phononic crystal is designed using a Helmholtz resonator with a membrane wall, in which the coupled vibration of air and membrane is utilized. The structure of the Helmholtz resonator is a two-dimensional structure. On the basis of the square Helmholtz resonator, a " W”-type outlet is used as a cavity outlet to increase the air quality involved in resonance, and the cavity wall is replaced with a membrane with distribution mass to increase the number of resonance units. </sec><sec> The finite element method is used to calculate the band gaps and transmission loss of sound below 1700 Hz. The results show that the starting frequency of the first band gap of the structure is further reduced. At the same time, it is lower than the starting frequency of ordinary Helmholtz structure and the natural frequency of membrane under the same conditions. Then, a new peak of transmission loss is obtained, and its value is greater than the original structure’s. And although the width of the first band gap is reduced, some new band gaps appear in the low-frequency range, so that the total band gap width is improved. </sec><sec> By analyzing the vibration mode of the membrane and sound pressure distribution, it is found that the sum of the sound pressures of the outer cavity is zero at the starting frequencies of the band gaps, and the sound pressure of the inner and outer cavity are respectively positive and negative at the cut-off frequency. With the increase of frequency, the vibration mode of the membrane gradually turns from low-order to high-order, but no anti-symmetric-type mode participation is found at the starting and cut-off frequency. </sec><sec> The components of the structure can be made equivalent to corresponding ones, respectively, i.e. air in the outlet is equivalent to uniform flexible rod, and the air in the inner and outer cavity are equivalent to a spring. So that the structure can be equivalent to a series system consisting of a rod, a spring and a membrane at starting frequency of the band gap, and a loop system consisting of a rod, two springs and a membrane at cut-off frequency. Thus, by the transfer matrix method and the Rayleigh-Ritz method considering the influence of tension and elastic modulus, it is possible to calculate the range of band gap which is extremely close to the result from the finite element method. Through the analysis of the formulas, it can be found that the new band gap is caused by the new vibration mode produced by the membrane or the air in the cavity outlet, and the lower starting frequency of the first band gap is due to the reduction of the equivalent extent of the system by the membrane. </sec><sec> By adjusting the relevant parameters of the membrane and the cavity outlet respectively, it can be found that the band gaps of the structure correspond to the modes of different orders of the air in the cavity outlet and the membrane. In other words, the change of the natural frequency of a certain mode of air in the outlet or membrane only has a greater influence on the corresponding band gap but has less influence on other band gaps, also, the trends of change are the same, and the change values are very close to each other. But, changing the volume of the inner cavity and the outer cavity has a great influence on all the band gaps. Therefore, it is possible to adjust some band gaps through this method. </sec>

Investigation of wide-range refractive index sensor based on asymmetric metal-cladding dielectric waveguide structure

A refractive index (RI) sensor based on an asymmetric metal-cladding dielectric waveguide structure, in which an analyte is used as the guiding layer, is proposed in this report. The angular scanning attenuated total reflection spectra of the sensor and optical field distributions under waveguide mode resonance conditions were simulated, which confirmed that multiple waveguide modes can be used for sensing, and that different waveguide modes exist in different analyte RI ranges. The curves of both waveguide mode resonance angles and the sensitivity as a function of the analyte RI demonstrated that a wide sensing range can be achieved by combining waveguide modes of different orders. The proposed sensor could be applied to numerous fields, such as biological and chemical detection.

Acoustic wave propagation along AlN on Bi12GeO20 structures

Using Bi12GeO20 plate coated by AlN film, the main characteristics of the A0, S0, A1, and S1 Lamb waves are numerically calculated and compared with those of SAW in the same structure. It is shown that (i) there exist only one SAW mode for this material combination, (ii) the mode exist in a narrow range of AlN thickness, (iii) it can be excited using four different transducer configurations, (iv) the velocity and coupling constants of the mode are, respectively, higher and lower as compared with the values for the uncoated Bi12GeO20 substrate. On the other hand, for Lamb waves (i) there are several modes of different orders, (ii) the modes exist in a wide range of the AlN thickness, (iii) they may be excited using twelve IDT configurations, (iv) the velocities and coupling constants of the waves may be larger and smaller as compared with the values for the uncoated plate, (v) both characteristics of the modes depend on the mode order, film thickness, plate thickness, and transducer configuration, and (vi) the depth profiles of the modes are neither symmetrical nor anti-symmetrical. These properties make Lamb waves more attractive than SAW. Application of the waves and the structure for some applications are suggested.

Self-hybridization within non-Hermitian localized plasmonic systems

Common intuition in physics is based on the concept of orthogonal eigenmodes. Those are well de- fined solutions of Hermitian equations used to describe many physical situations, from quantum mechanics to acoustics. A large variety of non-Hermitian problems, including gravitational waves close to black holes or leaky electromagnetic cavities require the use of bi-orthogonal eigenbasis. Physical consequences of non-Hermiticity challenge our physical understanding. However, the usual need to compensate for energy losses made the few successful attempts to probe non-Hermiticity extremely complicated. We show that this issue can be overcome considering localized plasmonic systems. Indeed, since the non-Hermiticity in these systems does not stem from from temporal invariance breaking but from spatial symmetry breaking, its consequences can be observed more easily. We report on the theoretical and experimental evidence of non-Hermiticity induced strong coupling between surface plasmon modes of different orders within silver nano-crosses. The sym- metry conditions for triggering this counter-intuitive self-hybridization phenomenon are provided. Similarly, observable effects are expected to be observed in any system subtending bi-orthogonal eigenmodes.

Waves and rays in plano-concave laser cavities: I. Geometric modes in the paraxial approximation

Eigenmodes of laser cavities are studied theoretically and experimentally in two companion papers, with the aim of making connections between undulatory and geometric properties of light. In this first paper, we focus on macroscopic open-cavity lasers with localized gain. The model is based on the wave equation in the paraxial approximation; experiments are conducted with a simple diode-pumped Nd:YAG laser with a variable cavity length. After recalling fundamentals of laser beam optics, we consider plano-concave cavities with on-axis or off-axis pumping, with emphasis put on degenerate cavity lengths, where modes of different order resonate at the same frequency, and combine to form surprising transverse beam profiles. Degeneracy leads to the oscillation of so-called geometric modes whose properties can be understood, to a certain extent, also within a ray optics picture. We first provide a heuristic description of these modes, based on geometric reasoning, and then show more rigorously how to derive them analytically by building wave superpositions, within the framework of paraxial wave optics. The numerical methods, based on the Fox–Li approach, are described in detail. The experimental setup, including the imaging system, is also detailed and relatively simple to reproduce. The aim is to facilitate implementation of both the numerics and of the experiments, and to show that one can have access not only to the common higher-order modes but also to more exotic patterns.

Application of strongly overcoupled resonant modes of long-period fiber gratings to measure the adulteration of olive oil.

We have shown that strongly overcoupled resonant modes of long-period fiber gratings (LPFGs) can be used to assess the adulteration of olive oil. In this background, we investigate the response characteristics of strongly overcoupled resonant modes of different orders to a surrounding medium, the refractive index (RI) of which is greater than that of the fiber cladding, and in the range where a precise refractive index measurement is immensely useful for inspecting the quality of olive oils and other edible oils. A theoretical simulation that would help in designing a sensor with suitable sensitivity and range of measurement has been presented in detail and also validated with experimental results. It was interesting to observe that in a high RI surrounding, a lower order overcoupled resonant mode is much more sensitive as compared to a higher-order one having a similar coupling coefficient. A quantitative analysis demonstrates that for a particular LPFG, the sensitivity of a strongly overcoupled LP06 mode was found to be ∼2000 dB/RIU, while that of the LP07 mode having similar coupling strength was ∼550 dB/RIU in the surrounding refractive index range from 1.458 to 1.520. The results have been validated experimentally.

Direction-of-arrival estimation based on superdirective multi-pole vector sensor array for low-frequency underwater sound sources

With the advances of ship noise reduction technology, the working frequency of the passive sonar must be reduced in order to detect a target. For the conventional array, it requires a large array aperture, comparable to the wavelength, in order to achieve an acceptable angular resolution. Arrays of small physical size with high angular resolution are thus attractive for low-frequency direction-of-arrival estimation of underwater sound source. In this paper, we consider a 33 uniform rectangular array which consists of vector sensors with inter-sensor spacing much smaller than the wavelength. A broadband super-directive beamforming method is proposed for this vector sensor array, which extracts multi-pole modes of different orders from the spatial differentials of the sound field. By normalizing the amplitudes of the multi-pole modes, frequency invariant mode functions can be obtained, which are used to build the desired beam pattern, despite the Rayleigh limit on the achievable angular resolution. Vector sensors are used to replace the pressure difference operation, thus to achieve a desirable beam pattern, the order of spatial differential will be reduced. In other words, for the same array configuration, using the vector sensors provides higher directivity than using the pressure sensor. To concentrate on the sources, and to minimize all hindrances from around circumference, a suitable beam pattern is constructed as an example to analysis. To verify the algorithm, a prototype is built and tested in a water tank. Comparisons are carried out between the actually synthesized beam patterns and the theoretical ones. The experimental results show good agreement with the theoretical results, and that the directivity increases with the multi-pole mode order increasing, at the expense of lower robustness. The performances for different values of ka are also investigated, where k is the wave number and a denotes the inter-sensor spacing. Simulation results show that when the inter-sensor spacing is no more than one-sixth of the incident wave length, the error introduced by the approximations for muti-pole mode extraction can be neglected. It should be noted that this result of the inter-sensor spacing still applicable when considering array gain, showing that the array is insensitive to uncorrelated noise while preserving a relatively high array gain. Finally, the influence of the underwater acoustic waveguide on the array performance is analyzed. Simulations and experimental tests show that due to the small array aperture, the waveguide effects on the array performance are limited.

Analysis and design of multi-pole vector sensor arrays

Super-directive sensor arrays, due to their small size and enhanced directivity, are quite promising in signal-to-noise enhancement and target direction finding. An array design method, which can build a desired beam pattern using compact multi-pole sensor arrays, is presented in this paper. The proposed method extracts multi-pole modes of different order from the spatial differentials of the sound fields which are measured by a uniform square array composed with hydrophones and particle velocity sensors. It can achieve more benefits with the employment of particle velocity sensors because of their inherent dipole directivities. The spatial differential in the extraction procedure is approximated with spatial difference and the approximation decreases the beam pattern performance. The performance degradations in multi-pole beam patterns are investigated for different incident wave frequency. Simulation and analysis show that the array gain is to a certain extent robust to uncorrelated noise.

Wave turbulence in a rotating channel

AbstractThis paper describes wave-turbulence closure and its consequences for rapidly rotating (i.e. small Rossby number) turbulence confined by two infinite, parallel walls perpendicular to the rotation axis. Expressing the flow as a combination of inertial waveguide modes leads to a spectral matrix, whose diagonal elements express the distribution of energy over modes and whose off-diagonal elements represent correlations between modes of different orders. In preparation for wave-turbulence closure, the flow is decomposed into two-dimensional and wave components. The former is found to evolve as if it were a classical, two-dimensional, non-rotating flow, but with wall friction due to Ekman pumping by the boundary layers. Evolution equations for the wave-component elements of the spectral matrix are derived using a wave-turbulence approach. Detailed analysis of these equations shows that, surprisingly, the two-dimensional component has no effect on wave-component energetics. As expected for wave turbulence, energy transfer between wave modes is via resonant triads and takes place at times $O(\varepsilon ^{-2})$ multiples of the rotational period, where $\varepsilon $ is the Rossby number. Despite playing no role in wave-mode energetics, the two-dimensional component produces decay of the off-diagonal elements of the spectral matrix on a time scale that is small compared with $O(\varepsilon ^{-2})$ rotation periods. There are thus three asymptotically distinct stages in the evolution of the turbulence in the limit of small Rossby number: the two-dimensional flow begins to evolve at the usual large-eddy turnover time scale ($O(\varepsilon ^{-1})$ multiples of the rotation period) and continues to develop thereafter. This is followed by decorrelation of different wave orders and finally evolution of the wave energy spectra due to resonant interactions.

Mode control and mode conversion in nonlinear aluminum nitride waveguides

While single-mode waveguides are commonly used in integrated photonic circuits, emerging applications in nonlinear and quantum optics rely fundamentally on interactions between modes of different order. Here we propose several methods to evaluate the modal composition of both externally and device-internally excited guided waves and discuss a technique for efficient excitation of arbitrary modes. The applicability of these methods is verified in photonic circuits based on aluminum nitride. We control modal excitation through suitably engineered grating couplers and are able to perform a detailed study of waveguide-internal second harmonic generation. Efficient and broadband power conversion between orthogonal polarizations is realized within an asymmetric directional coupler to demonstrate selective excitation of arbitrary higher-order modes. Our approach holds promise for applications in nonlinear optics and frequency up/down-mixing in a chipscale framework.

Modeling lasing of a chalcogenide glass microcylinder

Theoretical modeling of the interaction between electromagnetic radiation and microresonators in the shape of homogeneous microcylinders has shown that the morphology of the microresonators affects the transformation of optical radiation in the microlaser modes. We established that in a laser with microresonators of cylindrical symmetry made from chalcogenide glass, two-mode lasing regimes can be realized in modes of different orders for gain-to-loss ratios <1. The region of existence for two-mode lasing is determined by the gain-to-loss ratio and the magnitude of the electromagnetic fi eld for the spatial structure of morphological resonances. Introduction. Currently microresonators have been at the center of extensive research due to the possibility of using them as a convenient object for realization of miniature devices based on single dielectric microspheres and microfi bers. This is connected with the fact that weakly absorbing dielectric microparticles act as optical resonators, and morphological resonances are realized in them. A microresonator based on a dielectric microparticle of a certain symmetry (spherical, cylindrical, ellipsoidal, etc.) has additional advantages, including signifi cant lowering of the threshold for realization of various nonlinear phenomena, strong focusing of radiation, low sensitivity to fracture and overheating due to the strength scale effect, and convenient matching between microspheres or microcylinders and optical fi bers. One of the fi rst investigations of modes in a dye-impregnated cylindrical microlaser and their use for measuring the parameters of microresonators are described in (1, 2). Microfi bers enable nearly lossless excitation of high-Q modes in optical cavities, which are extremely sensitive to a change in the environment and can also be used as sensors (3, 4). Optical processes in cylindrical microresonators are the object of active experimental studies (5-8). However, these papers basically just demonstrate potential ways such systems may be used. At the same time, optical phenomena observed in this case need theoretical substantiation; in particular, the characteristic features of the effect of the spatial structure of the electromagnetic fi eld on the process of low-threshold stimulated emission have not yet been suffi ciently studied. Fabrication of chalcogenide glass microcylinders is of special interest (9). This material is widely used in spectroscopy for fast processing of optical signals in fi ber-optic communication lines, video camera tubes, and also for recording holograms. The high refractive index of chalcogenide glass (2.2-2.7) lets it be used for fabrication of microresonators, strongly concentrating the radiation within the microcylinders. This is why the corresponding theoretical studies were done using microcylinders made of chalcogenide glass. Theoretical Model. In microresonators of cylindrical symmetry, certain sets of eigenmodes are realized. Each mode is characterized by a certain frequency and Q factor, which depend both on the diffraction parameter (ρ = 2πR/λ, R is the radius of the microcylinder, λ is the wavelength of the laser radiation) for the microcylinder and on the optical constants of the microcylinder material. Along with the Q factor, an important electrodynamic characteristic of a microresonator is the spatial structure of the electromagnetic fi eld inside it, which is formed by summing the spatial structures of the fi elds for the eigenmodes. Methods for obtaining and solving the equations for determining the frequencies of the eigenmodes are well known and discussed in detail in (10). We should point out that each equation has an infi nite number of solutions. Each solution is characterized by the mode number l, which determines the number of peaks in the intensity distribution of the internal fi eld as the polar angle j varies from 0 to 180°, and the order s, equal to the number of peaks in the fi eld intensity distribution for

Locating a weakened interface in a laminated elastic plate

We study thickness-shear (TSh) free vibrations of an unbounded, laminated elastic plate with three layers of different materials. One of the two interfaces is slightly weakened as described by the shear-lag model that allows the displacement to be discontinuous across the interface. A frequency equation is obtained from the linear theory of elasticity. A perturbation solution of the frequency equation is obtained from which the frequency shifts of TSh modes due to the weakened interface can be calculated. It is shown that the frequency shifts of TSh modes of different orders are different, and they satisfy different conditions when different interfaces are weakened. These conditions are obtained which can potentially be used as criteria for determining specifically which interface is weakened.

Modal Analysis and Local Stiffness Optimization for Body-in-white of Car

AbstractUsing preprocessing software Hypermesh and FEA software MSC. Nastran, the detailed finite element model of car body-in-white (BIW) was established based on the theory of finite element. Finite element modal analysis and experimental modal analysis were conducted to get the modal parameters of BIW including natural frequencies and corresponding mode shapes respectively. With comparison of the result of the simulation with that of test, the validity of the FEA model was verified. From the view of vibration and strength, the impact of internal and external incentives on the BIW was analyzed. According to strain mode local characteristics, a new method to determine the position where the maximum elastic deformation takes place by using the strain energy distribution of the vibration modes of different orders was therefore proposed. The method can be used to improve the local rigidity of BIW. The result shows that this method is reasonable and practical for the car body design.

General properties of the Anisimkin jr. plate modes

Acoustic plate modes of different orders n, having equal velocities v(n) close to that of the longitudinal BAW v(L), are numerically studied in crystals of different symmetries. Three families of the modes with v(n) ≈ v(L), each at relevant plate thickness h/λ = (h/λ)n, are found (h is the thickness, λ is the wavelength): the generalized Lamb mode with comparable longitudinal u1, shear-horizontal u2, and shear-vertical u3 displacements, the Anisimkin Jr. (AN) mode with u1 >> u2 and u3, and u1 ≈ constant ≠ 0 at any depth, and the quasilongitudinal (QL) mode with u1 > u2, and u3, but u1 ≠ constant over the plate thickness. Existence of the families does not depend on anisotropy or piezoelectric properties of the plate, but on the closeness of the mode velocity v(n) to the BAW velocity v(L), the value of the dispersion slope dv(n)/d(h/λ) at v(n) = v(L) and h/λ = (h/λ)n, and the proximity of the plate thickness (h/λ)n supporting the mode, to the thickness (h/λ)R providing transverse BAW resonance between plate faces. The Lamb modes approach v(n) = v(L) at irregular (h/λ)n far from resonance (h/λ)R and at large dv(n)/d(h/λ) ~10(3) m/s. The two other modes are characterized by lower dispersion dv(n)/d(h/λ) ≤ 10(3) m/s and regular (h/λ)n close to the resonance (h/λ)R. Because both modes have small vertical displacement on plate faces and propagate almost entirely within the crystals, they are attractive for liquid sensing.

Geometric phases in astigmatic optical modes of arbitrary order

The transverse spatial structure of a paraxial beam of light is fully characterized by a set of parameters that vary only slowly under free propagation. They specify bosonic ladder operators that connect modes of different orders, in analogy to the ladder operators connecting harmonic-oscillator wave functions. The parameter spaces underlying sets of higher-order modes are isomorphic to the parameter space of the ladder operators. We study the geometry of this space and the geometric phase that arises from it. This phase constitutes the ultimate generalization of the Gouy phase in paraxial wave optics. It reduces to the ordinary Gouy phase and the geometric phase of nonastigmatic optical modes with orbital angular momentum in limiting cases. We briefly discuss the well-known analogy between geometric phases and the Aharonov–Bohm effect, which provides some complementary insights into the geometric nature and origin of the generalized Gouy phase shift. Our method also applies to the quantum-mechanical description of wave packets. It allows for obtaining complete sets of normalized solutions of the Schrödinger equation. Cyclic transformations of such wave packets give rise to a phase shift, which has a geometric interpretation in terms of the other degrees of freedom involved.

Oscillatory variations in the Q factors of high quality micropillar cavities

We report on the observation of oscillatory variations in the quality (Q) factor of quantum dot-micropillar cavities based on planar Bragg reflectors. The oscillatory behavior in the Q versus diameter dependence appears in the diameter range between 1.0 and 4.0 μm, has a characteristic period of a few hundred nanometers and increases in amplitude with increasing reflectivity of the planar microcavity structures. The experimental results are well reproduced by numerical calculations which support the interpretation that the Q oscillations are caused by coupling of propagating Bloch modes of different orders at the mirror interfaces.

Unitary spherical ESPRIT: 2-D angle estimation with spherical arrays for scalar fields

The authors consider the problem of two-dimensional (2-D) direction-of-arrival estimation of multiple plane waves incident on a spherical array. We propose a novel subspace technique that provides automatically paired source azimuth and elevation estimates relying on a spherical phase-mode excitation approach. The algorithm is remarkable for the low computational complexity and enhanced performance in correlated source scenarios by relying on a real beamspace approach, which allows forward-backward averaging. Moreover, the technique yields very accurate estimates by including spherical phase modes of different orders. This is demonstrated by comparing the mean error of the angle estimates obtained by spherical ESPRIT and the corresponding Cramer Rao bounds.

Tailoring of arbitrary optical vector beams

We present a robust interferometric method to generate arbitrary vector beam modes by diffracting a Gaussian laser beam from a spatial light modulator consisting of a high-resolution reflective nematic liquid crystal display. Vector beams may have the same intensity cross-section as the more common scalar Laguerre–Gaussian (LG) or Hermite–Gaussian (HG) beams, but with a spatially modulated polarization distribution. Special cases are the radially or azimuthally polarized ‘doughnut’ modes, which have superior focusing properties and promise novel applications in many fields, such as optical trapping, spectroscopy and super-resolution microscopy. Our system allows video rate switching between vector beam modes. We demonstrate the generation of high quality Hermite–Gaussian and Laguerre–Gaussian vector beam modes of different order, of vectorial anti-vortices, and of mode mixtures with interesting non-symmetric polarization distributions.