Cylindrical Wave Expansion Research Articles

Method of acoustic separation on the surface of cylindrical baffle

This paper suggests a statistically optimal near-field acoustic holography method based on the combination of plane waves and cylindrical waves, which is used for underwater cylindrical baffles. The method addresses the issue that the extraction of scattered acoustic waves from the surface of underwater cylindrical baffles requires too many measurement points in practical applications: separation of incident and dispersed sound fields and near-field measurements of sound fields at baffle surfaces.The underwater detection sound source is situated in the scatterer's far-field region, so in the actual application scenario for the surface acoustic scattering separation of the cylindrical baffle, the plane wave expansion is used to represent the surface incident acoustic wave and the cylindrical wave expansion is used to describe the scattered acoustic wave on the baffle surface. As a result, fewer expansion orders are required to describe the entire sound field on the baffle's surface. It is verified by simulation that the method is valid and efficient for near-field total sound field measurement and dispersed sound field separation of roughly cylindrical targets under underwater far-field incidence situations because the measurement holographic surface of the surface permits unlimited geometries.

Electromagnetic scattering of a plane wave by a perfectly conducting elliptic cylinder near a plane surface

In this paper, we present a rigorous analytical and numerical method to evaluate the scattering of a plane electromagnetic wave by an infinite, perfectly conducting elliptic cylinder parallel to a non-perfectly reflecting plane surface. The analytical solution is based on the spectral domain method and the plane-wave expansion of the elliptic cylindrical waves and is applicable in a broad variety of cases in which the properties of the plane surface may be expressed by means of a reflection coefficient. The proposed approach is suitably implemented in a specific computational code to achieve a numerical solution. Results of validation trials, performed by using finite element method commercial software, are provided.

Single-Cut Far-Field Pattern Determination of Polarized Linear Antenna Array via Quasi-Far-Field Mathematical Absorber Reflection Suppression

In this work, a method for rapidly and accurately determining the single-cut far-field (FF) patterns of any polarized linear antenna array, termed as quasi-far-field mathematical absorber reflection suppression (QFF-MARS), is introduced. The proposed QFF-MARS method takes advantage of the cylindrical wave expansion (CWE) theory and is able to exclude interference from scatters at the QFF range. In this method, probe compensation is not needed, owing to the fact that the measurement procedure is the same as that for FF. It is only required that the measurement distance for an antenna under test (AUT) satisfies the FF and QFF conditions for its smaller and larger dimensions, respectively. To validate the proposed method, a 45° polarized antenna array and a phased array have been simulated. The results obtained by the proposed approach are consistent with both theoretical and measured results. This indicates that the proposed approach can directly obtain the single-cut FF patterns of polarized linear antenna arrays at a QFF range, while identifying and extracting range-related multipath effects.

A modified convolution model for calculating the far field directivity of a parametric array loudspeaker.

The far field directivity is a straightforward indicator to describe the radiation pattern of the audio sound generated by a parametric array loudspeaker (pal), but its accurate and computationally efficient prediction is still challenging at present. This paper derives two-dimensional (2D), three-dimensional (3D), and 3D axisymmetric convolution models for calculating the far field directivity based on the quasilinear solution of Westervelt equation. The obtained expressions are expressed as linear and spherical convolutions of the ultrasound directivity and Westervelt directivity for 2D and 3D models, respectively. To improve prediction accuracy, the obtained expression is multiplied by an effective directivity resulted from the aperture factor of audio sound. The calculated directivities are compared against the exact solution obtained using the cylindrical and spherical wave expansions for 2D and 3D models, respectively. Numerical results with piston, apodized, and steerable profiles in both 2D and 3D models show that the proposed modified convolution model agrees well with the exact solution. It is also found that sidelobes appear in the audio sound directivity at large aperture sizes and high audio frequencies due to the aperture factor of audio sound, which can be predicted with the proposed method with a relatively low computational expenditure.

Borehole radiation and reception responses for azimuthal shear-wave reflection imaging with an off-centred dipole acoustic tool

Abstract Radiation and reception responses of a dipole acoustic logging tool placed eccentrically in borehole fluid are an interesting and important topic in acoustic reflection imaging. Herein, we present a thorough research study on these responses. We treat the wave incidence from the reflector as the radiation from a virtual source and use the cylindrical-wave expansion method to solve both the wave radiation and reception problems for the off-centred tool, which, by using the steepest-descent method, yields asymptotic solutions for modelling the radiation and reception wavefield characteristics. The modelling results from the analytical solution and the 3D finite-difference method were in good agreement. Specifically, we analysed the radiation directivity of an eccentric dipole source in a fluid-filled borehole. The results revealed that the radiation pattern was asymmetric with respect to the borehole, and the asymmetry was determined by the eccentric distance, source frequency and formation properties. In particular, for the typical 3 kHz dipole logging frequency, the radiation was stronger in the off-centred direction than in the opposite direction. The asymmetry of the eccentric radiation resulted in a significant amplitude difference relative to its centred counterpart, which provided a potential method for addressing the 180° azimuth ambiguity of the dipole source. We used a theoretical waveform modelling example to demonstrate this advantage. Therefore, the results of this study provide a theoretical foundation for the development and application of dipole shear-wave imaging technology.

Wave Propagation in Vegetation Field Using a Hybrid Method

In this article, we develop a hybrid method for studying wave scatterings and propagation in a vegetation field. The coherent multiple scatterings among the plants and the gaps in the vegetation field are considered through a two-step hybrid method. First, the T-matrix of a single plant is calculated using the Huygens principle and the vector cylindrical wave (VCW) expansion based on the full-wave solutions from the Ansys High Frequency Structure Simulator (HFSS). Second, the multiple scatterings between different plants are considered through the Foldy–Lax (FL) multiple-scattering equations using the calculated T-matrix and the translational addition theorem. The convergence and the accuracy of the hybrid method are verified with the HFSS by comparing the solutions of scatterings from four wheat plants. The full-wave Monte Carlo simulations of a field with 169 wheat plants at the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> -band are subsequently performed using the hybrid method. Numerical results show that the transmission in the gap region is significantly higher than that of the inner region with the difference given quantitatively. The transmissivity of microwave through the wheat field with a large vegetation water content (VWC) can be significantly underestimated by the classical radiative transfer equation (RTE) and distorted Born approximation (DBA) model.

Convergence analysis of various factorization rules in the Fourier-Bessel basis for solving Maxwell equations using modal methods.

The choice of factorization rule can strongly affect the convergence of solutions to Maxwell equations based on the orthogonal expansion of electromagnetic fields. While this issue has already been investigated thoughtfully for the Fourier basis (plane-wave expansion), for other bases it has not yet received much attention. Although there are works showing that, in the case of the Fourier-Bessel basis (cylindrical-wave expansion), the use of an inverse factorization rule can provide faster convergence than Laurent's rule, these works neglect the fact that other rules are also possible. Here, I mathematically demonstrate four different factorization rules for solving Maxwell equations in cylindrical coordinates using the Fourier-Bessel expansion in both infinite and finite domains. I compare their convergence for a step-index fiber (which has a known exact solution and thus enables the absolute numerical error to be determined), as well as for several VCSEL structures. I show that the cylindrical-wave expansion differs from the plane-wave expansion and that the application of an inverse factorization rule for the electric field component perpendicular to the discontinuities can result in deterioration of numerical convergence. Finally, I identify the factorization rule that gives the fastest convergence of the modal method using the Fourier-Bessel basis.

Fast Measurement of Single-Cut Far-Field Patterns at Quasi-Far-Field Distance for Linear Antenna Array of Any Polarization

Corrected method for fast measurement of single-cut Far-Field (FF) pattern at Quasi-Far-Field (QFF) distance for linear antenna array of any polarization by two orthogonal measurements has been proposed in this paper. The analytical equations were deduced based on cylindrical wave expansion (CWE). The required measurement distance of proposed method don’t need to satisfy the FF condition of the antenna under test (AUT), only need to satisfy the FF region of small dimension of the AUT, so it is in the Quasi-Far-Field (QFF) of another large dimension, which makes the process same as FF measurements. Therefore, the introduced method is especially applicable to arbitrary polarized antennas with one dimension large, and another dimension small. 45° linearly polarized and circularly polarized antenna array were simulated to prove the proposed method, the corrected results using the presented algorithm are consistent with the theoretical results.

Characterizing the borehole response for single-well shear-wave reflection imaging

Single-well shear-wave imaging using a dipole source-receiver system is an important application for detecting geologic structures away from the borehole. This development allows for determining the azimuth information of the structures. Existing analyses, however, focus on the data received at the borehole axis and use the elastic reciprocity theorem to model the borehole radiation and recording. We have extended the existing analyses to model the radiation, reflection, and recording response of the borehole for azimuthally spaced receivers off the borehole axis. By treating the mirror image of the borehole source with respect to the reflector plane as a virtual source, the borehole reception problem was shown to be equivalent to the response of the borehole to the spherical wave incidence from the virtual source, which can be solved using the cylindrical-wave expansion method. An asymptotic solution using the steepest-descent method is obtained if the virtual source is far from the borehole. The analytical solution allows us to analyze the borehole response for azimuthally spaced off-axis receivers. The analysis results agree well with those from 3D finite-difference simulations. With this analysis, one can further model multicomponent shear-wave reflection data from the cross-dipole acoustic tool and study the azimuthal variation characteristics of the data. Our results reveal that, whereas the data characteristics are dominated by those of a dipole, nondipole responses due to the off-axis reception can be observed, the magnitude of the responses depending on the off-axis distance and frequency and on the formation elasticity. The nondipole response characteristics have the potential to resolve the 180°-ambiguity problem in the azimuth determination for dipole shear-wave imaging. Our findings, therefore, provide new information for shear-wave reflection imaging analysis and development.

Analysis of the effects of viscosity on the SH-wave band-gaps of 2D viscoelastic phononic crystals by Dirichlet-to-Neumann map method

The effects of viscosity on the band-baps of two-dimensional (2D) viscoelastic phononic crystals are studied by the Dirichlet-to-Neumann (DtN) map method. The viscoelastic phononic crystals are composed of square or triangular lattices of elastic circular solid cylinders in a viscoelastic solid matrix. Time-harmonic anti-plane transverse (SH) waves are considered and the standard linear solid model is used to describe the viscoelastic behavior of the matrix material, in which the shear modulus depends on the frequency. Based on the DtN map constructed by using the cylindrical wave expansions, a linear eigenvalue problem related to the Bloch wave vector is formulated, which involves relatively small matrices. By using real frequencies and complex-valued wave vectors, the complex band structures are computed for different viscosity parameters, and the wave propagation and attenuation characteristics in dependence of the viscosity are analyzed. Then based on the storage shear modulus, real band structures for the Pb/epoxy phononic crystals with viscosity in the square and triangular lattices are studied, and the effects of the different viscoelastic material parameters and other relevant influencing parameters on the band-gaps are discussed. The results show that the viscosity of the matrix material affects the location and the width of the band-gaps. This fact provides an alternative way to tune the band-gaps of phononic crystals by adjusting the viscoelastic parameters combined with other influencing parameters.

Fast Determination of Single-Cut Far-Field Pattern of Base Station Antenna at a Quasi-Far-Field Distance

In this article, we present a new approach for fast determination of the single-cut far-field (FF) pattern and antenna gain of a base station antenna with remarkably unequal aperture length and width (such as linear antenna array) at a quasi- FF (QFF) range, by taking advantage of the cylindrical wave expansion (CWE). The proposed method is free of probe compensation since the probe is stationary and the antenna under test (AUT) is in the QFF range, which makes the measurements equivalent to FF measurements. The measurement distance required by the proposed approach only needs to satisfy the FF condition and QFF condition for the relatively smaller dimension and larger dimension of the AUT, respectively. To validate the proposed method, we simulated four types of antenna array, that is, an array with a uniform distribution, a low sidelobe array, a phased array, and a shaped-beam array. It is found that the results are consistent with our theoretical analysis.

Propagation of Waves in Randomly Distributed Cylinders Using Three-Dimensional Vector Cylindrical Wave Expansions in Foldy–Lax Equations

In this article, we develop a hybrid method to calculate the propagation of microwaves in randomly distributed dielectric cylinders. The hybrid method combines off-the-shelf techniques for single object and our developed techniques of Foldy–Lax (FL) method that include extracting the T-matrix for single object, vector translation addition theorem, and solving FL multiple scattering equations. For cylindrical scatterers such as tree trunks, the T-matrix in vector three-dimensional cylindrical waves are extracted from infinite cylinder approximation (ICA). In solving FL to calculate statistical moments, we iterate one order of multiple scattering at a time, with averaging over realizations performed after each order. This physically based iterative method of calculating statistical moments converges faster than the traditional iterative method of calculating the exact solution for each realization. The main purpose is to simulate tall tree trunks at the L-band and ICA is of sufficient accuracies. Numerical results are illustrated for a large number of cylinders of up to 196 and cylinder lengths of up to 94 wavelengths, which are typical of forests at the L-band. Results of the simulations of the hybrid method show that the transmission coefficients of waves are several times larger than that of the commonly used models of the radiative transfer equation and distorted Born approximation.

Inverse Scattering Analysis from Measurement Data of Total Electric and Magnetic Fields by Means of Cylindrical-Wave Expansion

Microwave tomography is an effective technique to estimate material distribution, where inverse scattering analysis is performed on the assumption that accurate information on the incident field is known for a measurement curve as well as in the target region. In reality, however, the information may often be unobtainable due to multiple scattering between the transmitting antenna and the target object, or existence of unwanted waves and obstacles. In this paper, a method to extract information on incident fields from measured total field data is proposed. The validity of the proposed method is verified on 2D TMz problems, where a cylindrical, a square, and an L-shape homogeneous object are employed as a target object. Furthermore, it is shown that the method is available even when there are unwanted obstacles outside the measurement curve.

Mimicking Localized Surface Plasmons with Structural Dispersion

AbstractOne major obstacle in developing plasmonic devices is dissipative loss. Structural waveguide dispersion offers a route to tackle this problem. Although long range propagation of surface waves using this concept is recently reported, experimental realizations of localized surface plasmon resonances with suppressed dissipative loss still remain elusive. In this paper, effective localized surface plasmons in a bounded waveguide filled with only positive dielectrics are modeled theoretically and demonstrated experimentally. Theoretical analysis based on cylindrical wave expansion shows that the effective surface modes are induced by structural dispersion of transverse electric modes. Owing to dramatically suppressed metallic loss, the designed structure can support multipolar sharp plasmonic resonances, which are difficult to attain with natural plasmons at optical frequencies. To probe the characteristics of these resonances in the experiment, a deep‐subwavelength open resonator is fabricated and the transmission spectrum at the boundary of the structure is measured. The results reveal that structured‐dispersion‐induced localized surface plasmons are quite sensitive to the background refractive index but relatively robust to the size and shape of the resonator. These findings open up a new avenue for designer localized surface waves at low frequencies and may find applications in miniaturization of microwave resonators, filters, and terahertz biosensors.

Hybrid method of plane-wave and cylindrical-wave expansions for distributed Bragg-reflector pillars: formalism and its application to topological photonics

A hybrid computational method of plane-wave and cylindrical-wave expansions for distributed Bragg-reflector (DBR) pillars is proposed. The plane-wave expansion is employed to represent the one-dimensional periodic structure of the DBR. The cylindrical-wave expansion is employed to describe the scattering by circular pillars with the DBR structure inside. This formalism enables us to calculate the radiation fields, $t$-matrices, scattering cross sections, photonic band structures, and quality factors of the DBR pillars. Furthermore, optical properties of arrayed DBR pillars are also investigated with the aid of the multiple-scattering method. Using this formalism, we demonstrate explicitly that high $Q$ photonic band modes including the so-called bound states in continuum are obtained both in isolated and arrayed DBR pillars. We also present a novel formation of gapless Dirac-cone surface states in a three-dimensional photonic crystal composed of a two-dimensional periodic arrangement of core-shell DBR pillars.

Accurate expansion of cylindrical paraxial waves for its straightforward implementation in electromagnetic scattering

The evaluation of vector wave fields can be accurately performed by means of diffraction integrals, differential equations and also series expansions. In this paper, a Bessel series expansion which basis relies on the exact solution of the Helmholtz equation in cylindrical coordinates is theoretically developed for the straightforward yet accurate description of low-numerical-aperture focal waves. The validity of this approach is confirmed by explicit application to Gaussian beams and apertured focused fields in the paraxial regime. Finally we discuss how our procedure can be favorably implemented in scattering problems.

Vertical mode expansion method for numerical modeling of biperiodic structures.

Due to the existing nanofabrication techniques, many periodic photonic structures consist of different parts where the material properties depend only on one spatial variable. The vertical mode expansion method (VMEM) is a special computational method for analyzing the scattering of light by structures with this geometric feature. It provides two-dimensional (2D) formulations for the original three-dimensional problem. In this paper, two VMEM variants are presented for biperiodic structures with cylindrical objects of circular or general cross sections. Cylindrical wave expansions and boundary integral equations are used to handle the 2D Helmholtz equations that appear in the vertical mode expansion process. A number of techniques are introduced to overcome some difficulties associated with the periodicity. The method is relatively simple to implement and highly competitive in terms of efficiency and accuracy.

A Rigorous Approach for the Modeling of Through-Silicon-Via Pairs Using Multipole Expansions

This paper presents a novel multipole expansion method for the rigorous analysis of wave propagation along a through-silicon-via (TSV) pair. TSV models are often constructed using equivalent circuits under the assumption that the interfaces between different media are equipotential. In comparison with that, the proposed method is a rigorous full-wave solution of the propagation mode. It takes advantage of cylindrical wave expansion functions and matches the boundary conditions at both the metal-to-insulator and insulator-to-silicon interfaces exactly. The method is validated against a full-wave finite-element method solver, and analysis examples using the method are demonstrated in a frequency range up to 100 GHz.

Pump-probe imaging of the fs-ps-ns dynamics during femtosecond laser Bessel beam drilling in PMMA.

A pump-probe shadowgraph imaging technique was used to reveal the femtosecond-picosecond-nanosecond multitimescale fundamentals of high-quality, high-aspect-ratio (up to 287:1) microhole drilling in poly-methyl-meth-acrylate (PMMA) by a single-shot femtosecond laser Bessel beam. The propagation of Bessel beam in PMMA (at 1.98 × 10⁸ m/s) and it induced cylindrical pressure wave expansion (at 3000-3950 m/s in radius) were observed during drilling processes. Also, it was unexpectedly found that the expansion of the cylindrical pressure wave in PMMA showed a linear relation with time and was insensitive to the laser energy fluctuation, quite different from the case in air. It was assumed that the energy insensitivity was due to the anisotropy of wave expansion in PMMA and the ambient air.

MATRIX METHOD FOR FAR-FIELD CALCULATION USING IRREGULAR NEAR-FIELD SAMPLES FOR CYLINDRICAL AND SPHERICAL SCANNING SURFACES

—A matrix method which takes into account the probe positioning errors in cylindrical and spherical near-field (NF) measurement techniques is proposed. The near-field irregularities made impossible the determination of the cylindrical or spherical wave expansion from the measured data using classical techniques based on 2D Discrete Fourier Transformation (2D-DFT) in cylindrical case (CC) and orthogonality properties in spherical case (SC). The irregularities can be randomly distributed but known and the matrix method expresses the linear relation between the measured near-field and the corresponding cylindrical or spherical modal expansion coefficients. Once the coefficients of the cylindrical and the spherical wave expansions are known the far-field of the antenna under test (AUT) is easily determined. Accuracy of the matrix method is numerically studied as a function of the irregularities magnitude and for different noise levels (data Signal to Noise Ratio). Also, experimental results have shown the efficiency of the proposed technique.