Spherical Wave Expansion Research Articles

Antenna pattern reconstruction for mid-band massive antenna array based on a single-cut field transformation algorithm in a compact multi-probe anechoic chamber setup

ABSTRACT It has become a common challenge on how to measure radiation patterns of massive antenna arrays working at high frequency with high efficiency and accuracy, which has attracted huge interest from both academia and industry. In this paper, a highly accurate and efficient far field (FF) pattern reconstruction strategy through conducting single cut near field (NF) to FF transformation algorithm based on spherical wave expansion (SWE) is proposed for fast measurements of large base stations (BSs) operating at mid-band. Our objective is to design a compact multi-probe setup to measure key radiation parameters of the massive array, with low-cost and high-efficiency. A BS of 16 × 16 elements operating at mid-band is adopted as the device under test (DUT) in this paper. Analysis is conducted on the impact of measurement distance, sampling range and sampling density on the reconstruction accuracy of the algorithm in the compact multi-probe anechoic chamber setup, demonstrating that the required key information of the main lobe of the DUT can be accurately reconstructed when the measurement range R = 1/20 FF distance, sampling resolution ∆θ = 3°, and sampling range θ ∈ [75°, 105°] are selected, respectively. The impact of phase center offset on the accuracy of FF pattern reconstruction is analyzed when only one sub-array is enabled, demonstrating the effectiveness of the algorithm for sub-array measurements.

Far-field directivity estimation with a cube-shaped array of 256 MEMS microphones

The far-field directivity function (FFDF) of a sound source characterises the angular dependence of the acoustic fields far away from this source. This function can be estimated by performing a spherical wave expansion (SWE) of the sound field from pressure measurements distributed around the source. The truncation order of the SWE is a critical parameter that depends on the number of measurement points and on the spatial sampling scheme used for the measurements. Unfortunately, for irregular sampling schemes, no theoretical result allows to determine a truncation order. In this work, a surrounding cuboid microphone array composed of 256 MEMS microphones is deployed. A cross-validation framework is used in order to determine an optimal truncation order for the SWE of the field radiated by a test source. The quality of the reconstructed field is assessed on the frequency range 100-5000 Hz. Finally, the estimated SWEs and far-field directivities are compared to an analytical model of the test source.

Extended King integral for modeling of parametric array loudspeakers with axisymmetric profiles.

Parametric array loudspeakers have been widely used in audio applications for generating directional audio beams. However, accurately calculating audio sound with a low computational load remains challenging, even for basic axisymmetric source profiles. This work addresses this challenge by extending the King integral in linear acoustics to incorporate both cumulative and local nonlinear effects, under the framework of the quasilinear solution without the paraxial approximation. The proposed method exploits the azimuthal symmetry in cylindrical coordinates to simplify modeling. To further improve computational efficacy, fast Hankel and Fourier transforms are employed for the radial and beam radiation directions, respectively. Numerical results with both uniform and focusing profiles demonstrate the advantages of the proposed approach over the traditional spherical wave expansion and direct integration methods, especially for larger aperture sizes. Specifically, for typical configurations with source aperture size of 0.2 m, we observe at least a 24-fold improvement in computational speed and a 227-fold reduction in memory requirements. These advancements allow us, for the first time, to present the sound field radiated by parametric array loudspeakers with a large aperture size of up to 0.5 m, without paraxial approximations. The implementation codes are available on https://github.com/ShaoZhe-LI/PAL_King.

Beam shape coefficients of the hollow vortex Gaussian beam and near-field scattering

The beam shape coefficients (BSCs) of the electromagnetic field of hollow vortex Gaussian beams (HVGBs) are formulated, based on the spherical wave expansion of the scalar function. The cylindrical wave spectrum decomposition is employed to expand the scalar function in the spherical coordinates. Numerical results on the beam field reproduced from the BSCs confirm that the BSC evaluation is efficient and reliable. The scattering in the near-field zone is calculated and discussed, revealing the dependence of the straight and curved photonic jets on the topological charge of the HVGB. The paper may be useful for studying the interaction between the HVGB and a spherical particle.

Phase‐reconstruction from magnitude‐only data of electrically small antennas for body‐centric wireless communication

AbstractA method for phase retrieval from magnitude‐only data for electrically small antennas intended for use in relation to body‐centric wireless communication, is presented. The method utilizes the spherical wave expansion (SWE) description of the electromagnetic field radiated by the antenna under test (AUT). The expansion coefficients of the SWE are optimized such that errors between the reconstructed and provided data magnitudes are as small as possible. The required additional phase information is acquired from additional magnitude‐only data sets of the AUT in different orientations. These orientations are defined by rotations of the AUT along the Eulerian angles. The performance of the method is evaluated for a realistic configuration, which consists of an inverted‐A antenna in the vicinity of an ear. The sensitivity of the method to the angular resolution of the field data, as well as to the Eulerian rotation accuracy, is investigated. Finally, the reconstruction method is tested on a set of practical antenna measurements conducted in an RF‐anechoic chamber.

An advanced light scattering imaging model for total internal reflection microscopy considering a stratified medium

An advanced light scattering model for Total Internal Reflection Microscopy (TIRM) is presented. The model considers the specific TIRM geometry and deals with the scattering by an axisymmetric particle of arbitrary orientation placed in a stratified medium and the imaging of the scattered field. The scattered field is computed by truncating the scattered and internal field expansions and by using spherical and plane wave expansions for the free-space dyadic Green’s function. While the first expansion is valid outside a sphere enclosing the particle, the second one is valid outside the tangent planes bounding the particle from above and below. We demonstrate that in both cases, the results are the same, and thus, that the restrictive condition according to which the interface should not intersect the particle’s circumscribed sphere is not relevant. The image of the scattered field is computed by using the Debye diffraction integral and fast Fourier transform, while for a better reconstruction of the particle orientation, an image processing step consisting in a contour extraction and ellipse fitting is considered. The numerical simulations dealing with scattering by a prolate spheroid provide evidence of the remarkably sensitivity of the geometric parameters of the image ellipse to the particle orientation angles, as well as, of the integral response of the detector to the distance between the particle and the interface.

On evanescent waves and blowing-ups of the finite series technique in spherical wave expansion of shaped beams

In generalized Lorenz-Mie theory, the beam shape coefficients of Gauss-like beams obtained with finite series method blow up when the wave order is high. The blowing-ups caused by the loss of significant digits may be suppressed by increasing numerical precisions. Once the elimination of such blowing-ups is carried out, blowing-up phenomena still occur whose origin is still not clear to us. The work presented here attempts to provide an explanation on this issue. Based on the angular spectrum representation of the beam field, it is found that the blowing-up originates from the contribution of evanescent waves.

Wireless Powering Efficiency of Deep-Body Implantable Devices

The wireless power transfer efficiency to implanted bioelectronic devices is constrained by several frequency-dependent physical mechanisms. Recent works have developed several mathematical formulations to understand these mechanisms and predict the optimal operating conditions. However, existing approaches rely on simplified body models, which are unable to capture important aspects of wireless power transfer. Therefore, this paper proposes the efficiency analysis approach in anatomical models that can provide insightful information on achieving the optimum operation conditions. First, this approach is validated with a theoretical spherical wave expansion analysis, and the results for a simplified spherical model and a human pectoral model are compared. The results show that although a magnetic receiver outperforms an electric one for near-field operation and both sources could be equally employed in far-field range, it is in mid-field that the maximum efficiency is achieved with an optimum frequency between 1-5 GHz depending on the implantation depth. The receiver orientation is another factor that affects the efficiency, with a maximum difference between the best and worst-case scenarios around five times for the electric source and over 13 times for the magnetic one. This approach is used to analyze the case of a deep-implanted pacemaker wirelessly powered by an on-body transmitter and subjected to stochastic misalignments. We evaluate the efficiency and exposure, and we demonstrate how a buffered transmitter can be tailored to achieve maximum powering efficiency. Finally, design guidelines that lead to optimal implantable wireless power transfer systems are established from the results obtained with the proposed approach.

Equivalence between radial quadrature and finite series for spherical wave expansion of Bessel beams.

The radial quadrature method was recently proposed for formulating the beam shape coefficients (BSCs) for shaped beams. A new deduction of BSCs using the R-quadrature method is presented in this paper, using the integral of the spherical Bessel functions in the interval ranging from zero to infinity. Based on the scalar description of the Bessel beam, the equivalence between the R-quadrature and the finite series (FS) method is confirmed. The spherical wave expansion of the scalar function allows us to simplify the formulation of the BSCs in the R-quadrature and the FS and to speed up the numerical BSC calculation. As a by-product, FS expansions of the associated Legendre functions are established, which we do not find in the literature.

Closed-Form Expression of Radiation Characteristics for Electrically Small Spherical Helix Antennas

To understand the radiation mechanism of an electrically small spherical helix antenna, we develop a theory on the radiation characteristics of the antenna. An analytical model of the antenna presuming a current on the wire to be sinusoidally distributed is proposed and analyzed with the spherical wave expansion. The radiation efficiency, radiation resistance, and radiation patterns are obtained in closed-form expression. The radiation efficiency evidently varies with the surface area of the wire and the radiation resistance depends on the square of the length of the wire. The obtained result for the radiation pattern illustrates the tilt of the pattern caused by the modes asymmetric to the z-axis. The radiation efficiency formula indicates a good agreement between the simulation and measurement result. In addition, the radiation resistance of the theoretical and simulation results exhibits good agreement. Considering the effect of the feeding structure of the fabricated antenna, the radiation resistance of the analytical model can be treated as a reasonable result. The result of radiation pattern also shows good agreement between the simulation and measurement results excluding a small contribution from the feeding cable acting as a scatterer.

Estimation of Absorbed Power Density Based on Spherical Wave Expansion to Plane Wave Expansion for Exposure Assessment at 30 GHz

The international guidelines and standard specified the area-averaged absorbed/epithelial power density (APD/EPD) as the basic restriction at frequencies above 6 GHz. Assessment methods for APD are current under the consideration of standardization organizations. In this study, we first showed the enhancement of the APD due to the antenna/body interaction using full-wave simulations. We then proposed a hybrid method combining the spherical near-field measurement and numerical simulation of the exposure dose in human body, using an equivalent source reconstructed on the total field/scattered field boundary using the spherical wave expansion to plane wave expansion. By comparing with the full-wave simulated APD at 30 GHz for various antenna arrays, the proposed method gives a rather accurate estimation of the upper bound of the area-averaged APD (with relative differences < 28%). It can be utilized as an efficient method for electromagnetic safety assessment as the need for specific measurement equipment is eliminated.

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.

Explicit Conversion Formulas Between Spherical Wave Expansions With Scalar and Vector Expansion Coefficients

Explicit formulas for converting spherical wave expansions with vector basis functions and scalar expansion coefficients into spherical wave expansions with scalar basis functions and vector expansion coefficients and vice versa are presented. The formulas are given in terms of Wigner-3- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula> -symbols. The conversion formulas are derived by spherical harmonics expansions of products of two spherical harmonics and by levering on recurrence relations of the associated Legendre functions. The expansions with vector coefficients are redundant and explicit formulas are given for the linear combination of vector coefficients of which the expanded fields cancel identically to zero. The redundancy in the expansion can be used to find compact expansions with a minimum expansion order. For every vector spherical wave function, two expansions using scalar spherical harmonics are presented: on the one hand, a radial-component free expansion and on the other hand, a minimum-order expansion. The correctness of all expansions is verified with a simple computer code up to a mode order of 65. The agreement of the expanded near and far fields in both types of expansions is better than 14 digits of accuracy for most cases.

Spherical near-field measurement of the offset vehicle antenna mounted on a vehicle

For the vehicle antenna measurement mounting on a vehicle, the offset mounting of the antenna will increase the maximum spherical wave expansion modes, with the increasing calculation time and the increasing risk of introducing environmental reflection noise. Therefore, basing on the matrix form of the traditional spherical wave expansion (TSWE) algorithm, this paper introduces the offset spherical wave expansion (OSWE) algorithm of the offset mounting antenna under test (AUT) aiming to decrease the maximum modes, shorten the calculation time and reduce the environmental noise with the method of spherical wave expansion. Taking LTE antenna and GPS antenna as examples, this paper analyzes the improved performance of OSWE algorithm comparing with TSWE algorithm on the vehicle antenna mounting on a vehicle, as well as the parameters like equivalent noise level (ENL), omnidirectional standard deviation (σ).

Determination of Excitation Amplitude and Phase for Wide-Band Phased Array Antenna Based on Spherical Wave Expansion and Mode Filtering

A new method for solving the excitation amplitude and phase of wide-band phased array antenna is presented, in which spherical wave expansion and mode filtering (SWEMF) techniques are applied for the first time. Different from the previous methods that are required of matrix inversion or optimization iteration, the proposed SWEMF method is a forward calculation process. Thus, the solution is unique, and the result is closer to the true value. On the other hand, the SWEMF method only needs the total radiated field data of the array antenna in a small angular domain to ensure that the operation is simple and efficient. The effectiveness of the SWEMF method is successfully verified by examples of low sidelobe planar and linear arrays. The mean square error of the excitation amplitude can reach −38.88 dB. The range of excitation amplitude error is 0.05 v, and the excitation phase error is within 5.2°. This method takes about 60 s to calculate amplitude and phase at any one time. The feed amplitude and phase can be only calculated with the data in a small angular domain, and when the amount of data is small.

A spherical wave expansion for a steerable parametric array loudspeaker using Zernike polynomials.

A steerable parametric array loudspeaker (PAL) can electronically steer highly directional audio beams in the desired direction. The challenge of modelling a steerable PAL is to obtain the audio sound pressure in both near and far fields with a low computational load. To address this issue, an extension of the spherical wave expansion is proposed in this paper. The steerable velocity profile on the radiation surface is expanded as Zernike polynomials which are an orthogonal and form a complete set over a unit circle. An expression for the radiated audio sound is then obtained using a superposition of Zernike modes. Compared to the existing methods, the proposed expansion is computationally efficient and provides a rigorous transformation of the quasilinear solution of the Westervelt equation without paraxial approximations. The proposed expansion is further extended to accommodate local effects by using an algebraic correction to the Westervelt equation. Numerical results for steering single and dual beams are presented and discussed. It is shown that the single beam can be steered in the desired direction in both near and far fields. However, dual beams cannot be well separated in the near field, which cannot be predicted by the existing far field models.

Surface-Wave Minimization Using Spherical Wave Expansion

This letter presents a method for reducing antenna-excited surface waves in mobile devices. Surface waves are canceled either by actively injecting compensating surface waves in opposite phase, or passively creating reactive conditions from which scattered, secondary surface waves cancel the primary waves. We use spherical wave expansion to calculate the amplitudes and phases of the canceling surface waves and show also how reactive loading near the antenna can launch these canceling waves. We verify the proposed method with simulations. Reduced surface waves result in a smoother radiation pattern and significantly increased gain.

EMI Minimization from Stacked Radiation Sources by Means of Multiple Objective Genetic Algorithm

The containment of unwanted electromagnetic radiation and interference is a relevant topic for the system design of any electrical system, and even more for data centers. In this context, the racks hosting piles of servers are one of the main sources of electromagnetic noise. Such unwanted radiation can couple and interact with other computing machines, but also with sensitive electronic devices needed for the management and/or maintenance of the center. The aim of this work is to show how the proper stack-up of the trays in the rack gives rise to a decrease in the unwanted physical electromagnetic radiation. Based on the application of the spherical wave expansion technique, a multi-objective genetic algorithm is developed to evaluate the optimal set of rack configurations that allows for a reduction of the external radiated field. The algorithm and the implemented constraints are described, and the results are discussed.

Scattering of Laguerre-Gauss light beam by a sphere: the angular spectrum decomposition method and a comparison with the localized approximation method

Evaluation of the beam shape coefficients (BSC) in the spherical wave expansion of the incident beam is vital in analyzing light scattering by spherical particles. Different methods have been developed for this purpose. In this work, the formulas of the BSCs for the Laguerre-Gauss beam (LGB) are derived in the angular spectrum decomposition (ASD) method. Under the paraxial condition, these ASD-BSCs are approximated to the expressions in either series or single integrals. Besides, the BSCs for the LGB are also obtained in the localized approximation (LA) method. It is found that the approximate ASD-BSCs are same as those LA-BSCs. Numerical calculation is implemented for verifying the BSC calculations. The validity of the approximate BSC calculation is discussed, together with the distribution of the angular spectrum of the beam. The work offers another example of the shaped beams for mathematically justifying the LA method and suggests another way to discuss the validity of the LA method.

Born approximation of acoustic radiation force used for acoustofluidic separation

Acoustofluidic separation often involves biological targets with specific acoustic impedance similar to that of the host fluid, and with dimensions on the order of the acoustic wavelength. This parameter range, combined with the use of standing waves to separate the targets, lends itself to use of the Born approximation for calculating the acoustic radiation force. Considered here is the configuration analyzed by Peng et al. [J. Mech. Phys. Solids 145, 104134 (2020)], in which two intersecting plane waves radiated into the fluid by a standing surface acoustic wave exert a force on a eukaryotic cell modeled as a multilayered sphere. The angle of intersection is determined by the velocity of the surface wave and the sound speed in the fluid. The acoustic field in this case is a standing wave parallel to the substrate and a traveling wave perpendicular to the substrate. For all parameter values considered by Peng et al., including spheres several wavelengths in diameter, the Born approximation of the acoustic radiation force parallel to the substrate is in good agreement with a full theory based on spherical wave expansions of the incident and scattered fields. [C.A.G. and T.S.J. were supported by ARL:UT McKinney Fellowships in Acoustics.]