Bessel Functions Of Integer Order Research Articles

Real-time tracking of moving objects from scattering matrix in real-world microwave imaging

<abstract><p>The problem of the real-time microwave imaging of small, moving objects from a scattering matrix without diagonal elements, whose elements are measured scattering parameters, is considered herein. An imaging algorithm based on a Kirchhoff migration operated at single frequency is designed, and its mathematical structure is investigated by establishing a relationship with an infinite series of Bessel functions of integer order and antenna configuration. This is based on the application of the Born approximation to the scattering parameters of small objects. The structure explains the reason for the detection of moving objects via a designed imaging function and supplies some of its properties. To demonstrate the strengths and weaknesses of the proposed algorithm, various simulations with real-data are conducted.</p></abstract>

A novel study on the orthogonality sampling method in microwave imaging without background information

Orthogonality sampling method (OSM) is a promising non-iterative technique for identifying small anomalies in microwave imaging. For a successful application, complete and accurate values of permittivity and conductivity of the background must be known. If one of values is unknown, inaccurate location and shape of anomaly will be retrieved by using the OSM. This phenomenon was examined through the simulation results but relevant theoretical result has not been established. Motivated by this, we investigate the structure of the indicator function of the OSM by establishing a relationship with an infinite series of the Bessel functions of integer order and applied values of permittivity, permeability, and conductivity. Revealed structure explains the theoretical reason why inaccurate location and shape of anomaly is retrieved through the OSM with inaccurate values of permittivity and conductivity. Various results of numerical simulations with synthetic data are illustrated to support the theoretical result.

Double and Square Bessel-Gaussian Beams.

We obtain a transform that relates the standard Bessel-Gaussian (BG) beams with BG beams described by the Bessel function of a half-integer order and quadratic radial dependence in the argument. We also study square vortex BG beams, described by the square of the Bessel function, and the products of two vortex BG beams (double-BG beams), described by a product of two different integer-order Bessel functions. To describe the propagation of these beams in free space, we derive expressions as series of products of three Bessel functions. In addition, a vortex-free power-function BG beam of the mth order is obtained, which upon propagation in free space becomes a finite superposition of similar vortex-free power-function BG beams of the orders from 0 to m. Extending the set of finite-energy vortex beams with an orbital angular momentum is useful in searching for stable light beams for probing the turbulent atmosphere and for wireless optical communications. Such beams can be used in micromachines for controlling the movements of particles simultaneously along several light rings.

On the Application of Orthogonality Sampling Method for Object Detection in Microwave Imaging

We consider a further development of the orthogonality sampling method (OSM) for a fast localization of small object in microwave imaging (MI). In contrast to the scenarios considered in traditional approaches, if the location of the transmitter and the receiver is the same, it is very hard to measure the scattering parameter data or to distinguish the weak scattered signal from the relatively high antenna reflection. Here, we set the unknown measurement data as a constant and design an indicator function for the OSM. To demonstrate the applicability of the OSM and its dependence on the constant, we show that the designed indicator function can be represented in terms of an infinite series of the Bessel functions of integer order, antenna configuration, and applied constant. To improve the imaging performance for a proper identification of the object, we design a new indicator function of the OSM with multiple sources, rigorously explore its mathematical structure, and discover some properties including the improvement and uniqueness. Simulation results with synthetic and real data are exhibited to support the theoretical results and to illustrate the pros and cons of the designed OSM.

A novel study on the MUSIC-type imaging of small electromagnetic inhomogeneities in the limited-aperture inverse scattering problem

We apply MUltiple SIgnal Classification (MUSIC) algorithm for the location reconstruction of a set of two-dimensional circle-like small inhomogeneities in the limited-aperture inverse scattering problem. Compared with the full- or limited-view inverse scattering problem, the collected multi-static response (MSR) matrix is no more symmetric (thus not Hermitian), and therefore, it is difficult to define the projection operator onto the noise subspace through the traditional approach. With the help of an asymptotic expansion formula in the presence of small inhomogeneities and the structure of the MSR-matrix singular vector associated with nonzero singular values, we define an alternative projection operator onto the noise subspace and the corresponding MUSIC imaging function. To demonstrate the feasibility of the designed MUSIC, we show that the imaging function can be expressed by an infinite series of integer-order Bessel functions of the first kind and the range of incident and observation directions. Furthermore, we identify that the main factors of the imaging function for the permittivity and permeability contrast cases are the Bessel function of order zero and one, respectively. This further implies that the imaging performance significantly depends on the range of incident and observation directions; peaks of large magnitudes appear at the location of inhomogeneities for permittivity contrast case, and for the permeability contrast case, peaks of large magnitudes appear at the location of inhomogeneities when the range of such directions is narrow, while two peaks of large magnitudes appear in the neighborhood of the location of inhomogeneities when the range is wide enough. The numerical simulation results via noise-corrupted synthetic data also show that the designed MUSIC algorithm can address both permittivity and permeability contrast cases.

Real-time detection of small anomaly from limited-aperture measurements in real-world microwave imaging

This study concerns a subspace migration technique used to identify the shape and location of an unknown anomaly from scattered parameter data collected within the scattering matrix for a limited-aperture inverse scattering problem. The mathematical theories of subspace migration are partially understood in terms of limited-aperture inverse scattering problems, but little investigation of real-world applications, such as microwave imaging, has taken place. Hence, we perform further research to analyze the subspace migration and explain the reasons for a number of unexplained phenomena. For theoretical corroboration, we prove that the imaging function of subspace migration is composed of an infinite series of integer-order Bessel functions of the first kind, and we outline the arrangement and the total number of antennas for transmitting and receiving signals. This is based on the application of the Born approximation to the identification of an anomaly and the uniform convergence of the Jacobi–Anger expansion formula. Various results of numerical simulations with synthetic and real data are exhibited to support the theoretical results of the imaging function and explain the reasons for a number of phenomena as well as the fundamental limitations. Further, we suggest a method of antenna arrangement to obtain better results and confirm the improvements through simulations.

Computational Study of Quantum Coherence from Classical Nonlinear Compton Scattering with Strong Fields

From the covariant formulation of radiation intensity of Hartemann-Kerman model entirely constructed in the classical electrodynamics scenario, a formulation of coherent states has been obtained in an explicit manner represented by the infinite sum of integer-order Bessel functions. Both linear and nonlinear Compton scattering are included, suggesting that Compton processes can be perceived as coherent states of light-matter interaction.

A study on the orthogonality sampling method corresponding to the observation directions configuration

Several theoretical and experimental studies have found that the orthogonality sampling method (OSM) is an effective, simple, and stable non-iterative technique for identifying the location of a set of well-separated small inhomogeneities in inverse scattering problems. However, most studies have been conducted assuming a uniform arrangement of the observation directions and a sufficiently large total number of such directions. The effect of the total number and the arrangement of the observation directions has not been studied. Therefore, this study investigates the effect of the configuration of the observation direction to identify the location of small inhomogeneities using OSM for the first time. We examine this interpretation using the mathematical structure that the indicator function of OSM can be expressed as an infinite series of the Bessel function of integer order and configuration of observation directions. We also demonstrate various simulation results with noisy data to support the constructed interpretation.

Thin-shell approximation of Mie theory for a thin anisotropic layer spaced away from a spherical core: Application to dye-coated nanostructures

We here develop a thin-shell approximation of the Mie scattering problem for a spherical multilayer structure consisting of central core, a spacer layer, and a thin layer with radially anisotropic dielectric function. This thin layer can, for example, represent a uniform layer of adsorbed dyes at a fixed distance from a spherical nanoparticle, with an effective anisotropic dielectric tensor to account for dye-orientation effects. This geometry with the spacer layer was recently shown to be necessary to precisely account for all electromagnetic effects in such systems [C. Tang, B. Augui\'e, and E. C. Le Ru, Phys. Rev. B 103, 085436 (2021)]. The Mie theory solution involves Bessel functions of complex order, which are not commonly available in many numerical calculations. We show that this hurdle is overcome in the thin-shell approximation, where the solution is of similar complexity to that of isotropic Mie theory with only spherical Bessel functions of integer order. We also apply this approximation to the calculation of the optical absorption in each individual layer. Using the dye-on-metal-nanoparticle system as illustration, we show that the thin-shell predictions agree extremely well with the full solution for experimentally relevant parameters, and can therefore be used instead. This speeds up the numerical implementation and will simplify further analytical developments.

Electric manifestations and social implications of bacteria aggregation from the Bessel-Keller-Segel equation

This paper proposes a fair extension of Keller-Segel equation based in the argument that bacteria exhibit proporties electric in their composition. The new mathematical form of this extension involves the integer-order Bessel functions. With this one can go through the electrodynamics of the representative scenarios in order to understand the social behavior of bacteria. From the theoretical side this paper demonstrates that, charged electrically, aggregation of bacteria would give rise to electric currents that hypothetically are the reasons for social organization and disruption among them. The electrical properties of bacteria from this mathematical proposal might be relevant in a prospective implementation of so-called Internet of Bio-Nano Things network, that aims to be characterized for having a very high signal/noise.

Accurate identification of multiple anomalies in microwave imaging via direct sampling method with multiple sources

Throughout various experimental results, it has been revealed that the direct sampling method (DSM) can be applied to microwave imaging and inverse problem for a fast identification of a small anomaly. However, in microwave imaging, it is very hard to identify multiple anomalies via DSM with a single source because the imaging performance is highly dependent on the location of source. In this paper, we design an improved indicator function of DSM with multiple sources for the successful identification of multiple anomalies from measured scattering parameter data. In order to show the feasibility of designed DSM, we investigate a mathematical theory of DSM by establishing a relationship with an infinite series of Bessel function of integer order and antennas setting. This is based on the representation formula of scattering parameters in the presence of small anomalies and the application of Born approximation. Simulation results with synthetic and real data are exhibited to assess the effectiveness of the designed DSM.

Theoretical Identification of Coupling Effect and Performance Analysis of Single-Source Direct Sampling Method

Although the direct sampling method (DSM) has demonstrated its feasibility in identifying small anomalies from measured scattering parameter data in microwave imaging, inaccurate imaging results that cannot be explained by conventional research approaches have often emerged. It has been heuristically identified that the reason for this phenomenon is due to the coupling effect between the antenna and dipole antennas, but related mathematical theory has not been investigated satisfactorily yet. The main purpose of this contribution is to explain the theoretical elucidation of such a phenomenon and to design an improved DSM for successful application to microwave imaging. For this, we first survey traditional DSM and design an improved DSM, which is based on the fact that the measured scattering parameter is influenced by both the anomaly and the antennas. We then establish a new mathematical theory of both the traditional and the designed indicator functions of DSM by constructing a relationship between the antenna arrangement and an infinite series of Bessel functions of integer order of the first kind. On the basis of the theoretical results, we discover various factors that influence the imaging performance of traditional DSM and explain why the designed indicator function successfully improves the traditional one. Several numerical experiments with synthetic data support the established theoretical results and illustrate the pros and cons of traditional and designed DSMs.

Application of MUSIC algorithm in real-world microwave imaging of unknown anomalies from scattering matrix

In this contribution, we consider MUltiple SIgnal Classification (MUSIC)-type algorithm for a non-iterative microwave imaging of small and arbitrary shaped extended anomalies located in a homogeneous media from scattering matrix whose elements are scattering parameters measured at dipole antennas. In order to explain the feasibility of MUSIC in microwave imaging, we investigate mathematical structure of MUSIC by establishing a relationship with an infinite series of Bessel function of integer order and antennas setting. This is based on the representation formula of scattering parameters in the presence of small anomalies and the application of Born approximation. Simulation results using real-data at f=925 MHz of angular frequency are exhibited to show the feasibility of designed algorithm and to support investigated structure of imaging function.

Direct sampling method for identifying magnetic inhomogeneities in limited-aperture inverse scattering problem

The direct sampling method (DSM) in limited-aperture inverse scattering problems for transverse electric (TE) polarization is considered. Based on the asymptotic expansion formula in the presence of small targets, we demonstrate that the indicator function of DSM can be represented by an infinite series of Bessel functions of integer order and correspondingly examine various properties of DSM. Simulation results with noisy data are created to support theoretical results. Based on the identified structure of the indicator function, we design a new indicator function of DSM, analyze its mathematical structure, and generate simulation results to demonstrate its effectiveness and improvement.

Localization of Small Anomalies via the Orthogonality Sampling Method from Scattering Parameters

We investigate the application of the orthogonality sampling method (OSM) in microwave imaging for a fast localization of small anomalies from measured scattering parameters. For this purpose, we design an indicator function of OSM defined on a Lebesgue space to test the orthogonality relation between the Hankel function and the scattering parameters. This is based on an application of the Born approximation and the integral equation formula for scattering parameters in the presence of a small anomaly. We then prove that the indicator function consists of a combination of an infinite series of Bessel functions of integer order, an antenna configuration, and material properties. Simulation results with synthetic data are presented to show the feasibility and limitations of designed OSM.

Application and analysis of direct sampling method in real-world microwave imaging

In this paper, a direct sampling method (DSM) is designed for a real-time detection of small anomalies from scattering parameters measured by a small number of dipole antennas. Applicability of the DSM is theoretically demonstrated by proving that its indicator function can be represented in terms of an infinite series of Bessel functions of integer order, Hankel function of order zero, and the antenna configurations. Experiments using real-data then demonstrate both the effectiveness and limitations of this method.

Real-time microwave imaging of unknown anomalies via scattering matrix

We consider an inverse scattering problem to identify the locations or shapes of unknown anomalies from scattering parameter data collected by a small number of dipole antennas. Most of researches does not considered the influence of dipole antennas but in the experimental simulation, they are significantly affect to the identification of anomalies. Moreover, opposite to the theoretical results, it is impossible to handle scattering parameter data when the locations of the transducer and receiver are the same in real-world application. Motivated by this, we design an imaging function with and without diagonal elements of the so-called scattering matrix. This concept is based on the Born approximation and the physical interpretation of the measurement data when the locations of the transducer and receiver are the same and different. We carefully explore the mathematical structures of traditional and proposed imaging functions by finding relationships with the infinite series of Bessel functions of integer order. The explored structures reveal certain properties of imaging functions and show why the proposed method is better than the traditional approach. We present the experimental results for small and extended anomalies using synthetic and real data at several angular frequencies to demonstrate the effectiveness of our technique.

Direct sampling method for anomaly imaging from scattering parameter

In this paper, we develop a fast imaging technique for small anomalies located in homogeneous media from S-parameter data measured at dipole antennas. Based on the representation of S-parameters when an anomaly exists, we design a direct sampling method (DSM) for imaging an anomaly and establishing a relationship between the indicator function of DSM and an infinite series of Bessel functions of integer order. Simulation results using synthetic data at f=1GHz of angular frequency are illustrated to support the identified structure of the indicator function.

Uniform analytic approximation of Wigner rotation matrices

We derive the leading asymptotic approximation, for low angle {\theta}, of the Wigner rotation matrix elements $d^j_{m_1m_2}(\theta)$, uniform in $j,m_1$ and $m_2$. The result is in terms of a Bessel function of integer order. We numerically investigate the error for a variety of cases and find that the approximation can be useful over a significant range of angles. This approximation has application in the partial wave analysis of wavepacket scattering.

Topological Derivative for Imaging of Thin Electromagnetic Inhomogeneity: Least Condition of Incident Directions

It is well-known that using topological derivative is an effective noniterative technique for imaging of crack-like electromagnetic inhomogeneity with small thickness when small number of incident directions are applied. However, there is no theoretical investigation about the configuration of the range of incident directions. In this paper, we carefully explore the mathematical structure of topological derivative imaging functional by establishing a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on this, we identify the condition of the range of incident directions and it is highly depending on the shape of unknown defect. Results of numerical simulations with noisy data support our identification.