Sommerfeld Integrals Research Articles

Efficient computation of the surface fields of a horizontal magnetic dipole located at the air‐ground interface

SummaryIn this paper, the classic problem of determining the fields of a vertically oriented current‐carrying small circular‐loop antenna located near a plane air–earth boundary is revisited. It is well known that the Sommerfeld integrals describing the field components generated by the loop source cannot be exactly evaluated and that their numerical integration is made difficult by the presence of highly oscillatory terms in the integrands. Here, for both source and observation points on the interface, a rigorous method is developed that allows analytical integration of the Sommerfeld integrals, providing fast‐convergent series representations for the surface fields. The obtained formulas permit to overcome all the drawbacks of the previously published approximate solutions to this problem, and to avoid the necessity of using purely numerical approaches when high computational accuracy is required. At the same time, the proposed series representations constitute an analytical benchmark for numerical procedures employed to solve electromagnetic boundary value problems, with applications in antenna design and close‐to‐the‐surface communication. Copyright © 2015 John Wiley & Sons, Ltd.

Performance of Dipole Antenna in Underwater Wireless Sensor Communication

The technology about undersea sensing and networking using electromagnetic wave has become one of the key topics in marine science and engineering. In such applications, the proper antenna types will significantly affect the performance of electromagnetic wave transmission in seawater or across the seawater-to-air interface. Therefore, in this paper, the model of dipole antennas in a three layered conducting media (air-seawater-sea bottom) is built. The electromagnetic fields in air produced by different kinds of undersea dipole antennas are derived. The field expressions in air with the form of Sommerfeld integral are rapidly and efficiently evaluated by the fast Fourier transform method. The results show that the performance of horizontal electric dipole is superior to that of vertical electric dipole, and the performance of magnetic dipole is superior to that of electric dipole. The approach and conclusions in this paper can provide a theoretical basis and technical foundations for the selection of antenna type when the electromagnetic wave is used for data transmission between seawater and air.

The Uniform geometrical Theory of Diffraction for elastodynamics: Plane wave scattering from a half-plane.

Diffraction phenomena studied in electromagnetism, acoustics, and elastodynamics are often modeled using integrals, such as the well-known Sommerfeld integral. The far field asymptotic evaluation of such integrals obtained using the method of steepest descent leads to the classical Geometrical Theory of Diffraction (GTD). It is well known that the method of steepest descent is inapplicable when the integrand's stationary phase point coalesces with its pole, explaining why GTD fails in zones where edge diffracted waves interfere with incident or reflected waves. To overcome this drawback, the Uniform geometrical Theory of Diffraction (UTD) has been developed previously in electromagnetism, based on a ray theory, which is particularly easy to implement. In this paper, UTD is developed for the canonical elastodynamic problem of the scattering of a plane wave by a half-plane. UTD is then compared to another uniform extension of GTD, the Uniform Asymptotic Theory (UAT) of diffraction, based on a more cumbersome ray theory. A good agreement between the two methods is obtained in the far field.

Electromagnetic Scattering of a Dipole Field by a Perfectly Conducting Wedge

The scattering problem of the electromagnetic field generated by an elementary dipole in the presence of a perfectly conducting wedge is here addressed in terms of a Sommerfeld integral which can be expressed by an infinite summation. The analytic solution presented is valid also in the case in which the dipole, the observation point, or both are close to the edge of the wedge. Approximations are also presented, and comparison with the uniform geometrical theory of diffraction (UTD) solution provided.

A Complex Image Reduction Technique Using Genetic Algorithm for the MoM Solution of Half-Space MPIE

The discrete complex image method (DCIM) has been widely used to analytically approximate the Sommerfeld integrals (SIs) of Green's functions in layered media. This method has been greatly improved in accuracy and generality in the past years, but has resulted in a large amount of complex images. It is found that using so many complex images for SI approximation in method of moment (MoM) applications is redundant and inefficient. To reduce the number of complex images, a technique by means of genetic algorithm is proposed and applied to the MoM solution of the half-space mixed-potential integral equation (MPIE).

Analytical Formulas for Dipole Radiation Buried inside Planar Stratified Media

In this paper, analytical formulas have been derived for the electromagnetic fields (EMFs) radiated from a vertical magnetic dipole (VMD) buried in planar stratified media. Three planar-layered conducting media model are adopted: air, seawater and ground. Expressions for the Hertz vector which are used to determine the electric and magnetic fieldcomponents of the dipole in the three planar-layered media are reduced to the integrals obtained by Sommerfeld previously. A simple technique with the aid of the complex image theory is used to obtain highly accurate analytical expressions for Sommerfeld integrals (SI) that arise in these calculations, which earlier were very tedious and complicated to evaluate. Closed-form expressions for the far field in the three regions due to a vertical magnetic dipole buried in second region (sea-water) are calculated and their physical meanings are discussed. The results are numerically evaluated and plotted. In addition, the results obtained in the three regions are compared with each other and with those mentioned elsewhere, and are proven that this method is very effective and simple. The formulas and computations can be appliedto the communication in lower frequencies region. The achieved results will also be useful for remote sensing of the ocean surface, especially when the transmitter is close to the surface.

A novel method for the design of diffractive optical elements based on the Rayleigh–Sommerfeld integral

The original design method for diffractive optical elements (DOEs) is limited to cases of small-angle diffraction due to the Fresnel or Fraunhofer diffraction integral. In this paper, we propose a new method based on the Rayleigh–Sommerfeld diffraction integral, which does not have this limit. In this method, the target intensity distribution is first modified via coordinate transformation and intensity adjustment. Then, the modified Gerchberg–Saxton algorithm is used to achieve the phase distribution of the DOE. To verify this method, simulations and experiments are performed. The results show that the original method is effective only when the full diffraction angle of the DOE is below 25°. Conversely, this method can achieve the target with both small and large diffraction angles.

Transmitted fields of a directional antenna in proximity of a wall

Transmission of a directional antenna in near proximity of a building wall modelled by a semi-finite dielectric slab above a perfectly conducting ground is investigated and an accurate and fast method to predict wall penetration loss (PL) once the source/observation points are in proximity of the wall is proposed. First, the antenna is replaced by an array of electric Hertzian dipoles with approximately the same half power beamwidth. Second, the dipole array radiation is calculated both numerically and asymptotically. They are based on computation of Sommerfeld integrals using Filon quadrature (FQ) technique and the steepest descent method, respectively. Third, PL of a typical x -polarised and y -polarised horn and a z -polarised wire antenna, simulated in FEKO using the method of moments (MoM) is compared with that of, respectively, the x -, y - and z -directed dipole arrays calculated using FQ technique. A good agreement between MoM and the dipole model results are obtained for relatively short stand-off distances for which estimation based on slab transmission coefficient is not an accurate approximation. The proposed dipole array model with FQ technique makes calculation at least 7.5 times faster than FEKO. Fourth, PL for different wall permittivity and conductivity is studied.

P-FFT Accelerated EFIE with a Novel Diagonal Perturbed ILUT Preconditioner for Electromagnetic Scattering by Conducting Objects in Half Space

A highly efficient and robust scheme is proposed for analyzing electromagnetic scattering from electrically large arbitrary shaped conductors in a half space. This scheme is based on the electric field integral equation (EFIE) with a half-space Green’s function. The precorrected fast Fourier transform (p-FFT) is first extended to a half space for general three-dimensional scattering problems. A novel enhanced dual threshold incomplete LU factorization (ILUT) is then constructed as an effective preconditioner to improve the convergence of the half-space EFIE. Inspired by the idea of the improved electric field integral operator (IEFIO), the geometrical-optics current/principle value term of the magnetic field integral equation is used as a physical perturbation to stabilize the traditional ILUT perconditioning matrix. The high accuracy of EFIE is maintained, yet good calculating efficiency comparable to the combined field integral equation (CFIE) can be achieved. Furthermore, this approach can be applied to arbitrary geometrical structures including open surfaces and requires no extra types of Sommerfeld integrals needed in the half-space CFIE. Numerical examples are presented to demonstrate the high performance of the proposed solver among several other approaches in typical half-space problems.

A Robust Hybrid Taguchi-Gradient Optimization Method for the Calculation of Analytical Green’s Functions of Microstrip Structures

A new hybrid technique is presented to find two-dimensional complex images of Green’s functions of microstrip structures. This technique combines the Taguchi algorithm with the BFGS gradient-based method. The analytical Green’s functions are valid over a wide range of frequencies and regions. The superiority of the technique is shown by comparing its results with the rigorous numerical integration of Sommerfeld integrals.

Sommerfeld integration for arbitrary, multi‐layer dielectric systems: a practical method for seawater antenna modelling

We present a formulation of the Sommerfeld integration method to calculate antenna radiation in an arbitrary stack of dielectric slabs. In the past, Sommerfeld integration, in the method of moments, has been used only for two semi-infinite layers (the lower layer often referred to as a groundplane). Here, we have returned to Sommerfeld's original formulation to extend the method for an arbitrary number of layers. When coded, our algorithm gives accurate results as proved by comparison to other simulations and approximate analytical formulas. Our motivation is to calculate antenna coupling in a seawater environment at ultra low frequencies, taking account of seawaves at the seabed and water surface. This is an important problem in marine engineering which cannot be address by finite element methods owing to the high apsect ratios and 106-order jumps in complex permittivity across the different layers. As well as validating the algorithm for multiple layers we show results from a realistic air/sea/seabed setup. We demonstrate that, if the antenna is close to the seabed, surface waves can form on the sea/seabed interface as well as the air/sea interface potentially improving reception.

An Efficient Algorithm of Both Fréchet Derivative and Inversion of MCIL Data in a Deviated Well in a Horizontally Layered TI Formation Based on TLM Modeling

In this paper, we set up an efficient Frechet derivative algorithm of inversion of multicomponent induction logging (MCIL) data in horizontal layered transversely isotropic (TI) formations based on transmission line method (TLM) in order to simultaneously reconstruct the model vector including both horizontal and vertical conductivities, horizontal interfaces, borehole dipping angle, and tool azimuth in deviated well from the MCIL data. First, MCIL responses in the TI model are efficiently determined by both the spectrum EM fields obtained by TLM and the semianalytical computation of Sommerfeld integrals based on the cubic spline interpolation. Then, the Born approximations are executed to derive the MCIL Frechet derivatives as the sixfold integrals of the products of two spectrum EM fields in infinite domains. By using the integral characters of the 2-D Dirac function, the sixfold integrals of Frechet derivatives are further simplified into twofold integrals: One is the Sommerfeld integral in the radial spectrum domain, and the other is the definite integral in the vertical spatial domain. They are efficiently computed by the semianalytical approach similar to the MCIL simulation. Therefore, we obtain the linear equations between changes in the MCIL responses and small perturbations in the model vector. After that, we iteratively modify all of the model parameters to realize the best fit between the input data and the synthetic data of the inverted model by the normalization of the Frechet derivative and singular value decomposition technique. Finally, we apply numerical results to investigate the characteristics of the Frechet derivatives and to validate the inversion method and its antinoise ability.

Photon field and energy flow lines behind a circular disc

In order to evaluate electromagnetic energy flow (EME) lines behind a circular disc illuminated by monochromatic laser light, the Rayleigh–Sommerfeld integral is transformed to a simpler form. EME flow lines, interpreted as photon trajectories, explain the change of the intensity pattern with the distance from a circular disc very well, including the Poisson–Arago spot in the center. Possible applications of these results on the study of Poisson–Arago spot experiments with matter beams, are indicated and explained.

Focal shift in tightly focused Laguerre–Gaussian beams

We study the nonparaxial propagation behavior of Laguerre–Gaussian (LG) beams under the tight focusing condition by using the Rayleigh–Sommerfeld integrals. We obtain an analytical formula and show the focal shift with respect to the geometric focus of a high numerical-aperture objective. An analytical expression for the focal shift of a tightly focused LG beam increases with the focal length while decreases with the beam waist. This approach can be extended to deal with the tight focusing field and the focal shift of LG vector fields with space-variant polarization distributions or other focusing behaviors such as the 4π focusing and the Fresnel zone plates.

High-order fast integration for earth-return impedance between underground and overhead conductors in Matlab

Purpose – The earth-return mutual impedances between underground and overhead conductors can be expressed by Pollaczek integrals. Many efforts have been exerted to calculating this kind of integrals. However, most of numerical methods turn out to be time-consuming as integrands become highly oscillatory and strongly singular. Therefore, efficient algorithms should be devised. The paper aims to discuss these issues. Design/methodology/approach – The paper separates the singularity from the whole integral and couple with the singularity and oscillation, respectively. A sinh transformation is applied for the finite part and complex integration method is used to calculate the tail. Findings – Numerical experiments show that the given method shares the property that the stronger the singularity and the higher the oscillation, the better the accuracy of the calculation. Originality/value – The sinh transformation is first proposed to calculate Pollaczek integrals. This efficient algorithm can be used to evaluate mutual impedances between conductors. Also, it provides a new aspect of the research on fast calculation of Pollaczek integrals and Sommerfeld integrals.

Radiation from a short vertical dipole in a metal-backed rod array

Based on an effective medium approach, we investigate the problem of radiation of a short vertical electric dipole embedded in a metal-backed dielectric rod array. We obtain the radiated electromagnetic field in the form of a Sommerfeld integral, and calculate the near and far electric fields for both lossless and lossy rod arrays. The characteristic equation for the guided modes is derived and solved. The guided modes are found to be slow waves and their propagation constants and modal distributions are extracted. All theoretical results are compared with full-wave simulations.

Beam Diffraction by a Wedge: Exact and Complex Ray Solutions

We explore and compare various solution techniques for the diffraction of a two-dimensional complex-source beam (2D-CSB) by a wedge. Specifically, we calculate the exact total field via the complex-source multipole expansion and via the Sommerfeld integral, and compare the results with the complex ray solution and with nonuniform and uniform diffraction solutions.

High-Efficient Evaluation of the Lightning Electromagnetic Radiation Over a Horizontally Multilayered Conducting Ground With a New Complex Integration Path

An efficient approach to evaluate the radiated electromagnetic field of a lightning channel over a horizontally stratified conducting ground is proposed in this paper. The electromagnetic field component produced by a lightning channel, expressed in the form of generalized Sommerfeld integral (GSI) with a complicated integrand, is calculated along a newly proposed path in the complex plane. Due to the change of the integration path, the integrand in the GSI turns out to decay dramatically fast and without any oscillation at all. As explained in this paper, the larger horizontal distance will become the benefit to achieve a quicker convergence, which is just opposite to the case that the integration path is along the real axis. The validity of the proposed methodology is justified by the extensive comparison with the results in other publications. As a matter of fact, the methodology in this paper can be generalized to evaluate stratified conducting ground with any number of layers.

On the inversion of sediment density profile by the image source method

Image Source Method is a recently developed method for geoacoustic inversion. Under the Born approximation, the reflection of a spherical wave above a stratified seafloor is modeled as the simultaneous emission of image sources, symmetric of the real one relatively to the layers. These image sources are detected and located by the use of Teager-Kaiser Energy Operator (TKEO), which amplifies sudden changes in signal amplitudes, and by a triangulation scheme. Time and arrival angle of the recorded signals coming from the image sources are the input values of the inversion algorithm which gives the thickness and sound speed of each detected sediment layer by the use of Snell-Descartes laws. The objective of the present work is to extend this method to the inversion of the sediment density profile. To this end, one supplementary data is required. Experimental amplitudes corresponding to arrival times of the reflected signals, detected by TKEO, are compared to theoretical ones computed by a numerical evaluation of the Sommerfeld integral. Having first inverted sound speed profile and neglecting absorption coefficient, density is the only remaining parameter and thus can be obtained. The effectiveness of this method is tested on both synthetic and real data.

Regularized inversion of controlled source audio-frequency magnetotelluric data in horizontally layered transversely isotropic media

We present an algorithm for inverting controlled source audio-frequency magnetotelluric (CSAMT) data in horizontally layered transversely isotropic (TI) media. The popular inversion method parameterizes the media into a large number of layers which have fixed thickness and only reconstruct the conductivities (e.g. Occam's inversion), which does not enable the recovery of the sharp interfaces between layers. In this paper, we simultaneously reconstruct all the model parameters, including both the horizontal and vertical conductivities and layer depths. Applying the perturbation principle and the dyadic Green's function in TI media, we derive the analytic expression of Fr?chet derivatives of CSAMT responses with respect to all the model parameters in the form of Sommerfeld integrals. A regularized iterative inversion method is established to simultaneously reconstruct all the model parameters. Numerical results show that the inverse algorithm, including the depths of the layer interfaces, can significantly improve the inverse results. It can not only reconstruct the sharp interfaces between layers, but also can obtain conductivities close to the true value.