Exact Image Theory Research Articles

Comments on “Transient Magnetic Shielding of a Planar Conductive Thin Screen via Exact Image Theory”

Comments on “Transient Magnetic Shielding of a Planar Conductive Thin Screen via Exact Image Theory”

Reply to “Comments on Transient Magnetic Shielding of a Planar Conductive Thin Screen Via Exact Image Theory”

Reply to “Comments on Transient Magnetic Shielding of a Planar Conductive Thin Screen Via Exact Image Theory”

Finite Range Integral Representation of Sommerfeld Integral for Homogeneous Dielectric Half-Space and Complete Uniform Asymptotic Expansion

Since the existing integral representations of the Sommerfeld integral (SI) are of the (semi) infinite type, a numerical error is inevitable due to the truncation of the integral range. In this article, a new finite range integral representation of the SI is formulated based on the exact image theory. To derive the new representation, a finite range integral expression of the exact image current is first derived. The radiated field by the exact image current is expressed in terms of a double integral in the complex domain where the exact image current flows. One of the two integrals is a semiinfinite line integral along the current path. This integral is deformed into a more convenient form as the exact image representation for the impedance plane. To convert the line source in the complex domain into a line source in the real domain, the path-deformation technique is again applied. The final line source consists of three line segments along which the exact image current flows vertically from the image source point to ±∞. In addition to the line source, a lateral wave component is also obtained. The impedance exact image representation can be evaluated as a closed-form expression in terms of an incomplete cylindrical function. Thus, the SI can be represented as the finite range integral of the incomplete cylindrical function. To efficiently evaluate the proposed integral, a complete uniform asymptotic expansion is formulated. All the proposed formulations are numerically verified, including, in some cases, near-Earth propagation, and their behaviors are investigated. Also, the electric field is computed for a vertical and horizontal infinitesimal dipole and compared by the exact Sommerfeld and known approximate formulations.

Analysis of numerical solutions to Sommerfeld integral relation of the half-space radiator problem

Sommerfeld integrals relate a spherical wave from a point source to a convolution set of plane and cylindrical waves. This relation does not have analytical solutions but it submits to a solution by numerical integration. Among others, it is significant for theoretical studies of many optical and radiation phenomena involving surfaces. This approach is preferred over discretized computational models of the surface because of the many orders of increased computations involved in the latter. One of the most widely used and accurate methods to compute these solutions is the numerical integration of the Sommerfeld integrand over a complex contour. We have analyzed the numerical advantages offered by this method, and have justified the optimality of the preferred contour of integration and the choice of two eigenfunctions used. In addition to this, we have also analyzed four other approximate methods to compute the Sommerfeld integral and have identified their regions of validity, and numerical advantages, if any. These include the high relative permittivity approximation, the short distance approximation, the exact image theory and Fourier expansion of the reflection coefficient. We also finally compare these five methods in terms of their computational cost.

Fast Frequency-Domain Modeling of Return Stroke Including Influence of Lossy Ground

A constitutive integral equation (CIE) formulation is proposed to model the lightning return stroke. From the CIE, the electromagnetic field radiated by the return stroke considering the lossy ground is determined by using the moment method (MM). To implement a fast numerical and rigorous approach, an exact image theory is first employed to accelerate the computation of Sommerfeld integral, and a translational mesh is then constructed under the thin-wire approximation. Such grid representation and the convolution properties of the integral kernel equip the fast Fourier transform to have the ability to efficiently solve the MM matrix equations. The feasibility and merits of the proposed method are positively confirmed by the comparison of its performances with those of an existing one on a case study.

Exact image theory for a three layer medium

In this work an exact formulation and solution for the image source amplitude in a three layer medium is given. To do so a novel generalization of the Sommerfeld integral is employed. Of particular interest in this paper is the time-domain solution for the impulse response between source and observation points in the middle layer. It is shown that global boundary conditions can be accommodated for multiple reflections from media of having infinite extent. A branch integral formulation of inverse Laplace transform of integral powers of the Fresnel reflection coefficient is given for this case. It is shown that only odd-powers of the branch point argument contribute thereby reducing computational effort required to numerically evaluate the impulse response.

Exact image theory for the slab problem

Exact image theory, recently introduced for the exact solution of problems involving homogeneous half spaces and microstrip-like geometries, is developed here for the problem of homogeneous slab of isotropic dielectric and/or magnetic material in free space. Expressions for image sources, creating the exact reflected and transmitted fields, are given and their numerical evaluation is demonstrated. Nonradiating modes, guided by the slab and responsible for the loss of convergence of the image functions, are considered and extracted. The theory can be applied, for example, in an analysis of finite ground planes in microstrip antenna structures.

A generalization of image theory to predict the interaction of multipole fields with plane surfaces

A derivation and computational scheme, based on exact image theory, for the field produced by the interaction of an outgoing vector wave harmonic with an infinite-extent plane surface is presented. The method represents the angular-dependent Fresnel reflection coefficients of the surface as Laplace transforms of a spatially dependent function, which results in the reflected field appearing as a superposition of image sources located at complex points along the normal axis within the surface medium. Exact, analytical formulas are given for the transformed reflection coefficients for arbitrary surface refractive index, and an efficient computation scheme for evaluation of the scattered field coupling between a particle and the surface is presented.

ELECTROMAGNETIC SOURCE EQUIVALENCE AND EXTENSION OF THE COMPLEX IMAGE METHOD FOR GEOPHYSICAL APPLICATIONS

In this work, source equivalence and computation of the re∞ected (induced) electromagnetic fleld in geophysical situations are studied. It is shown that the application of Huygens' principle allows for full generalization of Fukushima's equivalence theorem that applies only for magnetic fleld. The source equivalence is revisited for a vertical line current element, and it is shown that the equivalent charge required to replace the original source by a planar equivalent source together with the surface charge associated with the re∞ected fleld generates a purely vertical total electric fleld on the ground. Consequently, if the magnetic fleld and horizontal components of the total electric fleld on the ground are of interest, only equivalent currents need to be considered. The classical Complex Image Method (CIM) is derived from the exact image theory for planar impedance surfaces. The classical CIM is extended by considering a divergence-free source current that may have components also perpendicular to the ground plane. The extension is seen to generate a complex image charge not present in the classical CIM. Further, a generalized application of the extended CIM to geophysical situations having divergence-free volume source currents is introduced. The application involves decomposition of the source into linear current elements and rotations, translations and re∞ections of the electromagnetic fleld expressions associated with each element. The validity of the new approach is verifled for an

Application of the exact image theory to studies of ground effects of space weather

Summary An application of the electromagnetic exact image theory (EIT) for the calculation of electric and magnetic fields at the Earth's surface produced by an ionospheric line current with a non-zero divergence is presented. The complete EIT requires a large amount of computation, which can be reduced by using the two low-frequency approximations proposed here. Some examples are shown, with exact reference data obtained by using the Fast Hankel Transform for numerical computations. Limitations of the presented EIT method are discussed, e.g. the loss of accuracy of the other approximation at lower frequencies and the error in the horizontal electric field strength value. The EIT-based approximation is related to the well-known Wait's complex image method (CIM), but can be applied to currents with non-zero divergence, too, as demonstrated in this paper.

Efficient calculation of the fields of a dipole radiating above an impedance surface

In this paper, the classic problem of field computation for an infinitesimal dipole radiating above an impedance half-space is revisited. The expressions for the traditional solution consist of integrals of the Sommerfeld type that cannot be evaluated in closed form and due to their highly oscillatory nature are difficult to evaluate numerically. A method known as exact image theory, which has previously been applied to vertical electric and magnetic dipoles, is used to derive explicit expressions for dipoles of arbitrary orientation above impedance surfaces. Starting from the spectral representation of the field, the reflection coefficients are cast in the form of exact Laplace transforms and then by changing the order of integrations field expressions in terms of rapidly converging integrals are obtained. These expressions are exact, and valid for any arbitrary source alignment or observation position. It is shown that the formulation for a horizontal dipole contains an image in the conjugate complex plane resulting in a diverging exponential term not previously addressed in the literature. It is shown through further mathematical manipulations, that the diverging term is a contribution of the mirror image which can be extracted. Comparison of numerical results from exact image theory and the original Sommerfeld-type expressions shows good agreement as well as a speedup in computation time of many orders of magnitude, which depends on the distance between the transmitter and the receiver. This formulation can effectively replace the approximate asymptotic expressions used for predicting wave propagation over a smooth planar ground (having different regions of validity). The exact image formulation is also of practical use in evaluation of the Green's function for various applications in scattering problems where approximate solutions are not sufficient.

A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces

An efficient numerical solution for the scattering problem of inhomogeneous dielectric rough surfaces is presented. The inhomogeneous dielectric random surface represents a bare soil surface and is considered to be comprised of a large number of randomly positioned dielectric humps of different sizes, shapes, and dielectric constants above an impedance surface. Clods with nonuniform moisture content and rocks are modeled by inhomogeneous dielectric humps and the underlying smooth wet soil surface is modeled by an impedance surface. In this technique, an efficient numerical solution for the constituent dielectric humps over an impedance surface is obtained using Green's function derived by the exact image theory in conjunction with the method of moments. The scattered field from a sample of the rough surface is obtained by summing the scattered fields from all the individual humps of the surface coherently ignoring the effect of multiple scattering between the humps. The statistical behavior of the scattering coefficient /spl sigma//spl deg/ is obtained from the calculation of scattered fields of many different realizations of the surface. Numerical results are presented for several different roughnesses and dielectric constants of the random surfaces. The numerical technique is verified by comparing the numerical solution with the solution based on the small perturbation method and the physical optics model for homogeneous rough surfaces. This technique can be used to study the behavior of scattering coefficient and phase difference statistics of rough soil surfaces for which no analytical solution exists.

Exact image theory for vertical electromagnetic sources above a slightly chiral half space

Exact Image Theory, previously introduced to electromagnetic field problems involving arbitrary sources and planar interfaces of simple isotropic media, is extended to vertical electric and/or magnetic sources above a planar interface of a chiral medium with small chirality. The resulting image functions are shown to be generalizations of those encountered in the simple isotropic case and coincide with previously obtained quasistatic results. Far-field approximation of the image method is shown to lead to a simple reflection coefficient image principle.

Scattering by a small object close to an interface III Buried object

Exact-image theory is applied to the problem of electromagnetic-wave scattering from a small object below the interface separating two isotropic and homogeneous media. The object is assumed to be electrically small so that it can be approximated by an electric dipole with known dyadic polarizability. Scattering from a buried small dielectric sphere is considered as an example of the theory.

Exact Image Theory for Fields Reflected from Bi-Isotropic Impedance Surfaces

Conditions for replacing a bi-isotropic half-space by an impedance surface are formulated. The most general bi-isotropic surface impedance is a dyadic containing an antisymmetric part, which is non-null for nonreciprocal media. To formulate the general radiation problem in front of the bi-isotropic surface, the exact image theory has been extended. The image theory is verified by known special cases and demonstrated by an example of a vertical electric dipole in front of the surface.

Scattering from dielectric structures above impedance surfaces and resistive sheets

Interest in understanding of electromagnetic interaction with rough surfaces has prompted the study of scattering from typical dielectric humps over impedance surfaces. It is shown that the Green's function of the problem for a resistive sheet resembles that of the impedance surface. Hence both problems are considered here. A numerical solution for the scattered field of a two-dimensional dielectric object, possibly inhomogeneous, with arbitrary cross section above the impedance surface or resistive sheet is sought. First the Green's function of the problem is derived based on the exact image theory. This form of the Green's function is amenable to numerical computation. Then the induced polarization currents are calculated by casting the integral equations into a matrix equation via the method of moments. Numerical problems in calculation of the Green's function when both source and observation points are close to the surface are discussed. Comparison of numerical results with a perturbation solution shows excellent agreement between the two methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Time-domain study of arbitrary dipole in planar geometry with discontinuity in permittivity and permeability

Transient fields arising from a general impulsive dipole above ground characterized by permittivity and permeability are studied. A detailed analysis for arbitrary electric dipole excitation is given, and the fields arising from the analog magnetic sources can be obtained by applying the duality transform. The mathematical description of this problem is done using the Hertz vector, whose solution in the time domain is derived in a novel way using the exact image theory. From this solution, which involves a definite integral, the actual fields are obtained through simple differentiations. The efficiency of the method is demonstrated by solving 14 previously unknown closed-form results for various field components in certain specific cases. Seven of these are obtained for electric dipole excitation and listed, whereas the corresponding results in the case of magnetic excitation can be easily written by applying the duality transform. >

Analysis of the probe-fed annular microstrip antenna for hyperthermic applicators

The exact image theory is applied to obtain the exact field distribution of an annular microstrip radiator over a two layered biological medium. First, the image functions for the three layer structure (substrate-skin-fat) are derived by expanding the Fourier domain reflection coefficient into a sum of pole terms, known also as the Mittag-Leffler expansion. Secondly, the current distribution on the strip is solved by using the Galerkin’s method with continuous trigonometrical basis functions. In order to reduce the focus of the antenna, the dependence of the focus size on the radius of the antenna is calculated.

Scattering by a small object close to an interface I Exact-image theory formulation

Exact-image theory is applied to the problem of electromagnetic wave scattering from a small dielectric object above an interface separating two isotropic and homogeneous media. The object is assumed to be electrically small and far enough from the interface so that its internal field can be assumed to be uniform. The approach is applicable to any scatter that can be represented by an electric dipole.

Scattering by a small object close to an interface II Study of backscattering

Scattering by a small object located close to an interface is analyzed according to the exact-image theory formulation. The scatterer is assumed to be small compared with wavelength, permitting the electric-dipole approximation, and to have a scalar polarizability. After the derivation of the dipole moment, investigations concentrate on far-field scattering. Backscattering enhancement and reversal of linear polarization are confirmed through statistical averaging over scatterer height and system orientation.