Rigorous Integral Equation Research Articles

The Scattering of Electromagnetic Waves By Periodic Magnetodielectric Structures With Arbitrary Profiles and Inhomogeneous Media

The scattering of electromagnetic waves by periodic magnetodielectric structures with arbitrary profiles and inhomogeneous distribution of permittivity and permeability over a period is analyzed using rigorous integral equations of macroscopic electrodynamics and partitioning a structure into thin layers. The solution is found for E-polarized and H-polarized plane waves. Numerous examples of the problem solutions are compared with previously published results. Examples of approximation perfectly conducting periodic structures by magnetodielectric structures with the same geometry of a period are given. For illustration of the presented method effectiveness the analysis applies to a grating composed of circular magnetodielectric coaxial cylinders.

The use of shorting posts to improve the scanning range of probe-fed microstrip patch phased arrays

This paper presents a thorough investigation of the use of shorting posts to improve the scanning potential of microstrip probe-fed patch arrays. A rigorous spectral domain integral equation (SDIE) technique is used to analyze the scan impedance and related radiation characteristics of several proposed shorting post-patch array configurations. It is shown that by using shorting posts, potential scan blindnesses can be removed and the useful scanning range of the array can be significantly extended beyond 80/spl deg/ in the plane of interest. Comparisons of the scan performance of several microstrip patch arrays containing up to three shorting posts are presented. Novel hybrid shorting post configurations are also proposed and examined, which provide excellent scanning potential in the three principal planes (E-, H-, and D-planes). These modified microstrip patches incorporate switching diodes and provide minimal scan impedance variation for scan angles near 80/spl deg/ in all three planes, as well as very low cross-polarization levels.

An integrated quasi-planar leaky-wave antenna

A new quasi-planar leaky-wave antenna is presented. It consists of microstrip line on one side of the substrate and uniplanar circuit on the other side placed in a partially opened waveguide. The leakage is produced by the excitation of the first higher order (odd) microstrip mode coupled electromagnetically through a slotline on the opposite side of the substrate. Theoretic results based on rigorous Green's impedance integral equation method show that the new microstrip-slotline-coupled leaky-wave antenna has a broadband tuning range via structure parameters and is insensitive to the microstrip line width variation. Measured relative power absorbed (RPA) results indicate that the useful frequency bandwidth agrees with that predicted by rigorous field theory. The measured antenna radiation patterns also agree very well with the approximate theoretic computations. The theory and experiments show that the proposed leaky-wave antenna can interface to feeding structure easily and directly. The new antenna may become a good candidate for microwave and millimeter-wave integrated antenna design.

Polymers in Random Porous Materials: Structure, Thermodynamics, and Concentration Effects

The distribution of dissolved macromolecules between a bulk solution and the interior of a porous substrate occurs in a variety of technologically important systems. The partition coefficient K, which is the ratio of concentration in the pores to that in the bulk, is primarily determined by the effective size of the polymer molecules. For dilute solutions, K decreases rapidly as the effective polymer dimensions exceed the average pore size. We have used liquid-state theory to construct a rigorous integral equation model for the structure of concentrated flexible linear polymers in the presence of a rigid matrix of discrete repulsive obstacles. We have used the structural information to calculate the thermodynamic properties of the polymer in these model porous materials. In particular, we have been able to calculate the partition coefficient as a function of concentration. The model provides good agreement with thermodynamic properties obtained from previous computer simulations of bulk polymer solutions as well as with our own simulations of polymers in porous materials.

The scattering of electromagnetic waves by a periodic magneto dielectric layer

The scattering of electromagnetic waves by a periodic magnetodielectric layer is solved by the new method based on the rigorous integral equations of macroscopic electrodynamics. The general moments of the solution are given. Also the computer program for the numerical solution of the proposed scattering problem is developed. Taking into account permeability along with permittivity makes it possible to investigate the scattering by magnetodielectric structures approximating the structures made of perfect metal. There is an example of such approximation in the paper. The influence of modulation of magnetodielectric layer permittivity is displayed as an example.

The scattering of electromagnetic waves by two-dimensional gratings with magnetodielectric rods

The scattering of electromagnetic waves by two-dimensional grating with magnetodielectric rods is solved by the new method based on the rigorous integral equations of macroscopic electrodynamics. This method is considered in detail. The computer program for the numerical solution of the proposed scattering problem is created. The plots of the transmission coefficient for the gratings with dielectric, magnetodielectric, metallic rods are given as examples.

Reflection from a corrugated surface revisited

The problem of scattering of a plane sonic wave from a soft surface with periodic (sinusoidal) unevenness along one direction is examined by means of the Rayleigh plane-wave expansion and the Waterman extinction methods, numerically implemented by Fourier projection and expansion, respectively. The computations are done with real, double-precision, stochastic arithmetic instead of the usual complex, double-precision floating-point arithmetic in order to precisely evaluate the numerical accuracy of the results conditioned by round-off errors. It is shown that the low-order plane-wave coefficients obtained by the Rayleigh and Waterman methods are identical when obtained from matrix systems that are large enough to give ‘‘convergence’’ of these coefficients. For the same matrix size, the higher-order coefficients differ the higher the diffraction order. It is also shown that the Waterman (Fourier-series) computation of the near field is generally meaningful, whereas that of Rayleigh, involving summation of the plane waves is generally meaningless except near the points of the scattering surface first encountered by the incident wave (i.e., those in the valleys when the incident wave comes from below). Low-order plane-wave scattering coefficients, with at least two-to-three-digit accuracy, and which are identical (to this precision) to the plane-wave coefficients computed by the rigorous integral equation method, are obtained by both the Rayleigh and Waterman methods for scattering surfaces with slopes as large as 2.26 when the number of nonevanescent waves is 5. The number of significant digits decreases as the slope increases.

Stacked superconductor structure for microwave transmission

Propagation characteristics of microwaves along a stacked superconductor microstrip structure are investigated. The suitability of such a structure for practical applications is discussed. A rigorous integral equation formulation has been used for the analysis of such structure. The effect of thickness of superconducting strip on propagation characteristics has been studied. Results presented show that the slow wave propagation and low dispersive coupling are possible in a realistic stacked multi-dielectric and superconducting layers microstrip line package.

A general treatment of matched terminations using integral equations modeling and applications

This paper presents an original approach to simulate a broadband matched load in planar structures. Theoretically, the formulation appears as an additional boundary condition in the rigorous integral equations technique, and results in a partial rearrangement of the generalized impedance matrix in the moment method resolution. Practically, it enables the whole characterization of any planar multiport discontinuity in a procedure particularly realistic in regards to experimental measurement procedures. This new refinement is demonstrated to provide a versatile and powerful tool. Some typical applications are given which illustrate its numerous capabilities. The analysis of structures as different as a shielded microstrip step and an active receiving microstrip antenna is presented and successfully compared to experiments. >

FINITE-ELEMENT FORMULATIONS FOR ACCURATE CALCULATION OF RADIANT HEAT TRANSFER IN DIFFUSE-GRAY ENCLOSURES

Accurate formulations based on finite-element methods are developed for calculating radiant heat transfer among diffuse-gray surfaces in an enclosure. The method of Gebhart is compared with solutions of the rigorous integral equations using these formulations. A Swartz-Wendroff approximation applied to the finite-element discretizations yields higher-order-accurate analogs of view factors that depend on geometry only and can be readily employed for more accurate solution of general radiant heat transfer problems. The accuracy and efficiency of these approaches are presented in the context of a model problem representative of a high-temperature crystal growth system.

Identification of propagation regimes on integrated microstrip transmission lines

There has been a resurgence of interest in the propagation characteristics of open integrated microstrip transmission lines. This is due in part to the discovery of diverse propagation regimes for higher-order modes on open lines. In contrast to the dominant EH/sub 0/ mode, three distinct propagation regimes exist for higher-order modes on microstrip transmission lines. In this paper, a rigorous spectral-domain integral equation formulation is used to analyze propagation in all three regimes. This formulation provides a clear physical picture of the different propagation regimes based on the mathematical location of poles and branch points in the complex spectral-variable plane. As an illustration, the formulation is applied to the case of an isolated uniform microstrip transmission line. The integral equation is discretized via the method of moments, and entire-domain basis functions incorporating suitable edge behavior are utilized to provide convergence with relatively few terms. The results obtained are compared to the results of other workers, and good agreement is observed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Rigorous boundary integral equation solution for general isotropic and uniaxial anisotropic dielectric waveguides in multilayered media including losses, gain and leakage

Bounded and leaky eigenmodes of arbitrary shaped polygonal dielectric waveguides embedded in a multilayered medium are determined based on a rigorous full-wave analysis. The dielectric waveguides consist of isotropic or uniaxial anisotropic material. Losses and gain inside the layers and the waveguides are allowed. The eigenmodes are determined with a boundary integral equation technique in conjunction with the method of moments. Results for the propagation constants are presented for a number of waveguides and, where possible, compared with published data. Special attention is devoted to the transition from a dielectric waveguide to a perfectly conducting waveguide when the loss tangent of the waveguide material changes from zero to infinity. >

Semilocal kinetic analysis of the toroidal ion temperature gradient mode

The semilocal kinetic analysis based on appropriate norms of differential operators is able to recover the growth rates of the toroidal ion temperature gradient mode found from the rigorous integral equation formulation.

Design of transversely fed EMC microstrip dipole arrays including mutual coupling

Design techniques and procedures for microstrip dipole arrays transversely fed by proximity coupled microstrip lines are presented. Two design equations which include the effects of mutual coupling are developed, and the corresponding design curves are obtained by a rigorous integral equation solution. A seven-element standing wave linear array is designed to illustrate the developed design procedures. The design data are checked by a complete integral equation solution of the array, with good agreement. The measurements of radiation pattern and input impedance are found to be in good agreement with the design goal. >

Scattering of electromagnetic beams from rough surfaces.

In this paper a formalism, which leads to the exact solution of the scattering of beams of finite cross section from an arbitrary rough surface, is presented. With use of the Green theorem, a rigorous integral equation is obtained for the case where the wavelength of the incident radiation is comparable to the dimensions of the surface roughness (resonance region). Furthermore, a new explicit solution for the amplitude of the scattered field is introduced, which reduces to known expressions when the rough surface tends to a smooth surface and the beam is replaced by a plane wave. The explicit solution is applied to the particular case of gratings. Additionally, reciprocity relations for thick slit apertures in a rough-surface screen are obtained. These results are expected to be especially useful for calculating the electromagnetic field enhancement (s polarization) near rough surfaces and for the scattering of light atoms by nonperfect periodic hard surfaces.

Wave propagation in media with time‐dependent spatial inhomogeneities

The propagation of waves in a medium with time‐dependent spatial inhomogeneities in its velocity profile is given a rigorous integral equation formulation for source and incident field descriptions. This is in the spirit of Rayleigh, whose aim was to bypass complications arising from boundary conditions. A perturbation expansion, correct to all orders, is then obtained with the aid of a simple identity, for the two cases: (i) weak scatterers and (ii) strong scatterers. It is shown that for case (i) a correction term has to be added to previous results already in second order—this seems to have been overlooked until now. For case (ii), the expansion obtained here is new—the reason for this gap in the existing literature is discussed. With these results, expressions are derived for the wavefunctions when the inhomogeneity in the medium is due to distributions of moving particulates. These new closed‐form time‐dependent approximations exhibit explicitly the appropriate Doppler behavior. The application of this formulation to other cases, such as media with random inhomogeneities, is discussed.

Discrete element analysis of settlement of pile groups

Two discrete element models are described for the analysis of settlement of vertical pile groups. In both models, the piles are represented by discrete elements with an axial model of deformation, and the soil behaviour for the individual piles is represented by load-transfer curves. The essential difference between the models is the manner in which pile-soil-pile interaction is represented. The first model treats the soil as independent horizontal layers which permit interaction between piles to take place within the same layer only. Thus ignoring the continuity of the soil medium. The second utilizes Mindlin's solution which determines interaction effects in a homogeneous, isotropic elastic half-space, but an approximate procedure is used to account for soil inhomogeneity. Comparisons of both approaches with a rigorous integral equation method for pile groups in a homogeneous, isotropic elastic soil indicate that the second method generally gives good agreement while the first method tends to underestimate interaction, especially for shorter piles. For pile groups in nonhomogeneous soils, where rigorous theoretical solutions are generally not available, parametric studies comparing the two discrete element models solutions are presented. The studies performed using both methods on the nonlinear behaviour of an instrumented pile group compare favourably with the field measurements.

A method for the calculation of properties of wire antennas

A symmetrically-fed dipole antenna is considered in the paper. The rigorous integral equation is usually resolved by approximating the kernel, assuming a deltafunction feed, and iterating. To obtain consistent solutions, the sinusoidal current approximation needs to be modified («rounded») at the feed. A modified transmission line approach is equivalent to firstorder iteration by taking a certain value of characteristic impedance. The radiation impedance for a dipole contains the expression for the line reactance within it, and if this term is separated out, the remainder remains finite as the wire radius approaches zero. The iteration approach is equivalent to the usual induced EMF method result if a certain current amplitude is selected, but the result is unsuitable for the full wave dipole. An improved choice is indicated, and is suitable for any dipole length.

On Rayleigh's Theory of Sinusoidal Diffraction Gratings

It is shown that the Meecham-Yasuura numerical method is not a valid indicator of the possibilities of the Rayleigh theory. In this way, Petit showed that the Rayleigh theory is useful only if the amplitude/period (h/d) ratio does not exceed approximately 0·144. Actually, the application of the Fourier method, proposed by Rayleigh himself, enables the theory of the latter to be used, with success, for h/d attaining at least 0·35, regardless of the polarization. This is demonstrated by comparisons with results obtained by other authors from rigorous integral equation methods.

Molecular theory of fluid interfaces

A new description of fluid interfaces, the local correlation model, is developed in a gradient approximation to the interfacial density distribution but otherwise rests on fewer assumptions than either the modified Van der Waals model or Toxvaerd's mean field perturbation theory, with both of which it is compared. Its numerical predictions agree quantitatively with those of the modified Van der Waals model for a Lennard-Jones fluid. The differential equation for the density profile, which results from the gradient approximation, provides a particularly simple description of an interface: the intermolecular contributions to interface structure are divided among (1) interactions occurring in homogeneous fluid, and (2) interactions peculiar to inhomogeneous fluid, the latter being represented by an influence factor c and the total thermodynamic potential density ω of inhomogeneous fluid. Numerical results indicate that for a wide range of temperatures the differential equation gives interfacial density profiles and tensions in good agreement with the more rigorous integral equations.