Raphael Kastner

Finite difference time domain dispersion reduction schemes

The finite-difference-time-domain (FDTD), although recognized as a flexible, robust and simple to implement method for solving complex electromagnetic problems, is subject to numerical dispersion errors. In addition to the traditional ways for reducing dispersion, i.e., increasing sampling rate and using higher order degrees of accuracy, a number of schemes have been proposed recently. In this work, an unified methodology for deriving new difference schemes is presented. It is based on certain modifications of the characteristic equation that accompanies any given discretized version of the wave equation. The method is duly compared with existing schemes and verified numerically.

The time-domain discrete Green's function method (GFM) characterizing the FDTD grid boundary

For a given FDTD simulation space with an arbitrarily shaped boundary and an arbitrary exterior region, most existing absorbing boundary conditions become inapplicable. A Greens function method (GFM) is presented which accommodates arbitrarily shaped boundaries in close proximity to a scattering object and an arbitrary composition in the exterior of the simulation space. Central to this method is the numerical precomputation of a Greens function tailored to each problem which represents the effects of the boundary and the external region. This function becomes the kernel for a single-layer absorbing boundary operator, it is formulated in a manner which naturally incorporates numerically induced effects, such as the numerical dispersion associated with the FDTD scheme. The Greens function is an exact absorber in the discretized space. This property should be contrasted with other methods which are initially designed for the continuum and are subsequently discretized, thereby incurring inherent errors in the discrete space which cannot be eliminated unless the continuum limit is recovered. In terms of accuracy, the GFM results have been shown to be of a similar quality to the PML, and decidedly superior to the Mur (1981) condition. The properties of the GFM are substantiated by a number of numerical examples in one, two, and three dimensions.

A spectral domain approach for computing the radiation characteristics of a leaky-wave antenna for millimeter waves

A new method for evaluating the complex propagation constant \beta in a leaky-wave structure comprising thin metallic rectangular strips etched on a dielectric rod of rectangular cross section is described. The radiation pattern of the leaky-wave antenna can be determined once \beta is known, since Re( \beta ) governs the direction of the main beam and Im( \beta ) accounts for the beamwidth and aperture efficiency. In addition the knowledge of the dependence of \beta on frequency allows one to design the antenna for frequency-scanning applications. The method employed in this communication is based on the spectral domain approach that formulates the eigenvalue problem in the Fourier transform domain. Computed results are shown to be in very good agreement with experimental measurements.

Analysis of a dual circularly polarized microstrip antenna fed by crossed slots

A dual circularly polarized, wideband microstrip antenna fed by two perpendicular crossed slots is analyzed in this work. The cross shaped aperture in the ground plane provides coupling between the microstrip patch and a single microstrip line underneath the ground plane, traversing the four arms of the cross in a serial manner, thereby generating four coupling points in phase quadratures for both senses of circular polarization. The analysis of this antenna incorporates two stages: In the first one, the individual load impedances are computed via an integral equation based on reciprocity, assuming no interactions between the ports. The second stage employs a transmission line model where the four coupling points are considered as lumped loads along the feed line. An iterative scheme employs the transmission line model for the determination of the actual impedances of each coupling point, including the mutual interactions with the other coupling points. This procedure provides control over the overall level of radiated energy; hence it can be used in the design procedure of this antenna as a serial or parallel fed array element. The computed results agree very well with measurements. The antenna attains a -3 dB gain bandwidth of 13% to 15%, a 3-dB axial ratio bandwidth of 48%, and -15 dB return loss bandwidth of 60%. >

Spectral domain iterative analysis of single- and double-layered microstrip antennas using the conjugate gradient algorithm

An analysis for single- and double-layered microstrip antennas is described. The planarity of these structures makes it possible to construct a general spectral representation for any number of layers and to derive the spectral Greens dyad in a compact fashion using a transmission-line analogy. A formulation of the antenna problem in the spectral domain, incorporating this dyad, is coupled with the conjugate gradient algorithm, whose applicability as an efficient way of analyzing a number of microstrip antennas is studied. Results are quite accurate. Conclusions pertaining to the applicability of the method, including effects of problem parameters on convergence rates, are drawn. >

An 'add-on' method for the analysis of scattering from large planar structures

A so-called add-on procedure is proposed to deal with the data analysis problem resulting from the collection of scattering data from large planar structures. The computations (involving of the order of a few thousand unknowns) is undertaken in a gradual manner by building up the body from small patches which are added sequentially. The procedure is based on an initial expansion of the unknown current distribution into subsectional (pulse-type) basic functions. Each segment of the scatterer carries an unknown amplitude which is the response to an incident wave. However, rather than forming a matrix equation for these responses, they are computed in a gradual manner where the scatterer is built up from these segments as they are added one at a time. At the end of each addition of a segment, the result for scattering from a partial body is obtained. At each stage, the problem solved reflects the size of the small addition only, and the solution to an actual partial body is obtained. An important feature of this method is its ability to utilize a priori known information on a portion of the scatterer as an initial stage for the economic analysis of the entire structure. The process takes into account the interactions between all segments of the body. The process proves to be very efficient both in terms of computation time and storage requirements, as seen in the computed examples on of the order of 1000 to 6000 unknowns. >

Iterative analysis of finite-sized planar frequency selective surfaces with rectangular patches or perforations

Frequency selective surfaces (FSS) and other periodic gratings are often analyzed under the assumption that they are infinite in extent. Most existing methods for analyzing periodic structures are based on the use of a Floquet-type representation of the fields in a unit cell whose dimensions are typically comparable to the wavelength. In this work, a finite, truncated, version of an infinite periodic structure is dealt with directly, without the benefit of the assumption that the structure is periodic. This, in turn, requires the handling of a large number of unknowns and makes it difficult to solve the problem using conventional matrix methods. Two different iteration approaches to solving the finite FSS problem are discussed in the paper both of which employ the spectral domain formulation. The first of these employs the spectral iteration technique and the second one uses the conjugate gradient (CG) iteration algorithm. Convergence characteristics of both of these methods are investigated and the results are reported.

Stability analysis of the Green's function method (GFM) used as an ABC for arbitrarily shaped boundaries

The time-domain Greens function of the external region beyond a given boundary is an inherently discretized version of the impedance condition. It is incorporated within the framework of the FDTD as a single layer boundary condition, termed the Greens function method (GFM). The stability characteristics of this method are described. The analysis is based on the general representation of the method in matrix form, whose eigenvalues are investigated. This formulation helps detect and remove possible instabilities of the algorithm.

A spectral-iteration technique for analyzing a corrugated-surface twist polarizer for scanning reflector antennas

An analysis of the corrugated-surface twist polarizer which finds application in the design of scanning reflector antennas is presented. The spectraI-interation technique, a novel procedure which combines the use of the Fourier transform method with an iterative procedure is employed. The first step in the spectral-iteration method is the conversion of the original integral equation for the interface field into a form which is suitable for iteration using a method developed previously. An important feature of the technique is that it takes advantage of the discrete Fourier transform (DFT) type of kernel of the integral equation and evaluates the integral operators efficiently using the fast Fourier transform (FFT) algorithm. Thus, in contrast to the conventional techniques, e.g., the moment method, the spectral-iteration approach requires no matrix inversion and is capable of handling a large number of unknowns. Furthermore the method has a built-in check on the satisfaction of the boundary conditions at each iteration.

A Comprehensive New Methodology for Formulating FDTD Schemes With Controlled Order of Accuracy and Dispersion

Numerical dispersion errors in the wave-equation-finite-difference-time-domain (WE-FDTD) method have been treated by higher order schemes, coefficient modification schemes, dispersion relation preserving and non-standard schemes. In this work, a unified methodology is formulated for the systematic generation of WE-FDTD schemes tailored to the spectrum of the excitation. The methodology enables the scheme designer to gradually trade order of accuracy (OoA) for lower dispersion errors in a controlled manner at the cost of sacrificing low frequency behavior, that is not deemed critical for this type of excitation. The methodology is shown to encompass both existing and new schemes. Stability analysis is carried out concurrently with the generation of each scheme. Using a stencil size of 3 and 5 temporal and spatial samples, respectively, long term errors of a scheme designed for a specific pulse are compared with the standard (4,4) scheme that has the same computational complexity, via simulation of a modulated pulse that propagates over a million time steps.