Yair Shifman

On the use of spatio-temporal multiresolution analysis in method of moments solutions of transient electromagnetic scattering

A novel method of moments approach to the solution of time-domain integral-equation formulation of electromagnetic scattering problems is presented. The method is based on a spatio-temporal multiresolution analysis. This analysis facilitates a basis from which a small number of expansion functions is selected via an iterative procedure and utilized to model the unknown current distribution. In contrast to marching-on-in-time sequential procedures, the proposed method models the unknown current simultaneously at all the time steps within the time frame of interest. This new method is applied to a one-dimensional (1-D) problem of electromagnetic plane wave interaction with a dielectric slab. A comparison of the computed results with results based on the analytic solution demonstrates that the method is capable of attaining accurate results while achieving substantial reduction in computation time and resources.

Scattering by a groove in a conducting plane-a PO-MoM hybrid formulation and wavelet analysis

A novel method is presented to solve the two-dimensional (2-D) problem of scattering of an electromagnetic plane wave by a groove in a perfectly conducting infinite plane. In this method, the unknown induced current is expressed in terms of the known physical optics solution of the unperturbed problem of scattering by an infinite conducting plane plus a yet to be determined localized correction current placed in the vicinity of the groove. It is then shown that a good approximation of the induced current can be obtained using only a few dominant functions in the wavelet expansion of the correction current. Moreover, the same set of dominant wavelet functions serves the purpose of approximating the induced current at different angles of incidence. A numerical example demonstrates these various features of the proposed method of solution.

Analysis of transient interaction of electromagnetic pulse with an air layer in a dielectric medium using wavelet-based implicit TDIE formulation

The interaction of transient electromagnetic pulse with an air layer in a dielectric medium is formulated in terms of a time-domain integral equation and solved numerically via the method of moments. Previous related works pointed to the inherent inadequacy of the marching-on-in-time method in this case, but suggested no remedy. This paper explains why an implicit modeling scheme would work effectively in this case. It is also noted that the use of an implicit scheme would normally involve a solution of a very large and dense matrix equation. To alleviate this drawback of the implicit scheme, the use of a wavelet-based impedance-matrix-compression technique, which has facilitated in the very recent past solutions of time-domain problems with greater efficiency, is described.

On the use of spatio-temporal wavelet expansions for transient analysis of wire antennas and scatterers

To analyze a wire antenna excited by a time varying voltage source or a wire scatterer excitated by transient electromagnetic incident wave, the problem is formulated in terms of a time-domain integral equation for the induced current. To solve the integral equation, we reduce it to matrix equation via the method of moments using the known-to-be-stable implicit scheme. However, rather than directly constructing and solving the relatively large matrix equation, we propose an iterative procedure which allows us to gradually obtain a solution of refined accuracy both everywhere and simultaneously at any time instance. To render this procedure rapidly converging, we use a basis of spatio-temporal wavelet functions. This basis facilitates a good approximation of the induced current using far less basis functions than would be needed if other expansions, such as standard-pulse or Fourier basis functions were chosen. The use of this basis further enables the iterative procedure to increase the temporal and spatial resolutions where required without unnecessarily affecting their levels elsewhere.

Analysis of truncated periodic array using two-stage wavelet-packet transformations for impedance matrix compression

A novel method of moments procedure is applied to the problem of scattering by metallic truncated periodic arrays. In such problems, the induced current shows localized behavior within the unit cell and at the same time exhibits cell-to-cell periodicity. In order to select a set of expansion functions that may account for such behavior, a two-stage basis transformation, of which the first stage is an ordinary wavelet transformation performed independently on each unit-cell, has been applied to a pulse basis. The resultant basis functions at the first stage are regrouped and retransformed to reveal the periodicity of their coefficients. Expansion functions are then iteratively selected from this newly constructed basis to form a compressed impedance matrix. The compression ratios obtained in this manner are higher than the compression ratio achieved using a basis constructed via an ordinary single-stage wavelet transformation, where compression is the ratio between the number of nonzero elements in the matrix used to solve the problem and the number of elements in the original matrix. An even higher compression is attained by considering, in addition, functions that reveal array-end related features and iteratively selecting the expansion from an overcomplete dictionary.

Iterative selection of expansion functions from an overcomplete dictionary of wavelet packets for impedance matrix compression

The paper further develops the recently introduced idea of compressing the impedance matrix by an iterative selection of expansion functions. The improved algorithm uses an overcomplete dictionary comprising bases whose inherent properties match some features of the given scattering problem. Thus, the overcomplete dictionary offers a variety of expansion functions from which a solution-oriented spanning set for the unknown current is extracted. The number of iterations in the proposed algorithm is suppressed by selecting more than one expansion function at each iteration. This latter section is facilitated by a matching pursuit process. It is shown that the compression ratio of the resultant compressed impedance matrix is superior to the one achieved by the ordinary iterative matrix compression algorithm.

Transient analysis of EM pulse penetration into a conducting layer using wavelet-based implicit TDIE and iterative IMC technique

The penetration of a wide-band electromagnetic pulse into a conducting layer is formulated in terms of an implicit time-domain integral equation and cast into matrix form via the method of moments. Though such a wide-band problem indeed calls for a time-domain solution, one may argue that the frequency-dependent attenuation of the field penetrating the conductor could suggest a frequency-domain approach that would greatly reduce the number of unknowns. The proposed via media is to use spatio-temporal wavelet functions instead of the standard pulse basis. Owing to their multiresolution property, both in the spatial and temporal domains, these functions can span the field propagating inwardly with substantially fewer terms. The reduction in the number of basis functions used is effected by the impedance matrix compression technique, which automatically omits the basis functions whose coefficients would be insignificant due to the attenuation. Reducing the number of basis functions renders the matrix equation much smaller and the overall solution far more efficient.

Wavelet-based analysis of transient electromagnetic wave propagation in photonic crystals

Photonic crystals and optical bandgap structures, which facilitate high-precision control of electromagnetic-field propagation, are gaining ever-increasing attention in both scientific and commercial applications. One common photonic device is the distributed Bragg reflector (DBR), which exhibits high reflectivity at certain frequencies. Analysis of the transient interaction of an electromagnetic pulse with such a device can be formulated in terms of the time-domain volume integral equation and, in turn, solved numerically with the method of moments. Owing to the frequency-dependent reflectivity of such devices, the extent of field penetration into deep layers of the device will be different depending on the frequency content of the impinging pulse. We show how this phenomenon can be exploited to reduce the number of basis functions needed for the solution. To this end, we use spatiotemporal wavelet basis functions, which possess the multiresolution property in both spatial and temporal domains. To select the dominant functions in the solution, we use an iterative impedance matrix compression (IMC) procedure, which gradually constructs and solves a compressed version of the matrix equation until the desired degree of accuracy has been achieved. Results show that when the electromagnetic pulse is reflected, the transient IMC omits basis functions defined over the last layers of the DBR, as anticipated.

Transient analysis of distributed Bragg reflector using wavelet-based implicit TDIE and impedance matrix compression

Summary form only given. A common photonic device is the distributed Bragg reflector (DBR), which is a stack of alternating layers of two materials with different dielectric permittivity. It exhibits high reflectivity at certain frequencies, and is therefore used in state-of-the-art applications, such as mirrors for solid-state lasers and in hollow fibers. We explore a new and efficient approach to analyzing the transient interaction of a wideband electromagnetic pulse with such devices. Transient analysis of a DBR can be formulated in terms of a volume time-domain integral equation (TDIE) and, in turn, solved numerically via the method of moments (MoM) while resorting to the more stable implicit scheme. We seek a representation of the the unknown field within the DBR layers in terms of as compact as possible a set of basis functions. Towards that end, we use a multiresolution basis comprising spatio-temporal wavelet functions. We can effect an automatic selection of the dominant functions needed for the solution via an impedance matrix compression (IMC) procedure. Our results show that the IMC procedure incorporates or omits basis functions automatically, without assuming a priori that the fields are negligible anywhere or at any time. A comparison of the performance of the proposed procedure with that of a standard CG procedure shows that the new procedure is roughly twice its fast.

Transient analysis of air layer in a dielectric medium using wavelet-based implicit TDIE formulation

The interaction of a transient electromagnetic pulse with an air layer in a dielectric medium is formulated in terms of a time-domain integral equation and solved numerically via the method of moments. Application of the standard marching-on-in-time approach in this case can not yield a solution. Hence, we utilize an implicit modeling scheme, and, to reduce the computational complexity, resort to a previously proposed time-domain impedance matrix compression method. This method uses spatio-temporal wavelet basis functions to construct and solve a reduced-rank matrix equation. Furthermore, by modeling the problem simultaneously at all the time steps, we can obtain a solution which roughly has the same level of accuracy for all these times.