The singularity expansion method (SEM) for quantifying the transient electromagnetic (EM) scattering from targets illuminated by pulsed EM radiation is reviewed. SEM representations for both induced currents and scattered fields are presented. Natural-resonance-based target identification schemes, based upon the SEM, are described. Various techniques for the extraction of natural-resonance modes from measured transient response waveforms are reviewed. Particular attention is given to the aspect-independent (extinction) E-pulse and (single-mode) S-pulse discriminant waveforms which, when convolved with the late-time pulse response of a matched target, produce null or mono-mode responses, respectively, through natural-mode annihilation. Extensive experiment results for practical target models are included to validate the E-pulse target discrimination technique. Finally, anticipated future extensions and areas requiring additional research are identified. >

The singularity expansion method and its application to target identification

A unified treatment of the natural mode representations for induced currents and scattered fields is obtained by use of fundamental concepts regarding causality and superposition. The transient scattered field response is shown to have the form of a constant coefficient complex exponential sum only in the "late-time," after the last driven reponse is received from the object. Prior to this, the "early.time" response is found to be due to direct physical optics fields as well as a sum of temporally modulated natural modes produced by the progressive illumination of the incident wavefront. Alternate representations and s -plane behaviors are considered. The implications of these results on natural resonance target identification schemes are discussed.

Singularity expansion representations of fields and currents in transient scattering

The singularity expansion method (SEM) represents transient scattering by superposition of damped oscillatory fields corresponding to the complex resonant frequencies of the scatterer. The series of these global wave fields, which encompass the scattering object as a whole, is slowly convergent at early observation times and even deficient at very early times when portions of the object are as yet unexcited. Thus, the resonance series representation must generally be augmented by an entire function in the complex frequency domain. The choice of the entire function is relatively arbitrary but affects the excitation coefficients, called coupling coefficients, of individual resonances and also the "turn-on" and "switch-on" times of the SEM series. Moreover, it contains essential (intrinsic) and nonessential (removable) portions which have been subjected to various interpretations. By formulating the transient problem in terms of traveling (progressing) incident, reflected and diffracted wavefronts, these constructs in the SEM can be interpreted in a precise and physical manner. Furthermore, the analysis clarifies the evolution of resonances as collective summations of multiple wavefront fields which are caused by successive reflections or diffractions at the surfaces and scattering centers comprising the object. By combining wavefronts and resonances self-consistently, one may construct a hybrid field that avoids the difficulties at early times in the SEM formulation. The systematic exploration of the interplay between wavefronts and resonances is facilitated through use of a flow diagram, as introduced in system theory. These concepts are developed in broad generality and are illustrated for two-dimensional scattering by various special configurations.

A wavefront interpretation of the singularity expansion method

The singularity expansion method (SEM) arose from the observation that the transient response of complex electromagnetic scatterers appeared to be dominated by a small number of damped sinusoids. In the complex frequency plane, these damped sinusoids are poles of the Laplace-transformed response. The question is then one of characterizing the object response (time and frequency domains) in terms of all the singularities (poles, branch cuts, entire functions) in the complex frequency plane (hence singularity expansion method). Building on the older concept of natural frequencies, formulae were developed for the pole terms from an integral-equation formulation of the scattering process. The resulting factoring of the pole terms has important application consequences. Later developments include the eigenmode expansion method (EEM) which diagonalizes the integral-equation kernels and which can be used as an intermediate step in ordering the SEM terms. Additional concepts which have appeared include eigenimpedance synthesis and equivalent electrical networks. Of current interest is the use of the theoretical formulae to efficiently analyze and order experimental data, Related to this is the application of SEM results to target identification. This paper does not delve into the mathematical details; it presents an overview of the history and major concepts and results in SEM and EEM and related matters.

The singularity expansion method: Background and developments

The representation of the electromagnetic field scattered by a perfectly conducting finite-extent scatterer immersed in a lossless medium as a singularity expansion is considered. While the analytic properties of the temporal Laplace transform of the surface currents residing on such an object have received a great deal of attention, the properties of the scattered fields have not. It is shown that the representation of the transform of the scattered field must include an exponential entire function except for observation points in the forward-scattered direction. Explicit time domain representations that are counterpart to the Laplace domain representation are constructed and are shown to embody, in the early time, temporal variation besides that of the damped sinusoidal factors intrinsic to the singularity expansion. An important practical consequence of this more complicated time variation arises in connection with the application of the singularity expansion for target classification purposes and is commented upon herein.

A note on the representation of scattered fields as a singularity expansion

The most general form of the Singularity Expansion Method (SEM) representation of the transient scattering response of a finite extent object allows considerable freedom of choice as to the time at which one begins to include the sum of residue contributions into the transient response--i.e., the “turn-on” time. The issue of the chosen form of the coupling coefficient used in computing these residue constituents relates intimately with the choice of turn-on time. The practical range of choices for turn-on time is considerably more restricted than the theoretical one. In this paper limitations on turn-on time are established in terms of the maximum geometric extent of the object and of the geometric extent of the object projected in the direction of propagation of the incident wave. These limitations are dictated in order to insure the convergence of the SEM residue series, and are, in general, different for the Class 1 and for the Class 2 coupling coefficient. It is further argued that the conver...

L. W. Pearson

Papers

On the convergence and numerical sensitivity of the sem pole-series in early-time scattering response