Leopold B. Felsen

Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics.

Paraxial Gaussian beams (GBs) are collimated wave objects that have found wide application in optical system analysis and design. A GB propagates in physical space according to well-established quasi-geometric-optical rules that can accommodate weakly inhomogeneous media as well as reflection from and transmission through curved interfaces and thin-lens configurations. We examine the GB concept from a broad perspective in the frequency domain (FD) and the short-pulse time domain (TD) and within as well as arbitrarily beyond the paraxial constraint. For the formal analysis, which is followed by physics-matched high-frequency asymptotics, we use a (space-time)-(wavenumber-frequency) phase-space format to discuss the exact complex-source-point method and the associated asymptotic beam tracking by means of complex rays, the TD pulsed-beam (PB) ultrawideband wave-packet counterpart of the FD GB, GBs and PBs as basis functions for representing arbitrary fields, GB and PB diffraction, and FD-TD radiation from extended continuous aperture distributions in which the GB and the PB bases, installed through windowed transforms, yield numerically compact physics-matched a priori localization in the plane-wave-based nonwindowed spectral representations.

Asymptotic high-frequency Green's function for a planar phased sectoral array of dipoles

This paper deals with the derivation and physical interpretation of a uniform high-frequency Greens function for a planar right-angle sectoral phased array of dipoles. This high-frequency Greens function represents the basic constituent for the full-wave description of electromagnetic radiation from rectangular periodic arrays and scattering from rectangular periodic structures. The field obtained by direct summation over the contributions from the individual radiators is restructured into a double spectral integral whose high-frequency asymptotic reduction yields a series of propagating and evanescent Floquet waves (FWs) together with corresponding FW-modulated diffracted fields, which arise from FW scattering at the array edges and vertex. Emphasis is given to the analysis and physical interpretation of the vertex diffracted rays. The locally uniform asymptotics governing this phenomenology is physically appealing, numerically accurate, and efficient, owing to the rapid convergence of both the FW series and the series of corresponding FW-modulated diffracted fields away from the array plane. A sample calculation is included to demonstrate the accuracy of the asymptotic algorithm.

Wave propagation inside a two-dimensional perfectly conducting parallel-plate waveguide: hybrid ray-mode techniques and their visualizations

This work is intended as an educational aid, dealing with high-frequency (HF) electromagnetic wave propagation in guiding environments. It is aimed at advanced senior and first-year graduate students who are familiar with the usual engineering mathematics for wave equations, especially analytic functions, contour integrations in the complex plane, etc., and also with rudimentary saddle-point (HF) asymptotics. After an introductory overview of issues and physical interpretations pertaining to this broad subject area, detailed attention is given to the simplest canonical, thoroughly familiar, test environment: a (time harmonic) line-source-excited two-dimensional infinite waveguide with perfectly conducting (PEC) plane-parallel boundaries. After formulating the Greens function problem within the framework of Maxwells equations, alternative field representations are presented and interpreted in physical terms, highlighting two complementary phenomenologies: progressing (ray-type) and oscillatory (mode-type) phenomena, culminating in the self-consistent hybrid ray-mode scheme, which usually is not included in conventional treatments at this level. This provides the analytical background for two educational MATLAB packages, which explore the dynamics of ray fields, mode fields, and the ray-mode interplay. The first package, RAY-GUI, serves as a tool to compute and display eigenray trajectories between specified source/observer locations, and to analyze their individual contributions to wave fields. The second package, HYBRID-GUI, may be used to comparatively display range and/or height variations of the wave fields, calculated via ray summation, mode-field summation, and hybrid ray-mode synthesis.

A new algorithm for ground wave propagation based on a hybrid ray-mode approach

An efficient novel algorithm is introduced for ground wave propagation problems. First, ground wave propagation characteristics for a vertically polarized short electric dipole over a smooth spherical earth are reviewed, reducing the vector electromagnetic problem for the three-dimensional spherical geometry to an equivalent two-dimensional rectilinear scalar potential problem which is solved by spectral analysis and synthesis. Alternative evaluations of the spectral integral yield ray optical and normal mode solutions, which are conventionally referred to as the Norton and Wait formulations, respectively. Combining these formulations in an efficient manner yields a hybrid algorithm which is constructed so as to account adaptively for the characteristics of ground wave propagation in interference, intermediate and diffraction regions (including mixed paths) for various source and/or receiver heights. Numerical comparisons are made with reference results obtained via the parabolic equation (PE) method, in parametric ranges where PE is reliable; this permits assessment of the effectiveness of the hybrid approach.

Pulsed-beam propagation in lossless dispersive media. I. Theory

This first part of a two-part investigation is concerned with the effects of dispersion on the propagation characteristics of the scalar field associated with a highly localized pulsed-beam (PB) wave packet in a lossless homogeneous medium described by the generic wave-number profile k(ω)=ω/c(ω), where c(ω) is the frequency-dependent wave propagation speed. While comprehensive studies have been performed for the one-dimensional problem of pulsed plane-wave propagation in dispersive media, particularly for specific c(ω) profiles of the Lorentz or Debye type, even relatively crude measures tied to generic k(ω) profiles do not appear to have been obtained for the three-dimensional problem associated with a PB wave packet with complex frequency and wave-number spectral constituents. Such wave packets have been well explored in nondispersive media, and simple asymptotic expressions have been obtained in the paraxial range surrounding the beam axis. These paraxially approximated wave objects are now used to formulate the initial conditions for the lossless generic k(ω) dispersive case. The resulting frequency inversion integral is reduced by simple saddle-point asymptotics to extract the PB phenomenology in the well-developed dispersive regime. The phenomenology of the transient field is parameterized in terms of the space–time evolution of the PB wave-front curvature, spatial and temporal beam width, etc., as well as in terms of the corresponding space–time-dependent frequencies of the signal, which are related to the local geometrical properties of the k(ω) dispersion surface. These individual parameters are then combined to form nondimensional critical parameters that quantify the effect of dispersion within the space–time range of validity of the paraxial PB. One does this by performing higher-order asymptotic expansions beyond the paraxial range and then ascertaining the conditions for which the higher-order terms can be neglected. In Part II [J. Opt. Soc. Am. A15, 1276 (1998)], these studies are extended to include the transitional regime at those early observation times for which dispersion is not yet fully developed. Also included in Part II are analytical and numerical results for a simple Lorentz model that permit assessment of the performance of various nondimensional critical estimators.

High-frequency fields excited by truncated arrays of nonuniformly distributed filamentary scatterers on an infinite dielectric slab: parameterizing (leaky mode)-(Floquet mode) interaction

In previous studies, we have developed and tested observable-based parameterizations (OBP) of time-harmonic wavefield scattering by periodic or aperiodic finite arrays of planar strip and filament scatterers. The resulting algorithm is based on truncated Floquet modes and Floquet-modulated edge diffractions due to the truncations of the array. The corresponding robust wave processes link features (observables) in scattering data with geometrical features in the model configuration in such a manner as to be useful for subsequent application to target classification and identification. The present study extends these investigations to a finite array of filamentary scatterers located on the surface of an infinitely extended dielectric slab, thereby parameterizing (Floquet mode)-(leaky mode) interaction as a classifier of the more complicated phenomenology in this composite configuration. The outcome is an OBP with two separate constituents that can be interpreted, respectively, as slab-modified Floquet scattering by the truncated array and as truncated-Floquet-induced excitation of slab-guided leaky waves. This new OBP for the composite problem is validated by comparison with reference solutions generated numerically, its relevance to wave-oriented data processing is demonstrated in the companion paper.

Time-domain Green's function for an infinite sequentially excited periodic planar array of dipoles

The present paper is a continuation of previous explorations by the authors, aimed at gaining a basic understanding of the time domain (TD) behavior of large periodic phased (i.e., sequentially turned-on) array antenna and related configurations. Our systematic investigation of the relevant canonical TD dipole-excited Green’s functions has so far included those for infinite and truncated sequentially pulsedline periodic arrays, parameterized in terms of radiating (propagating) and nonradiating (evanescent)conical TD Floquet waves (FW) and truncation-induced TD FW-modulated tip diffractions. The present contribution extends these investigations to an infinite periodic sequentially pulsedplanar array, which generates pulsed planepropagating and evanescent FW. Starting from the familiar frequency domain (FD) transformation of the linearly phased element-by-element summation synthesis into summations of propagating and evanescent FWs, we access the time domain by Fourier inversion. The inversion integrals are manipulated in a unified fashion into exact closed forms, which are parameterized by the single nondimensional quantity = ( ) 1 , where ( ) and are the excitation phase speed along a preferred phasing direction 1 in the array plane and the ambient wave speed, respectively. The present study deals with the practically relevant rapidly phased propagating case 1, reserving the more intricate slowly phased 1 regime for a future manuscript. Numerical reference data generated via element-by-element summation over the fields radiated by the individual dipoles with ultrawide band-limited excitation are compared with results obtained much more efficiently by inclusion of a few TD–FWs. Physical interpretation of the formal TD–FW solutions is obtained by recourse to asymptotics, instantaneous frequencies and wavenumbers, and related constructs. Of special interest is the demonstration that the TD–FWs emerge along “equal-delay” ellipses from the array plane; this furnishes a novel and physically appealing interpretation of the planar array TD–FW phenomenology.

Ray theory for scattering by two-dimensional quasiperiodic plane finite arrays

Many scattering configurations of interest include finite portions with periodic or quasiperiodic features. Several recent investigations have dealt with this problem for the planar two-dimensional case and have developed high-frequency asymptotic solutions that include multibeam reflections obeying the Bragg condition and Bragg-modulated edge diffractions. These constituents have been interpreted as wave objects in a generalized geometrical theory of diffraction (GTD). The present investigation adds to these previous results and formalizes them into a ray theory. This allows the scattered fields due to a finite quasiperiodic array of obstacles, excited by an arbitrary incident field, to be constructed entirely by ray tracing. Scattered ray plots and caustics for various shapings of incident fields and array parameters illustrate the variety of phenomena associated with this class of scattering environments.

Multifrequency subsurface sensing in the presence of a moderately rough air–soil interface via quasi-ray Gaussian beams

[1]xa0An adaptive framework is presented for frequency-stepped ground-penetrating radar (GPR) imaging of low-contrast buried objects in the presence of a moderately rough air–soil interface, with potential applications intended in the area of humanitarian demining. The proposed approach, so far restricted to two-dimeansional (2-D) geometries, works with sparse data and relies on recently developed problem-matched narrow-waisted Gaussian beam (GB) algorithms as fast forward scattering predictive models to estimate and compensate for the effects of the coarse-scale roughness profile. Possible targets are subsequently imaged by inverting the Born-linearized subsurface scattering model via object-based curve evolution (CE) techniques. This frequency domain (FD) strategy implements a further step in our planned sequential approach toward a physics based, robust, and numerically efficient framework for rough surface underground imaging in both FD and time domain (TD). Numerical experiments indicate that the proposed framework is attractive from both computational and robustness viewpoints. The results in this paper could also be used for synthesis of TD illumination (in a previous study [Galdi et al., 2001b], we have dealt with wideband illumination directly in the TD).

Multifrequency reconstruction of moderately rough interfaces via quasi-ray Gaussian beams

In this paper, we present a new technique for determining the surface profile of a moderately rough interface between air and a homogeneous dielectric half-space. Based on sparsely sampled step-frequency ground penetrating radar measurements, the proposed inversion scheme uses a quasi-ray Gaussian beam fast forward model, coupled with a low-order parameterization of the surface profile in terms of B-splines. The profile estimation problem is posed as a parameter optimization problem, which is solved using a multiresolution continuation method via frequency hopping. Numerical experiments establish that the algorithm is efficient and yields accurate reconstructions throughout most of the illuminated region even in noisy environments, losing accuracy only in regions with very weak illumination.