Timor Melamed

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.

Local spectral analysis of short-pulse excited scattering from weakly inhomogeneous media. II. Inverse scattering

For Pt.I see ibid., vol.47, no.7, p.1208-17 (July 1999). This paper is concerned with the reconstruction of a weakly inhomogeneous scattering profile from data generated by a short-pulse incident plane wave, which is postprocessed so as to localize the interrogated region to a space-time resolved scattering cell, The phase-space localization due to postprocessing is brought about by applying local (i.e., windowed) slant-stack transforms to the time-dependent scattered fields. In the domain of the scatterer, this processing corresponds to applying windowed Radon transforms to the induced field distribution, which, in turn, generates pulsed-beam (PB) wave packets traveling toward the observer. The forward analysis parameterizing this new form of time-domain (TD) diffraction tomography has been performed in a companion paper and furnishes the framework for the investigation here. Via the forward parameterization, the three-dimensional (3-D) global scattering phenomenology has been reduced to scattering from an equivalent one-dimensional (1-D) scattering cell oriented along the bisector between the direction of the incident plane pulse and the direction of the scattered pulsed beam (PB) to the observer. For the inverse problem, this process is reversed by windowing the scattered field and backpropagating the resulting PBs so as to form local images of any selected region in the scattering domain. The phase-space signature of the scattering cell is related to the Radon transform of the medium in the cell so that the local profile function can be recovered by Radon inversion. An illustrative numerical example is included. Also discussed is the ultimate localization achieved by incident PB excitation and PB postprocessing of the scattered field.

Pulsed beam propagation in lossless dispersive media

Concerns the parameterization of the effects of frequency dispersion on the propagation characteristics of a paraxially approximated pulsed beam (PB) wavepacket in a lossless medium with generic wavenumber profile. Various nondimensional measures-critical parameters-have been defined in order to systematically assess and quantify: a) the effect of dispersion on various observables associated with the PB field and thereby on the resolution of these observables; and b) the range of validity of the paraxial approximation under these conditions.

Short-pulse inversion of inhomogeneous media: a time-domain diffraction tomography

Time-domain inversion of a three-dimensional inhomogeneous medium is formulated as a time-domain diffraction tomography. The scattered data are expanded into a spectrum of time-dependent plane waves using the slant-stack transform. It is then shown that each time-dependent plane-wave constituent in the data is directly related to the Radon transform of the mediums inhomogeneity along the direction that bisects the angle between the plane wave and the incident wave. This new tomographic relation provides the basis for two inversion approaches: a Radon-space reconstruction and a time-dependent filtered backpropagation. Finally, the reconstruction errors due to the limited spacetime aperture are identified via analysis and a numerical example.

PULSED-BEAM PROPAGATION IN LOSSLESS DISPERSIVE MEDIA. II. A NUMERICAL EXAMPLE

In Part I of this two-part investigation we presented a theory for propagation of pulsed-beam wave packets in a homogeneous lossless dispersive medium with the generic dispersion relation k(ω). Emphasis was placed on the paraxial regime, and detailed studies were performed to parameterize the effect of dispersion in terms of specific physical footprints associated with the PB field and with properties of the k(ω) dispersion surface. Moreover, critical nondimensional combinations of these footprints were defined to ascertain the space–time range of applicability of the paraxial approximation. This was done by recourse to simple saddle-point asymptotics in the Fourier inversion integral from the frequency domain, with restrictions to the fully dispersive regime sufficiently far behind the wave front. Here we extend these studies by addressing the dispersive-to-nondispersive transition as the observer moves toward the wave front. It is now necessary to adopt a model for the dispersive properties to correct the nondispersive high-frequency limit k(ω)=ω/c with higher-order terms in (1/ω). A simple Lorentz model has been chosen for this purpose that allows construction of a simple uniform transition function which connects smoothly onto the near-wave-front-reduced generic k(ω) profile. This model is also used for assessing the accuracy of the various analytic parameterizations and estimates in part I through comparison with numerically generated reference solutions. It is found that both the asymptotics for the pulsed-beam field and the nondimensional estimators perform remarkably well, thereby lending confidence to the notion that the critical parameter combinations are well matched to the space–time wave dynamics.

Space-Time Representation of Ultra Wideband Signals

Publisher Summary This chapter presents some analytic techniques and wave solutions, which are relevant to the analysis of ultra wideband short-pulse fields and data. To convey the ideas in the simplest format, the simple problem of short-pulse radiation from an aperture distribution is considered. The given field in the initial plane may be the physical time-dependent source (forward problem) or it may be a measured field because of a remote sensing or scattering experiment (inverse problem). In the forward problem one is interested in calculating the radiating field, and in its parameterization and optimization, whereas, in the inverse case the time-dependent data should be processed to extract the information on the sources (either real or induced sources). Typically, inverse processing involves the same radiation-type integrals used in forward problems. The chapter discusses the time-dependent plane-wave spectrum approach. To clarify the field structure and the numerical properties of the TD plane-wave integral, the approximate field solutions are considered. In the near zone the pulsed field propagates along space-time rays, which emerge from the aperture along directions that are determined by the gradient of the delay function.

Pulsed-beam propagation in dispersive media via pulsed plane wave spectral decomposition

This paper is concerned with the behavior of transient wavefields due to a pulsed-beam (PB) wavepacket launched obliquely from a hypothetical aperture plane in a medium with generic dispersion k(/spl omega/) where k and /spl omega/ are wavenumber and frequency, respectively. This generalizes our previous investigation of a PB launched normally from the hypothetical aperture plane. The problem is solved through spectral decomposition into plane waves in the frequency (/spl omega/) and spatial wavenumber (E) domains, followed by asymptotics on the spectral inversion integrals, with /spl omega/ synthesis performed before /spl xi/ synthesis. Special attention in the transient spectral domain is given to paraxial PB approximations and to criteria for their range of validity, which are expressed in terms of critical nondimensional estimators that contain the beam parameters as well as the dispersion parameters of the medium. The resulting PBs can be used to synthesize transient wavefields excited by arbitrary space-time source distributions of finite support on a specified aperture plane in the medium.

Exact Gaussian Beam Expansion of Time-Harmonic Electromagnetic Waves

The present contribution is concern with an exact frame-based expansion of planar initial time-harmonic electromagnetic fields. The propagating field is described as a discrete superposition of tilted and shifted electromagnetic beam waveobjects over the frame spatial-spectral lattice. Explicit asymptotic expressions for the electromagnetic Gaussian beam propagators are obtained for the commonly used Gaussian windows.

Parameterization of the Tilted Gaussian Beam Waveobjects

Novel time-harmonic beam fields have been recently obtained by utilizing a non-orthogonal coordinate system which is a priori matched to the field’s planar linearly-phased Gaussian aperture distribution. These waveobjects were termed tilted Gaussian beams. The present investigation is concerned with parameterization of these time-harmonic tilted Gaussian beams and of the wave phenomena associated with them. Specific types of tilted Gaussian beams that are characterized by their aperture complex curvature matrices, are parameterized in term of beam-widths, waist-locations, collimationlengths, radii of curvature, and other features. Emphasis is placed on the difference in the parameterization between the conventional (orthogonal coordinates) beams and the tilted ones.

Local spectrum analysis of field propagation in an anisotropic medium. Part I. Time-harmonic fields

The phase-space beam summation is a general analytical framework for local analysis and modeling of radiation from extended source distributions. In this formulation, the field is expressed as a superposition of beam propagators that emanate from all points in the source domain and in all directions. In this Part I of a two-part investigation, the theory is extended to include propagation in anisotropic medium characterized by a generic wave-number profile for time-harmonic fields; in a companion paper [J. Opt. Soc. Am. A 22, 1208 (2005)], the theory is extended to time-dependent fields. The propagation characteristics of the beam propagators in a homogeneous anisotropic medium are considered. With use of Gaussian windows for the local processing of either ordinary or extraordinary electromagnetic field distributions, the field is represented by a phase-space spectral distribution in which the propagating elements are Gaussian beams that are formulated by using Gaussian plane-wave spectral distributions over the extended source plane. By applying saddle-point asymptotics, we extract the Gaussian beam phenomenology in the anisotropic environment. The resulting field is parameterized in terms of the spatial evolution of the beam curvature, beam width, etc., which are mapped to local geometrical properties of the generic wave-number profile. The general results are applied to the special case of uniaxial crystal, and it is found that the asymptotics for the Gaussian beam propagators, as well as the physical phenomenology attached, perform remarkably well.