Beam Summation Representation Research Articles

Beam Summation Theory for Waves in Fluctuating Media. Part II: Stochastic Field Representation

In Part I of this two-part sequence, we introduced a novel beam summation (BS) scheme for tracking waves through fluctuating media. A key step has been the proof that the set of beam propagators constitutes a frame everywhere in the propagation domain. This beam frame (BF) provides a self-consistent framework for wave tracking, where the field is expanded using the BF and the local scattering of each beam is reexpanded using the same beam-set and expressed as beam-to-beam (B2B) scattering coefficients. This theory is used here to derive a BS representation for the stochastic-field for cases where the medium fluctuations are random with a given statistics. The stochastic B2B scattering moments are therefore expressed in terms of the local spectral statistics of the medium projected onto a phase space window formed by the intersection of the excitation and the scattered beams. Since the medium statistics is typically smooth, unlike a single realization, the resulting stochastic B2B scattering matrix is compact and smooth and may actually be calculated analytically. It is demonstrated, via numerical examples and a comparison with Monte Carlo simulations, that the formulation is not only computationally efficient, but also provides a compact representation for the scattering phenomenology.

Gaussian Beam Summation Representation of Beam Diffraction by an Impedance Wedge: A 3D Electromagnetic Formulation Within the Physical Optics Approximation

We present a beam summation (BS) representation for the field scattered by an impedance wedge illuminated by a general 3D electromagnetic Gaussian beam (EM-GB). The emphasis here is not only on the solution of the beam diffraction problem, but mainly on the BS representation. In this representation, the field is expressed as a beam optics (BO) term plus an edge field, described as a sum of diffracted EM-GB's emerging from a discrete set of points and directions along the edge. We introduce an edge-fixed set of EM-GB's that provides a basis for the edge field. The expansion coefficients (the beam's excitation amplitudes) account in a dyadic format for the polarization of the incident beam and also for its direction, displacement from the edge, collimation, and astigmatism. We derive exact expressions for these coefficients as well as simpler approximations that are valid uniformly as a function of the incident beam distance from the edge. The results of this paper provide essential building blocks for a BS representation of EM fields in complex configurations, where the source excited field is described as a sum of beam propagators, and the diffracted fields generated by propagators that hit near edges are also described using beams.

A Beam Summation Representation for 3-D Radiation From a Line Source Distribution

We introduce a Gaussian beams summation (GBS) scheme for radiation from a source distribution along a line in a 3-D configuration. The expansion is based on a discrete phase-space decomposition of the source distribution along the source axis, thus describing the field as a sum of Gaussian beams (GBs) emerging from a discrete set of points along the axis, at a discrete set of directions in a polar coordinate system. Applications to GBS representation of fields diffracted by edges is 3-D configurations are discussed.

Gaussian Beam Summation Representation of a Two-Dimensional Gaussian Beam Diffraction by a Half Plane

A Gaussian beam summation (GBS) representation for half plane diffraction of an incident two dimensional Gaussian beam (GB) that hits arbitrarily close to the edge is presented. The scattered field is expanded into an angular spectrum of GBs that emerge from the edge. We derive asymptotic expressions for the diffracted beams amplitudes, that are valid uniformly as a function of the distance of the incident beam from the edge and of the scattering beams angles. In the limiting cases when the incident GB hits far from the edge, these expressions reduce uniformly to a geometrical optics model plus a weak edge diffractions due to the off-axis beam field that hits the edge. Applications to GBS modeling of indoor propagation are discussed.

A phase-space Gaussian beam summation representation of rough surface scattering

A Gaussian beam (GB) summation representation for rough surface scattering is introduced. In this scheme, the coherent and incoherent scattered fields are described by a phase-space summation of GBs that emanate from the rough surface at discrete set of points and directions. It thus involves stochastic GB2GB scattering matrices for the coherent and incoherent fields, and deterministic GB propagators. It benefits from the simplicity and accuracy of the latter, and can be used in applications involving propagation in complex scenarios comprising inhomogeneous media with rough surface boundaries. The GB2GB matrices are calculated from the statistical moments of the scattering amplitude, which are given either analytically or empirically. An analytical and numerical example for weakly rough surface is presented and discussed. Applications to the more complicated propagation scenario of doubly rough surface waveguide with multiple reflection phenomena will be presented in a follow-up publication.

A phase-space beam summation formulation for ultrawide-band radiation. II: a multiband scheme

The ultrawide-band (UWB) beam summation representation introduced in Part I of this two-part sequence, is extended here to make it more efficient for excitations with bandwidths that are larger than one octave. The main features of the basic formulation in Part I were: 1) utilization of overcomplete windowed Fourier transform (WFT) frames to construct the beam lattice (or skeleton) that is independent of the frequency; 2) the use of isodiffracting Gaussian beams (ID-GBs) provides the snuggest frame representation, and thus stable and localized expansion coefficients, for all frequencies; 3) the ID-GBs can readily be tracked in the ambient medium and, due to the ID property, their propagation parameters are calculated only once and then can be used for all frequencies. Although the basic formulation can accommodate large bandwidths, it becomes increasingly less efficient at the low end of the frequency spectrum where the overcompleteness increases. The self-consistent multiband beam-summation scheme presented here extends and generalizes the basic formulation of Part I by dividing the excitation band into one-octave sub-bands and applying the UWB beam expansion of Part I in each band. This is done, though, via a novel self-consistent scheme wherein the beam sets at the lower bands are decimated subsets of those at the highest band, so that only the set of beam-propagators at the highest band needs to be traced, while for the lower bands one uses properly decimated subsets. This approach requires less beam elements at the lower frequency bands and thus keeps the overcompleteness (oversampling) in all the bands below a given level. As in Part I, we provide the guidelines for choosing the expansion parameters and then demonstrate the effectiveness of the new scheme via a numerical example of UWB focused aperture whose frequency band spans several octaves.

Frame‐based Gaussian beam summation method: Theory and applications

A discrete phase‐space Gaussian beam summation representation for electromagnetic radiation from a planar source is presented. The formulation is based on the theory of frames and removes the inherent difficulties of the Gabor representation for both monochromatic and ultra wideband (UWB) fields. For monochromatic fields the frame‐based representation leads to an efficient and flexible discrete Gaussian beam representation with local and stable expansion coefficients. For UWB fields a novel scaling of the frame overcompleteness parameter is introduced, leading to a new expansion that utilizes a discrete frequency‐independent set of beams over the entire relevant spectrum. It is demonstrated that the isodiffracting Gaussian beams provide the snuggest frame representation over the entire spectrum. The rules for choosing the “optimal” frame and beam parameters for a given problem are discussed and demonstrated on application examples.