## 1. Introduction

[2] Phase-space representations are a fundamental tool in wave theory as they provide a framework for the construction of spectrally uniform solutions in configurations of increasing complexity. Of particular interest are beam-based phase-space formulations, where the source field is expanded as a spectrum of beam propagators [*Steinberg et al.*, 1991a, 1991b; *Arnold*, 1995]. The advantages of the beam formulations over the more traditional distributed (Maslov-based) phase-space formulations are (1) the phase-space data is a priori localized in the vicinity of the Lagrange manifold that forms the phase-space skeleton of geometrical optics, (2) further localization is due to the fact that only those beam propagators that pass near the observation point actually contribute there, and (3) the beam propagators can be tracked locally in inhomogeneous media or through interactions with interfaces and, unlike ray or plane wave propagators, they are insensitive to transition zones. Thus the beam representations combine the algorithmic ease of geometrical optics with the uniform features of spectral representations and, therefore, have been used recently in various applications [*Červený*, 1985].

[3] The beam representations are based on windowed configuration-spectrum transforms of the source distributions, e.g., the local Fourier transform in the frequency domain and the local-slant-stack-Radon transform in the time domain [*Steinberg et al.*, 1991a, 1991b]. An important feature of these formulations is that they may be a priori discretized on a phase-space lattice. One example is the well known Gabor-based beam algorithms that have been utilized in various applications involving radiation and scattering in complex environments or for local inverse scattering [*Steinberg et al.*, 1991a; *Bastiaans*, 1980; *Einziger et al.*, 1986; *Maciel and Felsen*, 1990; *Galdi and Felsen*, 2001; *Burkholder and Pathak*, 1991; *Chou et al.*, 2001; *Rao and Carin*, 1998, 1999; *Maciel and Felsen*, 2003]. A major difficulty in these formulations is the nonlocality and instability of the expansion coefficients that follow from the highly irregular and distributed form of the “analysis function” (the Gabor dual or bi-orthogonal function) which is used to calculate the coefficients (see section 2.3). This difficulty has been circumvented recently by using a frame-based beam summation representation [*Lugara and Letrou*, 1998], which is considered in section 3. The overcomplete nature of this representation smoothes out and localizes the dual function, ending up with stable and local coefficients at the expense of having to calculate more coefficients and trace more beam propagators. This poses a tradeoff between the oversampling ratio and the stability of the representation. A reasonable solution is found at an oversampling of order 4/3 or larger for 1D problems ((4/3)^{2} for 2D).

[4] The Gabor representation suffers from another inherent difficulty when applied to wideband field representations. Since the phase-space lattice is constrained by the condition _{x} = 2π where and _{x} are the spatial and spectral unit-cell dimensions, respectively, the beam lattice (origins and directions) changes with frequency and hence a different set of beam axes needs to be tracked through the medium for each frequency [*Steinberg et al.*, 1991b]. Following [*Shlivinski et al.*, 2001a], we introduce in section 4 a novel scheme that accommodates this difficulty: using the overcomplete frame removes the Gabor constraint, hence by a proper scaling of the spatial overcompleteness with the temporal frequency, we construct a frequency-independent beam lattice so that the same discrete set of beam trajectories is used for the entire relevant frequency spectrum. Furthermore, with a proper choice of the parameters, the isodiffracting Gaussian-beams that have been introduced in a different context [*Heyman*, 1994; *Heyman and Melamed*, 1994; *Heyman and Felsen*, 2001] provide the “snuggest” frame basis for all frequencies and remove the coefficients' instabilities. This choice also simplifies the beam calculations since the calculation of the beam parameters along the propagation axes may be done only once for a representative frequency. Consequently, these beams can be transformed in closed form into the time domain where they give rise to the so-called isodiffracting pulsed beams [*Heyman*, 1994; *Heyman and Felsen*, 2001]. Based on this property, we have also introduced in the work of *Shlivinski et al.* [2001b] a new discrete phase-space beam summation representation for short-pulse fields directly in the time domain. Further details on these new formulations will be presented elsewhere.

[5] The frame theory can also provide guidelines for the method of moments (MOM) formulations [*Boag et al.*, 1996]. Sets of expansion and testing functions comprising snug frames ensure clustering of active eigenvalues of the MOM matrices, which in turn leads to rapid convergence, even for matrices with high condition numbers. In addition, the use of overcomplete sets of expansion functions makes the task of spanning the desired solution subspaces considerably easier and allows for accurate representation of the solutions. It was found that frame-based formulations can yield both sparse matrices facilitating fast matrix vector multiplication and convergence in a small number of iterations [*Lugara and Letrou*, 2002].

[6] The paper is organized as follows: section 2 presents a review of the relevant elements of the frame theory. We start with general frame properties (section 2.1) and then emphasize the windowed Fourier transform frames that form the basis of the beam summation representation (section 2.2). Our goal is to provide the user with guidelines for choosing the proper frame expansion for a particular problem. The frame-based Gaussian beam summation method is presented in section 3, including an application example of a dielectric lens analysis, while the wideband beam summation representation is finally presented in section 4. The paper ends with concluding remarks (section 5).