## 1. Introduction and Problem Scope

[2] Gaussian beam summation (GBS) formulations are an important tool in wave theory since they provide a systematic framework for ray-based construction of spectrally uniform local solutions in complex configurations. In these formulations, the source field is expanded into a spectrum of collimated beams that emanate at a given set of points and directions in the source domain, and thereafter are tracked locally in the medium. The advantages of the beam formulations are: (1) The beam propagators can be tracked locally in inhomogeneous media or through interfaces, and unlike rays, they are insensitive to the geometrical optics (GO) transition zones; and (2) the formulations are a priori localized in the vicinity of the GO skeleton, since only those beams that pass near the observation point actually contribute there. Thus beam representations combine the algorithmical ease of GO with the uniform features of spectral representations [*Červený*, 1985; *Babich and Popov*, 1989].

[3] An important feature of the continuous beam spectra is that they are overcomplete and thus may be a priori discretized, for example, via the critically complete Gabor series representation [*Bastiaans*, 1980; *Einziger et al.*, 1986; *Maciel and Felsen*, 1990; *Steinberg et al.*, 1991; *Arnold*, 1995]. The Gabor-based representations, however, are inconvenient for UWB applications for the following reasons [*Shlivinski et al.*, 2004]: (1) The expansion coefficients are nonlocal and unstable, and (2) the beam lattice is frequency-dependent (i.e., a different set of beams needs to be tracked for each frequency). The second reason makes the Gabor-based formulations inapplicable for UWB applications, while the first makes them inconvenient even for monochromatic fields [*Lugara and Letrou*, 1998].

[4] The UWB-GBS formulation [*Shlivinski et al.*, 2004] overcomes these difficulties by using an overcomplete frame representation [*Daubechies*, 1992] which removes the critical completeness constraints of the Gabor formulation. It has the following desirable features: (1) It utilizes a frequency-independent lattice of beams that emerge from a discrete set of points and orientations in the aperture (double-line arrows in Figure 1); (2) it utilizes analytically trackable isodiffracting Gaussian beams (ID-GB), whose propagation parameters are frequency-independent and need to be calculated only once and then used for all frequencies; and (c) the expansion coefficients are calculated via a Gaussian-windowed Fourier transform (WFT) of the source distribution and are therefore stable and localized for all frequencies. Furthermore, they extract the local spectral properties of the source and therefore enhance those propagators that conform with the local radiation properties of the source, whereas all others can be neglected a priori.

[5] These favorable properties are achieved since the expansion overcompleteness is inversely proportional to the frequency (see (6)). Consequently, the “basic” UWB-GBS formulation of *Shlivinski et al.* [2004] becomes increasingly less efficient at lower frequencies. The modified “multiband” formulation of *Shlivinski et al.* [2005a] overcomes this difficulty by dividing the signal bandwidth into a self-consistent hierarchy of one-octave subbands such that the beam sets at the lower-frequency bands are decimated subsets of those at the highest band. Consequently, only the set of beams at the highest band needs to be traced, while for the lower bands one uses properly decimated subsets (see Figure 2). Because of the decimation, the overcompleteness is maintained relatively small even when ω_{max}/ω_{min} is much larger than 2, where ω_{max} and ω_{min} are the highest and lowest frequencies in the band, respectively.

[6] In the present work, the two UWB-GBS formulations discussed above are reviewed and explored in the context of calculating and optimizing field focusing near curved interfaces. This problem is important in numerous applications, such as design of dielectric lens and radome antennas, cellular wireless communication, nondestructive testings, and geophysics, to name a few. Unlike the conventional ray representation, the UWB-GBS formulations provide uniform solutions in the focus zone. Uniform solutions may also be obtained by using a Kirchhoff-type Green's function integration over the aperture, where each Green's function is traced in the medium using rays. The Green's functions formulation requires a dense integration lattice, and a ray tracing for each observation points. In the UWB-GBS formulation, on the other hand, the aperture field is expanded using a relatively small number of GB (typically less than the Landau Pollak bound [*Shlivinski et al.*, 2004]). Once their excitation amplitudes are calculated for the desired frequencies, this set of GB needs to be tracked in the medium only once.

[7] The paper is organized as follows: The “basic” and the “multiband” UWB-GBS algorithms are presented in sections 2 and 3, respectively. The application example is then examined in section 4, where, referring to Figure 1, we first synthesize an aperture distribution that focuses at a desired point **r**_{f} and then use the UWB-GBS to calculate the resulting field in the focal zone. The presentation ends in section 5 with concluding remarks that also contain problem-oriented comparisons between the proposed method with its well-collimated wide-waisted GB propagators, and Kirchoff aperture-based methods with noncollimated narrow-waisted GB propagators.