Ultrawideband beam summation algorithms for focusing near curved interfaces



[1] Two ultrawideband Gaussian beam summation (UWB-GBS) algorithms are presented and then explored in the context of calculating UWB focusing by curved interfaces. The favorable features of the basic algorithm are: (1) The same lattice of beams is used for all frequencies, and (2) it utilizes isodiffracting Gaussian beams with frequency-independent propagation parameters, which also yield stable and localized expansion coefficients for all frequencies. We then present a modified multiband algorithm that is more efficient if the signal bandwidth is larger than one octave. Here the signal bandwidth is divided 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, so that only the set of GBs at the highest band needs to be traced, while for the lower bands one may use properly decimated subsets. The numerical example demonstrates the effectiveness, the localization, and the uniformity of the UWB-GBS representation in the focal zone. Concluding remarks contain comparisons between the proposed method and Kirchhoff aperture-based alternatives.

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.

Figure 1.

Physical configuration: the UWB-GBS formulation for focusing inside a dielectric sphere of radius a and dielectric constant n0. The UWB aperture source equation image0(x, ω) is designed to focus at rf, where we use the three-dimensional coordinate frame (x, z) = (x1, x2, z). The left-hand side of the figure sketches the UWB-GBS formulation, where the UWB field due to equation image0(x, ω) is expanded into a frequency-independent lattice of beams (double-line arrows) emerging at a discrete set of points and directions xm and ξn, where m = (m1, m2) and n = (n1, n2), respectively. The arrow lengths schematize the beams' excitation amplitudes. These beams are then tracked into the dielectric sphere. For the low-frequency bands, the beam lattice is decimated as schematized in Figure 2. The ray trajectories (single-line arrows) are used in the geometrical optics synthesis (30) of the aperture distribution equation image0(x, ω) that focuses at rf.

[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 ωmaxmin is much larger than 2, where ωmax and ωmin are the highest and lowest frequencies in the band, respectively.

Figure 2.

Beam lattices for the j = 1, 2, 3 bands (left, center, and right axis, respectively): (a) interlaced x-ξ decimation and (b) interlaced ξ-x decimation.

[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 rf 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.

2. Basic UWB PS-GBS Formulation

[8] The formulation is presented in the context of radiation from an extended aperture in the z = 0 plane in a three-dimensional coordinates frame r = (x, z) with x = (x1, x2). The field in the aperture is equation image0(x, ω) and the radiated field is equation image(r; ω), where the ˆ denotes a frequency domain constituent with eiωt time dependence. The field is constrained to the band Ω = (ωmin, ωmax) that may extend over several octaves. The conventional plane wave representation for the field is described by the transform pair

equation image
equation image

where, since the theory addresses UWB fields, we use a frequency-normalized spatial spectrum coordinate ξ = (ξ1, ξ2) = kx/k instead of the conventional wave number kx = (equation image, equation image), where k = ω/c. Consequently, ξ has a frequency-independent geometrical interpretation in terms of the plane wave direction (see (3)). Equation (1) extracts the plane wave amplitude equation image0 from the data equation image0, while (2) expresses the field at z ≥ 0 as a spectral superposition of plane waves, where ζ = equation image, with Im ζ ≥ 0, being the wave number in the z direction. The plane waves propagation direction for ∣ξ∣ < 1 is described therefore by the unit vector

equation image

with (θ, ϕ) being the conventional spherical angles (° denotes a unit vector). For ∣ξ∣ > 1, ζ is complex and the plane waves in (2) are evanescent.

2.1. General Expressions

[9] The UWB beam summation formulation is constructed in the framework of the windowed Fourier transform (WFT) frames, where the aperture field equation image0(x) is expanded using a set of WFT functions. The general theory of frames is given by Daubechies [1992], while a theoretical overview for the present context is given by Shlivinski et al. [2004]. The formulation is structured upon a discrete phase space lattice

equation image

where (equation imagej, equation imagej), j = 1, 2 are the unit cell dimensions in the xj coordinates, and the index μ = (m, n) = (m1, m2, n1, n2) is used to tag the lattice points. The unit cell dimensions in x1 and x2 need not be identical as long as they satisfy the overcompleteness criterion (6) in each coordinate, yet for the sake of simplicity we chose here equation image1 = equation image2 = equation image and equation image1 = equation image2 = equation image. Since each lattice point will be associated with a beam propagator emerging from xm in the aperture in a direction that is determined by ξn via (3), we use the same lattice for all the frequencies. In addition, Xμ should be overcomplete for all ω ∈ Ω (see (6)). To this end we choose a reference frequency equation image > ωmax (typical parameters are discussed after (14)) and define Xμ to be complete at equation image, that is, the unit cell area in the x-kx phase space is 2π:

equation image

Equation (5) implies that Xμ is overcomplete for all ω < equation image since

equation image

Here ν is the overcompleteness parameter: The lattice is overcomplete for ν < 1 and critically complete in the Gabor limit ν ↑ 1. Thus scaling ν with ω as in (6) yields a frequency-independent beam lattice [Shlivinski et al., 2004]. Having constructed the overcomplete lattice, we proceed by choosing a proper window equation image(x) and construct the WFT frame in L2(equation image2) for all ω < equation image, upon the phase space lattice Xμ,

equation image

Here equation image(x) are obtained as a Cartesian multiplication of two one-dimensional windows equation imagej(xj), j = 1, 2, each one yielding a proper WFT frame in L2(equation image) associated with the lattice (equation imagej, equation imagej). The frame equation imageμ(x) is associated with a “dual frame” equation imageμ(x) that has the same form in (7) but with the “dual window” equation image(x). The latter needs to be calculated for a given equation image and lattice (equation image, equation image) for each ω ∈ Ω (see Daubechies [1992] and Shlivinski et al. [2004] for calculation algorithms for equation image). For small ν, however, one may use the analytic approximation

equation image

The frame representations of equation image0(x) for ω < equation image are given by

equation image
equation image

Here equation imageμ are recognized as the WFT of equation image0 around (xm, ξn) with respect to the window equation image. In view of their role in (9) the frame functions equation image(x) and equation image(x) are thus identified as the “analysis” and “synthesis” windows, respectively. The radiated field equation image(r) for z > 0 is obtained by propagating the windows equation imageμ in (9a), giving

equation image

where the “beam propagator” equation imageμ(r) are the radiated fields due to the distributions equation imageμ(x) in the aperture. They may be described, for example, by the plane wave representation (2),

equation image

where equation imageμ = equation image(ξnequation image)eikξ·mequation image is the spectrum of equation imageμ, with equation image being the spectrum (1) of the “mother” window equation image. If equation image is wide on a wavelength scale then equation imageμ(r) behave like collimated beams emerging from xm in the z = 0 plane in the directions (θn, ϕn) defined by ξn via (3). Propagating beams occur only for ∣ξn∣ ≲ 1 − Δξ, where Δξ is the spectral width of equation imageξ ≪ 1 for collimated beams). For ∣ξn∣ ≳ 1, equation imageμ decay exponentially with z and are therefore neglected in (10). Thus, in practice (10) sums only terms with ∣ξn∣ < 1, thus expressing the field as a discrete superposition of beams emerging from all lattice points xm and real directions ξn. The number of terms in the beam summation (10) can be further reduced by applying source and observation localizations in the phase space. Source localization is affected by the WFT in (9b) that extracts the local feature of equation image0, and therefore enhances only the excitation amplitudes equation imageμ of the beams that match the local radiation direction of equation image0. Viewed from a phase space perspective, these equation imageμ reside near the Lagrange submanifold, which is the phase space skeleton of geometrical optics, and near the diffraction submanifold, which correspond to diffracted rays that emerge from discontinuity points in equation image0. These properties have been established theoretically by Steinberg et al. [1991] and Arnold [1995] via an asymptotic evaluation of the WFT (9b), as well as numerically [Shlivinski et al., 2004].

[10] The observation localization implies that the phase space summation in (10) is constrained to the vicinity of the observation manifold that defines the initiation points and directions of the beams whose axes pass near r. Actually, one needs to include beams passing within a three beam widths distance from r (see examples by Steinberg et al. [1991] and Shlivinski et al. [2004]).

2.2. Special Case: Isodiffracting Gaussian Beams (ID-GB)

[11] The ID-GB windows are our favorable windows for the following reasons: (1) They yield convenient GB propagators that are analytically tractable in inhomogeneous media; (2) the ID property implies that their propagation parameters (e.g., Γ in (17)) are frequency-independent and need to be calculated only once for all the frequencies in the band; (3) in view of the second reason they may be transformed into closed-form solutions in the time domain (the so-called ID pulsed beams [Heyman and Felsen, 2001; Heyman, 2002]); and (4) their width is scaled with ω so that they yield the snuggest frames for all ω (see (13)). The simplest isodiffracting window has the form

equation image

The frequency-independent parameter b is identified as the collimation (Rayleigh) distance of the resulting GB (see (20)), hence the designation isodiffracting (ID) since all frequency components diffract at the same distance. A snug frame is obtained if the window equation image is matched to the lattice in the sense that it provides a balanced spatial and spectral coverage of the unit cell, that is, Δxξ = equation image/equation image, where Δx and Δξ are the spatial and spectral widths of equation image. For the window in (12), Δx = (b/k)1/2 and Δξ = (kb)−1/2, giving

equation image

where in the second relation we used (5). Choosing b via (13) yields the snuggest frame for all ω ∈ Ω.

2.2.1. Choosing the Expansion Parameters

[12] The expansion above depends on four parameters equation image, equation image, equation image and b, but choosing two of them determines the other two via (5) and (13). Typically we choose equation image and b and then obtain equation image and equation image. Following (5), we set equation image = Pωmax, where P > 1 is a constant to be chosen as a trade-off between the desired accuracy and the numerical efficacy. From (6), ν(ω) = P−1equation image so that P should be as small as possible to minimize the oversampling and increase the efficiency. For analytic simplicity, on the other hand, P should be sufficiently large, making ν(ω) small enough for all ω < ωmax so that the dual ID window equation imageID can be approximated analytically by (8), giving

equation image

The error implied by (14) is studied by Shlivinski et al. [2004, Figure 4]. The formulation is still valid if one uses a smaller P, but this may require the use of the exact equation image instead of (14). To determine b we recall that the width of the window (12), W0 = equation image, should be large on a wavelength scale to yield collimated propagators, leading to the condition kminb ≫ 1. A more refined condition is obtained by accounting for the beam inclination angle θn, in which case the effective window width is W0 = equation image cos θn (see (20)), leading to

equation image

This condition should be satisfied even at the largest θn in the source spectral (directional) spread [Shlivinski et al., 2004, Figure 7]. In choosing b one should also consider the propagation environment, as it is required that the beam propagation could be modeled paraxially along its axis (e.g., if the beam hits an interface as in section 4, the projected beam width on that interface should be smaller than the interface radius of curvature).

2.2.2. GB Propagators

[13] If condition (15) is satisfied, the integral (11) for equation imageμ can be evaluated asymptotically [Melamed, 1997]. For a given phase space point μ, the result can be expressed by utilizing the beam coordinates (σ, η1, η2) associated with this point: σ is a coordinate along the beam axis that emerges from xm in the z = 0 plane in the direction equation imagen. The coordinates η = (η1, η2) transverse to the beam axis are chosen such that the projection of equation image1 on the z = 0 plane coincides with the direction of ξn, while equation image2 = equation image × equation image1. With this choice, the linear phase ξn · x implied by the window function in (7) is operative in the η1 direction but not in the η2 direction. These coordinates are related to the global coordinates via

equation image

where the rotation matrix is defined so that the transversal coordinates are rotated about the equation imagen axis so that ξn · η2 = 0. Along this axis, Bμ can be expressed in the standard form of an astigmatic GB field in a homogeneous medium

equation image

where Γ(σ) is a 2 × 2 complex symmetric matrix with Im Γ(σ) positive definite, so that the imaginary part of the quadratic form ηTΓ(σ)η = η12Γ11 + 2η1η2Γ12 + η22Γ22 increases away from the σ axis, leading to the Gaussian envelope of the beam field. In free space Γ(σ) is calculated from the initial condition Γ(0) via

equation image

where I is the unit matrix. One observes that indeed Im Γ(σ) is positive definite for all σ provided that Im Γ(0) is positive definite. In inhomogeneous medium, Γ(σ) is calculated by solving a proper Ricatti-type differential equation (the so-called “dynamic ray tracing equation”) along the ray trajectory that defines the beam axis [Červený, 1985; Babich and Popov, 1989; Heyman, 1994]. In the coordinate system defined above Γ(0), and thereby Γ(σ), is diagonal:

equation image

Substituting in (17)–(18), equation imageμ is given explicitly by

equation image

This expression is an astigmatic GB with principal axes η, ℓ = 1,2, a waist at σ = 0 and collimation distances F. The astigmatism is caused by the beam tilt which reduces the effective initial beam width in the η1 cut by a factor cos θn (see (15)), leading to the collimation distances F in (19). The beam characteristics in the η cuts are found by separating the corresponding phase term in (20) into real and imaginary parts in the form

equation image

giving the beam width equation image = equation imageequation image with waist equation image = equation image and the wave front radius of curvature R = σ + F2/σ.

[14] The generic ID-GB form in (20) can be extended to UWB propagation in inhomogeneous media or transmission through curved interfaces [Heyman, 1994, 2002; Heyman and Felsen, 2001]. The latter will be demonstrated in section 4, where we shall consider field tracking through curved interfaces using the GBS method.

3. Multiband Formulation

[15] As noted in connection with (14), the frame overcompleteness in our formulation increases at low frequencies like ν−1(ω) = Pequation image. Thus, although the basic formulation of section 2 can accommodate arbitrarily large bandwidths, it becomes increasingly less efficient at the low end of the frequency spectrum, where the overcompleteness increases. In this section we introduce a modified formulation, wherein the overcompleteness is maintained relatively small even when ωmaxmin is much larger than 2 [Shlivinski et al., 2005a]. This goal is achieved by dividing the excitation spectrum into subbands and then applying in each of them the basic formulation of section 2 using sparser beam sets at the lower subbands. The novel idea is to use a self-consistent scheme wherein the beam sets at the lower-frequency bands are decimated subsets of those at the highest band, so that only the set of beams at the highest band needs to be traced, while for the lower bands one may use properly decimated subsets (see Figure 2).

[16] We therefore divide their spectrum into J bands of one octave each:

equation image

where ⌊ ⌋ denotes the integer part of, and then construct a wideband beam summation formulation for each band as follows. For Ω(j) we choose the reference frequency equation image(j) to be P times the highest frequency of Ω(j), that is,

equation image

so that in each band, ν = ω/equation image(j) is bounded in the favorable range equation imageP−1 ≤ ν ≤ P−1. Following (6) we choose the phase space grid (equation image(j), equation image(j)) to be complete at equation image(j):

equation image

leading to

equation image

Equation (25) implies that (equation image(j), equation image(j)), and thereby the beam lattices for j > 1, can be constructed by an integer decimation of the lattice at the highest band j = 1 (see Figure 2). Two of the preferable decimation schemes following (25) are:

[17] 1. For interlaced x-ξ decimation, equation image(j) and equation image(j) are decimated for even and odd js, respectively (see Figure 2a), that is,

equation image

[18] 2. For interlaced ξ-x decimation, equation image(j) and equation image(j) are decimated for even and odd js, respectively (see Figure 2b), that is,

equation image

In each band, we use, again, a matched isodiffracting window equation imageID(j) with b(j) = equation image(j)/equation image(j) (see (12)–(13)). Thus, for the scheme (26a), b(j) = b or 2b for odd or even j, respectively, while for the scheme (26b), b(j) = b or b/2 for odd or even j, respectively. The dual window equation imageID(j) may be approximated using (14) with the parameters of the jth band. Finally, the field expansion for ω ∈ Ω(j) is as in (9)–(10) using the jth band frame constituents.

4. Application Example: UWB Focusing Near a Curved Interface

[19] We demonstrate the algorithm by considering UWB focusing near a curved dielectric interface. The example consists of two parts: (1) synthesis of an aperture field equation image0(x) that focuses at a given point rf (section 4.1) and (2) using the UWB-GBS algorithm to calculate the field generated by equation image0(x) at any desired region and in particular in the focal zone (section 4.2). The aperture synthesis is performed by propagating the field of a point source placed at rf back to the aperture, and then using phase and amplitude conjugation. Such approach is commonly used for propagation distortions compensation in laser optics, or in time-reversal algorithms of inverse scattering and inverse source applications (see, e.g., Fink [1992], Cassereau and Fink [1992], and Borcea et al. [2002] and the examples therein). The procedure is demonstrated on the configuration in Figure 1, where a sphere of radius a and dielectric constant n0 is placed in front of the aperture and is centered at z = a + s0 on the z axis. For simplicity we choose rf on the z axis at z = s0 + d0. The excitation band is between kmin = 0.25 and kmax = 1, and the configuration parameters are a = 300, n0 = 3, s0 = 200 and d0 = 400. The aperture in the z = 0 is circular with radius A = 300.

4.1. Aperture Synthesis

[20] To synthesize equation image0(x) we place a source at rf and calculate the GO field equation imagef(r) on the z = 0 plane via

equation image

where, referring to Figure 1, d and s are the distances along the ray from rf to the interface and from there to the point x on the z = 0 plane. Here αi and αt are the angle of incidence and transmission of the ray, which are related via Snell's law n0 sin αi = sin αt,

equation image

which is the ray transmission coefficient from the sphere to the outside (the boundary condition at the interface is continuity of the field and its normal derivative). Here ρ2,1 are the radii of curvature of the transmitted phase front at the interface in the plane of incident and normal to it, and are given by

equation image
equation image

The aperture field is now synthesized via

equation image

and 0 otherwise, where equation imaget) is the transmission coefficient into the sphere due to a ray impinging on it from the outside at an angle αt, and is given by (28) with n0 cos αi in the numerator replaced by cos αt. The aperture field equation image0 in (30) generates a field that, within the GO approximation, focuses at rf, such that all the rays have the same strength as they converge toward rf.

4.2. Applying the GBS Algorithm

[21] Following section 3 we divide the excitation into two bands: Ω(1) = (0.5, 1] and Ω(2) = (0.25, 0.5]. For Ω(1) we choose equation image = 3 and b(1) = 500, leading via (13) to (equation image, equation image) = (32.3, 0.0647). This value yields kb = 250 at the lowest frequency in that band, so that from Shlivinski et al. [2004, Figure 7] the error of the beams emerging from the aperture at angles smaller than 45° is bounded by 5%. For the lower band we have from (23)equation image(2) = 1.5 and, choosing the x-ξ decimation scheme in (26a), we obtain b(2) = 2b(1) = 1000 and (equation image(2), equation image(2)) = (64.6, 0.0647).

4.2.1. Expansion Coefficients

[22] The expansion amplitudes equation imageμ(ω) (the beam amplitudes) are calculated via the two-dimensional WFT (9b) using the dual window equation imageID of (14). To economize the calculations one may consider only those μ that are concentrated near the Lagrange and the diffraction manifolds discussed in section 2.1. All other equation imageμ(ω) are negligibly small and may be ignored a priori. To check the calculated coefficients, we calculated the reconstructed aperture field for k = 1 and 0.25 via (9a). Figure 3 depicts radial cuts of the reconstructed field at angles ϕ = 0, 30°, and 45° in the z = 0 plane, and compare them with the original radially symmetric distribution. Note that for k = 0.25 we used an x-decimated lattice as discussed in (26a).

Figure 3.

Radial cuts of the reconstructed field in the z = 0 plane at angles ϕ = 0, 30°, and 45° (dotted, dashed, and dash-dotted lines, respectively), compared with the original distribution (solid line). (a) k = 1. (b) k = 0.25.

[23] To demonstrate the phase space resolution of the method we show in Figure 4 the expansion coefficients ∣equation imageμ(j)∣ for k = 1 and 0.25. They are plotted in the (m1, n1) plane for slices of the phase space defined by specific values of (m2, n2). Actually the axes are normalized according to the physical space coordinates (x1, ξ1), wherein one readily observes the (m1, n1) lattice as well as the x-decimated lattice for k = 0.25.

Figure 4.

Expansion coefficients ∣equation imageμ(j)∣ in dB at k = 1 and at k = 0.25. For k = 1, the coefficients are shown in the (m1, n1) plane for (a) (m2, n2) = (0, 0) and (b) (m2, n2) = (6, 0). For k = 0.25, they are shown in the (m1, n1) plane for (c) (m2, n2) = (0, 0) and (d) (m2, n2) = (3, 0). The axes are the physical space coordinates (x1, ξ1). Note the decimated lattice at k = 0.25.

[24] For k = 1 the coefficients are shown in the (m1, n1) plane for (m2, n2) = (0, 0) and for (m2, n2) = (6, 0) (Figures 4a and 4b, respectively). They correspond to beams emerging, respectively, from the lines x2 = 0 and x2 = 6equation image(1) ≃ 194, with no ξ2 inclination (i.e., n2 = 0). In Figure 4a, the dominant coefficients are concentrated near the Lagrange submanifold corresponding to the ray trajectories in Figure 1, and near the diffraction submanifold m1 = ±9 (i.e., x1 ≈ ±300) corresponding to the edge-diffracted beams at the aperture truncation. In Figure 4b, the coefficients near the Lagrange manifold are weaker because of the fact that the dominant beams for x2 = 194 have both ξ1 and ξ2 inclination and therefore are not included in the phase space slice n2 = 0 depicted. The coefficients near m1 = ±7 correspond to the x1 ≃ ±226 truncation of the circular aperture at x2 ≃ 194. These coefficients are weaker than those in Figure 4a. This is consistent with Keller's geometrical theory of diffraction (GTD), which implies that the strongly diffracted beams here are normal to the edge axis and therefore have both ξ2 and ξ1 inclination and are not included in the phase space slice depicted. Similar observations are discerned from Figures 4c and 4d for k = 0.25, where the main difference is the x-decimated lattice. The coefficients are depicted in the (x1, ξ1) plane for (m2, n2) = (0, 0) and for (m2, n2) = (3, 0), respectively (m2 = 3 corresponds to x2 = 3equation image(2) ≃ 194, that is, the same x2 as in Figure 4b). Finally it should be noted that k = 0.25 could have been regarded as the highest frequency in band j = 3 (see (22)); hence we could have decimated here both in x and in ξ (see (26) and Figure 2a).

4.2.2. Propagators

[25] The beam propagators Bμ emerging from the aperture are given initially by (17). The beams Bμt(r) that are transmitted into the sphere are given by

equation image

Here Bμi(rs) is the value of the incident beam (17) at the point rs where its axis intersects the interface. equation imagei) is the transmission coefficient into the sphere discussed after (30), with θi being the angle of incidence of Bμi at rs. The last term in (31) has the canonical form (17) of a GB propagating in a uniform medium n0, with σt being the coordinate along the transmitted beam axis inside the sphere, measured from the interface, while ηt = (η1t, η2t) are the transversal coordinates. The only term to be determined is the 2 × 2 complex symmetric matrix Γt(0) at σt = 0 (on the interface), since for σt > 0 it is calculated from Γt(0) as in (18). It is calculated via a phase-matching procedure [Heyman, 1994, 2002; see also Deschamps, 1972], giving

equation image

where, as before the superscript T denotes a transpose. Here K is the 2 × 2 interface curvature matrix at rs with respect to the forward normal equation image, defined by equation image · equation imagei > 0, where equation imagei is the incident beam direction (here, K = −a−1I). Γi(0) is the matrix of the incident beam on the interface, as given by (18) with (19). The matrices Θi,t are the projections of the local Cartesian coordinates y = (y1, y2) of the interface at rs onto the beam coordinates ηi,t, that is,

equation image

Referring to Figure 5, simple expressions for these matrices are obtained if the y coordinates are chosen such that equation image1 is normal to the plane of incidence (equation image, equation image) and equation image2 = equation image × equation image1 is in that plane. For the transmitted beam we may use the plane of incidence coordinate system ηt depicted in Figure 5, where equation image1t = equation image1 and equation image2t = equation imaget × equation image1t, giving Θ1,1t = 1, Θ2,2t = cos θt and Θ1,2t = Θ2,1t = 0. For the incident beam, the principal coordinate system ηi of (16), wherein Γi is diagonal as in (19), does not coincide with the plane of incidence coordinate system as illustrated in Figure 5 (since the rectangular lattice of beams emerging from the aperture does not conform with the axial symmetry of the configuration); hence Θi is calculated here via (33).

Figure 5.

Beam transmission at dielectric interfaces.

[26] The beam transmission rule (32) is invalid for nearly grazing beams, but such beams are not excited by the present aperture distribution, where the beams that are strongly excited are near the geometrical optics skeleton in Figure 1. Otherwise, the accuracy of (32) has been verified by Lugara et al. [2003] by a comparison with numerical solutions for transmission from dielectric to air through a concave interface. Acceptable accuracy (error level lower than −20 dB) has been obtained with beam diameters at the interface as large as half the radius of curvature, and with incidence angles up to half the critical angle. Smaller errors have been obtained for narrower, more collimated beams.

4.2.3. Field Synthesis

[27] Figure 6 depicts the field in the focal zone near rf = (0, 0, 600), calculated using the UWB-GBS algorithm at two frequencies: k = 1 and k = 0.25. For k = 0.25 we used the decimated lattice of (26a).

Figure 6.

Field ∣equation image(r)∣ in the focal zone about rf calculated using the UWB-GBS algorithm. (a) k = 1. (b) k = 0.25.

5. Concluding Discussion

[28] We presented two ultrawideband Gaussian beam summation (UWB-GBS) algorithms and explored them in the context of UWB focusing by curved dielectric interfaces. The solution strategy and the trade-offs between the algorithms were summarized in section 1 and explored in detail in sections 2 and 3. In section 4 we have demonstrated that the proposed method works accurately and efficiently, and produces uniform solutions for the class of focusing problems under consideration. The methods and the numerical calculations in this paper can be easily extended to the time domain using the windowed Radon transform (WRT) frame-based pulsed beam summation method presented by Shlivinski et al. [2005b].

[29] The UWB-GBS approach should be contrasted with other ray-based formulations for aperture integration and field tracking. Clearly, conventional ray representations fail to describe the field in the focal zone, but ray summation algorithms, like those based on a Kirchhoff-type Green's function integration over the aperture, where each Green's function is tracked in the medium using rays, do provide uniform solutions. The latter, however, require a dense integration lattice, and a ray tracing for each source/observation point. A tight Kirchhoff-type representation is provided by the Gabor-based sampling algorithm of Felsen and Galdi [2003], where the sampling kernels are narrow Gaussians so that the propagators are Green's functions modulated by wide Gaussian radiation pattern. These kernels may be tracked in the medium along the rays that emerge from the sampling points in all directions, thus making this approach quite flexible in dealing with propagation scenarios that can be accommodated by ray techniques (e.g., complex undulating interfaces). It is required therefore in that approach that the local scale of the environment be sufficiently large on the wavelength scale and also on the scale of the ray's Fresnel zone (the latter is of order equation image, where equation image is typically given by (R−1 + R−1 + 2a−1 cos θi)−1 with R′ and R being the distances from the source and observer, respectively, and a is the surface local radius of curvature). Certain cases where the local features are very small compared to the wavelength can be treated by using diffraction theories like the uniformized geometrical theory of diffraction (UTD).

[30] Likewise, the collimated GBS representations in this paper are applicable only if the beam width is narrow on the medium's scale. Otherwise, one should develop diffraction models for the beam interaction with local features of the environment (see, for example, Felsen [1976], Green et al. [1979], and Suedan and Jull [1991] in the frequency domain and Heyman et al. [1993] and Heyman and Ianconescu [1995] in the time domain). Problems that involve rough or undulating surfaces whose features are smaller than the beam width may be addressed via homogenization or statistical methods that lead to a GB-to-GB scattering matrix representation of the surface [Gordon et al., 2005].

[31] An inherent difficulty in the Green's function integration approach is that the ray representation of the Green's function propagators may have caustics in the medium which disappear upon integration of the contributions from the entire aperture. This requires special care in the integration procedure since the total field should be regular. This problem does not exist in the collimated GBS representations since the GB propagators are insensitive to caustics.

[32] Finally we shall discuss certain algorithmical aspects of the UWB-GBS schemes presented here and of the Gabor-based Green's function integration approach of Felsen and Galdi [2003]. The UWB-GBS schemes require windowed Fourier transform (WFT) preprocessing of the source distribution to calculate the beam excitation amplitudes at all frequencies, followed by tracking of only the relevant ID-GB (those excited by the source). The results can then be used for all frequencies and all observation points. In the Green's function approach, on the other hand, the ray field should be retraced for each observation point, but this needs to be done only once for all frequencies. Furthermore, this approach does not require WFT preprocessing of the data since the Green's function amplitude is obtained essentially from pointwise sampling of the data (the source distribution).

[33] The discussion above clearly demonstrates that different environmental conditions, as well as different classes of source distributions (e.g., spatially or spectrally localized or distributed, narrow band or UWB), generally require different physics-based problem-matched methodologies. The UWB-GBS and the Felsen/Galdi algorithms are two representative examples.


[34] This work has been partially supported by the Israel Science Foundation under grant 216/02. E. Heyman would like to acknowledge very constructive discussions with L. B. Felsen in connection with section 5 and the overall assessment of the GBS methodology. Part of this work was carried out while one of the authors, A. Shlivinski, was a postdoctoral fellow at the Department of Electrical Engineering, University of Kassel, Germany.