Frame-based Gaussian beam summation method: Theory and applications



[1] 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.

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 equation imageequation imagex = 2π where equation image and equation imagex 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).

2. Frame Theory and Phase-Space Representation of Fields

[7] This section reviews the main concepts that are relevant for our frame-based method. Our aim is also to provide the user with guidelines for choosing the proper frame basis for a particular problem. Section 2.1 presents some general properties of frames, while section 2.2 provides further details of the windowed Fourier transform frames which are used in our field representation. The readers are referred to Daubechies [1990, 1992] for a detailed presentation of frame theory.

2.1. Frame Representation of Functions

[8] Definition 2.1 (see equation 3.2.1 of Daubechies [1990]): a family of functions {ψμ}, μ ∈ equation image, in a Hilbert space ℋ, is called a frame if there exist constants A > 0, and B < ∞ such that for all fequation image

equation image

A and B are called frame bounds. In the context of this paper, equation image is the function space L2(equation image) with the inner product 〈f, g〉 = ∫dx f(x)g*(x) for xequation image.

[9] If the set {ψμ} is a frame, it is generally overcomplete and hence it is not a basis composed of linearly independent elements. For a frame to be an orthonormal basis, its bounds A and B must be equal to 1.

[10] The frame operator &#55349;&#56494; is defined as

equation image

&#55349;&#56494; is a self adjoint operator. Rephrasing (1) in operator conventions as Af2 ≤ 〈&#55349;&#56494;f, f〉 ≤ Bf2 yields bounds on &#55349;&#56494;, Aℐ ≤ &#55349;&#56494; ≤ Bℐ where ℐ is the identity operator. Since the lower bound of &#55349;&#56494; is greater than zero, equation image has an inverse. Applying &#55349;&#56494;−1 to {ψμ} yields the set {φμ}:

equation image

which is also a frame with bounds B−1, A−1 and frame operator equation image. {φμ} is the dual frame of {ψμ} and &#55349;&#56494;−1 is the dual frame operator.

[11] The frame representation for f is then given by either of the following expressions as

equation image

The coefficient calculation with the aid of the dual function provides minimal sum of coefficients squared [Daubechies, 1992].

[12] Once an appropriate frame {ψμ} has been chosen, numerical implementation of the frame representation requires the calculation of its dual {φμ} via (3). This computation involves the inversion of the frame operator. The most common approach is an iterative procedure which is briefly reviewed here since its convergence properties add important insight into the properties of the frame.

[13] Let us first rewrite &#55349;&#56494; as

equation image

where C is a constant to be defined below and ℐ is the identity operator. Expanding &#55349;&#56494;−1 in a Neumann series yields

equation image

[14] This series converges if ∥ℛ∥ ≤ 1. Since from (1), ∥ℛ∥ ≤ max{∣1 − CA∣; ∣1 − CB∣} [Gröchenig, 1993] it follows that 0 ≤ C ≤ 2B−1. Using the frame definition (1), the optimal value is found to be Copt = 2/(A + B) giving

equation image

Thus the series converges fast if the ratio B/A ↓ 1 so that ∥ℛopt∥ ↓ 0. Such a frame is called snug whereas if B/A = 1 the frame is called tight. For a snug frame, one has φμCoptψμ. In general, A and B are not known analytically, but it is possible to use an analytic approximate value for Copt.

2.2. Windowed Fourier Transform Frames

[15] In L2(equation image), two types of frames are mostly used: the wavelet frames and the Weyl-Heisenberg frames (also termed the windowed Fourier transform frames or the Gabor frames if a Gaussian window is used). In this paper we shall be interested in the latter. These frames are constructed from a single window function, which is translated both in the spatial and spectral domains. Choosing a proper window function ψ(x) in L2(equation image),with xequation image, the frame elements ψμ are given by:

equation image

where (equation image, equation imagex) are the spatial and spectral dimensions of the phase-space unit cell, respectively. For the set {ψμ} to be a frame it is necessary that the unit-cell area equation imageequation imagex satisfies

equation image

The parameter ν describes the overcompleteness or the redundancy of the frame and is called the oversampling factor. The frame is overcomplete for ν < 1 and it is complete in the limit ν ↑ 1.

[16] In general, it is desired for ψ to be localized in both the spatial and the spectral domains, so that the frame {ψμ} will consist of localized constituents in the phase space. From the Balian-Low theorem [Balian, 1981; Low, 1985; Daubechies, 1992; Feichtinger and Strohmer, 1998] it then follows that in this case ν must be strictly less than 1.

[17] The properties of the frame depend also on the “match” parameter ρ that describes how well the spatial and spectral distributions of ψ fit into the phase-space lattice. This parameter can be expressed as

equation image

where Δx = ∥xψ(x)∥/∥ψ(x)∥ and Δkx = ∥kxequation image(kx)∥/∥equation image(kx)∥ are the spatial and spectral widths of ψ, with the tilde denoting the Fourier transform. If ρ = 1, the spatial and spectral window coverage of the unit cell are balanced, and we shall say that the frame is balanced.

[18] Finally, the dual frame (3) has the form

equation image

and the frame representation for f(x) is then given by

equation image

2.3. Windowed Fourier Transform Frames With Gaussian Windows

[19] According to the Balian-Low theorem, the use of a Gaussian window function to construct a frame leads to the oversampling constraint ν < 1. In the limit ν = 1, the dual series has the form as in the left expression in (11) with φ being now a biorthogonal function, generally denoted γ [Bastiaans, 1980]. Unfortunately, as it appears in Figure 1f, this function is nonlocalized, very irregular and, in fact, it is not L2-integrable, even in the weak sense. These properties of γ seriously affect the localization and the numerical stability of the coefficients aμ = 〈f, γμ〉 in (12).

Figure 1.

The dual functions φ(x) and φ(0) (solid and dotted lines) corresponding to a normalized Gaussian window ψ(x) = π−1/4ex2/2 for six different values of ν: (a) ν = 0.25, (b) ν = 0.375, (c) ν = 0.5, (d) ν = 0.75, (e) ν = 0.95, and (f) ν = 1. In all cases the phase-space grid is chosen to match to the window with ρ = 1, giving equation image = equation imagex = equation image, and the unit cell boundaries at ±equation image/2 are depicted as two vertical lines. The dotted line is the large oversampling (small ν) approximation φ(x) ∼ (ν/∥ψ∥2)ψ(x) [Shlivinski et al., 2001a]. As ν increases, the support of φ expands and for ν ↑ 1, φ tends to the Gabor biorthogonal function which is irregular and covers the entire domain.

[20] In the frame decomposition with (ν < 1) on the other hand, the dual functions appear to be smooth and well localized, leading to stable and localized coefficients aμ. Figure 1 depicts the dual functions for different values of the oversampling factor ν. For each case, the Gaussian windows are chosen to match the lattice with ρ = 1. For smaller ν the dual function appears to be more localized and closer to the Gaussian window function. In this case, the frame tends to be snugger and the iterative algorithm (6) for the dual function converges faster. The effect of the match parameter ρ for a given ν < 1 is illustrated in Figure 2: the dual function is the smoothest and most localized for ρ = 1. Balanced oversampling is thus the preferable choice in view of computational efficiency.

Figure 2.

The dual function φ corresponding to the Gaussian window in Figure 1 for frames all with ν = 0.5 but with three values of ρ: ρ = 0.5, 1, and 2 (dotted, solid, and dash-dotted lines, respectively). The corresponding unit cell dimensions are described by equation imageequation image = equation imagex/equation image = equation image.

[21] The choice of the oversampling factor value ν, however poses a tradeoff: for smaller ν, the dual function calculation is faster and the expansion coefficients become more localized, yet the sampling density and the number of elements to compute are larger. Parameter ν can therefore be used as a free parameter and adjusted according to the constraints of the problem to solve, as will be illustrated in sections 3.1 and 4. This gives the frame decomposition method a degree of flexibility, which is one of its most interesting features.

3. Frame-Based Gaussian Beam Summation Method

[22] The windowed Fourier transform frames presented in the previous section provide a rigorous framework for a beam summation representation for time-harmonic electromagnetic radiation problems. In this context, Gaussian windows are favored not only because of their mathematical properties, but also since they give rise to Gaussian beam fields that can be propagated locally in the medium. We shall briefly review the formulation in the context of radiation from 2D aperture distributions located in the z = 0 plane in a 3D coordinate space r = (x, z), with x = (x1, x2) being the transverse coordinate. For simplicity, we consider the scalar field u(r) and denote the aperture distribution as u0(x). A time harmonic dependence eiωt is assumed and suppressed.

[23] The 2D phase-space lattice in the aperture plane is defined by:

equation image

where equation imagej and equation imagexj, j = 1, 2 are the translation steps in the spatial and spectral domains, along the xj axes, and we use conveniently the vector index notation m = (m1, m2), n = (n1, n2) and μ = (m, n). Note that the x1 and x2 lattices need not be identical, but both should satisfy equation imagejequation imagexj = 2πνj with νj < 1.

[24] Extending the 1D frame set of section 2.2, we introduce now a 2D frame set

equation image
equation image

where the window ψ(x) and its dual φ(x) are obtained from a Cartesian multiplication of the 1D functions, i.e., ψ(x) = ψ1(x12(x2).

[25] Referring to the synthesis equation (12), the frame representation of the field distribution u0(x) in the z = 0 plane is given by

equation image

The field radiated into the half-space z > 0 is obtained by replacing ψμ by the beam propagator Bμ(r), resulting in

equation image

where propagating beams occur only in the visible spectrum ∣kxn∣ < k with k = ω/c, c being the speed of light. The beam propagators Bμ are

equation image

where the plane wave spectrum of ψμ is given by equation imageμ(kx) = equation image(kxkxn)eikx·xm with equation image being the spectrum of ψ, and kz = equation image is the spectral wave number in the z direction. Since ψμ are localized about (xm, kxn) in the phase space, Bμ describes a beam field that emerges from xm in the z = 0 plane in the direction

equation image

where (θ, ϕ) are the sperical angle coordinates with respect to the z axis.

[26] To be specific, we choose an square lattice with equation image1 = equation image2 = equation image and equation imagex1 = equation imagex2 = equation imagex and a symmetrical Gaussian window that is expressed as

equation image

where b > 0 is a real parameter so that ψ has a Gaussian decay with increasing ∣x∣. From (10) the match parameter is readily found to be ρ = bequation imagex/kequation image, hence for a balanced window with ρ = 1, the optimal value of b should be related to (equation image, equation imagex) via

equation image

Two important characteristics are the e−1 waist-width and diffraction angle of the resulting Gaussian beam, given by

equation image

respectively. For collimated beam, it is therefore required that the Gaussian window will be wide on the wavelength scale, i.e., kb ≫ 1. In this case, the integral (16b) can be evaluated asymptotically via saddle point integration as detailed by Melamed [1997] and Heyman and Melamed [1998]. One finds that Bμ is an astigmatic Gaussian beam that can be described in the most physically appealing format by utilizing the beam coordinates (xb1, xb2, zb) defined for a given phase-space point μ. The beam axis zb emerges from xm in the z = 0 plane in the direction κn. Along this axis, the astigmatic Gaussian beam field can be expressed in the form of a complex ray fields using the same formalism as in geometrical optics but with complex curvature matrices:

equation image

where Γ(zb) is the 2 × 2 complex symmetrical curvature matrix of the complex ray, so that xbTΓ(zb)xb is a quadratic form. If the transverse coordinates are rotated about the zb axis such that xb2 · kxn = 0, then Γb is diagonal: Γ−1(zb) = Γ−1(0) + zbI where I is the identity matrix and the diagonal elements of Γ−1(0) are equal to ib(kzn/k)2 and ib.

[27] Closed form expressions for tracking the Gaussian beam fields (21) through complex environments are well known. In smoothly inhomogeneous media, the beam follows the curved ray trajectory while the complex curvature matrix Γ satisfies a matrix Ricatti equation along this ray (and thereby determines the final field solution). Reflection and transmission at curved interfaces are treated by local matching of the Γ matrices corresponding to the incident, reflected, and transmitted beams, taking into account the curvature matrix of the interface and the direction of incidence. The result can be expressed in terms of the ABCD law applied to the complex curvature matrix. The accuracy of beam transmission rules has been verified by comparison with the MOM [Lugara et al., 2003]. It was found that in the case of transmission from dielectric to air through a concave interface, acceptable accuracy (error level lower than −20 dB) was 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. More generally, the critical angle of incidence should be sufficiently far from the spectral range [θi − Θ, θi + Θ] of the incident beam, θi being the angle of incidence of the beam axis. Once the Γ matrices of the reflected and transmitted beams are found at the interface, they are tracked through the medium (whether uniform or nonuniform) as described above. In addition, the amplitudes should be multiplied by Fresnel reflection and transmission coefficients corresponding to the local polarization and angle of incidence. All these rules are given by Heyman and Felsen [2001].

3.1. Application: Lens Antennas Analysis and Design

[28] We consider a planar antenna covered with a homogenous dielectric lens with a specified curved interface with air. Such configuration is used, for example, in a substrate-lens as shown in Figure 3. We start by performing Gaussian beam decomposition of the source field in the (x1, x2) plane at the back of the lens where the planar antenna is printed. These beams are tracked through the lens and across the curved interface, following the procedure described after (21), and finally they are recombined to form the total field at a desirable distance from the lens.

Figure 3.

Physical configuration of the substrate lens antenna, and illustration of beam tracking for a single frame beam in the x10z plane.

[29] The results presented here have been obtained for the following configuration: f = 110 GHz, εr = 3.8 (fused quartz), R = 25 mm and E = 22.5 mm (quasi-elliptical configuration). The source is a double-slot antenna, whose dimensions are optimized to provide a radiation pattern with good cylindrical symmetry [Popescu, 1996]. Its aperture field is x2-polarized and can be approximated accurately enough by the following expression:

equation image

where ΠL(x) represents a square aperture function that equals 1 for ∣x∣ ≤ L/2 and zero otherwise. Lo = 1.3 mm is the length of the slots, h = 0.7 mm the distance between the slots and d = 0.04 mm the widths of the slots. The effective wave number km is equal to kequation image, where k is the wave number in the air.

[30] The radiated fields calculated with the frame-based method and with the physical optics are presented in Figure 4. For the frame based calculations, the Gaussian window parameter b was chosen to be 36λd = 50.4 mm, with λd = 1.4 mm being the wavelength within the lens. This choice guarantees spectral localization (collimation) and thus paraxiality since the 1/e beam width of the propagators at the planar source is given by W0 = equation image = 6λd/equation image = 3.35 mm (kd = 2π/λd) and is larger than λd. It also guarantees spatial localization since W0 is comparable with the source dimensions. Furthermore, b defines the collimation (Rayleigh) length of the beam propagators, hence they remain collimated as they arrive at the external interface of the lens. Their beam width there is equation image and is much smaller than the radius of curvature of the interface, thereby allowing the use the paraxial expressions for beam transmission through curved interfaces (see discussion following (21)).

Figure 4.

E plane copolar far field radiated by the substrate lens: (a) amplitude and (b) phase.

[31] The planar source field distribution (22) being a separable function of x1 and x2, its frame decomposition is the product of two 1D frame decompositions. For brevity, Figure 5 shows frame coefficients only for the x1 variation of the source field. Choosing the same frames (same window function ψ, oversampling factor ν and match parameter ρ) for the x1 and x2 variations, we get very similar frame coefficients for both 1D frames. The oversampling factor ν has been taken equal to 0.25 and to 0.75, and the corresponding frames have been chosen to be balanced. From (19), the discretization unit cell is then given by equation image and = equation image. For both values of ν, the frame coefficients appear to be well localized. Due to the relatively large width of the frame window, as compared to the planar antenna dimensions, significant values of the frame coefficients are obtained only for very few values of the spatial translation index m1: m1 = 0 for ν = 0.75, m1 = −1, 0, 1 for ν = 0.25.

Figure 5.

Frame coefficients for the double-slot antenna field representation for (a) ν = 0.75 and (b) ν = 0.25.

[32] The choice ν = 0.75 is reasonable and numerically efficient for aperture radiation representation in an unbounded half-space. In the present problem, however, total reflection at the interface limits the number of beams that can be tracked across the curved interface, and leads to a truncation of the spectral translation index (n1) summations. The beam axes of the nonnegligible propagating beams (see m1 values above) launched from the source plane and tracked through the interface (not being totally reflected) are represented in Figures 6a and 6b for ν = 0.75 and ν = 0.25, respectively. In the ν = 0.75 frame representation, the value of the frame coefficients for the first truncated beams (n1 = 4) is rather high (see Figure 5a). The choice ν = 0.25 guarantees a lower truncation level (see Figure 5b for the first truncated beams: (m1, n1) = (0, 7), (1, −9), (−1, 9)), and a better coverage of the curved interface by tracked beams (see Figure 6a). The consequence of this observation is illustrated in Figure 6c where the tangent field incident at the interface is calculated with both frame decompositions, taking into account the not totally reflected beams, and compared with a reference field calculated from the usual far-field approximation of the planar source radiation, whose spectrum is known analytically. The field incident on the curved interface of the lens is reasonably well reconstructed for ν = 0.25, which is not the case for ν = 0.75. In the latter case, oscillations are significative of a diffraction effect induced by abrupt truncation of the frame summations.

Figure 6.

The beam axes of the dominant propagating, not totally reflected, beams (a) for ν = 0.75: m = 0, n = [−3,3]; (b) for ν = 0.25: m = −1, n = [−5,8]; m = 0, n = [−6,6]; m = 1, n = [−8,5]. (c) The tangent incident field at the curved interface, calculated by Gaussian beam summation with ν = 0.75 and ν = 0.25 and by far-field approximation (reference).

[33] The copolarized far field radiated by the lens and calculated with the frame-based method (ν = 0.25), and with the physical optics (PO), are presented in Figure 4. The results obtained with the frame-based method are in good agreement with the PO integration results. This example illustrates the fact that the frame-based Gaussian beam summation method provides accurate enough results when Gaussian beams are tracked through curved dielectric interface via asymptotic approximations. It also emphasizes the capability of the method to accommodate discontinuities produced by the environment by adjusting the oversampling in order to propagate fields in a physically relevant way. It must be noted that the computation time of the frame-based method is very fast as compared with the direct PO algorithm using adaptive integration routines, but employing none of the recently developed fast algorithms [Boag and Letrou, 2002]. Also, the Gaussian beam summation method could take into account internal reflections inside the lens in a very natural and numerically efficient way. In fact, this method has been proposed for field computations in the context of indoor propagation modeling, and has been proven to be numerically efficient in that context, too [Tahri and Fluerasu, 2000].

4. Wideband Discretized Phase-Space Beam Summation Formulation

[34] A fundamental requirement in a wideband beam summation representation is to have the same beam lattice at all frequencies, so that the beam trajectories need to be calculated only once. This is not the case in the conventional Gabor expansion where the phase-space discretization lattice is constrained by the relation equation imageequation imagex = 2π. The overcomplete expansion introduced in section 3, however, not only localizes and stabilizes the expansion coefficients, but it also adds a degree of freedom to the phase-space discretization that will be utilized here to generate a frequency-independent beam lattice. Furthermore, as will be shown, using the wideband isodiffracting Gaussian window provides the snuggest frame representation for all frequencies. Another important advantage of this window is that the complex curvature matrices Γ of the beam propagators in (21) are frequency-independent and thus need to be solved for only once (this is important in particular in inhomogeneous media where these matrices are found via a matrix Ricatti equations along the curved beam trajectories).

4.1. Frequency-Independent Beam Lattice

[35] We start by introducing the normalized spectral coordinate

equation image

[36] This coordinate defines the spectral direction via the frequency independent relation (ξ1, ξ2) = sin θ(cos ϕ, sin ϕ). For a wideband beam representation it is required to have a frequency-independent discretization lattice in the (x, ξ) phase space. To construct this lattice we first choose a reference frequency equation image such that equation image > ωmax, where ωmax is the highest relevant frequency in the excitation u0(x, ω). The (x, ξ) phase-space lattice is then defined to be complete at ω = equation image, i.e., it satisfies the Gabor criterion there

equation image

where equation image = equation image/c (for simplicity we choose the same discretization along the x1 and x2 coordinates). Then we choose the same (equation image, equation image) for all ω < equation image, giving an overcomplete lattice with

equation image

where, referring to (9), ν is the oversampling parameter. The resultant phase-space lattice

equation image

is frequency independent, thereby defining the same beam trajectories for all ω < equation image.

[37] Referring to (14), the wideband frame set is given by

equation image
equation image

where ψ(x; ω) and φ(x; ω) are the 2D window function and its dual. In general ψ(x; ω) depends on ω, while for each ω, φ(x; ω) should be calculated from ψ(x; ω) subject to the oversampling parameter ν = ω/equation image at that ω. Special attention will be given next to the isodiffracting window.

4.2. Isodiffracting Gaussian Windows

[38] The isodiffracting Gaussian windows are the preferable frame functions for ultra wideband phase-space analysis in the frequency or time domain. These windows have the general form [Heyman and Melamed, 1994]

equation image

where Γ is a complex symmetric frequency independent matrix whose imaginary part Im Γ is positive definite. The exponent in (28) is of a quadratic form xTΓx = x12Γ11 + 2x1x2Γ12 + x22Γ22. The positive definiteness of Im (Γ) guarantees that (28) has a Gaussian decay as the distance ∣x∣ from the origin increases. These windows have the following favorable properties with respect to wideband applications. (1) For a given lattice (xm, ξn), Γ may be chosen to obtain the snuggest frame for all frequencies. (2) The beam parameters which are found by tracking the beam field through the medium (e.g., by solving a Ricatti equation for the complex radii of curvature along the beam axis) need to be calculated only once for all the frequencies [Heyman, 1994; Heyman and Felsen, 2001; Heyman and Melamed, 1998]. (3) This window has a convenient time domain counterpart and the resulting propagators are also known in closed form in terms of the so-called isodiffracting pulsed beams [Heyman and Melamed, 1998].

[39] For the sake of simplicity, we choose here the window ψ(x; ω) in (18) which is a real and symmetrical special case of (28). The frequency independent parameter b > 0 there defines the the collimation (or Rayleigh) length of the resultant Gaussian beams, hence the term isodiffracting. Recalling the discussion in (20), in order to obtain collimated beams that can be tracked analytically via paraxial models, we choose kb ≫ 1 for all ω in the relevant frequency band. Another condition on b is obtained if we calculate the match parameter in (10). Recalling the discussion in (19), one finds that ρ = bequation image/equation image, hence for ρ = 1

equation image

where (24) has been used in the second and third expressions. Under this relation the window is matched to the lattice for all ω < equation image.

[40] This leads to a set of rules for choosing the frame parameters (equation image, equation image), equation image and b. First, referring to Figures 1 and 2, it is preferable not to exceed ν = 0.75 (or even ν = 0.5) since φ becomes poorly localized and quite different from ψ. For a given value of ωmax, this implies that equation image = equation imageωmax. Next, in many applications it is desirable that the beam propagators remain collimated even at the lowest end (ωmin) of the frequency spectrum. In view of (20), b is determined so that Θ(ωmin) = 1/equation image ≪ 1. On the other hand, b should not be too large in order to retain a certain degree of spatial localization at the lowest frequency, where W0min) = equation image. Once equation image and b are determined according to these constraints, the discretization lattice is derived via (29), giving: equation image(equation image.

4.3. Phase-Space Localization

[41] An important feature of the phase-space representation is the a priori localization of both aμ and Bμ around well defined phase-space regions. This can be achieved if the window function φμ is wide enough on the wavelength scale so that it senses the local radiation properties of the field u0 in the vicinity of xm. Assuming that the field distribution has the wideband form

equation image

where A0 and Φ0 are slowly varying functions of x on a wavelength scale, one finds that the coefficients aμ in (15) will be strong only for indices μ corresponding to phase-space points (xn; ξm) in the vicinity of the phase-space constraint

equation image

This condition defines the local radiation direction and is termed the Lagrange manifold (the phase-space manifold whereon geometrical optics resides) [Steinberg et al., 1991a; Arnold, 1995; Heyman and Felsen, 2001]. Note that the discretized Lagrange manifold ℒ(m, n) is frequency independent. Its “width”, i.e., the number of nonnegligible elements near the manifold depends on the width of φ and thereby on ω. The effective range of summation in (16a) is constrained further since only those beams that pass near r actually contribute to the field. For a given r = (x, z) the relevant indices μ are those corresponding to phase-space points in the vicinity of the observation constraint defined by [Steinberg et al., 1991a; Heyman and Melamed, 1998]

equation image

The simultaneous constraints ℒ(m, n) and equation image(m, n) implement the a priori localization.

[42] As an example, we show in Figure 7 the expansion coefficients for a wideband aperture distribution u0(x) = eikx∣2/2RΠL(x1L(x2). We choose here L = 400 and R = −400, representing a field that focuses to a cusp at z = 400. One readily observes that in the (m, n) domain the amplitude coefficients aμ are concentrated along the frequency-independent constraint n = m(equation image/Requation image) = m(b/R) where the second expression holds for a balanced frame as in (29). One also observes large amplitudes along the line mj = ±L/2equation image, representing diffracted fields that emerge from the aperture boundaries.

Figure 7.

The coefficients aμ in the (m1, n1) plane for (m2, n2) = (0, 0). The beam parameter: b = ∣R∣/2 = 200. The phase-space lattice (equation image, equation image) is determined by b via (29), giving equation image = (2πb/equation image)1/2 = 20π1/2 and equation image = (2π/equation imageb)1/2 = π1/2/10. Frequencies: (a) k = 0.25, (b) k = 0.5, and (c) k = 0.75.

5. Conclusion

[43] This paper has dealt with alternative formulations for discretized phase-space beam summation representations within the context of radiation from aperture source distributions. The conventional Gabor representation suffers from well known problems of nonlocal and unstable coefficients, originating in the distributed and irregular shape of the biorthogonal function. For wideband fields, further inconvenience is caused by the fact that the propagation trajectories of the beam propagators in the Gabor scheme are frequency dependent.

[44] It has been demonstrated that the new frame-based representation overcomes these difficulties. For monochromatic fields, the use of an overcomplete phase-space lattice smoothes the dual function and thus stabilizes and localizes the expansion coefficients (section 3). For ultra wideband (UWB) fields, the degree of freedom added by the overcomplete phase-space lattice has been used here to construct a frequency-independent phase-space lattice of beams (section 4). For each class of problems we have established the rules for choosing the window parameters in order to obtain a smooth frame representation, while accounting for the physics of the problem. In particular the snuggest representation is obtained when the window is matched to the lattice, as quantified by the parameter ρ in (10). For wideband fields, the snuggest representation for all frequencies is provided by the so-called isodiffracting Gaussian windows. These windows also simplify the calculation of the beam propagators since the beam parameters along the beam axes are frequency independent and hence need to be solved at a single representative frequency. Consequently, these propagators can be transformed in closed form into the time domain, giving the so-called isodiffracting pulsed beams [Heyman, 1994; Heyman and Felsen, 2001]. This leads to a new discretized phase-space representation for short-pulse fields directly in the time domain, which is based on a discretized local slant-stack Radon transform. Initial results of these new formulations have been reported by Shlivinski et al. [2001b], but full details will be published separately.

[45] Following a comment by one of the reviewers, we would like to point out that the representation of an aperture field by an overcomplete series of tilted Gaussian beams can also be performed numerically via an optimization procedure such as the matching pursuit algorithm [Mallat and Zhang, 1994] (see also a more heuristic approach by Chou et al. [2001]). Unlike the frame algorithm, these algorithms do not impose constraints on the aperture sampling, but this is not necessarily an advantage. Indeed, one of the important features of our new algorithm is that the sampling is constrained in such a way that it generates the same beam lattice for all frequencies. In the search algorithms, on the other hand, the beam lattice at each frequency depends on the search results, implying that a different set of beams needs to be traced through the medium at each frequency. Furthermore, the new algorithm is fast as it does not require a search routine and the series obtained has a minimum ℓ2 norm of the coefficient series (see (4)). Finally, the frame algorithm can be used in conjunction with asymptotic phase-space methods, e.g., for fast calculation of the expansion coefficients (see e.g., section 4.3).

[46] Another approach to overcome the nonlocality and instability of the expansion coefficients in the Gabor approach is to use the Daubechies-Wilson orthogonal basis as has been suggested recently by Arnold [2002]. It has been demonstrated there that the expansion coefficients in this formulation are localized in the high frequency regime, yet this may come at the expense of more complicated beam propagators. Furthermore, since the sampling rate in this formulation is fixed (two times oversampling), it cannot be used as a basis for UWB phase-space formulation like the one introduced in section 4 of the present work.

[47] It is also relevant to contrast the present method to the narrow waisted Gaussian beam method, which is reviewed in this issue [Maciel and Felsen, 2002]. The latter is basically a physical optics representation of the aperture field about relatively sparse sampling of the aperture (of the order of one fifth of a wavelength). The propagators there are therefore Green's functions multiplied by a wide radiation pattern, and cannot be considered as beam fields. The corresponding wide-angle fields can be tracked along ray trajectories, thus making this approach quite flexible in dealing with rather complex propagation scenarios that can be accommodated by the ray techniques, e.g., complex interfaces with rapidly varying curvature, provided that the radius of curvature is large on the wavelength scale. In the “beam” method presented in the present work, the propagators are well collimated hence the method is applicable only to configuration that admit this class of solutions. In particular it is required that the medium variations (e.g., changes of the interface curvature) will be slow on the scale of the beam width. Problems such as beam transition around critical angles or convex to concave transition should be addressed as isolated canonical problems [Heyman et al., 1993]. Yet for large problems, and in particular for UWB problems, this method leads to enhanced phase-space compactness and numerical saving as discussed in section 4.3.


[48] A. Boag acknowledges partial support by the Israel Science Foundation under grant 577/00. E. Heyman acknowledges partial support by the Israel Science Foundation under grant 404/98 and by the AFOSR under grant F49620-01-C-0018.