We have previously developed computationally efficient Gabor-based, narrow-waisted (NW), discretized Gaussian beam (GB) algorithms for the determination of the 3D vector fields radiated by 2D arbitrary vector planar aperture distributions. In the present paper, these algorithms are extended to tracking the aperture-generated GBs through polarization-sensitive reflection/transmission interactions with arbitrarily shaped 3D dielectric layers. The resulting tracking scheme is applied to predicting the performance of rotationally symmetric hemispherical and ogival missile radomes, which cover 2D plane truncated apertures on gimbaled (rotatable) platforms. Comparisons with measured data show satisfactory agreement with the analytic predictions.
 The efficient and accurate numerical calculation of wave interaction with large scale complex environments poses a major challenge in diverse forward and inverse wave propagation and scattering scenarios. It is widely acknowledged that analytic physics-based problem-matched modeling plays an important role in the construction of the relevant working algorithms. In a sequence of prototype studies [Maciel and Felsen, 1989, 1990a, 1990b, 2002a, 2002b; Galdi et al., 2001; Felsen and Galdi, 2001], we have explored a class of problems wherein the excitation is a specified frequency or time domain planar aperture distribution; the excitation due to any other given source distribution can generally be projected onto such a reference plane. For parameterization of the aperture-generated radiated fields which impinge on a complex propagation or scattering environment, we have explored a Gabor-based discretized superposition of narrow-waisted (NW) Gaussian beams (GBs), both in the high-frequency time-harmonic and short-pulse transient regimes. Although each individualized ray-like NW beam emits wide-angle radiation, these wave objects, when acting collectively, are found to have only a paraxial domain of influence, and they synthesize ray optically parameterized wave fields without the failure of conventional ray theory in shadow boundary and other transition regions. In two recent publications [Maciel and Felsen, 2002a, 2002b], we have formulated and validated the frequency domain Gabor-based NW GB algorithm for the calculation of the 3D vector electromagnetic fields radiated by coordinate-separable truncated 2D vector aperture field distributions. In the present paper, such fields are allowed to impinge on arbitrarily shaped 3D dielectric layers, with subsequent specific application to the performance of rotationally symmetric hemispherical and ogival missile radomes covering an arbitrarily placed truncated planar aperture platform.
 Recently, two new rigorous beam-based algorithms have emerged which avoid the biorthogonality difficulties associated with evaluation of the amplitude coefficients in the rigorous conventional Gabor expansion [Maciel and Felsen, 2002a], on which our NW GB scheme is based: the orthogonal Wilson-based beam expansion [Arnold, 2001], and the frame-based beam expansion [Letrou and Lugara, 2001]. The appearance of these new formulations suggests that our pragmatic approximate (but calibrated) NW GB procedure, which avoids the conventional Gabor biorthogonality problem for the relevant expansion coefficients through Gabor sampling of the aperture field profile in the NW beam limit [Maciel and Felsen, 2002b], should be reexamined from the perspective of these new rigorous orthogonal schemes. To do this will require passage to the NW limit in these new algorithms since our algorithm (which works well and efficiently in our cited problem scenarios of interest) is specifically restricted to the NW beam regime. We shall look forward to meaningful comparisons after the NW limit of the new schemes will have been established.
 The layout of this paper is as follows. Section 2 deals with previous results [Maciel and Felsen, 2002a, 2002b] required for the NW GB parameterization of the vector aperture and radiated fields. In section 3, the individual GBs are tracked through an arbitrarily shaped dielectric layer, with full account taken of the polarization-dependent 3D vector field characteristics. The tracking algorithm is structured around slightly complex paraxial ray and beam shooting methods. It is used in section 4 to synthesize the 3D vector fields transmitted from an off center, rotated rectangular aperture distribution through a hemispherical dielectric layer radome, and also from non rotated centered circular aperture distributions through two different ogival dielectric layer radomes. No independent reference solutions are available for these complex 3D propagation scenarios. Therefore, we assumed that field synthesis based on our thoroughly calibrated beam lattice “scrambling” criteria for stable and accurate 2D beam tracing from 1D aperture distributions through 2D complex environments [see Maciel and Felsen, 1990a, 1990b] can be applied locally, via the algorithm in section 3, to the 3D field/2D aperture problem for 3D complex environments. The numerical beam tracking results presented in these three examples have been verified to be stable according to the scrambling criteria, with explicit listing of the number of beams employed. These analytic predictions have been compared with experimental data for various performance measures (boresight error, power transmission) in practical missile radome configurations. Conclusions are presented in section 5.
2. NW Gabor-Based Ray-Like GB Algorithm for 3D Vector Fields Radiated by 2D Vector Planar Truncated Apertures
2.1. The Aperture Field
 In the presentation to follow, we shall employ two coordinate systems (see Figure 1): (1) a global coordinate system (x, y, z) which serves to identify a fixed observation point as well as a fixed layer configuration whose inner and outer surfaces are defined by z = g1(x, y) and z = g2(x, y), respectively, and (2) a local coordinate system (, , ) which defines an arbitrarily polarized (transverse to ) planar aperture distribution located in the (, ) plane, centered at (xa, ya, za) and rotated by (θa, φa) with respect to the (x, y, z) layer and observer coordinate system. The (, , ) coordinate system allows the arbitrary placement of the aperture with respect to the layer as is typically found in practical antenna/radome configurations with gimbaled (i.e., rotatable) antennas.
 In the 3D Gabor-based field representation, with bold face symbols denoting vector quantities, the 2D vector aperture electric field f(, ) is parameterized through the 4D (, , ) infinite phase space lattice with spatial and spectral periods and , respectively, subject to the self-consistency constraint . The spatial and spectral lattice points are indexed by (m, p) and , respectively, where , and m, n, p, q = 0, ±1, … [Maciel and Felsen, 2002a, equation (4)]. For symmetrical , NW (L/λ < 1, λ = wavelength) Gabor window functions, only the nontilted (n = q = 0) elements contribute to the radiated field; all n or q ≠ 0 spatial contributions are evanescent away from the aperture plane [Maciel and Felsen, 2002a, equation (15)]. The corresponding approximate representation of the vector aperture field in terms of a discretized set of NW coordinate-separable Gabor-based normalized Gaussian window functions is given by [Maciel and Felsen, 2002b, equation (18)]:
Furthermore, for a lowest-order approximation, we retain only the finite array of (m, p) beams that fills the aperture domain. The vector Gabor amplitude coefficients can be determined approximately by sampling the aperture profile function f(, ) at the lattice points m = mL, p = pL [Maciel and Felsen, 2002b, equation (19)]
 The n = q = 0 GBs generated by the initial aperture field distribution can be represented in the asymptotic high-frequency range by a complex source point (CSP) model [Maciel and Felsen, 2002b, equation (16)]. At an observation point r = (x, y, z), in the paraxial far zone of each basis field emanating from the aperture, this results in a CSP beam function , where denotes the CSP located at the Gabor lattice points in the aperture (, , ) coordinate system and the tilde ~ denotes an analytically continued complex quantity [Maciel and Felsen, 2002b, equation (15)]. In particular,
where the complex displacement is , and b represents the Fresnel length corresponding to L in the aperture domain. The subscript 0 on a bold face symbol denotes a unit vector.
 For subsequent beam tracking through the layered environment, it is necessary to relate the CSP locations in the (, , ) aperture coordinate system in (3a) to those in the (x, y, z) observer coordinate system. This is accomplished via the transformation
where a doubly underlined bold symbol denotes a matrix, a superscript † denotes vector or dyadic (matrix) transposition and is a 3 × 3 coordinate transformation matrix. For rotation of φa about the axis followed by rotation of θa about the transformed axis, the matrix elements are the direction cosines
whereas for rotation of θa about the axis followed by rotation of φa about the transformed axis, the matrix elements are
2.2. The Radiated Field
 The CSP beam functions mp can be more compactly represented in terms of the CSP free space Green's function f (r; ′mp) [see Maciel and Felsen, 2002b, equation (14)]. The resulting Gabor expansion for the radiated free space electric field, valid in the paraxial far zone of each beam element, can then be expressed in the (x, y, z) coordinate system in terms of the NW GB (slightly complex ray) fields as follows [Maciel and Felsen, 2002b, equation (16)]
where is a vector potential, and f is the free space Green's function
with . This expansion serves as the incident field for subsequent tracking through complex environments.
3. Reflection From, and Transmission Through, an Arbitrarily Shaped 3D Dielectric Layer
 In the presence of an arbitrarily shaped dielectric layer, the basic algorithm for tracking each of the individual NW beam fields (6b) consists of first choosing the beam Fresnel length b such that the layer lies in the far zone of each (m,p) beam element. Each beam element is then tracked through the layer by either a complex ray tracing technique or by a more computationally efficient beam shooting–paraxial approximation procedure. Initially, we present the case of real ray tracing from real point sources. The beam solution is obtained subsequently via analytic continuation of the source coordinates to complex values.
3.1. Real Ray Tracing
 The ray paths connecting a source at r′ = r(x′, y′, z′) and an external observer at P(x, y, z) in the presence of a perturbing environment are constructed using geometrical optics (see Figure 1). A ray from r′ in the (, ) plane with the initial departure angle (α, β) can be tracked recursively through the layer to some observation point Pe(xe, ye, ze) along a path consisting of ℓ external reflections at the boundary Σ1 and h internal reflections between the boundaries at Σ1 and Σ2. By a numerical search, (α, β) is varied in the above procedure until Pe(xe, ye, ze) coalesces with the observation point P(x, y, z). The resulting optical length ψN consists of N ray segments with length Sj and directions defined by the unit vector S0j which are ordered in a j-indexed sequence ((ℓ + 1) segments due to ℓ external reflections plus (2h + 1) segments due to h internal reflections plus 1 segment to the observer from the layer exit point),
The function ψN satisfies the stationary phase condition
where ej = 1 or ej = ≡ nj if Sj is outside or inside the layer, respectively; here nj is the refractive index. Moreover, the length Sj is given by
where (xj, yj, zj) is the jth impact point (1 ≤ j ≤ N − 1), (x0, y0, z0) ≡ (x′, y′, z′) is the source location and (xN, yN, zN) ≡ (x, y, z) is the observation point. At each j-indexed impact point (xj, yj, zj), the reflected (Γ) or transmitted (T) beam field is decomposed into polarized components in a local coordinate system tied to the plane of incidence which contains the incident unit vector u0ji and the unit vector N0j perpendicular to the boundary (see Figure 2). The parallel component (′) extends along the unit vector u01 which is perpendicular to u0ji in the plane of incidence; the perpendicular component (″) extends along the unit vector u02 = u0ji × u01. Each polarization has its local Fresnel reflection (Γj) or transmission (Tj) coefficient, together with divergence coefficients DjΓ or DjT. Phase knjSj accumulates along the beam axis direction u0ji from the prior impact point (j − 1) in the incident medium with refractive index nj. Accordingly, the reflected or transmitted vector potential at the jth impact point is given by
with Γ and T denoting polarization-dependent Fresnel reflection and transmission coefficients corresponding to the vector potential parallel (′) and perpendicular (″) to the plane of incidence
Here, n and n1 are the refractive indexes of the incident and transmitted media, respectively, while γ and γ1 are the incidence angles (with respect to the layer normal) for the incident and transmitted rays, respectively (see Figure 2). The effect of ray tube spreading at the (j − 1)th impact point along the jth ray segment from (xj−1, yj−1, zj−1) to (xj, yj, zj) is accounted for by the divergence coefficient DjΓ or DjT when reflection or transmission occurs, respectively, and it can be expressed in terms of local surface and wave front curvature matrices and [Deschamps, 1972]. The divergence coefficients for reflection and transmission at the jth impact point are given by
where the incident wave front curvature at point j, when reflection occurs, can be determined recursively as
with the reflected wave front curvature at the (j − 1)th impact point given by (14a). The superscript −1 denotes matrix inversion. When transmission occurs, the same equation applies, with Γ replaced by T, and use of (14b). In (12a), “det” denotes the determinant, = diag [1, 1] is the 2 × 2 identity matrix and the initial spherical wave front curvature at the first impact point is (x1, y1, z1) = S1−1. After reflection or refraction at the curved interface, one has the generally astigmatic matrices
where v = n1cosγ1 − n cosγ, and is the incident wave front curvature at the jth impact point. The surface curvature matrix can be expressed in terms of the surface function g(x, y) as [Stoker, 1969]:
where gx denotes the first partial derivative of the surface function g(x, y) with respect to x, etc. Finally, the sign conventions associated with the wave front curvature for a diverging (converging) ray tube are positive (negative); they are positive (negative) for the surface curvature for an interface which is locally convex (concave) with respect to the outward normal.
3.2. Complex Ray Tracing for Paraxial NW Ray-Like GBs
 The real ray field in (10a) and (10b) is transformed to a beam type field via the complex source point technique through replacing the real source points r′(x′, y′, z′) by the CSP ′(′, ′, ′) in (4). With the exception of the real observation point, the resulting ray path connecting source and observer generally lies entirely in complex configuration space, and it interacts with the complex extension of the perturbing environment, (, ) [Wang and Deschamps, 1974]. Implementation of the resulting rigorous algorithm, which is prohibitively difficult, can be avoided for the NW quasi-ray GBs. Motivated by our previous studies of GB-synthesized radiation from 1D apertures [Maciel and Felsen, 1990a, 1990b], which have been structured around a beam shooting and paraxial scheme, we have invoked the principle of locality to extend this algorithm to the 2D aperture/3D radiation problem. The beam shooting and paraxial scheme synthesizes the field accurately near the beam axis (paraxial region) of the NW ray-like GBs. Away from the interface, at an observation point (x, y, z) in the paraxial region near the reflected or transmitted beam axis, the vector potential field P can be approximated in terms of the field Pc at the real ray on-axis central observation point (xc, yc, zc) and a phase correction δp as (see Figure 1)
where δp is a second-order phase correction dependent on the wave front curvature matrix at the central observation point with wave front coordinates (u01, u02), and Δ is the perpendicular displacement vector between (x, y, z) and (xc, yc, zc), with † denoting the vector transpose. The calculation of the on-axis potential field Pc is performed along a ray path which proceeds from the CSP in complex configuration space up to the (real) intersection of the real beam axis with the layer. After the first impact point, the ray path follows the real beam axis and lies entirely in real configuration space. As in section 2.1, the field can be tracked successively along these real trajectories until the central observer is reached. The final field transmitted through the radome is composed of the aggregate of multiple internal/external, reflected/transmitted constituents. This algorithm has been used for the field predictions in the examples discussed below.
4. Numerical Examples and Results
 Numerical results have been generated for an extended 2D plane aperture distribution in the presence of spherical and ogival dielectric layers, using various aperture field profiles and various physical parameters. In our previous studies of 2D fields radiated from 1D layer-covered aperture distributions [Maciel and Felsen, 1990a, 1990b], the results obtained through the NW GB algorithm have been calibrated against reference solutions generated by direct numerical integration of the rigorous Kirchhoff integrals (when available). These numerical experiments demonstrated that reliable reference solutions can be furnished by NW beam synthesis subject to the “pragmatic” criterion that the outcome remains unaffected by different choices of beam/lattice parameters (this was referred to as “scramblings”). This procedure has been followed here because no other Green's function-based technique is available for reference data pertaining to the spherical and ogival layer radome. The scrambling test has been applied to all numerical results reported below to ensure their reliability.
 In the first example, the beam shooting–paraxial algorithm described in section 3.2 has been applied to the determination of the boresight error (BSE) (i.e., the displacement of the radiation pattern null) for a 2D aperture covered by a spherical dielectric layer radome of relative permittivity ε1 with inner and outer surface functions g1 (ρ) = and g2 (ρ) = , where a1 and a2 are the inner and outer radii, respectively (see Figure 3). For BSE applications, an appropriate aperture field distribution f(, ) is one that has a null along the aperture coordinate y = 0 and is discontinuous along the aperture edges ∣x∣ = dx/2. We have chosen f(, ) to be
In Figure 4, the far zone x and y component electric field patterns are synthesized about the null region in the y–z plane (ϕ = 90°). The rectangular aperture dimensions are dx = 5λ and dy = 3λ, displaced 5λ from the center of the radome to the off-axis location (xa, ya, za) = (0, 5λ, 0). The entire aperture platform is rotated mechanically through 30 degrees from the normal to the aperture in the y–z plane. The spherical dielectric layer has a relative permittivity ε1 = 5, with inner radius a1 = 10λ and outer radius a2 = 10.224λ, matched for normal incidence (i.e., n1(a2 − a1) = 0.5λ). Using different sets of NW paraxial CSP beams which fill the entire aperture (L = 0.5λ, ∣m∣ ≤ 5, ∣p∣ ≤ 2 and L = 0.25λ, ∣m∣ ≤ 10, ∣p∣ ≤ 5), or which surround only the null region (L = 0.5λ, ∣m∣ ≤ 5, p = +1) yields the same BSE, but with different depth of nulls. Note in Figure 4 that the offset and rotation of the aperture produces both x and y electric field components having different BSEs due to the polarization-dependent Fresnel reflection and transmission coefficients. In the principal plane, the 2D aperture BSE here and the 1D aperture BSE in the work of Maciel and Felsen [1990b] for the corresponding cylindrical layer radome agree as would be expected.
 The second numerical example involves a linear y-polarized 2D circular cosine monopulse antenna aperture with diameter Da = 9.6λ, and with f(, ) given by
Here, and sgn(x) = −1 or +1 for x < 0 or x > 0 is the signum function that yields the monopulse nulling pattern. The aperture is enclosed within the axisymmetric dielectric layer sketched in Figure 1, and is constructed of slip cast fused silica (ε = 3.44, tanδ = 0.001). The outer length of the layer is Lr = 23.8λ, and consists of a forward conical section of length 6.2λ tangent to a rear ogive section of length 17.6λ which is specified by gi(ρ) = , i = 1, 2. The outer base diameter is Dr = 11.9λ, and the wall thickness is constant at 0.31λ. The fineness ratio is Lr/Dr = 2.0. E-plane and H-plane boresight error predictions were generated via the beam shooting and paraxial techniques described above, using a stabilized set of NW GBs that fills the aperture (here L = 0.44λ, ∣m∣ ≤ 11, ∣p∣ ≤ 11). The beam-synthesized results were compared with measured BSE as shown in Figure 5. One notes adequate agreement between the beam-synthesized and measured results; measurement errors are estimated to be ±0.05°.
 The final numerical example involves the same configuration as the previous one but with the following change of parameters: Da = 5.2λ and f(, ) given by (18) with omission of sgn(x) (no pattern null). The axisymmetric dielectric layer now is ogival (without the conical section as in the previous example) and constructed of Pyroceram (ε = 5.5, tanδ = 0.0008). The length of the outside ogive layer is Lr = 15.7λ with an outside base diameter of Dr = 7.0λ, yielding a fineness ratio Lr/Dr = 2.2. The wall thickness is tapered from nose to base (average thickness of approximately 0.25λ) so as to optimize performance with respect to the antenna gimbal angle. Predicted E-plane and H-plane power transmission results were generated as before via the beam shooting and paraxial technique, employing a stabilized set of NW GBs filling the aperture (L = 0.33λ, ∣m∣ ≤ 8, ∣p∣ ≤ 8). The beam-synthesized results were compared with measured power transmission, as shown in Figure 6. Again, adequate agreement is observed between the beam-synthesized and measured results; measurement errors are estimated to be ±0.1 dB.
 Although the examples here deal with the moderate-sized apertures in operational Raytheon-designed missiles, radome concept designs using the NW beam scrambling algorithm have also been carried out, and validated experimentally, for linearly and circularly polarized larger-sized apertures in the 35–50 wavelength range [Maciel, 1993].
 In this paper, we have presented yet another example of the utility of the Gabor-based, NW, discretized GB algorithm for the interaction of plane aperture-generated time-harmonic and wideband transient fields with complex propagation environments, which we have exploited in a recent sequence of investigations [Maciel and Felsen, 2002a, 2002b; Galdi et al., 2001, 2002a, 2002b, 2002c; Felsen and Galdi, 2001; Galdi and Felsen, 2002; V. Galdi et al., Moderately rough surface underground imaging via short-pulse quasi-ray Gaussian beams, submitted to IEEE Transactions on Antennas and Propagation, October 2001]. The application here has been to the fully vectorized 3D field behavior observed when arbitrary 2D planar vector aperture field excitations are transmitted through arbitrary 3D dielectric layer configurations, like those encountered in practical missile radome design. Although the specific examples presented here are not the most general that can be envisioned (we have used linearly polarized coordinate-separable rectangular or circular aperture field distributions), the favorable comparisons achieved between the highly efficient analytic scrambling-stabilized GB algorithms and selected experimental comparisons suggest that the basic methodology works, and that it can be extended to more general cases and other classes of complex environments (for interaction with moderately rough dielectric interfaces between two media, see the work of Galdi et al. ). In each such generalization, comparisons with independently generated (purely numerical or experimental) data are essential.
 L. B. Felsen acknowledges partial support by ODDR&E under MURI grants ARO DAAG55-97-1-0013 and AFOSR F49629-96-1-0028, by the Engineering Research Centers Program of the National Science Foundation under award EEC-9986821, by grant 9900448 from the U.S.–Israel Binational Science Foundation, Jerusalem, Israel, and by Polytechnic University.