Two-dimensional pulsed propagation from extended planar aperture field distributions through a planar dielectric layer via quasi-ray Gaussian beams

A previously developed Gabor-based quasi-ray narrow-waisted (NW) Gaussian beam (GB) algorithm for time-harmonic propagation of aperture-excited two-dimensional (2-D) electromagnetic fields through a planar dielectric layer is extended here to the time domain (TD) to deal with short-pulse excitation. The dielectric layer is assumed to be nondispersive; however, slight Ohmic losses can be accommodated. The frequency domain (FD) algorithm is based on a self-consistent discretization of the aperture field distribution in terms of basis NW-GBs in conjunction with an efficient quasireal ray tracing scheme for tracking the individual basis beams. The TD results are obtained by analytic Fourier inversion from the FD in terms of pulsed beam wavepackets, following a procedure similar to that utilized by Galdi et al. (see IEEE Trans. Antennas Propagat., vol.49, p.1322-32, Sept. 2001) in connection with free-space aperture radiation. The proposed algorithm is validated and calibrated against a rigorous numerical reference solution via an extensive series of numerical experiments. A priori accuracy assessments in terms of critical nondimensional estimators, and computational costs, are also given attention.     

Complex rays for radiation from discretized aperture distributions

Many electromagnetic propagation problems require tracking of fields radiated by large actual or induced aperture distributions through complicated environments before reaching the observer. For a systematic approach to this problem area, it is desirable to represent the aperture field in terms of basis functions which are physically informative and well adapted to traversing the propagation path. At high frequencies, Ganssian beam-type basis functions meet these requirements. After referring to a rigorous aperture discretization scheme, various quasi-Gaussian basis field profiles are examined, with a special view toward expressing their radiation properties in terms of complex rays; complex ray tracing is promising for field tracking in complicated surroundings. By comparing reference solutions from numerical integration of radiation integrals with complex ray asymptotics, it is concluded that the true Gaussian has the most favorable attributes for matching aperture discretization, propagation requirements, and complex ray tracing. Thus, the analysis here may point the way toward systematic treatment of the above-noted class of propagation problems.     

Gaussian beam analysis of propagation from an extended planeaperture distribution through dielectric layers. II. Circularcylindrical layer

For pt.I see ibid., vol.38, no.10, p.1607-17 (1990). The theory developed in pt.I of this paper for transmission of fields from an extended planar aperture through a plane dielectric layer is applied to transmission from such an aperture through an arbitrarily located cylindrical dielectric layer. By the self-consistency considerations enunciated previously, it is argued that the narrow-beam algorithm for near and far zone fields is truly predictive, although no other reference solution is now available for an independent check. As in pt.I, the aperture distributions are smoothly tapered or abruptly truncated, and they may generate beam tilts and focusing in the layer or its vicinity     

Discretized Gabor-based beam algorithm for time-harmonic radiation from two-dimensional truncated planar aperture distributions .I. Formulation and solution

In this first part of a two-paper sequence, we develop a Gabor-based Gaussian beam (GB) method for the representation of three-dimensional (3-D) time-harmonic vector electromagnetic fields excited by two-dimensional (2-D) truncated arbitrarily polarized planar aperture field distributions. The biorthogonal Gabor basis is tied to a lattice in the discretized four-dimensional (4-D) [configuration (space)]-[spectrum (wavenumber)] phase space which spans the 2-D aperture plane. This study generalizes previous investigations of the simpler corresponding procedure for 2-D fields excited by one-dimensional (1-D) apertures. By subsequent specialization, in the 1-D aperture case, to narrow-waisted 2-D ray-like GBs, we have shown that tracking such beams through interactions with complex environments and recombining them to synthesize the total 2-D field produces robust, efficient and accurate algorithms that are useful for a variety of forward and inverse scattering scenarios. Extension to the time domain via narrow-waisted pulsed GBs has likewise been considered. These potential applications have motivated the extension here to general 3-D EM fields excited by time-harmonic 2-D truncated apertures. The presentation relates each step in the analytic development to a corresponding step in the 1-D aperture case, thereby highlighting the complications (in the parameterizing phase space) associated with the 2-D aperture problem. The outcome is the formal exact solution of the problem under consideration.     

Two-dimensional diffraction by half-planes and wide slits nearradiating apertures

With the complex source-point method used to produce the basis elements of an array of linearly and directionally equispaced two-dimensional (2-D) beams, the fields of any aperture distribution at any range to any degree of accuracy can be obtained. For efficiency a limited number of significant beams and beam directions is required. Approximately twice as many beams as the aperture width in wavelengths, with all beam directions normal to the aperture, is found to be sufficient here for simple uniform and cosinusoidal distributions in apertures of moderate size at ranges outside the evanescent field zone of the aperture. Now the exact solution for the far field of a line source, or here a beam source in the presence of a conducting half-plane, is used as our basis element to give the solution for antenna pattern diffraction by a local half-plane. Antenna pattern diffraction by an aperture near a wide slit is presented as simply a superposition of the solutions for two coplanar half-planes with separated parallel edges. Antenna pattern distortion by various other local obstacles can be obtained similarly     

An approach to analysis of waveguide arrays with shaped dielectricinserts and protrusions

The classical moment method solution of the waveguide-array problem is extended to allow for generally shaped dielectric matching inserts in the waveguide-to-free-space transition region. The aperture electric field is represented in terms of waveguide modes. To account for the presence of the matching inserts, the aperture fields are numerically propagated through the dielectric regions. Novel matching configurations, which extend the scanning range of waveguide elements or can be used to shape the element pattern in limited-scan applications, are proposed and analyzed     

Discretized Gabor-based beam algorithm for time-harmonic radiation from two-dimensional truncated planar aperture distributions .II. Asymptotics and numerical tests

For pt.I see ibid., vol.50, no.12, p.1751-59 (2002). In this second part of the two-part sequence dealing with Gabor-based Gaussian beam (GB) representations for the excitation of time-harmonic three-dimensional (3-D) vector electromagnetic fields excited by two-dimensional (2-D) arbitrarily polarized truncated planar aperture distributions (set in a discretized [configuration (space)]-[spectrum (wavenumber)] phase space), we employ high-frequency asymptotic approximations to reduce the formal solutions developed in part I to efficient algorithms for implementation. The resulting explicit expressions for the 3-D GB propagators are applied to the species of narrow-waisted GBs, which possess ray-like features without the failures of ray fields in ray-optical transition regions. The potential utility of such GBs in the synthesis of wave interactions with complex environments has been discussed previously. The narrow-waisted GB algorithms for the aperture and radiated near-to-far zone fields are calibrated for robustness, accuracy and efficiency by comparison with numerically generated reference solutions in a series of tests involving coordinate-separable rectangular aperture distributions with cosine amplitude tapers.     

Systematic study of fields due to extended apertures by Gaussianbeam discretization

Gaussian beams are used as basis elements in field representations. To gain insight into how the choice of beam parameters affects the final representation, a systematic study for the simple test case of a one-dimensional linearly phased cosine-aperture distribution has been undertaken. By successively adding individual displaced and/or tilted beams with large, narrow, or matched waists, one can assess how the elements in various portions of the lattice contribute to the build-up of the actual field in the aperture, near zone, and far zone. Adding enough beams always guarantees homing in on the exact solution, as is verified here by independent comparison. Different beam choices imply different modeling of the radiation process. The understanding gained thereby is helpful for selecting beam parameters in subsequent applications where it is necessary to balance requirements of good convergence, ease of computation, and ability to track the beams through perturbing environments like a radome. Indications are that the narrow beams provide the most robust and versatile formulation to deal with these generalized conditions     

On the role of the geometrical optics field in aperture diffraction

The asymptotic solution of a high-frequency electromagnetic field transmitted through a finite aperture is studied. In applying Keller's geometrical theory of diffraction (GTD), a basic yet unanswered question for an observation point in the lit region is: "Should the geometrical optics field on a direct incident ray be included in the total field solution?" By studying a test problem and utilizing the newly developed uniform asymptotic theory (UAT), we have deduced simple and explicit rules for the role of the geometrical optics field and the regions of validity for GTD in a general aperture diffraction. The rules reveal that the success of GTD in treating aperture problems in the literature depends critically on the assumption that both the source and the observation points are infinitely far away from the aperture. Had either point been a finite distance away, Keller's GTD, in general, would fail and UAT must be used. The paper also demonstrates a physical phenomenon: the diffusion of the incident field as the observation point moves away from the aperture.     

Quasi-ray Gaussian beam algorithm for time-harmonic two-dimensionalscattering by moderately rough interfaces

Gabor-based Gaussian beam (GB) algorithms, in conjunction with the complex source point (CSP) method for generating beam-like wave objects, have found application in a variety of high-frequency wave propagation and diffraction scenarios. Of special interest for efficient numerical implementation is the noncollimated narrow-waisted species of GB, which reduces the computationally intensive complex ray tracing for collimated GB propagation and scattering to quasi-real ray tracing, without the failure of strictly real ray field algorithms in caustic and other transition regions. The Gabor-based narrow-waisted CSP-GB method has been applied previously to two-dimensional (2-D) propagation from extended nonfocused and focused aperture distributions through arbitrarily curved 2-D layered environments. In this 2-D study the method is applied to aperture-excited field scattering from, and transmission through, a moderately rough interface between two dielectric media. It is shown that the algorithm produces accurate and computationally efficient solutions for this complex propagation environment, over a range of calibrated combinations of the problem parameters. One of the potential uses of the algorithm is as an efficient forward solver for inverse problems concerned with profile and object reconstruction     

空心高斯光束通过光阑透镜系统的传输特性

为了研究空心光束通过硬边光阑透镜傍轴ABCD光学系统后的传输特性,利用柯林斯衍射积分公式,推导出了空心高斯光束通过受圆孔硬边光阑限制的傍轴ABCD光学系统的传输公式,所得公式可用来描述空心高斯光束通过任意傍轴ABCD光学系统的传输.研究了光阑、透镜以及光阑透镜系统对空心高斯光束传输特性的影响,并用数值例做了详细说明.结果表明,光阑和透镜均会使光束的光场分布向源平面前移,光阑会破坏空心高斯光束的空心性,而单纯的透镜只改变光场分布,不会破坏光束的空心性.这一结果对于空心光束的产生和应用有理论指导意义.     

Asymptotic analysis of mutual coupling in concave circularcylindrical arrays. II. Near-zone

For pt.I, see ibid., vol.41, no.2, p.121-36 (1993). This work evaluates the near-zone coupling coefficients in large concave arrays. A canonical model of a circular cylindrical concave array of open-ended rectangular waveguides is analyzed. It is shown that in the paraxial region (which also includes the area close to the excited element) the coupling coefficients are identical, to lowest asymptotic order, to those in an equivalent infinite planar array. In the transition region, which lies between the paraxial and the far zone, or the ray region, two representations were developed: one, a transition function, expressed in terms of a canonical integral which must be numerically evaluated, and the other a uniform representation, which consists of a superposition of a planar array contribution and a few periodic structure rays. The uniform representation is valid in the near zone (which includes both the paraxial and the transition region) as well as in the far zone. This form is simple and may be immediately generalized to concave arrays with slowly varying curvature and periodicity     

Rays, Beams, and Modes Pertaining to the Excitation of Dielectric Waveguides

The two-dimensional problem of excitation of an inhomogeneous dielectric layer by a Gaussian beam is considered, with emphasis on useful representations that treat the field either in terms of multiple reflections or in terms of guided modes. A recently developed method is employed whereby the beam fields are generated from line source fields by assigning a complex value to the source coordinates. When applied to the asymptotic solution for the line source field, this procedure furnishes a simple and quantitative relation between line-source-excited ray optics and paraxial beam optics. It also clarifies the role of lateral ray and beam shifts for reflection at a boundary with incidence-angle-dependent reflection coefficient, especially when multiply reflected fields are converted into modal form. Results are given for beams which are reflected at both boundaries, reflected at one boundary and refracted before reaching the other boundary, and trapped by refraction without reaching either boundary. In the first case, conversion to modal form is more convenient at large distances whereas in the latter case, paraxial beam tracking is preferable.     

Enhanced compact range reflector concept using an R-card fence:two-dimensional case

Compact range reflector edge diffraction can be reduced by using an R-card fence. Well-designed R-cards (resistive sheets) placed in front of reflector edges reduce the field variations in the test zone. The keys to successful R-card design are proper choices of both the geometry and resistance profile. In this paper, a two-dimensional (2-D) version of the problem is attacked to show the basic concepts and essential considerations in designing R-cards. The complexity of a design is simplified by separating the different mechanisms associated with R-cards. Undesired mechanisms can be visualized by ray tracing and then eliminated by choosing the correct R-card geometry. The useful impact of this treatment is illustrated as it controls the transmitted energy level through the R-card; thus, the resistance of the R-card is defined based on an optimum aperture taper. Excellent performance in both the cross-range and down-range directions is presented, and the validity of the design is verified over a wide band of frequencies. The simplicity, flexibility, and low cost of this R-card fence concept provides a viable alternative to other edge treatments     

Modal analysis of periodic planar phased arrays of apertures

A general method is established for the evaluation of the driving point admittance of a radiating aperture, fed by a waveguide of the same cross section as the aperture, in an infinite periodic planar phased array. The array may have an arbitrary element lattice and it may be covered by a dielectric layer. The coefficients of the waveguide modal expansion and of the Floquet series representing the electromagnetic field in the waveguide and in the radiation half-space, respectively, are determined by approximately enforcing the boundary conditions in the array plane through an application of Galerkin's method. By eliminating from the set of equations thus obtained the complex amplitudes of the waveguide modes and of the Floquet harmonics, the driving point admittance can be expressed as the ratio of two determinants of order N and N-1 (N being the number of the waveguide modes utilized), whose elements contain truncated bidimensional series, structurally similar to the well-known grating-lobe series. The expression allows relatively simple numerical computations if the Fourier transforms of the waveguide vector mode functions are known in closed form (as they are for rectangular or circular elements). The variation of the power reflection loss with scan angle has been numerically calculated for various array configurations. The results are in some cases substantially different from those predicted through the conventional grating-lobe series technique, which is based on the assumption of one-mode elements.     

A comparison among 1-, 3-, and 7-horn feeds for a 37-beam MBA

A very common multiple beam antenna (MBA) configuration consists of a collimating device illuminated by an array of feeds. The collimating device is usually a reflector or a lens. The feeds are usually horn antennas with a circular aperture. The reflector is usually offset-fed to eliminate aperture blockage; the lens is center fed. The antenna's feeds are excited to produce a finite number of beams, so as to provide contiguous coverage of the field of view. The designer is forced to minimize the angular spacing between adjacent beams in order to maximize the minimum gain over the antenna's field of view. On the other hand, the feed horn's aperture gain is maximized when the feed horn spacing and its aperture diameter are equal. This results in antenna efficiency of the order of 30% when a single feed horn is excited to produce a beam. When a cluster of 3 or 7 adjacent feed horns are excited to produce a single beam, antenna efficiency can be increased to 50%. When it is tolerable, several identical antenna apertures can be used to replace a single aperture configuration. In this case, each of M apertures produces approximately N/M beams of an MBA that produces N beams. Horns producing adjacent beams do not illuminate the same aperture. This permits the use of a much larger horn aperture for a given beam spacing. This results in reduced spillover, higher gain of each beam, and increased antenna efficiency of each aperture. This paper investigates the maximization of gain for several lens antennas. It shows that antenna gain is increased as its focal length is decreased. That is, a focal length-to-diameter ratio (F/D) less than 1 is preferred     

用轴棱锥实现可改变的局域空心光束

理论和实验研究了利用轴棱锥-透镜系统产生的局域空心光束，分析了局域空心光束的演变过程，由简单的几何光学结构和近轴光线追迹的方法进行模拟。研究结果表明，局域空心光束的暗中空区域不仅与聚焦透镜的焦距有关，而且与光阑半径以及聚焦透镜到轴棱锥间的距离有关；对于给定的轴棱锥，局域空心光束的横向暗斑尺寸与焦距成正比，并且其纵向暗斑尺寸与聚焦透镜到轴棱锥间的距离以及光阑半径成反比。这些结果表明，改变聚焦透镜的焦距和聚焦透镜到轴棱锥间的距离以及光阑半径，可有效地改变局域空心光束的暗中空区域。     

部分相干光通过多个硬边光阑ABCD光学系统传输的快速算法

部分相干光通过受光阑限制近轴ABCD光学系统传输是实际工作中一个十分有意义却是很困难的问题。虽然使用“Mathematia”编制的程序可对Collins公式作数值积分，但随着光阑数目的增加，计算变得非常繁琐和耗时。用有限个复高斯函数之和来逼近硬边光阑窗函数的方法，推导出了部分相干光通过多个硬边光阑ABCD光学系统传输的递推公式，提供了一种模拟部分相干光通过硬边光学系统传输的快速算法。计算示例表明，当离光阑不太近时，使用解析公式和直接对Collins公式作数值积分的结果符合甚好，并具有快速计算的主要优点，因此大为节约了机时。此外，由于将单光阑推广到多光阑的ABCD光学系统，并考虑了光束的部分空间相干性，因此所得结果更具一般性，且对多模部分相干激光是适用的。     

High frequency fields in the presence of a curved dielectric interface

A high-frequency line source in a dielectric medium that is separated by a concave cylindrical boundary from an exterior medium with lower dielectric constant generates a variety of wave phenomena which have been explored extensively. This problem is reexamined here with a view toward clarifying relevant reflection and transmission characteristics within the framework of ray optics, with emphasis on the more complicated transmitted field. The exterior domain is divided into illuminated and shadow regions separated by the transmitted tangent ray launched by a ray incident at the critical angle. Conventional ray optics is valid far from the tangent ray shadow boundary on the illuminated side. The shadow boundary is surrounded by transition regions wherein Fock type integrals and Weber functions yielding local lateral waves provide alternative representations. On the shadow side, not too far from the shadow boundary, the field can be interpreted via "tunneling" and subsequent radiation along a ray from a virtual caustic to the observer. The tunneling is associated with the initial evanescent decay of the transmitted field excited by a totally reflected incident ray. However, deeper inside the shadow, this mechanism is inapplicable, and the field is expressed either in terms of the Fock integrals or a creeping wave-type residue series. The results are presented in a format that permits insertion into a geometrical theory of diffraction (GTD) user's manual.     

Aperture feed for a spherical reflector

The problem of designing a feed system for illuminating a spherical reflector is examined. A method is proposed for specifying the required field distribution over the aperture of the feed system, and the primary illumination and gain resulting from this distribution are derived. The results indicate that a significantly smaller feed aperture can be employed than would be indicated by conventional ray tracing methods. Specific numerical results are obtained by taking the Arecibo antenna as an example, for which a calculated aperture efficiency of 67.5 percent is possible with approximately a 38-foot-diameter aperture feed.