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In an earlier paper by V. Galindo-Israel et al. (see ibid., vol.AP-35, p.887-96, August 1987) the geometrical optics (GO) principles, constraints, and requirements of the dual- and single-offset-shaped reflector synthesis problem were collected and developed into a set of nonlinear first-order partial differential equations (PDEs). Methods of solving these PDEs numerically were illustrated, as were certain problems that may arise. An extension of the methods by which solutions to the PDEs can be obtained is presented, together with several case examples. These examples are independently analyzed by GO and physical optics diffraction methods. The starting point for the integration over each reflector can be taken on the outer rim, at the center, or at an intermediate point-the intermediate starting point being the more general case. The utility of the speed of this synthesis method is demonstrated. This makes practical the incorporation of the synthesis into a search algorithm that can optimize one or more parameters of the reflector system. As an example, the optimization of the mapping equations for low cross polarization is discussed. >

The finding of smooth analytic representations for antenna reflector surfaces which are prescribed only by discretized data obtained by various synthesis methods is examined. Frequently the data are distributed in a nonuniform grid and contain noise. The smoothness required is to C sub 1 for physical optics diffraction analysis and to C sub 2 for geometrical theory of diffraction (GTD) analysis. The GTD analysis approach requires a surface description which returns data very rapidly. Two methods of interpolation, the global and the local methods, are discussed. They each have advantages and disadvantages; characteristics are discussed and examples are presented.

The dual-shaped reflector synthesis problem was first solved by Galindo and Kinber in the early 1950's for the circularly symmetric-shaped reflectors. Given an arbitrary feed pattern, it was shown that the surfaces required to transform this feed pattern by geometrical optics into any specified phase and amplitude pattern in the specified output aperture are found by the integration of two simultaneous nonlinear ordinary differential equations. For the offset noncoaxial geometry, however, it is shown that the equations found by this method are partial differential equations which, in general, do not form a total differential. Hence the exact solution to this problem is generally not possible. It is also shown, however, that for many important problems the partial differential equations form a nearly total differential. It thus becomes possible to generate a smooth subreflector by integration of the differential equations and then synthesize a main reflector which gives an exact solution for the specified aperture phase distribution. The resultant energy (or amplitude) distribution in the output aperture as well as the output aperture periphery are then approximately the specified values. A representative group of important solutions are presented which illustrate the very good quality that frequently results by this synthesis method. This includes high gain, low sidelobe, near-field Cassegrain, and different ( f/D ) ratio reflector systems.

A numerical analysis of an infinite phased array of open rectangular waveguides has been made which includes the effects of wall thickness. Two planes of scan, the H and quasi- E planes, have been considered. In these cases, the general vector problem can be expressed in the form of a scalar one-dimensional Fredholm integral equation of the first kind. The approximate fields obtained numerically from the integral equation are used for the evaluation of the input complex reflection coefficient. A variational expression for the reflection coefficient is developed and used for improving the accuracy. Numerical results for the H and quasi- E plane scans are presented as a function of wall thickness and scan angle. Agreement with experimental results is very good.

It has been observed that numerical difficulties may arise when the exact synthesis techniques are applied, especially in the vicinity of the center region. The authors investigate the possibility of circumventing these difficulties by starting the synthesis procedure not from the periphery, as is done conventionally, but from the center point instead. Numerical results are presented, and it is shown that the periphery condition, namely, that the projected aperture of the main reflector be circular, is approximated very well by the solution obtained via the synthesis procedure presented here. >

For a given feed illumination and desired aperture field distribution, an exact formulation of the problem of simultaneously synthesizing the shapes of the main and subreflectors exists in terms of a set of nonlinear first-order differential equations. A numerical approach to solving these equations is reported. >

It is well-known that the far field of an arbitrary antenna may be calculated from near-field measurements. Among various possible nearfield scan geometries, the planar configuration has attracted considerable attention. In the past the planar configuration has been used with a probe scanning a rectangular geometry in the near field, and computation of the far field has been made with a two-dimensional fast Fourier transform (FFT). The applicability of the planar configuration with a probe scanning a polar geometry is investigated. The measurement process is represented as a convolution derivable from the reciprocity theorem. The concept of probe compensation as a deconvolution is then discussed with numerical results presented to verify the accuracy of the method. The far field is constructed using the Jacobi-Bessel series expansion and its utility relative to the FFT in polar geometry is examined. Finally, the far-field pattern of the Viking high gain antenna is constructed from the plane-polar near-field measured data and compared with the previously measured far-field pattern. Some unique mechanical and electrical advantages of the plane-polar configuration for determining the far-field pattern of large and gravitationally sensitive space antennas are discussed. The time convention exp ( j \omega r ) is used but is suppressed in the formulations.

Given the true or any approximate current on a reflector, the radiated far-field is determined from a rapidly convergent series representation of the radiation integral. The leading term is a well-shaped J_{1}(x)/x beam pointing in a desired direction. Higher order terms provide perturbations to the leading term. The coefficients of the series are independent of the observation angles. Hence, once they are computed, the field may be determined very rapidly at large numbers of points. Initially, a suitable small angle approximation is made that places the radiation integral in the form of a Fourier transform on a circular disk. The theory is then extended such that the results are valid in both the near and the wide angle regions. Application to a rotationally symmetric paraboloid is presented herein. Other applications include the offset and dual reflectors and near- to far-field integrations. A modified form of the series can also be used for Fresnel zone computations.

In the paper by V. Galindo-Israel and R. Mittra it is claimed that a dual shaped subreflector (DSS) feed can be used to upgrade the efficiency of classical Cassegrain antennas. In this comment it is shown that upgrading is not possible unless the diameter of the DSS feed is extremely large. In fact, if the Cassegrain antenna at Goldstone, CA, used as an example in the above paper, is fed by the DSS feed, the efficiency in X -band will be about the same as or less than if it is fed directly by an optimized scalar horn. The low efficiency is caused by diffraction. Instead, the efficiency can be upgraded by using a scalar horn with shaped lens.

The diffraction analysis of reflector surfaces which are described only at a discrete set of locations usually leads to the requirement of an interpolation to determine the surface characteristics over a continuum of locations. Two methods of interpolation, the global and the local methods, are presented. The global interpolation representation is a closed-form or series expression valid over the entire surface. The coefficients of a series expression are found by an integration of all of the raw data. Since the number of coefficients used to describe the surface is much smaller than the number of raw data points, the integration effectively provides a smoothing of the raw data. The local interpolation provides a closed-form expression for only a small area of the reflector surface. The subreflector is divided into sectors each of which has constant discretized data. Each area segment is then locally described by a two-dimensional quadratic surface. The second derivative data give the desired smoothed values. >

The physical optics approximation is employed to derive the radiation integral for a doubly curved offset reflector antenna illuminated by an arbitrary source. A novel procedure is presented for expressing the radiation integral in terms of a summation of Fourier transforms of an "effective" aperture distribution which includes the effect of the curvature of the surface. The Jacobi-Bessel series is used to evaluate the Fourier transforms. The vector nature of the far-field pattern is studied by evaluating its three Cartesian components in a unified fashion. The rapid numerical evaluations of the expressions obtained are demonstrated via extensive test cases. In particular, the scattering characteristics of symmetric and offset parabolic, spherical, and shaped reflectors are studied in detail, and comparisons are made with other available data.

Many useful applications exist for the efficient computation of Fresnel and near zone fields of large antennas. Even small antennas in beam waveguide systems must be evaluated in the Fresnel zone. Far zone fields computed from measured near zone measurements can be verified by both the measurement and the computation of the Fresnel zone fields. The authors start with the premise that the far field has been computed by a Jacobi-Bessel series. These results are used then to determine the higher order terms of a Barrar-Kay 1/R^{p} expansion of the fields. The leading term of the 1/R^{p} series is the far zone field. Classically, the higher order terms are found by repetitive differentiation, a laborious and often inaccurate procedure particularly since the 1/R^{P} series is slowly convergent-as D^{2}/R ( D is the diameter of antenna source). The approach of the authors via the Jacobi-Bessel 1/R^{P} series determines the higher order terms by simple algebraic recursion. The only restriction on the method is that it be used within the range of validity of the Fresnel small angle (FSA) approximation. However, since the Fresnel approximation is a second order approximation in terms of ( D/R ), the range of validity is quite large. This is demonstrated in detail. The method is applicable to reflector as well as aperture field sources.

Since Kinber (Radio Technika and Engineering-1963) and Galindo (IEEE Trans. Antennas Propagat.-1963/1964) developed the solution to the circular symmetric dual shaped synthesis problem, the question of existence (and of uniqueness) for offset dual (or single) shaped synthesis has been a point of controversy. Many researchers thought that the exact offset solutions may not exist. Later, Galindo-Israel and Mittra (IEEE Trans. Antennas Propagat.-1979) and others formulated the problem exactly and obtained excellent and numerically efficient but approximate solutions. Using a technique similar to that first developed by Schruben for the single reflector problem (Journal of the Optical Society-1973), Brickell and Westcott (Proc. Institute of Electrical Engineering-1981) developed a Monge-Ampere (MA) second-order nonlinear partial differential equation for the dual reflector problem. They solved an elliptic form of this equation by a technique introduced by Rall (1979) which iterates, by a Newton method, a finite difference linearized MA equation. The elliptic character requires a set of finite difference equations to be developed and solved iteratively. Existence still remained in question. Although the second-order MA equation developed by Schruben is elliptic, the first-order equations from which the MA equation is derived can be integrated progressively (e.g., as for an initial condition problem such as for hyperbolic equations) a noniterative and usually more rapid type solution. In this paper, we have solved, numerically, the first-order equations. Exact solutions are thus obtained by progressive integration. Furthermore, we have concluded that not only does an exact solution exist, but an infinite set of such solutions exists. These conclusions are inferred, in part, from numerical results.

The asymptotic decay of coupling with distance from a single excited element it an infinite linear array has been analyzed for several special cases and is shown to behave according to exp (-jkr)/r3/2where r is the distance from the excited element. This behavior is also characteristic of propagation over a lossy surface. By geometric extrapolation one would expect an asymptotic decay of coupling for planar arrays to behave according to exp (-jkr)/r2. This extension is carried out for an arbitrary infinite periodic planar current sheet with the expected result. In addition, this behavior is shown to be valid for an infinite parallel plate array immersed in a magnetized cold plasma. In all cases, the asymptotic behavior is shown to depend on the form of the singularity of the derivative of the reflection coefficient with scan angle. This singularity occurs at grazing incidence of the radiated beams.

Electromagnetic scattering problems, including waveguide discontinuity, phased array, and scattering (exterior type) problems, are frequently described by integral equations that can be solved by the Ritz-Galerkin or generalized method of moments. Under appropriate conditions, it has been shown that reciprocity and variational properties are, in fact, preserved in the approximate solutions. It is shown here that in the Ritz-Galerkin method, energy is also conserved under certain conditions, even in those scattering problems where reciprocity does not exist. Hence energy conservation cannot serve as a check for accuracy of a numerical solution obtained by the Ritz method or other related methods.

Dual-shaped reflectors have been used for many years. Thus, these reflectors have been used as high gain antennas on Voyagers 1 and 2. The objectives of the geometrical optics (GO) dual shaped synthesis are considered. Concerning the synthesis of dual shaped reflectors, it has been shown for circular symmetric reflectors that an exact GO solution can be found to the problem of transforming, by two reflections, any feed pattern into any aperture distribution. This problem involves solving two simultaneous nonlinear ordinary differential equations. The same approach for offset geometry leads to two simultaneous nonlinear partial differential equations. It is shown that these equations could also be integrated numerically, except that in general these equations are not total and therefore, in general, they do not have a 'smooth' solution. It is further shown that the offset partial differentials often very nearly form a total differential in many cases of practical importance.

The unitary condition, or the constraint imposed by the conservation of probability, is examined for certain approximate solutions to physical scattering problems. The unitary condition is shown to be preserved for an approximate nonrelativistic S matrix derived by a rather generalized moments method. The unitary condition, or conservation theorem, is shown to be satisfied independent of the accuracy of the approximate solution under certain specifically defined conditions.

The integral equations for infinite phased arrays have generally been constructed with complex nonsymmetric kernels. Nevertheless, it has been shown earlier that these nonsymmetric forms can generally be transformed into a complex-symmetric form with consequent stationary properties. It is shown here that the phased array must possess certain symmetry properties for the complex-symmetric form to exist.