V. Galindo-Israel

A new series representation for the radiation integral with application to reflector antennas

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.

A plane-polar approach for far field reconstruction from near-field measurements

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.

An efficient technique for the computation of vector secondary patterns of offset paraboloid reflectors

A series approach for the rapid computation of the vector secondary pattern of offset paraboloid reflectors wherein the feed is displaced is presented. We show that the Jacobi polynomial series method, which has been demonstrated to provide an efficient means for evaluating the radiation integral of symmetric paraboloid reflectors, can be extended to the case of an offset paraboloid without compromising the ease or speed of computation. The analysis leading to the series formula is also useful for deriving an analytic expression for the optimum scan plane for the displacement of the feed. Representative numerical results illustrating the application of the method and the properties of the offset paraboloid are presented.

Aperture amplitude and phase control of offset dual reflectors

The dual-shaped reflector synthesis problem was first solved by Galindo and Kinber in the early 1950s 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.

On the theory of the synthesis of single and dual offset shaped reflector antennas

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.

On the reflectivity of complex mesh surfaces (spacecraft reflector antennas)

Poorer than expected surface reflectivity was observed in an early Tracking and Data Relay Satellite System antenna utilizing a tricot mesh weave. This poor reflectivity was determined to be caused by inadequate electrical contact at wire crossover points. A proper mathematical and numerical approach to assess the impact of wire junctions on reflectivity performance is developed. A mathematical method is presented for computing the surface reflectivity of complex mesh configurations like those on unfurlable-type spacecraft antennas. The method is based on the Floquet mode expansion to establish an integral equation for mesh wire currents. The equation is solved using the method of moments with triangular basis functions. It is observed that it is necessary to give special attention to the junction treatment among different branches of the mesh configurations. A vector junction current approach that resulted in satisfactory solutions for the current is described. The results of numerical simulations are compared against measured data and excellent agreement is observed. >

A new look at Fresnel field computation using the Jacobi-Bessel series

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.

On the theory of the synthesis of offset dual-shaped reflectors-case examples

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. >

Design considerations for beamwaveguide in the NASA Deep Space Network

A generalized solution is found for retrofitting a large dual-shaped reflector antenna for a beamwaveguide. The design is termed as a bypass beamwaveguide. Both highpass design feed imaging and bandpass design feed imaging are considered. Each design was studied using geometrical optics, Gaussian wave analysis, and both low-frequency and high-frequency diffraction analysis. An important extension of the Mizusawa-Kitsuregawa criteria was discovered (M.M. Zusama and T. Kitsuregawa, ibid., vol.AP-21, pp.844-8, Nov. 1973). The principle revealed shows how a two-reflector cell, although in itself distorting, may be combined with a second cell which compensates for the first and delivers an output beam which is a good image of the input beam. >

Scanning properties of large dual-shaped offset and symmetric reflector antennas

The scanning properties of shaped reflectors, both offset and circularly symmetric, are examined and compared to conic section scanning characteristics. Scanning of the pencil beam is obtained by lateral and axial translation of a single point source feed. The feed is kept pointed toward the center of the subreflector. The effects of power spillover and aperture phase error as a function beam scanning are examined for several different types of large reflector design including dual-offset, circularly symmetric large f/D, and smaller f/D dual reflector antenna system. It is shown that the Abbe-sine condition for improved scanning of an optical system cannot, inherently, be satisfied in a dual-shaped reflector system that is shaped for high gain and low feed spillover. The gain loss, with scanning, of a high-gain shaped reflector pair is demonstrated to be due to both aperture phase error loss and power spillover loss. >