A. Prata’s research while affiliated with University of Southern California and other places

An analytical formula capable of describing, with high accuracy, the offset paraboloidal umbrella reflector mesh shape is presented. This formula takes full account of the pillowing effect, and also applies to the front-end geometry, which is simply a degenerate case of an offset paraboloid. The basic steps used in deriving this formula parallel the ones used for the front-fed geometry. The validity of the analytical solution presented has been confirmed by comparison against a numerical solution obtained using the finite-difference technique.

Some of the design aspects of a practical millimeter-wave compact range employing an axially symmetric main-reflector illuminated by a shaped subreflector are presented. Although in principle any reasonable surface shape can be used for the main reflector (all that is required is that it provides a reasonably small caustic region when illuminated by an incoming plane wave), a paraboloid of revolution is used here. This choice is partially motivated by the fact that axially symmetric paraboloids are readily available on the antenna market, and hence provide a cost-effective alternative. Once the subreflector surface has been determined, the geometrical optics field produced at an arbitrary aperture location by a given feed element can be calculated using ray optics. The final co- and cross-polarized levels of the aperture field of the compact range are shown. This system covers a frequency range from 18 to 170 GHz and boasts a measured quiet-zone region with dimensions exceeding half the main reflector diameter.

A numerically efficient algorithm for expanding a function in a series of Zernike polynomials is presented. The algorithm evaluates the expansion coefficients through the standard 2-D integration formula derived from the Zernike polynomials’ orthogonal properties. Quadratic approximations are used along with the function to be expanded to eliminate the computational problems associated with integrating the oscillatory behavior of the Zernike polynomials. This yields a procedure that is both fast and numerically accurate. Comparisons are made between the proposed scheme and a procedure using a nested 2-D Simpson’s integration rule. The results show that typically at least a fourfold improvement in computational speed can be expected in practical use.

Zernike polynomials form a complete orthogonal set that provides a convenient way of expanding an arbitrary function, defined over a circular area, into an infinite series. They provide a numerically efficient way to evaluate the diffraction characteristics of circular aperture antennas and can also be used for surface interpolation. In these applications, a basic step is to expand an appropriate function in a series of Zernike polynomials. Each expansion coefficient is normally determined by evaluating a two-dimensional integral derived using the polynomials' orthogonal properties. In the present work, an algorithm for numerically performing the integration is presented. This algorithm factors out the oscillatory behavior of the polynomials to provide a fast and accurate procedure. The use of the algorithm to evaluate radiation patterns of reflector antennas is discussed.

Zernike polynomials were used to perform the physical optics scattering integral of umbrella reflectors. A computationally efficient numerical procedure was obtained for the umbrella reflector. To demonstrate the method, it was applied to a typical offset umbrella reflector. The radiation patterns for a focused and a scanned feed are shown. Three patterns are shown superposed: one was computed using a conventional two-dimensional integration, and the other two using the Zernike expansion procedure. The agreement between the Zernike approach and the conventional method is good, provided that enough gore-related expansion terms are included in the computations.