International Journal of Microwave Science and Technology

Volume 2014, Article ID 849194, 8 pages

http://dx.doi.org/10.1155/2014/849194

## A New Method of Designing Circularly Symmetric Shaped Dual Reflector Antennas Using Distorted Conics

Department of Electrical and Electronic Engineering, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh

Received 30 July 2014; Accepted 1 December 2014; Published 17 December 2014

Academic Editor: Ramon Gonzalo

Copyright © 2014 Mohammad Asif Zaman and Md. Abdul Matin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new method of designing circularly symmetric shaped dual reflector antennas using distorted conics is presented. The surface of the shaped subreflector is expressed using a new set of equations employing differential geometry. The proposed equations require only a small number of parameters to accurately describe practical shaped subreflector surfaces. A geometrical optics (GO) based method is used to synthesize the shaped main reflector surface corresponding to the shaped subreflector. Using the proposed method, a shaped Cassegrain dual reflector system is designed. The field scattered from the subreflector is calculated using uniform geometrical theory of diffraction (UTD). Finally, a numerical example is provided showing how a shaped subreflector produces more uniform illumination over the main reflector aperture compared to an unshaped subreflector.

#### 1. Introduction

Reflector antennas are widely used in radars, radio astronomy, satellite communication and tracking, remote sensing, deep space communication, microwave and millimetre wave communications, and so forth [1–3]. The rapid developments in these fields have created demands for development of sophisticated reflector antenna configurations. There is also a corresponding demand for analytical, numerical, and experimental methods of design and analysis techniques of such antennas.

The configuration of the reflectors depends heavily on the application. The dual reflector antennas are preferred in many applications because they allow convenient positioning of the feed antenna near the vertex of the main reflector and positioning of other bulky types of equipment behind the main reflector [3]. Also, the feed waveguide length is reduced [4]. They also have some significant electromagnetic advantage over single reflector systems [5]. Although many dual reflector configurations exist, the circularly symmetric dual reflector antennas remain one of the most popular choices for numerous applications [1].

One of the most common circularly symmetric dual reflector antennas is the Cassegrain antenna. The Cassegrain antenna is composed of a hyperboloidal subreflector and a paraboloidal main reflector. A feed antenna (usually a horn antenna) illuminates the subreflector which in turn illuminates the main reflector. The main reflector produces the radiated electric field that propagates into space. The radiation performance of the dual reflector antennas depends on the radiation characteristics of the feed and the geometrical shapes of the main reflector and the subreflector. Modern wireless communication and RADAR applications enforce stringent requirements on the far-field characteristics of the antenna. For example, satellite communications impose limitations on maximum beamwidth and maximum sidelobe levels of the antenna to avoid interference with adjacent satellites [2]. The traditional Cassegrain antennas have fixed geometries and offer limited flexibilities to antenna designers. As a result, the maximum performance that can be extracted from these antennas is limited by geometrical constraints.

For high performance applications, the traditional hyperboloid/paraboloidal geometry must be changed. Reflector shaping is the method of changing the shape of the reflecting surfaces to improve the performance of the antenna. Shaped reflector antennas outperform conventional unshaped reflector antennas. Reflector shaping allows the designers additional flexibility. The antenna designers have independent control over relative position of the reflectors, diameter of the reflectors, and the curvature of the reflectors when shaped reflectors are used instead of conventional reflectors. This makes reflector shaping an essential tool for designing high performance reflector antennas.

Many methods of designing shaped reflectors are present in literature. One of the first major articles related to reflector shaping was published by Galindo in 1964 [6]. The method is based on geometrical optics (GO). Galindo’s method required solution of multiple nonlinear differential equations, which sometimes may be computationally demanding. A modified version of this method was presented by Lee [7]. Lee divided the reflector surfaces into small sections and assumed the sections to be locally planar. This assumption converted the differential equations to algebraic equations, which are much easier to solve. However, the reflector surface must be divided into a large number of sections to increase the accuracy of this method.

Another popular method for designing shaped reflectors involves expanding the shaped surfaces using a set of orthogonal basis functions [8, 9]. Rahmat-Samii has published multiple research papers on this area [8–10]. The expansion coefficients determine the shape of the surface. A small number of terms of the expansion set are sufficient to accurately describe a shaped surface. So, a few expansion coefficients must be determined to define the surface. The differential equation based methods determine the coordinates of the points on the reflector surfaces, whereas the surface expansion based method only determines the expansion coefficients. Due to the decrease in number of unknowns, the surface expansion method is computationally less demanding. The surface expansion method can be incorporated with geometrical theory of diffraction (GTD) or its uniform version, uniform theory of diffraction (UTD), to produce an accurate design algorithm [10]. These design procedures are known as diffraction synthesis [8–10]. This method has been successfully used in many applications. Recently, a few new efficient methods for designing circularly symmetric shaped dual reflector design have been developed. One of the first significant works on this method was reported by Kim and Lee in 2009 [11]. This method divides the shaped reflector surfaces into electrically small sections. Each section is assumed to be a conventional unshaped dual reflector system. Since well-established methods for analyzing conventional dual reflector system exist, the radiation characteristics of each section can easily be evaluated. The shaped surface is defined by combining all the local conventional surfaces. The method requires solutions of several nonlinear algebraic equations. So it is computationally convenient. Another method based on the same principle was proposed by Moreira and Bergmann in 2011 [12]. This method also divides the shaped surface into small local sections. The local sections are represented by unshaped conics. Each conic section is optimized to produce a desired aperture distribution, which is formulated by GO method. As these methods have recently appeared in literature, most of the advantages and drawbacks of the method have not been investigated. Reduction in computational complexity is an obvious advantage. The proposed work concentrated on presenting an alternative method rather than improving the existing methods.

A design method that requires lesser number of parameters to define the shaped reflector surfaces without decreasing accuracy of the obtained results is a challenging goal for reflector antenna designers. In this paper, the surface of the shaped reflector is defined using a novel equation. The shaped subreflector surfaces are assumed to be distorted forms of unshaped surfaces. As most shaped subreflectors resemble their unshaped counterparts [11–13], the assumption is logical. The shaped surfaces can therefore be represented by modified versions of the equations that represent the conventional unshaped surfaces. This method of visualizing the shaped surface as perturbed/distorted form of unshaped surface has not been reported in literature yet. As the general form of the surface is generated from the conventional conics, only a small number of parameters need be used to represent the shaped surface. Once the subreflector surface is defined, the main reflector surface is defined using GO method and equal optical path length criterion.

The paper is organized as follows. Section 2 describes the geometry of the shaped subreflector. Surface equations and differential geometric analysis are presented in this section. The synthesis method of the main reflector is discussed in Section 3. Section 4 covers the numerical results. Concluding remarks are made in Section 5.

#### 2. Geometry of the Shaped Subreflector

##### 2.1. Surface Equations

In a conventional Cassegrain geometry, the subreflector is hyperboloidal. In shaped-Cassegrain geometry, the subreflector is shaped to provide a desired illumination over the aperture of the main reflector. However, the geometrical features of this shaped subreflector are very similar to the geometrical features of the unshaped hyperboloid. Due to these similarities, the shaped surfaces can be considered a distorted form of the unshaped surfaces. So, to define the geometry of the shaped subreflector, the conventional Cassegrain geometry must be described first.

The geometry of a Cassegrain dual reflector antenna is shown in Figure 1. The parameters describing the geometry of the Cassegrain system are : diameter of the main reflector = 10 m, : focal length of the main reflector = 5 m, : diameter of the subreflector = 1.25 m, : distance between the foci = 4 m, = c/a: subreflector eccentricity = 1.4261, : depth of the paraboloid = 1.25 m, : distance from feed to paraboloid vertex = 1 m, , : opening half angle of the main reflector and subreflector = 53.13° and 10.037°, respectively.