The antenna pattern synthesis problem is of the utmost importance in almost every kind of antenna applications. Therefore, a very large number of contributions have appeared on this subject. But virtually all of them deal with simplified versions of the complete synthesis problem, wherein the degrees of freedom available in principle are strongly reduced, and/or idealized design criteria or requirements are considered. In this paper we present a formulation which allows us to overcome this fragmentation of the synthesis problem. A clear and direct description of the performance actually required by the antenna and a representation of the radiating properties of the antenna as a system allows us to formulate the synthesis problem as an intersections finding problem, i.e., to find a common element between a number of sets, each one containing elements fulfilling part of the requirements. This allows a completely general and flexible formulation of the problem, independent of the actual structure of the antenna. Then the practical implementation of this formulation is widely discussed, showing how an efficient solution procedure can be devised. The implications of the well-known ill-conditioning of the synthesis problem are also discussed. In order to show how the approach works and to assess its flexibility and power, a couple of significant examples are included, namely, a phase-only reconfigurable array and a shaped reflector synthesis. These examples are unconventional since no a priori choice of the intensity distribution (for the array case) or of the feed cluster (for the reflector case) is required. The method presented is able to exploit all the available degrees of freedom in order to fulfill the design requirements. >

Antenna pattern synthesis: a new general approach

Stringent requirements on reflector antenna performances in modern applications such as direct broadcast satellite (DBS) communications, radar systems, and radio astronomy have demanded the development of sophisticated synthesis techniques. Presented in the paper is a generalized diffraction synthesis technique for single- and dual-reflector antennas fed by either a single feed or an array feed. High versatility and accuracy are achieved by combining optimization procedures and diffraction analysis such as physical optics (PO) and physical theory of diffraction (PTD). With this technique, one may simultaneously shape the reflector surfaces and adjust the positions, orientations, and excitations of an arbitrarily configured array feed to produce the specified radiation characteristics such as high directivity, contoured patterns, and low sidelobe levels, etc. The shaped reflectors are represented by a set of orthogonal global expansion functions (the Jacobi-Fourier expansion), and are characterized by smooth surfaces, well-defined (superquadric) circumferences, and continuous surface derivatives. The sample applications of contoured beam antenna designs and reflector surface distortion compensation are given to illustrate the effectiveness of this diffraction synthesis technique. >

A generalized diffraction synthesis technique for high performance reflector antennas

We consider the problem of recovering a reflecting surface such that for a given point source of light the directions of the reflected rays cover a prescribed region of a far field sphere and the density of the distribution of the reflected rays is a function prescribed in advance, where the aperture of the incident ray cone is also prescribed in advance. Mathematically this problem requires one to solve a nonlinear partial differential equation of Monge - Ampere type subject to a nonlinear boundary condition. Numerical computations for the problem have been carried out by several authors. In this paper we study the existence, uniqueness, and smoothness of the reflecting surfaces for the above problem.

https://iopscience.iop.org/article/10.1088/0266-5611/12/3/013/pdf

On the design of a reflector antenna

Numerous optical and electromagnetic applications require synthesis of reflecting and refracting surfaces capable of reshaping the energy radiation intensity of a given source into a prescribed output irradiance distribution. Determination of such surfaces requires investigation of nonlinear, second order partial differential equations of Monge-Ampere type and development of computational algorithms for constructing their numerical solutions. These equations are very far from being standard and it is quite remarkable that geometric ideas not only provide natural means for their analysis but also means for computing solutions numerically. In this paper we survey some of these problems and describe the current progress in their study.

Mathematical Aspects of Design of Beam Shaping Surfaces in Geometrical Optics

An offfset imaging antenna system with compensated optical aberrations comprises a main a paraboloid reflector (4), a first paraboloid sub reflector (6), and a first feeding array (8) as an arrangement of first feed array elements to illuminate or to be illuminated by the first sub reflector. The main reflector (4) and the first sub reflector (6) are confocal by sharing a common focal point. The first feeding array (8) has a curved shape that corresponds to a first equivalent array (12) of magnified image feed elements (14) lying on a plane (16) crossing the main optical centre and perpendicular to the main bore-sight axis, all the first feed array elements (10) being positioned as to provide a planar distribution of the positions of the image feed elements (14) onto the first equivalent array (12) after a second reflection by the main reflector (4) with a central image point coinciding with the main reflector optical centre under a maximum illumination condition.

Imaging antenna systems with compensated optical aberrations based on unshaped surface reflectors

The synthesis of a dual offset reflector system with uniform aperture phase is formulated using mathematically exact equations under the assumptions of geometrical optics. We demonstrate the existence of numerical solutions to problems of practical significance. The key to the numerical method is the ability to solve the Monge-Ampere equation as a boundary value problem. It is shown that the complexity of the analysis is no greater than that involved in single reflector synthesis and both reflectors are produced simultaneously.

Go synthesis of offset dual reflectors

The mathematics of reflector design under the geometrical optics approximation is treated using complex coordinates. Conformal transformations occur naturally in the new formalism and their application to reflector design appears to have great potential. Some detailed illustrations are included.

https://iopscience.iop.org/article/10.1088/0305-4470/10/2/014/pdf

The geometrical optics design of reflectors using complex coordinates

Complex coordinates are used under the assumptions of geometrical optics to study the relations between phase and power density distributions on plane apertures of reflector antenna systems. No symmetry assumptions are made and the results form a generalisation of existing work in the field. The synthesis of a dual reflector system to produce given aperture phase and power distributions when illuminated by a point source of prescribed power density is shown to depend on the solution of a particular partial differential equation of the Monge-Ampere type. Finally it is shown that even in cases of symmetry the use of complex coordinates simplifies the design equations.

http://iopscience.iop.org/article/10.1088/0305-4470/11/4/019/pdf

Phase and power density distributions on plane apertures of reflector antennas

The problem of shaping dual offset reflectors, fed by a linearly polarised rotationally symmetric feed horn, to produce a parallel beam over a non-circular aperture with zero cross-polarisation, is considered under the assumptions of geometrical optics. The condition for zero cross-polarisation leads to a partial differential equation, which hitherto has been solved numerically, but is shown here to possess an analytical solution. For uniqueness an additional constraint is put on systems with a plane of symmetry by prescribing the power distribution along the aperture line of symmetry. A suitable choice yields a uniform aperture edge illumination.

Dual offset reflectors shaped for zero cross-polarisation and prescribed aperture illumination

The design of a reflector capable of producing a generalized two-variable beam shape when illuminated by a point source is investigated under the geometric-optics approximation. The problem is formulated as a mapping problem between points on a unit sphere in which areas are related by energy considerations. The solution of a resulting set of nonlinear partial differential equations is required. An investigation of the hyperbolic form of these equations shows that they can be reduced to a set of quasi-linear first-order partial differential equations which can be solved subject to initial conditions.

https://iopscience.iop.org/article/10.1088/0305-4470/9/1/017/pdf

F. Brickell

Papers

Go synthesis of offset dual reflectors

The geometrical optics design of reflectors using complex coordinates

Phase and power density distributions on plane apertures of reflector antennas

Dual offset reflectors shaped for zero cross-polarisation and prescribed aperture illumination

Reflector design for two-variable beam shaping in the hyperbolic case