Study of Modelling Methods for Large Reflector Antennas

The Radiocommunications Agency of the Department of Trade and Industry has commissioned a study into modelling methods for large reflector antennas. In particular, the requirement relates to a number of large radioastronomy antennas operating in frequency bands from 37 MHz to 40 GHz, which can be shared with active services.

In performing co-ordination studies between radioastronomy and active services, it is desirable to know the antenna sidelobe gain patterns. Radioastronomy antennas are, in general, physically large and so it is difficult, or even impossible, to measure their sidelobe gain patterns. The sidelobe gain patterns can be estimated, but as the typical co-ordination distances are often within the near-field of the radioastronomy antenna the results can be difficult to evaluate.

In relation to this problem, ITU-R Study Group 7 has published Question 127/7 which calls for studies into the methodology for determining the radiation patterns and sidelobe levels of antennas both in the near-field and far-field. The Agency, therefore, has commissioned a research project to provide the necessary expertise in the field of antenna modelling.

The objectives of the programme of work undertaken by ERA were to study the technical issues relating to the modelling of large reflector antennas and to provide technical input to the Agency which can then be used by the ITU-R Correspondence Group which is considering this issue.

Techniques for analysing the effects of struts, feed or subreflector blockage, reflector surface irregularities etc. have also been addressed.

Even though most of the methods can be applied to determine the radiation pattern of reflector antennas, the validity and accuracy of the techniques is limited for specific geometries or certain regions of space. The most widely used technique for calculating the radiation patterns of large antennas utilises physical optics integration for the main beam and first few near-in sidelobes and then switches to the geometrical theory of diffraction and its derived extensions to account for diffraction and to calculate the wide angle sidelobe radiation patterns. This combination is the recommended analysis tool for calculating the radiation patterns of large reflector antennas.

Experimental techniques such as photogrammetry and holography can also be used in an indirect way to evaluate radiation patterns. These techniques are primarily diagnostic providing information concerning the surface accuracy state as well as the alignment state of feed and subreflector systems. It is these geometrical information that can be used in conjunction with the appropriate method in order to arrive at a theoretical estimate of the radiation properties of the radiotelescope.

Depending on the details of a particular installation, one should also consider effects due to the environment in which the antenna operates. Radome effects, reflection from the ground and surrounding buildings are factors that should be taken into account. Due to the obvious complications of a composite radiotelescope environment scenario, it is advisable to use the simplest possible technique. As such, ray-based methodology may be the only practical approach to evaluate the interaction of a large radioastronomical antenna with its immediate environment.

CPU		Computer Processor Unit
ECM		Equivalent Current Method
FFT		Fast Fourier Transform
GBM		Gaussian Beam Mode
GO		Geometrical Optics
GTD		Geometrical Theory of Diffraction
IFR		Induced Field Ratio
MIFR		Magnetic Current Induced Field Ratio
MoM		Method of Moments
NEC		Numerical Electromagnetics Code
PO		Physical Optics
PTD		Physical Theory of Diffraction
rms		Root mean square
UAT		Uniform Asymptotic Theory
UTD		Uniform Theory of Diffraction

In performing co-ordination studies between radioastronomy and active services, it is desirable to know the antenna sidelobe gain patterns. Radioastronomy antennas are, in general, physically large and so it is difficult, or even impossible, to measure their sidelobe gain patterns. The sidelobe levels can be estimated, but as the typical co-ordination distances are often within the near-field of the radioastronomy antenna the results can be difficult to evaluate.

The objectives of the programme of work undertaken by ERA were to study the technical issues relating to the modelling of large reflector antennas and to provide technical input to the Agency which then can be used by the ITU-R Correspondence Group which is considering this issue.

The scope of work was to provide recommendations as to the most appropriate methods for predicting the gain patterns of large reflector antennas.

Radioastronomical observations take place over an extended range within the radio frequency, microwave and millimetrewave parts of the spectrum. In order to be able to discriminate between individual and weak cosmic radio signals, radioastronomical observatories must be equipped with antennas offering significant values of gain and resolution. This usually implies that the radioastronomical antennas are very large both mechanically and electrically. For mechanically large radioastronomical antennas, direct measurement of the radiation pattern is at best an extremely difficult and costly problem, and often an impossible task.

Under these circumstances, the problem of assessing the radiation patterns of large radioastronomical antennas can be approached in two ways:

The various methods are described in Chapter 2 with the recommendations for their applicability to large reflectors discussed in Chapter 3. As a useful preamble, the rest of this section provides a general overview of the problem of modelling large reflector antennas.

The theoretical prediction methods can be divided into numerically exact methods and high frequency approximations.

The Method of Moments (MoM) is an example of a theoretically exact method. Using this approach, one can accurately evaluate the full radiation properties of any antenna provided that it is wholly metallic and can be subdivided into a wire or patch grid. The main drawback of this method is the large computational resources required. As a result the MoM is not a practical approach for antennas larger than a few wavelengths.

A number of analytic and numerical techniques have been developed, several of these over the past four decades, for the analysis and design of electrically small and large reflectors. These range from the traditional aperture field method that involves integrating the electric fields scattered by the reflector onto a projected planar aperture, to more modern hybrid methods that employ a variety of techniques each of which is applicable over different regions of the radiation pattern. Some of these yield exact solutions but require computational resources proportional to the square of the size of the reflector and are, therefore, not viable methods except for antennas up to a few wavelengths in diameter. Larger reflectors, over 100 wavelengths in size, need to be analysed using high-frequency techniques which are approximations based on asymptotic solutions to canonical problems. These have proved to be quite successful in predicting both far and near-field patterns that compare very favourably against measured data.

Circularly symmetric reflectors with axial feeds, and earlier analytic techniques, are reviewed in some detail by Clarricoats and Poulton [Ref. 1]. A comprehensive study indicating a number of configurations and theoretical options in reflector systems may be found in the collection of papers edited by Love[Ref. 2]. Other developments are described by Wood [Ref. 3], Lo and Lee [Ref. 4], and Scott [Ref. 5].

It should be emphasised that even though the reflector antennas of concern to the present study are large, techniques applicable to smaller structures (eg. MoM) can be applied to solve canonical problems that may be part of the structure, including the struts, the subreflector or the feed.

The prediction techniques for reflector radiation pattern calculations assume that the primary radiation patterns from the feed horn have been accurately modelled. Any approximations in the feed horn radiation patterns assumed will influence the results of the reflector pattern predictions. The modelling of the feed horn patterns is not seen as a major problem, since not only can the feed be measured directly but also accurate analytical techniques (such as mode matching) are available. Mode matching is a waveguide discontinuity analysis technique where the fields in each constant cross-section on either side of the discontinuity are expanded as a summation of the waveguide modes. The levels of the waveguide modes are computed by applying the boundary conditions at the discontinuity cross-section. Cascading a series of discontinuities can provide means of analysing structures such as waveguide networks and feed horns.

This chapter will treat in some detail the analytical and numerical methods currently employed in reflector analysis and outline their relative advantages and drawbacks, and also indirectly their regions of validity in modelling electrically large reflector systems. Experimental methods and their use in such radiation pattern predictions are also described.

These cover all the various techniques normally used, although minor variants exist.

The method of moments is well documented in the literature with Harrington [Ref. 6], Moore and Pizer [Ref. 7] and Miller et al [Ref. 8] providing classical textbooks that describe the technique.

Strictly the method of moments is a mathematical technique for solving inhomogeneous linear equations of the type

where L is usually a linear integro-differential operator, and the functions f and g are elements of Hilbert spaces. In this equation, g is known and the idea is to invert L to obtain the unknown function f =L^-1g. The procedure involves a technique that transforms the operator Equation (1) to a system of linear algebraic equations. To this end, the unknown function f is expanded in a series of basis functions {fn} with unknown constant coefficients {Cn}. Substituting this back into Equation (1), and taking the inner product of both sides with a set of known testing functions {wm} reduces Equation (1) to a simple matrix equation of the form:

where A and b are given by the inner products Amn = á wm, Lfnñ , bm = á wm, gñ , and x is the vector of unknown coefficients {Cn}. Equation (2) is easily solved for x using elementary numerical methods which then yields f.

In order to apply this technique to reflector analysis, it is necessary to formulate the problem in the form of Equation (1). This is accomplished by expressing the field scattered by the antenna as an integral of the unknown surface currents on the reflecting surface. Invoking the electromagnetic boundary condition that the tangential component of the total electric field be zero on a perfect conductor yields an equation for the unknown surface current density JS in the form of equation (1) above:

which is a Fredholm integral equation of the first kind. Here, un is the unit normal to the surface,

is the unit dyadic given by

with r´ and r distances for the source and observation points respectively. Ei is the incident electric field, and k=2p/lo is the free space wave number . Equation (3a) can be solved by dividing the surface into small patches over each of which JS is expanded as a sum of current components along two orthogonal directions, Wilton and Butler [Ref. 9]. Alternatively, the reflector may be modelled in the form of a wire grid. This has the advantage that the scattered field can then be expressed as a one dimensional integral of current flowing along the wire. For the case of a thin wire segment along the z-direction defined by the unit vector uz , the appropriate equation of the form (1) is given by Popovic et al [Ref. 10] as follows:

where the prime denotes the derivative. Equation (3b) is solved for the unknown current distribution by expanding it in a suitable set of basis functions.

In principle, this is the most accurate of all known methods used in electromagnetic scattering analysis. The formulation of the governing equation is exact, and extremely accurate solutions can be obtained by a suitable choice of basis and testing functions. In addition, struts, feed, subreflector and supporting structures can all be integrated into the problem. Well-defined surface irregularities on the reflector can be similarly modelled. The technique essentially fragments the complete structure into tiny linear or planar segments, on each of which a boundary condition directly derived from Maxwell's equations is applied by brute force. This results in a coupled system of equations in which electromagnetic interaction of every segment with every other segment is automatically accounted for. The method is, therefore, capable of predicting the complete antenna pattern at all points in space, taking into account the effect of antenna support and related sub-systems. Herein lies the difficulty: assuming a wire grid solution, if the reflector is modelled with M wire segments, and the current in each is represented by N basis functions, this would, in general, lead to a system of MN linear equations in as many unknowns, requiring the numerical evaluation of (MN)2 integrals to obtain the elements of the coefficient matrix. Typically, ten to twenty segments per wavelength with three basis functions per segment are needed for an accurate representation of the currents, leading to a system with over 650 unknowns per square wavelength of the reflecting surface.

In practice, however, some simplifications can be effected. In the case of focus-fed axi-symmetric reflectors, circular symmetry can be exploited to significantly reduce the number of unknown coefficients. In addition, Kirchoff's current law can be invoked at wire junctions to relate some of the unknown constants. In Numerical Electromagnetics Code (NEC), Burke and Poggio [Ref. 55], Poggio and Miller [Ref. 11], a well-known commercially available suite of moment method software that uses Equation (3b), the current I(z) in each segment is represented as the sum of three terms - a constant, a sine and a cosine. Of the three coefficients, two are eliminated by the conditions that charge and current be continuous at wire junctions leaving only one constant, which determines the current amplitude, to be determined by matrix methods. For this representation to be adequate, the length of each wire segment needs to be less than l/10, producing over 220 segments per square wavelength of the reflecting surface.

For a 100l diameter reflector, in the absence of symmetry, this method would require determination of about 1.8 million elements in the coefficient matrix A, followed by the inversion of a 1340 x 1340 complex matrix. If the sub-systems and support structures are also modelled, it would result in a significantly larger system of equations. Apart from CPU time, computer memory resources also increase rapidly with reflector size. The method is, therefore, computationally intensive, and is not a viable technique for electrically large reflectors. Typical maximum size for which the method of moments can be successfully applied is 10l. If circular symmetry is exploited, reflectors as large as 25lmay be analysed. These limits are continuously extended with powerful computer machines becoming available but it is doubtful whether they can be applied to large reflector antennas, at least in the near future.

The aperture field method described by Silver [Ref. 12], is based on a theorem which states that if S is a closed surface enclosing a finite collection of sources S , then the field due to Sat any arbitrary point exterior to S can be expressed in terms of integrals of field vectors Ea and Ha over S, where the subscript a refers to the tangential component. Thus, if S is chosen to be a sphere enclosing the antenna, then a spherical near-field scanning set-up can be used to measure the magnitude and phase of Ea and Ha over S, and from this the field of the antenna at every point in space outside S can be computed. However, measuring the near-field over a complete spherical surface surrounding a large reflector is very difficult, if not impossible, to carry out in practice. An alternative is to determine the fields over S by analytical techniques, but with complex sub-systems this is often an intractable problem and various approximations need to be invoked.

One such approximation, called the aperture field method (see Fig. 1a), is based on the assumption that Ea and Ha are non-zero over only a finite region of S. This is justified in the case of a large class of focus-fed convex reflectors where there exists a finite closed contour GA which circumscribes the family of all specularly reflected rays from the illuminated side of the reflector. The projection along the reflected ray paths on S defines a region AÎ S , bounded by GA, over which Ea and Ha are computed using the laws of geometrical optics, with Ea = 0 and Ha = 0 over S - A. This prescription specifies a sharp discontinuity along GAwhich is inconsistent with Maxwell's equations. To overcome this difficulty, electric and magnetic charge densities are postulated along GA in accordance with the equation of continuity. With this, the field scattered by the reflector is given by the expression:

where un is the outward unit normal to A, and G is the free-space scalar Green's function.

Equation (4) forms the fundamental result of the aperture field method, and applies equally to both near- and far-fields exterior to S. In the far-field region of the antenna, some simplifications can be effected in Equation (4) which significantly eases its computational complexity. Its main drawback, however, is the discontinuity postulated along GA which is then overcome by a purely artificial construct. Apart from making the formula consistent with Maxwell's equations, the addition of electric and magnetic charge densities along GA does not make it any more accurate. In actual usage, however, Equation (4) is often reduced to a scalar integral by a suitable choice of S, as discussed in the next section. It is in this form that the method is better known.

The Projected Aperture method, (see Fig.1b) again described by Silver [Ref. 12], is essentially a simplification of the aperture field method discussed in the previous section. The surface S is taken to be made up of an infinite plane P (chosen on the radiating side of the reflector) closed at infinity by an infinite hemisphere on the source side, thereby enclosing the antenna. The field over the hemispherical region vanishes (in view of the radiation condition) and the right hand side of Equation (4) reduces to a surface integral over P. With some mathematical manipulation this can be transformed into a scalar radiation integral:

where F stands for any Cartesian component of the aperture electric field, and ¶ /¶n is the normal derivative. Equation (5) can be written in a more usable form by taking P to coincide with the x-y plane as shown inFig. 1bwith the sources confined to the region z < 0. This gives the scattered field Es (x, y, z) at any arbitrary point, Q(x, y, z), as

where r is the distance from the point (x, h, 0) on the aperture to the field point, Q(x, y, z), uS is a unit vector normal to the wavefront at (x, h, 0), and uretc. are unit vectors along directions indicated by the corresponding subscripts. In Equation (6), the integral has been truncated to a finite aperture AÎ P with the implicit assumption that F(x, h) = 0 over P - A. Region A is the surface enclosed by the curve of intersection of the reflection shadow boundary with P.

In the far-field region, along the direction given by (J , f ), Equation (6) simplifies further to:

Equation (7) is the well-known scalar diffraction integral which expresses the far-field in terms of the tangential electric field over a planar aperture. In its derivation it is assumed that the phase of F varies little over A although this fact is often overlooked.

Equation (7) is widely used in the prediction of far-field patterns. The aperture field F (x , h ) is determined using geometrical optics in the region of specularly reflected rays. The field truncates along the reflection shadow boundary G resulting in a discontinuity in F (x , h ) along G . This is, of course, not true in reality. Nevertheless, Equation (7) has been widely used in the past and correctly predicts the main beam and near-in sidelobes.

The integral in Equation (7) can be evaluated in explicit closed form for a large class of aperture fields. Since F is assumed zero outside A, the limits of integration can be set to -¥to ¥ without any loss of accuracy, whence it takes the form of a double Fourier integral. Fast numerical algorithms like the Fast Fourier Transform (FFT) can then be employed in its numerical evaluation.

This method is comparatively fast, and efficient codes that employ this technique are available for a wide variety of aperture type antennas. If the edge of the reflector forms a planar contour G , A can be chosen to be the surface circumscribed by G. In such cases, the radiation integral of Equation (7) offers a distinct computational advantage over the physical optics approximation (discussed in the next section) since the integral in Equation (7) is over a planar surface (unlike in the latter where suitable curvilinear co-ordinate systems need to be employed over curved reflectors). To a first order, this can predict deformities on the reflector surface that are large in terms of wavelength, typically those with minimum radii of curvature over five wavelengths in size for geometrical optics to be valid. Qualitative effects of aperture blockage can be accounted for by suitably tracking the rays. Effect of struts can be included in the analysis by a technique known as IFR (Induced Field Ratio) which is described in Section 2.4 of this report. As mentioned earlier, the projected aperture method can correctly predict only the main beam and first few sidelobes, and of necessity only the pattern in the forward hemisphere. However, it fails to predict the cross polar pattern with sufficient degree of accuracy, and it can be shown that the method gives symmetric patterns even in cases where there is asymmetry in the feed structure Rahmat-Samii [Ref. 15]. In modern analysis, the projected aperture field method is always used in conjunction with GTD techniques (Section 2.2.5).

Physical Optics (PO) is essentially an approximation that relates the surface current on a conductor with the incident electromagnetic field. The scattered field E_s in an unbounded region, due to a collection of electric and magnetic current sources J and J_m respectively, confined to within a finite volume V, is given by

If the source consists of simply an induced current density J_son a perfectly conducting surface S, Equation (8) transforms to a surface integral over S:

Equation (9) is exact and is valid at all points in space exterior to the source region (on the actual sources the Green's function G has a singularity). If the surface current density J_s were known at every point on the reflector surface, the scattered field in both near and far-field zones could be determined from Equation (9). Unfortunately J_S is `not known, and its determination involves the solution of a complex boundary value problem. (In fact, method of moments is an attempt in this direction.)

Physical Optics is an approximation that expresses J_Sat any point on the reflector in terms of the incident magnetic field intensity H_i at that point. Specifically, it is assumed that

where u_n is the unit normal to S. This implies zero current on portions of the reflector surface not directly illuminated by the feed. Strictly, Equation (10) is valid only for an infinite perfectly conducting plane. The actual current distribution is modified (from that given by Equation (10)) by the finiteness of the reflector as well as by its curvature. If the radius of curvature is large in terms of wavelength, Equation (10) is very accurate except near the edges and in the shadow zone. Fringe currents along edges, Hwang, et al [Ref. 16] can be added to improve prediction, although this significantly increases computational complexity.

where

is the unit dyadic. Equation (11) is the standard far-field expression used in the PO approximation. In terms of CPU time and storage requirements, PO is comparable to the aperture field method. It is, however, generally more accurate than the latter and correctly predicts the main beam and close in sidelobes. It also gives a better prediction of the cross polar pattern. Smooth surface deformations can be easily modelled. In addition, the effect of struts, feed and other sub-systems can be added if the currents flowing on their surfaces (obtained by the PO expression of Equation (10)) are taken into account but the interactions between various sources are not included, and as a result such predicted effects may have only a qualitative value. Physical optics is generally used in all cases except where the radiated field can be projected onto a planar aperture comparable in size to the reflector itself, in which case the projected aperture method is computationally significantly superior. As with the aperture field method, PO is now always used in tandem with high frequency diffraction techniques.

Several approaches have been proposed for the efficient numerical evaluation of the double integral in Equation (11). The earliest of these is the so-called Ludwig algorithm [Refs. 17 and 18] where the integrand is first split into three complex scalar components that are then expressed locally as linear functions of position with unknown constant coefficients over a suitable grid. The coefficients are then determined by imposing conditions of continuity at the grid points and solving the resulting system by a linear least squares algorithm. The advantage of the method is that once the integrand is expanded in this manner, the resulting double integral of Equation (11) can be evaluated in explicit closed form. For a complete pattern prediction, up to three grid points per wavelength may be required. Similar schemes that employ higher order polynomials to piece-wise expand the magnitude and/or the phase part of the integrand have also been exploited,Pogorzelski [Ref. 19 and 20].

In contrast to the above numerical solution, the integral in Equation. (11) may be evaluated analytically after some simplification of the phase term, Rusch and Potter [Ref. 21]; Wong [Ref. 22]. For a parabolic reflector fed by a linearly polarised feed, this leads to a closed-form expression involving Bessel functions. This is an efficient technique for focus-fed axi-symmetric reflectors.

The integral of Equation (11) involves both source and field co-ordinates, a serious handicap in the case of large reflectors where a double integral needs to be numerically evaluated over a large area for each far-field direction. If the integrand could somehow be split into two sets of functions, one involving just the source co-ordinates, while the other contains only field variables, this would significantly enhance computational efficiency. The principle is that the functions containing source co-ordinates can be computed once and stored; then for each new far-field direction, we need to determine only the quantities involving far-field variables. This offers a major computational advantage, especially in cases where a full three-dimensional characterisation of the antenna pattern is required.

There are two methods that implement this scheme for reflectors with a projected circular aperture. Both yield the field co-ordinate functions in closed form, so that these are easily computed. The first technique, called the Jacobi-Bessel method Galindo-Israel and Mittra [Ref. 23]; Rahmat-Samii and Galindo-Israel [Ref. 24]; Galindo-Israel and Rahmat-Samii [Ref. 25], factors the integrand and expands an awkward exponential factor obtained in a Taylor series. Each term in the resulting expansion can be further expanded into a double series summation, using modified Jacobi polynomials for the radial direction, and a standard Fourier series for the circumferential variation. This finally yields a triple series where the integrals involving both source and field co-ordinates can be carried out in closed form, thereby realising the goal of completely decoupling the two sets of functions. For a parabolic reflector of focal length f and projected radius a, the far-field along (J , f ) is given in an infinite series of the form:

where w = cos J and w₀ indicates the approximate direction of the main beam. The functions

and

involve only the field co-ordinates and are available in closed form (these contain Bessel functions of the first kind, of order 2m + n + 1). The coefficients and are double integrals of the current density with a kernel containing only source variables and are, therefore, independent of the field co-ordinates. Hence, once C_mn and D_mnare determined, the field in any direction can be calculated from the series (12). The method can also be successfully used with GO/GTD techniques also Narasimhan and Philips [Ref. 26].

There is nothing unique about the expansion of the integrand in terms of modified Jacobi polynomials. These were chosen simply to obtain terms which could be integrated in closed form. Other expansions have been tried and are equally suitable. One of these, the Fourier-Bessel method Scott [Ref. 5], extends the projected circular aperture to the circumscribing square and expresses the integrand as a Fourier series, thereby allowing the fast FFT algorithm to compute the coefficients.

Yet another technique, particularly adapted to large reflectors, is based on sampling the far-field along a small number of discrete directions, in order to obtain the fields in other directions by interpolation Bucci, et al [Ref. 27 and 28]. This method is based on the Whittaker-Shannon sampling theorem, and expresses the integral in Equation. (11) in the form of a sampling series analogues to that used in communication theory. Sampling density is l /D, where D is the reflector diameter. This shows that about one sampling point per lobe is sufficient to completely characterise the reflector pattern, yielding a very sparse sampling grid. The field values E_mn at the sampling points may be determined by direct numerical integration of Equation (11). The field in any other direction given by the direction cosines (u, v) is then simply

Physical optics combined with several high-frequency techniques forms the basis of most modern reflector analysis methods. Its comparative accuracy has resulted in the development of a variety of schemes for the efficient evaluation of its radiation integral. Physical optics must, therefore, be the central basis of any analysis method, at least for the prediction of patterns in the forward hemisphere.

Geometrical Theory of Diffraction (GTD) is a high-frequency technique suitable for the analysis of antennas that are large in terms of wavelength. It was originally formulated by Joseph Keller [Refs. 29 and 30] as an extension of geometrical optics (GO) to account for the non-zero fields in the shadow region. This is accomplished by introducing a set of diffracted rays analogous to the reflected and transmitted rays of GO. Diffracted rays (Fig. 2) arise from edges, corners and from any other similar discontinuities in the surface curvature. Like GO, diffraction is assumed to be a strictly local phenomenon: this means the diffracted field depends only on the strength of the incident field at the point of diffraction, and on the local geometry of the diffracting wedge.

In GO, the reflected field is obtained by multiplying the incident field by a reflection coefficient. In a similar manner, the diffracted field is determined by multiplying the incident field by a diffraction coefficient; the latter is found as an asymptotic solution of a suitable canonical problem.

GTD along with its extensions is the most widely used high-frequency technique in reflector analysis, as witnessed by the huge amount of literature based on it. The theory is discussed in detail by James [Ref. 31], and a number of applications can be found in the collection of papers edited by Hansen [Ref. 32].

where

is the dyadic reflection coefficient (made up of the Fresnel coefficients for reflection from an infinite dielectric planar interface Stratton, [Ref. 33]), E_i is the incident field at the point of reflection Q_R , H is a divergence factor Deschamps [Ref. 34] which depends on the principal radii of curvature of the incident wavefront, and those of the reflecting surface at Q_R , and s is the distance from Q_Rto the field point. The diffracted field is, similarly,

where

is the dyadic diffraction coefficient, and L is a similar divergence factor. Keller's expression for

, however, fails in the transition regions close to the reflection and shadow boundaries, as well as at the caustics. To overcome this difficulty, "uniform" theories have been formulated that yield continuous functions across the transition regions. These include the Uniform Geometrical Theory of Diffraction (UTD) Kouyoumjian and Pathak [Ref. 35], and the Uniform Asymptotic Theory (UAT) Ahluwalia, et. al. [Ref. 36]. Both theories give values of

that have no singularity in the transition regions, and have been successfully applied in far-field prediction for over two decades now. The diffraction coefficients contain Fresnel integrals that are easily evaluated, thus providing a fast and efficient algorithm for the analysis of large reflectors. In computation with GTD, most of the time is actually used in locating the points of reflection and diffraction on the reflector, given the source and field points. With multiple reflectors and complex geometrical shapes, this can sometimes be quite time- consuming although not nearly as much as in evaluating double integrals over large surfaces.

Uniform theories, however, fail along caustics, defined as regions where a family of rays converge to form focal points or focal lines. Such regions can be analysed by the Equivalent Current Method (ECM) Ryan and Peters [Ref. 37], which works back from the GTD solution away from caustics to obtain an equivalent current that would produce identical fields there. This current is then used to extrapolate the field at the caustics. GTD, UTD, UAT and ECM all fail in regions where GTD caustics and the transition regions overlap. Such regions can be treated by the Physical Theory of Diffraction (PTD) Ufimtsev [Ref. 38], which is a systematic extension of the PO approach, just as GTD is an extension to GO. PTD formulates electric and magnetic edge currents from the GO fields tangential to the edge; to evaluate the diffracted field, it is necessary to integrate these currents along the length of the edge. In regions where PTD and UTD/UAT both apply, it can be shown that the leading terms of the latter can be recovered from the PTD solution. However, application of PTD involves an additional integration along the edge, and numerical computations indicate that it does not improve accuracy moreover UTD/UAT techniques outside regions where GTD caustics overlap GO shadow boundaries.

The main advantages of GO/UTD/UAT are that it renders itself to fast computation and that it can be used on arbitrarily shaped surfaces with arbitrary contours, provided the surfaces and contours have radii of curvature that are large in terms of wavelength. This means well- defined surface irregularities and surface deformations (caused, for example, by gravitational effects) can be treated by this method. Feeds and sub-reflectors can be accounted for through multiple reflections. In some cases imperfectly conducting surfaces and dielectric media can be included in the analysis. The GO/GTD method fails along caustics and alternative formalisms need to be invoked in such regions.

Most of the techniques outlined above are capable of predicting accurately the far-field patterns in their respective regions of validity. In general, a complete characterisation of the antenna pattern is best achieved by a suitable combination of various methods. Two standard approaches are commonly employed. In the first one, the field over a closed surface is obtained by a combination of GO and UTD or GO and UAT (along with ECM where these fail), and the far-field is obtained by the aperture field method. In the second approach, PO and PTD are used together, along with the radiation integral given by Equation (11). Both yield the main beam and the complete sidelobe structure, including backlobes. The latter technique, however, can more easily incorporate sub-systems into the analysis by including the PO currents on their illuminated sides.

Going back to the aperture field method, the reflector system can be completely enclosed by a closed surface S. The geometrical optics (GO) field in region A can now be supplemented by edge-diffracted fields emanating from the rim of the reflector, obtained by GTD and its extensions (UTD and UAT). Effect of feed and sub-reflectors can be incorporated by multiple reflected rays within S. This yields smooth continuous functions for both E_aand H_a over the entire surface S. If there are no GTD caustics over S, the aperture field method can correctly predict the main beam and the entire sidelobe structure, including backlobes. In modern analysis, hybrid methods involving the PO formulation is generally favoured except in instances where S can be reduced to a plane (which can be done for a wide class of reflectors). Its main application is in cases where this reduction cannot be achieved (as when a GO/GTD formulation directly provides non-zero field components over a complete closed surface Philips, et. al [Ref. 39]. It also finds occasional application with symmetric reflectors such as those formed by surfaces of revolution, but in such cases the spherical wave expansion method,Wood [Ref. 3] is often computationally more efficient if struts and other structures are excluded.

As pointed out earlier, in most cases where the aperture field method plus GTD is actually employed, S is chosen to be planar and GTD based diffracted rays are added to cover a finite surface A. Area A is chosen such that F decays to a negligible value along its periphery. This leads to a smooth and continuous function F over the entire plane P and a planar integration quickly yields the pattern in the entire forward hemisphere. If A is considerably larger than the normal projection A' of the source antenna on P, then this method fails since the phase variation of F outside A' increases rapidly as we move further away from its boundary. If the rim of the reflector is planar, A can be chosen to cap the reflector.

The PO method is similarly used with PTD techniques and also predicts the complete three dimensional pattern of the antenna, both in the near and far zones. A numerically efficient way of incorporating the currents along edges has been given in the literature Michaeli [Refs. 40 and 41].

In many systems, the reflector is in the near-field of the feed/sub-reflector. This requires accurate evaluation of the near-field patterns of the feed and/or feed sub-reflector combination. Method of moments (MoM) and hybrid MoM/GTD techniques can be applied to obtain feed patterns that take into account distortions due to the supporting structure. Gaussian beam modes discussed in Section 2.2.7 below, can be used to represent the near-field of horns accurately. Spherical mode theory can also be exploited for a rigorous expansion of the feed near-field pattern. The spherical mode theory expands the fields in the neighbourhood of the antenna into spherical vector wave functions which are elementary solutions to the vector wave equation in spherical co-ordinates for a source free unbounded space, Stratton [Ref. 33].

In the approximate techniques described above, their region of validity is usually around the main beam (paraxial direction) or along the far out sidelobe directions. For the paraxial direction the radiation patterns are evaluated as an integration of PO currents or postulated aperture fields. This integration process can be slow depending on the electrical size of the radiotelescope surface. An alternative fast analytical way for evaluating the radiative properties of the antenna is the Gaussian Beam Mode (GBM) analysis, Goldsmith [Ref. 42], Lamb [Ref. 43], Bogush [Ref. 56] and Lesurf [Ref. 57]. The analysis is valid in the near-field, the far-field and also the transition region between. The GMB forms a complete solution spectrum for Maxwell's equations under the approximation of paraxial propagation; i.e. that there exists a main propagation direction and the fields around that direction are required. Gaussian Beam Modes follow simple propagation laws and have an analytic description; thus rapid calculations are possible.

The application of GBM to large reflectors and quasioptical systems has distinct advantages. For an oversize quasioptical system illuminated by a primary feed, we need only to know the GBM expansion for the small primary feed as the total radiated field is facilitated by simple manipulation of the GBM parameters for the representation of the reflection or transmission from the quasioptical system. Furthermore, the expansions are valid in both the near and far-field points along the paraxial directions with equally rapid and straightforward evaluations.

However, GBM cannot predict the far-out sidelobe region and blockage due to subreflector or struts should be either negligible or avoided for valid results.

For efficient computation the quasioptical system needs to be "oversized" so that one expansion is used for a range of incidence angles along the quasioptical subsystem without the need to re-expand the GBM. The quasioptical systems should have adequate electrical size so that they do not significantly truncate the illuminating field of the individual modes considered.

In conclusion, the GBM analysis technique provides a rapid evaluation of the fields around the main beam for very large electrically reflectors that are blockage free.

Apart from relying in purely theoretical techniques, the radiation pattern of a radiotelescope antenna can be assessed when intermediate experimentally retrieved data are processed. The methods considered here will involve experimental data associated with:

In general, the first two methods can be considered electrically conjugate operations. The last two methods can also be considered to be similar as they relate to the processing of optically derived information with the primary aim of assessing the mechanical, and not directly the electrical, state of the radiotelescope structure.

Holography and near-field probing are considered conjugate since they are broadly based on the dual relationship ( in the Fourier transform sense ) between radiation pattern and aperture fields for a given antenna. In near-field probing, aperture fields are measured and patterns are evaluated, in holography radiative properties are recorded and aperture fields derived. Holography on the other hand has a similarity with the last two methods in the sense that the derived aperture field can be used to assess the mechanical state of the radiotelescope surface and associated systems such as feed and subreflector.

The theoretical foundation of near-field measurements is based on the enclosing surface theorem as this is expressed by Equation (4) stated in the previous aperture field chapter of this report. The only difference is that now symbol A is used to describe the full surface S. We can see that if the tangential fields over a surface enclosing an antenna are known, then the radiation pattern of the antenna can be found everywhere. Here however, rather than using a theoretical technique to evaluate the surface fields, direct measurements are involved. Detailed procedures for the determination of the radiation pattern from near-field measurement can be found in Lo and Lee [Ref. 4] when the enclosing surface is planar, cylindrical or spherical. These procedures can also include probe compensation i.e. the removal of the effects that the measurement probe can have on the accuracy of the calculated pattern.

Planar near-field probing can be used to evaluate the radiated fields over most of a hemisphere. On the other hand cylindrical and most importantly spherical near-field metrology can provide patterns for directions within most or all of the complete radiation sphere.

For these canonical geometries, the radiation pattern can be cast as an integration or summation of elemental plane wave components, cylindrical or spherical harmonics. Numerical algorithms exist which can carry out all the necessary arithmetic manipulation in a very efficient manner. However, the problems associated with the applicability of near-field metrology for the determination of the radiation pattern of a radiotelescope are due to the large amounts of near-field data required as well as the practical problems faced in order to acquire these data in a reliable way. The amount of near-field data required for an accurate pattern determination is defined by the necessary sampling interval. In the case of planar scanning this sampling interval should be

or less. For the case of a cylindrical scanning the sampling distance along the axis of the cylinder should be

or less whilst the angular sampling interval along the circular cross section should be l/(2R) in radians or less, where R is the radius of enclosing cylindrical surface. Similarly for a spherical near-field scanner the angular sampling intervals in azimuth and elevation should be at no more than l/(2R) radians.

From these sampling values it can be deduced that for common radiotelescope dimensions the amount of near-field data required is extremely large. In addition, there are significant mechanical, electrical and financial problems associated with devising suitable arrangements for the actual near-field data acquisition. These problems include the requirements for tight mechanical accuracies for the probe position when this moves over the large distances required to sample the field of the radiotelescope antenna. Also the RF source and metrology equipment should be sufficiently stable ( or their variations should be reliably compensated ) over the large time used for data acquisition. Environmental conditions can severely affect these near-field measurements which, due to practical considerations, have to be performed in open space.

For all these reasons a full near-field determination of a radiotelescope pattern cannot be a serious practical proposition. On the other hand, near-field probing can be used in order to evaluate the radiative properties of critical radiotelescope antenna subsystems such as the feed, compact feed-subreflector arrangements, or if used, beam waveguide feeding subsystems. These measured parameters can be used in a PO or GO + GTD like methodology to analyse the radiotelescope performance.

Microwave holography includes the measurement of the radiation performance of the antenna; extra complex measurements are performed from which the state of the telescope reflector profiles and alignment can be deduced. Holography can be used to identify the location and the magnitude of surface distortions. Additional information that can be retrieved from the holographic process can relate to possible feed displacement from the nominal focal position. The holographic process, Bennett et al, [Ref. 44], Rahmat-Samii, [Ref. 45], Godwin et al, [Ref. 46], G James et al, [Ref. 47], Baars et al, [Ref. 48], Wright et al, [Ref. 49], Kitsuregawa, [Ref. 50], achieves its goals when an accurate picture of the phase variation of the radiotelescope aperture field is established. Once this has been done, it is possible to employ ray tracing in order to interpret the aperture phase variation as surface profile information, Wright, [Ref. 49], Kitsuregawa, [Ref. 50].

The holographic process starts with the recording of the antenna pattern not only in amplitude but also in phase. In order that reliable phase information can be recorded, a second (reference) antenna is usually employed which remains stationary throughout the measurement process. The pattern is recorded when a distant or close-by source is illuminating the radiotelescope antenna. In the latter case, the derived aperture phase distribution should be corrected for the quadratic type phase error resulting from the inadequate separation between radiotelescope antenna and source.

The actual sources used can be located in the terrestrial environment, can be geostationary satellite-borne, or even cosmic sources of microwave radiation. The latter two types are very useful for radiotelescope holography as they allow the pattern measurement to be performed at elevation ranges similar to those used in the actual operation of the radiotelescope. Hence, the gravitational distortion effects will have much more representative values than in the case when a terrestrial illuminating source was employed. Holography based on space-borne sources offers the advantage over the use of cosmic sources in the sense that the measurement can be performed with a sufficiently large value of signal to noise ratio. It is, however, necessary to employ algorithms in order to compensate for the satellite movements throughout the measurement period, and special-to-type receivers are normally required.

Once the radiation pattern is recorded, the Fourier transform relationships are employed in order to derive the aperture field. Sampling theorem dictates that the radiation pattern should be recorded at discrete directions with a sampling interval d_UV in the U-V space (U = sin q cos f , V = sin q sin f) such that:

where D is the diameter of the radiotelescope. Fourier transform theory can be employed to prove that if an N x N measurement array is the U-V space is processed, then the aperture distribution can be recovered with pixel resolution having dimensions:

The recovered amplitude and phase value for a pixel represents the average values for the actual fields that exists in the corresponding area. This is the result of convolution type effects, due to the fact that only a truncated part of the total radiation pattern is recorded and subsequently processed. In this sense, holography can only record local averages and not point information for the radiotelescope surface profile.

Polynomial fitting of the aperture phase information can recover deterministic errors such as gravitational and global thermal distortion and aberration due to axial and lateral displacement of the feed. If all these factors are subtracted then what remains can be attributed to a pseudo-random deformation of the surface profile. These random deformations can usually be attributed primarily to manufacturing errors, and if panels are used, to panel misalignment. In the latter case, these retrieved surface data can be used in order to improve the alignment state of the panels that form the radiotelescope surface. In any case, the statistical properties of the radiotelescope surface are usually described by s which is the rms value of the surface deviation from a specified (usually parabolic) reflector profile. The value of s that can be unambiguously determined from the holographic process depends on a number of factors including the operating wavelength, the signal to noise ratio, the extent of the measured pattern record, phase instabilities on the receiving systems etc. Practical measurements, Baars et al, [Ref. 48], have indicated that the highest achievable D/s values that can be reliably measured are of the order of 250,000.

Once holographic information for the radiotelescope is retrieved it is possible to provide analytical expressions for the reflector surface using spline interpolation functions, zernike polynomials etc. This is primarily the value of holography as a pattern prediction tool. The analytical surface description can now be used in conjunction with any technique such as PO, PO + PTD, or GO + GTD to provide radiation pattern data over angular sectors not covered in the original holographic data acquisition or for frequencies other than the one used in the holographic process. Due to the convolution phenomena explained earlier, certain information on the surface profile variation is inevitably lost. This means that the employed surface description may not be entirely correct. Hence some ambiguity may exist as far as the accuracy of the predicted far out sidelobe level is concerned. However, this situation may not be as bad since the far out sidelobes are also strongly influenced by the accuracy with which the reflector rim geometry and associated illumination is reproduced in the theoretical model.

The holographic approach may not however be entirely successful in providing surface profile information when a multireflector radiotelescope is considered. In this case, it is not straightforward to correctly and unambiguously associate retrieved aperture phase information with deterministic or random profile errors for the main reflector or the subreflector(s).

As it can be seen from the above, microwave holography is not normally needed if the only requirement is to assess the antenna RF performance rather than to also identify the profile errors causing performance degradation. However, the technique can be used to measure the antenna at an intermediate range and derive the far-field.

Measurements using theodolites, as well as the photogrammetry methods which will be discussed later, are optically based approaches which utilises the geometrical nature of optical propagation in order to establish the geometrical co-ordinates of selected points (targets). These targets can be set on the surface of the main reflector, the subreflector, the backing structure, the struts or even on the primary feed of the radiotelescope antenna.

The theodolite is an instrument that basically measures two angles, one elevation and one azimuth, as it aims at the target position. There are basically two variants of theodolite measurements that can be made:

In the theodolite and tape method, Kitsuregawa, [Ref. 50], Kesteven et al, [Ref. 51], the co-ordinates of a point are expressed in a co-ordinate system associated with the theodolite. The instrument itself basically offers the two polar angles of a target. The radial distance of the target from a theodolite reference point can be measured with either a steel tape or more accurately with a laser rangefinder.

The theodolite and tape system is assumed to be calibrated prior to the actual start of the measurements. Also, there appears to be a potential problem at non-zenith locations as gravity disturbs the optical instrument as it tilts (as it will if mounted on the reflector) and so it must be specially designed.

In multi-theodolite station triangulation, the targets are surveyed by two or more (usually between 2 and 4) instruments placed at different locations in front of the radiotelescope. At this stage only angular measurements are recorded. It is possible to set up a universal co-ordinate system so that the rectangular co-ordinates of every target can be expressed unambiguously; the precise location of every theodolite station should therefore be accurately known in advance. This is the calibration phase (which is always the first task during a theodolite based measurements) and essentially consists of surveying known points, such as end points of a reference marker, bars, etc.

The measurement accuracy of theodolite systems is of course affected by the basic accuracy of the instrument. Nowadays, highly accurate digital theodolites can offer significantly reduced errors of about 0.5 arc sec. This means that the final accuracy of the theodolite measurement will be more dominated by the accuracy of the calibration process and less by the basic accuracy of the measurement itself. The accuracies experience in actual measurements expressed on the ratio of D/s explained earlier is of the order of 120,000 for two theodolite station and 250,000 for a four theodolite stations, Fraser, [Ref. 52].

Photogrammetry is a process of obtaining reliable geometrical information for an object by measuring its photographic images, Fraser, [Ref. 52], Brown, [Ref. 53], Fraser, [Ref. 54]. Again a number of target points are defined on the radiotelescope structure. Using specialised photogrammetric cameras, the radiotelescope is photographed from different directions such that the given targets will appear in more than one photograph. The co-ordinates x,y of a target image expressed in a photographic plate co-ordinate system are determined. Using the "projective equations", Brown, [Ref. 53], one can relate the image co-ordinates of a given target to its true 3-D co-ordinates associated with a universal system. The set of projective equations for all targets on the radiotelescope, and for every photograph taken, forms the basis of the photogrammetric triangulation process.

The projective equations for every target are dependent on a set of parameters (projective parameters) which are constant for every individual photograph. The determination of these constants is the goal of the calibration phase. Calibration can be performed along similar lines as those used in theodolite based measurements, ie. surveying a number of control points. However, the ability of photogrammetry to record a number of targets within a single photograph can enable an alternative calibrating procedure which runs concurrently with the actual target position determination. In essence, the possibility of having a given target recorded in a number of photographs leads to an over-determined system which can be solved by least square methods. This approach yields the projective parameters through a reduced set of equations. The co-ordinates of the target points can be deduced afterwards. Such an approach is called the "bundle method", Brown, [Ref. 53].

There is a similarity between multi-theodolite system method and photogrammetry as both methods utilise the triangulation method for establishing the target co-ordinates. There is however, an important difference in the sense that in photogrammetry there is no need to know the precise position of the instrument station. This point can be retrieved from the measurements as the bundle of rays representing the targets imaged in a photograph are always assumed to be stigmatic at the reference station position.

The accuracy of photogrammetry increases as the number of photographs increase and photogrammetry makes it easier to survey more points and at more different location than in the case of theodolite measurements. Photogrammetry is also very "quick", which is good if time variant parameters are present, as they usually are. (Mechanical measurements on large antennas often have to be performed at night with good cloud cover for constant temperature.) These advantages combined with the above mentioned powerful calibration procedure makes photogrammetry a more accurate co-ordinate determination procedure than when theodolites are employed. According to Fraser, [Ref. 52 and 54] D/s accuracies of 250,000 are routinely available from photogrammetry with figures such as 500,000 or even 1,000,000 entirely feasible from a careful photogrammetric survey of a radiotelescope structure. However, since the antenna is usually tilted in elevation to almost horizontal during the measurements, unrepresentative gravitational deformations may be introduced.

Since surveying a large number of points is possible with photogrammetry, a better expression of the radiotelescope main or subreflector surface can be found than by other measurement techniques. Processing this data with any technique such as PO or GO + GTD can yield a good estimate for the radiation pattern over an extended angular region.

The radiotelescope feed, the subreflector, if any, as well as the associated mechanical support structures and cable feeding may block some of the energy that would normally arrive at the antenna aperture. This problem is obviously much more important to axisymmetric systems that to offset arrangements. Nevertheless, these blockage problems reduce the performance of the radiotelescope. The associated effects manifest as a reduction in gain, a significant increase in sidelobe levels (at least in certain directions) as well as deterioration in the polarisation purity of the antenna system.

It is possible to include the blockage effects when we analyse the radiotelescope pattern with any technique such as PO, aperture integration or GO/UTD.

Subreflector and feed blockage can usually be taken into account by considering the optical shadowing effects on the PO surface currents or aperture fields. This is usually adequate since it allows us to evaluate the drop in main beam and the changes in the near-in sidelobe levels. Equally well the radiation pattern due to the original understated illumination could be subtracted from the illumination that would exist hypothetically over the blocked portion of the structure. If a scheme is employed by which PO is switched to GO/UTD along certain directions then this subtraction approach could still be employed with the only difference that the "unobstructed" pattern will be calculated by the GO/UTD approach. As the main beam due to the "blocked illumination" is usually much wider than the main beam of the total radiotelescope pattern, a PO or aperture integration can still be employed to calculate the blocked pattern over much wider angular regions than the range of validity of the corresponding methods for the core of the main reflector patterns. Effects due to struts can be also handed in a similar fashion, i.e. by considering this optical shadowing effects. This can be a successful approach provided that the strut cross-section is electrically large. If this is not true accurate predictions of strut effects are still possible with the aid of the Induced Field Ratio (IFR) concept Rush et al [Ref. 13], Shavit et al [Ref. 14], P S Kildal et al [Ref. 58]. The underlying philosophy behind the IFR concept is simple. The primary feed, after reflection from the main reflector, illuminates the struts and is then blocked; this illumination field has locally the character of a plane wave. The next step is to find the two dimensional (per length) "forward scattering" properties of an infinity long object having the same cross-section as the strut under plane wave illumination. Finally, the actual strut blockage field, and in particular the forward scattered field, can be calculated by scaling by the length and weight appropriate to the local illumination. The IFR concept allows the definition of two corresponding values for the two independent polarisations into which local incident plane waves can be decomposed; hence this methodology also allows the evaluation of polarisation related effects due to strut scattering.

The IFR is a quantity which relates the actual forward strut scattered field to the strut blockage field under geometrical optics shadowing conditions. The formal definition of IFR for a cylindrical object under plane wave illumination states that it is a ratio between the forward scattered field to a hypothetical field radiated in the forward direction by a plane wave having a width equal to the optical shadow size of the geometrical cross section of the cylinder. The useful aspect of the IFR concept lies in the fact that the actual IFR quantity can be determined with analytical, numerical (Method of Moments, finite elements etc.) or even experimental techniques so as the strut effects can be predicted accurately. The analytical or numerical evolution of the IFR also permits, as an intermediate step, the determination of real or equivalent currents on the struts themselves. In principle, the full fan-shaped scattered pattern of the strut can therefore be calculated. In this case, the above process correctly includes the strut blockage and scattering effects over a large angular region of the radiotelescope pattern. Some useful numerical results concerning gain loss, sidelobes and cross-polarisation due to strut blockage have been given by Kildal [Ref. 58].

When the radiotelescope surface is made of panels, interpanel gap diffraction can affect the radiation pattern as well as the boresight gain. These effects are naturally stronger as the frequency of operation increases and should be taken into account for radiotelescopes intended to be operational at millimetrewave frequencies. The concept of optical blockage is applicable if the gaps have an electrical large width. In the opposite case we can define a Magnetic Current Induced Field Ratio (MIFR) much along the same lines as the traditional IFR concept Shavit et al [Ref. 59].

Radomes can also be used to cover large radiotelescope antennas, mainly in order to protect the installation from adverse weather conditions such as strong winds, rain or snow. The radome however, is likely to cause degradation in the basic performance of the radiotelescope unit. These effects are usually:

Gain loss can be due to mismatch or transmission losses in the radome's dielectric or due to blockage by individual frame struts as in the case of a space frame radome configuration Kay [Ref. 60]. Sidelobe level increase can be also due to blockage or due to multiple reflections inside a radome due to dielectric mismatch effects. Polarisation purity degradation and beam aberration can also be caused due to similar reasons. It is usually very difficult to perform an accurate analysis of a radome enclosed radiotelescope since a number of complicated interactions do occur.

The prediction of radome effects usually rely on ray tracing approaches from the radiotelescope apertures to either a projected plane just outside the radome or to the far-field directly. In either case, the ray tracing can include effects due to the transmission through the dielectric walls of the radome as well as blockage effects due to metallic or dielectric radome frame members.

Finally, an additional factor to consider is the impact of the surrounding environment on the radiation pattern, such as the effects of near-in buildings or close-by geographical features that may obstruct the primary radio-view of the radiotelescope. Again due to the complexity of the situation only ray tracing combined possibly with UTD (from the obstacle edges) should be used to assess their environmental effects.

The review of the analytical and numerical methods presented above highlighted the validity and limitations of the various techniques. The Method of Moments is an example of a theoretically exact method that can take into account all the features of the structure. However, it is computationally intensive and as a result it cannot be applied to antennas larger than a few wavelengths. The technique can be used for canonical problems that can be considered small which can then be used to enhance other techniques.

The physical optics integration is the most widely used technique for large reflector analysis. It provides accurate results in the main beam region and near-in sidelobes. Enhancing the PO solution with GTD and its derivatives (UTD etc.) or with PTD provides full description of the antenna patterns in all directions including the wide angle illumination and back radiation. Feed spillover can be included by straightforward superposition of the fields. Subreflector or feed blockage can be included as a shadow or projected shadow on the main reflector surface. The IFR technique can be used to estimate the strut contributions to the patterns of the antennas. The IFR technique is limited to the canonical problems that can be solved approximately or efficiently and additional work may be required to establish the strut radiation patterns for a specific installation. However, general approximations can provide acceptable estimation to the levels of strut scattering which may be sufficient for the purposes of the Agency.

The physical optics integration technique can include the reflector deformation or feed/subreflector misalignments. The reflector profile can be determined experimentally by measurements or mechanical analysis. Microwave holography, Theodolite based measurements and photogrammetry are diagnostic techniques that can be used to assess the mechanical state of the reflector surface and associated systems such as feed and subreflector, the latter generally provides the best results, although it requires significant post-processing time.

The objectives of the programme of work undertaken by ERA were to study the technical issues relating to the modelling of large reflector antennas and to provide technical input to the Agency which can be used by the ITU-R Correspondence Group which is considering this issue.

as well as hybrid techniques. Techniques for struts, feed or subreflector blockage, reflector surface irregularities etc. have also been addressed.

Even though most of the methods can be applied to determine the radiation pattern of reflector antennas, the practical validity and accuracy of the techniques is usually limited to specific geometries or certain regions of space.

The Method of Moments (MoM) is an example of a theoretically exact method. Using this approach, one can accurately evaluate the full radiation properties of any antenna provided that it is wholly metallic and can be subdivided into a wire or patch grid. Additionally, this equivalent wire or patch grid is assumed to sit over a canonical environment, such as free space or infinite dielectric half plane. The latter choice is interesting since preliminary effects due to the ground are automatically taken into account, which is important for radiotelescopes operating into the HF/VHF/UHF spectrum. The main drawback of this method is the large computational resources required. As a result the MoM cannot be practically used for antennas larger than a few wavelengths.

For electrically large quasioptical antennas, the radiation problem can be solved by applying high frequency methods. These methods can be subdivided into those based on the Kirchoff's approximation (Projected aperture method, Scalar diffraction, Physical Optics, Gaussian Beam Mode Optics), on Ray tracing methods (Geometrical Optics, Geometrical - Uniform Theory of Diffraction) and corrected Kirchoff's methods (Physical Optics and Physical Theory of Diffraction).

The high frequency methods are well suited for the analysis of electrically large reflector antennas as well as for the modelling of secondary effects such as the interaction of the main antenna with adjacent structures such as building or terrain obstacles. Far-field and near-fields can be predicted using high frequency methods. Methods based on Kirchoff's approximation are well suited for the estimation of the antenna boresight, main lobe and few adjacent sidelobes. The most widely used variant is Physical Optics which involves the vectorial summation of radiated fields due to individual currents induced over the structure by the illuminating fields which have been evaluated by considering the Geometrical Optics propagation due to the primary feed. Methods based on the Kirchoff's approximation suffer from the inability to provide accurate predictions at angles widely displaced from the antenna's main beam direction. Although the computational requirements for the summation of the Physical Optics current field or projected field when the total field is evaluated are not as severe as in the case of the MoM, it is nevertheless a significant factor to consider taking into account that the sampling interval on the reflector surface needs to be of the order of a wavelength or less.

The Gaussian Beam Mode optics analysis can offer computational advantages when an axisymmetric quasi-optical system is considered. Using this approach, the field due to the prime feed can be considered as the sum of few Gaussian beam modes. One of the properties of Gaussian beam modes is that they are equally valid in the near and the far-field. Relations also exist that describe the transformation of the basic Gaussian beam-mode parameters as the beam propagates, reflects and refracts inside the quasi-optical system and the space beyond. The Gaussian beam mode fields are given by simple analytical expressions, hence this formulation offers advantages due to the simplicity and the speed of evaluations.

The problem of wide angle and non-paraxial direction prediction can be solved when the Physical Optics method is augmented with Physical Theory of Diffraction (PTD) contribution. The PTD correction consists of a fringe current acting along the rim of the quasioptical system. The value of these fringe currents is such that the 'edge condition' is satisfied, hence the diffraction phenomenon is accurately described. Since wide angle radiation for a well focussed reflector is entirely due to spillover and diffraction then the composite PO+PTD scheme can provide accurate near and far-field information along any direction. The PO+PTD method when applied to electrically large systems may require large amounts of memory and computer time in order that the PO+PTD surface and line integration can be evaluated. This problem is more severe than the simple PO case since the wide angle potential offered by the PO+PTD combination can only materialise when fine sampling grids are used. The sampling requirement can be as small as l/10.

Considerable improvement in terms of calculation speed and memory requirements can be made when we introduce a ray description for the diffracted field. This is accomplished with the introduction of diffracted rays within the framework of Geometrical Theory of Diffraction (GTD), or its uniform versions such as the Uniform Theory of Diffraction (UTD), which provides diffracted field expressions valid across the shadow boundaries of the incident or reflected fields. Radiation pattern predictions can be accomplished with the PO + UTD or the GO+ UTD scheme. It must be emphasised that any ray description fails at caustics and so a purely ray technique cannot predict the far-field characteristics of a radioastronomical antenna near boresight or indeed, the field in the vicinity of any real focal region.

The PO + UTD scheme effectively switches from PO to UTD; the PO is used to predict the radiation characteristics in the boresight region and the UTD the wide angle characteristics of the radiation pattern. Prediction of the antenna backlobe is also interesting as it corresponds to a caustic of the diffracted rays. This can be calculated by defining an equivalent edge current using the basic UTD formulation. This current is integrated in the usual far-field sense to provide the radiation pattern in the rearward direction. The implementation of a UTD scheme offers significant speed enhancement combined with minimal memory requirements. The only real requirement is the determination of the diffraction points along the antenna's rim. The location of diffracted points can be determined either analytically for simple rim shapes or as a result of a minimisation process. This is possible as diffracted rays obey the Fermat principle which states that rays travel along minimum path lengths in a homogenous medium. In view of the above mentioned advantages, a PO + UTD scheme may well be the first option for evaluating the radiated field from a quasioptical antenna. The GO + UTD combination can also provide very rapid field calculations, but as has been stated, fails to predict the main beam characteristics. It can however, be used to provide full near-field predictions. It can also be used to provide the main reflector illumination in a dual reflector configuration, even when employing a beam waveguide feed.

Pattern characteristics and gain are also significantly influenced from other factors such as strut, subreflector or feed blockage as well as deterministic or random surface distortions, including interpanel gaps.

Feed and subreflector blockage can be incorporated by including shadowing coefficient on the PO currents on reflected and diffracted rays. Strut blockage can be handled in a similar way provided that their projected width is electrically significant. However, strut blockage effects predicted in this way often do not lead to accurate results for wide angles. A more preferred approach is to use the principle of Induced Field Ratio (IFR) for the case of struts having a cross-section susceptible to analytical or numerical 2D-solution.

Surface distortion and other irregularities are likely to have significant effects on the gain as well as the sidelobe envelope of a radioastronomical antenna pattern. The associated effects depend on whether these irregularities are random or deterministic and, if random, on their statistical properties such as mean value, standard deviation, correlated interval etc. Random irregularities include panel misalignments and surface roughness. Deterministic irregularities may include gravitational distortions, thermal effects, manufacturing errors etc. Feed and subreflector misalignment may also fall into either of the above categories. Reflector surface irregularities can be accounted for analytically or numerically depending on their properties. If an analytical model describing the deformed surface exists then the methods described above can be used to calculate the effects on the radiated pattern. Feed and subreflector misalignment effects can be theoretically assessed in the same way. Simple quasioptical models can be combined with detailed statistical analyses to arrive at purely analytical expressions relating to gain and sidelobe envelopes. Although these expressions may not have been derived for a system identical to a given radioastronomical antenna, it may, nevertheless, provide an accurate estimate on the likely degradation of a given facility under the influence of surface distortion. Interpanel gaps may also degrade the performance of a large radioastronomical antenna. Interpanel gap effects can be accounted for in two ways. One is to employ a rigorous UTD or PTD formulation for every interpanel gap edge. The other way is to introduce the principle of Magnetic current IFR relying on the canonical problem of a gap between two flat panels. The former approach is well suited for gaps with width of a wavelength or more, where the latter method better suits gaps with width less than a wavelength.

Radioastronomical antennas are usually very large in mechanical and electrical size so a direct pattern measurement in the far-field is not possible. Direct application of conventional near to far-field metrology has also to be excluded, due to the obvious practical difficulties. However, conventional near-field to far-field metrology can be applied to the feed and subreflector system and the derived composite feed radiation pattern can be used in a theoretical investigation of the total main reflector radiation properties.

Other experimental techniques such as photogrammetry and holography can also be used in an indirect way to evaluate radiation patterns. These techniques are primarily diagnostic and can provide information concerning the surface accuracy state as well as the alignment state of feed and subreflector systems; the geometrical information can then be used in conjunction with the previously discussed methods in order to arrive at a theoretical estimate of the radiation properties of the radiotelescope.

The scope of the work was to provide recommendations as to the most appropriate methods for predicting the gain patterns of large reflector antennas.

Feed spillover past the reflector can be included by straightforward superposition of the fields, although GTD is also required to provide an accurate model close to the shadow boundary. Subreflector or feed blockage can be included as a shadow or projected shadow on the main reflector surface. The IFR technique can be used to estimate the strut contributions. The IFR technique is limited to measured values or to the canonical problems that can be solved approximately or exactly and additional work may be required to establish the strut radiation patterns for a specific installation. However, general approximations can provide acceptable estimation to the levels of strut scattering which usually occurs at a significant angle from boresight (dependent upon the strut configuration) and this may be sufficient for the purposes of the Agency. The Physical Optics integration technique can include the reflector deformation or feed/subreflector misalignments. The reflector profile can be determined experimentally by measurements or mechanical analysis. Microwave holography, theodolite based measurements and photogrammetry are diagnostic techniques that can also be used to assess the mechanical state of the reflector surface and associated systems such as feed and subreflector positioning.

The combination of Physical Optics integration with UTD or PTD is the recommended analysis tool for predicting the radiation pattern of large reflector antennas. The "real" profile of the reflectors can be obtained by diagnostic techniques such as holography or photogrammetry

Depending on the details of a particular installation, one should also take into account effects due to the environment in which the antenna operates. Radome effects, reflection from the ground and surrounding buildings are factors that should be taken into account. Due to the obvious complications of a composite radiotelescope environment scenario, it is advisable to use the simplest possible technique. As such, ray-based methodology is usually likely to be the only practical approach to evaluate the interaction of a large radioastronomical antenna with its immediate environment.

The recommendations are summarised in Fig. 3and Table 1below. The radiation pattern of a large reflector is divided into four regions.

	Main Mechanisms	Analysis Techniques
REGION I Forward axial region	- Feed performance - Overall antenna configuration - Reflector/feed alignment - Reflector distortions	PO
REGION II Far sidelobes	- Feed performance - Overall antenna configuration - Reflector edge diffraction - Struts	GTD/UTD and IFR
REGION III Backlobes	- Reflector edge diffraction	GTD/UTD
REGION IV Rear axial region	- Reflector edge illumination - Reflector edge geometry	Equivalent edge currents