Theoretical determination of the electromagnetic field in the neighbourhood of a parabolic cylinder antenna* J.C. Bolomey and A. Wirgin Indexing terms: Antenna radiation patterns, Antenna theory, Reflector antennas Abstract A critical evaluation is made of available asymptotic methods for the determination of the near field of parabolic cylinder reflector antennas. The analysis is restricted to reflectors fed at their focus by a line current radiating an E polarisation field. Reference results, obtained in a numerically precise manner from solutions of the integral equation associated with the rigorous formulation of the radiation problem, serve as the basis of comparison of the asymptotic methods. The latter are the aperture methods, physical optics method and the Keller geometrical optics method in its original and modified forms. This study revelas that: (a) the physical optics method is accurate and reliable for medium and large size reflectors as well as for reflectors with aperture diameters as small as two wavelengths (b) the aperture methods and the modified Keller geometrical optics method are satisfactory for medium and large size reflectors (c) all of the considered asymptotic methods (excepting the Keller method in its original form) lead to results that are practically identical to those of the rigorous theory for reflectors whose diameters exceed or are equal to ten wavelengths (d) the modified Keller method constitutes the best accuracy-calculational economy trade-off for the near field analysis of the parabolic cylinder reflector antennas ordinarily encountered in microwave applications. P x,y,z F(y) / Q D 0 co E o u0 8 A k v 0 X List of principal symbols = parabolic arc described by reflector in cross section plane = co-ordinates in cartesian system Oxyz = function which defines f focal length of reflector aperture segment diameter of reflector aperture electric field vector associated with source radiation angular frequency spatial component of £ 0 z component of E o Dirac distribution Laplacian wavenumber of source radiation in air velocity of source radiation in air wavelength of source radiation in air nth-order Hankel function of first kind distance between origin 0 and point P point whose co-ordinates are x, y electric-field vector associated with reradiated field spatial component of fi d z component of E d arbitrary point of F whose co-ordinates are x',y' two points situated at distance e along line normal to Fat?' infinitesimal distance between P' + or P'~ and P' x co-ordinate of aperture segment radiation half-space to right of line x = c entire x-y plane distance betweenPand/?'. induced current density at the point P' of F normal derivative at P' unit normal vector at P' gradient vector differential arc element centred on P' of F Euler constant nth subarc of F midpoint of F n arc length of F n number of subarcs of F asymptotic form of w 0 numerical constant associated with u 0 geometrical function associated with u 0 u g = geometrical optics approximation of u d * The paper was presented at the URSI symposium on electromagnetic wave theory, London, July 1974 Paper 7760E, first received 30th October 1975 and in revised form 28th May 1976 Dr. Bolomey and Dr. Wirgin ere with the Laboratoire des Signaux et Systtmes, CNRS-ESE, Ecole Supirieure d'Electriciti, Plateau de Moulon, 91190 Gifsur Yvette, France u d = P' = P 1 * ,P'~ = e = c = 3)> = R 2 R v{P r ) = dn, = n(P) = V = dS(P') = y = F n = P n = S n = TV = w 0 = i- = A(y) = amplitude function of u g \p = phase factor associated with u g 5) = lit subregion of U 2 in geometrical optics sense Q = point of M whose co-ordinates are x' = c,y M = linex = c dS(Q) = differential arc element centered on Q of M p = distance between P and Q d n = nth subsegment of fl Q n = midpoint of (?„ A n = length of Q n a = angle associated with points/', P', Q P\,2 = 7Ti j2 = u d ' 2 = u h = f = edge points of F tangent planes to F at P, 2 fields associated with diffracted rays coming from Pi field associated with Keller diffraction coefficient P°l ar co-ordinates of P with respect to the point Pj angle between incident ray and normal vector to TTI 2 distance between origin 0 and P x or P 2 normal vector to n lt2 field associated with modified diffraction coefficient Fresnel integral factors associated with u s numerical factors associated with u g at D k = D s = Keller diffraction coefficient modified diffraction coefficient Abbreviations r.t.m. p.o.m. a.m. k.g.o.m. m.k.g.o.m. = rigorous theory method = physical optics method = aperture method = Keller geometrical optics method = modified Keller geometrical optics method 1 Introduction Classical treatments of antenna problems are, in general, concerned with the prediction of the radiation pattern in the far field region. 1 " 3 ' 17 In the last few years increasing attention has centered on the near field region, 4 " 6 for instance, it is necessary to determine the near field radiated by an antenna in order to evaluate the influence on the far field pattern of a radome placed at a small distance from the reflector. Sl7 The question that arises is whether the well-known approximation methods used for the determination of the far field are also valid in the near-field region. It is clear that some of the approximations inherent in these methods are no longer appropriate; e.g. it is not possible to reduce the diffraction integral in the near-field region to a Fourier transform of the aperture field distribution. Other approximations associated with these methods are more specifically concerned with the induced current density on the reflector or with the aperture field and it may be that these approximations are PROC. IEE, Vol. 123, No. 11, NOVEMBER 1976 1201