A new method of image formation with annular apertures and an application in radio astronomy

Abstract

A new method is developed for obtaining images of high quality with an annular aperture. The principle is of general application to all forms of optics but has special application to the use of annular arrays of aerials for radio astronomy. The method allows any desired polar diagram within the diffraction limit of the aperture to be generated without recourse to the technically difficult concept of image convolution. It depends on two ideas: first, that by inserting appropriate phases into the transmission lines from each aerial element any polar diagram with the form of the Bessel function J 2 k (2πα r ) may be generated, where a is the radius of the annulus in wavelengths, r is a measure of angular distance from the axis of the annulus and k is any integer; secondly that any circularly symmetrical function F ( r ) whose Fourier spatial frequencies do not exceed the number 2 a can be expressed in the form F ( r ) = Σ t k J 2 k (2πα r ) k = 0 where the coefficients t k can be evaluated explicitly. The summation may be achieved technically by time-shared integration. The theory of the method, which is termed J 2 synthesis , is extended to the case when the annulus consists of discrete aperture elements with uniform spacing. ' The application of the principle to an annular radio telescope of diameter 3 km, at present under construction, is described.