Geometrical defects of a high gain reflector antenna can cause the radiation pattern of the antenna to fail to meet its specifications. These defects give rise to loss of gain, widening of the main beam and raising of sidelobes. The geometrical defects can be identified, and subsequently corrected, by utilizing information contained in the phase of the copolar aperture field distribution. For technical reasons, this phase can be difficult or inconvenient to measure directly. Therefore, indirect methods of deducing the phase are often preferred. This thesis introduces an iterative algorithm, called the modified Gerchberg-Saxton algorithm, which has been developed for retrieving the copolar aperture field phase distribution from the far field copolar amplitude pattern. In order to aid convergence of this algorithm, it incorporates information concerning the design and any known aspect of the antenna. The modified Gerchberg-Saxton algorithm is based on the conventional Gerchberg-Saxton algorithm, originally developed for electron microscopy, but incorporates features of Fienup's phase retrieval algorithms. This thesis reviews radio engineering theory with an emphasis on high gain reflector antennas. In particular, the Fourier transform relationship between the copolar aperture field distribution and the copolar radiation pattern is critically examined. The problem of retrieving the copolar aperture field distribution from the amplitude of its Fourier transform is called a Fourier phase problem. The Fourier phase problem, the uniqueness of its solutions and iterative algorithms for solving it are discussed. Other established methods for determining geometrical defects of an antenna are described and their relative advantages and disadvantages are assessed. The main advantage of the modified Gerchberg-Saxton algorithm is that it requires measurement of only a single copolar amplitude pattern. The modified Gerchberg-Saxton algorithm is evaluated by applying it to computer simulated data and to measured amplitude patterns of an acoustic antenna. This evaluation illustrates the relationship between the accuracy of the data to which the algorithm is applied and the accuracy of the retrieved copolar aperture field phase distribution. The performance of the algorithm appears to be insensitive to the location and dimensions of the geometrical defects of the antenna. The optimum form of the algorithm seems to be versatile and robust enough to offer real hope of being able to retrieve, to a useful level of accuracy, the phase of the aperture field from a single measured radiation pattern amplitude (i.e. there is no need to measure the phase of the radiation pattern).