A new transform relationship for interpreting axial fields near a focus

A new Fourier transform relationship is shown to apply between the axial distribution of a field near a focus and the field, averaged around the axis, in the aperture of the focussing device. If phase and amplitude are measured along the axis the transform gives the radial distribution in the aperture, averaged around the axis, of the field amplitude and of any deviations in phase from an ideally focussed wavefront. The transform is, therefore, useful in diagnosing surface errors in reflector antennas. It may also be of use in determining errors in optical aspheric lenses and mirrors where radial errors are more prevalent in manufacture than are azimuthal errors. If only axial power is measured, a Wiener-Khinchine relationship gives the autocorrelation of the averaged radial field distribution in the aperture. The autocorrelation by itself contains useful phase information and, in some instances, the averaged aperture field can be recovered from the autocorrelation through a process of modeling. Symmetry relationships are discussed together with a practical example where phase and amplitude in a radio telescope aperture are estimated by modeling from the autocorrelation function.