This technology is based on sampling considerations for a band-limited function, which has application to optical estimation generally, and to phase retrieval specifically. The analysis begins with the observation that the Fourier transform of an optical aperture function (pupil) can be implemented with minimal aliasing for Q values down to Q = 1. The sampling ratio, Q, is defined as the ratio of the sampling frequency to the band-limited cut-off frequency. The analytical results are given using a 1-d aperture function, and with the electric field defined by the band-limited sinc(x) function. Perfect reconstruction of the Fourier transform (electric field) is derived using the Whittaker-Shannon sampling theorem for 1<q
The Fourier transform is constructed by periodic extension, i.e., by spacing copies of the transform in a definite way, recognizing that no aliasing occurs for values of the sampling ratio such that 1<q<2, which can be used to advantage in the application of phase retrieval estimation. A method was developed for propagating the electromagnetic field with no aliasing, which has been extended to 2-d optical apertures.
This work was done by Bruce Dean, Jeffrey Smith, and David Aronstein of Goddard Space Flight Center. GSC-15947-1