With a view to using Phase Retrieval to accurately predict Terahertz antenna far-field from near-field intensity measurements, this paper reports on three fundamental advances that achieve very low algorithmic error penalties. The first is a new Gaussian beam analysis that provides accurate initial complex aperture estimates including defocus and astigmatic phase errors, based only on first and second moment calculations. The second is a powerful noise tolerant near-field Phase Retrieval algorithm that combines Anderson's Plane-to-Plane (PTP) with Fienup's Hybrid-Input-Output (HIO) and Successive Over-Relaxation (SOR) to achieve increased accuracy at reduced scan separations. The third advance employs teraflop Graphical Processing Units (GPUs) to achieve practically real time near-field phase retrieval and to obtain the optimum aperture constraint without any a priori information. © 1963-2012 IEEE.