<title>Linear Methods In Phase Retrieval</title>

Abstract

Abstract On-orbit wavefront sensing and active alignment control are essential features of many spaceborne optical systems currently being developed. Phase retrieval is an especially appropriate wavefront sensing technique for this application, because it directly monitors system image quality and eliminates or reduces the need for auxiliary wavefront sensors. Although the general phase retrieval problem is highly complex and requires sophisticated nonlinear estimation techniques, properly selected linear methods provide satisfactory and efficient solutions to a number of important special cases. This paper discusses the per­ formance of several such linear phase retrieval algorithms. One method yields noise-optimal estimates of small wavefront errors, while a second approach can be used with arbitrarily large errors but is much more sensitive to noise. These two phase retrieval algorithms are actually special cases of a general linear algorithm that can be tuned as a function of wavefront error characteristics, measurement noise statistics, and focal plane detector geometry.Introduction the utility of linear phase retrievalOptical systems with large apertures and high resolution over a wide field of view will be required for a variety of future infrared- and visible-wavelength space sensors. The optical forms selected for these systems frequently consist of fast, off-axis mirrors that are highly sensitive to misalignments. Passive structural approaches to achieving the required opto-mechanical tolerances in such telescopes are often not feasible because sensor weight must be kept low; periodic or active on-orbit alignment sensing and control are then necessary when conventional passive approaches are inadequate. Phase retrieval from focal plane imagery is an attractive approach to alignment error sensing for this application, because it controls alignment directly on the basis of optical system performance and avoids the physical complexity of auxiliary alignment sensors, which are subject to drift or failure.Although the general phase retrieval problem is highly complex and requires sophisticated nonlinear estimation methods for its solution,15 three aspects of the phase retrieval application summarized above suggest that a simpler approach may prove satisfactory in this case. For near- to mid-term applications, the wavefront errors to be estimated will arise from rigid body mirror misalignments or from low-frequency mirror thermal deformations. The number of wavefront degrees freedom will therefore vary from three to approximately twenty.The phase retrieval algorithm need not obtain an exact value for wavefront error in a single application, but must instead produce satisfactory optical alignment after several cycles of alignment error estimation and correction. Point sources in the form of stars or beacons can be assumed for many potential systems. Granted these simplifying assumptions, it appears that properly devised linear phase retrieval algorithms possess sufficient accu­ racy and range to fine tune the alignment of optical systems with wavefront errors of up to 0.3 to 0.4 waves rms. These linear algorithms impose comparatively modest signal processing requirements, and therefore would be valuable when (as is usual) sensor signal processing requirements must be minimized.The remainder of this paper describes the theory and application of linear phase retrieval algorithms in greater detail. First, the advantages and limitations of two early attempts at linear phase retrieval are described, and then an optimal linear algorithm that a) is tunable as a function of operating conditions, b) permits performance analysis via closed-form equations of expected estimation accuracy, and c) can be specialized to the two previously described methods as limit cases, is presented. Then several additional applica­ tions of this linear phase retrieval algorithm are discussed.90