Optical Wavefront Reconstruction: Theory and Numerical Methods

Abstract

Optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in NASA's Hubble Space Telescope (HST). NASA's success with reconstruction algorithms on the HST has lead to an effort to develop software that can aid and in some cases replace complicated, expensive and error-prone hardware. Among the many applications is HST's replacement, the Next Generation Space Telescope (NGST).This work details the theory of optical wavefront reconstruction, reviews some numerical methods for this problem, and presents a novel numerical technique which we call extended least squares. We compare the performance of these numerical methods for potential inclusion in prototype NGST optical wavefront reconstruction software. We begin with a tutorial of Rayleigh-Sommerfeld diffraction theory.

Vectorial Phase Retrieval of 1-D Signals

“…In particular, if the phases are known, a standard procedure is to compute the discrete signal values (2) where . In this paper, we restrict our study to the discrete phase retrieval problem, and consider the reconstruction of the signal values as the end goal 1 . For simplicity, we rescale time and frequency, such that , and consequently for .…”

“…As we originally proposed in [22], the key motivation underlying our work is that in various common measurement schemes, it is possible to sample more than just the absolute spectral intensity of a single signal. In this paper, we consider a setting where one measures not only the spectral intensities and , of 1 Note that since and are finite, , though the two values are close if and (in dimensionless units) [36].…”

“…This inverse problem, known as phase retrieval, is of fundamental importance in a broad range of scientific fields [1], [2]. Phase retrieval applications range from X-ray crystallography [3], nuclear magnetic resonance [4], astrophysics [5], lens-less imaging [6]- [11] and radar [12] to characterization of ultrashort pulses [13], [14].…”

“…Moreover, due to the non-convexity of the classical phase retrieval problem, relatively little is understood about the convergence or robustness to measurement noise of current algorithms [2], [19]. For a review of phase retrieval, we refer the reader to [1], [18], [20], [21] and references therein.…”

On relaxed averaged alternating reflections (RAAR) algorithm for phase retrieval with structured illumination

“…Those references point out that the solution is almost relatively unique in multidimensional cases, except for the above three "trivial associates". In literature [14], it is pointed out that, these uniqueness results are of fundamental importance, but these do not apply to numerical algorithms, in particular for the noisy data. The second issue is how to design efficient numerical algorithms.…”