What Is Phase Retrieval?

The PSF for an optical system is determined by the amplitude and phase of the (approximately) spherical wavefront as it converges on the point of focus. The amplitude measures the intensity of the wavefront at each point on the sphere and is usually approximately uniform across the entire pupil, except where it is obscured by objects in the light path such as the secondary mirror and its supports. The phase measures the deviation of the wavefront from the sphere (a perfectly focussed wavefront has zero phase error). Usually the phase error is measured in units of the wavelength of light being observed.

The PSF for the given amplitude and phase is

Stated simply, the PSF is the square of the amplitude of the Fourier transform of the complex pupil function, . Note that this equation assumes that the wavefront is not too strongly curved over the pupil; if the curvature of the wavefront is large one must use a Fresnel transform rather than a Fourier transform.

Phase retrieval is the process of trying to recover the wavefront error (and possibly the amplitude as well) given a measurement of the PSF. Phase-retrieval methods have been used since the discovery of the aberration in the HST primary mirror (Burrows et al. 1991, Fienup et al. 1993) to characterize the HST optical system. Previous phase-retrieval efforts have been aimed mainly at an accurate measurement of the spherical aberration. We are now using similar techniques (and some newly developed methods) to try to improve our understanding of the HST and WFPC/FOC optical systems. Our goal is to be able to compute better PSFs for use in image restorations.

Phase retrieval has much in common with deconvolution, and many of the techniques used for image restoration have counterparts for phase retrieval. However, the equation relating the phase and the observed PSF is non-linear in the phase-retrieval problem, which makes phase retrieval considerably more difficult than image restoration. A particular problem in phase retrieval is that maximum likelihood approaches to finding the phase tend to get stuck at local maxima of the likelihood rather than finding the globally best solution. Fienup and Wackerman (1986) discuss this and other phase-retrieval problems.