Adaptive Optics for Vision Science and Astronomy ASP Conference Series, Vol. **VOLUME**, **PUBLICATION YEAR** A. Quirrenbach Principles of Wavefront Sensing and Reconstruction Gary Chanan University of California, Irvine, Department of Physics and Astronomy, Irvine, California 92697 Abstract. A variety of approaches to wavefront sensing and recon- struction are surveyed as they are used in adaptive optics and related applications. These include the Gerchberg-Saxton algorithm; shearing interferometry; and Shack-Hartmann, curvature, and pyramid wavefront sensing. Emphasis is placed on the relevant optics and mathematics, which are developed in some detail for Shack-Hartmann and curvature sensing (currently the two most widely-used approaches) and also to a lesser extent for pyramid sensing. Examples are given throughout. 1. Introduction The basic goal of adaptive optics is easily stated: to measure the aberrations of an incoming wavefront and then cancel these out by applying compensating aberrations, all in real time. Of course, underlying this simple statement is a host of very challenging optical, mathematical, computational, and technological problems. In this work we concentrate on some of the optical and mathematical issues associated with the first task from the above statement, i.e. measuring the aberrations of an incoming wavefront. It is convenient to subdivide this into two separate tasks, which we refer to as wavefront sensing and wavefront reconstruction, where the distinction is articulated below. We do not measure wavefront aberrations or phases directly; in practice the di- rect measurements virtually always consist of intensity distributions on a CCD or other area detector. In this work, we use “wavefront sensing” to refer to a technique by which an arbitrary wavefront phase surface is converted into an uniquely defined intensity distribution — which in turn can be (more or less) readily inverted to yield the original phase. This latter inversion is referred to as “wavefront reconstruction”; this may involve inverting a matrix, solving Poisson’s equation subject to certain boundary conditions, or another inversion procedure. The wavefront sensing generally involves manipulation of the orig- inal wavefront to facilitate the subsequent inversion. This manipulation may take place in the aperture plane (the insertion of a lenslet array in the Shack- Hartmann procedure) or in the image plane (the insertion of knife edge in the knife edge test or of a pyramid in pyramid wavefront sensing). [For the pur- poses of this paper we consider the introduction of a deliberate focus error — as in curvature sensing — as a manipulation in the aperture plane because the mathematical condition corresponding to defocus is more easily expressed there 5