## Wavefront estimation in astronomical imaging.

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##### Author

##### Date

1999##### Permanent Link

http://hdl.handle.net/10092/12798##### Thesis Discipline

Electrical Engineering##### Degree Grantor

University of Canterbury##### Degree Level

Doctoral##### Degree Name

Doctor of PhilosophyThe challenge in building astronomical telescopes is to obtain the clearest possible image of a distant star, which should appear as a single point. Extended objects, such as galaxies and planets can be regarded as collections of points. However, turbulence in the atmosphere degrades any optical signal that passes through it. The optical effects of the atmospheric turbulence arise from random inhomogeneities in the temperature distribution of the atmosphere. As a consequence of these temperature inhomogeneities, the index of refraction distribution of the atmosphere is random. Plane waves striking the atmosphere from space objects acquire an aberration as they propagate through the atmosphere. The plane wave's surface of constant phase is no longer planar when intercepted by a,n a.stronornica.l telescope. The prnctica.l consequence of a.tmospheric turbulence is that resolution is generally limited by turbulence rather than by optical design and quality of a telescope. There are a number of approaches to solving this problem, ranging from an orbiting telescope (the Hubble Space Telescope), adaptive optics, and post detection processing. The latter approaches have applications to less expensive ground based telescopes and have been the subject of many years of research. Adaptive optics is a general term for optical components whose characteristics can be modified in real time so as to alter the phase of an incident optical wavefront. An adaptive optics system can be used to correct for atmospheric induced distortions. Before any corrections can be applied, however, some measurement must be made of the phase distortions. It is the aim of this study to estimate the degradation of the wavefronts phase. Two approaches to do so are presented. Firstly, through wavefront sensors, which many adaptive optics systems have been devised from. Among them the Shack-Hartmann sensor is the most commonly used. The sensor requires a subdivision of the receiving pupil by means of sub-apertures, wherein the lowestorder deformation of the wavefront phase is estimated. This linearizes the problem of phase retrieval to solving a linear system of equations. A new analysis is presented which differs from previously published work in the estimation of the noise inherent in the centroid calculation used in this sensor. This analysis is supported by computer simulations. Secondly, the nonlinear approach of phase retrieval is discussed. The problem becomes how to relate the phase and magnitude of the Fourier transform. It is thus necessary to estimate the phase distortion in the instrument solely from measurements made at the image plane of the telescope. The process of phase retrieval is then divided into two distinct steps. The expression for the covariance of the phase distortion using a Kolmogorov model for the turbulence is derived first. This covariance is then employed as part of a formal Bayesian estimate of the phase distortion. It is also shown that phase retrieval can be employed as a robust technique for estimating the wavefront distortion using a lenslet array. The results obtained compare favorably with the alternative approach of phase diversity. Furthermore, the introduction of prior information, in the form of statistical information of the distortion, is shown to considerably enhance the success of the phase retrieval especially for very low light levels. A comparative evaluation shows the superiority of phase retrieval to Shack-Hartmann sensing, only if the local maxima are overcome. The principal drawback of phase retrieval is the relatively long computing time required to find the solution when general-purposed computer is used.