## Phase-error balancing, nonlinear phase shifts, and entropy transformations in image formation optics

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https://hdl.handle.net/2144/7842##### Abstract

It is well known that the performance of an optical system is completely determined from either a knowledge of the system's of Green's function or its Fourier transform, the frequency response, these quantities being calculated from a knowledge of the amplitude and phase over the exit pupil of the system. Of particular interest is the fact that primary coma introduces a nonlinear phase shift; however, there has appeared in the literature some ambiguity concerning the nature of this phase shift. Nonlinear phase curves with and without abrupt phase jumps have been reported in the literature for the case of pure primary coma. One result of this report has been to investigate this case in both one and two dimensions, and to show that no abrupt phase jumps occur. It is also well known that for an ideal lens out of focus, one can observe abrupt phase jumps; i.e., the phenomenon of "spurious resolution." In an attempt to observe nonlinear phase shifts with abrupt jumps, the case of coma and defocussing was analyzed in detail. The second result of this report, once again, indicates that no abrupt phase shifts occur. Finally, although much research work has been performed on both measuring and calculating the frequency response of optical systems, the problem of the least ambiguous method of representing the phase of this function has not been resolved. The final purpose of this section is, then, to present one such method.
The problem of image evaluation in the range of physical optics (small phase-errors) by means of curves representing the general Seidel aberrations is well known, e.g., the Maréchal method of phase-error balancing. For the case of large phase-errors, the more appropriate ray density method has been suggested in the literature; however, a rigorous method for calculating a quality factor for image evaluation in this region is not known. Using the radius of gyration as a measure, the ray density method of averaging over the image plane is shown to be equivalent to the simpler and more physical process of averaging over the exit pupil using the radius as a weighting factor. In addition, a proof for the justification of the ray density method is given along with the geometrical intensity law. This proof stems from an analysis of Liouville's Theorem. Finally, the two limits are compared with each other and with results published in the literature for various combinations of aberrations.
Any attempt to observe the physical world will invariably result in the compilation of data from series of observations. The subsequent analysis and report of these data must necessarily be accompanied by the quotation of some degree of confidence. This usually takes the form of an error which specifies the precision of our observations, and there are many ways to represent this. One such measure has been introduced in the field of Information Theory, and it is referred to as probabilistic entropy. The purpose of this section is to clarify certain inherent difficulties associated with the definition of the entropy of a continuous probability density distribution by returning to the basic definition of information associated with a measurement and developing a formalism which eliminates these difficulties. The resulting formalism is then applied to the entropy of a Gaussian distribution. The application is then described in terms of a simple laboratory experiment in which a length is measured.

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Thesis (Ph.D.)--Boston University