There are two topics in this dissertation: one is to develop new phase reduction algorithms for test interferograms especially of large optics and the other one is to find more accurate analytical expression of surface deflection due to gravity when the mirror is supported in the axial direction. Two new algorithms for generating phase maps from interferograms are developed. Both methods are sensitive to small-scale as well as large-scale surface errors. The first method is designed to generate phase from an interferogram that is sampled and digitized only along fringe centers, as in the case of manual digitization. A new interpolation algorithm uses the digitized data more efficiently than the fitting of Zernike polynomials, so the new method can detect small-scale surface error better than Zernike polynomial fitting. The second algorithm developed here is an automatic phase reduction process which works on test interferograms recorded by CCD camera and transferred digitally to a personal computer through a frame grabber. The interferogram results from interference of the test wavefront with a tilted reference wave-front. Phase is generated by assuming it to be proportional to the intensity of the interferogram, apart from changes of sign and offset occurring every half fringe so as to make the phase increase monotoically. The error of the new algorithm is less than 1/20 waves in the wavefront, which can be reduced further by averaging several phase maps which are generated by interferograms with random phase shifts. The new algorithm is quick and involves no smoothing, so it can detect surface errors on large mirrors on a scale of several centimeters. A new model is developed to calculate analytically the surface deflection of a mirror supported axially on multiple points. It is based on thin plate theory, but considerations of thickness variation of a curved mirror, lightweight honeycomb structure and shear are included. These additions improve the accuracy of the calculated surface deflection, giving results close to those obtained from the accurate but computer intensive finite element model.

The effects of alignment and surface figure errors and their compensation with each other in optical systems are analyzed based on computer simulations with exact ray tracing data. These effects are included in the prediction of system performance and the testing of optics. Several simple systems are used as examples. In the prediction of system performance, a Ritchey-Chretian telescope and a Reflaxicon system are studied. A correct alignment can be found to compensate certain surface figure errors in the system. This will allow larger surface figure errors to be tolerated in the system. In the testing of optics, a method to separate the figure errors from the alignment error contributions is discussed and an off-axis test configuration, the Ritchey-Common test, is studied thoroughly. A figure design approach is suggested and compared with other approaches for reduction of the measured wavefront data in the Ritchey-Common test.

A parametric design study of light weight mirror shapes with various support conditions was performed utilizing the finite element program NASTRAN. Improvements in the mirror performance were made based on the following design criteria: (1) minimization of the optical surface wavefront variations, (2) minimization of the self-weight directly related to cost of manufacturing, and (3) optimal location of support points. A pre-processor to automatically generate a finite element model for each mirror geometry was developed in order to obtain the structural deformations systematically. Additionally, a post-processor, which prepares an input data file for FRINGE (an optical computer code) was developed for generating the optical deflections that lead to the surface wavefront variations. Procedures and modeling techniques to achieve the optimum (the lightest and stiffest mirror shape due to self-weight) were addressed. Fundamental natural frequency analyses, for contoured back mirror shapes for a variety of support conditions, were performed and followed by comparisons of the results which were obtained from NASTRAN and a closed-form approximate solution. In addition, element validity and sensitivity studies were conducted to demonstrate the behavior of the element types provided in the NASTRAN program when used for optical applications. Scaling Laws for the evaluations of the optical performances and the fundamental frequencies were established.

Unobscured reflective optical systems can be more transmissive and of higher diffraction quality than classical systems. Unobscured systems are generated by decentering symmetric systems, tilting elements to correct coma or astigmatism along a real ray, or by cross-tilting elements to control astigmatism. Such a system of relatively high quality may be further corrected with a general spline surface. For spline surfaces, optical aberration coefficients are undefined. This study developed real ray analysis and design techniques for general optical systems. A decentered symmetric system with a field correcting spline surface was designed. The optical design program ACCOS V was used for most design and analysis tasks. Design and analysis of general systems are considered first. Basic system quantities of image location, scaling, and irradiation are defined with real rays. Spline surfaces are discussed with special emphasis on features important in optical design. Real ray analytical techniques of composite spot diagrams across the image, footprints on spline surfaces, wavefront aperture maps, and spline surface maps are described. The use of these tools in general system design procedures is discussed. Standard telescope objectives of f/8.5 were considered as base designs for systems with spline surfaces. A spline surface was added to the decentered Schmidt-Cassegrain. Optimization yielded diffraction-limited performance across a 0.85 degree square field. The spline system was compared to the Galileo spacecraft narrow angle lens and a three-mirror decentered design. It had a far wider field than the Galileo but at a lower quality. Diffraction quality was better than that of the three-mirror system. Simple tolerances were considered for the spline system. The allowable effect of a thermal gradient was estimated by bending the reference axis. Decentration and figure tolerances for the spline were commensurate with classical surfaces. Techniques presented were shown to be useful for design and analysis of general systems. Spline surfaces were found to be useful in optimization of such systems. This work was supported by the Director's Discretionary Fund, Jet Propulsion Laboratory, California Institute of Technology.

The primary distinction between an ordinary optical system and one which is both unobscured and reflective is that the elements of the latter must be tilted or decentered with respect to one another. In general, this results in an optical system which has no axis of rotational symmetry, and therefore the classical aberration theory of symmetric systems is no longer applicable. Furthermore, the image becomes anamorphic and keystone distorted, due to the relative tilt between the object and the optical surfaces. The first part of this work is the development of a semi-analytic treatment of the properties (through third order) of systems possessing large tilts and decentrations. The Gaussian properties of both the image and pupil are described in terms of tilt, decentration, magnification, keystone distortion, and anamorphic distortion parameters. In computing these parameters, it is important to take into account the transferred components of the parameters, which are due to the Gaussian properties of the preceding surfaces. The third order properties are computed using the fact that each optical surface, together with its associated entrance pupil, form an optical subsystem which possesses an axis of approximate symmetry. About this axis, the aberration contributions of that surface may be described in the classical wave aberration expansion. The coefficients in this expansion may be corrected for the non-circular appearance of both the object and pupil, using the parametric description of their Gaussian form. the aberration fields due to the various surface contributions are then combined vectorally to yield the resultant aberration field in the image plane. The vector theory is applied to the analysis of several optical systems, and the accuracy of the theory verified by comparison with raytrace data. As a demonstration of the usefulness of the theory to an optical designer, a three mirror unobscured system was designed using a methodology derived from the theory. At each step, the design options are explained, and the probable results are discussed. The residual aberrations of the resulting system are studied, and the selection of another design starting point is discussed from the point of view of the theory.

The purpose of this investigation is to examine synthesis for reconstruction of electrostatic lenses having an axial potential distribution four times continuously differentiable. The solution of the electrode and pole piece reconstruction is given. Spline functions are used to approximate a continuous function to fit a curve. The present method of synthesis is based on cubic spline functions, which have only two simultaneous continuous derivatives, and all the other higher derivatives are ignored. The fifth-order or quintic spline is introduced simply because it has four simultaneous continuous derivatives. So the reconstruction program would have three terms appearing in the series expansion of the off-axis potential distribution, with regard to two terms when using cubic functions.

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