The motivation for this research stems from the optical design problem. From a mathematical perspective the problem can be stated as follows: given a starting optical configuration and a set of variable parameters, determine the specific configuration which yields the global minimum of the merit function which represents the imaging quality of the system. Currently, no satisfactory solution to this problem has been found, although a process called "simulated annealing" has shown some potential. The idea behind this research is that perhaps a merit function can be constructed in such a way that information contained in higher order polychromatic aberration coefficients can be used to indicate the region of the global minimum. In pursuit of this, the construction of two physically significant merit functions (the wavefront variance and the mean square ray aberration) is formulated in such a way as to allow the segregation of aberration coefficients by order within the merit function. This suggests a sequence of merit "subfunctions" can be constructed in such a way that each member of the sequence is associated with a particular order of aberration, and that the sequence itself converges to the complete merit function. In order to compute the polychromatic aberration coefficients needed to construct the merit functions, an algorithmic approach to proximate ray tracing is developed. This is shown to be an extension of the original form of proximate ray tracing and has proved highly successful in the computation of polychromatic aberration coefficients. The behavior of three optical systems with respect to their effective design parameters is then investigated. The investigation takes the form of topographic maps of the merit subfunctions. A study of the maps reveals that the global topography of the subfunctions remains relatively invariant with respect to order. Also, any minima present tend to remain relatively stationary with respect to order, although any particular one can slowly migrate within some small region of parameter space.

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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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