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# Dissertation/Thesis Abstract

Orthogonal Polynomials and Mathematical Surface Descriptions in Freeform Optical Design
by Takaki, Nick, Ph.D., University of Rochester, 2021, 160; 28317279
Abstract (Summary)

As the freeform optics revolution continues to grow and develop, techniques for design, fabrication, and metrology enable creating systems involving freeform surfaces in increasingly broad applications. However, the difficulties and cost of fabricating and testing the surfaces in these systems continue to inhibit the widespread adoption of freeform optics. Further, existing methods for design for manufacture are often regime-specific and can involve time consuming calculations, such as sampling the surfaces over a large grid of points, that prohibit their inclusion from the beginning of the design process. Consequently, it is paramount to develop efficient, general methods for designing systems with suitable optical performance and improvements in freeform surface manufacturability.

In this work, we explore the application of the mathematics of orthogonal polynomials and freeform surface descriptions, including base surfaces, to the task of design for manufacture for freeform optics. Mathematical interactions between the sag departure variables and the base surfaces and system geometry parameters are discussed, including harmful interactions such as degeneracy and constraints for mitigating these interactionsâ€™ adverse effects.

Manufacturability estimates are then incorporated into the merit function during design by leveraging the association between the sum of the squares of the orthogonal polynomial coefficients and the physical quantity associated with the orthogonality of those polynomials. The impact of this incorporation is explored via two design examples.

Similarly, the use of off-axis conics, as more complex but still null testable base surfaces, are examined. Two design examples are shown that improve surface interferometric-testability estimates by as much as an order of magnitude via base off axis conic parameters in design.

Finally, because efficient evaluation of numerical integrals is central to the merit functionâ€™s calculation during design, schemes for numerical integration over the unit disk are explicitly constructed that achieve greater accuracy or reduced numbers of sample points versus those in the literature. Trends in those schemes are also discussed.

Indexing (document details)
 Advisor: Rolland, Jannick P., Lambropoulos, John C. Commitee: Bentley, Julie L., Stone, Bryan D. School: University of Rochester Department: Hajim School of Engineering and Applied Sciences School Location: United States -- New York Source: DAI-A 82/8(E), Dissertation Abstracts International Source Type: DISSERTATION Subjects: Optics, Design, Mechanical engineering Keywords: Cubature, Design-for-manufacture, Freeform optical design, Orthogonal polynomials Publication Number: 28317279 ISBN: 9798582542506