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Mathematical and Computational Methods for Freeform Optical Shape Description

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Date Issued:
2013
Abstract/Description:
Slow-servo single-point diamond turning as well as advances in computer controlled small lap polishing enable the fabrication of freeform optics, specifically, optical surfaces for imaging applications that are not rotationally symmetric. Freeform optical elements will have a profound importance in the future of optical technology. Orthogonal polynomials added onto conic sections have been extensively used to describe optical surface shapes. The optical testing industry has chosen to represent the departure of a wavefront under test from a reference sphere in terms of orthogonal ?-polynomials, specifically Zernike polynomials. Various forms of polynomials for describing freeform optical surfaces may be considered, however, both in optical design and in support of fabrication. More recently, radial basis functions were also investigated for optical shape description. In the application of orthogonal ?-polynomials to optical freeform shape description, there are important limitations, such as the number of terms required as well as edge-ringing and ill-conditioning in representing the surface with the accuracy demanded by most stringent optics applications. The first part of this dissertation focuses upon describing freeform optical surfaces with ? polynomials and shows their limitations when including higher orders together with possible remedies. We show that a possible remedy is to use edge clustered-fitting grids. Provided different grid types, we furthermore compared the efficacy of using different types of ? polynomials, namely Zernike and gradient orthogonal Q polynomials. In the second part of this thesis, a local, efficient and accurate hybrid method is developed in order to greatly reduce the order of polynomial terms required to achieve higher level of accuracy in freeform shape description that were shown to require thousands of terms including many higher order terms under prior art. This comes at the expense of multiple sub-apertures, and as such computational methods may leverage parallel processing. This new method combines the assets of both radial basis functions and orthogonal phi-polynomials for freeform shape description and is uniquely applicable across any aperture shape due to its locality and stitching principles. Finally in this thesis, in order to comprehend the possible advantages of parallel computing for optical surface descriptions, the benefits of making an effective use of impressive computational power offered by multi-core platforms for the computation of ?-polynomials are investigated. The ?-polynomials, specifically Zernike and gradient orthogonal Q-polynomials, are implemented with a set of recurrence based parallel algorithms on Graphics Processing Units (GPUs). The results show that more than an order of magnitude speedup is possible in the computation of ?-polynomials over a sequential implementation if the recurrence based parallel algorithms are adopted.
Title: Mathematical and Computational Methods for Freeform Optical Shape Description.
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Name(s): Kaya, Ilhan, Author
Foroosh, Hassan, Committee Chair
Rolland, Jannick, Committee CoChair
Turgut, Damla, Committee Member
Thompson, Kevin, Committee Member
Ilegbusi, Olusegun, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2013
Publisher: University of Central Florida
Language(s): English
Abstract/Description: Slow-servo single-point diamond turning as well as advances in computer controlled small lap polishing enable the fabrication of freeform optics, specifically, optical surfaces for imaging applications that are not rotationally symmetric. Freeform optical elements will have a profound importance in the future of optical technology. Orthogonal polynomials added onto conic sections have been extensively used to describe optical surface shapes. The optical testing industry has chosen to represent the departure of a wavefront under test from a reference sphere in terms of orthogonal ?-polynomials, specifically Zernike polynomials. Various forms of polynomials for describing freeform optical surfaces may be considered, however, both in optical design and in support of fabrication. More recently, radial basis functions were also investigated for optical shape description. In the application of orthogonal ?-polynomials to optical freeform shape description, there are important limitations, such as the number of terms required as well as edge-ringing and ill-conditioning in representing the surface with the accuracy demanded by most stringent optics applications. The first part of this dissertation focuses upon describing freeform optical surfaces with ? polynomials and shows their limitations when including higher orders together with possible remedies. We show that a possible remedy is to use edge clustered-fitting grids. Provided different grid types, we furthermore compared the efficacy of using different types of ? polynomials, namely Zernike and gradient orthogonal Q polynomials. In the second part of this thesis, a local, efficient and accurate hybrid method is developed in order to greatly reduce the order of polynomial terms required to achieve higher level of accuracy in freeform shape description that were shown to require thousands of terms including many higher order terms under prior art. This comes at the expense of multiple sub-apertures, and as such computational methods may leverage parallel processing. This new method combines the assets of both radial basis functions and orthogonal phi-polynomials for freeform shape description and is uniquely applicable across any aperture shape due to its locality and stitching principles. Finally in this thesis, in order to comprehend the possible advantages of parallel computing for optical surface descriptions, the benefits of making an effective use of impressive computational power offered by multi-core platforms for the computation of ?-polynomials are investigated. The ?-polynomials, specifically Zernike and gradient orthogonal Q-polynomials, are implemented with a set of recurrence based parallel algorithms on Graphics Processing Units (GPUs). The results show that more than an order of magnitude speedup is possible in the computation of ?-polynomials over a sequential implementation if the recurrence based parallel algorithms are adopted.
Identifier: CFE0005012 (IID), ucf:49993 (fedora)
Note(s): 2013-12-01
Ph.D.
Engineering and Computer Science, Electrical Engineering and Computing
Doctoral
This record was generated from author submitted information.
Subject(s): optical freeform shape description -- shape description -- freeform shape -- parallel processing -- local shape description
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0005012
Restrictions on Access: public 2013-12-15
Host Institution: UCF

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