Measuring optical freeform surfaces using a coupled reference data method
Identifieur interne : 004712 ( Istex/Corpus ); précédent : 004711; suivant : 004713Measuring optical freeform surfaces using a coupled reference data method
Auteurs : L B Kong ; C F Cheung ; S. To ; W B Lee ; K W ChengSource :
- Measurement Science and Technology [ 0957-0233 ] ; 2007.
Abstract
Flat optical freeform surfaces usually possess non-rotational symmetry with a small curvature and lack of strong features for surface alignment. Due to the lack of strong features and small curvature, it is difficult to align the design and measured surfaces for characterizing the surface quality of flat optical freeform surfaces with sub-micrometre form accuracy. The traditional least squares method (LSM) generally produces large errors as there is a lack of strong features as reference for the alignment of the design and measured surfaces. This paper proposes a novel and practical method named the coupled reference data method (CRDM) to evaluate flat optical freeform surfaces with high efficiency and precision in the nanometre scale. The method couples reference data to the workpiece of the freeform surface designed model and the concerning reference features are machined together with the workpiece. By aligning the reference data, the proposed CRDM carries out fast surface matching. This makes good preparation for the next matching optimization which is conducted by the least-squares and minimax zone method. After the precise surface matching, the flat optical freeform surface can be evaluated by 3D form error topography and parameters. As compared with a traditional freeform measurement method such as LSM, it is interesting to note that the accuracy and the stability of the measurement can be significantly enhanced by the CRDM.
Url:
DOI: 10.1088/0957-0233/18/7/060
Links to Exploration step
ISTEX:41E153D63871CAEFECC894AE2A02BBBD0B0ACC6BLe document en format XML
<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title>Measuring optical freeform surfaces using a coupled reference data method</title><author><name sortKey="Kong, L B" sort="Kong, L B" uniqKey="Kong L" first="L B" last="Kong">L B Kong</name><affiliation><mods:affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail:Benny.cheung@inet.polyu.edu.hk</mods:affiliation></affiliation></author><author><name sortKey="Cheung, C F" sort="Cheung, C F" uniqKey="Cheung C" first="C F" last="Cheung">C F Cheung</name><affiliation><mods:affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation></author><author><name sortKey="To, S" sort="To, S" uniqKey="To S" first="S" last="To">S. To</name><affiliation><mods:affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation></author><author><name sortKey="Lee, W B" sort="Lee, W B" uniqKey="Lee W" first="W B" last="Lee">W B Lee</name><affiliation><mods:affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation></author><author><name sortKey="Cheng, K W" sort="Cheng, K W" uniqKey="Cheng K" first="K W" last="Cheng">K W Cheng</name><affiliation><mods:affiliation>Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B</idno><date when="2007" year="2007">2007</date><idno type="doi">10.1088/0957-0233/18/7/060</idno><idno type="url">https://api.istex.fr/document/41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">004712</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a">Measuring optical freeform surfaces using a coupled reference data method</title><author><name sortKey="Kong, L B" sort="Kong, L B" uniqKey="Kong L" first="L B" last="Kong">L B Kong</name><affiliation><mods:affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail:Benny.cheung@inet.polyu.edu.hk</mods:affiliation></affiliation></author><author><name sortKey="Cheung, C F" sort="Cheung, C F" uniqKey="Cheung C" first="C F" last="Cheung">C F Cheung</name><affiliation><mods:affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation></author><author><name sortKey="To, S" sort="To, S" uniqKey="To S" first="S" last="To">S. To</name><affiliation><mods:affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation></author><author><name sortKey="Lee, W B" sort="Lee, W B" uniqKey="Lee W" first="W B" last="Lee">W B Lee</name><affiliation><mods:affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation></author><author><name sortKey="Cheng, K W" sort="Cheng, K W" uniqKey="Cheng K" first="K W" last="Cheng">K W Cheng</name><affiliation><mods:affiliation>Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Measurement Science and Technology</title><title level="j" type="abbrev">Meas. Sci. Technol.</title><idno type="ISSN">0957-0233</idno><idno type="eISSN">1361-6501</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2007">2007</date><biblScope unit="volume">18</biblScope><biblScope unit="issue">7</biblScope><biblScope unit="page" from="2252">2252</biblScope><biblScope unit="page" to="2260">2260</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint><idno type="ISSN">0957-0233</idno></series><idno type="istex">41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B</idno><idno type="DOI">10.1088/0957-0233/18/7/060</idno><idno type="PII">S0957-0233(07)40482-9</idno><idno type="articleID">240482</idno><idno type="articleNumber">060</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0957-0233</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">Flat optical freeform surfaces usually possess non-rotational symmetry with a small curvature and lack of strong features for surface alignment. Due to the lack of strong features and small curvature, it is difficult to align the design and measured surfaces for characterizing the surface quality of flat optical freeform surfaces with sub-micrometre form accuracy. The traditional least squares method (LSM) generally produces large errors as there is a lack of strong features as reference for the alignment of the design and measured surfaces. This paper proposes a novel and practical method named the coupled reference data method (CRDM) to evaluate flat optical freeform surfaces with high efficiency and precision in the nanometre scale. The method couples reference data to the workpiece of the freeform surface designed model and the concerning reference features are machined together with the workpiece. By aligning the reference data, the proposed CRDM carries out fast surface matching. This makes good preparation for the next matching optimization which is conducted by the least-squares and minimax zone method. After the precise surface matching, the flat optical freeform surface can be evaluated by 3D form error topography and parameters. As compared with a traditional freeform measurement method such as LSM, it is interesting to note that the accuracy and the stability of the measurement can be significantly enhanced by the CRDM.</div></front></TEI><istex><corpusName>iop</corpusName><author><json:item><name>L B Kong</name><affiliations><json:string>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</json:string><json:string>E-mail:Benny.cheung@inet.polyu.edu.hk</json:string></affiliations></json:item><json:item><name>C F Cheung</name><affiliations><json:string>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</json:string></affiliations></json:item><json:item><name>S To</name><affiliations><json:string>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</json:string></affiliations></json:item><json:item><name>W B Lee</name><affiliations><json:string>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</json:string></affiliations></json:item><json:item><name>K W Cheng</name><affiliations><json:string>Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</json:string></affiliations></json:item></author><subject><json:item><lang><json:string>eng</json:string></lang><value>optical freeform surface</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>form characterization</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>surface matching</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>least squares method</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>minimax zone method</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>reference data</value></json:item></subject><language><json:string>eng</json:string></language><abstract>Flat optical freeform surfaces usually possess non-rotational symmetry with a small curvature and lack of strong features for surface alignment. Due to the lack of strong features and small curvature, it is difficult to align the design and measured surfaces for characterizing the surface quality of flat optical freeform surfaces with sub-micrometre form accuracy. The traditional least squares method (LSM) generally produces large errors as there is a lack of strong features as reference for the alignment of the design and measured surfaces. This paper proposes a novel and practical method named the coupled reference data method (CRDM) to evaluate flat optical freeform surfaces with high efficiency and precision in the nanometre scale. The method couples reference data to the workpiece of the freeform surface designed model and the concerning reference features are machined together with the workpiece. By aligning the reference data, the proposed CRDM carries out fast surface matching. This makes good preparation for the next matching optimization which is conducted by the least-squares and minimax zone method. After the precise surface matching, the flat optical freeform surface can be evaluated by 3D form error topography and parameters. As compared with a traditional freeform measurement method such as LSM, it is interesting to note that the accuracy and the stability of the measurement can be significantly enhanced by the CRDM.</abstract><qualityIndicators><score>7.492</score><pdfVersion>1.4</pdfVersion><pdfPageSize>595 x 842 pts (A4)</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>6</keywordCount><abstractCharCount>1455</abstractCharCount><pdfWordCount>4328</pdfWordCount><pdfCharCount>24815</pdfCharCount><pdfPageCount>9</pdfPageCount><abstractWordCount>222</abstractWordCount></qualityIndicators><title>Measuring optical freeform surfaces using a coupled reference data method</title><genre.original><json:string>paper</json:string></genre.original><pii><json:string>S0957-0233(07)40482-9</json:string></pii><genre><json:string>research-article</json:string></genre><host><volume>18</volume><publisherId><json:string>MST</json:string></publisherId><pages><total>9</total><last>2260</last><first>2252</first></pages><issn><json:string>0957-0233</json:string></issn><issue>7</issue><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><eissn><json:string>1361-6501</json:string></eissn><title>Measurement Science and Technology</title></host><publicationDate>2007</publicationDate><copyrightDate>2007</copyrightDate><doi><json:string>10.1088/0957-0233/18/7/060</json:string></doi><id>41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B</id><score>1</score><fulltext><json:item><original>true</original><mimetype>application/pdf</mimetype><extension>pdf</extension><uri>https://api.istex.fr/document/41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B/fulltext/pdf</uri></json:item><json:item><original>false</original><mimetype>application/zip</mimetype><extension>zip</extension><uri>https://api.istex.fr/document/41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a">Measuring optical freeform surfaces using a coupled reference data method</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>IOP Publishing</publisher><availability><p>IOP</p></availability><date>2007</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a">Measuring optical freeform surfaces using a coupled reference data method</title><author><persName><forename type="first">L B</forename><surname>Kong</surname></persName><affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation><affiliation>E-mail:Benny.cheung@inet.polyu.edu.hk</affiliation></author><author><persName><forename type="first">C F</forename><surname>Cheung</surname></persName><affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation></author><author><persName><forename type="first">S</forename><surname>To</surname></persName><affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation></author><author><persName><forename type="first">W B</forename><surname>Lee</surname></persName><affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation></author><author><persName><forename type="first">K W</forename><surname>Cheng</surname></persName><affiliation>Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation></author></analytic><monogr><title level="j">Measurement Science and Technology</title><title level="j" type="abbrev">Meas. Sci. Technol.</title><idno type="pISSN">0957-0233</idno><idno type="eISSN">1361-6501</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2007"></date><biblScope unit="volume">18</biblScope><biblScope unit="issue">7</biblScope><biblScope unit="page" from="2252">2252</biblScope><biblScope unit="page" to="2260">2260</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint></monogr><idno type="istex">41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B</idno><idno type="DOI">10.1088/0957-0233/18/7/060</idno><idno type="PII">S0957-0233(07)40482-9</idno><idno type="articleID">240482</idno><idno type="articleNumber">060</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2007</date></creation><langUsage><language ident="en">en</language></langUsage><abstract><p>Flat optical freeform surfaces usually possess non-rotational symmetry with a small curvature and lack of strong features for surface alignment. Due to the lack of strong features and small curvature, it is difficult to align the design and measured surfaces for characterizing the surface quality of flat optical freeform surfaces with sub-micrometre form accuracy. The traditional least squares method (LSM) generally produces large errors as there is a lack of strong features as reference for the alignment of the design and measured surfaces. This paper proposes a novel and practical method named the coupled reference data method (CRDM) to evaluate flat optical freeform surfaces with high efficiency and precision in the nanometre scale. The method couples reference data to the workpiece of the freeform surface designed model and the concerning reference features are machined together with the workpiece. By aligning the reference data, the proposed CRDM carries out fast surface matching. This makes good preparation for the next matching optimization which is conducted by the least-squares and minimax zone method. After the precise surface matching, the flat optical freeform surface can be evaluated by 3D form error topography and parameters. As compared with a traditional freeform measurement method such as LSM, it is interesting to note that the accuracy and the stability of the measurement can be significantly enhanced by the CRDM.</p></abstract><textClass><keywords scheme="keyword"><list><head>keywords</head><item><term>optical freeform surface</term></item><item><term>form characterization</term></item><item><term>surface matching</term></item><item><term>least squares method</term></item><item><term>minimax zone method</term></item><item><term>reference data</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="2007">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><original>false</original><mimetype>text/plain</mimetype><extension>txt</extension><uri>https://api.istex.fr/document/41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus iop not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="ISO-8859-1"</istex:xmlDeclaration><istex:docType SYSTEM="http://ej.iop.org/dtd/iopv1_5_1.dtd" name="istex:docType"></istex:docType><istex:document><article artid="mst240482"><article-metadata><jnl-data jnlid="mst"><jnl-fullname>Measurement Science and Technology</jnl-fullname><jnl-abbreviation>Meas. Sci. Technol.</jnl-abbreviation><jnl-shortname>MST</jnl-shortname><jnl-issn>0957-0233</jnl-issn><jnl-coden>MSTCEP</jnl-coden><jnl-imprint>IOP Publishing</jnl-imprint><jnl-web-address>stacks.iop.org/MST</jnl-web-address></jnl-data><volume-data><year-publication>2007</year-publication><volume-number>18</volume-number></volume-data><issue-data><issue-number>7</issue-number><coverdate>July 2007</coverdate></issue-data><article-data><article-type type="paper" sort="regular"></article-type><type-number type="paper" artnum="060">60</type-number><article-number>240482</article-number><first-page>2252</first-page><last-page>2260</last-page><length>9</length><pii>S0957-0233(07)40482-9</pii><doi>10.1088/0957-0233/18/7/060</doi><copyright>2007 IOP Publishing Ltd</copyright><ccc>0957-0233/07/072252+09$30.00</ccc><printed>Printed in the UK</printed><features colour="global" mmedia="no" suppdata="no"></features></article-data></article-metadata><header><title-group><title>Measuring optical freeform surfaces using a coupled reference data method</title><short-title>Measuring optical freeform surfaces using a coupled reference data method</short-title><ej-title>Measuring optical freeform surfaces using a coupled reference data method</ej-title></title-group><author-group><author address="mst240482ad1" email="mst240482ea1"><first-names>L B</first-names><second-name>Kong</second-name></author><author address="mst240482ad1"><first-names>C F</first-names><second-name>Cheung</second-name></author><author address="mst240482ad1"><first-names>S</first-names><second-name>To</second-name></author><author address="mst240482ad1"><first-names>W B</first-names><second-name>Lee</second-name></author><author address="mst240482ad2"><first-names>K W</first-names><second-name>Cheng</second-name></author></author-group><address-group><address id="mst240482ad1"><orgname>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University</orgname>, Hung Hom, Kowloon, Hong Kong, <country>People's Republic of China</country></address><address id="mst240482ad2"><orgname>Department of Electrical Engineering, The Hong Kong Polytechnic University</orgname>, Hung Hom, Kowloon, Hong Kong, <country>People's Republic of China</country></address><e-address id="mst240482ea1"><email mailto="Benny.cheung@inet.polyu.edu.hk">Benny.cheung@inet.polyu.edu.hk</email></e-address></address-group><history received="13 December 2006" finalform="29 April 2007" online="19 June 2007"></history><abstract-group><abstract><heading>Abstract</heading><p indent="no">Flat optical freeform surfaces usually possess non-rotational symmetry with a small curvature and lack of strong features for surface alignment. Due to the lack of strong features and small curvature, it is difficult to align the design and measured surfaces for characterizing the surface quality of flat optical freeform surfaces with sub-micrometre form accuracy. The traditional least squares method (LSM) generally produces large errors as there is a lack of strong features as reference for the alignment of the design and measured surfaces. This paper proposes a novel and practical method named the coupled reference data method (CRDM) to evaluate flat optical freeform surfaces with high efficiency and precision in the nanometre scale. The method couples reference data to the workpiece of the freeform surface designed model and the concerning reference features are machined together with the workpiece. By aligning the reference data, the proposed CRDM carries out fast surface matching. This makes good preparation for the next matching optimization which is conducted by the least-squares and minimax zone method. After the precise surface matching, the flat optical freeform surface can be evaluated by 3D form error topography and parameters. As compared with a traditional freeform measurement method such as LSM, it is interesting to note that the accuracy and the stability of the measurement can be significantly enhanced by the CRDM.</p></abstract></abstract-group><classifications><keywords print="yes"><keyword>optical freeform surface</keyword><keyword>form characterization</keyword><keyword>surface matching</keyword><keyword>least squares method</keyword><keyword>minimax zone method</keyword><keyword>reference data</keyword></keywords></classifications></header><body numbering="bysection"><sec-level1 id="mst240482s1" label="1"><heading>Introduction</heading><p indent="no">Freeform optics is an emerging field usually applied in photonics and telecommunication products such as laser printers, hand held scanners, flat panel displays and broad band optical fibre connectors [<cite linkend="mst240482bib01">1</cite>] which have a rapidly growing market. The fabrication of high quality optical freeform optics demands advanced machining technology capable of machining freeform surfaces with nanometric surface roughness and a form error in the submicrometre range.</p><p>Optical freeform surfaces possess non-rotational symmetry and the general principle of form characterization of the freeform surfaces has been analysed by some researchers [<cite linkend="mst240482bib02" range="mst240482bib02,mst240482bib03,mst240482bib04,mst240482bib05,mst240482bib06,mst240482bib07">2–7</cite>] based on the best matching method, the minimum principle and the least squares method. Although some research work [<cite linkend="mst240482bib05">5</cite>, <cite linkend="mst240482bib07">7</cite>, <cite linkend="mst240482bib13">13</cite>] has taken place in the form characterization of optical freeform surfaces, they were not developed for the characterization of ultra-precision freeform surfaces with sub-micrometre form accuracy. Some research groups such as MIT, Stanford, Carnegie Mellon and Georgia Tech. also conducted some research work in reverse engineering, and some commercial software packages such as Geomagic [<cite linkend="mst240482bib14">14</cite>], Rapidform [<cite linkend="mst240482bib15">15</cite>] and Metris [<cite linkend="mst240482bib16">16</cite>] are available for CAD design and reverse engineering with surface reconstruction for applications in sheet metal forming, casting, forging, etc. Some research work is also done on the characterization of freeform surfaces based on reverse engineering [<cite linkend="mst240482bib17" range="mst240482bib17,mst240482bib18,mst240482bib19,mst240482bib20,mst240482bib21,mst240482bib22">17–22</cite>]. Reverse engineering focuses on the conversion of a discrete set of data into a piecewise smooth and continuous model [<cite linkend="mst240482bib23">23</cite>, <cite linkend="mst240482bib24">24</cite>]. Tasks of reverse engineering try to represent the original surface as closely as possible, which is usually achieved by surface fitting. However, the fitting error usually remains in mm scale [<cite linkend="mst240482bib25">25</cite>]. Huang <italic>et al</italic> [<cite linkend="mst240482bib26">26</cite>] tried an approach to compare two freeform surfaces but the accuracy is not better than 1 mm. To improve the accuracy of the form characterization, an integrated freeform characterization method [<cite linkend="mst240482bib27">27</cite>] and a robust form characterization method [<cite linkend="mst240482bib28">28</cite>] have been proposed by the authors, which allow the measurement of the form accuracy of ultra-precision freeform surfaces with sub-micrometre form accuracy.</p><p>However, flat optical freeform surfaces usually possess non-rotational symmetry with a small curvature and lack of strong features for surface alignment. Due to the lack of strong features and the small curvature, traditional freeform measurement methods such as the least-squares method generally produce large errors as there is a lack of strong features as a reference for the alignment of the design and measured surfaces. As a result, it is difficult to align the design and measured surfaces for characterizing the surface quality of flat optical freeform surfaces with sub-micrometre form accuracy. There is a need to develop this new method to perform high-precision characterization of the form error of flat optical freeform surfaces with sub-micrometre form accuracy. In this paper, a novel method named the coupled reference data method (CRDM) is proposed for the characterization of the form error for flat optical freeform surfaces. The method mainly employs three steps to evaluate a flat optical freeform surface, which includes data measurement, surface matching and surface evaluation. The surface matching is carried out based on the coupled reference data (CRD), LSM and minimax zone optimizing method (3M).</p></sec-level1><sec-level1 id="mst240482s2" label="2"><heading>Coupled reference data method (CRDM)</heading><sec-level2 id="mst240482s2-1" label="2.1"><heading>Systematic error of surface measurement</heading><p indent="no">The coordinate system for the design of an optical freeform surface is called the design coordinate system. After the optical freeform surface is fabricated, measurement is taken to evaluate the quality of the surface in terms of the form accuracy and surface roughness. The coordinate system for the measured surface is called the measurement coordinate system. To remove the misalignment between the design and measured surfaces, surface matching is firstly undertaken to remove the system alignment error between the two coordinate systems.</p></sec-level2><sec-level2 id="mst240482s2-2" label="2.2"><heading>Coupled reference data</heading><p indent="no">Due to the lack of strong features of the flat optical freeform surface, it is very difficult to find a remarkable feature to determine the precise corresponding position from the measured surface to the designed one. However, some reference data are used to help the establishment of the corresponding relationship between the design and measured surfaces. In the present study, coupled reference data are used to carry out the rough alignment for the measured and designed surfaces.</p><p>After the optical freeform surface is designed according to the functional and optical specifications, the design data can be represented in the form of an equation or a cloud of scattered data points. Reference data are then added to the design surface model. These reference data can be machined in the workpieces which are fixed in relative positions on the workpiece surface.</p><p>In the present study, there are three types of CRD, namely PNP, PPP and NPN, as shown in figure <figref linkend="mst240482fig01">1</figref>:
<ordered-list id="mst240482ol1" type="roman" pattern="2"><list-item id="mst240482ol1.1" marker="(i)"><p indent="no">PNP: two data points and one directional vector;</p></list-item><list-item id="mst240482ol1.2" marker="(ii)"><p indent="no">PPP: three data points which are required not to be in the same line;</p></list-item><list-item id="mst240482ol1.3" marker="(iii)"><p indent="no">NPN: one data point and two directional vectors while the two vectors should not be parallel to each other.</p></list-item></ordered-list>Spherical and planar surfaces are employed as the reference data coupled with the surface of the workpiece, which is referred to as the work surface. The centre of the spherical surface is referred to as the data point of coupled reference data (CRD), and the normal vector of a plane is taken as the directional vector of the CRD. As shown in figure <figref linkend="mst240482fig02">2</figref>, surface 1 is the work surface which is evaluated. Surfaces 2 and 4 are spherical surfaces while surface 3 is a planar surface. Surfaces 2–4 serve as the CRD, which are designed and machined with a predefined relative position on the work surface.
<figure id="mst240482fig01"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig01.eps" width="15pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig01.jpg"></graphic-file></graphic><caption id="mst240482fc01" label="Figure 1"><p indent="no">Coupled reference data (CRD).</p></caption></figure><figure id="mst240482fig02"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig02.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig02.jpg"></graphic-file></graphic><caption id="mst240482fc02" label="Figure 2"><p indent="no">Surface with PNP type of CRD.</p></caption></figure></p></sec-level2><sec-level2 id="mst240482s2-3" label="2.3"><heading>Surface matching by the CRD method</heading><p indent="no">The optical freeform surface is measured together with the coupled reference data. They are used to match the measured surface with the designed one. The matching proceeds in two steps in which the surfaces are first matched based on the CRD and then the matching is optimized. With the use of the CRDM, the measured surface can be efficiently matched with the design surface which allows preparation for the evaluation of the form error of the flat optical freeform surface. The surface matching is conducted using the homogeneous transformation matrix (HTM). There are two HTMs involved in the CRDM: one is the HTM around a spatial point by three transitional parameters and three rotational parameters while the other is the HTM around a spatial line.</p><p>For the HTM by transition and rotation in coordinate systems [<cite linkend="mst240482bib27">27</cite>], <italic>t<sub>x</sub></italic>, <italic>t<sub>y</sub></italic> and <italic>t<sub>z</sub></italic> denote the transition distances while α, β and γ denote the rotation angles along the <italic>X</italic>, <italic>Y</italic> and <italic>Z</italic> axes, respectively. Hence, the transformation matrix can be expressed in equation (<eqnref linkend="mst240482eqn01">1</eqnref>):
<display-eqn id="mst240482eqn01" lines="multiline" eqnnum="1" eqnalign="left"></display-eqn>The HTM around a spatial line [<cite linkend="mst240482bib28">28</cite>] employed in the proposed method is the rotation around the spatial line. It is assumed that the line is across the point (<italic>x<sub>L</sub></italic>, <italic>y<sub>L</sub></italic>, <italic>z<sub>L</sub></italic>) with the directional vector [<italic>e<sub>x</sub></italic>, <italic>e<sub>y</sub></italic>, <italic>e<sub>z</sub></italic>]. Hence, the transformation matrix can be described by rotating by an angle &thetas; around this line as expressed in equation (<eqnref linkend="mst240482eqn02">2</eqnref>):
<display-eqn id="mst240482eqn02" eqnnum="2" eqnalign="center"></display-eqn>where
<display-eqn id="mst240482eqn03" lines="multiline" eqnnum="3" eqnalign="left" numalign="dropped"></display-eqn>To proceed with the matching of PPP by CRD, the first task is to recognize the three reference data, which can be carried out by data fitting. The three reference surfaces are identified by their different radii by data fitting. As shown in figure <figref linkend="mst240482fig01" override="yes">1(<italic>a</italic>)</figref>, the centres of the three reference spherical surfaces are <italic>P</italic><sub>1</sub>, <italic>P</italic><sub>2</sub> and <italic>P</italic><sub>3</sub>.</p><p>It is assumed that the points <italic>P</italic>′<sub valign="yes">1</sub>, <italic>P</italic>′<sub valign="yes">2</sub> and <italic>P</italic>′<sub valign="yes">3</sub> are the design points corresponding to <italic>P</italic><sub>1</sub>, <italic>P</italic><sub>2</sub> and <italic>P</italic><sub>3</sub>. Then, a fast matching process is carried out according to the following steps.
<itemized-list id="mst240482il1" type="none"><list-item id="mst240482il1.1"><p indent="no"><italic>Step 1</italic>. The centre point <italic>P</italic><sub>1</sub> is made coincident with the corresponding design point <italic>P</italic>′<sub valign="yes">1</sub>, which is simply a translation process.</p></list-item><list-item id="mst240482il1.2"><p indent="no"><italic>Step 2</italic>. The reference data are rotated around <italic>P</italic><sub>1</sub> to make <italic>P</italic><sub>2</sub> coincident with the corresponding point <italic>P</italic>′<sub valign="yes">2</sub>.</p></list-item><list-item id="mst240482il1.3"><p indent="no"><italic>Step 3</italic>. The reference data are rotated along the line <italic>P</italic><sub>1</sub><italic>P</italic><sub>2</sub> to make <italic>P</italic><sub>3</sub> coincident with <italic>P</italic>′<sub valign="yes">3</sub>.</p></list-item></itemized-list></p><p>Thus, the matching of the workpiece surface is carried out. Meanwhile, the reference data are matched at every step. Figure <figref linkend="mst240482fig03">3</figref> shows the flow chart of the matching process by CRD of PPP.
<figure id="mst240482fig03"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig03.eps" width="18pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig03.jpg"></graphic-file></graphic><caption id="mst240482fc03" label="Figure 3"><p indent="no">Surface matching by PPP type of CRD.</p></caption></figure></p><p>If the type of CRD is PNP, there are two position points and one directional vector and the surfaces are matched by CRD as shown in figure <figref linkend="mst240482fig04">4</figref>. The directional vector <italic>N</italic>′ is obtained by fitting the planar surface data to find the normal vector. If the CRD is NPN, there are one point data and two normal vectors (not parallel to each other) and the surfaces are matched by CRD as shown in figure <figref linkend="mst240482fig05">5</figref>. Basically, the surface matching by PNP and NPN is similar to the matching of the PPP type of CRD except for the different types of reference surfaces.
<figure id="mst240482fig04"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig04.eps" width="18pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig04.jpg"></graphic-file></graphic><caption id="mst240482fc04" label="Figure 4"><p indent="no">Surface matching by PNP type of CRD.</p></caption></figure><figure id="mst240482fig05"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig05.eps" width="18pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig05.jpg"></graphic-file></graphic><caption id="mst240482fc05" label="Figure 5"><p indent="no">Surface matching by NPN type of CRD.</p></caption></figure></p><p>For the optimization of the surface matching, the least squares method and MinMax method (3M) are employed [<cite linkend="mst240482bib27">27</cite>, <cite linkend="mst240482bib28">28</cite>]. The criterion function of the LSM is set up as follows:
<display-eqn id="mst240482eqn04" eqnnum="4" eqnalign="center"></display-eqn>where <italic>P<sub>i</sub></italic> are the data points of the measured surface and <italic>P</italic>′<sub valign="yes"><italic>i</italic></sub> are the data points of the design surface.</p><p>The 3M can be set up as follows:
<display-eqn id="mst240482eqn05" eqnnum="5" eqnalign="center"></display-eqn>where
<display-eqn id="mst240482eqn06" eqnnum="6" eqnalign="center"></display-eqn><italic>d<sub>i</sub></italic> denotes the distance between the design and measured surfaces.</p><p>Figure <figref linkend="mst240482fig06">6</figref> shows the schematic diagram of surface matching by the CRD method. After the measurement of the surface data, the measured data for the workpiece surface and the reference surfaces for the CRD are obtained. Different reference data are first recognized by different radii of the spherical surfaces. Then the measured reference data 1 are matched to the designed ones. Meanwhile, the workpiece surface is matched to the designed one. The other reference data can be handled similarly together with the matching of the workpiece surface. These matching steps can be carried out rapidly (e.g. a few seconds by the developed software). This is called fast matching. After the fast matching process, optimization matching processes based on the least squares method and MinMax method are conducted to eliminate the errors of measured reference data so as to achieve the optimum matching results.
<figure id="mst240482fig06"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig06.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig06.jpg"></graphic-file></graphic><caption id="mst240482fc06" label="Figure 6"><p indent="no">Schematic diagram of freeform surface matching by the CRDM.</p></caption></figure></p></sec-level2><sec-level2 id="mst240482s2-4" label="2.4"><heading>Characterization of form error</heading><p indent="no">After the matching of the design and measured surfaces, the form error of the optical freeform surface can be evaluated by comparing the measured data with the design data. The deviation from the measured data to the corresponding design data is determined and the surface parameters [<cite linkend="mst240482bib29">29</cite>] for form error characterization are given in table <tabref linkend="mst240482tab01">1</tabref>.
<table id="mst240482tab01" frame="topbot"><caption id="mst240482tc01" label="Table 1"><p indent="no">Parameters for form characterization of freeform surfaces.</p></caption><tgroup cols="3"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="left"></colspec><colspec colnum="3" colname="col3" align="left"></colspec><thead><row><entry>Parameter</entry><entry>Definition</entry><entry>Explanatory notes</entry></row></thead><tbody><row><entry><italic>S</italic><sub>a</sub></entry><entry><inline-eqn></inline-eqn></entry><entry>Arithmetic mean of the deviations</entry></row><row><entry><italic>S</italic><sub>q</sub></entry><entry><inline-eqn></inline-eqn></entry><entry>Quadratic mean of the deviations</entry></row><row><entry><italic>S</italic><sub>p</sub></entry><entry><italic>S</italic><sub>p</sub> = max(<italic>d<sub>i</sub></italic>)</entry><entry>Highest peak of the measured</entry></row><row><entry></entry><entry></entry><entry>surface</entry></row><row><entry><italic>S</italic><sub>v</sub></entry><entry><italic>S</italic><sub>v</sub> = min(<italic>d<sub>i</sub></italic>)</entry><entry>Deepest valley of the surface</entry></row><row><entry><italic>S</italic><sub>t</sub></entry><entry><italic>S</italic><sub>t</sub> = <italic>S</italic><sub>p</sub> − <italic>S</italic><sub>v</sub></entry><entry>Total height between the highest</entry></row><row><entry></entry><entry></entry><entry>peak and the deepest valley</entry></row></tbody></tgroup></table></p><p>The overall work flow of the evaluation of the optical freeform surface is shown in figure <figref linkend="mst240482fig07">7</figref>. The work flow mainly consists of two parts: the measurement part and the form evaluation part. In the measurement process, the reference surface data are measured in the same coordinate system as for the machined workpiece surface. The workpiece is fixed during the measurement process. This ensures a definite relative position between the workpiece surface and the reference surfaces of the CRD. After the measured data are obtained, surface matching is conducted with the aid of CRD. The initial matching of the reference data generates the transfer matrix while the measured workpiece surface is matched to the designed one by determining the transform matrix. Due to the uncertainty of the reference data which contain machining errors, there is a need for optimum matching by MinMax or minimum zone criteria and least square criteria. After the completion of matching, the form error of the machined surface can be determined.
<figure id="mst240482fig07" width="page"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig07.eps" width="26pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig07.jpg"></graphic-file></graphic><caption id="mst240482fc07" label="Figure 7"><p indent="no">Flow chart of the form evaluation by CRDM.</p></caption></figure></p></sec-level2></sec-level1><sec-level1 id="mst240482s3" label="3"><heading>Experimental verification</heading><p indent="no">For the verification of the CRDM, a freeform surface measurement system is built which is composed of a data acquisition system and a CRDM software package. In the present study, a Form Talysurf PGI1240 surface measurement system is used as the data acquisition system for acquiring the surface data from the measured freeform surface. The measured data are fed and analysed by the CRDM software package which is developed by MATLAB software. Figure <figref linkend="mst240482fig08">8</figref> shows a snapshot of the operations of the CRDM software. The coupled reference data and the optical freeform surfaces were raster milled by an ultra-precision freeform machining system (Precitech Freeform 700G). With the use of the CRDM, the surface alignment can be carried out for the sampled area within a minute.
<figure id="mst240482fig08" width="page"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig08.eps" width="26pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig08.jpg"></graphic-file></graphic><caption id="mst240482fc08" label="Figure 8"><p indent="no">Snapshot of the operations of CRDM software.</p></caption></figure></p><p>In the present study, the performance of the CRDM is experimentally verified through a series of simulation and measurement experiments which have been conducted in different optical freeform surfaces. Basically, the experimental verification is composed of two parts, i.e. Part I and Part II. Part I includes computer simulation experiments which were conducted to evaluate the theoretical accuracy of the CRDM. A flat optical freeform surface was designed as the design surface model while the measured surface was made by the introduction of known error (i.e. transition or rotation) in the design surface. The results of measurement were compared with those obtained by the traditional LSM. In Part II, actual measurement was undertaken to verify the accuracy of the actual measurement of CRDM when a real optical freeform surface was measured.</p><sec-level2 id="mst240482s3-1" label="3.1"><heading>Part I: simulation experiment</heading><p indent="no">A flat optical freeform surface is proposed as the design model which can be expressed as
<display-eqn id="mst240482eqn07" eqnnum="7" eqnalign="center"></display-eqn>With the dimensions −10 ⩽ <italic>x</italic> ⩽ 10 and −8 ⩽ <italic>y</italic> ⩽ 8, the sampled surface from the design model is defined as 0 ⩽ <italic>x</italic> ⩽ 8 and 0 ⩽ <italic>y</italic> ⩽ 6. Figure <figref linkend="mst240482fig09">9</figref> shows the sampled surface for the design surface model used in the simulation experiment.
<figure id="mst240482fig09"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig09.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig09.jpg"></graphic-file></graphic><caption id="mst240482fc09" label="Figure 9"><p indent="no">The designed surface model and the sample surface.</p></caption></figure></p><p>The sampled surface was transformed with a known error Err(1, 2, 3, 0.1, 0.2, 0.3) which served as the measured surface as shown in figure <figref linkend="mst240482fig10">10</figref>. Hence, the measured surface was matched to the normal model by the least squares method (LSM) and the proposed CRDM. Since there are only systematic errors (only with different position and attitude) between the measured surface and the designed surface, the ideal matching error should be zero. The matching results by the LSM are shown in figure <figref linkend="mst240482fig11">11</figref> while figure <figref linkend="mst240482fig12">12</figref> depicts the matching error of the 3D topography.
<figure id="mst240482fig10"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig10.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig10.jpg"></graphic-file></graphic><caption id="mst240482fc10" label="Figure 10"><p indent="no">Designed surface and measured surface.</p></caption></figure><figure id="mst240482fig11"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig11.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig11.jpg"></graphic-file></graphic><caption id="mst240482fc11" label="Figure 11"><p indent="no">Matching results by LSM.</p></caption></figure><figure id="mst240482fig12"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig12.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig12.jpg"></graphic-file></graphic><caption id="mst240482fc12" label="Figure 12"><p indent="no">Matching errors topography by LSM.</p></caption></figure></p><p>As shown in figure <figref linkend="mst240482fig11">11</figref>, it is interesting to note that the measured surface is not properly aligned with the original position in the design model. This is due to the fact that the optimization of the LSM is very sensitive to the initial value and tends to fall in the local optimum value. This results in the uncertainty of the LSM. For some flat optical freeform surfaces with very few strong geometrical features, or in the case when the measured surface is only part of the design surface, there are uncertainties to characterize the surface quality using the matching approach by the LSM.</p><p>With the aid of the coupled reference data, the surface matching can be performed in a more robust way. Figure <figref linkend="mst240482fig13">13</figref> shows the designed surface coupled with PPP type of reference data (three spherical surfaces with different radii). Figure <figref linkend="mst240482fig14">14</figref> shows the measured surface and the designed surface coupled with three reference data. By employing the proposed CRDM, the measured surface is matched to the designed model as shown in figure <figref linkend="mst240482fig15">15</figref>. Figure <figref linkend="mst240482fig16">16</figref> shows the 3D topography of the matching errors. The matching results are further evaluated by the error surface parameters. The capability of CDRM is realized by comparing the results between CDRM and LSM. Table <tabref linkend="mst240482tab02">2</tabref> summarizes the surface parameters determined by CDRM and LSM.
<figure id="mst240482fig13"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig13.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig13.jpg"></graphic-file></graphic><caption id="mst240482fc13" label="Figure 13"><p indent="no">Designed surface coupled with reference data.</p></caption></figure><figure id="mst240482fig14"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig14.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig14.jpg"></graphic-file></graphic><caption id="mst240482fc14" label="Figure 14"><p indent="no">Measured surface and designed surface model with coupled reference points.</p></caption></figure><figure id="mst240482fig15"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig15.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig15.jpg"></graphic-file></graphic><caption id="mst240482fc15" label="Figure 15"><p indent="no">Surface matching results by CRDM.</p></caption></figure><figure id="mst240482fig16"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig16.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig16.jpg"></graphic-file></graphic><caption id="mst240482fc16" label="Figure 16"><p indent="no">Matching error topography by the CRDM.</p></caption></figure><table id="mst240482tab02" frame="topbot"><caption id="mst240482tc02" label="Table 2"><p indent="no">Error parameters by LSM and CRDM.</p></caption><tgroup cols="6"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="left"></colspec><colspec colnum="3" colname="col3" align="left"></colspec><colspec colnum="4" colname="col4" align="left"></colspec><colspec colnum="5" colname="col5" align="left"></colspec><colspec colnum="6" colname="col6" align="left"></colspec><thead><row><entry>Method</entry><entry><italic>S</italic><sub>p</sub> (nm)</entry><entry><italic>S</italic><sub>v</sub> (nm)</entry><entry><italic>S</italic><sub>t</sub> (nm)</entry><entry><italic>S</italic><sub>a</sub> (nm)</entry><entry><italic>S</italic><sub>q</sub> (nm)</entry></row></thead><tbody><row><entry>LSM</entry><entry>0.448 07</entry><entry>−0.679 28</entry><entry>1.1273</entry><entry>0.333 03</entry><entry>0.377 98</entry></row><row><entry>CRDM</entry><entry>0.003 1362</entry><entry>−0.003 1362</entry><entry>0.006 272</entry><entry>0.001 8302</entry><entry>0.002 057</entry></row></tbody></tgroup></table></p><p>Comparing the matching results between the CRDM and LSM, it is interesting to note that the CRDM matches the measured surface to the most perfect position of the design surface model. In other words, the CRDM performs in a more robust way in which the systematic matching error due to the local optimum value in LSM is significantly reduced.</p><p>To further validate the robustness of the CRDM, a series of simulation experiments has been undertaken to obtain the matching error budgets. Table <tabref linkend="mst240482tab03">3</tabref> shows the CRDM error budgets. The simulation experiments are featured so that the sample surface has significant rotation deviations from the designed model, and the sample surface is only a part of the designed model. Under this environment, IFCM [<cite linkend="mst240482bib27">27</cite>] would fall short when surface matching is conducted. With further incorporation of the six degree deviations in the three linear directions, <italic>t<sub>x</sub></italic>, <italic>t<sub>y</sub></italic>, <italic>t<sub>z</sub></italic>, and the three rotational directions, α, β, γ, it is interesting to note that the proposed CRDM maintains the robust high precision of matching ability while this cannot be maintained by IFCM. In other words, CRDM's performance appears to be superior to IFCM under the situations of large deviations between the measured surface and the designed model.
<table id="mst240482tab03" frame="topbot"><caption id="mst240482tc03" label="Table 3"><p>Matching error budgets by CRDM (compared with IFCM [<cite linkend="mst240482bib27">27</cite>]).</p></caption><tgroup cols="11"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="left"></colspec><colspec colnum="3" colname="col3" align="left"></colspec><colspec colnum="4" colname="col4" align="left"></colspec><colspec colnum="5" colname="col5" align="left"></colspec><colspec colnum="6" colname="col6" align="left"></colspec><colspec colnum="7" colname="col7" align="left"></colspec><colspec colnum="8" colname="col8" align="left"></colspec><colspec colnum="9" colname="col9" align="left"></colspec><colspec colnum="10" colname="col10" align="left"></colspec><colspec colnum="11" colname="col11" align="left"></colspec><spanspec namest="col8" nameend="col9" spanname="8to9" align="center"></spanspec><spanspec namest="col10" nameend="col11" spanname="10to11" align="center"></spanspec><thead><row><entry></entry><entry></entry><entry></entry><entry></entry><entry></entry><entry></entry><entry></entry><entry spanname="8to9">CRDM</entry><entry spanname="10to11">IFCM</entry></row><row><entry></entry><entry></entry><entry></entry><entry></entry><entry></entry><entry></entry><entry></entry><entry spanname="8to9"></entry><entry spanname="10to11"></entry></row><row><entry>Deviation</entry><entry><italic>t<sub>x</sub></italic> (mm)</entry><entry><italic>t<sub>y</sub></italic> (mm)</entry><entry><italic>t<sub>z</sub></italic> (mm)</entry><entry>α (rad)</entry><entry>β (rad)</entry><entry>γ (rad)</entry><entry><italic>S</italic><sub>t</sub> (nm)</entry><entry><italic>S</italic><sub>q</sub> (nm)</entry><entry><italic>S</italic><sub>t</sub> (nm)</entry><entry><italic>S</italic><sub>q</sub> (nm)</entry></row></thead><tbody><row><entry>Shift (<italic>S</italic>)</entry><entry>1</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>4.227 × 10<sup>−3</sup></entry><entry>1.295 × 10<sup>−3</sup></entry><entry>3.588 × 10<sup>−2</sup></entry><entry>1.489 × 10<sup>−2</sup></entry></row><row><entry></entry><entry>0</entry><entry>1</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>1.061 × 10<sup>−3</sup></entry><entry>3.268 × 10<sup>−4</sup></entry><entry>1.794 × 10<sup>2</sup></entry><entry>4.784 × 10<sup>1</sup></entry></row><row><entry></entry><entry>0</entry><entry>0</entry><entry>1</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>1.482 × 10<sup>−3</sup></entry><entry>4.679 × 10<sup>−4</sup></entry><entry>5.468 × 10<sup>−4</sup></entry><entry>1.515 × 10<sup>−4</sup></entry></row><row><entry></entry><entry>1</entry><entry>1</entry><entry>1</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>4.107 × 10<sup>−3</sup></entry><entry>1.359 × 10<sup>−3</sup></entry><entry>4.004 × 10<sup>1</sup></entry><entry>1.041 × 10<sup>1</sup></entry></row><row><entry>Rotation (<italic>R</italic>)</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>1</entry><entry>0</entry><entry>0</entry><entry>1.901 × 10<sup>−3</sup></entry><entry>6.155 × 10<sup>−4</sup></entry><entry>1.034 × 10<sup>3</sup></entry><entry>3.505 × 10<sup>2</sup></entry></row><row><entry></entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>1</entry><entry>0</entry><entry>5.505 × 10<sup>−3</sup></entry><entry>1.857 × 10<sup>−3</sup></entry><entry>3.647 × 10<sup>1</sup></entry><entry>1.199 × 10<sup>1</sup></entry></row><row><entry></entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>1</entry><entry>4.981 × 10<sup>−3</sup></entry><entry>1.517 × 10<sup>−3</sup></entry><entry>1.463 × 10<sup>−1</sup></entry><entry>3.599 × 10<sup>−2</sup></entry></row><row><entry></entry><entry>0</entry><entry>0</entry><entry>0</entry><entry>1</entry><entry>1</entry><entry>1</entry><entry>4.117 × 10<sup>−3</sup></entry><entry>9.758 × 10<sup>−4</sup></entry><entry>1.022 × 10<sup>4</sup></entry><entry>3.116 × 10<sup>3</sup></entry></row><row><entry><italic>S</italic> + <italic>R</italic></entry><entry>1</entry><entry>1</entry><entry>1</entry><entry>1</entry><entry>1</entry><entry>1</entry><entry>5.023 × 10<sup>−4</sup></entry><entry>1.534 × 10<sup>−4</sup></entry><entry>3.807 × 10<sup>4</sup></entry><entry>9.403 × 10<sup>3</sup></entry></row></tbody></tgroup></table></p></sec-level2><sec-level2 id="mst240482s3-2" label="3.2"><heading>Part II: actual measurement of optical freeform surfaces</heading><p indent="no">In the present study, an F-theta surface for a laser scanner lens is designed as follows:
<display-eqn id="mst240482eqn08" eqnnum="8" eqnalign="center"></display-eqn>With the use of PNP type CRD, two spherical surfaces and one planar surface are added as follows.</p><p indent="no">Spherical surface 1:
<display-eqn id="mst240482eqn09" lines="multiline" eqnnum="9" eqnalign="left"></display-eqn>Spherical surface 2:
<display-eqn id="mst240482eqn10" lines="multiline" eqnnum="10" eqnalign="left"></display-eqn>Planar surface:
<display-eqn id="mst240482eqn11" eqnnum="11" eqnalign="center"></display-eqn>The 3D modelling design of the workpiece is shown in figure <figref linkend="mst240482fig17">17</figref> while the workpiece was machined by an ultra-precision freeform machining system (Precitech Freeform 705G from USA) as shown in figure <figref linkend="mst240482fig18">18</figref>. Table <tabref linkend="mst240482tab04">4</tabref> shows the machining conditions for the workpiece. The machined surface was measured by a Talysurf PGI 1240 freeform measurement system (see figure <figref linkend="mst240482fig19">19</figref>) and was analysed by the CRDM software package.
<figure id="mst240482fig17"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig17.eps" width="10pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig17.jpg"></graphic-file></graphic><caption id="mst240482fc17" label="Figure 17"><p indent="no">3D modelling design.</p></caption></figure><figure id="mst240482fig18"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig18.eps" width="12pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig18.jpg"></graphic-file></graphic><caption id="mst240482fc18" label="Figure 18"><p indent="no">F-theta workpiece with PNP type of CRD.</p></caption></figure><figure id="mst240482fig19"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig19.eps" width="18pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig19.jpg"></graphic-file></graphic><caption id="mst240482fc19" label="Figure 19"><p indent="no">Measurement of surface data using TalySurf PGI 1240 freeform measurement systems.</p></caption></figure></p><p>Figure <figref linkend="mst240482fig20">20</figref> shows the 3D measurement data (3D coordinate data) while the design surface was matched with the measured surface using the CRDM as shown in figure <figref linkend="mst240482fig21">21</figref>. Hence, the F-theta lens surface was characterized and the form error of 3D topography is shown in figure <figref linkend="mst240482fig22">22</figref>. Table <tabref linkend="mst240482tab05">5</tabref> shows the surface parameters of the form errors of the measurement. It is found that the CRDM can measure the F-theta surface with a form error in the submicrometre range.
<figure id="mst240482fig20"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig20.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig20.jpg"></graphic-file></graphic><caption id="mst240482fc20" label="Figure 20"><p indent="no">Measured surface and designed surface before matching.</p></caption></figure><figure id="mst240482fig21"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig21.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig21.jpg"></graphic-file></graphic><caption id="mst240482fc21" label="Figure 21"><p indent="no">Measured surface and designed surface after matching.</p></caption></figure><figure id="mst240482fig22"><graphic><graphic-file version="print" format="EPS" filename="images/mst240482fig22.eps" width="20.5pc"></graphic-file><graphic-file version="ej" format="JPEG" filename="images/mst240482fig22.jpg"></graphic-file></graphic><caption id="mst240482fc22" label="Figure 22"><p indent="no">F-theta workpiece form error 3D topography.</p></caption></figure><table id="mst240482tab04" frame="topbot"><caption id="mst240482tc04" label="Table 4"><p indent="no">Machining parameters for raster milling of the optical freeform surface.</p></caption><tgroup cols="5"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="left"></colspec><colspec colnum="3" colname="col3" align="left"></colspec><colspec colnum="4" colname="col4" align="left"></colspec><colspec colnum="5" colname="col5" align="left"></colspec><thead><row><entry>Feed rate, <italic>F</italic> (mm min<sup>−1</sup>)</entry><entry>Spindle speed, <italic>S</italic> (rpm)</entry><entry>Step interval, <italic>Y</italic><sub>s</sub> (mm)</entry><entry>Swing distance, <italic>R</italic><sub>s</sub> (mm)</entry><entry>Tool nose radius, <italic>r</italic> (mm)</entry></row></thead><tbody><row><entry>8</entry><entry>4000</entry><entry>0.05</entry><entry>26.405</entry><entry>2.51</entry></row></tbody></tgroup></table><table id="mst240482tab05" frame="topbot"><caption id="mst240482tc05" label="Table 5"><p indent="no">Form error parameters of an F-theta workpiece.</p></caption><tgroup cols="6"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="left"></colspec><colspec colnum="3" colname="col3" align="left"></colspec><colspec colnum="4" colname="col4" align="left"></colspec><colspec colnum="5" colname="col5" align="left"></colspec><colspec colnum="6" colname="col6" align="left"></colspec><thead><row><entry>Parameters</entry><entry><italic>S</italic><sub>p</sub> (µm)</entry><entry><italic>S</italic><sub>v</sub> (µm)</entry><entry><italic>S</italic><sub>t</sub> (µm)</entry><entry><italic>S</italic><sub>a</sub> (µm)</entry><entry><italic>S</italic><sub>q</sub> (µm)</entry></row></thead><tbody><row><entry>Values</entry><entry>3.0183</entry><entry>−2.6414</entry><entry>5.6596</entry><entry>0.388 68</entry><entry>0.637 89</entry></row></tbody></tgroup></table></p></sec-level2></sec-level1><sec-level1 id="mst240482s4" label="4"><heading>Conclusion</heading><p indent="no">Traditional freeform surface measurement method such as the least squares method is inadequate for precision measurement of the form error of flat optical freeform surfaces which possess non-rotational symmetry and submicrometre form accuracy. This is due to the fact that a flat optical freeform surface lacks strong geometrical features for surface alignment. In this paper, a novel and practical method, named the coupled reference data method is proposed to address the challenge of the precision alignment of the designed and measured flat optical freeform surfaces. The CRDM employs coupled reference data, surface transition by CRD and hence an optimum matching of the designed and measured surfaces by the least squares method and minimum zone method. The capability of the CRDM has been experimentally verified through a series of simulation and actual measurement experiments. The results indicate that the accuracy and robustness of the CRDM are significantly improved as compared to a traditional freeform surface measurement method such as LSM. The CRDM is also useful and effective for reducing the uncertainties in measurement of the freeform surface in which the measured surface is only a part of the designed surface.</p></sec-level1><acknowledgment><heading>Acknowledgment</heading><p indent="no">The authors would like to express their sincere thanks to the Innovation and Technology Commission of the Government of the Hong Kong Special Administrative Region of the People's Republic of China for the financial support of the research work (project no ITF/073/04).</p></acknowledgment></body><back><references><heading>References</heading><reference-list type="numeric"><journal-ref id="mst240482bib01"><authors><au><second-name>Ruckman</second-name><first-names>J</first-names></au><au><second-name>Pollicove</second-name><first-names>H</first-names></au><au><second-name>Golini</second-name><first-names>D</first-names></au></authors><year>1999</year><art-title>Advanced manufacturing generates conformal optics</art-title><jnl-title>Laser Focus World</jnl-title><volume>35</volume><issno>July</issno><pages>S13–5</pages></journal-ref><book-ref id="mst240482bib02"><authors><au><second-name>Xiong</second-name><first-names>Y L</first-names></au></authors><year>1989</year><book-title>The Mathematic Approach of Precision Measurement</book-title><publication><place>Beijing, China</place><publisher>Metrology Press</publisher></publication></book-ref><journal-ref id="mst240482bib03"><authors><au><second-name>Rentoul</second-name><first-names>A H</first-names></au><au><second-name>Medland</second-name><first-names>G</first-names></au></authors><year>1994</year><art-title>Interpretation of errors from inspection results</art-title><jnl-title>Comput. Integr. Manuf. Syst.</jnl-title><volume>7</volume><pages>173–8</pages><crossref><cr_doi>http://dx.doi.org/10.1016/0951-5240(94)90036-1</cr_doi><cr_issn type="print">09515240</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib04"><authors><au><second-name>Wang</second-name><first-names>P J</first-names></au><au><second-name>Chen</second-name><first-names>J H</first-names></au><au><second-name>Li</second-name><first-names>Z Q</first-names></au></authors><year>1997</year><art-title>A new algorithm for the profile error of a parameter surface</art-title><jnl-title>J. Huazhong Univ. Sci. Technol.</jnl-title><volume>25</volume><pages>1–4</pages></journal-ref><journal-ref id="mst240482bib05"><authors><au><second-name>Kase</second-name><first-names>K</first-names></au><au><second-name>Makinouchi</second-name><first-names>A</first-names></au><au><second-name>Nakagawa</second-name><first-names>T</first-names></au><au><second-name>Suzuki</second-name><first-names>H</first-names></au><au><second-name>Kimura</second-name><first-names>F</first-names></au></authors><year>1999</year><art-title>Shape error evaluation method of free-form surfaces</art-title><jnl-title>Comput. Aided Des.</jnl-title><volume>31</volume><pages>495–505</pages><crossref><cr_doi>http://dx.doi.org/10.1016/S0010-4485(99)00046-9</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib06"><authors><au><second-name>Qiu</second-name><first-names>H</first-names></au><au><second-name>Li</second-name><first-names>Y</first-names></au><au><second-name>Cheng</second-name><first-names>K</first-names></au><au><second-name>Li</second-name><first-names>Y</first-names></au></authors><year>2000</year><art-title>A practical evaluation approach towards form deviation for two-dimensional contours based on coordinate measurement data</art-title><jnl-title>Int. J. Mach. Tools Manuf.</jnl-title><volume>40</volume><pages>259–75</pages><crossref><cr_doi>http://dx.doi.org/10.1016/S0890-6955(99)00057-7</cr_doi><cr_issn type="print">08906955</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib07"><authors><au><second-name>Yu</second-name><first-names>Y</first-names></au><au><second-name>Lu</second-name><first-names>J</first-names></au><au><second-name>Wang</second-name><first-names>X C</first-names></au></authors><year>2001</year><art-title>Modeling and analysis of the best matching free-form surface measuring</art-title><jnl-title>Mech. Sci. Technol.</jnl-title><volume>20</volume><pages>469–71</pages></journal-ref><journal-ref id="mst240482bib08"><authors><au><second-name>Yau</second-name><first-names>H</first-names></au><au><second-name>Chen</second-name><first-names>J</first-names></au></authors><year>1997</year><art-title>Reverse engineering of complex geometry using rational B-splines</art-title><jnl-title>Int. J. Adv. Manuf. Technol.</jnl-title><volume>13</volume><pages>548–55</pages><crossref><cr_doi>http://dx.doi.org/10.1007/BF01176298</cr_doi><cr_issn type="print">02683768</cr_issn><cr_issn type="electronic">14333015</cr_issn></crossref></journal-ref><misc-ref id="mst240482bib09"><authors><au><second-name>ASME B46.1</second-name></au></authors><year>1995</year><misc-text>Surface texture (surface roughness, waviness, and lay)</misc-text></misc-ref><misc-ref id="mst240482bib10"><authors><au><second-name>ISO 11562</second-name></au></authors><year>1996</year><misc-text>Geometrical Product Specification (GPS)—Surface Texture: Profile Method—Metrological Characteristics of Phase Correct Filters (Geneva: International Organization for Standardization)</misc-text></misc-ref><misc-ref id="mst240482bib11"><authors><au><second-name>ISO/DTS 16610–32</second-name></au></authors><year>2002</year><misc-text>Geometrical Product Specifications (GPS)—Filtration—Part 32: Robust profile filters: Spline filters</misc-text></misc-ref><misc-ref id="mst240482bib12"><authors><au><second-name>ISO/DTS 16610–31</second-name></au></authors><year>2002</year><misc-text>Geometrical Product Specifications (GPS)—Filtration—Part 31: Robust profile filters: Gaussian regression filter</misc-text></misc-ref><journal-ref id="mst240482bib13"><authors><au><second-name>Li</second-name><first-names>Y</first-names></au><au><second-name>Gu</second-name><first-names>P</first-names></au></authors><year>2005</year><art-title>Inspection of free-form shaped parts</art-title><jnl-title>Robot. Comput.-Integr. Manuf.</jnl-title><volume>21</volume><pages>421–30</pages><crossref><cr_doi>http://dx.doi.org/10.1016/j.rcim.2004.11.015</cr_doi><cr_issn type="print">07365845</cr_issn></crossref></journal-ref><misc-ref id="mst240482bib14"><misc-text><webref url="http://www.geomagic.com/en/products/index.php">http://www.geomagic.com/en/products/index.php</webref></misc-text></misc-ref><misc-ref id="mst240482bib15"><misc-text><webref url="http://www.rapidform.com/index.htm">http://www.rapidform.com/index.htm</webref></misc-text></misc-ref><misc-ref id="mst240482bib16"><misc-text><webref url="http://www.metris.com/">http://www.metris.com/</webref></misc-text></misc-ref><journal-ref id="mst240482bib17"><authors><au><second-name>Au</second-name><first-names>C K</first-names></au><au><second-name>Yuen</second-name><first-names>M M F</first-names></au></authors><year>1999</year><art-title>Feature-based reverse engineering of mannequin for garment design</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>31</volume><pages>751–9</pages><crossref><cr_doi>http://dx.doi.org/10.1016/S0010-4485(99)00068-8</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib18"><authors><au><second-name>Sun</second-name><first-names>W</first-names></au><au><second-name>Bradley</second-name><first-names>C</first-names></au><au><second-name>Zhang</second-name><first-names>Y F</first-names></au><au><second-name>Loh</second-name><first-names>H T</first-names></au></authors><year>2001</year><art-title>Cloud data modelling employing a unified, non-redundant triangular mesh</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>33</volume><pages>183–93</pages><crossref><cr_doi>http://dx.doi.org/10.1016/S0010-4485(00)00088-9</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib19"><authors><au><second-name>Peternell</second-name><first-names>M</first-names></au><au><second-name>Steiner</second-name><first-names>T</first-names></au></authors><year>2004</year><art-title>Reconstruction of piecewise planar objects from point clouds</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>36</volume><pages>333–42</pages><crossref><cr_doi>http://dx.doi.org/10.1016/S0010-4485(03)00102-7</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib20"><authors><au><second-name>Yang</second-name><first-names>Z Y</first-names></au><au><second-name>Cheng</second-name><first-names>Y H</first-names></au></authors><year>2005</year><art-title>A reverse engineering method based on haptic volume removing</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>37</volume><pages>45–54</pages><crossref><cr_doi>http://dx.doi.org/10.1016/j.cad.2004.03.003</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib21"><authors><au><second-name>Ke</second-name><first-names>Y L</first-names></au><au><second-name>Zhu</second-name><first-names>W D</first-names></au><au><second-name>Liu</second-name><first-names>F S</first-names></au><au><second-name>Shi</second-name><first-names>X Q</first-names></au></authors><year>2006</year><art-title>Constrained fitting for 2D profile-based reverse modeling</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>38</volume><pages>101–14</pages><crossref><cr_doi>http://dx.doi.org/10.1016/j.cad.2005.07.004</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib22"><authors><au><second-name>Ke</second-name><first-names>Y L</first-names></au><au><second-name>Fan</second-name><first-names>S Q</first-names></au><au><second-name>Zhu</second-name><first-names>W D</first-names></au><au><second-name>Li</second-name><first-names>A</first-names></au><au><second-name>Liu</second-name><first-names>F S</first-names></au><au><second-name>Shi</second-name><first-names>X Q</first-names></au></authors><year>2006</year><art-title>Feature-based reverse modeling strategies</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>38</volume><pages>485–506</pages><crossref><cr_doi>http://dx.doi.org/10.1016/j.cad.2005.12.002</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib23"><authors><au><second-name>Várady</second-name><first-names>T</first-names></au><au><second-name>Martin</second-name><first-names>R R</first-names></au><au><second-name>Cox</second-name><first-names>J</first-names></au></authors><year>1997</year><art-title>Reverse engineering of geometric models—an introduction</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>29</volume><pages>255–68</pages><crossref><cr_doi>http://dx.doi.org/10.1016/S0010-4485(96)00054-1</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib24"><authors><au><second-name>Karniel</second-name><first-names>A</first-names></au><au><second-name>Belsky</second-name><first-names>Y</first-names></au><au><second-name>Reich</second-name><first-names>Y</first-names></au></authors><year>2005</year><art-title>Decomposing the problem of constrained surface fitting in reverse engineering</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>37</volume><pages>399–417</pages><crossref><cr_doi>http://dx.doi.org/10.1016/j.cad.2004.06.015</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib25"><authors><au><second-name>Milroy</second-name><first-names>M J</first-names></au><au><second-name>Bradley</second-name><first-names>C</first-names></au><au><second-name>Vickers</second-name><first-names>G W</first-names></au><au><second-name>Weir</second-name><first-names>D J</first-names></au></authors><year>1995</year><art-title>G1 continuity of B-spline surface patches in reverse engineering</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>27</volume><pages>471–8</pages><crossref><cr_doi>http://dx.doi.org/10.1016/0010-4485(95)00020-R</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib26"><authors><au><second-name>Huang</second-name><first-names>X</first-names></au><au><second-name>Gu</second-name><first-names>P</first-names></au><au><second-name>Zernicke</second-name><first-names>R</first-names></au></authors><year>1996</year><art-title>Localization and comparison of two free-form surfaces</art-title><jnl-title>Comput.-Aided Des.</jnl-title><volume>26</volume><pages>1017–22</pages><crossref><cr_doi>http://dx.doi.org/10.1016/0010-4485(96)00011-5</cr_doi><cr_issn type="print">00104485</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib27"><authors><au><second-name>Cheung</second-name><first-names>C F</first-names></au><au><second-name>Li</second-name><first-names>H F</first-names></au><au><second-name>Lee</second-name><first-names>W B</first-names></au><au><second-name>To</second-name><first-names>S</first-names></au><au><second-name>Kong</second-name><first-names>L B</first-names></au></authors><year>2007</year><art-title>An integrated form characterization method for measuring ultra-precision freeform surfaces</art-title><jnl-title>Int. J. Mach. Tools Manuf.</jnl-title><volume>47</volume><pages>81–91</pages><crossref><cr_doi>http://dx.doi.org/10.1016/j.ijmachtools.2006.02.013</cr_doi><cr_issn type="print">08906955</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib28"><authors><au><second-name>Cheung</second-name><first-names>C F</first-names></au><au><second-name>Li</second-name><first-names>H F</first-names></au><au><second-name>Kong</second-name><first-names>L B</first-names></au><au><second-name>Lee</second-name><first-names>W B</first-names></au><au><second-name>To</second-name><first-names>S</first-names></au></authors><year>2006</year><art-title>Measuring ultra-precision freeform surfaces using a robust form characterization method</art-title><jnl-title>Meas. Sci. Technol.</jnl-title><volume>17</volume><pages>488–94</pages><crossref><cr_doi>http://dx.doi.org/10.1088/0957-0233/17/3/S05</cr_doi><cr_issn type="print">09570233</cr_issn><cr_issn type="electronic">13616501</cr_issn></crossref></journal-ref><journal-ref id="mst240482bib29"><authors><au><second-name>Kong</second-name><first-names>L B</first-names></au><au><second-name>Cheung</second-name><first-names>C F</first-names></au><au><second-name>Gao</second-name><first-names>D</first-names></au><au><second-name>Lee</second-name><first-names>W B</first-names></au><au><second-name>To</second-name><first-names>S</first-names></au></authors><year>2006</year><art-title>Theoretical and experimental evaluation of surface quality for ultra-precision freeform surfaces</art-title><jnl-title>Proc. Inst. Mech. Eng.</jnl-title><part>B</part><volume>220</volume><pages>1439–48</pages></journal-ref></reference-list></references></back></article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>Measuring optical freeform surfaces using a coupled reference data method</title></titleInfo><titleInfo type="abbreviated"><title>Measuring optical freeform surfaces using a coupled reference data method</title></titleInfo><titleInfo type="alternative"><title>Measuring optical freeform surfaces using a coupled reference data method</title></titleInfo><name type="personal"><namePart type="given">L B</namePart><namePart type="family">Kong</namePart><affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation><affiliation>E-mail:Benny.cheung@inet.polyu.edu.hk</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">C F</namePart><namePart type="family">Cheung</namePart><affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">S</namePart><namePart type="family">To</namePart><affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">W B</namePart><namePart type="family">Lee</namePart><affiliation>Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">K W</namePart><namePart type="family">Cheng</namePart><affiliation>Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="paper"></genre><originInfo><publisher>IOP Publishing</publisher><dateIssued encoding="w3cdtf">2007</dateIssued><copyrightDate encoding="w3cdtf">2007</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType><note type="production">Printed in the UK</note></physicalDescription><abstract>Flat optical freeform surfaces usually possess non-rotational symmetry with a small curvature and lack of strong features for surface alignment. Due to the lack of strong features and small curvature, it is difficult to align the design and measured surfaces for characterizing the surface quality of flat optical freeform surfaces with sub-micrometre form accuracy. The traditional least squares method (LSM) generally produces large errors as there is a lack of strong features as reference for the alignment of the design and measured surfaces. This paper proposes a novel and practical method named the coupled reference data method (CRDM) to evaluate flat optical freeform surfaces with high efficiency and precision in the nanometre scale. The method couples reference data to the workpiece of the freeform surface designed model and the concerning reference features are machined together with the workpiece. By aligning the reference data, the proposed CRDM carries out fast surface matching. This makes good preparation for the next matching optimization which is conducted by the least-squares and minimax zone method. After the precise surface matching, the flat optical freeform surface can be evaluated by 3D form error topography and parameters. As compared with a traditional freeform measurement method such as LSM, it is interesting to note that the accuracy and the stability of the measurement can be significantly enhanced by the CRDM.</abstract><subject><genre>keywords</genre><topic>optical freeform surface</topic><topic>form characterization</topic><topic>surface matching</topic><topic>least squares method</topic><topic>minimax zone method</topic><topic>reference data</topic></subject><relatedItem type="host"><titleInfo><title>Measurement Science and Technology</title></titleInfo><titleInfo type="abbreviated"><title>Meas. Sci. Technol.</title></titleInfo><genre type="journal">journal</genre><identifier type="ISSN">0957-0233</identifier><identifier type="eISSN">1361-6501</identifier><identifier type="PublisherID">MST</identifier><identifier type="CODEN">MSTCEP</identifier><identifier type="URL">stacks.iop.org/MST</identifier><part><date>2007</date><detail type="volume"><caption>vol.</caption><number>18</number></detail><detail type="issue"><caption>no.</caption><number>7</number></detail><extent unit="pages"><start>2252</start><end>2260</end><total>9</total></extent></part></relatedItem><identifier type="istex">41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B</identifier><identifier type="DOI">10.1088/0957-0233/18/7/060</identifier><identifier type="PII">S0957-0233(07)40482-9</identifier><identifier type="articleID">240482</identifier><identifier type="articleNumber">060</identifier><accessCondition type="use and reproduction" contentType="copyright">2007 IOP Publishing Ltd</accessCondition><recordInfo><recordContentSource>IOP</recordContentSource><recordOrigin>2007 IOP Publishing Ltd</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Ticri/CIDE/explor/HapticV1/Data/Istex/Corpus
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 004712 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Istex/Corpus/biblio.hfd -nk 004712 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien
|wiki= Ticri/CIDE
|area= HapticV1
|flux= Istex
|étape= Corpus
|type= RBID
|clé= ISTEX:41E153D63871CAEFECC894AE2A02BBBD0B0ACC6B
|texte= Measuring optical freeform surfaces using a coupled reference data method
}}
| This area was generated with Dilib version V0.6.23. |  |