Generalization of Zernike polynomials for regular portions of circles and ellipses

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit circle. Here, we present a generalization of this Zernike basis for a variety of important optical apertures. On the contrary to ad hoc s...

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Autores: Navarro, Rafael, López García, José Luis, Díaz, José A., Pérez Sinusía, Ester
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2014
País:España
Institución:Universidad Pública de Navarra
Repositorio:Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
OAI Identifier:oai:academica-e.unavarra.es:2454/32041
Acceso en línea:https://hdl.handle.net/2454/32041
Access Level:acceso abierto
Palabra clave:Zernike polynomials
Circular optical apertures
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spelling Generalization of Zernike polynomials for regular portions of circles and ellipsesNavarro, RafaelLópez García, José LuisDíaz, José A.Pérez Sinusía, EsterZernike polynomialsCircular optical aperturesZernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit circle. Here, we present a generalization of this Zernike basis for a variety of important optical apertures. On the contrary to ad hoc solutions, most of them based on the Gram-Schmidt orthonormalization method, here we apply the diffeomorphism (mapping that has a differentiable inverse mapping) that transforms the unit circle into an angular sector of an elliptical annulus. In this way, other apertures, such as ellipses, rings, angular sectors, etc. are also included as particular cases. This generalization, based on in-plane warping of the basis functions, provides a unique solution and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for most common, elliptical and annular apertures are provided.This research was supported by the Spanish Ministry of Economía y Competitividad and the European Union, grant FIS2011-22496, and by the Government of Aragón, research group E99.Optical Society of AmericaIngeniería Matemática e InformáticaMatematika eta Informatika Ingeniaritza2014info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2454/32041reponame:Academica-e. Repositorio Institucional de la Universidad Pública de Navarrainstname:Universidad Pública de NavarraInglésinfo:eu-repo/grantAgreement/MICINN//FIS2011-22496© 2014 Optical Society of America. Users may use, reuse, and build upon the article, or use the article for text or data mining, so long as such uses are for non-commercial purposes and appropriate attribution is maintained. All other rights are reserved.info:eu-repo/semantics/openAccessoai:academica-e.unavarra.es:2454/320412026-06-17T12:41:47Z
dc.title.none.fl_str_mv Generalization of Zernike polynomials for regular portions of circles and ellipses
title Generalization of Zernike polynomials for regular portions of circles and ellipses
spellingShingle Generalization of Zernike polynomials for regular portions of circles and ellipses
Navarro, Rafael
Zernike polynomials
Circular optical apertures
title_short Generalization of Zernike polynomials for regular portions of circles and ellipses
title_full Generalization of Zernike polynomials for regular portions of circles and ellipses
title_fullStr Generalization of Zernike polynomials for regular portions of circles and ellipses
title_full_unstemmed Generalization of Zernike polynomials for regular portions of circles and ellipses
title_sort Generalization of Zernike polynomials for regular portions of circles and ellipses
dc.creator.none.fl_str_mv Navarro, Rafael
López García, José Luis
Díaz, José A.
Pérez Sinusía, Ester
author Navarro, Rafael
author_facet Navarro, Rafael
López García, José Luis
Díaz, José A.
Pérez Sinusía, Ester
author_role author
author2 López García, José Luis
Díaz, José A.
Pérez Sinusía, Ester
author2_role author
author
author
dc.contributor.none.fl_str_mv Ingeniería Matemática e Informática
Matematika eta Informatika Ingeniaritza
dc.subject.none.fl_str_mv Zernike polynomials
Circular optical apertures
topic Zernike polynomials
Circular optical apertures
description Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit circle. Here, we present a generalization of this Zernike basis for a variety of important optical apertures. On the contrary to ad hoc solutions, most of them based on the Gram-Schmidt orthonormalization method, here we apply the diffeomorphism (mapping that has a differentiable inverse mapping) that transforms the unit circle into an angular sector of an elliptical annulus. In this way, other apertures, such as ellipses, rings, angular sectors, etc. are also included as particular cases. This generalization, based on in-plane warping of the basis functions, provides a unique solution and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for most common, elliptical and annular apertures are provided.
publishDate 2014
dc.date.none.fl_str_mv 2014
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2454/32041
url https://hdl.handle.net/2454/32041
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/MICINN//FIS2011-22496
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Optical Society of America
publisher.none.fl_str_mv Optical Society of America
dc.source.none.fl_str_mv reponame:Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
instname:Universidad Pública de Navarra
instname_str Universidad Pública de Navarra
reponame_str Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
collection Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.812429

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