000076957 001__ 76957 000076957 005__ 20210121114514.0 000076957 0247_ $$2doi$$a10.1364/AO.54.006575 000076957 0248_ $$2sideral$$a92769 000076957 037__ $$aART-2015-92769 000076957 041__ $$aeng 000076957 100__ $$0(orcid)0000-0002-3698-6719$$aFerreira, C.$$uUniversidad de Zaragoza 000076957 245__ $$aZernike-like systems in polygons and polygonal facets 000076957 260__ $$c2015 000076957 5060_ $$aAccess copy available to the general public$$fUnrestricted 000076957 5203_ $$aZernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In Opt. Lett. 32, 74 (2007)] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piecewise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. 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 a typical example of telescope optical aperture are provided. 000076957 536__ $$9info:eu-repo/grantAgreement/ES/DGA/E99$$9info:eu-repo/grantAgreement/ES/MINECO/FIS2011-22496$$9info:eu-repo/grantAgreement/ES/MINECO/FIS2014-58303$$9info:eu-repo/grantAgreement/ES/MINECO/MTM2014-52859 000076957 540__ $$9info:eu-repo/semantics/openAccess$$aAll rights reserved$$uhttp://www.europeana.eu/rights/rr-f/ 000076957 590__ $$a1.598$$b2015 000076957 591__ $$aOPTICS$$b45 / 91 = 0.495$$c2015$$dQ2$$eT2 000076957 592__ $$a0.837$$b2015 000076957 593__ $$aElectrical and Electronic Engineering$$c2015$$dQ1 000076957 593__ $$aEngineering (miscellaneous)$$c2015$$dQ1 000076957 593__ $$aAtomic and Molecular Physics, and Optics$$c2015$$dQ2 000076957 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/submittedVersion 000076957 700__ $$aLópez, J.L. 000076957 700__ $$0(orcid)0000-0002-1328-1716$$aNavarro, R. 000076957 700__ $$0(orcid)0000-0002-8021-2745$$aPérez Sinusía, E.$$uUniversidad de Zaragoza 000076957 7102_ $$12005$$2595$$aUniversidad de Zaragoza$$bDpto. Matemática Aplicada$$cÁrea Matemática Aplicada 000076957 773__ $$g54, 21 (2015), 6575-6583$$pAppl. opt. (2004)$$tApplied Optics$$x1559-128X 000076957 8564_ $$s466132$$uhttps://zaguan.unizar.es/record/76957/files/texto_completo.pdf$$yPreprint 000076957 8564_ $$s70786$$uhttps://zaguan.unizar.es/record/76957/files/texto_completo.jpg?subformat=icon$$xicon$$yPreprint 000076957 909CO $$ooai:zaguan.unizar.es:76957$$particulos$$pdriver 000076957 951__ $$a2021-01-21-10:58:47 000076957 980__ $$aARTICLE