Zernike-like systems in polygons and polygonal facets
Ver/
Fecha
2015Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión aceptada / Onetsi den bertsioa
Impacto
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10.1364/AO.54.006575
Resumen
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 disk. In
[Diaz et all, 2014] 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 ...
[++]
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 disk. In
[Diaz et all, 2014] 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 piece-wise 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. [--]
Materias
Zernike polynomials,
Orthonormal systems,
Polygonal facets,
Segmented mirror telescopes
Editor
Optical Society of America
Publicado en
Applied Optics Vol. 54, Issue 21, pp. 6575-6583 (2015)
Departamento
Universidad Pública de Navarra. Departamento de Ingeniería Matemática e Informática /
Nafarroako Unibertsitate Publikoa. Matematika eta Informatika Ingeniaritza Saila
Versión del editor
Entidades Financiadoras
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.
The financial support of the State University of Navarra is also acknowledged.