(Optical Society of America, 2014) Navarro, Rafael; López García, José Luis; Díaz, José A.; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
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.
