  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-Smith 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.
