Complete Modal Representation with Discrete Zernike Polynomials - Critical Sampling in Non Redundant Grids

Zernike polynomials (ZPs) form a complete orthogonal basis on a circle of unit radius. This is useful in optics, since a great majority of lenses and optical instruments have circular shape and/or circular pupil. The ZP expansion is typically used to describe either optical surfaces or distances between surfaces, such as optical path differences (OPD), wavefront phase or wave aberration. Therefore, applications include optical computing, design and optimization of optical elements, optical testing (Navarro & Moreno-Barriuso, 1999), wavefront sensing (Noll, 1978)(Cubalchini, 1979), adaptive optics (Alda & Boreman, 1993), wavefront shaping (Love, 1997) (Vargas-Martin et al., 1998), interferometry (Kim, 1982)(Fisher et al., 1993)(van Brug, 1997)(Chen & Dong, 2002), surface metrology topography (Nam & Rubinstein, 2008), corneal topography (Schwiegerling et al., 1995)(Fazekas et al., 2009), atmospheric optics (Noll, 1977) (Roggemann, 1996), etc. This brief overview shows that the modal description provided by ZPs was highly successful in a wide variety of applications. In fact, ZPs are embedded in many technologies such as optical design software, large telescopes, ophthalmology, communications , etc. The modal representation of a function (wavefront, OPD, surface, etc.) over a circle in terms of ZPs is: