Orthogonal Basis Functions over the Binocular Pupil

Abstract

AbstractSets of orthogonal basis functions over circular areas - pupils in optical applications - are known in the literature for the full circle (Zernike or Jacobi polynomials) and the annulus. Here, an orthogonal set is proposed if the area is two non-overlapping circles of equal size. The geometric master parameter is the ratio of the pupil radii over the distance between both circles. Increasingly higher order aberrations - as defined for a virtual larger pupil in which both pupils are embedded - are fed into a Gram-Schmidt orthogonalization to distill one unique set of basis functions. The key effort is to work out the overlap integrals between a full set of primitive basis functions of hyperspherical type centered at the mid-point between both pupils. Constructed from the same primitive basis, the orthogonal Karhunen-Loève modes of spatially filtered Kolmogorov phase screens are computed for this shape of mask. Matrix elements of the covariance matrix - an established intra-circle and a special inter-circle category - are worked out in wavenumber space.