Wavefront aberration function in terms of R. V. Shack's vector product and Zernike polynomial vectors

Previous papers have shown how, for rotationally symmetric optical imaging systems, nodes in the field dependence of the wavefront aberration function develop when a rotationally symmetric optical surface within an imaging optical system is decentered and/or tilted. In this paper, we show how Shack's vector product (SVP) can be used to express the wavefront aberration function and to define vectors in terms of the Zernike polynomials. The wavefront aberration function is then expressed in terms of the Zernike vectors. It is further shown that SVP fits within the framework of two-dimensional geometric algebra (GA). Within the GA framework, an equation for the third-order node locations for the binodal astigmatism term that emerge in the presence of tilts and decenters is then demonstrated. A computer model of a three-mirror telescope system is used to demonstrate the validity of the mathematical development.