Nonlinear Differential Equations for the Wavefront Surface at Arbitrary Hartmann-Plane Distances

In the Hartmann test, a wave aberration functionW is estimated from the information of the spot diagram drawn in an observation plane. The distance from a reference plane to the observation plane, the Hartmann-plane distance, is typically chosen as z = f , where f is the radius of a reference sphere. The functionW and the transversal aberrations fX;Y g calculated at the plane z = f are related by two well-known linear differential equations. Here, we propose two nonlinear differential equations to denote a more general relation betweenW and the transversal aberrations fU;V g calculated at any arbitrary Hartmann-plane distance z = r. We also show how to directly estimate the wavefront surface w from the information of fU;V g. The use of arbitrary r values could improve the reliability of the measurements of W, or w, when finding difficulties in adequate ray identification at z = f . © 2016 Optical Society of America.