Distortion properties and correction abilities of axis-symmetric aspherical surfaces of arbitrary shape

Abstract

The real aberration properties (all orders) and correction abilities of optical aspherical surfaces of arbitrary shape are insufficiently investigated due to the lack of exact aberration theory. Here we derive and investigate an exact analytical distortion function for axis-symmetric aspherical surfaces of arbitrary shape that describes correctly image distortion for the whole object space without any approximations. We prove that in the object and image spaces of every aspherical refracting or reflecting surface at fixed stop position there is in general one orthoscopic object surface and one conjugate image surface. In addition we offer formulae for determination of object and image orthoscopic surface coordinates in every refracting or reflecting axis-symmetric optical surface with known coordinates and continuous first derivative on the whole profile. To verify the distortion correction we use the commercial program OSLO. The differences between our results and these obtained by OSLO are less than 1.10^-8 mm. Using these exact formulae we investigate the real shape and location of the orthoscopic object and image surfaces as a function of stop position in different reflecting and refracting aspherics. The results can be used by optical designers in the process of synthesis of orthoscopic optical systems.