Mapping geometry in Fizeau transmission spheres with a small f-number

The ideal mapping geometry in a Fizeau interferometer is to map equal height increments on a flat object and equal angle increments on a spherical surface to equal heights on the detector. So the initial intent of the optical design of Fizeau transmission spheres (TSs) is to provide R-θ mapping geometry for equal angle increments. The corresponding unequal heights mapping will introduce retrace error as coma when linear carrier fringes exist. On the contrary, equal heights mapping with R-sin θ mapping geometry will avoid linear carrier fringes induced coma error. These two different mapping geometries conflict especially for the TS with a small f-number. In this paper, we will first explore the design and the performance of the f/0.75  TS according to the two different mapping geometries, and then evaluate the mapping geometry for the commercial ZYGO f/0.75  TS, and give some engineering notes for the designers, the metrologists, and the fabricators in the optical laboratory.