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.