Novel analysis of alignment error on spherical Fizeau interferometer and uncertainty evaluation of sphericity calibration system based on random ball test

The National Metrology Institute of Japan (NMIJ) has developed a high-precision sphericity calibration system utilizing a Fizeau interferometer and conducted a thorough evaluation of the system's measurement uncertainty. Generally, there are two principal sources of uncertainty in the measurement outcomes when utilizing a spherical Fizeau interferometer. First, the accuracy of the system is significantly influenced by the prior knowledge of the reference sphere surface's absolute profile. To address this, the study established a straightforward yet effective calibration system for the reference sphere surface using a random ball test method. Second, the system's precision is affected by misalignment aberration, which occurs when there is any lateral or longitudinal displacement of the test sphere from the confocal position relative to the reference sphere. This misalignment can introduce both high-order shape errors (misalignment aberrations) and low-order shape errors (alignment errors). Through analytical consideration of an observation coordinate system on the camera plane, this study delves into misalignment aberrations, suggesting that the impact of misalignment should be determined experimentally for each reference sphere unit due to potential imperfections that may be revealed by misalignment. Furthermore, the study proposes that uncertainties related to misalignment aberrations are theoretically confirmed to be smaller than previous studies by conducting an in-depth uncertainty analysis of the misalignment, with a focus on the observation coordinate system on the camera plane. Notably, the research demonstrates that a measurement uncertainty level of λ/100 is achievable, maintaining a broader tolerance for misalignment than previously reported studies. The uncertainty of calibration of the reference sphere unit's absolute profile was 4.2 nm and the uncertainty of sample measurement was determined to be 4.6 nm with misalignment tolerance of $\pm \lambda /10$. This advancement marks a stride toward improving the accuracy and reliability of sphericity measurements, offering potential for widespread application in precision engineering and metrology.