Infinitesimal optical singularity ruler for three-dimensional picometric metrology

Optical metrology with picometer-scale precision in three-dimensional space is of considerable importance in modern physics and state of the art technology, optical interference is an effective method, but techniques with rapid spatial variation have the potential to enhance measurement precision, which will be required as measurement dimensions decrease. Here, the concept of the vanishingly small optical phase singularity ruler is introduced. Inspired by the well-known plumb-line technique used to locate the centroid, an analogous singularity line technique is proposed to locate the optical singularity with a precision of ~4.5 pm (~λ/140000) in the transverse direction and ~24.2 pm (~λ/26000) in the longitudinal direction. This precisely positioned singularity can serve as a ruler to detect displacement signals with an accuracy approaching ~60 pm.