Robust linear ruler near a longitudinal optical vortex for sub-nanometer displacement sensing of a single nanoparticle

We present a method for nanoscale displacement sensing based on robust spin–orbit conversion in tightly focused beams carrying net spin angular momentum. This mechanism naturally defines a two-dimensional linear ruler in the vicinity of the zeros of the complex longitudinal field component E z , which are surrounded by a first-order topological phase vortex. In this region, the real and imaginary parts of E z vary linearly and orthogonally with lateral displacements along the x and y directions without any beam shaping or resonant nanostructures. Experiments using a single gold nanoparticle and polarization-resolved back-focal-plane imaging confirm displacement sensing with a resolution below 1 nm. We further show that this linear ruler remains functional under strong optical aberrations. Our approach, therefore, offers a universal, resonance-free, and experimentally simple route to displacement metrology in complex and noisy environments