Localization of nanoscale objects with light singularities

Unprecedented atomic-scale measurement resolution has recently been demonstrated in single-shot optical localization measurements based on deep-learning analyses of diffraction patterns of topologically structured light scattered from objects. Here, we show that variations in the diffraction patterns caused by positional changes of an object depend upon the spatial derivatives of the amplitude and phase of the incident field, most strongly around phase singularities. Despite lower intensity near the singularity, an orders-of-magnitude increase in Fisher information contained in the diffraction patterns can be achieved when a nano-object is illuminated by light containing phase singularities, rather than a plane wave. Our work provides a fundamental explanation and motivation for singularity-based metrology with deeply subwavelength precision.