Three-dimensional particle localization techniques based on phase modulation and digital vortex imaging

Three-dimensional localization methods are crucial for accurately resolving particle positions and trajectories in complex systems, enabling deeper insights into dynamic processes in diverse scientific and engineering fields. Owing to its compact configuration and high space-bandwidth product, in-line digital holography has been widely used for three-dimensional particle localization and tracking. However, the accuracy of axial localization is often compromised by intrinsic twin image artifacts, leading to ambiguities in the reconstruction process. Additionally, multi-plane iterative reconstruction methods increase the complexity and computational efficiency of imaging. To address these two challenges, we propose a defocus-encoded coherent framework that simultaneously encodes defocus direction and distance. During holographic recording, a vortex phase with topological charge l=1 is introduced in the frequency domain, encoding the defocus direction into spiral interference patterns in the hologram. During numerical reconstruction, a digital vortex interferometry encoding is then used to quantitatively map the defocus distance to the rotation angle of the vortex pattern, enabling precise retrieval of the axial offset. Experiments and simulations demonstrate that the strategy enables volumetric imaging within a 10 mm³ field with precise axial resolution. The single-shot and non-iterative characteristics of this method enables robust and efficient tracking of dynamic particles.