Multiscale axial displacement measurement based on radially segmented double-helix PSF and multi-feature hierarchical demodulation

Abstract

The double-helix point spread function (DH-PSF) inherently suffers from a trade-off between axial range and sensitivity, which limits its use in optical alignment and focal-plane tracking tasks that require both wide-range acquisition and high-precision localization. To overcome this limitation, we propose a fast axial displacement sensing method that integrates a radially segmented double-helix point spread function (RS-DHPSF) with multi-feature hierarchical demodulation. By tailoring the phase responses of the inner and outer radial zones, the RS-DHPSF simultaneously provides a wide axial capture range and high sensitivity in the central region. We further establish a non-ideal optical model incorporating phase quantization, physical noise, and residual aberrations to evaluate axial localization performance. To efficiently decode the complex RS-DHPSF patterns evolving with axial displacement, we further develop a hierarchical region-of-interest chirp z-transform normalized cross-correlation (ROI-CZT-NCC) algorithm constrained by multi-feature candidate intervals. Simulation and experimental results show that the proposed method reduces the per-frame localization time by approximately 90% while achieving a continuous axial measurement range of approximately 7 mm, with a mean absolute error (MAE) of 0.771 μm over the full range and a reduced MAE of 0.419 μm within the central ±250 μm region.