Direction-constrained phase reconstruction for nanoscale displacement measurement with a weak-feedback self-mixing grating interferometer

Laser self-mixing interferometry (SMI) and its derivative, self-mixing grating interferometry (SMGI), offer compact, self-aligned, single-channel architectures for nanoscale displacement metrology. Built on a Cr atom-lithography grating (ALG) whose period, d = (212.7779 ± 0.0021) nm ( k = 1), is directly traceable to an atomic transition frequency with picometre-level accuracy, an SMGI acquires a length scale anchored in this traceable standard. In this grating-based configuration, however, the limited diffraction efficiency keeps the SMGI under weak feedback, where the direction-dependent fringe asymmetry is subtle, making phase reconstruction near motion turning points susceptible to noise, and direction misjudgement. This paper proposes a direction-constrained phase reconstruction (DCPR) method: Hilbert analytic phase provides continuous phase increments; a direction function constructed from piecewise integration of the differential envelope's midline imposes a physical constraint on phase increments; and piecewise cubic Hermite interpolating polynomial (PCHIP) interpolation suppresses spurious increments near turning points. Validated on a Cr-ALG-based SMGI system against a Fabry–Pérot reference, the DCPR reconstruction yielded cycle-wise forward–return RMS deviations relative to the reference of 8.4 ± 1.7 nm and 12.4 ± 0.6 nm for the nominal 8- and 12.5-μm motion datasets, respectively. DCPR converts weak directional asymmetry into a cumulative physical constraint, providing a robust phase-reconstruction scheme for compact SMI/SMGI displacement sensing.