Linearizing phase retrieval: the theory of near-focus interference

Phase retrieval from square-law detectors in common path is traditionally a nonlinear, nonconvex inverse problem requiring complex iterative algorithms. Near-focus interference (NFI) uniquely enables direct linearization of self-referential phase retrieval. NFI exploits the interference between the longitudinal evolution of a coherent wavefield over a short free-space propagation-just a few Rayleigh distances-and the unpropagated self. NFI equations assume nothing about the complex fields except that they are bandlimited and fully represented on a sampling grid. Rigorous theory shows the reconstructed complex field is unique and depends continuously on the measured interference term and a single intensity, ensuring stability. Numerically, NFI is applied to bright-field and pure dark-field microscopy via projection onto a discrete hexagonal basis and a coordinate-free Krylov subspace least-squares analysis (LSA). In bright field, the phase locks to the known background field. In dark field, a data-driven bootstrap provides an initial guess for the LSA, resolving the global phase factor ambiguity common to self-referential measurements. Numerical simulations with 8-bit noise show recovery of 900 independent scatterers in 5 iterations. Experimental bright-field results are shown for samples much larger than the defocused diffraction spot. Beyond these guarantees of uniqueness and continuity, the principal practical advantage of NFI over state-of-the-art quantitative phase imaging is self-calibration: the propagation operator and the complex field are recovered directly from the noisy measurement, without external calibration standards, and the same reconstruction is obtained even after realignment of the instrument. These properties provide a foundation for standardized quantitative phase imaging across instruments and laboratories, facilitating adoption in medical diagnostics.