A Concept of Two-Point Propagation Field of a Single Photon: A Way to Picometer X-ray Displacement Sensing and Nanometer Resolution 3D X-ray Micro-Tomography
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Abstract
We introduce the two-point propagation field (TPPF) -- a real-valued, phase-sensitive quantity defined as the functional derivative of the single-photon detection probability with respect to an infinitesimal opaque perturbation placed between source and detection slits. The TPPF is analytically derived and shown to exhibit stable, high-frequency sinusoidal structure with periods reaching 4 to 7 nm near the detection slit for both soft and hard X-rays. This structure enables shot-noise-limited displacement sensing at $\sim200$ pm precision for 10 keV X-rays using total photon counts $2.1\times 10^{6}$ and detector photon counting as low as 120, readily achievable with routinely available synchrotron fluxes and practical nanofabricated comb/slit geometries, requiring mechanical stability only over the final 0.5 mm. Beyond displacement sensing, the TPPF physically performs a Fourier-Radon transformation of the projection data, providing a foundational pathway to deterministic, non-iterative frequency-domain tomography. Two conceptual strategies, a central blocker and off-axis multi-slit arrays, are estimated to lower the required incident photon budget by more than one order of magnitude each, yielding combined reductions of two to three orders of magnitude with near-term detector development. The TPPF concept, originally developed in a perturbative study of single-particle propagation, thus bridges fundamental quantum measurement questions with practical high-resolution X-ray metrology and imaging.