Analytic Fourier ptychotomography for volumetric refractive index imaging

Three-dimensional (3D) refractive index (RI) tomography offers label-free, quantitative volumetric imaging but faces limitations due to optical aberrations, limited resolution, and the computational complexity inherent to existing approaches. To overcome these barriers, we propose Analytic Fourier Ptychotomography (AFP), a new computational microscopy technique that analytically reconstructs aberration-free, complex-valued 3D RI distributions without iterative optimization or axial scanning. AFP incorporates a new concept called the finite sample thickness (FST) prior, and analytically solves the inverse scattering problem through three sequential steps: complex-field reconstruction via the Kramers-Kronig relation, linear aberration correction using overlapping spectra, and analytic spectrum extension into the darkfield region. Unlike iterative reconstruction methods, AFP does not require parameter tuning or computationally intensive optimizations, which are often error-prone and non-generalizable. We experimentally demonstrate that AFP significantly enhances image quality and resolution under various aberration conditions across a range of applications. AFP corrected aberrations associated with 25 Zernike modes (with a maximal phase difference of 2.3$\pi$ and maximal Zernike coefficient value of 4), extended the synthetic numerical aperture from 0.41 to 0.99, and provided a two-fold resolution enhancement in all directions. AFP's simplicity and robustness make it an attractive imaging technology for quantitative 3D analysis in biological, microbial ecological, and medical studies.