Analytic Fourier ptychotomography (AFP) for volumetric refractive index imaging

Zhenyu Dong1,*, Haowen Zhou1,*, Ruizhi Cao1,*,†,
Oumeng Zhang1, Shi Zhao1, Panlang Lyu1, Reinaldo Alcalde2, Changhuei Yang1
1Department of Electrical Engineering, California Institute of Technology
2Division of Biology and Biological Engineering, California Institute of Technology
*Equal contribution   †Correspondence
Under review, 2025
arXiv Code Data

System overview

System_overview
Fig. 1. Overview of the AFP concept.

Abstract

Three-dimensional (3D) refractive index (RI) tomography offers label-free, quantitative volumetric imaging but faces limitations due to optical aberrations, limited resolution, and 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 named finite sample thickness (FST) prior, thereby simplifying the inverse scattering problem into solving linear equations. AFP consists of 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 and computationally intensive optimizations – which are slow to converge, error-prone, parameter-sensitive and non-generalizable across samples and systems.

We experimentally demonstrated that AFP significantly enhances image quality and resolution under various aberration conditions across a range of applications. AFP corrected aberrations associated with maximal phase difference of 2.3π, 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 makes it an attractive imaging technology for quantitative 3D analysis in biological, microbial ecological, and medical studies.

Simulation

Video 1. Simulation results with a 16-μm thickness lymph node vascular phantom. GT: Ground truth; DF: Darkfield; w/: with; w/o: without; AC: Aberration correction.
Simulation_Crosssection
Fig. 2. Cross-section comparisons in simulation.

Polystyrene bead imaging

polysterene_bead
Fig. 3. Polystyrene bead reconstruction under defocus, spherical and random aberrations. w/o: without; w/: with; AC: Aberration correction; NAm: NA-matching; DF: Darkfield. Scale bar: 10 μm.

Mouse embryo imaging

Video 2. Label-free RI tomography of fixed mouse embryos at two-cell, four-cell, eight-cell and blasto stages. MIP: Maximum intensity projection.
Video 3. Label-free RI tomography of live mouse embryos at four-cell and eight-cell stages.

Wheat root and bacteria imaging


1. Large FOV wheat root imaging

fig_root_largeFOV
Fig. 4. Large volume (4.18 × 2.40 × 0.140 mm3) wheat root imaging (MIP with color-coded depth).

2. Root hair skeletons

fig_roothair_skeleton
Fig. 5. Root hair skeletons extracted from RI reconstruction (depth is color-coded).

3. 3D RI Stack

Video 4. 3D RI stack containing root hair, cortex, and vascular cylinder regions.

4. Root-bacteria interaction

fig_root_bacteria
Fig. 6. Root and bacteria bi-modality imaging with fluorescence (Fluo) and AFP (RI). MIP: Maximum intensity projection. AIF: All in focus. Pink arrows mark some of the bacteria locations. Scale bar: 25 μm.

3D Digital pathology


1. Complex-valued RI reconstruction

Video 5. 20-μm thick human gastric adenocarcinoma cancer pathological slide imaging with DUV AFP. MIP rendering of RI, Absorption and the composite. RI relates to cytoplasm and extracellular structures, and Absorption relates to nuclei.

2. Aberration correction for field-dependent aberration

Fig. 7. Field-dependent aberration in FOV 1.
w/o AC
w/ AC
z-plane
Fig. 8. Field-dependent aberration in FOV 2.
w/o AC
w/ AC
z-plane

BibTeX


      @misc{dong2025analyticfourierptychotomographyvolumetric,
        title={Analytic Fourier ptychotomography for volumetric refractive index imaging}, 
        author={Zhenyu Dong and Haowen Zhou and Ruizhi Cao and Oumeng Zhang and Shi Zhao and Panlang Lyu and Reinaldo Alcalde and Changhuei Yang},
        year={2025},
        eprint={2504.16247},
        archivePrefix={arXiv},
        primaryClass={physics.optics},
        url={https://arxiv.org/abs/2504.16247}, 
      }