Fig. 1. Workflow of the proposed method. (a) An illustration of T2oFu’s experimental setup. Green light from an addressable LED array is circularly polarized with a left circular polarizer to illuminate the sample. Sequential illuminations from various angles are used to scan the object in the spatial frequency domain. The sample is then imaged with an infinity-corrected optical system. The polarized light intensities at 0 deg, 45 deg, 90 deg, and 135 deg are recorded with a polarization-sensitive CMOS camera as images illustrated in (b). (c) Those intensity measurements are then fused to form volumetric sample permittivity tensor reconstructions by solving the corresponding inverse problem.

Fig. 2. Reconstruction results for polystyrene microspheres. (a) and (b) 0 deg-polarization intensity images of the sample illuminated with center LED. (a) The data are captured when focused at z=0μm. (b) Images were captured when focused at −5 and 7μm by mechanically moving the sample. These images serve as a reference to be compared with the reconstruction. (c)–(e) The reconstructed refractive index and birefringence. All columns in (c) and (d) share the same scale bar; all the refractive index reconstructions (c), as well as all birefringence reconstructions (d) and all cross sections (e) share the same color bar. (e) Cross sections of the reconstruction in places color labeled in (a). (f) The profile of reconstruction averaged over 10 microspheres.

Fig. 3. Reconstruction results of potato starch grain. (a) Polarized intensity measurements from the center LED. (b) Images taken when focused at −9 and 6μm serve as a reference for validating the reconstruction, from which we can see adjacent 2×2pixels that have different intensities due to sample-introduced polarization changes. (c) Tomographic reconstruction of the sample. The orientation is coded in color, while the birefringence is represented as brightness. All the images in (c) share the same color bar.

Fig. 4. Reconstructions of MSU crystals. (a) The reconstructed polarization orientation of the crystals at two different depths, labeled with their respective structural directions. (b) The birefringence reconstruction of the sample in 3D.

Fig. 5. Reconstructions of a muscle fiber. (a) The image of a muscle fiber with the center LED illumination. The imaging system is focused in the middle of the muscle fiber. (b) The reconstructed birefringence. The zoom-in region resembles the structures of healthy muscle fibers reported in literature.³⁵ (c) The image of the same muscle fiber at a different region, where a non-muscle fiber with a 90-deg bend is placed below the muscle fiber [see panel (d1)]. The imaging system is focused between this and the muscle fiber. (d) The reconstructed orientation at different depths, with a zoom-in showing the fine sarcomere structure of muscle tissue. (e) Histogram of reconstructed orientation shown in (d).

Fig. 6. Images of a heart tissue sample with cardiac amyloidosis. (a) A brightfield image. (b) A cross-polarized image was taken with a color microscope. (c) and (d) Reconstructed refractive index and birefringence, along with zoom-ins of the boxed region depicted in (e)–(h). (e) and (g) Lateral slices at different depths; (f) and (h) cross sections of the region highlighted with dashed lines in (c) and (d). Panels (c), (e) and (f) share the same color bar, while (d), (g), and (h) share another common color bar.