According to Huygen's principle, we can collect the same information by scanning a line-focused beam across a sample as that acquired with varying the illumination angle of a plane wave onto the sample. In the earlier approcah, we measure the angular spectra of scattered light whereas in the second method we directly measure distorted wave fronts after the sample. Importantly, from the measured angular spectra corresponding to the varying locations of the line-focused beam, we can obtain the depth-resolved refractive-index map. This technique is known as synthetic-aperture tomography, and was first implemented in the ultrsound regime [1].
First implementation of synthetic aperture tomography in the optical regime was also realized in LBRC [2]. A line-focused beam (or a light sheet) was created by placing a cylindrical lens in the illumination path. The complex angular spectra were measured by phase shifting interferometry. Since multiple interferometric measurements were to be made at each location of the sample, the sample was moved across the light sheet in discrete steps using translational stage. Later, the LBRC developed a scalar diffraction theory based method to acquire single-shot angular spectra of samples continuoulsy flowing in a microfluidic channel [3].
Figure 1(A) shows the Mach-Zehnder interferometer based synthetic aperture tomography setup. A helium-neon laser beam (λ = 633 nm) is split into sample and reference arms. The line-focused beam is created using a high-NA condenser lens (Olympus, NA = 1.4) and a cylindrical lens (f = 100 mm). The angular spectra are collected using a high NA microscope objective as the samples flow (across the light sheet) in a microfluidic channel. After the tube lens, three additional cylindrical lenses are used to deliver the image on a high-speed CMOS camera. The reference plane wave at the camera plane is tilted relative to the sample beam for off-axis interferometric detection of the sample field. The camera is triggered to capture the interferograms at 5000 frames/sec. The flow speed is adjusted to ~150 µm/sec, which allows acquiring ~500 images per cell (with 15 µm diameter).
The acquired angular spectra corresponding to each cell are analyzed using the mathematical framework described in Ref. [3]. Figure 1(B) shows the kx-ky, kx-kz, and ky-kz cross sections of spatial frequency coverage. Notice the empty region resembling an apple core near the center of the image in Fig. 1B (i)), which indicates that the spatial resolution and optical sectioning capability of synthetic aperture tomography (in Fig. 1(A)) are the same as those for plane-wave tomography adopting one-axis scanning. The empty region in Fig. 1B(ii) indicates limited angular coverage of the condenser and objective lenses. Overall, these empty regions in 3-D spatial frequency map leads to missing angle artifacts, i.e., elongation of the object shape in the reconstructed image, and underestimation of its refractive index. We adopt contraints such as non-negativity of mass density and piecewise smoothness of the refractive-index profile during the reconstruction process to improve refractive index accuracy and spatial resolution. Figure 1(C) shows the refractive-index map of an RKO human colon cancer cell after 200 iterations.