This article presents a method to perform diffraction tomography in a standard microscope that includes an LED array for illumination. After acquiring a sequence of intensity-only images of a thick sample, a ptychography-based reconstruction algorithm solves for its unknown complex index of refraction across three dimensions. The experimental microscope demonstrates a spatial resolution of 0.39 @math m and an axial resolution of 3.7 @math m at the Nyquist-Shannon sampling limit (0.54 @math m and 5.0 @math m at the Sparrow limit, respectively), across a total imaging volume of 2.2 mm @math 2.2 mm @math 110 @math m. Unlike competing methods, the 3D tomograms presented in this article are continuous, quantitative, and formed without the need for interferometry or any moving parts. Wide field-of-view reconstructions of thick biological specimens demonstrate potential applications in pathology and developmental biology.
Here, we first outline a solid foundation for the application of ptychographic phase retrieval to DT. Unlike approaching the problem from a projection-based or multi-slice perspective, the framework of DT (under the first Born approximation) follows directly from the scalar wave equation. It offers a clear picture of achievable resolution in 3D, spells out sampling and data redundancy requirements for an accurate reconstruction, and presents a clear path forward for future extensions to account for multiple scattering @cite_6 . Furthermore, our method does not require the arbitrary assignment of the number slices in the 3D volume, or their location, or for us to select a particular order in which to address each slice as iterations proceed. Instead, it simply inserts the measured data into its appropriate location in 3D Fourier space and ensures phase consistency between each measurement, given a sufficient amount of data redundancy (just like ptychography). From the initial starting point of solving for the first term in the Born expansion, we aim this approach as a general framework to eventually solve the challenging problem of forming tomographic maps of volumetric samples, at sub-micrometer resolution, in the presence of significant scattering.
@cite_6
2003147468
A new method, based on an iterative procedure, for solving the two-dimensional inverse scattering problem is presented. This method employs an equivalent Neumann series solution in each iteration step. The purpose of the algorithm is to provide a general method to solve the two-dimensional imaging problem when the Born and the Rytov approximations break down. Numerical simulations were calculated for several cases where the conditions for the first order Born approximation were not satisfied. The results show that in both high and low frequency cases, good reconstructed profiles and smoothed versions of the original profiles can be obtained for smoothly varying permittivity profiles (lossless) and discontinuous profiles (lossless), respectively. A limited number of measurements around the object at a single frequency with four to eight plane incident waves from different directions are used. The method proposed in this article could easily be applied to the three-dimensional inverse scattering problem, if computational resources are available.