# Optical Diffraction Tomography Using Nearly In-Line Holography with a Broadband LED Source
## 1 Introduction  
Optical Diffraction Tomography (ODT) is an emerging tool for label-free imaging of semi-transparent samples in three-dimensional space [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Being semi-transparent, such objects do not strongly alter the amplitude of the illuminating field. However, the total phase delay at a particular wavelength is a function of the refractive index of the sample and also the thickness of the sample. Due to this ambiguity, one cannot distinguish between those parameters from 2D projections. Hence, to reconstruct the 3D refractive index (RI) map of semi-transparent samples, a holographic detection is needed to extract the phase of the field after passing through the sample. Then, by acquiring different holograms at different illumination angles, the 3D RI map can be reconstructed using inverse scattering models [10]. Holographic detection was introduced by [PERSON] who used \"in-line\" holography. He showed that the intensity image retrieved from the in-line holography is composed of an \"in-focus\" image in addition to an \"out-of-focus\" image (i.e., \"Twin\" image) [11]. Due to this \"Twin\" image problem, in-line holography usually encounters problems in retrieving the phase of the object. [PERSON] and [PERSON] proposed an \"off-axis\" holography [12]. In this configuration, a small tilt is introduced between the reference arm and the sample arm, which results in shifting in the Fourier domain the \"out-of-focus\" image with respect to the \"in-focus\". Since then, \"off-axis\" interferometry has been widely used in ODT by first extracting the phase before using the inverse models [13, 14]. Several limitations remain that limit the use of ODT in biological imaging. These limitations include phase instability due to interferometry and laser fluctuations and speckle artifacts due to the high coherence of the laser source. To overcome these limitations, [PERSON], [PERSON], and co-workers used a relatively broadband source (i.e., LED illumination) to illuminate the sample for Fourier ptychographic and 3D imaging [15, 16, 17, 18]. In particular, in [18], [PERSON] and [PERSON] used LED illumination and demonstrated an iterative reconstruction scheme with a multi-slice forward model to estimate the 3D complex RIdistribution by minimizing an error function between the intensity patterns estimated from the forward model. Their approach showed in-focus reconstruction at different depths while taking multiple scattering phenomena into account. In recent years, this approach was demonstrated both in reflection and transmission configurations [19; 20; 21; 22; 23; 24]. For example, a motion-free illumination scanning scheme was demonstrated for 3D RI reconstruction using an LED ring that mimics a circular scanning approach [25]. Other approaches to intensity diffraction tomography include the use of iterative schemes to reconstruct the 3D RI map from 2D intensity images using nonlinear iterative schemes that minimize an error function [26; 27]. These models usually start with an initial guess of the 3D RI that it keeps modifying by minimizing the error between the actual measurement from the experiment and the intensity profile from the forward physical model. In the earlier work [18], the phase and absorption transfer functions were calculated in the spatial domain using the intensity images as a function of the illumination angle. In this work, we use LED illumination and apply the Fourier diffraction theorem or Wolf Transform [1; 28] in order to reconstruct the 3D scattering potential in the 3D Fourier space followed by a 3D inverse Fourier transform to produce the 3D RI distribution. The reconstructions showed higher resolution, lower speckle noise, and high contrast reconstructions compared to the results we presented earlier for a Wolf transform reconstruction applied to projections obtained with a laser illumination [28]. We begin by discussing the use of the Fourier diffraction theorem on the \"in-line\" intensity data for the retrieval of the 3D RI map of the sample.