Clarification and unification of the obliquity factor in diffraction and scattering theories: discussion

Recent Advances and Current Trends in Transmission Tomographic Diffraction Microscopy

Optical microscopy techniques are among the most used methods in biomedical sample characterization. In their more advanced realization, optical microscopes demonstrate resolution down to the nanometric scale. These methods rely on the use of fluorescent sample labeling in order to break the diffraction limit. However, fluorescent molecules' phototoxicity or photobleaching is not always compatible with the investigated samples. To overcome this limitation, quantitative phase imaging techniques have been proposed. Among these, holographic imaging has demonstrated its ability to image living microscopic samples without staining. However, for a 3D assessment of samples, tomographic acquisitions are needed. Tomographic Diffraction Microscopy (TDM) combines holographic acquisitions with tomographic reconstructions. Relying on a 3D synthetic aperture process, TDM allows for 3D quantitative measurements of the complex refractive index of the investigated sample. Since its initial proposition by Emil Wolf in 1969, the concept of TDM has found a lot of applications and has become one of the hot topics in biomedical imaging. This review focuses on recent achievements in TDM development. Current trends and perspectives of the technique are also discussed.

Efficient and accurate intensity diffraction tomography of multiple-scattering samples

Optical Diffraction Tomography (ODT) is a label-free method to quantitatively estimate the 3D refractive index (RI) distributions of microscopic samples. Recently, significant efforts were directed towards methods to model multiple-scattering objects. The fidelity of reconstructions rely on accurately modelling light-matter interactions, but the efficient simulation of light propagation through high-RI structures over a large range of illumination angles is still challenging. Here we present a solution dealing with these problems, proposing a method that allows one to efficiently model the tomographic image formation for strongly scattering objects illuminated over a wide range of angles. Instead of propagating tilted plane waves we apply rotations on the illuminated object and optical field and formulate a new and robust multi-slice model suitable for high-RI contrast structures. We test reconstructions made by our approach against simulations and experiments, using rigorous solutions to Maxwell's equations as ground truth. We find the proposed method to produce reconstructions of higher fidelity compared to conventional multi-slice methods, especially for the challenging case of strongly scattering samples where conventional reconstruction methods fail.

In-silico clearing approach for deep refractive index tomography by partial reconstruction and wave-backpropagation

Refractive index (RI) is considered to be a fundamental physical and biophysical parameter in biological imaging, as it governs light-matter interactions and light propagation while reflecting cellular properties. RI tomography enables volumetric visualization of RI distribution, allowing biologically relevant analysis of a sample. However, multiple scattering (MS) and sample-induced aberration (SIA) caused by the inhomogeneity in RI distribution of a thick sample make its visualization challenging. This paper proposes a deep RI tomographic approach to overcome MS and SIA and allow the enhanced reconstruction of thick samples compared to that enabled by conventional linear-model-based RI tomography. The proposed approach consists of partial RI reconstruction using multiple holograms acquired with angular diversity and their backpropagation using the reconstructed partial RI map, which unambiguously reconstructs the next partial volume. Repeating this operation efficiently reconstructs the entire RI tomogram while suppressing MS and SIA. We visualized a multicellular spheroid of diameter 140 µm within minutes of reconstruction, thereby demonstrating the enhanced deep visualization capability and computational efficiency of the proposed method compared to those of conventional RI tomography. Furthermore, we quantified the high-RI structures and morphological changes inside multicellular spheroids, indicating that the proposed method can retrieve biologically relevant information from the RI distribution. Benefitting from the excellent biological interpretability of RI distributions, the label-free deep visualization capability of the proposed method facilitates a noninvasive understanding of the architecture and time-course morphological changes of thick multicellular specimens.

Nonparaxial optical transfer function for arbitrary illumination in partially coherent imaging systems and the oblique source application

Recent research in quantitative phase and refractive index microscopy showed promising results with methods using a partially coherent imaging setup, such as partially coherent optical diffraction tomography. For these methods, the phase optical transfer function (POTF), which describes the transmission of spatial frequencies by the imaging system, is crucial. Here, a one-dimensional integral representation of the POTF for imaging systems with arbitrary illumination is derived. It generalizes the existing expression, which is limited to axially symmetric setups. From the general integral form, an analytical solution is derived for the case of oblique homogeneous disk-shaped illumination. This demonstrates the potential of the general representation by offering an additional approach for illumination design in quantitative phase and refractive index microscopy.

Diffraction as scattering under the Born approximation

Light diffraction at an aperture is a basic problem that has generated a tremendous amount of interest in optics. Some of the most significant diffraction results are the Fresnel-Kirchhoff and Rayleigh-Sommerfeld formulas. These theories are based on solving the wave equation using Green's theorem and result in slightly different expressions depending on the particular boundary conditions employed. In this paper, we show that the diffraction by a thin screen, which includes apertures, gratings, transparencies etc, can be treated more generally as a particular case of scattering. Furthermore, applying the first order Born approximation to 2D objects, we obtain a general diffraction formula, without angular approximations. Finally, our result, which contains no obliquity factor, is consistent with the 3D theory of scattering. We discuss several common approximations and place our results in the context of existing theories.

Quantitative 3D refractive index tomography of opaque samples in epi-mode

Three-dimensional (3D) refractive index (RI) tomography has recently become an exciting new tool for biological studies. However, its limitation to (1) thin samples resulting from a need of transmissive illumination and (2) small fields of view (typically ~50 μm × 50 μm) has hindered its utility in broader biomedical applications. In this work, we demonstrate 3D RI tomography with a large field of view in opaque, arbitrarily thick scattering samples (unsuitable for imaging with conventional transmissive tomographic techniques) with a penetration depth of ca. one mean free scattering path length (~100 μm in tissue) using a simple, low-cost microscope system with epi-illumination. This approach leverages a solution to the inverse scattering problem via the general non-paraxial 3D optical transfer function of our quantitative oblique back-illumination microscopy (qOBM) optical system. A theoretical analysis is presented along with simulations and experimental validations using polystyrene beads, and rat and human thick brain tissues. This work has significant implications for the investigation of optically thick, semi-infinite samples in a non-invasive and label-free manner. This unique 3D qOBM approach can extend the utility of 3D RI tomography for translational and clinical medicine.

Three-dimensional phase optical transfer function in axially symmetric microscopic quantitative phase imaging

Three-dimensional quantitative phase imaging (3D QPI) is widely recognized as a potentially high-impact microscopic modality. Central to determining the resolution capability of 3D QPI is the phase optical transfer function (POTF). The magnitude of the POTF over its spatial frequency coverage (SFC) specifies the intensity of the response for each allowed spatial frequency. In this paper, a detailed analysis of the POTF for an axially symmetric optical configuration is presented. First, a useful geometric interpretation of the SFC, which enables its visualization, is presented. Second, a closed-form 1D integral expression is derived for the POTF in the general nonparaxial case, which enables rapid calculation of the POTF. Third, this formulation is applied to disk, annular, multi-annuli, and Gaussian illuminations as well as to an annular objective. Taken together, these contributions enable the visualization and simplified calculation of the 3D axially symmetric POTF and provide a basis for optimizing QPI in a wide range of applications.

High-fidelity optical diffraction tomography of multiple scattering samples

We propose an iterative reconstruction scheme for optical diffraction tomography that exploits the split-step non-paraxial (SSNP) method as the forward model in a learning tomography scheme. Compared with the beam propagation method (BPM) previously used in learning tomography (LT-BPM), the improved accuracy of SSNP maximizes the information retrieved from measurements, relying less on prior assumptions about the sample. A rigorous evaluation of learning tomography based on SSNP (LT-SSNP) using both synthetic and experimental measurements confirms its superior performance compared with that of the LT-BPM. Benefiting from the accuracy of SSNP, LT-SSNP can clearly resolve structures that are highly distorted in the LT-BPM. A serious limitation for quantifying the reconstruction accuracy for biological samples is that the ground truth is unknown. To overcome this limitation, we describe a novel method that allows us to compare the performances of different reconstruction schemes by using the discrete dipole approximation to generate synthetic measurements. Finally, we explore the capacity of learning approaches to enable data compression by reducing the number of scanning angles, which is of particular interest in minimizing the measurement time.

Iterative optimization in tomographic deconvolution phase microscopy

Tomographic deconvolution phase microscopy (TDPM) is a three-dimensional (3D) quantitative phase imaging (QPI) method using partially coherent light that can be implemented on a commercial microscope platform. However, the measurement procedure is relatively time-consuming because it requires many illumination angles. In the present work, an edge-preserving iterative optimization algorithm is presented and applied to TDPM, so that the required number of illumination angles is reduced from 15 to 3, while the measurement accuracy remains high. In addition, the iterative algorithm does not require matrix representation of operators, so the memory requirement is correspondingly reduced.