Reconstruction of C. Elegans Worm using Coherent Diffraction Imaging


Purpose and Introduction

The purpose of this lab is to familiarize ourselves with working in a biophysical setting, and to better understand diffraction microscopy. When a coherent plane wave is projected onto an object, the light is reflected and an image is observed of the amplitude squared of the Fourier transform. A phase difference between the individual photons also occurs, however this part of the image is harder to observe and leads to the “phase problem”. We were able solve this problem by isolating our sample and taking several images and using algorithms in Matlab to reconstruct our image of the worm. CDI is of great use in biology. It allows high quality imaging on a micro and nano scale that typical optical microscopes would need several lens to accomplish with lackluster results due to aberrations from the geometry of the lens. This experiment serves as a proof of concept within the limitations of the lab and materials provided to us. In an ideal situation, a much longer (\(\sim\) 1 mile) accelerator would be used to generate a far-field diffraction pattern of an object using X-rays or high energy electrons [3]. Of course we do not have such resources, but it can be shown that we can generate a diffraction pattern of a  1 mm sample and reconstruct the image with the tools given to us in a standard optics laboratory \cite{Thibault_2007}. Once a diffraction pattern is obtained, a Fourier transform of the object is taken to reproduce the original image. This requires a high over sampling ratio, an isolated sample, and several images at different exposures so one can take the Fourier transform over several and average the values to reproduce the original image.


The equation for a diffraction pattern is

\[2d\sin{\theta}=\lambda\] where d is the, \(\theta\) is the angle of the image with the detector, and \(\lambda\) is the wavelength. An oversampling ratio is found by \[\frac{\lambda}{O\,a}=\frac{\Delta p}{z}\] where \(\lambda\) is again the wavelength, \(O\) is the linear oversampling ratio, \(a\) is the size of the sample, \(\Delta p\) is the difference in length in pixels along the detector, and \(z\) is the distance between the sample and the detector.
Since we are dealing with far-field diffraction patterns, we want to use Fraunhofer diffraction equation \[FN = \frac{x^{2}}{z\lambda}\] where \(FN\) is the Fresnel number, \(x\) is the distance along the detector, \(z\) is the distance between the sample and the detector, and \(\lambda\) is the wavelength. To get a far-field diffraction patter, \(FN << 1\) [4].
To find the find the focal point of ours lens and the proper placement of our sample and detector, we use the equation \[\frac{1}{f_{0}}=\frac{1}{d_{0}}+\frac{1}{d_{1}}\] where \(f_{0}\) is the focal length of the particular lens, and \(d_{0}\) and \(d_{1}\) are the distances of the sample and the detector relative to the lens [2].

In this experiment, there was also an attempt to perform a three dimensional reconstruction. This was done using Equally Sloped Tomography method (EST). This will be elaborated on in the 3D reconstruction section in the results.