Coherent Diffraction Imaging of a C. Elegans Larvae and Reconstruction Experiment Proposal


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 hope to 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 limitation of the lab and materials provided to us. In an ideal situation, a much longer (  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 [1]. 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 \[\frac{1}{f_{o}}=\frac{1}{d_{o}}+\frac{1}{d_{1}}\] where \(f_{o}\) is the focal length of the particular lens, and \(d_{o}\) and \(d_{1}\) are the distances of the sample and the detector relative to the lens [2].

Schematic of the experimental setup. (L) is the laser, (P) is the pinhole, (C1) is the converging lens, (I) is the iris, (S) is the sample used, (C2) is the objective lens, and (D) is the detector. The schematic is not set to scale.

Experimental Setup

The schematic for the experimental setup is shown in figure 1. Our setup uses a 635 nm laser as the light source, which is attenuated by a combination of neutral density filters in order to prevent over saturation. The light is then projected onto a 100 \(\mu\)m pinhole, generating a high quality uniform wavefront. To create a collimated beam, a converging lens is placed one focal length after the pinhole (in our case 15 mm). To eliminate higher order lobes, an iris is placed after the converging lens to only allow the central disk of the beam. After a creating a collimated plane wave, it is illuminated onto the sample, which generates a diffraction pattern. An objective lens is placed after the sample to focus the image onto the Charged Couple Device detector. The objective lens also increases the optical distance of the image, allowing us to acquire a far-field diffraction pattern within the size limitations