Diffraction Modeling

This is the mathematica file for the first mathematica worksheet.

This is the mathematica file for a delta function potential. Edit 2/15: Delta Function File has been updated.

Project Proposal

My proposal is to work off the existing research that was performed by Jenny Magnes, Kathleen M. Raley-Susman, Alicia Sampson, Margo Kinneberg, Rahul Khakurel, and Rebecca Eells to model an orientation for an organism, specifically the C. elegans or nemotodes. The existing research has shown these organisms in certain positions, such as almost overlapping or orthogonal, so by using their existing research into the ways light diffracts off of these organisms, I can model a different shape or size using Matlab, which would permit me to add to this body of research.  This project will specifically model physical size of the organisms using diffraction.

Quantum Cryptography Assignment

This is an encrypted message using quantum cryptography.  On the “key” page is a demo of what happens when there is an eavesdropper.

Wave Interference Assignment

This is the mathematica file for the wave interference assignment

This is the mathematica file for the Fourier Transform assignment.

Diffraction Models

This project involves taking the Fourier Transform of a mathematical function that represents a small, thin organism(s) and determining what diffraction pattern one would see, should a light source, such as a laser, pass around the organism(s).  This modeling builds off of the research performed by Jenny Magnes, Kathleen M. Raley-Susman, Alicia Sampson, Margo Kinneberg, Rahul Khakurel, Rebecca Eells in 2009.  Magnes et al. worked with C. elegans, which have a thickness similar to a single hair, and passed laser light around the C. elegans, which produced diffraction patterns.  Due to the size of these organisms, it is impossible to see with the naked eye the number or orientation of the C. elegans.

Our first question is “what is diffraction?”  When light interacts with an obstacle, light will appear to bend around it and will produce a pattern.  When light passes through a small slit whose size is on the same order of magnitude as the wavelength of the light, the effects of diffraction are more pronounced.  For this project, light will be passing around C. elegans and it is necessary to determine how the C. elegans are aligned when the light interacted with them.  Mathematically, it is possible to determine what a diffraction pattern would look like when light interacts with the C. elegans.  To do this, we will use Fourier Transforms.

Fourier Transforms come from the idea that any given function can be created as a sum of sines and cosines.  The basic Fourier Transform equation is

where f(ξ) is our normal function and f(x) is a sum of sines and cosines.

Why do we use Fourier Transforms?  Fourier Transforms are mathematically equal, to a certain approximation, to Fraunhofer Diffraction (Rodenburg).  Fraunhofer Diffraction is also known as far-field diffraction.  Fraunhofer diffraction occurs when light passes through a slit and causes only the size of an observed aperture image to change due to the distant location of observation, and the planar nature of outgoing diffracted waves passing through the aperture.  The equations for Fraunhofer diffraction can be applied when a^2/Lλ much smaller than 1, where a is  the size of the aperture, L is the length from the aperture to the screen, and λ is the wavelength of the light source.

In order to use Fourier Transforms, we first need a function to use.  In this first example, a function that defines a rectangle was used.


By taking the Fourier Transform of a shape  like this rectangleDiffraction Structure rectangle we get thisRectangle Fourier

If the pattern of light seen by the researchers looks similar to the Fourier Transform of the rectangle, then it is likely that the C. elegans were positioned such that there was a single C. elegan who is extended.  The size and number of rectangles can be changed in the modeling program, which permits researchers to adjust the equations until a diffraction pattern is created that looks similar to experimental data.

Magnes et al.’s research was produced using MatLab, so this project worked to create various shapes in mathematica to further future research.  In my modeling efforts I created the following shapes: A single rectangle, which you can see above.  Here is the mathematica file for it.

A pair of rectangles:  The rectangles were created using this mathematica code.

This graph is of the two rectanglesDiffraction Structure 2 rectangles

This is the Fourier Transform of those two rectangles 2 Rectangle Fourier

I was also able to create a Cross, which was made of two intersecting rectangles.  This is the mathematica file for the Cross.

This is the graph of the Cross:Diffraction Structure cross

This is the Fourier Transform of that Cross:Cross Fourier

I also created shapes that were asymmetric.This is a .pdf with two rectangles off center from each other.

While this is the transform2 off center and this is the mathematica file for it.

Here we have 4 rectangles.

And this is their TransformTetris Mathematica

And their mathematica file.

The successful shapes were created using sums of Unit Step functions.  These functions are 0 when the argument is less than 0, and 1 when the argument is greater than or equal to zero.

Problems arose in my attempts to create shapes that were angled, or non-rectangular.  Shapes that were angled were difficult to create, and anything that resembled them created Fourier Transforms that were imaginary, making it impossible to graph.  However, despite this problem, the results obtained permit research into microscopic organisms using diffraction to now use mathematica to attempt analysis.  The plots of these shapes have whitelines throughout them.  These lines appear to be a product of mathematica and do not impact the project.

As a result of this research, I have realized the complexity and difficulties that are innate to understanding microscopic structures.  We cannot see them with our naked eyes, which means we have to rely on elaborate ways “seeing” these structures.  This procedures require a lot of fine tuning and models, and we have to take each structure on a case by case basis, which is astounding.

Citation

The Fourier Transform.  John Rodenburg.  http://www.rodenburg.org/theory/y1300.html

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2 thoughts on “Diffraction Modeling

  1. mieshete

    Hello Kyle, I enjoyed reading through you blog. Unlike Rahul, I am kinda unfamiliar with the experiment so a brief description of the procedure would have made it much clearer for me. Also, you have taken the fourier transforms of these shapes. but they appear to be in 2D. is there another dimension in the z direction that I’m not seeing? or are the shapes simply rectangles and you are only investigating their orientation instead of their volume?
    great read
    Peace

  2. rakhakurel

    Hey Kyle. I like your simulations. I think its impressive how you’ve tried to use unit step functions to create all your functions, because it doesn’t seem easy.
    I am wondering why there are these white bands that separate your functions into many different rectangles. Is this just how mathematica represents sums of step functions or are those actually regions that get light?
    Also, I have a few recommendations regarding the problems you stated. You said you had problems plotting the imaginary outputs. I think you will be able to plot them if you plot the Real Parts and the Imaginary Parts of the Fourier Transforms separately. Also, try plotting the Amplitude of the Transforms because this takes both the Real and Imaginary parts into account and it represents the actual diffraction pattern. And as for the problem you had making angled shapes, maybe Mathematica’s drawing tool could work. http://reference.wolfram.com/mathematica/guide/GraphicsInteractivityAndDrawing.html
    It was interesting going through your results. Good luck !

    -Rahul

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