the single slit, the incident pattern is similar to a step function in
that only the light from the center passes through.
This effect can be seen in the schematic of the single slit found
(Left) Schematic for the single sit of width b and length L. (Right) The step funtion produced by the slit.
To find the Fraunhofer diffraction pattern for the Single Slit, we used
the Fresnel-Kirchhoff formula to find the following equation:
where and Cí is a constant. The intensity of the integral can then be calculated as the following:
Where Io is the Intensity of the system when q = 0.
When the function U is plugged into Mathematica, we receive the following theoretical Fraunhofer pattern:
Fraunhofer pattern for the single slit.
The center of the peak is located at the origin and is more intense than the outer regions. This is due to constructive and destructive interference. In the center of the pattern, the light is focused straight through the lens and as a result, all in phase. The farther out from the center, the waves become increasingly out of phase and the destructive intererence causes the intensity to decrease.
The image projected onto the screen is observed by taking the Fourier Transform of our Fraunhofer pattern. To find this result we used the following equation:
For our system, we saw the following image:
Image observed for the single slit.
The image viewed is the same as the object we originated with. This allows to to believe that our calculations are correct.
To filter the sides of the Fraunhofer pattern theoretically, we used a condition statement that eliminated the sides. With that function, we could obtain the following results:
(Left) Fraunhofer pattern of the single slit when the sides are blocked. (Right) Image observed from the Fraunhofer pattern when the sides were filtered. The image of the step without filtering is superimposed on the image. (Not drawn to scale)
When the sides were removed from the Fraunhofer plane, it left a large peak in the center. When the Fourier Transform was taken of the new function, the result is a less defined peak. This is a result of the outer regions of the Fraunhofer plane containing the information for the sides of the image.
When the center of the Fraunhofer pattern for the single slit was filtered, we observed the following theoretical results:
(Left) Fraunhofer pattern for the single slit when the center is removed. (Right) Image observed from the Fraunhofer pattern when the center was filtered. The image of the step without filtering is superimposed on the image. (Not drawn to scale)
The elimination of the center information of the Fraunhofer pattern resulted in a loss of information for the center of the image.