| The double slit is
comparable to two single slits each of width b and separated by a distance
incident pattern is similar to the single slit, but in this case there are
(Left) Schematic for the double slit of width b and separated by a distance h. (Right) Step function produced by the double slit.
The Fraunhofer diffraction integral can be calculated with the following equation:
where , and . The intensity of the integral can then be calculated as the following:
(Fowles 120)Where Io is the Intensity of the system when q = 0. The difference between the single slit and the double slit is in the presence of the cosine factor. This term gives the Fraunhofer pattern the fringes within the pattern. The factor acts as the envelope for the diffraction pattern.
When the intensity equation was calculated in Mathematica, we received the following Fraunhofer pattern:
Fraunhofer pattern for the double slit. The inner function is the pattern for the double slit while the outer envelope is the factor.
The observed image on the screen for the double slit can be calculated by taking the Fourier Transform of the Fraunhofer effect. We used the same equation as in the single slit experiment to get the following pattern.
Image observed from the FFT of the Fraunhofer Patern
This is once again the same function we began with. This allows us to presume that our calculations are correct.
When the sides of the Fraunhofer plane were removed, we calculated the following results:
(Left) Fraunhofer pattern for the double slit when the sides have been filtered. (Right) Image observed from the FFT of the filtered Fraunhofer pattern. The unfiltered image of the double slit has been superimposed onto the frame for reference. (Not drawn to scale.)
The elimination of the sides of the Fraunhofer pattern created a lack of definition in the edges of the image.
When the center of the Fraunhofer plane is removed, we can expect the following results:
(Left) Fraunhofer pattern for the double slit when the center has been filtered. (Right) Image observed from the FFT of the filtered Fraunhofer pattern. The unfiltered image has been superimposed onto the image as a reference. (Not drawn to scale)
The elimination of the center of the Fraunhofer pattern created a lack of central peak in the image. The center peak in the Fraunhofer pattern seems to contain all the information for the centers of the image. This result is consistent with the results for the single slit when the center was blocked in the Fraunhofer pattern.