Angular Resolution

In the first part of this problem, we were asked to reconstruct Young's double slit experiment.  We drew this picture, with the curves representing the light waves coming from behind the slits.

We were told to convince ourselves that the monochromatic light hits the screen in a cosine function.  We drew this picture:

which represented a situation in which the light waves are interfering in one spot on the screen.  We figured that in order for the waves to interfere constructively, the line labeled $\lambda$ would have to be equal to the wavelength multiplied by an integer. That way, the two waves would be in phase together. The waves are interfering constructively in the picture, creating a peak at a distance, d, from the center of the screen.  If the two waves aren't in phase, they will interfere each other destructively.  We knew that the waves hit the screen in a periodic wave function, so we were able to convince ourselves it was a cosine function that looked kind of like this:

Because we were told that L>>D, the lengths of all of the waves going to the screen from the two slits are almost equal and the angle formed where the line L meets the screen is a right angle.  With this information, we made the following equation:

$\frac{d}{L}=\frac{n\lambda}{D}\rightarrow d=\frac{nL\lambda }{D}$

The second part of the problem asked us to imagine what would happen to the brightness pattern on the screen if there were two more slits closer together than the already existing two.  The distance, D, between these two slits would be less than in the first part of the problem.  According to the equation above D and d are inversely proportional, so as D decreases, d increases.  An increase in d results in a brightness pattern on the screen with more space between each peak where the waves interfere constructively.  The light coming in through the four slits would result in a more intense peak in the middle of the screen where the two waves interfere constructively and lower amplitudes elsewhere on the screen.  The resulting wave function, relative to the first, would look like this:

In the third part of the problem, we were asked to find the brightness pattern on the screen if there were a continuous set of slits with ever increasing D.  So, essentially we were asked to find the brightness pattern on the screen if there were a giant gap in front of the light source instead of a wall with slits.  Again, using the equation above, as D decreases, d increases, thus increasing the distance between peaks in the brightness pattern.  As more and more slits are added, the peak in the middle of the screen will get higher because of the constructive interference of all of the light waves. The resulting brightness pattern, again relative to the first two, will look like this:

In the next part of the problem, we were asked to compare the double slit experiment to the top hat Fourier transform function.  The top hat transform is a function that shows a frequency spike at 0 on the x-axis.  I find it easier to picture the graphical representation of the top hat function from an aerial view, which looks like a ripple in water with a big splash in the middle and diminishing waves radiating out.

http://www.radiantzemax.com/content_images/apodization/ap2.jpg

The final part of the problem asked us to relate what we had done to a telescope primary mirror.  Working with the slit experiment and convincing myself that a brightness pattern left by light on a surface will be a cosine function.  All of the light waves interfere constructively and destructively in such a way that there is a peak of information in the center of the primary mirror.