Young's Double Slit Experiment - Brightness Patterns

In this post I'll be doing #1a from this weeks worksheet on optics. I chose this problem because it let me familiarize myself with optics and why Fourier transforms are used in the field. I struggled particularly hard on this concept and working through this problem helped me understand the ideas in the worksheet better. Here's the problem:

Convince yourself that the brightness pattern of light emerging from the two slits as a function of angle is a cosine function, and that the brightness pattern on the screen is the square of this function.

The two slits refer to the two slits used in Young's double slit experiment, and the screen is the sheet that the light passing through the two slits is projected onto. Here's a picture showing the setup:

(from Wikipedia)

In this setup, we assume L to be much greater than d. So, how do we determine that the brightness pattern on the screen is the square of a cosine function of angle?

First, let's talk about diffraction and interference. Light diffracts, so upon entering the slit, the incident light will spread out in all directions. Light also interferes, so if light from one slit hits a spot, it will interfere with the light from the other slit in a way that depends on the relative phases of the two light waves.

Looking at the above diagram, we have light drawn from the two slits to the same arbitrary point. Since L is much greater than d, we can assume that the two rays of light are essentially parallel near the slits. So, we can modify the diagram as follows:

Since the distance between the screen and the slits is so large, we can effectively say that the distance from the bottom slit to the arbitrary point is equal to the distance from the top slit to the point plus some "extra" bit x. Using trigonometry and a small angle approximation (which we can do, since d is much smaller than L), we have:

So, what is x? Let's think about interference. Constructive interference is when two waves add together in the same phase. In other words, the extra distance that the ray from the bottom slit travels must be equal to an integer multiple of the wavelength of light - then the two rays will be "in phase" at the point on the screen:

What about destructive interference? Our two rays of light will interfere destructively when they produce a brightness of zero - thus, they must be out of phase by half a wavelength:

This means that points of maximum brightness on our screen our spaced apart with a distance of the wavelength of light divided by the distance between the two slits, and points of zero brightness are spaced apart with the same distance, exactly in between the brightest spots. Thus, our brightness pattern is a periodic function that is continuous, since the light rays are continuous waves. At an angle of zero, exactly in between the two slits, we obviously get constructive interference (since the waves travel the same amount), so there must be a maximum at the zero. This is described perfectly by the cosine function.

I'm actually mildly incorrect. The cosine function describes the electric field of the two light rays, not the intensity on the screen. Intensity is actually the square of the electric field, so our brightness pattern at the screen is really a cosine squared function.

It turns out that this behavior is described by the Fourier Transform, which makes it extremely useful to use the Fourier Transform when talking about optics in astronomy.