Ch0223 - Chapter 23 Single-Slit Diffraction 23 Single-Slit...

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Chapter 23 Single-Slit Diffraction 211 23 Single-Slit Diffraction Single-slit diffraction is another interference phenomenon. If, instead of creating a mask with two slits, we create a mask with one slit, and then illuminate it, we find, under certain conditions, that we again get a pattern of light and dark bands. It is not the same pattern that you get for two-slit interference, but, it’s quite different from the single bright line in the straight-ahead direction that you might expect. Here’s how it comes about. Firstly, here’s the setup: Again, we get a bright fringe in the straight-ahead position on the screen. From there, working out to either side, we get bands that alternate between dark and bright. The first maximum to the right or left of the central maximum is not nearly as bright as the central maximum. And each maximum after that is less bright than the maximum preceding it. As far as the analysis goes, I want to start with the minima. Consider an imaginary line extending out from the midpoint of the slit all the way to the screen. Incoming Plane Waves Screen Mask with single slit
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Single-Slit Diffraction 212 So now the question is, “Under what conditions will there be completely destructive interference along a line such as the one depicted to be at angle θ above?” To get at the answer, we first divide the slit in half. I’m going to enlarge the mask so that you can see what I mean. Now I imagine dividing side A up into an infinite number of pieces and side B up the same way. When the slit is illuminated by the light, each piece becomes a point source. Consider the first point source (counting from the left) on side A and the first point source (again counting from the left) on side B. These two point sources are a distance w /2 apart, where w is the width of the slit. If the light from these two point sources (which are in phase with each other because they are really both part of the same incoming plane wave), interferes completely destructively, at some angle with respect to the straight-ahead direction, then the light from the second point source on side A and the second point source on side B will also interfere with each other completely destructively because these two point sources are also w /2 apart. The same goes for the third-from-the-left point sources on both sides, the fourth, the fifth, and so on, ad infinitum. So, all we need is to establish the condition that makes the light from the leftmost point source on side A (overall, the leftmost point of the slit) interfere completely destructively with the leftmost point source on side B (overall, essentially the midpoint of the slit). So, consider any point P on a proposed line of minima. 1
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Ch0223 - Chapter 23 Single-Slit Diffraction 23 Single-Slit...

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