Diffraction, Babinet and Optical Transforms

In an elementary wave mechanics course, the subject of Fraunhofer diffraction is usually addressed within the context of single-slit and double-slit interference. This is usually followed up with a discussion of diffraction from a grating. In these discussions, one usually has the picture that light is “coming through the slits” like in the image below:

Now, if you take a look at Ashcroft and Mermin or a book like Elements of Modern X-ray Physics by Als-Nielsen and McMorrow, one gets a somewhat different picture. These books make it seem like X-ray diffraction occurs when the “scattered radiation from the atoms add in phase”, as in the image below (from Ashcroft and Mermin):

So in one case it seems like the light is emanating from the free space between obstacles, whereas in the other case it seems like the obstacles are scattering the radiation. I remember being quite confused about this point when first learning X-ray diffraction in a solid-state physics class, because I had already learned Fraunhofer diffraction in a wave mechanics course. The two phenomena seemed different somehow. In their mathematical treatments, it almost seemed as if for optics, light “goes through the holes” but for X-rays “light bounces off the atoms”.

Of course, these two phenomena are indeed the same, so the question arises: which picture is correct? Well, they both give correct answers, so actually they are both correct. The answer as to why they are both correct has to do with Babinet’s principle. Wikipedia summarizes Babinet’s principle, colloquially, as so:

the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam intensity.

To get an idea of what this means, let’s look at an example. In the images below, consider the white space as openings (or slits) and the black space as obstacles in the following optical masks:

What would the diffraction pattern from these masks look like? Well, below are the results (taken from here):

Apart from minute differences close to the center, the two patterns are basically the same! If one looks closely enough at the two images, there are some other small differences, most of which are explained in this paper.

Hold on a second, you say. They can’t be the exact same thing! If I take the open space in the optical mask on the left and add it to the open space on the mask to the right, I just have “free space”. And in this case there is no diffraction! You don’t get the diffraction pattern with twice the intensity. This is of course correct. I have glossed over one small discrepancy. First, one needs to realize that intensity is related to amplitude as so:

$I \propto |A|^2$

This implies that the optical mask on the left and the one on the right give the same diffraction intensity, but that the amplitudes are 180 degrees out of phase. This phase doesn’t affect the intensity, though, as in the formula above intensity is only related to the magnitude of the amplitude. Therefore the masks, while giving the same intensity, are actually slightly different. The diffraction pattern will then cancel when the optically transparent parts of the two masks are added together. It’s strange to think that “free space” is just a bunch of diffraction patterns cancelling each other out!

With this all in mind, the main message is pretty clear though: optical diffraction through slits and the Ashcroft and Mermin picture of “bouncing off atoms” are complementary pictures of basically the same diffraction phenomenon. The diffraction pattern obtained will be the same in both cases because of Babinet’s principle.

This idea has been exploited to generate the excellent Atlas of Optical Transforms, where subtleties in crystal structures can be manipulated at the optical scale. Below is an example of such an exercise (taken from here). The two images in the first row are the optical masks, while the bottom row gives the respective diffraction patterns. In the first row, the white dots were obtained by poking holes in the optical masks.

Basically, what they are doing here is using Babinet’s principle to image the diffraction from a crystal with stacking faults along the vertical direction. The positions of the atoms are replaced with holes. One can clearly see that the effect of these stacking faults is to smear out and broaden some of the peaks in the diffraction pattern along the vertical direction. This actually turns out to gives one a good intuition of how stacking faults in a crystal can distort a diffraction pattern.

In summary, the Ashcroft and Mermin picture and the Fraunhofer diffraction picture are really two ways to describe the same phenomenon. The link between the two explanations is Babinet’s principle.