near-field to far-field transition

or "what does a Fourier transform look like?"

Many educational videos have tried to come up with some kind of animation to represent the Fourier transform. This is hard to do since there is no obvious mathematical way to represent a space that is halfway between "real space" and "reciprocal space". However, in reality there is a continuum of shapes between atoms and diffraction patterns.

Consider a theoretical detector with small enough pixels that it could resolve the distance between two "point atoms". Now, these "atoms" could be a few Angstrom apart, or they could be several inches apart. The important thing to diffraction is the ratio of this distance to the wavelength of the light being used. If we ignore the direct beam (and the huge flare it would produce on the detector), then we can consider these two atoms to be point emitters of light, and if they are both experiencing the "same" incident beam, then their scattering will be "in phase".

Now, if our theoretical detector is virtually in contact with these two atoms, then the image we will record is two distinct points, created by the light being emitted by each atom. The relative phase of the light doesn't matter since the pixels pick it up long before they can interact with light from the other atom. However, if we move the detector further away, the light from each atom will start to interfere, and in the limit of infinite distance we will see the classic "interference pattern" of light and dark bands with a spacing equal to the reciprocal distance between the atoms. What does the transition between two points and a row of bands look like? See below.

The animation below was rendered with nearBragg2D. The detector is 512x512 pixels and the pixels are 0.1 Angstrom wide, making this a ~5 nm wide detector. The wavelength of light being used is 1 Angstrom, and the atoms are 10 Angstrom apart. The green text in the upper left corner is the "detector distance" inasmuch as it is the distance between the detector pixel plane and the plane normal to the incident beam that contains the two atoms. It is expressed in multiples of the distance between the atoms (1 nm). The detector size is fixed for all frames in the movie.

One can see that the two point atoms are clearly resolved in the first frame, but as the detector starts to move away (the near field), one can see bow-shaped lines where the sum of the distance from one atom to the other atom via the pixel becomes an integral multiple of wavelengths. This is the definition of an interference line. At this distance, however, the lines are bowed because the detector is very close to the atoms, relative to the wavelength of the light. At longer detector distances, the pattern rapidly converges to bands (the far field). These bands also rapidly begin to move apart. This is because our detector is still only 5 nm wide, but it is the angle between the bands that is actually fixed. In this way, it can be seen that "reciprocal space" is really more of an angle space than it is a "reciprocal distance space", although the latter is a useful tool in the far field.

The colors indicate the phase of the wave arriving at the detector relative to that of a single atom located at the origin.
Also available in black and white swf wmv mov

So, what does it look like with three atoms? Like this: black and white or color-by-phase: gif swf wmv mov
Where a third atom has been thrown in at an arbitrary position.

What if you "zoom" the pixels to kind of "follow" the far-field pattern? : black and white color-by-phase: gif swf wmv mov
This movie is the same as the previous (near2far3), but once the detector has reached a far-field distance, I start making the pixels bigger so that the resolution on the edge of the detector is fixed depsite the increasing distance. This effectively changes the pixel coordinates from positions to angles, making it easier to see how the pattern evolves in reciprocal space.

How was these movies made? With these tcsh shell scripts: near2far.csh near2far3.csh