What makes diffraction spikes move with the focus?

Question

This is a bit of a follow up on The Bahtinov focusing mask and it came with this question on Astronomy. But I think the effect can also be observed when looking through a hazy atmosphere or a stained window pane at a distant point light source.

My question is: which physical effect makes a spike move over the light source when adjusting the focus or in other words why does the spike move when the focus changes, while the central point stays still ?

The point is at infinity, usually, while the obstruction is very close. And though the point's Airy disk changes size with focus, the diffraction pattern stays sharp.

Answer 1

The imaging happens as follows: The star imaged is very far away, so it reaches the telescope nearly as a plane wave. Depending on the position of the star in the image field, the wavefronts are incoming at a specific angle. The angle of course is small, but thats what telescopes are for - differentiating sources that have small angular separation in the sky. This angle corresponds to a wavefront error: the planes are tilted with regard to the entrance pupil of the telescope. The resulting wavefront deviation from an ideal plane wave is a linear function of the distance to the optical axis, and as is known through abberation theory such a wavefront error causes a shift in the image field. Thus, the diffraction pattern is displaced from the optical axis by an amount depending on the telescopes focal length.

What you see with the telescope is thus only the diffraction of the mask, shifted to the position of the star, and most importantly, you cannot say that the center point of the diffracted pattern is the image of the star, and the outer part is the diffraction pattern - both together are the diffraction pattern.

Regarding your actual question on why precisely this happens: I really tried to find a rigorous answer that was easy to follow, which I was unable to do. So I will not speculate here or try a hand wavy explanation.

What I want to point out though is, when looking at simulations of the effect done by proper scalar diffraction theory with aberrations (see this animation on Wikipedia), for one, the diffraction spikes in fact do defocus (in contrast to your statement above) and second - for me at least - it looks like the central peak of the diffracted image also moves, or at least changes shape (it is just harder to see). This is why the mask is used in the first place: to improve the visibility of defocus. So I at least would be questioning if the premise of your question (that it happens, not how) is actually true in the first place.

Just a guess: I suppose the answer why the shifting occurs is similar to this Question where part of the optical path of light from a source is cut of only on one side and thus shifts in the image plane. This is the case here, because the diffracting slits are not placed on the optical axis, but decentered.

Answer 2

The diffraction spikes are image and object independent and the mask which is divided into thirds (like a pie cut in 3 pieces) divides the rays into say an upper third and a bottom right and left thirds. Under-focused the top pie rays can't make it to the centre .... when over focused the rays go towards the bottom. Note that the same is happening to the central star ... it blurs as focus is lost.

Another way to think about it is that the object (star) has rays that cover the entire lens whereas the mask only lets rays cover 1/3 of the lens of each pattern. When the lens is out of focus this spatial bias becomes visible, photons from each third are shifted (blur is small) until focused.