# How do Airy disks happen in focused vs unfocused light?

In articles about Airy disks, I can't find whether they are talking about an aperture with a lens or mirror that focuses the light, or about a plain aperture.

Most of the time they are talking about imaging systems, where the light is getting focused, but their explanations, mathematical formulae and diagrams are the same as for slits (where there are no lenses or mirrors), and they are not taking in account the effects of focusing.

Asked another way, articles about diffraction say that there's an interference pattern (=Airy pattern) on any slit without focusing, on the other hand, articles about imaging systems say that there's an Airy pattern after focusing. So what did the focusing actually do?

Asked more general, what effect do the shape of the wavefront have on interference? What would be with a diverging wavefront?

People told me that without focusing the Airy pattern occurs at infinity. My question is what occurs in the near distance, and why?

The most helpful would be to show diagrams like this: (the part under L is what is important) but with a focusing aperture, so the difference will be obvious. So please do it if you can, or explain how would such a diagram look like.

My problem is mainly because I don't understand Huygens' principle. So I added a question directly about that: Understanding Huygens' principle: How is the direction of wave propagation determined? And why there is not destructive interference in every wave?

Let's start with the following initial condition: A circular aperture is filled with a propagating field, and the phase front is planar. Another way to say it: start with a plane wave at the location of the aperture, but set the field outside of the aperture equal to zero. I'm trying to avoid the question of how an aperture works. Assume it works.

The field is no longer a plane wave. A plane wave has infinite cross section and its phase fronts are planes. A plane wave can be characterized by a single wave vector. The field at our aperture might appear to be planar, but actually it's not: it's truncated. It cannot be characterized by a single wave vector. However, it can be constructed by superimposing an infinite number of plane waves, each having the same frequency as our original field, but with each constituent plane wave having a different direction (wave vector). The magnitude of the wave vectors are all the same because the frequencies are all the same. But the direction of the wave vectors are all different. The effect of the aperture is to produce a set of plane waves that propagate in directions other than "straight ahead". The field spreads out.

The "Airy disk" is distribution of wave vectors. It's best to think of diffraction not as a pattern on a screen or at infinity, but as a pattern of wave vectors. The interference pattern is not a distribution of intensity on a plane, but rather a distribution of intensity in angle.

But suppose we want to observe the pattern with our eyes. We can't see the waves themselves or their wave vectors. We can see something if we let the light hit a screen. The scattered light from the screen can enter our eyes and be seen. Put a simple screen behind the aperture and the unwieldy set of diverging plane waves (wave vectors) illuminates the entire screen. No pattern is evident. This is "the near distance", what we call the near field.

How can we tame a set of unwieldy plane waves? A lens. A lens takes parallel rays (a plane wave) and causes the rays to meet (focus) at a point. The point lies on a plane a focal length's distance away from the lens. Put a screen there, and your eyes see a dot. In this sense a lens maps plane waves to dots on a screen. A plane wave that hits the lens at another angle is mapped to a dot in a different location. The distribution of plane waves (wave vectors) after the aperture maps to a distribution of dots on the screen. The pattern on the screen is what we normally call the diffraction pattern; it is an image of the distribution of wave vectors.

So what is meant when we say that in an absence of a lens the image of the diffraction pattern appears at infinity? Take that lens and increase it's focal length. That's done by reducing the curvature of the lens. The plane where the pattern appears moves away from the lens. Continue to reduce the curvature, and the focal plane continues to move away. When the curvature is completely gone, and the lens is just a slab of glass having flat parallel faces, the focal plane is at infinity. Well, that slab of glass does nothing. The lens has been effectively removed. In the absence of a lens "the pattern appears at infinity".

But of course there's no way to view a pattern at infinity. If you actually want to see the diffraction pattern (an image of the distribution of plane wave directions) you have to use a lens, or set up a screen far enough away that neighboring wave vectors interfere to form a pattern.

• From your answer looks like there are some bright plane waves and some dark. But from what I read all over I understand that all waves are identical and the pattern is due to interference between waves. So either without or with a lens there are waves from every point of the aperture to every point of the lens, the only difference is which wave goes where. How does it effect interference? Sep 16, 2020 at 18:14
• The image in my question is of an aperture without a lens, and we do see an Airy pattern. But you said that we can't. What's going on? Sep 16, 2020 at 18:19
• I skipped over a step or two. Indeed, the field down stream of the aperture can be calculated by assuming interference of spherical waves sourced at every point in the aperture. Whatever the result of that interference is, it can be represented by a sum of plane waves. The distribution of wave vectors that results from the interference is responsible for the interference patterns we see. We can't see plane waves (unless we look into the source) so we need a way of visualizing. Either use a lens or set up a screen far away. (So my last sentence isn't quite right. I'll edit it.) Sep 18, 2020 at 19:23
• @GeorgeLee "there are some bright plane waves and some dark. But from what I read all over I understand that all waves are identical" — what differs is the amplitude of corresponding plane waves. "The image in my question is of an aperture without a lens, and we do see an Airy pattern" — after you put the screen far enough from the source—in the far field region—the field angular distribution no longer changes much with distance, just reduces in amplitude. So you see "almost" an Airy pattern even though the exact one is supposed to be at infinity. Aug 5, 2022 at 18:19