Why should I care? The smallest spot of light that a perfect lens or mirror can focus is the Airy disk.
What is the Airy disk? Light going through a circular opening forms a spot of light surrounded by ever fainter rings. A telescope lens or mirror bends or reflects light to a focus. The aperture of the lens or mirror is the circular opening.
Why a discrete spot size and rings? Light diffracts because it has wave properties. Think of the famous slit experiment where light or electrons, anything that has wave properties travel through a slit. Projected on the wall will be a fuzzy image of the slit (the equivalent of the Airy disk), surrounded with ever fainter straight bands. Make the slit a circular opening and now the fuzzy image of the slit becomes a fuzzy disk surrounded by ever fainter rings. Incidentally this demonstrates that pattern or shape of the diffraction is caused by the shape of the opening, not caused by the edge of the opening.
What is the size of the Airy disk? There are two ways to measure
the Airy disk. The first is the linear size as if we had an incredibly
precise ruler. For green light, the diameter is 0.00124 mm times the focal
ratio. For example, an F8 scope has an Airy disk diameter of 0.00124*8=
0.01 mm. In inches it is 0.00005 times the focal ratio. For example, an F8
scope has an Airy disk diameter of 0.00005*8= 0.0004 inches.
The second is measuring the angular diameter of the Airy disk. For green light, the diameter of the Airy disk in arc seconds is 260 divided by the lens or mirror diameter in millimeters. For example, the Airy disk’s angular size for a 150mm diameter lens or mirror is 260/150= 1.8 arc-seconds. In inches it is 11 divided by the lens or mirror diameter in inches. For example, the Airy disk’s angular size for a 6 inch diameter lens or mirror is 11/6= 1.7 arc-seconds.
How can it be that the linear size of the Airy disk depends only
on the focal ratio? Because that's what remains when the math terms
cancel. But why is that? Let's make up the following example.
Scope 'A' is a 6 inch [15cm] F8,
scope 'B' is a 12 inch [30cm] F4,
scope 'C' is a 6 inch [15cm] F4.
Scopes 'A' and 'B' have the same focal length,
scopes 'A' and 'C' have the same aperture,
scopes 'B' and 'C' have the same focal ratio.
Scope 'B', with twice the aperture resolves twice that of scope 'A' so the Airy disk is half the size of scope 'A'.
Scope 'C' with the same aperture of scope 'A' resolves the same but with half the focal length, the Airy disk diameter is half the size. So I see that the Airy disk diameter depends solely on the focal ratio where faster focal ratios result in smaller Airy disk diameters.
Scope 'B' resolves twice that of scope 'C' but with twice the focal length the Airy disk ends up the same linear size.
How does resolution relate to the Airy disk? Resolution is typically taken as half the Airy disk. For a 15cm, 6 inch lens or mirror, the resolution is 0.8 arc-seconds.
Why do I care about the size of the Airy disk? If I am an imager, then I want to match my camera’s resolution to the telescope’s resolution. I’ll alter the camera or telescope to make the match. If I am a visual observer, then I want to know the resolution limit of my telescope. Visually I will select eyepieces that allow my eye to resolve the telescope’s theoretical resolution limit.
How does the Airy disk play into mirror making? For a smooth mirror the optician wants all the geometric light rays to pass through the Airy disk. The optician needs to know the size of the Airy disk when testing with an artificial star or say the Foucault test.
Who was George Biddell Airy? Sir Airy was an English astronomer and mathematician who became Astronomer Royal, reorganizing the national observatory. He’s famous for many accomplishments including establishing the prime meridian and measuring the density of the Earth. In 1831 he described diffraction of circular apertures and a complete theory of the rainbow. He also discovered the 243 year Venus cycle. Controversially he is blamed for losing the race to discover Neptune to the French.
References https://en.wikipedia.org/wiki/Airy_disk https://en.wikipedia.org/wiki/George_Biddell_Airy