You are seeing the effects of polycrystalline diffraction. If you were to pass the light through a single crystal, you would see something like a spot or line pattern (kikuchi bands).

Because the orientation of the ice crystals are random in the atmosphere, you get a random orientation during the light scattering. summing up all these many spot patterns from each individual scattering at random rotations gives you a ring.

You could emulate the process with a stencil of dots at whatever pattern you like. Rotate it randomly and then draw in the dots from the stencil. If you do this enough, you will get a ring.

Technically there are many processes going on. Single scattering (which produces the dot pattrn) requires very small amounts of material. As the material (here ice) gets thicker, you will see lines forming (multiple scattering).