Light rings are sometimes seen when a streetlight or the sun is viewed through wet tree branches and twigs. Scratches on glass can form them. They might even be found in the fine hairs of thistle seeds.
In most cases the scratches or other reflectors are more or less randomly oriented. We see the light rings because only a few select scratches glint light towards our eye. Even then, only a part of a scratch does so. We can think of the sides of a scratch as mirrors.
The positions of the glints are from points on the scratch/mirror perpendicular or tangential to radial lines extending back to the light source (or its direct reflection in for example a bowl).
A mathematical simulation tests this arm waving assertion. At left are computer drawn scratches of random length and orientation shown as dark grey lines. Sections of the scratches ‘light’ where they fulfil the condition for specular reflection towards the eye.
When a reflection was allowed only when the scratch was almost perpendicular to the light/eye axis, the bright streaks were too short. To obtain the longer streaks seen in John’s bowl it was necessary to allow reflections from scratches up to 15° from the perpendicular condition. This is reasonable because the scratches are likely to have imperfections and irregularities that reflect and scatter light over a range of angles. Some scratches can even have sufficient micro irregularities that they give diffraction grating effects. John Adam's rings, above, are not circular because the bowl surface was curved.
Download the light ring simulator to experiment with different conditions. For it to run you need to have previously installed HaloSim, IRIS or TiltingSun to have the necessary files on your PC.