Rainbow Caustics ~ Image by Vincent Favre (Photos Crystal de givre) taken at Drôme, Rhône-Alpes, France    ©Vincent Favre, shown with permission.
Atmospheric
Optics

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To say a rainbow is formed by reflection inside raindrops does not explain its amazing intensity and sharpness because reflected sun rays emerge from raindrops in many directions.  

The bow sharpness and intensity arise because the internally reflected rays cross and cluster to form a caustic sheet where classically the intensity is infinite.    Caustics are regions of space marking discontinuities in ray behaviour.   Crossing rays cluster on one side of a caustic giving intense light that falls off rapidly a short distance from it.

The emerging primary rainbow caustic forms a cone with a half angle of ~42°.    More caustics (thick yellow lines at left) are inside the drop.    They form surfaces of revolution around the red axis.

Most rays emerge from a drop without being reflected.   They cross close to the drop surface but then diverge without forming a distant caustic.   These rays give no rainbow, instead they form a soft zero order glow around the sun.