Well the OP doesn’t say anything much about the nature of the “fiber optic” cable. There’s a whole range of things describable as fiber optic cables, from cut up fishing lines, glued as a bundle onto an LED to make an ornament, up to transoceanic single mode high bandwidth cables. Let’s assume (for different reasons) that we are not talking about either of those; the latter because it’s quite complex, and the former is just a toy.
So let’s consider just cables of up to a few meters, intended for practical signal transmission, from audio signals in hi-fi equipment of common computer data signal (digital). The shorter multimode types could be made from plastic fibers, but these are fairly lossy. More serious multi-mode cables would be glass fibers. For a glass refractive index of 1.5, the critical angle for Total Internal Reflection (TIR) at the glass-air interface would be 41.8 degrees (from the surface normal) or 48.2 degrees from the fiber axis. Light at this angle would have the longest possible path of 1/cos48.2.
Well this comes out to 1.5 times the fiber length; exactly the refractive index (no surprise). On top of that you would have the light speed slowdown, also a factor of 1.5, so the longest travel time would be the square of the index times the cable length divided by c of course. So most of the light would take between a factor of n and a factor of n^2 times the air length time.
Well there’s a complication I haven’t mentioned. Serious fiber optic cables ; even multimode fibers made of glass, are almost certain to be “clad” fibers; consisting of a core glass with a refractive index of perhaps 1.6, coated with an outer layer of lower index, perhaps 1.5. The OP may want to research what actual values are found in available FO cables. This cladding separates the environment and contamination from the reflecting interface that confines the radiation. So the light is travelling in the core glass, of index 1.6, but the effective critical angle is now arcsin (1.5/1.6)=69.6 degrees, so the maximum path length is now 1.6/1.5 (1.0667) times the fiber length, not 1.5 times.
So the angles shown in OP’s diagram have greatly exaggerated off axis paths, and the path extension is only a few percent. But the higher core index, also means a slower light speed in the fiber.
You can actually determine much of this without knowing the fiber index, if the manufacturer specifies the “numerical aperture” of the fiber, which is a measure of the maximum acceptance angle of the cone of light which can propagate along the cable.
I don’t want to belabor this too much, because there are so many variables. Unclad fibers are too sensitive to contamination to be of much use. Even touching them with your fingers and getting sweat on the surface, will cause serious loss of light out the side of the fiber.