Assumed fiber cores are straight and uniform, but they could also be tapered (although not over long distances).
The figure shows what happens to a light ray entering a tapered fiber at an angle θ1. If the ray meets criteria for total internal reflection, it is confined in the core. However, it meets the core-cladding boundary at different angles on each bounce so each total internal reflection is at different angles from the axis. The result is that it emerges from the fiber at a different angle, θ2. If input core diameter is d1 and output core diameter is d2, the relationship between input and output angles is
The same relationship holds for the fiber’s outer diameter as long as core and outer diameter change by the same factor, d1/d2.
As a numerical example, suppose the input angle is 30° and the taper expands diameter by a factor of 2. The sine of the output angle θ2 would be
Thus θ2 would be about 14.5° and light exiting the broad end of a taper would emerge at a smaller angle to the fiber axis than it entered. Conversely, light going from the broad end to the narrow end would emerge at a broader angle.
Tapered bundles of fused fibers can be used as magnifiers if the narrow end is placed on a page and you look at the top side. Each fiber expands or shrinks the spot of the image it transmits by the same amount. The eye sees this as each spot being spread over a larger area at the large end of the taper. This increases the size of the image, but not the clarity,
because the transmitted image has only as many picture elements as the narrow end of the bundle.
*Tapered fibers magnify or demagnify objects seen through them. Tapered fibers are used in bundles.