Numerical Aperture: A relationship between the acceptance angle and the refractive indices of the three media involved, namely the core, cladding and air (or water).
A light ray incident on the fiber at an angle theta one to the fiber axis which is less than the acceptance angle for the fiber theta a. The light ray enters the fiber from a medium (air) of refractive index n0, and the fiber core has a refractive index n1, which is slightly greater than the cladding refractive index n2. Assuming the entrance face at the fiber core to be normal to the axis, then considering the refraction at the air-core interface and using Snell’s law given by n0 sin theta1=n1 sin theta 2
and we use the triangle ABC property, simply it and find the result keeping in mind the limiting case for total internal reflection, where phi angle becomes equal to the critical angle for the core cladding interface.
Also in this limiting case theta 1 becomes the acceptance angle for the core theta a, now we combine these two concepts and find the result for Numerical Aperture.
Since the Numerical Aperture of Optical Fiber is often used with the fiber in air where n0 is the unity, it is simply equal to sin theta a. It may also be noted that incident meridional rays over the range 0 less than or equal to theta 1 less than or equal to theta a will be propagated within the fiber.