Fibers are labeled by their numerical apertures.  The numerical aperture takes into account not only the cone of acceptance, but the effect on that cone of different media outside the fiber.  Let us consider once again the fiber with a core index n₁=1.5 and a cladding index of n₂=1.4.

We can find the cut-off angle θ_0max  by working backward from the point where light strikes the upper edge of the fiber.  In order for total internal reflection (TIR) to occur, the angle at this edge must be greater than the critical angle:

θ'₁ is defined as the angle the light makes with respect to the normal at the entrance (left side) of the fiber.  Looking at the figure, we see that θ'₁ and θ₁ are complementary.  Thus

Snell's law applies at the entrance to the fiber, so

We can use some trig identities (click here for detailed calculation) to arrive at the result

The quantity on the lefthand side of the equation is called the numerical aperture (NA) of the fiber.  Notice that the expression for NA on the lefthand side of the equation seems to depend on the material outside the fiber, but the expression in the center of the equation does not.  Numerical aperture does not depend on the material outside the fiber but depends only on the indices of the core and cladding of the fiber.  If the fiber is placed in a material other than air, the index n₀ will indeed increase, but the cut-off angle θ₀ will decrease to keep the product n₀ sin θ_0maxconstant.  If NA were really dependent upon the material outside of the fiber, it would not be a very good way of categorizing fibers.
 

That's all for this Module!  Click on Next to see a Summary and links to Supporting Materials