Bessel function: The functions are finite for all values of .

Neumann Function: The functions start from at and have finite value for all other values of r.

For the core represents the axis of fiber. Therefore if Neumann function is chosen as a solution, the field strength would be at the axis of the fiber which is inconsistent with the physical conditions. The fields must be finite all over the cross section of the core. So the Neumann functions cannot be the solution if point is included in the region under consideration.

Therefore we conclude that only is the appropriate solution for the modal fields inside the core of an optical fiber.

cladding. We hence should have

where is the refractive index of the cladding.
Since is negative, let us define a real quantity such that .

Let us now look at the modified Bessel's functions, as shown in figures 6 & 7. For modified Bessel's functions of the kind, as increases, that is, as we move away from the axis of the fiber the field monotonically increases and when field goes to infinity. Since the energy source is inside the core, the fields cannot grow indefinitely away from the core. The only acceptable situation is that the field decays away from the core i.e., for larger values of . This behavior is correctly given by the Modified Bessel function of second kind, .
So we conclude that the modified Bessel function of 1 st kind is not appropriate solution in the cladding. The correct solution would be only Modified Bessel function of 2 nd kind, .