Hence, is also a real quantity albeit it can be positive or negative. In other words, can be real or imaginary depending upon whether is greater or lesser than . Hence for a lossless situation, solution cannot be given by the Hankel functions.

For a lossless case, we have a solution which is a linear combination of either Bessel and Neumann functions or modified Bessel functions of first and second kind.

cladding the field must decay monotonically. Therefore it is obvious that inside the core the Modified Bessel functions is not the proper solution. Only Bessel function or Neumann function could be solutions inside the core.

Let us take dielectric constant for the core. Then we have

For oscillatory type of solution inside the core, is positive. Therefore for a guided mode we must have

i.e.,

Where is the wave number in the free space and is nothing but the wave number in an unbound medium of refractive index .