Comparison of Vector and Scalar Modes in a Lightguide with a HyperbolicSecant Index Distribution

As is well known, the index distribution N(x) = N0 sech gx (1) has an ideal focusing property for a two-dimensional medium.1"4 The group velocities of the various modes are equalized, and if the medium is dispersion-free the group velocities are independent of frequency as well. It is believed that, for a cylindrical lightguide where the index is a function of the radius, such an optimum distribution does not exist.1 In particular, a distribution of the sort N(r) = No sech gr, (2) where r is the radius, will not be expected to have ideal focusing properties. In fact, in the presence of material dispersion, the distribution described by eq. (2) is expected to be similar to a parabolic distribution. The properties of the distribution described by eq. (2) can most easily be inferred from an analysis of the scalar modes in an infinite lens-like medium.1,5"9 The purpose of this work is to review these 1681 properties and then compare them with an exact numerical analysis of the vector modes in a self-focusing fiber with an infinite homogeneous cladding.