Consider a self-focusing optical fiber discussed in Ex. 7.8, in which the refractive index is

where r = |x|.

(a) Write down the Helmholtz equation in cylindrical polar coordinates and seek an axisymmetric mode for which ψ = R(r)Z(z), where R and Z are functions to be determined, and z measures distance along the fiber. In particular, show that there exist modes with a Gaussian radial profile that propagate along the fiber without spreading.

(b) Compute the group and phase velocities along the fiber for this mode.

Data from Exercises 7.8

Optical fibers in which the refractive index varies with radius are commonly used to transport optical signals. When the diameter of the fiber is many wavelengths, we can use geometric optics. Let the refractive index be

where n₀ and α are constants, and r is radial distance from the fiber’s axis.

n(x) = n(1-ar) 1/2, (8.41)