Modelling Axi-symmetric Travelling Waves in a Dielectric with Nonlinear Refractive Index

We consider an isotropic dielectric with a nonlinear refractive index. The medium may be inhomogeneous but its spatial variation has an axial symmetry. We characterize all monochromatic axi-symmetric travelling waves as solution of a system of six second order differential equations on (0, ∞). Boundary conditions at 0 ensure the regularity of the fields on the axis. Guided waves satisfy additional conditions at ∞. Special solutions of this system correspond to what are normally referred to as TE and TM modes.