Ray invariants and wave equations for transverse modes in three-dimensional graded-index waveguides

A theory of the second ray invariant is proposed using the theory of plane Frenet curves. Its existence requires that the coordinate dependence of the refractive index in the waveguide cross section should satisfy the regularity condition: curves of equal refractive index differ only by an amount which can be obtained using an isotropic scaling transformation. The theoretical conclusions are illustrated using the example of waveguides having the generalized refractive index distribution n ( r ) = n [ (x/ a)^q + (y/ b)^q].