Scalar and vector mode relations in gradient-index light guides

Abstract

The relations between scalar-wave-equation modes, which represent approximate solutions for the transverse Cartesian electric and magnetic fields Ex, Ey, Hx, Hy, and the modes that are solutions of a pair of equations that we call the cylindrical scalar equations, which represent approximate solutions for the transverse cylindrical electric and magnetic fields Er, Eθ, Hr, Hθ, are derived. These relations allow construction of the natural vector modes of the inhomogeneous cylindrical geometry, matching of boundary conditions, and loss calculations in cylindrical geometries from knowledge of the conventional scalar-wave-equation solutions.