An Exact Numerical Solution to Maxwell's Equations for Lightguides

A possible first step in analyzing production line MCVD (modified chemical vapor deposition) lightguide preforms1 is to calculate the effective indices for a propagating wave in a fiber assumed to be produced from the preform. A knowledge of the effective indices for various wavelengths permits one to calculate the group indices. Using these, we can calculate the group delay of the modes. In this way, we can ascertain the reproducibility of the preforms and ultimately increase yields of the products. The solution of Maxwell's equations for a radially inhomogeneous medium such as is found in lightguides is by no means easy. This is because we must deal with fourth-order differential equations with variable coefficients. Various approximations can, of course, be made.2"8 For example, a popular one is to neglect the gradient of the logarithm of the dielectric constant with respect to radial distance9 and then employ WKB methods.10 These approximations require that discontinuities in index be of little importance and that the refractive index vary slowly with distance. Solutions of this sort have had a substantial impact on lightguide engineering.11 1175