Dielectric-waveguide-modal properties: A new analysis of the one-dimensional wave equation

Abstract

This paper studies the modal character of optical, dielectric waveguides with arbitrary spatial variation (in one dimension) of the dielectric coefficient, a subject vital to the development of optical fibers, strip waveguides, semiconductor lasers, etc. Within (but not limited to) this context, a new mathematical tool is presented for analysis of solutions of the general second-order, linear, homogeneous differential equation, which, when put into its canonical form, is the wave equation: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d^{2}Y/dx^{2} + u(\eta, x)Y = 0</tex> . The central theme is the development and study of the uses of a "mantle function" which simultaneously describes two, complementary solutions. It is shown how the method provides new analytical insights; new, more convenient techniques for numerical evaluation of eigensolutions; and a firm basis for evaluating the merits of existing approximation methods, e.g., the Wentzel-Kramer-Brillouin (WKB) method.