Matrix method for analysis of optical waveguides and quantum well structures

Abstract

The one dimensional wave equation $$ \frac{{d^2 \Psi }} {{dx^2 }} + \Gamma ^2 (x)\Psi (x) = 0 $$ appears in quantum mechanics, optical waveguide theory, plasma physics and many other diverse areas. For example, in waveguide theory $$ \Gamma ^2 (x) = [k_0^2 n^2 (x) - \beta ^2 ] $$ where k 0 is the free space wave number, n(x) is the refractive index distribution and β is the propagation constant. In quantum mechanics $$ \Gamma ^2 (x) = \frac{{2\mu }} {{n^2 }}[E - V(x)] $$ where the symbols have their usual meaning. In this paper we will describe the matrix method for the solution of the wave equation and apply it to obtaining the eigenvalues. The method can also be used for the analysis of leaky structures.