An Application of Matrix Theory to the Evolution of Coupled Modes

In order to overcome loss in optical fibers, experimentalists are interested in employing parametric amplifiers using four-wave mixing. Upon linearizing the nonlinear Schrodinger equation typically used as a model for such amplifiers, a system of ODEs results for the complex amplitude. The solution can also be expressed as the product of transfer matrices and the initial condition and its conjugate. Physical insight about the fiber-optic system can be gained by examining the theoretical properties of the matrices in the mathematical system. This module, suitable for inclusion in an advanced undergraduate or graduate linear algebra course, explores these properties and should provide a good physical motivation for the theoretical explorations in such a course.