Fundamentals of Photonics, 2 Volume Set, 3rd Edition

APPENDIX C MODES OF LINEAR SYSTEMS

This Appendix provides a brief overview of modes of linear systems that are described explicitly by input–output relations that take the form of a matrix or integral operation, or implicitly by a linear ordinary or linear partial differential equation.

Consider first a linear system described by an explicit input–output relation characterized by a linear operator L that operates on an input vector X to generate a corresponding output vector Y:

(C.1-1)

The vector X may be an array of complex numbers, represented by a column matrix, or a complex function of one or more variables. The modes of such a system are those special inputs that remain unaltered (except for a multiplicative constant) upon passage through the system. They thus obey

(C.1-2) Eigenvalue Problem

where q is an index that labels the mode. The vector X_q is known as the eigenvector, and the associated multiplicative constant λ_q, which is generally a complex number, is called the eigenvalue. The condition set forth in (C.1-2) is known as an eigenvalue problem.

Consider next a linear dynamical system whose state is described by N continuous variables constituting a vector X(t). The evolution of any of the N variables of this N-dimensional vector is, in general, dependent on all N