Characteristic mode theory of coupled waveguide arrays with periodically varying properties

A method is presented for determining the eigenvalues and eigenvectors (characteristic mode shapes) for coupled waveguide arrays in which the interguide coupling is uniform but where the dephasing varies periodically from guide to guide. It is demonstrated that if the dephasing alternates from guide to guide, a quadratic equation must be solved for the eigenvalues, whatever the number of guides. For dephasing functions of larger period, the eigenvalue equation is of correspondingly higher order. The effect of such functions is to insert additional band gaps into the eigenvalue distribution and to couple together spatial harmonics. Numerical examples illustrating the method are presented where relevant.