RIGOROUS ANALYSIS OF GUIDED-WAVES IN DOUBLY PERIODIC STRUCTURES

An exact formulation is presented for the boundary-value problem of a class of doubly periodic structures, each of which is composed of a uniform dielectric layer sandwiched between two gratings with two different periods. The general dispersion properties of doubly periodic structures as waveguides are rigorously established and both global and local behaviors of the Brillouin diagram are analyzed to illustrate the guiding characteristics of doubly periodic structures, regardless of the grating profiles. The effects of the commensurability of the two grating periods are carefully studied; in particular, it is shown that the commensurate case will still support purely bounded waves, but the noncommensurate case will not. Furthermore, it is shown that, in a doubly periodic structure, strong couplings may involve two or more space harmonics that may come from more than one mode. Numerical results are given for the special case of a dielectric layer with rectangular corrugations of different periods on its two boundary surfaces. The stop-band behaviors are examined in detail for various cases involving one, two, and three modes. © 1990 Optical Society of America.