Second- and higher order resonant gratings with gain or loss - Part I: Green's function analysis

This paper addresses a systematic method for obtaining the coupling coeffic ients due to diffraction orders less than the grating order of second- and higher order resonant complex gratings in a multilayer structure. This meth od is based on an improved Green's function, and its distinguishing feature is its use of the transfer matrix method to simplify the theoretical analy sis for obtaining the Green's function in multilayer structures. More impor tantly, it is also shown that by introducing gain or loss in second- and hi gher order resonant gratings, the range of controlling the total coupling c oefficient becomes wider than those in first-order complex gratings or seco nd-order index gratings, Concepts like in-phase and anti-phase gratings are generalized in this type of grating. Finally, guidelines for designing com plex-coupled DFB lasers with second- and third-order grating are also prese nted.