Second- and higher-order resonant gratings with gain or loss-Part 1: Green´s function analysis

We describe a systematic method for obtaining the coupling coefficients due to diffraction orders less than the grating order of second- and higher order resonant complex gratings in a multilayer structure. This method is based on an improved Green´s function, and its distinguishing feature is its use of the transfer matrix method to simplify the theoretical analysis for obtaining the Green´s function in multilayer structures. More importantly, it is also shown that by introducing gain or loss in second- and higher order resonant gratings, the range of controlling the total coupling coefficient becomes wider than those in first-order complex gratings or second order index gratings. Concepts like in-phase and anti-phase gratings are generalized in this type of grating. Finally, guidelines for designing complex-coupled DFB lasers with second- and third-order grating are also presented.