An asymmetric discrete-time approach for the design and analysis of periodic waveguide gratings

A discrete-time approach is introduced for the analysis of periodic waveguide gratings with gain (or loss) extending concepts developed for transfer matrix and Gel´fand-Levitan-Marchenko (GLM) inverse scattering techniques. The periodic waveguide grating with gain (or loss) is modeled as a lossy layered dielectric that allows for a digital signal processing (DSP) formulation of the forward and inverse scattering problem. It is shown that the DSP forward scattering formulation as an asymmetric two-component wave system is equivalent to the impedance matching matrix method. A numerical example is presented to emphasize this result. The DSP formulation is an exact discrete design, not just an approximation to a continuous design, and includes all multiple reflections, transmission scattering losses, and absorption effects. A comparison of the continuous GLM, discrete GLM, and discrete Krein inverse problem formulations for a medium with gain (or loss) is presented. The discrete lossy formulations generalize previous lossless results and are found from two different types of reflection data. Since slab gratings are discrete (not continuous) structures, the integral equations used to describe the continuous inverse problem are shown to become matrix equations. Thus, our result enables fast algorithms to be used to solve the inverse problem. A fast algorithm is presented allowing for the complete reconstruction of the grating parameters from its two-sided response in a recursive (slab by slab) fashion