Calculation of optical-waveguide grating characteristics using Green's   functions and the Dyson's equation
  We present a method for calculating the transmission spectra, dispersion, and time delay characteristics of optical-waveguide gratings based on Green's functions and Dyson's equation. Starting from the wave equation for transverse electric modes we show that the method can solve exactly both the problems of coupling of counter-propagating waves (Bragg gratings) and co-propagating waves (long-period gratings). In both cases the method applies for gratings with arbitrary dielectric modulation, including all kinds of chirp and apodisation and possibly also imperfections in the dielectric modulation profile of the grating. Numerically, the method scales as O(N) where N is the number of points used to discretize the grating along the propagation axis. We consider optical fiber gratings although the method applies to all 1D optical waveguide gratings including high-index contrast gratings and 1D photonic crystals. 
Lars Rindorf and Niels Asger Mortensen