Lineshape and polarization mode dispersion of waves diffracted by apodized and chirped fiber Bragg gratings in reflection

The amplitude and phase of a light wave propagating in an apodized and chirped Bragg grating is given by the superposition of two orthogonal coupled wave functions that are the solution of a non-linear Riccati equation. An analytical solution, in the form of an integral transform, that predicts the amplitude, phase, and intensity profile of the light diffracted by grating, in which the local Bragg wavelength is asymmetric, and the forward and counter propagating modes are non-degenerate is presented. For even apodization and odd chirp the lineshape has an asymmetrical Gaussian form the result of mixing of the real and imaginary components of the apodized grating wave function. The effect of chirping is to shift the resonance wavelength, increase the grating bandwidth and polarization mode dispersion. The dependence of linewidth and dispersion on parameters that alter apodization and chirp after fabrication, is also discussed.