Design and characterization of finite-length fiber gratings

A rigorous analysis of the response of fiber Bragg gratings of finite length is presented. For the discrete grating model, we find necessary and sufficient conditions for the response to be realizable as a grating of finite length. These conditions are used to develop a general method for designing gratings with a prescribed length. The design process is divided into two parts. First, we find a realizable reflection spectrum which approximates the target spectrum. Once the spectrum is found, one can determine the associated grating profile by straightforward layer-peeling inverse-scattering or transfer matrix factorization methods. As an example, a dispersionless bandpass filter is designed and compared to the results when the layer-peeling algorithm is applied directly to a windowed impulse response. We also discuss potential applications to grating characterization including regularization and finding the absolute reflection spectrum from a measured, normalized version.