Modified F-matrix calculation of Bragg grating spectra and its use with a novel nonlinear index growth law

Reflection spectra and index structures created by the growth of fiber Bragg gratings (FBGs) are modeled using a modified piecewise-uniform approach that can accommodate realistic index growth behavior. Because grating formation generally involves nonlinear index growth, models that assume sinusoidal modulation shapes do not accurately predict the evolution of the grating spectra during the writing exposure. The authors first present a generally applicable treatment of arbitrarily shaped index modulations such that their reflection spectra can be accurately treated with an established modeling technique. This approach examines the actual photoinduced index modulation shape at each subregion of the grating and identifies the ac and dc coupling coefficients (from coupled-mode theory) of an equivalent sinusoidal modulation at the fundamental Bragg resonance. These derived coupling coefficients are then used to compute the grating spectrum via the fundamental matrix (F-matrix) method. Given an accurate description of index at each point along the grating, the modified F-matrix method can efficiently model grating spectra that result from complex exposure schemes including scanned exposures, various apodization profiles, chirp, and postexposures with fringeless light. Additionally, this paper presents a method for determining the detailed index profiles formed by arbitrary exposures. To obtain realistic index modulation profiles, a new index growth model consisting of a three-dimensional (3-D) surface of induced index (versus exposure time and intensity) and a rule for linking complex sequences of index growth under differing intensities is introduced. Using the index growth surface, the compound growth rule, and the modified F-matrix technique, the spectra of weak FBGs similar to those found in distributed fiber sensor systems are numerically synthesized.