Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings

We demonstrate a new inverse scattering algorithm for reconstructing the structure of highly reflecting fiber Bragg gratings. The method, called integral layer-peeling (ILP), is based on solving the Gel´fand-Levitan-Marchenko (GLM) integral equation in a layer-peeling procedure. Unlike in previously published layer-peeling algorithms, the structure of each layer in the ILP algorithm can have a nonuniform profile. Moreover, errors due to the limited bandwidth used to sample the reflection coefficient do not rapidly accumulate along the grating. Therefore, the error in the new algorithm is smaller than in previous layer peeling algorithms. The ILP algorithm is compared to two discrete layer-peeling algorithms and to an iterative solution to the GLM equation. The comparison shows that the ILP algorithm enables one to solve numerically difficult inverse scattering problems, where previous algorithms failed to give an accurate result. The complexity of the ILP algorithm is of the same order as in previous layer peeling algorithms. When a small error is acceptable, the complexity of the ILP algorithm can be significantly reduced below the complexity of previously published layer-peeling algorithms.