source: Â© 2003 IEEE Journal of Quantum Electronics

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.. [Read more…]

Fig. 1. Reconstructed modulation index n1(z) of a uniform grating with a refractive index modulation amplitude n1=6.5Ã—10âˆ’4, a length of 4 mm, and a maximum reflectivity of 99.99%, calculated using the ILP algorithm (solid line), the FDLP algorithm (dashed line), and iterative solution to the GLM equation with 70 iterations (dotted line). The reflection spectrum was sampled over a bandwidth of 40 nm with a resolution of 0.01 nm. The figure shows that an excellent reconstruction of the grating was obtained using the ILP algorithm, while the FDLP algorithm and the iterative solution to the GLM equation gave a large error. The inset of the figure shows a zoom on the profile close to the input end of the grating.