Inverse scattering for the one-dimensional Helmholtz equation: fast numerical method

The inverse scattering problem for the one-dimensional Helmholtz wave equation is studied. The equation is reduced to a Fresnel set that describes multiple bulk reflection and is similar to the coupled-wave equations. The inverse scattering problem is equivalent to coupled Gel'fand-Levitan-Marchenko integral equations. In the discrete representation its matrix has Töplitz symmetry, and the fast inner bordering method can be applied for its inversion. Previously the method was developed for the design of fiber Bragg gratings. The testing example of a short Bragg reflector with deep modulation demonstrates the high efficiency of refractive-index reconstruction.