The Inverse Scattering Problem


The reconstruction method uses a layer peeling algorithm, a complete description is
found in reference [1]. This method can be implemented after the problem is
approximated by a series of discrete layers, each with a constant coupling coefficient.
At the beginning of the problem, all these coupling coefficients are unknown. The
method uses an iterative approach, in which the first N layers of the profile are
assumed known, and in the next iteration, the coupling coefficient for the N+1 layer is
deduced. The layers are chosen to be of uniform width so that the time it takes for a
wavefront to cross any layer is a constant, Δ. The method uses the fact that the
impulse response at the time 2(N+1)Δ must be independent of all the layers
following the layer N+1. By the property of causality the impulse response at this
instant must be independent of the coupling coefficients to be found in the layers N+2,
N+3, and so on. On the other hand, the coupling coefficients in the first N layers are
assumed known, so by solving the scattering problem with these N layers, the
impulse response for a grating truncated at the Nth layer can be found. Furthermore,
the truncated impulse response from the first N+1 layers can be found if only the
single coupling coefficient at N+1 were known. The truncated impulse response is
then compared to the desired impulse response at time 2(N+1)Δ. By choice of a
suitable value of coupling coefficient in the N+1 layer, the impulse response of the
truncated grating can be made the same as the desired impulse response, from time
0 up to time 2(N+1)Δ. In this way the coupling coefficients in the first N layers are
used to find the coefficient in the N+1 layer. The unknown layers are “peeled” from the
grating one at a time.