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.