Inverse problems for scattering by periodic structures (Q1905666)

The authors consider the following two-dimensional inverse scattering problem which aims to determine the location of the interfaces \(S_1\) and \(S_2\) if the total field is known on the lines \(x_2= b\) and \(x_2= -b\): \[ [\Delta+ k^2(x_1, x_2)] u(x_1, x_2)= 0,\quad x_{1, 2}\in (- \infty, \infty), \] \[ k^2(x_1, x_2)= k_1\text{ above the line } S_1, = k_2\text{ below the line } S_2, = k_0\text{ between } S_1\text{ and } S_2, \] where \(k_0\), \(k_1\) and \(k_2\) are known (complex) constants. They consider the case, where the shapes of the interfaces \(S_1\) and \(S_2\) are \(2\pi\)-periodic with respect to \(x_1\) and the incident field is a linearly polarized plane wave. The main result of the paper is that if \((\widetilde S_1, \widetilde S_2)\) is another pair of periodic interfaces close to \((S_1, S_2)\), then, for any \(\delta> 0\), if the measurements for these two pairs are \(\delta\)-close, then \((\widetilde S_1, \widetilde S_2)\) is \(O(\delta)\)-close to \((S_1, S_2)\). The cases where \(u\) is the unique nonzero component of the electric or magnetic field are treated separately. Moreover, the cases where \(k_0= k_2\) and \(k_0= k_2= \infty\) are also considered. Notice that the introduction of some conditions, for example \(T^\alpha_1\) in (1.7) and \(B_1\) in (1.11) are not clear.