Inverse problems for scattering by periodic structures

AbstractSuppose the 3-dimensional space is filled with three materials having dielectric constants ɛ1 above S1={x2=f1(x1), x3 arbitrary}, ɛ2 below S2 = {x2 =f2(x1), x3 arbitrary} and ɛo in {f2(x1) <x2 <f1(x1), x3 arbitrary} where f1f2 are periodic functions. Suppose for a plane wave incident on S1 from above we can measure the reflected and transmitted waves of the corresponding time-harmonic solution of the Maxwell equations, say at x2=±b,b large. To what extent can we infer from these measurements the location of the pair (S1, S2 ? In this paper, we establish a local stability: If ( $$\tilde S_1 ,\tilde S_2$$ ) is another pair of periodic curves “close” to (S1, S2), then, for any δ>0, if the measurements for the two pairs are δ-close, then $$\tilde S_1$$ and $$\tilde S_2$$ are 0(δ)-close to S1 and S2, respectively.