Field sensitivity to L variations of a scatterer

For the problem of diffraction of harmonic scalar waves by a lossless periodic slab scatterer, we analyze field sensitivity with respect to the material coefficients of the slab. The governing equation is the Helmholtz equation, which describes acoustic or electromagnetic fields. The main theorem establishes the variational (Fréchet) derivative of the scattered field measured in the H1 (root-mean-square-gradient) norm as a function of the material coefficients measured in an Lp (p-power integral) norm, with 2 < p < ∞, as long as these coefficients are bounded above and below by positive constants and do not admit resonance. The derivative is Lipschitz continuous. We also make rigorous the variational derivative of the transmitted energy with respect to the material coefficients in Lp, which was previously derived formally by Lipton, Shipman, and Venakides using the adjoint method.