Inverse and optimal design problems in diffractive optics

Consider a time-harmonic electromagnetic plane wave incident from the top on a periodic structure in R(3). The periodic structure separates two regions. Above the structure, the index of refraction k is assumed to be a fixed constant. Below the structure is a perfectly reflecting material. Given the incident field, the inverse diffraction problem is to determine the periodic structure or the shape of the interface from the scattered field. The optimal design problem in this context is to create grating profiles which yield some specified diffraction patterns. In this paper, recent significant progress on the characterization of uniqueness and stability for the inverse problem is discussed. Optimal design of diffractive structures is also addressed. The paper is concluded with discussions on future research directions in the field.