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This paper concerns the problem of designing a periodic interface between two homogene0us 'materials in such a way that normally incident time-harmonic plane waves scattered from the structure have a specified far-field phase and intensity pattern. The motivation comes from applications in micro-optics where such structures are increasingly used in advanced devices. The design problem can be formulated in various ways as a mathematical optimization problem, with the Helmholtz equation as the underlying model for wave propagation. We consider a "relaxed" formulation of the design problem in which "mixtures" of the two materials are included as admissible designs. A key feature of the problem is the extreme ill-posedness: we are trying to determine a function (the refractive index) from a few scalar values (the far-field pattern). In principle there are infinitely many solutions. However, in contrast to the typical situation in inverse problems, for the design problem non-uniqueness and instability are in some sense an asset: they allow some flexibility to choose designs which are more desirable from an engineering point of view. Exploiting this flexibility presents two primary challenges: first, to characterize the source of the ill-posedness and second, to devise appropriate computational schemes. We briefly discuss both issues and present a particular computational scheme--based on the minimization of the total variation of the design-along with some numerical results.
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