Appl Math Optim 43:103–116 (2001)
2001 Springer-Verlag New York Inc.
Optimal Design of Periodic Diffractive Structures
and E. Bonnetier
Department of Mathematics, Michigan State University,
East Lansing, MI 48824-1027, USA
Centre de Math´ematiques Appliqu´ees, CNRS UMR 7641,
Ecole Polytechnique, 91128 Palaiseau, France
Communicated by I. Lasiecka
Abstract. The problem of designing a periodic interface between two different
materials, which gives rise to a speciﬁed far-ﬁeld diffraction pattern for a given
incoming plane wave, is considered. The time harmonic waves are assumed to
be TM (transverse magnetic) polarized. The diffraction problem is modeled by a
generalized Helmholtz equation with transparent boundary conditions. In this paper
the design problem is relaxed to include highly oscillatory proﬁles. Existence of
an optimal design is established. The principal method is based on the theory of
homogenization for the model equation.
Key Words. Optimal design, Diffractive optics, Periodic structure, Generalized
Helmholtz equations, Homogenization.
AMS Classiﬁcation. Primary 78A45, 49J20, Secondary 35R30, 35J05.
Weconsiderthe problemof designing aperiodic interface between twodifferentmaterials
which gives rise to a speciﬁed far-ﬁeld diffraction pattern for a given incoming plane
wave. Throughout, the medium is assumed to be nonmagnetic and invariant in the y-
direction. We study the diffraction problem in TM (transverse magnetic) polarization,
This research was supported in part by the CNRS–NSF International Programs INT 98-15798. The
research of the ﬁrst author was also partially supported by NSF Grant DMS 98-03604, and NSF University–
Industry Cooperative Research Programs Grants DMS 98-03809 and DMS 99-72292.