We will consider two dimensional periodic structures with the
homogeneous Dirichlet boundary condition. We discuss the
determination of shapes of structures by near-field observations of the
scattered fields generated by incident plane waves.
Then the governing equation of the scattered field is
the Helmholtz equation and we attach also quasiperiodicity
and a radation condition to a solution.
We consider the Lipschitz curves as well as smooth
ones as periodic structures
and consider the determination of the curve by a finite
number of incident waves.
Our main results are
(i) uniqueness in the determination within piecewise
linear curves.
(ii) conditional stability in determination.
(iii) reconstruction by a nonlinear optimization
problem.
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