  In this paper, we consider the inverse problem of recovering a doubly
periodic Lipschitz structure through the measurement of the scattered field
above the structure produced by point sources lying above the structure. The
medium above the structure is assumed to be homogenous and lossless with a
positive dielectric coefficient. Below the structure is a perfect conductor
partially coated with a dielectric. A periodic version of the linear sampling
method is developed to reconstruct the doubly periodic structure using the near
field data. In this case, the far field equation defined on the unit ball of
R^3 is replaced by the near field equation which is a linear integral equation
of the first kind defined on a plane above the periodic surface.
