Shape derivatives for the scattering by biperiodic gratings

Light diffraction by biperiodic grating structures can be simulated by a boundary value problem of the equation ∇×∇×u−k2u=0 for the electric field u. To optimize the geometry parameters of the grating, a quadratic functional of u is defined. The minimization of this functional by gradient based optimization schemes requires shape derivatives of the functional with respect to the geometry parameters. However, a simple application of classical shape calculus is not possible since the energy space for the electric fields is not invariant with respect to the transformation of geometry. In a recent paper, Hettlich (2012) [15] has proposed to replace the electric field by a simple transform which leads to a differentiable vector field in the energy space. We follow here a different approach. For constant magnetic permeability, the magnetic field is piecewise in [H1]3. Applying the shape calculus to the magnetic field equation, substituting the magnetic field by the curl of the electric field, and employing some technical transformations, we derive stable formulas for the material derivatives depending on the electric field. Numerical tests confirm the formulas.