Calderón Preconditioned Spectral-Element Spectral-Integral Method for Doubly Periodic Structures in Layered Media

An efficient and accurate spectral-element spectral-integral method is developed for rigorously modeling arbitrary inhomogeneous objects with doubly periodicity embedded in a layered medium background. It is an improvement over the traditional finite-element boundary-integral (BI) method. In the proposed approach, the domain decomposition between the interior spectral-element domain and the outer surface BI domain enables the BI domain to be solved by a far more efficient spectral integral method using a uniform mesh; without any loss of generality, it leaves the interior spectral-element domain with flexibility to model any inhomogeneous objects. For the BI domain, with a uniform mesh and rooftop basis functions, we can exploit the Toeplitz structure of the BI matrices with the fast Fourier transform algorithm and apply the Calderón preconditioner to achieve a high iteration convergence rate that will not slow down as the problem gets larger. With the periodic layered medium Green's functions, the layers without objects can be treated as background and the discretization is avoided. We show that this method is very promising for large-scale problems, because the CPU time and memory required with respect to the number of unknowns grow slowly. We validate this method with a commercial finite element solver and show its ability by solving a large-scale optical lithography problem.