Spatial spectral Maxwell solver
In electrical engineering, it is necessary to have accurate and reliable insight into the electromagnetic scattering from structures in terms of their material and geometrical properties for applications such as scatterometry for wafer manufacturing or for designing metamaterial lenses. To this end, the spatial spectral Maxwell solver was developed to accurately and rapidly analyze the scattering by complex geometrical dielectric objects in a dielectric layered medium. The core of the spatial spectral method is a Gabor expansion in the plane parallel to the layer interfaces. It allows us to form a domain integral equation that is solved partly in the spatial domain and partly in the spectral domain, thus avoiding the heavy workload of solving tedious Sommerfeld integrals.
In scatterometry for wafer metrology, a Maxwell solver is tasked with computing electromagnetic scattering data from the structures under test. This data forms the basis for estimating the material and geometric properties of the structures under test. Ideally, scatterometry-based computation are executed in a fast and accurate manner. Hence, effective scatterometry hinges on a Maxwell solver that makes use of computationally efficient modeling techniques.
The spatial spectral Maxwell solver is specifically developed for the accurate and rapid analysis of scattering problems by (non-periodic) complex geometrical dielectric objects in a dielectric layered medium: a description that closely matches that of the structures under test in scatterometry. This solver鈥檚 efficiency originates from the Gabor expansion in the plane parallel to the layer interfaces of the layered background medium. It leads to a domain integral equation, which is simultaneously formulated in the spatial and spectral domain. Further, it becomes possible to avoid the tedious Sommerfeld integrals.
Recently, the modeling techniques within spatial spectral Maxwell solver have been improved by parallel computing strategies to significantly reduce the computation time for scattering problems, while maintaining accuracy. We have tested the spatial spectral Maxwell solver with parallel computing capabilities in several test cases. The first figure above contains one of the test cases and it shows a small part of a grating consisting of 529 scattering objects, where there are 23 repeating objects in both the x- and y-directions. This grating has a volume of approximately 1300 times the wavelength and it is illuminated by a plane wave with a wavelength of 13.5 nanometer. In other words, it is an electrically large problem and, in spite of this size, the electromagnetic scattering has to be computed in an accurate manner for all 529 objects.
In the second figure, the wall-clock time is shown for computing the electromagnetic scattering data is shown as a function of CPU cores. Here, the capabilities of the spatial spectral Maxwell solver are clearly shown by reducing the wall-clock time from approximately 4000 seconds to less than 500 seconds, while still providing accurate scattering data as shown in the third figure. This third figures is a diffraction image, which displays the magnitude of the scattering data in log10 scale for various observation kx- and ky-angles.