This is so because physical boundary conditions have to be neglected with a scalar approach. Real electromagnetic corrugations are vectorial waves. As structure sizes decreases particularly in semiconductor technology, patterns with dimensions comparable to the wavelenght of visible light become focus of research. There interaction effects of light and structure strongly influence diffraction spectra of reflected and trasmitted light. These contributions can be merely considered by rigorous calculations applying full vectorial Maxwell equations. In contrast to scalar optics there are only few analytical solutions for rigorous diffraction so you have to use numerical methods in practice. ITO has been active in the field of rigorous numerical simulation of diffraction on periodic structures since end of 1990. Since then, our simulation tool MICROSIM, which is powered by a rigorous coupled wave approximation (RCWA), has been continuously used and improved.
At the beginning MICROSIM was used for simulating high resolution microscopy but in recent years it was adopted also for diffractometry. Now diffraction of cross gratings with arbitrary unit cells and unequal periodicity in both directions is possible. Furthermore convergence enhancement methods are implemented which are directly based on boundary conditions of the Maxwell equations. Calibrated on the basis of various simulation technologies and tested by the use in numerous projects MICROSIM is well-proven. It is used by national and international companies and research facilites for R&D by now. Beside RCWA finite elements (FE) and finite difference time domain methods (FDTD) are applied as well in our workgroup.
MICROSIM is based on a Fourier expansion of electromagnetic fields in x and y direction in the presence of periodic structures and a fragmentation with rectangular cuboids of constant refraction index in z direction. In this way a solution of the Maxwell equations can be achieved by eigenvalue decomposition. The simulation results give a complete characterisation of light-structure interaction. These results will be processed further in order to represent the full microscopic imaging.
Simulation results
The simplified procedure is done in the following way: The results of diffraction calculation, which includes light from various directions on the illumination pupil, is recomposed after diffraction according to Abbe's theory of image formation. You can get either coherent or incoherent microscopic images. As the complete information of the structure is available in the exit pupil, various microscopic imaging techniques can be applied by manipulating diffraction data prior to recomposing the image. Besides polarisation resolved optical imaging, model based reconstruction techniques like diffractometry or scatterometry become focus of interest. These methods are used for nanostructure metrology by fitting measured diffraction spectra to simulated ones. It's possible to get detailed information about structure size if the profile of the structure is a priori known.
References
- M. Totzeck, "Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields", Optik, 112 (2001) 381-390
- Reinig P., Dost R., Mört M., Hingst, T., Mantz U., Schuster, T. Kerwien, N., Kaufmann J., Osten W.: "Potential and limits of scatterometry: A study on bowed profiles and high aspect ratios", Scatterometry Workshop 2004, 3.-5.5.2004 Porquerolles, Frankreich
- R. Berger, J. Kauffmann, N. Kerwien, W. Osten, H.J. Tiziani: Rigorose Beugungssimulation: Ein Vergleich zwischen RCWA, DTD und der Finiten Elemente Methode, 105. DgaO-Tagung 2004 P59
- Kerwien N., Schuster T., Rafler S., Osten W., "Semi-rigorous Diffraction Theory: Realization of Classical Concepts in the Framework of Electrodynamics", J. Opt. Soc. Am. A 24 (2007) No. 4 1074-1084