The formulation of a matrix-vector product with linear complexity for layer-coupling is discussed in the context of scattering by periodic dielectric scatterers embedded in a layered medium and formulated as a spectral-domain volume integral. It is shown how a traditional formulation in terms of reflection and transmission coefficients can be modified to arrive at an algorithm of linear complexity if used as a matrix-vector product. The computational performance scheme is demonstrated for stacks in which scattering objects are distributed over hundreds of layers.
2. Micron, 176-Layer NAND, https://www.micron.com/products/nand-flash/176-layer-nand, 2020,[Online, Accessed June 2021].
3. SK hynix newsroom, "SK hynix Unveils the Industry's Most Multilayered 176-Layer 4D NAND Flash,", https://news.skhynix.com/sk-hynix-unveils-the-industrys-highest-layer-176-layer-4d-nand-flash/, 2020, [Online, Accessed June 2021].
4. Lapedus, M., "3D NAND flash Wars Begin," Semiconductor Engineering, Aug. 2018. [Online, Accesed June 2021]..
5. Maas, J., M. Ebert, K. Bhattacharyya, H. Cramer, A. Becht, S. Keij, R. Plug, A. Fuchs, M. Kubis, T. Hoogenboom, and V. Vaenkates, "YieldStar: A new metrology platform for advanced lithography control," Proc. SPIE. 7985, 27th European Mask and Lithography Conference, 2011.
doi:10.1109/TAP.1966.1138693
6. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, No. 3, 302-307, 1966.
doi:10.1364/JOSA.72.001385
7. Moharam, M. G. and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am., Vol. 72, No. 10, 1385-1392, 1982.
doi:10.1364/JOSAA.13.001870
8. Li, L., "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A., Vol. 13, No. 9, 1870-1876, 1996.
doi:10.1364/JOSAA.14.002758
9. Li, L., "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A., Vol. 14, No. 10, 2758-2767, 1997.
doi:10.1109/TAP.2008.2009695
10. Valerio, G., P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, "Regularization of mixed-potential layered-media Green's functions for efficient interpolation procedures in planar periodic structures," IEEE Transactions on Antennas and Propagation, Vol. 57, 122-134, 2009.
doi:10.2528/PIERB15071703
11. Jorna, P., V. Lancellotti, and M. C. van Beurden, "Computational aspects of 2D-quasi-periodic-Green-function computations for scattering by dielectric objects via surface integral Equations," Progress In Electromagnetics Research PIER B, Vol. 63, 49-66, 2015.
doi:10.1364/JOSAA.23.000638
12. Chang, Y. C., G. Li, H. Chu, and J. Opsal, "Efficient finite-element Green's function approach for critical-dimension metrology of three-dimensional gratings on multilayer films," J. Opt. Soc. Am. A, Vol. 23, No. 3, 638-645, 2006.
doi:10.1109/TAP.2005.861527
13. Magath, T., "Coupled integral equations for diffraction by profiled, anisotropic, periodic structures," IEEE Transactions on Antennas and Propagation, Vol. 54, No. 2, 681-686, 2006.
doi:10.1364/JOSAA.27.000308
14. Shi, Y. and C. H. Chan, "Multilevel Green's function interpolation method for analysis of 3-D frequency selective structures using volume/surface integral equation," J. Opt. Soc. Am. A, Vol. 27, 308-318, 2010.
doi:10.1364/JOSAA.28.002269
15. Van Beurden, M. C., "Fast convergence with spectral volume integral equation for crossed block-shaped gratings with improved material interface conditions," J. Opt. Soc. Am. A, Vol. 28, No. 11, 2269-2278, 2011.
doi:10.2528/PIERB11100307
16. Van Beurden, M. C., "A spectral volume integral equation method for arbitrary bi-periodic gratings with explicit Fourier factorization," Progress In Electromagnetic Research B, Vol. 36, 133-149, 2012.
doi:10.1109/8.52240
17. Michalski, K. A. and D. Zheng, "Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part 1: Theory," IEEE Transactions on Antennas and Propagation, Vol. 38, No. 3, 335-344, 1990.
doi:10.1109/8.558666
18. Michalski, K. A. and J. R. Mosig, "Multilayered media Green's functions in integral equation formulations," IEEE Transactions on Antennas and Propagation, Vol. 45, No. 3, 508-519, 1997.
doi:10.1109/9780470546307
19. Felsen, L. B. and N. Marcuvitz, Radiation and Scattering of Waves, Wiley-IEEE Press, 1994.
doi:10.1007/BF01389711
20. Gohberg, I. and I. Koltracht, "Numerical solution of integral equations, fast algorithms and Krein-Sobolev equation," Numerical Mathematics, Vol. 47, No. 2, 237-288, 1985.
21. Van Beurden, M. C., T. Zacharopoulou, A. Roc'h, and M. G. M. M. van Kraaij, "A pseudo-spectral longitudinal expansion in a spectral domain integral equation for scattering by periodic dielectric structures," Proceedings of the 2015 International Conference on Electromagnetics in Advanced Applications (ICEAA), 1419-1422, Torino, Italy, September 7-11, 2015.
22. Kreyszig, E., Advanced Engineering Mathematics, John Wiley & Sons, 1988.
doi:10.1364/JOSAA.5.001863
23. Ko, D. Y. K. and J. R. Sambles, "Scattering matrix method for propagation of radiation in stratified media: Attenuated total reflection studies of liquid crystals," J. Opt. Soc. Am. A, Vol. 5, No. 11, 1863-1866, 1988.
24. Van Kraaij, M. G. M. M., Forward diffraction modelling: Analysis and application to grating reconstruction, Ph.D. thesis, Ch. 5, Eindhoven University of Technology, 2011.
25. Karawia, A., "A computational algorithm for solving periodic penta-diagonal linear systems," Appl. Math. Comput., Vol. 174, No. 1, 613-618, 2006.
Submit
