Parallel computing for the three-dimensional spatial spectral volume integral equation method is presented for the computation of electromagnetic scattering by finite dielectric scatterers in a layered medium. The first part exploits the Gabor-frame expansion to compute the Gabor coefficients of scatterers in a parellel manner. The second part concerns the decomposition and restructuring of the matrix-vector product of this spatial spectral volume integral equation into (partially) independent components to enable parallel computing. Both capitalize on the hardware to reduce the computation time by shared-memory parallelism. Numerical experiments in the form of solving electrically large scattering problems, namely volumes up to 1300 cubic wavelengths, in combination with a large number of finite scatterers show a significant reduction in wall-clock time owing to parallel computing, while maintaining accuracy.
2. Dasari, P., J. Li, J. Hu, Z. Liu, O. Kritsun, and C. Volkman, "Scatterometry metrology challenges of EUV," Proc. of SPIE, Vol. 8324, 83240M, 2012.
doi:10.1117/12.916006
3. Ansuinelli, P., W. M. J. Coene, and H. P. Urbach, "Automatic feature selection in EUV scatterometry," Applied Optics, Vol. 58, No. 22, 5916-5923, 2019.
doi:10.1364/AO.58.005916
4. Kumar, N., P. Petrik, G. K. P. Ramanandan, O. El Gawhary, S. Roy, S. F. Pereira, W. Coene, and H. Urbach, "Reconstruction of sub-wavelength features and nano-positioning of gratings using coherent Fourier scatterometry," Optics Express, Vol. 22, No. 20, 24678-24688, 2014.
doi:10.1364/OE.22.024678
5. Gross, H., A. Rathsfeld, F. Scholze, and M. Bar, "Profile reconstruction in extreme ultraviolet (EUV) scatterometry: Modeling and uncertainty estimates," Meas. Sci. Technol., Vol. 20, No. 10, 105102, 2009.
doi:10.1088/0957-0233/20/10/105102
6. Dilz, R. J., M. G. M. M. van Kraaij, and M. C. van Beurden, "A 3D spatial spectral integral equation method for electromagnetic scattering from finite objects in a layered medium," Opt. Quant. Electron., Vol. 50, No. 206, 2018.
7. Dilz, R. J., A spatial spectral domain integral equation solver for electromagnetic scattering dielectric layered media, Ph.D. Dissertation, Eindhoven University of Technology, 2017.
8. 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.
doi:10.1007/BF01389711
9. Eijsvogel, S., L. Sun, F. Sepehripour, R. J. Dilz, and M. C. van Beurden, "Describing discontinuous finite 3D scattering objects in Gabor coefficients: Fast and accurate methods," J. Opt. Soc. Am. A, Vol. 39, No. 1, 86-96, 2022.
doi:10.1364/JOSAA.438866
10. Dilz, R. J. and M. C. van Beurden, "Computational aspects of a spatial-spectral domain integral equation for scattering by objects of large longitudinal extent," 2017 International Conference on Electromagnetics in Advanced Applications (ICEAA), 637-640, 2017.
doi:10.1109/ICEAA.2017.8065327
11. Levinson, H. J., Principles of Lithography, 4th Ed., SPIE Press, 2019.
doi:10.1117/3.2525393
12. Bakshi, V., EUV Lithography , 2nd Ed., SPIE Press, 2018.
doi:10.1117/3.2305675
13. Van den Berg, P. M., Forward and Inverse Scattering Algorithms Based on Contrast Source Integral Equations, 1st Ed., Wiley, 2021.
doi:10.1002/9781119741602
14. Solis, D., F. Obelleiro, and J. Taboada, "Surface integral equation-domain decomposition scheme for solving multiscale nanoparticle assemblies with repetitions," IEEE Photonics Journal, Vol. 8, No. 5, 1-14, 2016.
doi:10.1109/JPHOT.2016.2614895
15. Chanaud, M., L. Giraud, D. Goudin, J. J. Pesque, and J. Roman, "A parallel full geometric multigrid solver for time harmonic Maxwell problems," SIAM J. Sci. Comput., Vol. 36, No. 2, C119-C138, 2014.
doi:10.1137/130909512
16. Ruiz-Cabello, M., M. Abelenkovs, L. M. Diaz Angulo, C. Cobos Sanchez, F. Moglie, and S. G. Salvadore, "Performance of parallel FDTD method for shared-and distributed-memory architectures: Application to bioelectromagnetics," PLOS ONE, Vol. 15, No. 9, e0238115, 2020.
doi:10.1371/journal.pone.0238115
17. Vaccari, A., A. Lesina, L. Cristoforetti, and R. Pontalti, "Parallel implementation of a 3D subgridding FDTD algorithm for large simulations," Progress In Electromagnetics Research, Vol. 120, 263-292, 2011.
doi:10.2528/PIER11063004
18. Pan, X. M., W. C. Pi, and X. Q. Sheng, "On OpenMP parallelization of the multilevel fast multipole algorithm," Progress In Electromagnetics Research, Vol. 112, 199-213, 2011.
doi:10.2528/PIER10120802
19. Yang, M. L., H. W. Gao, X. M. Sun, and X. Q. Sheng, "Fast domain decomposition methods of FE-BI-MLFMA for 3D scattering/radiation problems (invited paper)," Progress In Electromagnetics Research, Vol. 155, 39-52, 2016.
doi:10.2528/PIER15102802
20. MacKie-Mason, B., A. Greenwood, and Z. Peng, "Adaptive and parallel surface integral equation solvers for very large-scale electromagnetic modeling and simulation (invited paper)," Progress In Electromagnetics Research, Vol. 154, 143-162, 2015.
doi:10.2528/PIER15113001
21. Wait, J. R., Electromagnetic Waves in Stratified Media, Pergamon Press, 1970.
22. Dilz, R. J. and M. C. van Beurden, "A domain integral equation approach for simulating two dimensional transverse electric scattering in a layered medium with a Gabor frame discretization," Journal of Computational Physics, Vol. 345, 528-542, 2017.
doi:10.1016/j.jcp.2017.05.034
23. Dilz, R. J., M. G. M. M. van Kraaij, and M. C. van Beurden, "The 2D TM scattering problem for finite objects in a dielectric stratified medium employing Gabor frames in a domain integral equation," Journal of the Optical Society of America A, Vol. 8, No. 34, 1315-1321, 2017.
doi:10.1364/JOSAA.34.001315
24. Barzegar, E., S. J. L. Eijndhoven, and M. C. van Beurden, "Scattered field in random dielectric inhomogeneous media: A random resolvent approach," Progress In Electromagnetics Research B, Vol. 62, 29-47, 2015.
doi:10.2528/PIERB14111304
25. Van Beurden, M. C. and I. D. Setija, "Local normal vector field formulation for periodic scattering problems formulated in the spectral domain," J. Opt. Soc. Am. A, Vol. 34, No. 2, 224-234, 2014.
doi:10.1364/JOSAA.34.000224
26. Li, L., "Use of fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A, Vol. 13, 1019-1023, 1996.
27. Popov, E. and M. Neviere, "Maxwell equations in fourier space: Fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media," J. Opt. Soc. Am. A, Vol. 18, 1019-1023, 2001.
28. Bastiaans, M., "Gabor's expansion and the Zak transform for continuous-time and discrete-time signals: Critical sampling and rational oversampling," EUT Report. E, Vol. 95-E-295, Fac. of Electrical Engineering, Eindhoven University of Technology, 1995.
29. Strohmer, T., "Approximation of dual gabor frames, window decay, and wireless communications," Applied and Computational Harmonic Analysis, Vol. 11, No. 2, 243-262, 2001.
doi:10.1006/acha.2001.0357
30. Janssen, A. and P. Sondergaard, "Iterative algorithms to approximate canonical gabor windows: Computational aspects," Journal of Fourier Analysis and Applications, Vol. 31, No. 1, 211-241, 2007.
doi:10.1007/s00041-006-6069-y
31. Janssen, A., "Some Weyl-Heisenberg frame bound calculations," Indagationes Math, Vol. 7, No. 7, 165-183, 1996.
doi:10.1016/0019-3577(96)85088-9
32. Jenkins, W., Digital Signal Processing Handbook, CRC, 2010.
33. Christensen, O., Frames and Bases: An Introductory Course, Birkhauser, 2008.
doi:10.1007/978-0-8176-4678-3
34. Von Praun, C. and D. Padua, Encyclopedia of Parallel Computing, Springer, 2011.
35. Yang, C., Introduction to GIS Programming and Fundamentals with Python and ArcGIS, CRC, 2017.
doi:10.1201/9781315156682
36. Hermanns, M., Parallel programming in Fortran 95 using OpenMP, 2002, https://www.openmp.org/wp-content/uploads/F95_OpenMPv1 v2.pdf.
37. Sleijpen, G. L. G. and D. R. Fokkema, "BiCGstab(ℓ) for linear equations involving unsymmetric matrices with complex spectrum," Electron. Trans. Numer. Anal., Vol. 1, No. 1, 11-32, 1993.
38. Chandra, R., R. Menon, L. Dagum, D. Kohr, D. Maydan, and J. Mcdonald, , Parallel Programming in OpenMP, Morgan Kaufmann, 2001.
39. Hugonin, J. P. and P. Lalanne, RETICOLO software for grating analysis, 2022, https://arxiv.org/abs/2101.00901.
40. EMVA, Standard for characterization of image sensors and cameras, 2016, https://www.emva.org/wp-content/uploads/EMVA1288-3.0.pdf.
41. Yu, C.-Y., C.-Y. Lin, S.-C. Yang, and H.-Y Lin, "Eight-scale image contrast enhancement based on adaptive inverse hyperbolic tangent algorithm," MDPI Journal of Algorithms, Vol. 7, 597-607, 2014.
doi:10.3390/a7040597
42. Versaci, M., F. C. Morabito, and G. Angiulli, "Adaptive image contrast enhancement by computing distances into a 4-dimensional fuzzy unit hypercube," IEEE Access, Vol. 5, 26922-26931, 2017.
doi:10.1109/ACCESS.2017.2776349
43. Kulkarni, A., F. Franchetti, and J. Kovacevic, "Algorithm design for large scale FFT-based simulations on CPU-GPU platforms," 47th International Conference on Parallel Processing, 2018.
44. Hanounik, B. and X. Hu, "Linear-time matrix transpose algorithms using vector register file with diagonal registers," Proceedings 15th International Symposium on Parallel and Distributed Processing, 8, 2001.
doi:10.1109/IPDPS.2001.924973
Submit
