We describe a surface integral-equation (SIE) method suitable for computation of electromagnetic fields scattered by 2D-periodic high-permittivity and plasmonic scatterers. The method makes use of fast evaluation of the 2D-quasi-periodic Green function (2D-QPGF) and its gradient using a tabulation technique in combination with tri-linear interpolation. In particular we present a very efficient technique to create the look-up tables for the 2D-QPGF and its gradient where we use to our advantage that it is very effective to simultaneously compute the QPGF and its gradient, and to simultaneously compute these values for the case in which the role of source and observation point are interchanged. We use the Ewald representation of the 2D-QPGF and its gradient to construct the tables with pre-computed values. Usually the expressions for the Ewald representation of the 2D-QPGF and its gradient are presented in terms of the complex complementary error function but here we give the expressions in terms of the Faddeeva function enabling efficient use of the dedicated algorithms to compute the Faddeeva function. Expressions are given for both lossy and lossless medium parameters and it is shown that the expression for the lossless case can be evaluated twice as fast as the expression for the lossy case. Two case studies are presented to validate the proposed method and to show that the time required for computing the method of moments (MoM) integrals that require evaluation of the 2D-QPGF becomes comparable to the time required for computing the MoM integrals that require evaluation of the aperiodic Green function.
2. Solís, D. M., M. G. Araújo, L. Landesa, S. García, J. M. Taboada, and F. Obelleiro, "MLFMA-MoM for solving the scattering of densely packed plasmonic nanoparticle assemblies," IEEE Photonics Journal, Vol. 7, No. 3, 4800709, Jun. 2015.
3. 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, No. 1, 122-134, Jan. 2009.
4. Celepcikay, F. T., D. R. Wilton, and D. R. Jackson, "Interpolation of 2D layered-medium periodic Green's function," Antennas and Propagation Society International Symposium (APSURSI), Toronto, Jul. 11-17, 2010.
5. Wilton, D. R., D. R. Jackson, and F. T. Celepcikay, "Efficient computation of periodic, layered media Green's functions," 6th European Conference on Antennas and Propagation (EuCAP 2012), Prague, Mar. 26-30, 2012.
6. Celepcikay, F. T., "Efficient calculation of layered-medium periodic Green's function,", PhD Thesis, University of Houston, Houston, Texas, Aug. 2010.
7. Ylä-Oijala, P., M. Taskinen, and J. Sarvas, "Surface integral equation method for general composite metallic and dielectric structures with junctions," Progress In Electromagnetics Research, Vol. 52, 81-108, 2005.
8. Solís, D. M., J. M. Taboada, and F. Obelleiro, "Surface integral equation-method of moments with multiregion basis functions applied to plasmonics," IEEE Transactions on Antennas and Propagation, Vol. 63, No. 5, 2141-2152, May 2015.
9. Dardenne, X. and C. Craeye, "Method of moments simulation of infinitely periodic structures combining metal with connected dielectric objects," IEEE Transations on Antennas and Propagation, Vol. 56, No. 8, 2372-2380, Aug. 2008.
10. Gallinet, B., A. M. Kern, and O. J. F. Martin, "Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach," Journal of the Optical Society of America A, Optics and Image Science, Vol. 27, No. 10, 2261-2271, Oct. 2010.
11. Jorna, P., V. Lancelotti, and M. C. van Beurden, "Formulation and implementation of boundary integral equations for scattering by doubly periodic plasmonic and dielectric structures of infinite lateral extent," International Conference on Electromagnetics in Advanced Applications (ICEAA), 1423-1426, Torino, Sep. 7-11, 2015.
12. Ewald, P. P., "Die berechnung optischer und elektrostatischer gitterpotentiale," Annalen der Physik IV, Vol. 64, 253-287, 1921.
13. Jordan, K. E., G. R. Richter, and P. Sheng, "An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures," Journal of Computational Physics, Vol. 63, No. 1, 222-235, Mar. 1986.
14. Kambe, K., "Theory of electron diffraction by crystals, I. Green's function and integral equation," Z. Naturforschg., Vol. 22a, 422-431, 1967.
15. Stevanović, I., P. Crespo-Valero, K. Blagović, F. Bongard, and J. R. Mosig, "Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation," IEEE Transactions on Microwave Theory and Techniques, Vol. 54, No. 10, 3688-3697, Oct. 2006.
16. Li, S., D. A. van Orden, and V. Lomakin, "Fast periodic interpolation method for periodic unit cell problems," IEEE Transactions on Antennas and Propagation, Vol. 58, No. 12, 4005-4014, Dec. 2010.
17. 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," Journal of the Optical Society of America A, Optics and Image Science, Vol. 27, No. 2, 308-318, 2010.
18. Poggio, A. J. and E. K. Miller, "Integral equation solutions of three dimensional scattering problems," Computer Techniques for Electromagnetics, R. Mittra, editor, Pergamon Press, Elmsford, New York, 1973.
19. Chang, Y. and R. Harrington, "A surface formulation for characteristic modes of material bodies," IEEE Transactions on Antennas and Propagation, Vol. 25, No. 6, 789-795, Jun. 1977.
20. Wu, T. K. and L. L. Tsai, "Scattering from arbitrarily-shaped lossy dielectric bodies of revolution," Radio Science, Vol. 12, No. 5, 709-718, 1977.
21. Harrington, R. F., Field Computation by Moment Methods, Wiley-IEEE Press, 1993.
22. Rao, S. M., D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Transactions on Antennas and Propagation, Vol. 30, No. 3, 409-418, May 1982.
23. Kustepeli, A. and A. Q. Martin, "On the splitting parameter in the Ewald method," IEEE Transactions on Microwave and Guided Wave Letters, Vol. 10, No. 5, 168-170, May 2000.
24. Oroskar, S., D. R. Jackson, and D. R. Wilton, "Efficient computation of the 2D periodic Green's function using the Ewald method," Journal of Computational Physics, Vol. 219, No. 2, 899-911, Dec. 2006.
25. Celepcikay, F. T., D. R. Wilton, D. R. Jackson, and F. Capolino, "Choosing splitting parameters and summation limits in the numerical evaluation of 1-D and 2-D periodic Green's functions using the Ewald method," Radio Science, Vol. 43, RS6S01, Sep. 2008.
26. Stevanović, I. and J. R. Mosig, "Green's function for planar structures in periodic skewed 2-D lattices using Ewald transformation," 1st European Conference on Antennas and Propagation (EuCAP 2006), Nice, France, Nov. 6-10, 2006.
27. Abramowitz, M. and I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1965.
28. Gautschi, W., "Efficient computation of the complex error function," SIAM J. Numer. Anal., Vol. 7, No. 1, 187-198, Mar. 1970.
29. Poppe, G. P. M. and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Transactions on Mathematical Software, Vol. 16, No. 1, 38-46, Mar. 1990.
30. Zaghloul, M. R. and A. N. Ali, "Algorithm 916: Computing the Faddeyeva and Voigt functions," ACM Transactions on Mathematical Software, Vol. 38, No. 2, 1-22, Dec. 2011.
31. Van Beurden, M. C., "A spectral volume integral equation method for arbitrary bi-periodic gratings with explicit Fourier factorization," Progress In Electromagnetics Research B, Vol. 36, 133-149, 2012.
32. Van Beurden, M. C., "Fast convergence with spectral volume integral equation for crossed block-shaped gratings with improved material interface conditions," Journal of the Optical Society of America A, Vol. 28, No. 11, 2269-2278, 2011.
33. Jorna, P., V. Lancelotti, and M. C. van Beurden, "SIE approach to scattered field computation for 2D periodic diffraction gratings in 3D space consisting of high permittivity dielectric materials and plasmonic scatterers," International Conference on Electromagnetics in Advanced Applications (ICEAA), 143-146, Aruba, Aug. 3-9, 2014.
34. Van Kraaij, M. G. M. M., "Forward diffraction modelling: analysis and application to grating reconstruction,", PhD thesis, Eindhoven University of Technology, Mar. 2011.