Abstract
We review guidelines to obtain fast and accurate solutions—based on integral equations—of the Helmholtz equation with mixed boundary values in two dimensions, a crucial issue when modeling electromagnetic phenomena in complex photonic media. We solve the electric- and magnetic-field integral equations (EFIE and MFIE) to treat scattering of electromagnetic waves from transverse magnetic (TM)- and transverse electric (TE)-excited impedance cylinders represented with smoothly parameterized cross-section contours. We show that it is possible to obtain superalgebraic convergence with accurate calculations of the kernels of the integral equations whose singularities vary from weak to hypersingular. These Fredholm equations of the second kind are subject to stable discretization procedures. However, for various values of impedance, numerical stability can be maintained only via analytical regularization. Finally, we provide numerical results that support our conclusions.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig5_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig6_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig7_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig8_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig9_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10825-017-1073-9/MediaObjects/10825_2017_1073_Fig10_HTML.gif)
Similar content being viewed by others
Change history
31 January 2022
A Correction to this paper has been published: https://doi.org/10.1007/s10825-022-01855-3
References
Gagnon, D., Dub, L. J.: Tutorial; Lorenz-Mie theory for 2D scattering and resonance calculations. arXiv:1505.07691v2 [physics.optics] 22 Sep (2015)
Alvarado-Rodriguez, I., Yablonovitch, E.: Separation of radiation and absorption losses in two-dimensional photonic crystal single defect cavities. J. Appl. Phys. 92(11), 6399 (2002)
Ivanova, O.V., Hammer, M., Stoffer, R., van Groesen, E.: A variational mode expansion mode solver. Opt. Quant. Electron. 39(10–11), 849–864 (2007). ISSN 0306-8919
Gopinath, A., Boriskina, S.V., Feng, N.-N., Reinhard, B.M., Dal Negro, L.: Photonic-plasmonic scattering resonances in deterministic aperiodic structures. Nano Lett. 8(8), 2423–2431 (2008)
Bezus, E.A., Bykov, D.A., Doskolovich, L.L.: Antireflection layers in low-scattering plasmonic optics. Photonics Nanostruct. Fundam. Appl. 14, 101–105 (2015)
Ivanova, O.V., Stoffer, R., Hammer, M.: A dimensionality reduction technique for 2D scattering problems in photonics. J. Opt. 12(3), 035502 (2010). ISSN 2040-8978
M, Hammer: Hybrid analytical/numerical coupled-mode modeling of guided-wave devices. J. Lightwave Technol. 25(9), 2287–2298 (2007). ISSN 0733-8724
Naghizadeh, S., Kocabas, Ş.E.: Guidelines for designing 2D and 3D plasmonic stub resonators. J. Opt, Soc. Am. B 34(1), 207 (2017)
Morita, N., Kumagai, N., Mautz, J.R.: Integral Equ. Methods Electromagn. Artech House, Boston (1991)
Senior, T.B.A., Volakis, J.L.: Approximate Boundary Conditions In Electromagnetics, Iee Electromagnetic Waves Series, vol. 41. IEE Press, London (1995)
Idemen, M.: Discontinuities In The Electromagnetic Field. Wiley, Hoboken (2011)
Tuchkin, Y. A.: Electromagnetic Wave Scattering by Smooth Imperfectly Conductive Cylindrical Obstacle, Book Chapter in Ultra-Wideband, Short-Pulse Electromagnetics, vol. 5, pp 137-142. Kluwer Academic/Plenum Publishers, New York(2002) Tuchkin, Y. A.: Electromagnetic Wave Scattering by Smooth Imperfectly Conductive Cylindrical Obstacle, Book Chapter in Ultra-Wideband, Short-Pulse Electromagnetics, vol. 5, pp 137-142. Kluwer Academic/Plenum Publishers, New York(2002)
Sirenko, Y.K., Strom, S.: Modern Theory Of Gratings Resonant Scattering Analysis Techniques And Phenomena. Springer, New York (2010)
Sekulic, I., Ubeda, E., Rius, J.M.: Versatile and accurate schemes of discretization in the scattering analysis of 2D composite objects with penetrable or perfectly conducting regions. IEEE Trans. Antennas Propaga. 65(5), (2017). doi:10.1109/TAP.2017.2679064
Sever, E., Dikmen, F., Tuchkin, Y.A., Sabah, C.: Numerically stable algorithms for scattering by impedance cylinders. Int. J. Mech. 11, 64–68 (2017). ISSN: 1998-4448
Tsalamengas, J.L.: Exponentially converging Nystöm method in scattering from infinite curved smooth strips, part 1, part 2. IEEE Trans. Antenna Propag. 58–10, 3265–3281 (2010)
Sever, E., Dikmen, F., Suvorova, O., Tuchkin, Y.A.: An analytical formulation with ill-conditioned numerical scheme and its remedy: scattering by two circular impedance cylinders. Turk. J. Electr. Eng. Comput. Sci. 24, 1194–1207 (2016)
Colton, D.L., Kress, R.: Integral Equations Methods In Scattering Theory. Krieger Publishing Company, Malabar (1992)
Hutson, V., Pym, J.S., Cloud, M.J.: Applications Of Functional Analysis And Operator Theory, 2nd edn. Elsevier Science, Amsterdam (2005). ISBN 0-444-51790-1
Poyedinchuk, A.E., Tuchkin, Y.A., Shestopalov, V.P.: New numerical-analytical methods in diffraction theory. Math. Comput. Model. 32, 1029–1046 (2000)
Vinogradov, S.S., Vinogradova, E.D., Wilson, C., Sharp, I., Tuchkin, Yu.: Scattering of E-polarized plane wave by 2-D airfoil. Electromagnetics 29(3), 268–282 (2009)
Dikmen, F., Tuchkin, Y.A.: Analytical regularization method for electromagnetic wave diffraction by axially symmetrical thin annular strips. Turk. J. Electr. Eng. Comput. Sci. 107–124 (2009). doi:10.3906/elk-0811-10
Dallas, A.G., Hsiao, G.C., Kleinman, R.E.: Observations on the numerical stability of the Galerkin method. Comput. Math. 9, 37–67 (1998)
Wandzura, S.: slides of the talk, fast methods for fast computers-, within the program. In: Fast Multipole Method, Tree-Code and Related Approximate Algorithms. Trading Exactness for Efficiency, CSCAMM Program Spring, April 19–30. http://www2.cscamm.umd.edu/programs/fam04/FastTalk_wandzura_fam04.pdf, (2004). Accessed 15 June 2017
Şimşek, E., Liu, J., Liu, Q.H.: A spectral integral method (SIM) for layered media. IEEE Trans. Antennas Propag. 54(6), 1742–1749 (2006)
Hsiao, G.C., Wendland, W.L.: Boundary integral methods in low frequency acoustics. J. Chin. Inst. Eng. 23(3), 369–375 (2000). doi:10.1080/02533839.2000.9670557
Shestopalov, V., Tuchkin, Yu., Poyedinchuk, A., Sirenko, Yu.: New Methods Of Solution For Direct And Inverse Scattering Problems. Osnova, in Russian, Kharkiv (1997)
Abramowitz, M., Stegun, I.A.: Handbook Of Mathematical Functions With Formulas, Graphs, And Mathematical Tables. Dover Publications, New York (1972)
Tikhonov, J.A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Winston, New York (1977)
Kishk, A., Parrikar, R.P., Elsherbeni, A.Z.: Electromagnetic scattering from an eccentric multilayered circular cylinder. IEEE Trans. Antennas Propag. 40, 295–303 (1992)
Acknowledgements
This work was supported by the Scientific and Technical Research Council of Turkey (TUBITAK) under research grant 114E927.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sever, E., Tuchkin, Y.A. & Dikmen, F. On a superalgebraically converging, numerically stable solving strategy for electromagnetic scattering by impedance cylinders. J Comput Electron 17, 427–435 (2018). https://doi.org/10.1007/s10825-017-1073-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10825-017-1073-9