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Application of the Generalized Method of Eigenoscillations to the Solution of the Problems of Scattering on Nanostructures

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We study the problem of scattering of electromagnetic waves on small-sized dielectric bodies by using the generalized method of eigenoscillations. The inhomogeneous problem is formulated on the basis of a version of the generalized method of eigenoscillations in which the dielectric permittivity plays the role of an eigenvalue. The corresponding homogeneous problem is solved with the use of Debye potentials. We obtain a system of linear algebraic equations for the unknown expansion coefficients. For the solution of this system, we use the method of successive approximations. The results of the performed numerical analyses confirm the efficiency of the proposed method.

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References

  1. N. N. Voitovich, “Homogeneous problems of the generalized method of eigenoscillations for bodies of revolution,” Radiotekh. Élektron., 25, No. 7, 1526–1529 (1980).

    Google Scholar 

  2. N. N. Voitovich, B. Z. Katselenenbaum, and A. N. Sivov, Generalized Method of Eigenoscillations in the Theory of Diffraction [in Russian], Nauka, Moscow (1977).

  3. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Introduction to Statistical Radiophysics. Part 2: Random Fields [in Russian], Nauka, Moscow (1978).

  4. M. I. Andriychuk, B. Z. Katsenelenbaum, V. V. Klimov, and N. N. Voitovich, “Application of generalized method of eigenoscillations to problems of nanoplasmonics,” in: Proc. of 16th Internat. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED) (September 26–29, 2011, Lviv), Lviv (2011), pp. 165–169.

  5. M. Andriychuk, “Investigation of radiation properties of nanoparticles by generalized eigenoscillation method,” in: Proc. of the 9th Int. Conf. on Advanced Computer Information Technologies (ACIT) (June 5–7, 2019), Ceske Budejovice, Czech Republic (2019), pp. 109–112; https://doi.org/10.1109/ACITT.2019.8779900.

  6. M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory, Wiley–VCH, Berlin (1999).

    MATH  Google Scholar 

  7. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature, 424, No. 6950, 824–830 (2003); https://doi.org/10.1038/nature01937.

    Article  Google Scholar 

  8. F. Bertó-Roselló, E. Xifré-Pérez, J. Ferré-Borrull, and L. F. Marsal, “3D-FDTD modelling of optical biosensing based on goldcoated nanoporous anodic alumina,” Results Phys., 11, 1008–1014 (2018); https://doi.org/10.1016/j.rinp.2018.10.067.

  9. S. I. Bozhevolnyi (editor), Plasmonic Nanoguides and Circuits, Pan Stanford Publ., Singapore (2008).

  10. M. L. Brongersma and P. G. Kik (editors), “Surface plasmon nanophotonics,” in: Springer Series in Optical Science, Vol. 131, Springer, New York (2007).

  11. M. A. Garcia, “Surface plasmons in metallic nanoparticles: fundamentals and applications,” J. Phys. D: Appl. Phys., 44, No. 28, Article 283001 (2011); https://doi.org/10.1088/0022-3727/44/28/283001.

  12. J. I. Hage, J. M. Greenberg, and R. T. Wang, “Scattering from arbitrarily shaped particles: Theory and experiment,” Appl. Optics, 30, No. 9, 1141–1152 (1991); https://doi.org/10.1364/AO.30.001141.

    Article  Google Scholar 

  13. U. S. Inan and R. A. Marshall, Numerical Electromagnetics: The FDTD Method, Cambridge Univ. Press, Cambridge–New York (2011).

  14. B. Z. Katsenelenbaum, High-Frequency Electrodynamics, Wiley-VCH, Berlin (2006).

    Book  Google Scholar 

  15. V. Klimov, Nanoplasmonics, CRC Press, Boca Raton (2014).

    Book  Google Scholar 

  16. A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic-fields,” Int. J. Mod. Phys. C., 3, No. 3, 583–603 (1992); https://doi.org/10.1142/S0129183192000385.

  17. J. M. Luther, P. K. Jain, T. Ewers, and P. A. Alivisatos, “Localized surface plasmon resonances arising from free carriers in doped quantum dots,” Natur. Mater., 10, No. 5, 361–366 (2011); https://doi.org/10.1038/nmat3004.

    Article  Google Scholar 

  18. S. A. Maier, Plasmonics: Fundamentals and Applications, Springer, New York (2007).

    Book  Google Scholar 

  19. D. Sarid and W. Challener, Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling, and Applications, Cambridge Univ. Press, Cambridge (2010).

    Book  Google Scholar 

  20. V. M. Shalaev and S. Kawata (editors), Nanophotonics with Surface Plasmons, Elsevier, Amsterdam (2007).

    Google Scholar 

  21. S. Sheikholeslami, Y. Jun, P. K. Jain, and A. P. Alivisatos, “Coupling of optical resonances in a compositionally asymmetric plasmonic nanoparticle dimer,” Nano Lett., 10, No. 7, 2655–2660 (2010); https://doi.org/10.1021/nl101380f.

    Article  Google Scholar 

  22. W. Sun, N. G. Loeb, G. Videen, and Q. Fu, “Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference time-domain algorithm,” Appl. Optics, 43, No. 9, 1957–1964 (2004); https://doi.org/10.1364/AO.43.001957.

    Article  Google Scholar 

  23. A. Taflove, A. Oskooi, and S. G. Johnson, Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, Artech House, Boston (2013).

    Google Scholar 

  24. S. Tretyakov, Analytical Modeling in Applied Electromagnetics, Artech House, Boston (2003).

    MATH  Google Scholar 

  25. M. Wahbeh, Discrete-Dipole-Approximation (DDA) Study of the Plasmon Resonance in Single and Coupled Spherical Silver Nanoparticles in Various Configurations, Master Thesis, Concordia University, Montreal (2011).

  26. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Anten. Propag., 14, No. 3, 302–307 (1966).

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

    М. І. Andriychuk, М. М. Voitovych & V. P. Tkachuk

Authors
  1. М. І. Andriychuk
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  2. М. М. Voitovych
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  3. V. P. Tkachuk
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Correspondence to М. І. Andriychuk.

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М. М. Voitovych is deceased.

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 63, No. 2, pp. 59–71, April–June, 2020.

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Andriychuk, М.І., Voitovych, М.М. & Tkachuk, V.P. Application of the Generalized Method of Eigenoscillations to the Solution of the Problems of Scattering on Nanostructures. J Math Sci 272, 64–79 (2023). https://doi.org/10.1007/s10958-023-06400-6

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  • DOI: https://doi.org/10.1007/s10958-023-06400-6

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