Two-Dimensionally Periodic Gratings: Pulsed and Steady-State Waves in an Irregular Floquet Channel

Part of the Springer Series on Atomic, Optical, and Plasma Physics book series (SSAOPP, volume 91)


This chapter presents a series of analytical results that serves as theoretical basis for numerical study of electromagnetic wave transformations in two-dimensionally periodic structures. Among them is the solution of the important problem of truncation of the computational space by artificial boundaries. The author establishes and analyzes fundamental characteristics of transient and steady-state fields in the regular part of the rectangular Floquet channel. For the first time, strict corollaries of Poynting’s complex power theorem and Lorentz’s lemma (the energy-balance equations and reciprocity relations) is presented for two-dimensionally periodic gratings of finite thickness illuminated by transverse-electric or transverse-magnetic plane waves. The method of transport operators (a space-time analogue of the generalized scattering matrices), developed in the chapter, can significantly reduce the computational resources required for calculation of wave scattering by multilayer periodic structures or by the structures on thick substrates. A number of questions concerning the spectral theory of two-dimensionally periodic gratings is answered—it is the result, which is essential for a reliable physical analysis of the resonant scattering of pulsed and monochromatic waves.


Periodic Structure Initial Boundary Free Oscillation Periodic Grating Magnetic Field Vector 
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  1. 1.
    Shestopalov, V.P., Lytvynenko, L.M., Masalov, S.A., Sologub, V.G.: Wave Diffraction by Gratings. Kharkov State University Press, Kharkov (1973) (in Russian)Google Scholar
  2. 2.
    Petit, R. (ed.): Electromagnetic Theory of Gratings. Springer, New York (1980)Google Scholar
  3. 3.
    Shestopalov, V.P., Kirilenko, A.A., Masalov, S.A., Sirenko, Y.K.: Diffraction gratings. In: Resonance Wave Scattering, vol.1. Naukova Dumka, Kiev (1986) (in Russian)Google Scholar
  4. 4.
    Shestopalov, V.P., Sirenko, Y.K.: Dynamic Theory of Gratings. Naukova Dumka, Kiev (1989) (in Russian)Google Scholar
  5. 5.
    Neviere, M., Popov, E.: Light Propagation in Periodic Media: Differential Theory and Design. Dekker, New York (2003)Google Scholar
  6. 6.
    Sirenko, Y.K., Strom, S., Yashina, N.P.: Modeling and Analysis of Transient Processes in Open Resonant Structures: New Methods and Techniques. Springer, New York (2007)MATHGoogle Scholar
  7. 7.
    Sirenko, Y.K., Strom, S. (eds.): Modern Theory of Gratings: Resonant Scattering: Analysis Techniques and Phenomena. Springer, New York (2010)Google Scholar
  8. 8.
    Ladyzhenskaya, O.A.: The Boundary Value Problems of Mathematical Physics. Springer, New York (1985)CrossRefMATHGoogle Scholar
  9. 9.
    Taflove, A., Hagness, S.C.: Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House, Boston (2000)MATHGoogle Scholar
  10. 10.
    Sirenko, K., Pazynin, V., Sirenko, Y., Bagci, H.: An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures. Prog. Electromagnet. Res. 111, 331–364 (2011)CrossRefGoogle Scholar
  11. 11.
    Liu, M., Sirenko, K., Bagci, H.: An efficient discontinuous Galerkin finite element method for highly accurate solution of Maxwell equations. IEEE Trans. Antennas Propag. 60(8), 3992–3998 (2012)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Rothwell, E.J., Cloud, M.J.: Electromagnetics. CRC Press, New York (2001)CrossRefGoogle Scholar
  13. 13.
    Vladimirov, V.S.: Equations of Mathematical Physics. Dekker, New York (1971)MATHGoogle Scholar
  14. 14.
    Sirenko, K.Y., Sirenko, Y.K.: Exact ‘absorbing’ conditions in the initial boundary value problems of the theory of open waveguide resonators. Comput. Math. Math. Phys. 45(3), 490–506 (2005)MathSciNetMATHGoogle Scholar
  15. 15.
    Kravchenko, V.F., Sirenko, Y.K., Sirenko, K.Y.: Electromagnetic Wave Transformation and Radiation by the Open Resonant Structures. Modelling and Analysis of Transient and Steady-State Processes. Fizmatlit, Moscow (2011) (in Russian)Google Scholar
  16. 16.
    Titchmarsh, E.: Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1948)MATHGoogle Scholar
  17. 17.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, San Diego, London (2000)MATHGoogle Scholar
  18. 18.
    von Hurwitz, A.: Allgemeine Funktionentheorie und Elliptische Funktionen. von Courant, R.: Geometrische Funktionentheorie. Springer, Berlin (1964) (in German)Google Scholar
  19. 19.
    Sirenko, Y.K., Velychko, L.G., Erden, F.: Time-domain and frequency-domain methods combined in the study of open resonance structures of complex geometry. Prog. Electromagnet. Res. 44, 57–79 (2004)CrossRefGoogle Scholar
  20. 20.
    Velychko, L.G., Sirenko, Y.K., Velychko, O.S.: Time-domain analysis of open resonators. Analytical grounds. Prog. Electromagnet. Res. 61, 1–26 (2006)CrossRefGoogle Scholar
  21. 21.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York (1978)MATHGoogle Scholar
  22. 22.
    Hokhberg, I.Z., Seagul, Y.I.: Operator generalization of the theorem about logarithmic residue and the Rouche theorem. Matematicheckiy Sbornik 84(4), 607–629 (1971) (in Russian)Google Scholar
  23. 23.
    Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Wiley-Interscience, New York (1983)MATHGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.O.Ya. Usikov Institute for Radiophysics and Electronics, National Academy of SciencesKharkivUkraine

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