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On the Completeness of Normal Waves in an Inhomogeneous Partially Shielded Dielectric Layer

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Abstract

The problem on normal TE-waves in an inhomogeneous partially shielded dielectric layer is reduced to eigenvalue problem for the component of the electromagnetic field in Sobolev space. We formulate the definition of solution using variational relation. The variational problem is reduced to the study of an operator pencil. We investigate properties of the operators of the pencil for the analysis of its spectral properties. We prove theorem of discrete spectrum and theorem of localization of eigenvalues of the operator pencil on complex plane. The main result of the paper is the proof of double completeness of eigen- and associated vectors of the pencil in Sobolev space, which leads to existence of infinitely many waves in the waveguiding structure.

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REFERENCES

  1. M. J. Adams, An Introduction to Optical Waveguide (Wiley, New York, 1981).

    Google Scholar 

  2. A. Snyder and J. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

    Google Scholar 

  3. L. A. Vainshtein, Electromagnetic Waves (Sovetskoe Radio, Moscow, 1957) [in Russian].

    Google Scholar 

  4. D. Marcuse, Light Transmission Optics (Krieger, Malabar, 1989).

    Google Scholar 

  5. L. Brekhovskikh, Waves in Layered Media (Academic, New York, 2012).

    Google Scholar 

  6. J. Polo, T. Mackay, and A. Lakhtakia, Electromagnetic Surface Waves: A Modern Perspective (Elsevier, Amsterdam, 2013).

    Google Scholar 

  7. F. Chiadini et al., ‘‘Compound guided waves that mix characteristics of surface-plasmon-polariton, Tamm, Dyakonov-Tamm, and Uller–Zenneck waves,’’ J. Opt. Soc. Am. B 33, 1197–1206 (2016).

    Article  Google Scholar 

  8. R. E. Collin, Field Theory of Guided Waves (Wiley-IEEE, New York, 1990).

    Book  Google Scholar 

  9. Yu. G. Smirnov, ‘‘On the completeness of the system of Eigen-waves and joined waves of partially filled waveguide with non-regular boundary,’’ Dokl. Akad. Nauk SSSR 297, 829–832 (1987).

    Google Scholar 

  10. Yu. G. Smirnov, ‘‘Application of the operator pencil method in the Eigenvalue problem for partially filled waveguide,’’ Dokl. Akad. Nauk SSSR 312, 597–599 (1990).

    Google Scholar 

  11. Yu. G. Smirnov, ‘‘The method of operator pencils in the boundary transmission problems for elliptic system of equations,’’ Differ. Uravn. 27, 140–147 (1991).

    Google Scholar 

  12. Yu. G. Smirnov, Mathematical Methods for Electromagnetic Problems (Penza Gos. Univ. Press, Penza, 2009) [in Russian].

    Google Scholar 

  13. Yu. V. Shestopalov and Yu. G. Smirnov, ‘‘Eigenwaves in waveguides with dielectric inclusions: Spectrum,’’ Applic. Anal. 93, 408–427 (2014).

    Article  MathSciNet  Google Scholar 

  14. Yu. V. Shestopalov and Yu. G. Smirnov, ‘‘Eigenwaves in waveguides with dielectric inclusions: Completeness,’’ Applic. Anal. 93, 1824–1845 (2014).

    Article  MathSciNet  Google Scholar 

  15. Yu. G. Smirnov and E. Yu. Smolkin, ‘‘On the existence of an infinite number of leaky complex waves in a dielectric layer,’’ Dokl. Math. 101, 53–56 (2020).

    Article  Google Scholar 

  16. Yu. Smirnov and E. Smolkin, ‘‘Complex waves in dielectric layer,’’ Lobachevskii J. Math. 41 (7), 1396–1403 (2020).

    Article  MathSciNet  Google Scholar 

  17. Yu. G. Smirnov and E. Smolkin, ‘‘Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide,’’ Differ. Equat. 53, 1168–1179 (2018).

    Article  MathSciNet  Google Scholar 

  18. Yu. G. Smirnov, E. Smolkin, and M. O. Snegur, ‘‘Analysis of the spectrum of azimuthally symmetric waves of an open inhomogeneous anisotropic waveguide with longitudinal magnetization,’’ Comput. Math. Math. Phys. 58, 1887–1901 (2018).

    Article  MathSciNet  Google Scholar 

  19. Yu. G. Smirnov and E. Smolkin, ‘‘Operator function method in the problem of normal waves in an inhomogeneous waveguide,’’ Differ. Equat. 54, 1262–1273 (2017).

    Article  MathSciNet  Google Scholar 

  20. Yu. G. Smirnov and E. Smolkin, ‘‘Investigation of the spectrum of the problem of normal waves in a closed regular inhomogeneous dielectric waveguide of arbitrary cross section,’’ Dokl. Math. 97, 86–89 (2017).

    Article  Google Scholar 

  21. T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1980).

    MATH  Google Scholar 

  22. R. A. Adams, Sobolev Spaces (Academic, New York, 1975).

    MATH  Google Scholar 

  23. I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space (Am. Math. Soc., Philadelphia, 1969).

    Book  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation, project no. 20-11-20087.

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Authors and Affiliations

  1. Penza State University, 440026, Penza, Russia

    Yury Smirnov

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  1. Yury Smirnov
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Correspondence to Yury Smirnov.

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(Submitted by E. E. Tyrtyshnikov)

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Smirnov, Y. On the Completeness of Normal Waves in an Inhomogeneous Partially Shielded Dielectric Layer. Lobachevskii J Math 42, 1445–1452 (2021). https://doi.org/10.1134/S1995080221060287

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  • DOI: https://doi.org/10.1134/S1995080221060287

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