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Resonances for Maxwell’s equations in a periodic structure

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Abstract

This paper deals with the existence and the distribution of the resonances associated to the Maxwell system in a periodic infinite structure. By boundary integral operators method, we reduce our problem to the existence and distribution of the characteristic values of a family of periodic non-self adjoint integral operators. The distribution of resonances is completely shown and an algorithm to compute them is proposed.

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References

  1. M. Ando, A. Kondo and K. Kagoshima, Scattering of an arbitrary wave by a thin strip grating reflector. IEEE Proceedings,133, No.3 (1986), 203–208.

    Google Scholar 

  2. H. Begehr, Complex Analytic Methods for Partial Differential Equations. An Introductory Text. World Scientific, Singapore, 1994.

    Google Scholar 

  3. A.S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Meth. Appl. Sci.,17 (1994), 305–338.

    Article  MATH  MathSciNet  Google Scholar 

  4. A.S. Bonnet-BenDhia, Mathematical analysis of conductive and superconductive transmission lines. Proceedings of the Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation, SIAM Publications, 1998, 12–21.

  5. J.M. Bony, Cours d’Analyse. Cou rs de l’Ecole Polytechnique, 1994.

  6. O.P. Bruno and F. Reitich, Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain. Proc. Roy. Soc. Edinburgh,122A (1992), 317–340.

    MathSciNet  Google Scholar 

  7. X. Chen and A. Friedman, Maxwell equations in a periodic structure. Trans. Amer. Math. Soc,323 (1991), 465–507.

    Article  MATH  MathSciNet  Google Scholar 

  8. R.E. Collin, Field Theory of Guided Waves, Second Edition. IEEE Press, New York, 1991.

    MATH  Google Scholar 

  9. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory. John Wiley, New York, 1983.

    MATH  Google Scholar 

  10. E. Danicki, B. Langli and K. Bløtekjaer, Spectral theory of EM wave scattering by periodic strips. IEEE Trans. Antennas Propogat.,43 (1995), 97–103.

    Article  Google Scholar 

  11. R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Volume 6, Chapitre XIII (Méthodes intégrales et numériques). Masson, 1988.

  12. I.T.S. Gohberg and E.I. Sigal, Operator extension of the logarithmic residue theorem and Rouche’s theorem. Math. USSR Sbornik,84, No.4 (1971), 607–642.

    MathSciNet  Google Scholar 

  13. J.D. Hanfling, G. Jerinic and L.R. Lewis, Twist reflector design usingE-type andH-type modes. IEEE Trans. Antennas Propagat.,29 (1981), 622–629.

    Article  Google Scholar 

  14. Y. Hayashi, The Dirichlet problem for the two-dimensional Helmholtz equation for an open boundary. J. Math. Anal. Appl.,44 (1973), 489–530.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Jacobsen, Analytical, numerical and experimental investigation of guided waves on a periodically strip-loaded dielectric slab. IEEE Trans. Antennas Propagat.,18 (1970), 379–388.

    Article  Google Scholar 

  16. H.A. Kalhor and M. Ilyas, Scattering of plane electromagnetic waves by a grating of conducting cylinders embedded in a dielectric slab over a ground plane. IEEE Trans. Antennas Propagat.,30 (1982), 576–579.

    Article  Google Scholar 

  17. H.A. Kalhor, Electromagnetic scattering by a dielectric slab loaded with a periodic array of strips over a ground plane. IEEE Trans. Antennas Propagat.,36 (1988), 147–151.

    Article  Google Scholar 

  18. A. Kirsch, Diffraction by periodic structures. Inverse Problems in Mathematical Physics, Lecture Notes in Physics,422, Springer-Verlag, 1993.

  19. A. Morelot, Etude d’une méthode numérique de simulation de la diffraction d’une onde électromagnétique par un réseau bi-périodique. Thèse de l’École Polytechnique, 1992.

  20. P.M. Morse and H. Feshbach, Methods of Theoretical Physics. McGraw Hill.

  21. L.A. Muravei, Analytic continuation with respect to a parameter of the Green’s functions of exterior boundary value problems for the two-dimensional Helmholtz equations III. Math. USSR Sbornik,34, No.1 (1978), 55–98.

    Article  MATH  Google Scholar 

  22. N.I. Muskhelishvili, Singular Integral Equations, Noordhoff International Publishing, 1977.

  23. J.C. Nédélec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell equations. SIAM J. Math. Anal.,22 (1991), 1679–1701.

    Article  MATH  MathSciNet  Google Scholar 

  24. D.M. Pozar and D.H. Schaubert, Scan blindness in infinite phased arrays of printed dipoles. IEEE Trans. Antennas Propagat.,32 (1984), 602–610.

    Article  Google Scholar 

  25. T. Rozzi and G. Cerri, Radiation modes of open microstrip with applications. IEEE Trans. Microwave Theory and Tech.,43, No.6 (1995), 1364–1369.

    Article  Google Scholar 

  26. V.P. Shestopalov and Yu. V. Shestopalov, Spectral Theory and Excitation of Open Structures. IEEE Press, London, 1996.

    MATH  Google Scholar 

  27. R.A. Sigelmann and A. Ishimaru, Radiation from periodic structure excited by an aperiodic source. IEEE Trans. Antennas Propagat.,13 (1965), 354–364.

    Article  Google Scholar 

  28. R.A. Sigelmann, Surface waves on a grounded dielectric slab covered by a periodically slotted conducting plane. IEEE Trans. Antennas Propagat.,15 (1976), 672–676.

    Article  Google Scholar 

  29. J.C. Nédélec and F. Starling, Integral equation methods in a quasiperiodic diffraction problem for the time-harmonic Maxwell Equations. SIAM J. Math. Anal.,22 (1991), 1679–1701.

    Article  MATH  MathSciNet  Google Scholar 

  30. S. Steinberg, Meromorphic families of compact operators. Arch. Rat. Mech. Anal.,31 (1968), 372–380.

    Article  MATH  MathSciNet  Google Scholar 

  31. Te-Kao Wu, Fast convergent integral equation solution of strip gratings on dielectric substrate. IEEE Trans. Antennas Propagat.,35 (1987), 205–207.

    Article  Google Scholar 

  32. Y. Xu, Scattering of acoustic waves by an obstacle in a stratified medium. Partial Differential Equations with Real Analysis, Pitman Res. Notes Math. Ser.,263, 1992, 147–168.

    Google Scholar 

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Ammari, H., Béreux, N. & Nédélec, J.C. Resonances for Maxwell’s equations in a periodic structure. Japan J. Indust. Appl. Math. 17, 149 (2000). https://doi.org/10.1007/BF03167342

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  • DOI: https://doi.org/10.1007/BF03167342

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