Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

Chesnel, Lucas; Claeys, Xavier; Nazarov, Sergei A.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1285-1313.

We investigate the eigenvalue problem −div(σ∇u) = λu (P) in a 2D domain Ω divided into two regions Ω_±. We are interested in situations where σ takes positive values on Ω₊ and negative ones on Ω₋. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [L. Chesnel, X. Claeys and S.A. Nazarov, Asymp. Anal. 88 (2014) 43–74], we highlighted an unusual instability phenomenon for the source term problem associated with (P): for certain configurations, when the interface between the subdomains Ω_± presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem (P). We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.

Classification : 35B25, 65N25
Mots clés : Negative materials, corner, asymptotic analysis, plasmonic, metamaterial, sign-changing coefficients

[1] F.L. Bakharev and S.A. Nazarov, On the structure of the spectrum for the elasticity problem in a body with a supersharp peak. Siberian Math. J. 50 (2009) 587–595. | DOI | MR | Zbl

[2] W.L. Barnes, A. Dereux and T.W. Ebbesen, Surface plasmon subwavelength optics. Nature 424 (2003) 824–830. | DOI

[3] M.Sh. Birman and M.Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series). Translated from the 1980 Russian original by S. Khrushchëv and V. Peller, D. Reidel Publishing Co., Dordrecht, (1987). | DOI | MR | Zbl

[4] A. Boltasseva, V.S. Volkov, R.B. Nielsen, E. Moreno, S.G. Rodrigo and S.I. Bozhevolnyi, Triangular metal wedges for sub-wavelength plasmon-polariton guiding at telecom wavelengths. Opt. Express 16 (2008) 5252–5260. | DOI

[5] A.-S. Bonnet-Ben Dhia, C. Carvalho, L. Chesnel and P. Ciarlet Jr., On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients. J. Comput. Phys. 322 (2016) 224–247. | DOI | MR | Zbl

[6] A.-S. Bonnet-Ben Dhia, L. Chesnel and P. Ciarlet Jr., T-coercivity for scalar interface problems between dielectrics and metamaterials. Math. Mod. Num. Anal. 46 (2012) 1363–1387. | DOI | Numdam | MR | Zbl

[7] A.-S. Bonnet-Ben Dhia, L. Chesnel and P. Ciarlet Jr., T-coercivity for the Maxwell problem with sign-changing coefficients. Commun. in PDEs 39 (2014) 1007–1031. | DOI | MR | Zbl

[8] A.-S. Bonnet-Ben Dhia, L. Chesnel and P. Ciarlet Jr., Two-dimensional Maxwell’s equations with sign-changing coefficients. Appl. Num. Math. 79 (2014) 29–41. | DOI | MR | Zbl

[9] A.-S. Bonnet-Ben Dhia, L. Chesnel and X. Claeys, Radiation condition for a non-smooth interface between a dielectric and a metamaterial. Math. Models Meth. App. Sci. 23 (2013) 1629–1662. | DOI | MR | Zbl

[10] A.-S. Bonnet-Ben Dhia, P. Ciarlet Jr. and C.M. Zwölf. Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234 (2010) 1912–1919. Corrigendum J. Comput. Appl. Math. 234 (2010) 2616. | DOI | MR | Zbl

[11] A.-S. Bonnet-Ben Dhia, M. Dauge and K. Ramdani, Analyse spectrale et singularités d’un problème de transmission non coercif. C. R. Acad. Sci. Paris, Ser. I 328 (1999) 717–720. | DOI | MR | Zbl

[12] A.-S. Bonnet-Ben Dhia and K. Ramdani, A non elliptic spectral problem related to the analysis of superconducting micro-strip lines. Math. Mod. Num. Anal. 36 (2002) 461–487. | DOI | Numdam | MR | Zbl

[13] E. Bonnetier and F. Triki, On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D. Arch. Rational Mech. Anal. (2013) 1–27. | MR | Zbl

[14] L. Chesnel and P. Ciarlet Jr., T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients. Numer. Math. 124 (2013) 1–29. | DOI | MR | Zbl

[15] L. Chesnel, X. Claeys and S.A. Nazarov, A curious instability phenomenon for a rounded corner in presence of a negative material. Asymp. Anal. 88 (2014) 43–74. | MR | Zbl

[16] L. Chesnel, X. Claeys and S.A. Nazarov, Spectrum of a diffusion operator with coefficient changing sign over a small inclusion. Z. Angew. Math. Phys. 66 (2015) 2173–2196. | DOI | MR | Zbl

[17] M. Costabel and E. Stephan, A direct boundary integral method for transmission problems. J. Math. Anal. Appl. 106 (1985) 367–413. | DOI | MR | Zbl

[18] M. Dauge and B. Texier, Problèmes de transmission non coercifs dans des polygones. Technical Report 97–27, Université de Rennes 1, IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France. Available at: http://hal.archives-ouvertes.fr/docs/00/56/23/29/PDF/BenjaminT_arxiv.pdf (1997).

[19] P. Fernandes and M. Raffetto, Well posedness and finite element approximability of time-harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials. Math. Models Meth. App. Sci. 19 (2009) 2299–2335. | DOI | MR | Zbl

[20] D. Grieser, The plasmonic eigenvalue problem. Rev. Math. Phys. 26 (2014) 1450005, 26 | DOI | MR | Zbl

[21] J. Helsing and K.-M. Perfekt, On the polarizability and capacitance of the cube. Appl. Comput. Harmon. Anal. 34 (2013) 445–468. | DOI | MR | Zbl

[22] A. M. Il’In, Matching of asymptotic expansions of solutions of boundary value problems, Vol. 102 of Translations of Mathematical Monographs. AMS, Providence, RI (1992). | DOI | Zbl

[23] I.V. Kamotskii and S.A. Nazarov, Spectral problems in singularly perturbed domains and selfadjoint extensions of differential operators. Amer. Math. Soc. Transl. Ser. 199 (2000) 127–182. | MR | Zbl

[24] T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Reprint of the 1980 edition Springer Verlag, Berlin (1995). | MR | Zbl

[25] D. Khavinson, M. Putinar and H.S. Shapiro, Poincaré’s variational problem in potential theory. Arch. Rational Mech. Anal. 185 (2007) 143–184. | DOI | MR | Zbl

[26] V.A. Kondratiev, Boundary-value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967) 227–313. | Zbl

[27] V.A. Kozlov, V.G. Maz’Ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities. Vol. 52 of Math. Surveys and Monographs. AMS, Providence (1997). | MR | Zbl

[28] J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications. Dunod (1968). | MR | Zbl

[29] V.G. Maz’Ya, S.A. Nazarov and B.A. Plamenevskiĭ, On the asymptotic behavior of solutions of elliptic boundary value problems with irregular perturbations of the domain. Probl. Mat. Anal. 8 (1981) 72–153. | MR | Zbl

[30] V.G. Maz’Ya, S.A. Nazarov and B.A. Plamenevskiĭ, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. 1, 2. Birkhäuser, Basel (2000). | MR | Zbl

[31] V.G. Maz’Ya and B.A. Plamenevskiĭ, On the coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points. Math. Nachr. 76 (1977) 29–60. Engl. Transl. Amer. Math. Soc. Transl. 123 (1984) 57–89 | MR | Zbl

[32] J. Meixner, The behavior of electromagnetic fields at edges. IEEE Trans. Antennas and Propagat. 20 (1972) 442–446. | DOI

[33] S.A. Nazarov, Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions. In Proc. of the St. Petersburg Mathematical Society, Vol. V, Vol. 193 of Amer. Math. Soc. Transl. Ser. 2 Providence, RI AMS (1999) 77–125 | MR | Zbl

[34] S.A. Nazarov and B.A. Plamenevskiĭ, Elliptic problems in domains with piecewise smooth boundaries. Vol. 13 of Expositions in Mathematics. De Gruyter, Berlin, Germany (1994). | MR | Zbl

[35] S.A. Nazarov and J. Taskinen, On the spectrum of the Steklov problem in a domain with a peak. Vestnik St. Petersburg Univ. Math. 41 (2008) 45–52. | DOI | MR | Zbl

[36] S.A. Nazarov and J. Taskinen, Radiation conditions at the top of a rotational cusp in the theory of water-waves. Math. Model. Numer. Anal. 45 (2011) 947–979. | DOI | Numdam | MR | Zbl

[37] H.-M. Nguyen, Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients. J. Math. Pures Appl. 106 (2016) 342–374. | DOI | MR | Zbl

[38] S. Nicaise and J. Venel, A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients. J. Comput. Appl. Math. 235 (2011) 4272–4282. | DOI | MR | Zbl

[39] P. Ola, Remarks on a transmission problem. J. Math. Anal. Appl. 196 (1995) 639–658. | DOI | MR | Zbl

[40] G. Oliveri and M. Raffetto, A warning about metamaterials for users of frequency-domain numerical simulators. IEEE Trans. Antennas Propag 56 (2008) 792–798. | DOI | MR | Zbl

[41] K.-M. Perfekt and M. Putinar, Spectral bounds for the Neumann-Poincaré operator on planar domains with corners. J. Anal. Math. 124 (2014) 39–57. | DOI | MR | Zbl

[42] K. Ramdani. Lignes supraconductrices: analyse mathématique et numérique. Ph.D. thesis, Université Paris 6, France (1999).

[43] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier analysis, self-adjointness. Academic Press, New York London (1975). | MR

[44] Ya.A. Roĭtberg and Z.G. Šeftel’, General boundary-value problems for elliptic equations with discontinuous coefficients. Dokl. Akad. Nauk. SSSR 148 (1963) 1034–1037. | MR | Zbl

[45] M. Schechter, A generalization of the problem of transmission. Ann. Scuola Norm. Sup. Pisa 14 (1960) 207–236. | Numdam | MR | Zbl

[46] M. Stockman. Nanofocusing of optical energy in tapered plasmonic waveguides. Phys. Rev. Lett. 93 (2004) 137–404. | DOI

[47] A.A. Sukhorukov, I.V. Shadrivov and Y.S. Kivshar, Wave scattering by metamaterial wedges and interfaces. Int. J. Numer. Model. 19 (2006) 105–117. | DOI | Zbl

[48] M. Van Dyke, Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford, Calif. (1975). | MR | Zbl

[49] H. Wallén, H. Kettunen and A. Sihvola, Surface modes of negative-parameter interfaces and the importance of rounding sharp corners. Metamaterials 2 (2008) 113–121. | DOI

[50] A.V. Zayats, I.I. Smolyaninov and A.A. Maradudin, Nano-optics of surface plasmon polaritons. Phys. Reports 408 (2005) 131–314. | DOI