×

On the analysis of perfectly matched layers for a class of dispersive media and application to negative index metamaterials. (English) Zbl 1404.35427

This paper is concerned about the Perfectly Matched Layers (PMLs) in dispersive media, especially in Negative Index Metamaterials (NIMs). The motivation is due to the discovery that the classical PMLs are unstable in simulating wave propagation in NIMs. The main goal of this paper is to propose a generalized way in constructing stable PMLs for a class of dispersive electromagnetic media including the Drude materials. Numerical simulation by using the FDTD method is presented to demonstrate the stability of the newly constructed PML.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35B35 Stability in context of PDEs
35L05 Wave equation
35L40 First-order hyperbolic systems
35Q61 Maxwell equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
78A25 Electromagnetic theory (general)
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] Appel\"o, Daniel; Hagstrom, Thomas; Kreiss, Gunilla, Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability, SIAM J. Appl. Math., 67, 1, 1-23 (2006) · Zbl 1110.35042 · doi:10.1137/050639107
[2] B\'ecache, E.; Fauqueux, S.; Joly, P., Stability of perfectly matched layers, group velocities and anisotropic waves, J. Comput. Phys., 188, 2, 399-433 (2003) · Zbl 1127.74335 · doi:10.1016/S0021-9991(03)00184-0
[3] B\'ecache, \'Eliane; Joly, Patrick; Kachanovska, Maryna; Vinoles, Valentin, Perfectly matched layers in negative index metamaterials and plasmas. CANUM 2014-42e Congr\`“es National d”Analyse Num\'erique, ESAIM Proc. Surveys 50, 113-132 (2015), EDP Sci., Les Ulis · Zbl 1341.78013 · doi:10.1051/proc/201550006
[4] Berenger, Jean-Pierre, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 2, 185-200 (1994) · Zbl 0814.65129 · doi:10.1006/jcph.1994.1159
[5] Bouchitt\'e, Guy; Bourel, Christophe; Felbacq, Didier, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris, 347, 9-10, 571-576 (2009) · Zbl 1177.35028 · doi:10.1016/j.crma.2009.02.027
[6] Bouchitt\'e, Guy; Schweizer, Ben, Homogenization of Maxwell’s equations in a split ring geometry, Multiscale Model. Simul., 8, 3, 717-750 (2010) · Zbl 1228.35028 · doi:10.1137/09074557X
[7] correia20043d D. Correia and J.-M. Jin, 3d-fdtd-pml analysis of left-handed metamaterials, Microwave and optical technology letters 40 (2004), no. 3, 201-205.
[8] cui2010metamaterials T. J. Cui, D. R. Smith, and R. Liu, Metamaterials: Theory, Design, and Applications, Springer, 2010.
[9] cummer2004perfectly S. A. Cummer, Perfectly matched layer behavior in negative refractive index materials, Antennas and Wireless Propagation Letters, IEEE 3 (2004), no. 1, 172-175.
[10] Demaldent, E.; Imperiale, S., Perfectly matched transmission problem with absorbing layers: application to anisotropic acoustics in convex polygonal domains, Internat. J. Numer. Methods Engrg., 96, 11, 689-711 (2013) · Zbl 1352.74342 · doi:10.1002/nme.4572
[11] Diaz, J.; Joly, P., A time domain analysis of PML models in acoustics, Comput. Methods Appl. Mech. Engrg., 195, 29-32, 3820-3853 (2006) · Zbl 1119.76046 · doi:10.1016/j.cma.2005.02.031
[12] dong2004perfectly X. T. Dong, X. S. Rao, Y. B. Gan, B. Guo, and W. Y. Yin, Perfectly matched layer-absorbing boundary condition for left-handed materials, Microwave and Wireless Components Letters, IEEE 14 (2004), no. 6, 301-303.
[13] Duru, Kenneth; Kreiss, Gunilla, A well-posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation, Commun. Comput. Phys., 11, 5, 1643-1672 (2012) · Zbl 1373.35302 · doi:10.4208/cicp.120210.240511a
[14] Evans, Lawrence C., Partial Differential Equations, Graduate Studies in Mathematics 19, xviii+662 pp. (1998), American Mathematical Society, Providence, RI · Zbl 0902.35002
[15] Halpern, L.; Petit-Bergez, S.; Rauch, J., The analysis of matched layers, Confluentes Math., 3, 2, 159-236 (2011) · Zbl 1263.65088 · doi:10.1142/S1793744211000291
[16] Jackson, John David, Classical Electrodynamics, xxii+848 pp. (1975), John Wiley & Sons, Inc., New York-London-Sydney · Zbl 0997.78500
[17] Joly, Patrick, An elementary introduction to the construction and the analysis of perfectly matched layers for time domain wave propagation, SeMA J., 57, 5-48 (2012) · Zbl 1331.35003
[18] Kato, Tosio, Perturbation Theory for Linear Operators, Classics in Mathematics, xxii+619 pp. (1995), Springer-Verlag, Berlin · Zbl 0836.47009
[19] Kreiss, Heinz-Otto; Lorenz, Jens, Initial-boundary Value Problems and the Navier-Stokes Equations, Classics in Applied Mathematics 47, xviii+402 pp. (2004), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA · Zbl 1097.35113 · doi:10.1137/1.9780898719130
[20] loh2009fundamental P.-R. Loh, A. F. Oskooi, M. Ibanescu, M. Skorobogatiy, and S. G. Johnson, Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backward-wave structures, Physical Review E 79 (2009), no. 6, 065601.
[21] brien2002photonic S. O’Brien and J. B. Pendry, Photonic band-gap effects and magnetic activity in dielectric composites, Journal of Physics: Condensed Matter 14 (2002), no. 15, 4035.
[22] pendry2000negative J. B. Pendry, Negative refraction makes a perfect lens, Physical review letters 85 (2000), no. 18, 3966.
[23] shi2006perfectly Y. Shi, Y. Li, and C.-H. Liang, Perfectly matched layer absorbing boundary condition for truncating the boundary of the left-handed medium, Microwave and optical technology letters 48 (2006), no. 1, 57-63.
[24] smith2004metamaterials D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Metamaterials and negative refractive index, Science 305 (2004), no. 5685, 788-792.
[25] tip2004linear A. Tip, Linear dispersive dielectrics as limits of drude-lorentz systems., Physical review. E, Statistical, nonlinear, and soft matter physics 69 (2004), no. 1 Pt 2, 016610-016610.
[26] veselago1968electrodynamics V. G. Veselago, The electrodynamics of substances with simultaneously negative values epsilon and mu, Soviet physics uspekhi 10 (1968), no. 4, 509.
[27] vinoles2016problemes V. Vinoles, Probl\`“emes d”interface en pr\'esence de m\'etamat\'eriaux: mod\'elisation, analyse et simulations, Ph.D. thesis, Universit\'e Paris-Saclay, 2016.
[28] ziolkowski2001wave R. W. Ziolkowski and E. Heyman, Wave propagation in media having negative permittivity and permeability, Physical review E 64 (2001), no. 5, 056625.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.