# American Institute of Mathematical Sciences

December  2013, 8(4): 857-878. doi: 10.3934/nhm.2013.8.857

## Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings

 1 Université de Toulon, IMATH, EA 2134, 83957 La Garde, France 2 Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund

Received  November 2012 Revised  July 2013 Published  November 2013

We investigate the transmission properties of a metallic layer with narrow slits. Recent measurements and numerical calculations concerning the light transmission through metallic sub-wavelength structures suggest that an unexpectedly high transmission coefficient is possible. We analyze the time harmonic Maxwell's equations in the $H$-parallel case for a fixed incident wavelength. Denoting by $\eta>0$ the typical size of the complex structure, effective equations describing the limit $\eta\to 0$ are derived. For metallic permittivities with negative real part, plasmonic waves can be excited on the surfaces of the channels. When these waves are in resonance with the height of the layer, the result can be perfect transmission through the layer.
Citation: Guy Bouchitté, Ben Schweizer. Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings. Networks and Heterogeneous Media, 2013, 8 (4) : 857-878. doi: 10.3934/nhm.2013.8.857
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [2] G. Bouchitté and C. Bourel, Multiscale nanorod metamaterials and realizable permittivity tensors, Commun. in Comput. Phys., 11 (2012), 489-507. doi: 10.4208/cicp.171209.110810s. [3] G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris, 347 (2009), 571-576. doi: 10.1016/j.crma.2009.02.027. [4] G. Bouchitté and D. Felbacq, Negative refraction in periodic and random photonic crystals, New J. Phys., 7 (2005). [5] G. Bouchitté and D. Felbacq, Homogenization of a wire photonic crystal: The case of small volume fraction, SIAM J. Appl. Math., 66 (2006), 2061-2084. doi: 10.1137/050633147. [6] G. Bouchitté and B. Schweizer, Cloaking of small objects by anomalous localized resonance, Quart. J. Mech. Appl. Math., 63 (2010), 437-463. doi: 10.1093/qjmam/hbq008. [7] G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations with split rings, SIAM Multiscale Modeling and Simulation, 8 (2010), 717-750. doi: 10.1137/09074557X. [8] Q. Cao and P. Lalanne, Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits, Phys. Rev. Lett., 88 (2002), 057403. doi: 10.1103/PhysRevLett.88.057403. [9] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, volume 93 of Applied Mathematical Sciences. Springer-Verlag, Berlin, second edition, 1998. [10] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio and P. A. Wolff, Extraordinary optical transmission through sub-wavelength hole arrays, Letters to Nature, 391 (1998), 667-669. [11] D. Felbacq, Noncommuting limits in homogenization theory of electromagnetic crystals, J. Math. Phys., 43 (2002), 52-55. doi: 10.1063/1.1418013. [12] R. Kohn, J. Lu, B. Schweizer and M. Weinstein, A variational perspective on cloaking by anomalous localized resonance, Comm. Math. Phys. (accepted). [13] P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru and K. D. Möller, One-mode model and airy-like formulae for one-dimensional metallic gratings, Journal of Optics A: Pure and Applied Optics, 2 (2000), 48. doi: 10.1088/1464-4258/2/1/309. [14] P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier and P. Chavel, Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures, Physical Review B, (2003). doi: 10.1103/PhysRevB.68.125404. [15] A. Lamacz and B. Schweizer, Effective maxwell equations in a geometry with flat rings of arbitrary shape, SIAM J. Math. Anal., 45 (2013), 1460-1494. doi: 10.1137/120874321. [16] A. Mary, S. G. Rodrigo, L. Martin-Moreno and F. J. Garcia-Vidal, Holey metal films: From extraordinary transmission to negative-index behavior, Physical Review B, 80 (2009). doi: 10.1103/PhysRevB.80.165431. [17] G. Milton and N.-A. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 3027-3059. doi: 10.1098/rspa.2006.1715. [18] S. O'Brien and J. Pendry, Magnetic activity at infrared frequencies in structured metallic photonic crystals, J. Phys. Condens. Mat., 14 (2002), 6383-6394. [19] J. A. Porto, F. J. Garcia-Vidal and J. B. Pendry, Transmission resonances on metallic gratings with very narrow slits, Phys. Rev. Lett., 83 (1999), 2845-2848. doi: 10.1103/PhysRevLett.83.2845. [20] L. Schwartz, Théorie des Distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X. Nouvelle édition, Entiérement corrigée, Refondue et Augmentée. Hermann, Paris, 1966. [21] T. Vallius, J. Turunen, M. Mansuripur and S. Honkanen, Transmission through single subwavelength apertures in thin metal films and effects of surface plasmons, J. Opt. Soc. Am. A, 21 (2004), 456-463. doi: 10.1364/JOSAA.21.000456.

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [2] G. Bouchitté and C. Bourel, Multiscale nanorod metamaterials and realizable permittivity tensors, Commun. in Comput. Phys., 11 (2012), 489-507. doi: 10.4208/cicp.171209.110810s. [3] G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris, 347 (2009), 571-576. doi: 10.1016/j.crma.2009.02.027. [4] G. Bouchitté and D. Felbacq, Negative refraction in periodic and random photonic crystals, New J. Phys., 7 (2005). [5] G. Bouchitté and D. Felbacq, Homogenization of a wire photonic crystal: The case of small volume fraction, SIAM J. Appl. Math., 66 (2006), 2061-2084. doi: 10.1137/050633147. [6] G. Bouchitté and B. Schweizer, Cloaking of small objects by anomalous localized resonance, Quart. J. Mech. Appl. Math., 63 (2010), 437-463. doi: 10.1093/qjmam/hbq008. [7] G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations with split rings, SIAM Multiscale Modeling and Simulation, 8 (2010), 717-750. doi: 10.1137/09074557X. [8] Q. Cao and P. Lalanne, Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits, Phys. Rev. Lett., 88 (2002), 057403. doi: 10.1103/PhysRevLett.88.057403. [9] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, volume 93 of Applied Mathematical Sciences. Springer-Verlag, Berlin, second edition, 1998. [10] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio and P. A. Wolff, Extraordinary optical transmission through sub-wavelength hole arrays, Letters to Nature, 391 (1998), 667-669. [11] D. Felbacq, Noncommuting limits in homogenization theory of electromagnetic crystals, J. Math. Phys., 43 (2002), 52-55. doi: 10.1063/1.1418013. [12] R. Kohn, J. Lu, B. Schweizer and M. Weinstein, A variational perspective on cloaking by anomalous localized resonance, Comm. Math. Phys. (accepted). [13] P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru and K. D. Möller, One-mode model and airy-like formulae for one-dimensional metallic gratings, Journal of Optics A: Pure and Applied Optics, 2 (2000), 48. doi: 10.1088/1464-4258/2/1/309. [14] P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier and P. Chavel, Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures, Physical Review B, (2003). doi: 10.1103/PhysRevB.68.125404. [15] A. Lamacz and B. Schweizer, Effective maxwell equations in a geometry with flat rings of arbitrary shape, SIAM J. Math. Anal., 45 (2013), 1460-1494. doi: 10.1137/120874321. [16] A. Mary, S. G. Rodrigo, L. Martin-Moreno and F. J. Garcia-Vidal, Holey metal films: From extraordinary transmission to negative-index behavior, Physical Review B, 80 (2009). doi: 10.1103/PhysRevB.80.165431. [17] G. Milton and N.-A. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 3027-3059. doi: 10.1098/rspa.2006.1715. [18] S. O'Brien and J. Pendry, Magnetic activity at infrared frequencies in structured metallic photonic crystals, J. Phys. Condens. Mat., 14 (2002), 6383-6394. [19] J. A. Porto, F. J. Garcia-Vidal and J. B. Pendry, Transmission resonances on metallic gratings with very narrow slits, Phys. Rev. Lett., 83 (1999), 2845-2848. doi: 10.1103/PhysRevLett.83.2845. [20] L. Schwartz, Théorie des Distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X. Nouvelle édition, Entiérement corrigée, Refondue et Augmentée. Hermann, Paris, 1966. [21] T. Vallius, J. Turunen, M. Mansuripur and S. Honkanen, Transmission through single subwavelength apertures in thin metal films and effects of surface plasmons, J. Opt. Soc. Am. A, 21 (2004), 456-463. doi: 10.1364/JOSAA.21.000456.
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