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 & 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.  doi: 10.1137/0523084.  Google Scholar

[2]

G. Bouchitté and C. Bourel, Multiscale nanorod metamaterials and realizable permittivity tensors,, Commun. in Comput. Phys., 11 (2012), 489.  doi: 10.4208/cicp.171209.110810s.  Google Scholar

[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.  doi: 10.1016/j.crma.2009.02.027.  Google Scholar

[4]

G. Bouchitté and D. Felbacq, Negative refraction in periodic and random photonic crystals,, New J. Phys., 7 (2005).   Google Scholar

[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.  doi: 10.1137/050633147.  Google Scholar

[6]

G. Bouchitté and B. Schweizer, Cloaking of small objects by anomalous localized resonance,, Quart. J. Mech. Appl. Math., 63 (2010), 437.  doi: 10.1093/qjmam/hbq008.  Google Scholar

[7]

G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations with split rings,, SIAM Multiscale Modeling and Simulation, 8 (2010), 717.  doi: 10.1137/09074557X.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.88.057403.  Google Scholar

[9]

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory,, volume 93 of Applied Mathematical Sciences. Springer-Verlag, (1998).   Google Scholar

[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.   Google Scholar

[11]

D. Felbacq, Noncommuting limits in homogenization theory of electromagnetic crystals,, J. Math. Phys., 43 (2002), 52.  doi: 10.1063/1.1418013.  Google Scholar

[12]

R. Kohn, J. Lu, B. Schweizer and M. Weinstein, A variational perspective on cloaking by anomalous localized resonance,, Comm. Math. Phys. (accepted)., ().   Google Scholar

[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).  doi: 10.1088/1464-4258/2/1/309.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1137/120874321.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1098/rspa.2006.1715.  Google Scholar

[18]

S. O'Brien and J. Pendry, Magnetic activity at infrared frequencies in structured metallic photonic crystals,, J. Phys. Condens. Mat., 14 (2002), 6383.   Google Scholar

[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.  doi: 10.1103/PhysRevLett.83.2845.  Google Scholar

[20]

L. Schwartz, Théorie des Distributions,, Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1966).   Google Scholar

[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.  doi: 10.1364/JOSAA.21.000456.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.  doi: 10.1137/0523084.  Google Scholar

[2]

G. Bouchitté and C. Bourel, Multiscale nanorod metamaterials and realizable permittivity tensors,, Commun. in Comput. Phys., 11 (2012), 489.  doi: 10.4208/cicp.171209.110810s.  Google Scholar

[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.  doi: 10.1016/j.crma.2009.02.027.  Google Scholar

[4]

G. Bouchitté and D. Felbacq, Negative refraction in periodic and random photonic crystals,, New J. Phys., 7 (2005).   Google Scholar

[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.  doi: 10.1137/050633147.  Google Scholar

[6]

G. Bouchitté and B. Schweizer, Cloaking of small objects by anomalous localized resonance,, Quart. J. Mech. Appl. Math., 63 (2010), 437.  doi: 10.1093/qjmam/hbq008.  Google Scholar

[7]

G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations with split rings,, SIAM Multiscale Modeling and Simulation, 8 (2010), 717.  doi: 10.1137/09074557X.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.88.057403.  Google Scholar

[9]

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory,, volume 93 of Applied Mathematical Sciences. Springer-Verlag, (1998).   Google Scholar

[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.   Google Scholar

[11]

D. Felbacq, Noncommuting limits in homogenization theory of electromagnetic crystals,, J. Math. Phys., 43 (2002), 52.  doi: 10.1063/1.1418013.  Google Scholar

[12]

R. Kohn, J. Lu, B. Schweizer and M. Weinstein, A variational perspective on cloaking by anomalous localized resonance,, Comm. Math. Phys. (accepted)., ().   Google Scholar

[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).  doi: 10.1088/1464-4258/2/1/309.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1137/120874321.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1098/rspa.2006.1715.  Google Scholar

[18]

S. O'Brien and J. Pendry, Magnetic activity at infrared frequencies in structured metallic photonic crystals,, J. Phys. Condens. Mat., 14 (2002), 6383.   Google Scholar

[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.  doi: 10.1103/PhysRevLett.83.2845.  Google Scholar

[20]

L. Schwartz, Théorie des Distributions,, Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1966).   Google Scholar

[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.  doi: 10.1364/JOSAA.21.000456.  Google Scholar

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