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Sub- metallic surfaces : a microscopic analysis Philippe Lalanne INSTITUT d'OPTIQUE, Palaiseau - France Haitao Liu (Nankai Univ.) Jean-Paul Hugonin La TEO a été decouverte il y a une dixaine d'annees. Elle est rapidement devenue un phénomène emblématique de la plasmonique qui a suscites des querelles, des polémiques, des passions souvent trop fortes pour expliquer.

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z exp(- 1 z) exp(- 2 z) z x SPPs are localized electromagnetic modes/ charge density oscillations at the interfaces, which exponentially decay on both sides c dmdm d m ( ) 1/2 k SP = x Dielectric Metal Surface plasmon polariton

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Dielectric x Drude model : | 2 z k SP exp(-z/ 1 ) = 1/2 /2 >> Surface plasmon polariton k Re( /c) p/2p/2 =ck k SP exp(-z/ 2 ) = 1/2 /2 cte

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1.The emblematic example of the EOT -extraordinary optical transmission (EOT) -limitation of classical "macroscopic" grating theories -a microscopic pure-SPP model of the EOT 2.SPP generation by 1D sub- indentation -rigorous calculation (orthogonality relationship) -slit example -scaling law with the wavelength 3.The quasi-cylindrical wave -definition & properties -scaling law with the wavelength 4.Multiple Scattering of SPPs & quasi-CWs -definition of scattering coefficients for the quasi-CW

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The extraordinary optical transmission T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998). De quoi s'agit t'il? Avant tout, il s'agit de la transmission a travers une matrice de petits trous (plus petits que ld) qui presente un peak de transmission et un minimum a une longueur d'onde plus petite. De quoi s'agit t'il? Avant tout, il s'agit de la transmission a travers une matrice de petits trous (plus petits que ld) qui presente un peak de transmission et un minimum a une longueur d'onde plus petite.

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1/ (µm -1 ) k // transmittance ( ,k // ) Two branches EOT peak SPP of the flat interface minimum

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T (%) (nm) 0 1 2 800640720 Black : experiment red : Fano fit C. Genet et al. Opt. Comm. 225, 331 (2003) The extraordinary optical transmission

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T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998). -The effect is accompagnied by a field surprising funneling of light at the resonance transmission and of a strong squeezing of light in small volumes. Hhow do we understand that for specific wavelength, no light goes through, and for some other one, more light is transmitted than the light directly impinging onto the hole apertures? The initial vision proposed by ebessen and his coworkers to explain the EOT is the excitation of SPPs on the metallic surface. Since 10 years, there have been a considerable amount of works to explain the EOT, experimental numerical, and theoretical works. I just peak out two beautiful contributions leading to an anlytical formula for the transmission. -The effect is accompagnied by a field surprising funneling of light at the resonance transmission and of a strong squeezing of light in small volumes. Hhow do we understand that for specific wavelength, no light goes through, and for some other one, more light is transmitted than the light directly impinging onto the hole apertures? The initial vision proposed by ebessen and his coworkers to explain the EOT is the excitation of SPPs on the metallic surface. Since 10 years, there have been a considerable amount of works to explain the EOT, experimental numerical, and theoretical works. I just peak out two beautiful contributions leading to an anlytical formula for the transmission. ?

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The extraordinary optical transmission a grating scattering problem =

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Phil. Mag. 4, 396-402 (1902). What is learnt from grating theory?

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Wire grid polarizer Inductive-capacitive grids Hertz (1888) first used a wire grid polarizer for testing the newly discovered radio wave. J.T. Adams and L.C. Botten J. Opt. (Paris) 10, 109–17 (1979). R. Ulrich, K. F. Renk, and L. Genzel, IEEE Trans. Microwave Theory Tech. 11, 363 (1963). C. Compton, R. D. McPhedran, G. H. Derrick, and L. C. Botten, Infrared Phys. 23, 239 (1983). Nearly 100% of the incident energy is transmitted at resonance frequencies for TM polarization

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Poles and zeros of the scattering matrix tFtF 700750800 0 0.1 0.2 (nm) |tF|2|tF|2 E. Popov et al., PRB 62, 16100 (2000). tF tF - z - p rFrF -Global analysis. -Why does the pole exist? Why does the zero exist? Why are they close or not to the real axis? The Fano-type formula is very elegant but does not explain why the pole exist, why it is close or not to the real axis.

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tF =tF = t A 2 exp(ik 0 nd) r A 2 exp(2ik 0 nd) The surface-mode interpretation you assume that inside the grating, the light is transmitted from the top to the bottom interfaces only via the fundamental evanescent mode of the hole array. (Doing that, one obtains a FP formula, where the exponential factor is small because the effective index is imaginary. The resonant transmission is then explained as a resonance of the rA and tA coefficients.) Doing that you obtain a FP-like analytical expression of the transmission coefficient. This is great but not completely satisfactory since the resonance of the transmission coefficient is explained through the resonance of another scattering coefficient. This is something like shifting the real problem. Le probleme avec cette interpretation, c'est que peu de temps apres de nouvelles manips ont montre que la TEO pouvait etre reproduite a des ld bcp plus grandes, ds l'infrarouge puis ds le domaine THz. La le metal peut etre considerer comme quasi parfait et l'interpretation à base de plasmon de surface you assume that inside the grating, the light is transmitted from the top to the bottom interfaces only via the fundamental evanescent mode of the hole array. (Doing that, one obtains a FP formula, where the exponential factor is small because the effective index is imaginary. The resonant transmission is then explained as a resonance of the rA and tA coefficients.) Doing that you obtain a FP-like analytical expression of the transmission coefficient. This is great but not completely satisfactory since the resonance of the transmission coefficient is explained through the resonance of another scattering coefficient. This is something like shifting the real problem. Le probleme avec cette interpretation, c'est que peu de temps apres de nouvelles manips ont montre que la TEO pouvait etre reproduite a des ld bcp plus grandes, ds l'infrarouge puis ds le domaine THz. La le metal peut etre considerer comme quasi parfait et l'interpretation à base de plasmon de surface L. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001). Resonance-assisted tunneling 650700750800 0 10 20 30 (nm) rArA rArA tAtA tFtF

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1/ P. Lalanne, J.C. Rodier and J.P. Hugonin, J. Opt. A 7, 422 (2005). k // Mode of the perforated interface = pole of r A or t A The surface-mode interpretation SPP of the flat interface Black board : flat SPP versus surface mode of the corrugated surface Black board : flat SPP versus surface mode of the corrugated surface |H y | Hybrid character

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tF =tF = t A 2 exp(ik 0 nd) r A 2 exp(2ik 0 nd) The surface-mode interpretation Reinforce the initial SPP vision L. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001). Resonance-assisted tunneling 650700750800 0 10 20 30 (nm) rArA reinforcement of the initial vision of a SPP assisted effect rArA tAtA tFtF

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Theory : J. Pendry, L. Martín-Moreno & F. Garcia- Vidal, Science 305, 847 (2004). Experimental verification : P. Hibbins et al., Science 308, 670 (2004). The surface-mode also exists as perfect conductor "SPOOF" SPP

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tF =tF = t A 2 exp(ik 0 nd) r A 2 exp(2ik 0 nd) Weaknesses of classical grating theories -The resonance of t F is explained by the resonance of another scattering coefficients. -In reality, nothing is known about the waves that are launched in between the hole and that are responsible for the EOT -The resonance of t F is explained by the resonance of another scattering coefficients. This is unsatisfactory. -In reallity, nothing is known about the waves that are launched inbetween the hole and that are responsible for the EOT -The resonance of t F is explained by the resonance of another scattering coefficients. This is unsatisfactory. -In reallity, nothing is known about the waves that are launched inbetween the hole and that are responsible for the EOT L. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001). Resonance-assisted tunneling rArA 650700750800 0 10 20 30 (nm) rArA

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M. C. Hutley and D. Maystre, “Total absorption of light by a diffraction grating,” Opt. Commun. 19, 431-436 (1976). D. Maystre, General study of grating anomalies from electromagnetic surface modes, in: A.D. Boardman (Ed.), Electromagnetic Surface Modes, Wiley, NY, 1982, (chapter 17). /30 R( , )=0 r - z - p

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Indeed what is missing is a microscopic (or mesoscopic) theory, which explicitely considers the excitation of surface modes inbetween the holes, their further scattering with nearby holes, coupling their energy to free space and to the holes themselves. Thinking that way one gets a microscopic of light scattering by metallic gratings. This is in opposition with the usual macroscopic description, where we use global physical quantities, such as plane-wave expansion above and below the grating, and Bloch-mode expansion in the grating region. Indeed what is missing is a microscopic (or mesoscopic) theory, which explicitely considers the excitation of surface modes inbetween the holes, their further scattering with nearby holes, coupling their energy to free space and to the holes themselves. Thinking that way one gets a microscopic of light scattering by metallic gratings. This is in opposition with the usual macroscopic description, where we use global physical quantities, such as plane-wave expansion above and below the grating, and Bloch-mode expansion in the grating region.

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Microscopic pure-SPP model H. Liu, P. Lalanne, Nature 452, 448 (2008).

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SPP coupled-mode equations Coupled-mode equations A n = w 1 w 2 …w n (k x ) + u n A n 1 + u n B n+1 B n = w 1 w 2 …w n ( k x ) + u n+1 B n-1 + u n A n 1 c n = w 1 w 2 …w n t( k x ) + u n A n-1 + u n+1 B n+1 with u n = exp(ik SP a n ), w n = exp(ik x a n ) Periodicity is not needed! Coupling only with the nearest neighbors Tight-binding approach Numerical solution consists in solving a linear system with N unknowns (N is the numer of hole rows) If periodic, then it is analytical Coupling only with the nearest neighbors Tight-binding approach Numerical solution consists in solving a linear system with N unknowns (N is the numer of hole rows) If periodic, then it is analytical

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only non-resonant quantities rArA tAtA tFtF tF =tF = t A 2 exp(ik 0 nd) 1- r A 2 exp(2ik 0 nd) t A = t + 2 u -1 ( + ) rA =rA = 2222 u -1 ( + ) Microscopic pure-SPP model

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|u| 1 u=exp(ik SP a) |u | -1 slightly larger than 1 | | 1 slightly smaller than 1 resonance condition Re(k SP )a+arg( ) 0 modulo 2 SPP coupled-mode equations (k x =0) Microscopic interpretation Mention the strong difference with slits.

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holes slits holes slits tF =tF = t A 2 exp(ik 0 nd) r A 2 exp(2ik 0 nd) n complex |exp(2ik 0 nd)| <<1 pole of t F = pole of r A (for k 0 d>>1) n real |exp(2ik 0 nd)|=1 no relation between the pole of t F and that of r A FP condition: arg(r A )+k 0 nd=2m

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holes slits holes slits tF =tF = t A 2 exp(ik 0 nd) r A 2 exp(2ik 0 nd) 11.2 0 10 20 30 /a |r A | 11.2 0 1 /a |r A |

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|u| 1 u=exp(ik SP a) |u | -1 slightly larger than 1 | | 1 slightly smaller than 1 resonance condition Re(k SP )a+arg( ) 0 modulo 2 SPP coupled-mode equations (k x =0) Microscopic interpretation Mention the strong difference with slits.

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resonance condition : Re(k SP )a + arg( ) k x a (modulo 2 Microscopic interpretation the macroscopic surface Bloch mode superposition of many elementary SPPs scattered by individual hole chains that fly over adjacent chains and sum up constructively

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=0° 0.9511.051.11.15 0 0.1 0.2 a=0.68 µm a=0.94 µm a=2.92 µm Transmittance /a RCWA SPP model Influence of the metal conductivity H. Liu & P. Lalanne, Nature 452, 448 (2008).

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