Presentation on theme: "Sub-l metallic surfaces : a microscopic analysis"— Presentation transcript:
1Sub-l metallic surfaces : a microscopic analysis 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.Philippe LalanneINSTITUT d'OPTIQUE, Palaiseau - FranceJean-Paul HugoninHaitao Liu (Nankai Univ.)
2Surface plasmon polariton zzexp(-k1z)Dielectricxxexp(-k2z)MetalSPPs are localized electromagnetic modes/ charge density oscillations at the interfaces, which exponentially decay on both sideswcedemed+em( )1/2kSP=
41.The emblematic example of the EOT -extraordinary optical transmission (EOT)-limitation of classical "macroscopic" grating theories-a microscopic pure-SPP model of the EOT2.SPP generation by 1D sub-l indentation-rigorous calculation (orthogonality relationship)-slit example-scaling law with the wavelength3.The quasi-cylindrical wave-definition & properties4.Multiple Scattering of SPPs & quasi-CWs-definition of scattering coefficients for the quasi-CW
5The extraordinary optical transmission 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.T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998).
6transmittance (w,k//) 1/l (µm-1) k// SPP of the flat interface minimum EOT peakTwo branchesk//
8The extraordinary optical 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.?T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998).
9The extraordinary optical transmission =a grating scattering problem
10What is learnt from grating theory? Phil. Mag. 4, (1902).
11Inductive-capacitive grids Wire grid polarizerInductive-capacitive gridsNearly 100% of the incident energy is transmitted at resonance frequencies for TM polarizationHertz (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).
12Poles and zeros of the scattering matrix l-lztF |tF|2l-lprFThe Fano-type formula is very elegant but does not explain why the pole exist, why it is close or not to the real axis.0.20.1700l (nm)750800-Global analysis.-Why does the pole exist? Why does the zero exist? Why are they close or not to the real axis?tFE. Popov et al., PRB 62, (2000).
13The 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 surfaceResonance-assisted tunnelingtAtF =tA2 exp(ik0nd)1- rA2 exp(2ik0nd)rAL. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001).tF102030rA650700750800l (nm)
14The surface-mode interpretation 1/lSPP of the flat interfaceBlack board :flat SPP versus surface mode of the corrugated surface|Hy|Mode of the perforated interface = pole of rA or tAHybrid characterk//P. Lalanne, J.C. Rodier and J.P. Hugonin , J. Opt. A 7, 422 (2005).
15The surface-mode interpretation Resonance-assisted tunnelingtAtF =tA2 exp(ik0nd)1- rA2 exp(2ik0nd)rAReinforce the initial SPP visionL. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001).tF102030rA reinforcement of the initial vision of a SPP assisted effect650700750800l (nm)
16The surface-mode also exists as l perfectconductorTheory :J. Pendry, L. Martín-Moreno & F. Garcia-Vidal, Science 305, 847 (2004).Experimental verification :P. Hibbins et al., Science 308, 670 (2004)."SPOOF" SPP
17Weaknesses of classical grating theories -The resonance of tF 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 EOTResonance-assisted tunnelingtF =tA2 exp(ik0nd)1- rA2 exp(2ik0nd)rAL. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001).30rA-The resonance of tF 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 EOT2010650700750800l (nm)
18r l-lzl-lpR(q0,l0)=0l/30M. C. Hutley and D. Maystre, “Total absorption of light by a diffraction grating,” Opt. Commun. 19, (1976).D. Maystre, General study of grating anomalies from electromagnetic surfacemodes, in: A.D. Boardman (Ed.), Electromagnetic Surface Modes, Wiley, NY,1982, (chapter 17).
19Indeed 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.
20Microscopic pure-SPP model H. Liu, P. Lalanne, Nature 452, 448 (2008).
21SPP coupled-mode equations Coupling only with the nearest neighborsTight-binding approachNumerical solution consists in solving a linear system with N unknowns (N is the numer of hole rows)If periodic, then it is analyticalCoupled-mode equationsAn = w1w2…wn b(kx) + untAn1 + unrBn+1Bn = w1w2…wn b(kx) + un+1tBn-1 + unrAn1cn = w1w2…wn t(kx) + unaAn-1 + un+1aBn+1with un = exp(ikSPan) , wn = exp(ikxan)Periodicity is not needed!
22Microscopic pure-SPP model tArAtFtF =tA2 exp(ik0nd)1- rA2 exp(2ik0nd)tA = t +2abu-1 - (r+t)rA =2a2only non-resonant quantities
23Microscopic interpretation SPP coupled-mode equations (kx=0)Mention the strong difference with slits.|u|1u=exp(ikSPa) |u |-1 slightly larger than 1|t|1t slightly smaller than 1resonance conditionRe(kSP)a+arg() 0 modulo 2p
24holes slits tA2 exp(ik0nd) tF = 1- rA2 exp(2ik0nd) n complex n real pole of tF = pole of rA (for k0d>>1)n real|exp(2ik0nd)|=1no relation between the pole of tF and that of rAFP condition: arg(rA)+k0nd=2mp
25holes slits tA2 exp(ik0nd) tF = 1- rA2 exp(2ik0nd) |rA| |rA| l /a l /a 301201011.211.2l /al /a
26Microscopic interpretation SPP coupled-mode equations (kx=0)Mention the strong difference with slits.|u|1u=exp(ikSPa) |u |-1 slightly larger than 1|t|1t slightly smaller than 1resonance conditionRe(kSP)a+arg() 0 modulo 2p
27Microscopic interpretation resonance condition :Re(kSP)a + arg() kxa (modulo 2p)the macroscopic surface Bloch modesuperposition of many elementary SPPs scattered by individual hole chains that fly over adjacent chains and sum up constructively
28Influence of the metal conductivity RCWASPP modela=0.68 µmq=0°a=0.94 µm0.2Transmittancea=2.92 µm0.1l/a0.9511.051.11.15H. Liu & P. Lalanne, Nature 452, 448 (2008).