Subl metallic surfaces a microscopic analysis Philippe Lalanne
Sub-l metallic surfaces : a microscopic analysis Philippe Lalanne INSTITUT d'OPTIQUE, Palaiseau - France Jean-Paul Hugonin Haitao Liu (Nankai Univ. )
Surface plasmon polariton z z exp(-k 1 z) Dielectric x x exp(-k 2 z) Metal SPPs are localized electromagnetic modes/ charge density oscillations at the interfaces, which exponentially decay on both sides w k. SP= c ( e de m ed+em 1/2 )
Surface plasmon polariton exp(-z/d 1) d 1=l e 1/2/2 p >> l k. SP z k. SP Dielectric exp(-z/d 2) d 2=l e 1/2/2 p cte Re(w/c) w=ck wp/ 2 Drude model : |e| l 2 k x
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 1 D sub-l 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
The extraordinary optical transmission T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio and P. A. Wolff, Nature 391, 667 (1998).
transmittance (w, k//) 1/l (µm-1) SPP of the flat interface minimum EOT peak Two branches k//
The extraordinary optical transmission T (%) 2 1 0 640 720 l (nm) Black : experiment 800 red : Fano fit C. Genet et al. Opt. Comm. 225, 331 (2003)
The extraordinary optical transmission ? T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio and P. A. Wolff, Nature 391, 667 (1998).
The extraordinary optical transmission = a grating scattering problem
What is learnt from grating theory? Phil. Mag. 4, 396 -402 (1902).
Wire grid polarizer Inductive-capacitive grids Nearly 100% of the incident energy is transmitted at resonance frequencies for TM polarization • 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. Mc. Phedran, G. H. Derrick, and L. C. Botten, Infrared Phys. 23, 239 (1983).
Poles and zeros of the scattering matrix z r. F |t. F|2 l-l t. F l-lp 0. 2 0. 1 0 t. F E. Popov et al. , PRB 62, 16100 (2000). 700 750 l (nm) 800 -Global analysis. -Why does the pole exist? Why does the zero exist? Why are they close or not to the real axis?
The surface-mode interpretation Resonance-assisted tunneling t. A 2 exp(ik 0 nd) t. F = 1 r. A 2 exp(2 ik 0 nd) r. A 30 20 L. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001). t. F r. A 10 0 650 700 750 l (nm) 800
The surface-mode interpretation 1/l SPP of the flat interface |Hy| Mode of the perforated interface = pole of r. A or t. A Hybrid character P. Lalanne, J. C. Rodier and J. P. Hugonin , J. Opt. A 7, 422 (2005). k//
The surface-mode interpretation Resonance-assisted tunneling t. A 2 exp(ik 0 nd) t. F = 1 r. A 2 exp(2 ik 0 nd) r. A 30 20 L. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001). t. F r. A reinforcement of the initial 10 0 650 vision of a SPP assisted effect 700 750 l (nm) 800
The surface-mode also exists as l perfect conductor Theory : J. Pendry, L. Martín-Moreno & F. Garcia. Vidal, Science 305, 847 (2004). Experimental verification : P. Hibbins et al. , Science 308, 670 (2004). "SPOOF" SPP
Weaknesses of classical grating theories Resonance-assisted tunneling t. A 2 exp(ik 0 nd) t. F = 1 r. A 2 exp(2 ik 0 nd) r. A L. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001). r. A 30 20 10 0 650 700 750 l (nm) 800 -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
r l-lz l-lp R(q 0, l 0)=0 l/30 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,
Microscopic pure-SPP model H. Liu, P. Lalanne, Nature 452, 448 (2008).
SPP coupled-mode equations Coupled-mode equations • An = w 1 w 2…wn b(kx) + un An 1 + unr. Bn+1 • Bn = w 1 w 2…wn b( kx) + un+1 Bn-1 + unr. An 1 • cn = w 1 w 2…wn t( kx) + una. An-1 + un+1 a. Bn+1 with un = exp(ik. SPan) , wn = exp(ikxan) Periodicity is not needed!
Microscopic pure-SPP model t. A r. A t. F = t. A 2 exp(ik 0 nd) 1 - r. A 2 exp(2 ik 0 nd) 2 ab t. A = t + -1 u (r+ ) 2 a 2 r. A = -1 u (r+ ) only non-resonant quantities
Microscopic interpretation SPP coupled-mode equations (kx=0) |u| 1 u=exp(ik. SPa) |u |-1 slightly larger than 1 | | 1 slightly smaller than 1 resonance condition Re(k. SP)a+arg( ) 0 modulo 2 p
holes slits t. A 2 exp(ik 0 nd) t. F = 1 r. A 2 exp(2 ik 0 nd) • n complex • |exp(2 ik 0 nd)| <<1 • pole of t. F = pole of r. A (for k 0 d>>1) • n real • |exp(2 ik 0 nd)|=1 • no relation between the pole of t. F and that of r. A FP condition: arg(r. A)+k 0 nd=2 mp
holes slits t. A 2 exp(ik 0 nd) t. F = 1 r. A 2 exp(2 ik 0 nd) |r. A| 1 30 20 10 0 1 l /a 1. 2
Microscopic interpretation SPP coupled-mode equations (kx=0) |u| 1 u=exp(ik. SPa) |u |-1 slightly larger than 1 | | 1 slightly smaller than 1 resonance condition Re(k. SP)a+arg( ) 0 modulo 2 p
Microscopic interpretation resonance condition : Re(k. SP)a + arg( ) kxa (modulo 2 p) the macroscopic surface Bloch mode superposition of many elementary SPPs scattered by individual hole chains that fly over adjacent chains and sum up constructively
Influence of the metal conductivity Transmittance RCWA SPP model a=0. 68 µm q=0° a=0. 94 µm 0. 2 a=2. 92 µm 0. 1 0 0. 95 l/a 1 1. 05 H. Liu & P. Lalanne, Nature 452, 448 (2008). 1. 15
- Slides: 28