1. Sub-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 Lalanne
INSTITUT d'OPTIQUE, Palaiseau - France
Jean-Paul Hugonin
Haitao Liu (Nankai Univ.)

2. Surface plasmon polaritonz z
exp(-k1z)
Dielectric x x
exp(-k2z)
Metal
SPPs are localized electromagnetic modes/ charge density oscillations at the interfaces, which exponentially decay on both sides w c
edem
ed+em
( )
1/2
kSP=

3. Surface plasmon polariton
exp(-z/d1)
d1=l e1/2/2p >> l
kSP z
kSP
Dielectric x
exp(-z/d2)
d2=l e-1/2/2p cte k
Re(w/c)
wp/2
w=ck
Drude model : |e| l2

4. 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-l indentation
-rigorous calculation (orthogonality relationship)
-slit example
-scaling law with the wavelength
3.The quasi-cylindrical wave
-definition & properties
4.Multiple Scattering of SPPs & quasi-CWs
-definition of scattering coefficients for the quasi-CW

5. The 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).

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

7. The extraordinary optical transmission2
T (%) 1
640
720
800
l (nm)
Black : experiment red : Fano fit
C. Genet et al. Opt. Comm. 225, 331 (2003)

8. The 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).

9. The extraordinary optical transmission=
a grating scattering problem

10. What is learnt from grating theory?
Phil. Mag. 4, (1902).

11. Inductive-capacitive grids
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. McPhedran, G. H. Derrick, and L. C. Botten, Infrared Phys. 23, 239 (1983).

12. Poles and zeros of the scattering matrix
l-lz
tF 
|tF|2
l-lp rF
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.
0.2
0.1
700
l (nm)
750
800
-Global analysis.
-Why does the pole exist? Why does the zero exist? Why are they close or not to the real axis? tF
E. Popov et al., PRB 62, (2000).

13. 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
Resonance-assisted tunneling tA
tF =
tA2 exp(ik0nd)
1- rA2 exp(2ik0nd) rA
L. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001). tF 10 20 30 rA
650
700
750
800
l (nm)

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

15. The surface-mode interpretation
Resonance-assisted tunneling tA
tF =
tA2 exp(ik0nd)
1- rA2 exp(2ik0nd) rA
Reinforce the initial SPP vision
L. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001). tF 10 20 30 rA
 reinforcement of the initial vision of a SPP assisted effect
650
700
750
800
l (nm)

16. 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

17. Weaknesses 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 EOT
Resonance-assisted tunneling
tF =
tA2 exp(ik0nd)
1- rA2 exp(2ik0nd) rA
L. Martín-Moreno, F. Garcia-Vidal & J. Pendry, Phys. Rev. Lett. 86, 1114 (2001). 30 rA
-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 EOT 20 10
650
700
750
800
l (nm)

18. r 
l-lz
l-lp
R(q0,l0)=0
l/30
M. 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 surface
modes, in: A.D. Boardman (Ed.), Electromagnetic Surface Modes, Wiley, NY,
1982, (chapter 17).

19. 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.

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

21. SPP coupled-mode equations
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
Coupled-mode equations
An = w1w2…wn b(kx) + untAn1 + unrBn+1
Bn = w1w2…wn b(kx) + un+1tBn-1 + unrAn1
cn = w1w2…wn t(kx) + unaAn-1 + un+1aBn+1
with un = exp(ikSPan) , wn = exp(ikxan)
Periodicity is not needed!

22. Microscopic pure-SPP modeltA rA tF
tF =
tA2 exp(ik0nd)
1- rA2 exp(2ik0nd)
tA = t +
2ab
u-1 - (r+t)
rA =
2a2
only non-resonant quantities

23. Microscopic interpretation
SPP coupled-mode equations (kx=0)
Mention the strong difference with slits.
|u|1
u=exp(ikSPa) |u |-1 slightly larger than 1
|t|1
t slightly smaller than 1
resonance condition
Re(kSP)a+arg()  0 modulo 2p

24. holes 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)|=1
no relation between the pole of tF and that of rA
FP condition: arg(rA)+k0nd=2mp

25. holes slits tA2 exp(ik0nd) tF = 1- rA2 exp(2ik0nd) |rA| |rA| l /a l /a30 1 20 10 1
1.2 1
1.2
l /a
l /a

26. Microscopic interpretation
SPP coupled-mode equations (kx=0)
Mention the strong difference with slits.
|u|1
u=exp(ikSPa) |u |-1 slightly larger than 1
|t|1
t slightly smaller than 1
resonance condition
Re(kSP)a+arg()  0 modulo 2p

27. Microscopic interpretation
resonance condition :
Re(kSP)a + arg()  kxa (modulo 2p)
the macroscopic surface Bloch mode
superposition of many elementary SPPs scattered by individual hole chains that fly over adjacent chains and sum up constructively

28. Influence of the metal conductivity
RCWA
SPP model
a=0.68 µm
q=0°
a=0.94 µm
0.2
Transmittance
a=2.92 µm
0.1
l/a
0.95 1
1.05
1.1
1.15
H. Liu & P. Lalanne, Nature 452, 448 (2008).