Sub-l metallic surfaces : a microscopic analysis

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 polariton
z
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
3
Surface plasmon polariton
exp(-z/d1) d1l e1/2/2p gtgt l
kSP
z
kSP
Dielectric
x
exp(-z/d2) d2l e-1/2/2p ?cte
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 -scaling law with
the wavelength 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)
SPP of the flat interface
minimum
EOT peak
Two branches
k//
7
The extraordinary optical transmission
2
T ()
1
0
800
640
720
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, 396-402 (1902).
11
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,
  10917 (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 ?
tF2
l-lp
rF
0.2
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.
0.1
0
700
750
800
l (nm)
-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, 16100 (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
rA
L. Martín-Moreno, F. Garcia-Vidal J. Pendry,
Phys. Rev. Lett. 86, 1114 (2001).
tF
rA
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
rA
L. Martín-Moreno, F. Garcia-Vidal J. Pendry,
Phys. Rev. Lett. 86, 1114 (2001).
Reinforce the initial SPP vision
tF
rA
? reinforcement of the initial vision of a SPP
assisted effect
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). Experiment
al 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
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
0
650
700
750
800
l (nm)
18
R(q0,l0)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, 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-bi
nding 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 w1w2wn b(kx) untAn?1 unrBn1
• Bn w1w2wn b(?kx) un1tBn-1 unrAn?1
• cn w1w2wn t(?kx) unaAn-1 un1aBn1
• with un exp(ikSPan) , wn exp(ikxan)

Periodicity is not needed!
22
Microscopic pure-SPP model
tA
rA
tF
only non-resonant quantities
23
Microscopic interpretation
SPP coupled-mode equations (kx0)
Mention the strong difference with slits.
u?1 uexp(ikSPa) ?u -1 slightly larger than
1 t?1 t slightly smaller than 1 resonance
condition Re(kSP)aarg(?) ? 0 modulo 2p
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holes slits

• n real
• exp(2ik0nd)1
• no relation between the pole of tF and that of rA
• FP condition arg(rA)k0nd2mp

• n complex
• exp(2ik0nd) ltlt1
• pole of tF pole of rA (for k0dgtgt1)

25
holes slits
rA
rA
1
30
20
10
0
0
1
1.2
1
1.2
l /a
l /a
26
Microscopic interpretation
SPP coupled-mode equations (kx0)
Mention the strong difference with slits.
u?1 uexp(ikSPa) ?u -1 slightly larger than
1 t?1 t slightly smaller than 1 resonance
condition Re(kSP)aarg(?) ? 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
a0.68 µm
q0
a0.94 µm
0.2
Transmittance
a2.92 µm
0.1
l/a
0

0.95
1
1.05
1.1
1.15
H. Liu P. Lalanne, Nature 452, 448 (2008).