Pr

1
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) -the important example of
slit -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
2
How to know how much SPP is generated?
3
General theoretical formalism
Tableau Preciser la decomposition modale avec
a(x) qui inclut le exp(ikx). Dire qu'on est dans
la convention Hy,-Ez pour le backward-propagating
mode de Hy,Ez dire que l'orthogonalite n'est pas
definie au sens de Poynting dire que c'est les
champs transverses Hy et Ez qui sont concernes.

• 1) make use of the completeness theorem for the
  normal modes of waveguides
• Hy 
• Ez 
• 2) Use orthogonality of normal modes

PL, J.P. Hugonin and J.C. Rodier, PRL 95, 263902
(2005)
4
General theoretical formalism
exp-Im(kspx)
5
General theoretical formalism applied to gratings
x
eSP
x/a
6
General theoretical formalism applied to grooves
ensembles

• 11-groove optimized SPP coupler with 70
  efficiency.
• The incident gaussian beam has been removed.

7
General theoretical formalism
0.4
0.3
0.2
0.1
0

• 11-groove optimized SPP coupler with 70
  efficiency.
• The incident gaussian beam has been removed.

8
General theoretical formalism
0.4
0.3
0.2
0.1
0

• 11-groove optimized SPP coupler with 70
  efficiency.
• The incident gaussian beam has been removed.

9
1.Why a microscopic analysis? -extraodinary
optical transmission (EOT) -limitation of
classical "macroscopic" grating theories -a
microscopic pure-SPP model of the EOT 2.SPP
generation by sub-l indentation -rigorous
calculation (orthogonality relationship) -the
important 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
Important case for plasmonics analytical
expressions
10
SPP generation
n1
n2
11
SPP generation
n1
a(x)
a -(x)
n2
12
SPP generation
n1
b(x)
b -(x)
n2
13
Analytical model
1) assumption the near-field distribution in
the immediate vicinity of the slit is weakly
dependent on the dielectric properties 2)
Calculate this field for the PC case 3) Use
orthogonality of normal modes
n1
a(x)
a -(x)
n2
Valid only at x ?w/2 only!
PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608
(2006).
14
a2
b2
solid curves (model) marks (calculation)
total SP excitation probability
Question how is it possible that with a perfect
metallic case, ones gets accurate results?
15
Analytical model
n1
a(x)
a -(x)
n2
describe geometrical properties
-the SPP excitation peaks at a value w0.23l and
for visible frequency, a2 can reach 0.5, which
means that of the power coupled out of the slit
half goes into heat EXP. VERIFICATION H.W. Kihm
et al., "Control of surface plasmon efficiency by
slit-width tuning", APL 92, 051115 (2008) S.
Ravets et al., "Surface plasmons in the Young
slit doublet experiment", JOSA B 26, B28 (special
issue plasmonics 2009).
16
Analytical model
1) assumption the near-field distribution in
the immediate vicinity of the slit is weakly
dependent on the dielectric properties 2)
Calculate this field for the PC case 3) Use
orthogonality of normal modes
n1
a(x)
a -(x)
n2
describe material properties

• Immersing the sample in a dielectric enhances the
  SP excitation (? n2/n1)
• The SP excitation probability a2 scales as
  e(l)-1/2

PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608
(2006).
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18
Grooves
S
S
S  b  t0 a exp(2ik0neffh) / 1-r0
exp(2ik0neffh)
PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608
(2006).
19
groove
S
S
S  b  t0 a exp(2ik0neffh) / 1-r0
exp(2ik0neffh)
3
gold
l0.8 µm
2
S
l1.5 µm
1
0
0
0.2
0.4
0.6
0.8
h/l
Can be much larger than the geometric aperture!
20
Grooves
S
S
Mention the alpha squared dependence.
Mode-to-mode reciprocity has been used for that
(dessine au tableau). Mention that is is a weak
process in general, since alpha is always smaller
than 1.
S  b  t0 a exp(2ik0neffh) / 1-r0
exp(2ik0neffh)
r
r  r?  a2 exp(2ik0neffh) / 1-r0
exp(2ik0neffh)
PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608
(2006).
21
1.Why a microscopic analysis? -extraodinary
optical transmission (EOT) -limitation of
classical "macroscopic" grating theories -a
microscopic pure-SPP model of the EOT 2.SPP
generation by sub-l indentation -rigorous
calculation (orthogonality relationship) -the
important 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
22
- All dimensions are scaled by a factor G - The
incident field is unchanged Hinc(l)Hinc(l')
H'inc Hinc
r'
l'
Hinc
r'/G
l
?G
e
e'
If ee', the scattered field is unchanged
E'(r')E(r'/G)
The difficulty comes from the dispersion e' ?
e e'/e G2 (for Drude metals)
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A perturbative analysis fails
(E' H')
Unperturbed system ?? E' jwµ0H' ??H'
-jwe(r)E' Perturbed system ??E jwµ0H ??H
-jwe(r)E -jwDeE EsE-E' HsH-H' ??Es jwµ0Hs
??Hs -jwe(r)Es jwDeE
(E H)
De
Indentation acts like a volume current source
J -jwDeE
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A perturbative analysis fails
wDeE d(r-r0)
c1
c?1
One may find how c1 or c?1 scale.
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"Easy" task with the reciprocity
J d(r-r0)
c1
c?1
Source-mode reciprocity c1 ?ESP(-1)(r0)?J c?1
?ESP(1)(r0)?J provided that the SPP
pseudo-Poynting flux is normalized to 1 ½ ?dz
ESP(1) ? HSP(1))?x 1 ESP N1/2
exp(ikSPx)exp(igSPz), with N?e1/2/(4we0)
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A perturbative analysis fails
wDeE d(r-r0)
c1
c?1
What is wrong with that. Simply that the E-field
on the particle is not the incident field. There
is a local field effect which is very strong for
large De's. This is difficuly in general, and a
solution can be provided only case by case.
One may find how c1 or c?1 scale. Then, one may
apply the first-order Born approximation (E
E'?Einc /2). Since Einc 1, one obtains that
dielectric and metallic indentations scale
differently.
De 1 for dielectric ridges De e for
dielectric grooves De e for metallic ridges
De
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Numerical verification
e-1/2
HSP
l (µm)
(results obtained for gold)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special
issue)
28
Numerical verification
e-1/2
HSP
l (µm)
(results obtained for gold)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special
issue)
29
Numerical verification
e-1/2
HSP
l (µm)
(results obtained for gold)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special
issue)
30
Numerical verification
e-1/2
HSP
l (µm)
(results obtained for gold)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special
issue)
31
Local field corrections small sphere (Rltltl)
J.D. Jackson, Classical Electrodynamics
De eh-eb
eb



eh
E


incident wave E0
3eb
E E0 E E0 for small De
only !
De3eb
Just plug De in perturbation formulas fails for
large De.
32
H'inc Hinc
l'
Hinc
l
?G
e
e'
assumptions the field distribution in the plane
in the immediate vicinity of the slit becomes
independent of the metallic permittivity (as
e??). (the particle dimensions are larger than
the skin depth)
33
H'inc Hinc
l'
Hinc
l
?G
e
e'
assumptions the field distribution in the plane
in the immediate vicinity of the slit becomes
independent of the metallic permittivity (as
e??). (the particle dimensions are larger than
the skin depth)
34
H'inc Hinc
l'
Hinc
l
?G
e
e'
assumptions the field distribution in the plane
in the immediate vicinity of the slit becomes
independent of the metallic permittivity (as
e??). (the particle dimensions are larger than
the skin depth)
35
H'inc Hinc
l'
Hinc
l
?G
e
e'
assumptions the field distribution in the plane
in the immediate vicinity of the slit becomes
independent of the metallic permittivity (as
e??). (the particle dimensions are larger than
the skin depth)
36
H'inc Hinc
d1 ? l (e')1/2
l'
Hinc
l
?G
e
e'
The initial launching of the SPP changes HSP ?
e-1/2