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
l
How to know how much SPP
is generated?
General theoretical formalism
1) make use of the completeness theorem for
the normal modes of waveguides
(z)
Hy= α(x) α(x) HSP(z)  σaσ(x)H(rad)
σ
Ez= α(x) α(x) ESP(z)   aσ(x)E(rad)
(z)
σ
σ
x
2) Use orthogonality of normal modes




dz Hy(x,z) ESP(z) 2 α(x) α(x)

dz Ez(x,z) HSP(z) 2 α(x) α(x)

PL, J.P. Hugonin and J.C. Rodier, PRL 95, 263902 (2005)
General theoretical formalism
x
exp[-Im(kspx)]
General theoretical formalism
applied to gratings
eSP
x

eSP

eSP
x/a
General theoretical formalism
applied to grooves ensembles
• 11-groove optimized SPP coupler with 70% efficiency.
• The incident gaussian beam has been removed.
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.
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.
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
SPP generation
n1
n2
SPP generation
n1
a -(x)
a+(x)
n2
SPP generation
n1
b -(x)
b+(x)
n2
Analytical model
1) assumption : the near-field distribution
in the immediate vicinity of the slit
is weakly dependent on the
dielectric properties
n1
2) Calculate this field for the PC case
a
a+(x)
-(x)
n2
3) Use orthogonality of normal modes



dz Hy(x,z) ESP(z) 2 α(x) α(x)


dz Ez(x,z) HSP(z) 2 α(x) α(x)

Valid only at x = w/2 only!
1/2
wI
 n

ε
l 1

α  α   4 2
 π n1 εn22  1n2 n1 w' I0


PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608 (2006).
probability
excitation
total SPSP
efficiency
excitation
Total
|a|2
results obtained for gold
solid curves (model)
marks (calculation)
Question: how is
it possible that
with a perfect
metallic case,
ones gets
accurate
results?
|b|2
Analytical model
1) assumption : the near-field distribution
in the immediate vicinity of the slit
is weakly dependent on the
dielectric properties
n1
2) Calculate this field for the PC case
a
a+(x)
-(x)
n2
3) Use orthogonality of normal modes
1/2
wI
 n

ε
l 1

α  α   4 2
 π n1 εn22  1n2 n1 w' I0


describe geometrical properties
-the SPP excitation peaks at a value w=0.23l and for visible frequency, |a|2 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).
Analytical model
1) assumption : the near-field distribution
in the immediate vicinity of the slit
is weakly dependent on the
dielectric properties
n1
2) Calculate this field for the PC case
a
a+(x)
-(x)
n2
3) Use orthogonality of normal modes
1/2
wI
 n

ε
l 1

α  α   4 2
 π n1 εn22  1n2 n1 w' I0


describe material properties
-Immersing the sample in a dielectric enhances the SP excitation ( n2/n1)
-The SP excitation probability |a|2 scales as |el|-1/2
PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608 (2006).
a-(x)
a+(x)
b-(x)
b+(x)
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).
groove
S
S
S = b + [t0 a exp(2ik0neffh)] / [1-r0 exp(2ik0neffh)]
3
gold
l=0.8 µm
2
S
l=1.5 µm
1
0
0
0.2
0.4
0.6
0.8
h/l
Can be much larger than the geometric aperture!
Grooves
S
S
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).
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
- All dimensions are scaled by a factor G
- The incident field is unchanged Hinc(l)=Hinc(l')
H'inc= Hinc
r'
l'
Hinc
l
r'/G
e
G
e'
If e=e', the scattered field is unchanged : E'(r')=E(r'/G)
The difficulty comes from the dispersion : e'  e
e'/e = G2 (for Drude metals)
A perturbative analysis fails
Unperturbed system
(E' H')
 E' = jwµ0H'
H' = -jwe(r)E'
Perturbed system
E = jwµ0H
H = -jwe(r)E -jwDeE
(E H)
De
Es=E-E' & Hs=H-H'
Es = jwµ0Hs
Hs = -jwe(r)Es –jwDeE
Indentation acts like a volume current source : J= -jwDeE
A perturbative analysis fails
c1
wDeE d(r-r0)
One may find how c1 or c1 scale.
c1
"Easy" task with the reciprocity
c1
J d(r-r0)
c1
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|e|1/2/(4we0)
A perturbative analysis fails
wDeE d(r-r0)
c1
c1
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
De ~ 1 for dielectric ridges
De ~ e for dielectric grooves
De ~ e for metallic ridges
Numerical verification
0
10
|HSP|
e-1/2
-1
10
0
10
1
l (µm)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)
10
(results obtained for gold)
Numerical verification
0
10
|HSP|
e-1/2
-1
10
0
10
1
l (µm)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)
10
(results obtained for gold)
Numerical verification
0
10
|HSP|
e-1/2
-1
10
0
10
1
l (µm)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)
10
(results obtained for gold)
Numerical verification
0
10
|HSP|
|e|-1/2
-1
10
0
10
1
l (µm)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)
10
(results obtained for gold)
Local field corrections: small
sphere (R<<l)
De  eh-eb
J.D. Jackson, Classical Electrodynamics
eb
-
eh
-
+
+
-
+
E
-
+
-
+
incident wave E0
E=
3e b
E0
De+3eb
E = E0 for small De only !
Just plug De in perturbation formulas fails for large De.
H'inc= Hinc
l'
Hinc
l
e
G
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)
H'inc= Hinc
l'
Hinc
l
e
G
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)
H'inc= Hinc
l'
Hinc
l
e
G
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)
H'inc= Hinc
l'
Hinc
l
e
G
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)
H'inc= Hinc
l'
Hinc
d1  l e')1/2
l
e
G
The initial launching of the SPP changes : HSP 
e'
|e|-1/2