1.The emblematic example of the EOT extraordinary optical transmission (EOT) limitation of classical "macroscopic" grating theories a microscopic pureSPP model of the EOT 2.SPP generation by 1D sub l indentation rigorous calculation (orthogonality relationship)
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1.The emblematic example of the EOT
extraordinary optical transmission (EOT)
limitation of classical "macroscopic" grating theories
a microscopic pureSPP model of the EOT
2.SPP generation by 1D subl indentation
rigorous calculation (orthogonality relationship)
the important example of slit
scaling law with the wavelength
3.The quasicylindrical wave
definition & properties
scaling law with the wavelength
4.Multiple Scattering of SPPs & quasiCWs
definition of scattering coefficients for the quasiCW
How to know how much SPP is generated?
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 backwardpropagating 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.
PL, J.P. Hugonin and J.C. Rodier, PRL 95, 263902 (2005)
0.4
0.3
0.2
0.1
0
0.4
0.3
0.2
0.1
0
extraodinary optical transmission (EOT)
limitation of classical "macroscopic" grating theories
a microscopic pureSPP model of the EOT
2.SPP generation by subl indentation
rigorous calculation (orthogonality relationship)
the important slit example
scaling law with the wavelength
3.The quasicylindrical wave
definition & properties
scaling law with the wavelength
4.Multiple Scattering of SPPs & quasiCWs
definition of scattering coefficients for the quasiCW
Important case for plasmonics
analytical expressions
1) assumption : the nearfield 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, JOSAA23, 1608 (2006).
a2
b2
results obtained for gold
Total SP excitation efficiency
solid curves (model)
marks (calculation)
total SP excitation probability
Question: how is it possible that with a perfect metallic case, ones gets accurate results?
1) assumption : the nearfield 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
Analytical model
n1
a(x)
a+(x)
n2
describe geometrical properties
the SPP excitation peaks at a value w=0.23l 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 slitwidth 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).
1) assumption : the nearfield 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
PL, J.P. Hugonin and J.C. Rodier, JOSAA23, 1608 (2006).
S
S
S = b+ [t0a exp(2ik0neffh)]/ [1r0exp(2ik0neffh)]
PL, J.P. Hugonin and J.C. Rodier, JOSAA23, 1608 (2006).
S
S
S = b+ [t0a exp(2ik0neffh)]/ [1r0exp(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!
S
S
Mention the alpha squared dependence. Modetomode 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+ [t0a exp(2ik0neffh)]/ [1r0exp(2ik0neffh)]
r
r = r +a2exp(2ik0neffh)]/ [1r0exp(2ik0neffh)]
PL, J.P. Hugonin and J.C. Rodier, JOSAA23, 1608 (2006).
extraodinary optical transmission (EOT)
limitation of classical "macroscopic" grating theories
a microscopic pureSPP model of the EOT
2.SPP generation by subl indentation
rigorous calculation (orthogonality relationship)
the important slit example
scaling law with the wavelength
3.The quasicylindrical wave
definition & properties
scaling law with the wavelength
4.Multiple Scattering of SPPs & quasiCWs
definition of scattering coefficients for the quasiCW
 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 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)
(E' H')
Unperturbed system
E' = jwµ0H'
H' = jwe(r)E'
Perturbed system
E = jwµ0H
H = jwe(r)E jwDeE
Es=EE' & Hs=HH'
Es = jwµ0Hs
Hs = jwe(r)Es–jwDeE
(E H)
De
Indentation acts like a volume current source : J= jwDeE
"Easy" task with the reciprocity
Jd(rr0)
c1
c1
Sourcemode reciprocity
c1= ESP(1)(r0)J
c1= ESP(1)(r0)J
provided that the SPP pseudoPoynting flux is normalized to 1
½ dz [ESP(1) HSP(1))]x = 1
ESP = N1/2 exp(ikSPx)exp(igSPz), with Ne1/2/(4we0)
wDeEd(rr0)
c1
c1
What is wrong with that. Simply that the Efield 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 c1 scale.
Then, one may apply the firstorder Born approximation (E = E'Einc /2).
SinceEinc~ 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
e1/2
HSP
l (µm)
(results obtained for gold)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)
e1/2
HSP
l (µm)
(results obtained for gold)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)
e1/2
HSP
l (µm)
(results obtained for gold)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)
e1/2
HSP
l (µm)
(results obtained for gold)
H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)




Local field corrections: small sphere (R<<l)De = eheb
J.D. Jackson, Classical Electrodynamics
eb
+
+
+
eh
E
+
+
incident wave E0
3eb
E = E0 E = E0 for small De only !
De+3eb
Just plug De in perturbation formulas fails for large De.
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)
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)
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)
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)
H'inc= Hinc
d1 l (e')1/2
l'
Hinc
l
G
e
e'
The initial launching of the SPP changes : HSP e1/2