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 -the importance of the
quasi-CW -definition properties -scaling law
with the wavelength 4.Microscopic theory of
sub-l surfaces -definition of scattering
coefficients for the quasi-CW -dual wave picture
microscopic model
2
What is due to SPP in the EOT
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
3
Tableau mention that far away from the surface
the wave is cylindrical with (1/r)1/2
dependence. mention on the surface, it is
different, and this is why we will call that a
quasi-cylindrical wave. Tableau plot a hole,
says that the SPP has a (1/r)1/2 damping with a
exp(ikspr), and that the quasi cylindrical wave
becomes a quasi-spherical wave. But this is
another story.
Dual wave picture
l 940 nm
exp(ikSPx)
? x-m exp(ik0x)
P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556
(2006).
4
Tableau mention that far away from the surface
the wave is cylindrical with (1/r)1/2
dependence. mention on the surface, it is
different, and this is why we will call that a
quasi-cylindrical wave. Tableau plot a hole,
says that the SPP has a (1/r)1/2 damping with a
exp(ikspr), and that the quasi cylindrical wave
becomes a quasi-spherical wave. But this is
another story.
Dual wave picture
exp(ikSPx)
? x-m exp(ik0x)
P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556
(2006).
5
Dual wave picture
Tableau mention that far away from the surface
the wave is cylindrical with (1/r)1/2
dependence. mention on the surface, it is
different, and this is why we will call that a
quasi-cylindrical wave. Tableau plot a hole,
says that the SPP has a (1/r)1/2 damping with a
exp(ikspr), and that the quasi cylindrical wave
becomes a quasi-spherical wave. But this is
another story.
exp(ikSPx)
quasi-
Why calling it a quasi-CW?
P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556
(2006).
6
Youngs slit experiment
L1
F
L2
PBS
?/2
S
CCD
laser
Au
titanium
glass
N. Kuzmin et al., Opt. Lett. 32, 445 (2007).
7
Youngs slit experiment
q
0
10
20
30
40
l850 nm
0
10
20
30
40
q () in air
N. Kuzmin et al., Opt. Lett. 32, 445 (2007). S.
Ravets et al., JOSA B 26, B28 (2009).
8
Computational results
l0.6 µm
SPP mainly
l1 µm
Quasi-CW SPP
l3 µm
Far-field intensity (a.u.)
l10 µm
Quasi-CW mainly
PC
Quasi-CW only
q ()
S. Ravets et al., JOSA B 26, B28 (2009).
9
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 -the importance of the
quasi-CW -definition properties -scaling law
with the wavelength 4.Microscopic theory of
sub-l surfaces -definition of scattering
coefficients for the quasi-CW -dual wave picture
microscopic model
10
Es, Hs scattered field E actual field ??Es
jwµ0Hs ??Hs -jwe(r)Es jwDeE
Hinc
1/Hypothesis the sub-l indentation can be
replaced by an effective dipole ppxxpyy. When
is it reliable? Polarizability
tensor px(axxEx,inc axzEz,inc)x pz(azxEx,inc
azzEz,inc)z
11
Es, Hs scattered field E actual field ??Es
jwµ0Hs ??Hs -jwe(r)Es jwDeE
Hinc
1/Hypothesis the sub-l indentation can be
replaced by an effective dipole ppxxpyy. 2/The
effective dipoles px and py are unknown. They
probably depend on many parameters especially for
sub-l indentation that are not much smaller than
l (such as resonant grooves)
12
Es, Hs scattered field E actual field ??Es
jwµ0Hs ??Hs -jwe(r)Es jwDeE
Hinc
1/Hypothesis the sub-l indentation can be
replaced by an effective dipole ppxxpyy.
2/The effective dipoles px and py are
unknown. 3/ We solve Maxwell's equation for both
dipole source ??E jwµ0H ??H -jwe(r)E (Es,xx
Es,yy) d(x,y)
13
The bad scenario
14
The actual scenario
The two dipole sources approximately generate the
same field
15
Es, Hs scattered field E actual field ??Es
jwµ0Hs ??Hs -jwe(r)Es jwDeE
Hinc
1/Hypothesis the sub-l indentation can be
replaced by an effective dipole ppxxpyy. 2/The
effective dipoles px and py are unknown. 3/ The
field scattered (in the vicinity of the surface
for a given frequency) has always the same shape
F(x,y) aSPEinc FSP(x,y) aCWEinc
FCW(x,y)
16
Analytical expression for the quasi-CW
Cauchy theorem
H HSP HCW
HCW Integral over a single real variable
Dominated by the branch-point singularity
A single pole singularity SPP contribution Two
Branch-cut singularities Gm and Gd
17
Maths have been initially developped for the
transmission theory in wireless telegraphy with a
Hertzian dipole radiating over the earth I.
Zenneck, Propagation of plane electromagnetic
waves along a plane conducting surface and its
bearing on the theory of transmission in wireless
telegraphy, Ann. Phys (1907) 23, 846-866. R.W.P.
King and M.F. Brown, Lateral electromagnetic
waves along plane boundaries a summarizing
approach, Proc. IEEE (1984) 72, 595-611. R. E.
Collin, Hertzian dipole radiating over a lossy
earth or sea some early and late 20th-century
controversies, IEEE Antennas Propag. Mag. (2004)
46, 64-79.
Sommerfeld
18
0/H. J. Lezec and T. Thio, "Diffracted evanescent
wave model for enhanced and suppressed optical
transmission through subwavelength hole arrays",
Opt. Exp. 12, 3629-41 (2004). 1/G. Gay, O.
Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J.
Weiner, H. J. Lezec, "The optical response of
nanostructured surfaces and the composite
diffracted evanescent wave model", Nature Phys.
2, 262-267 (2006). 2/PL and J.P. Hugonin, Nature
Phys. 2, 556 (2006). 3/B. Ung, Y.L. Sheng,
"Optical surface waves over metallo-dielectric
nanostructures", Opt. Express (2008) 16,
9073?9086. 4/Y. Ravel, Y.L. Sheng, "Rigorous
formalism for the transient surface Plasmon
polariton launched by subwavelength slit
scattering", Opt. Express (2008) 16,
21903?21913. 5/W. Dai C. Soukoulis,
"Theoretical analysis of the surface wave along a
metal-dielectric interface", PRB accepted for
publication (private communication). 6/L. Martin
Moreno, F. Garcia-Vidal, SPP4 proceedings
2009. 7/PL, J.P. Hugonin, H. Liu and B. Wang, "A
microscopic view of the electromagnetic
properties of sub-l metallic surfaces", Surf.
Sci. Rep. (review article under proof
corrections, see ArXiv too)
19
Analytical expression for the quasi-CW

• Maxwell's equations
• ??E jwµ0H
• ??H -jwe(r)E (Es,xx Es,yy) d(x,y)
• Analytical solution
• Hz,CW
• cm/emEs,x nd/eSEs,y ? Ex,CW
• Ey,CW
• eS is either ed or em, whether the Dirac source
  is located in the dielectric material or in the
  metallic medium
• Under the Hypothesis that
• em gtgt ed
• z lt l
• x gt l/2p

20
quasi-CW for gold at l940 nm
Independant of the indentation, as long as it is
subwavelength and can be considered as an
effective dipole Independant of the incident
illumination. You could illuminate by a plane
wave at normal incidence, at oblique incidence,
by a SPP, you always get this, just the complex
amplitude is varying!!
dominant
Hy
z
Ex
x
dominant
Ez
0
10
20
30
x/?
21
Intrinsic properties of quasi-CW
Hy,CW
z
x

• 
  Hy,CW Hy,0
• 
  Ex,CW F(x)   Ex,0
• 
  Ez,CW Ez,0
• F(x) is a slowly-varying envelop
• Hy,0, Ex,0, Ez,0 is the normalized field
  associated to the limit case of the reflection
  of a plane-wave at grazing incidence

• Hypothesis
• z lt l
• x gt l/2p

PL et al., Surf. Sci. Rep. (review article under
production, 2009)
22
Grazing plane-wave field
Hy,CW Hy,0 Ex,CW
F(x)   Ex,0 Ez,CW Ez,0
Hy,CW
z
x

• linear z-dependence for z lt l
• Main fields are almost null for z ? (l /2p)
  em1/2
• nearly an exp(ik0x) x-dependence for x gtgt l

PL et al., Surf. Sci. Rep. (review article under
production, 2009)
23
Closed-form expression for F(x)
100
10-2
F(x) (a.u.)
10-4
10-6
silver _at_ l1 µm
102
100
x/l
PL et al., Surf. Sci. Rep. (review article under
production, 2009)
24
Closed-form expression for F(x)
Highly accurate form for any x F(x) exp(ik0x)
W2p(nSP-nd)x/l (x/l)3/2 with W(t) an Erf-like
function
Highly accurate for x lt 10l F(x) exp(ik0x)
(x/l)-m m varies from 0.9 in the visible to 0.5
in the far IR
m0.83 for silver _at_ 852 nm
25
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 -the importance of the
quasi-CW -definition properties -scaling law
with the wavelength 4.Microscopic theory of
sub-l surfaces -definition of scattering
coefficients for the quasi-CW -dual wave picture
microscopic model
26
Scaling law
(result for silver)
PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)
27
Scaling law
(result for silver)
PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)
28
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 -the importance of the
quasi-CW -definition properties -scaling law
with the wavelength -experimental
evidence 4.Microscopic theory of sub-l surfaces
-definition of scattering coefficients for the
quasi-CW -dual wave picture microscopic model
29
Slit-Groove experiment
Fall off for d lt 5l
silver
l852 nm
d
frequency 1.05 k0 kSPk0 1-1/(2eAg) ? 1.01k0
S2
S02
promote an other model than SPP quasi-CW (CDEW
model)
G. Gay et al. Nature Phys. 2, 262 (2006)
30
Near field validation
l975 nm
gold
experiment
G. JULIE, V. MATHET IEF, Orsay
computation
L. AIGOUY ESPCI, Paris
slit
slit
TM
31
Field recorded on the surface
E?
E//
gold
l974 nm
32
Ez  ASP sin(kSPx)  Ac
ik0-m/(xd) - Ac ik0m/(x-d)
standing SPP
right-traveling cylindrical wave
left-traveling cylindrical wave
fitted parameter ASP (real) Ac (complex) (m0.5)
33
total field
SPP
cylindrical wave
z
x
L. Aigouy et al., PRL 98, 153902 (2007)
34
A direct observation?
If one controls the two beam intensity and phase
(or the separation distance) so that there is no
SPP generated on the right side, do I observe a
pure quasi-CW on the right side of the doublet?
35
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 -the importance of the
quasi-CW -definition properties -scaling law
with the wavelength -experimental
evidence 4.Microscopic theory of sub-l surfaces
-definition of scattering coefficients for the
quasi-CW -dual wave picture microscopic model
36
Defining scattering coefficients for CWs
The SPP is a normal mode
It takes almost one year to solve that problem.
SPP
? elastic scattering coefficients like the SPP
transmission may be easily defined. You may also
define inelastic scattering coefficients with
other modes like the radiated plane waves You
may use mode orthogonality and reciprocity
relationships.
?
37
CW-to-SPP cross-conversion
tc
CW
SPP
SPP
rc
CW
38
Other scattering coefficients
bCW
bSP
CW
SPP
aCW
aSP
bCW bSP aCW aSP
X. Yang et al., Phys. Rev. Lett. 102, 153903
(2009)
39
Cross-conversion scattering coefficients
Because the SPP mode and the CW have similar
characteristics at the metal surface (nearly
identical propagation constants, similar
penetration depth in the metal ) The same
causes produce the same effects. We refer here
to a form of causality principle where equal
causes have equal effects
CW
SPP
Ansatz THE SCATTERED FIELDS ARE IDENTICAL. (if
the two waves are normalized so that they have
identical amplitudes at the slit)
X. Yang et al., Phys. Rev. Lett. 102, 153903
(2009)
40
Scaling law for rc and tc
tc
rc2 tc2
rc
-1
Im(rc)
Im(tc)
10
Re(rc)
Re(tc)
-2
10
em-1
-4
10
0
1
10
10
l (µm)
l (µm)
l (µm)
41
Other scattering coefficients
bCW
bSP
CW
SPP
aCW
aSP
bCW bSP aCW aSP
X. Yang et al., Phys. Rev. Lett. 102, 153903
(2009)
42
Mixed SPP-CW model for the extraordinary optical
transmission
pure SPP model
CW
mixed SPP-CW model
2a2
2ab
rA
tA t
SPP
(P-11) - (rt)
(P-11) - (rt)
P 1/(u-1 - 1) Sn1,? HCW(na)
H. Liu et al. (submitted)
43
Pure SPP-model prediction of the EOT
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).
44
Mixed SPP-CW model for the extraordinary optical
transmission
0.3
a0.68 µm
a0.94 µm
0.2
Transmittance
q0
a2.92 µm
0.1
0

0.95
1
1.05
1.1
1.15
l/a
45
RCWA
CW-SPP model
a900 nm
CW
SPP
46
SPPquasi-CW coupled-mode equations
It is not necessary to be periodic It is not
necessary to deal with the same indentations
An-1
An
Bn-1
Bn
47
Conclusion
Two different microscopic waves, the SPP mode and
the quasi-CW, are at the essence of the rich
physics of metallic sub-l surfaces Their
relative weights strongly vary with the metal
permittivity They echange their energy through a
cross-conversion process, whose efficiency scales
as em-1 The local SPP elastic or inelastic
scattering coefficients are essential to
understand the optical properties, since they
apply to both the SPP and the quasi-CW
48
SPP, quasi-CW, C-SPP, quasi-SP
ld.8k02pi/ldnh1nbretindice(ld,2)nsnbS
1,0,0,0,0,0 st3ldst11/10ldxlinspace(-st,
-st1,31) linspace(st1,st,31)yxz0 e,e_oc,e_r
esretoc(nh,nb,ns,S,0,0,0,x,y,z,k0,struct('z0
_varie',0)) Ezsqueeze(e_oc(1,,,3)) figure(1)
retcolor(x/ld,y/ld,real(Ez)),shading interp, axis
equal caxis(caxis/10) Ezsqueeze(e_res(1,,,3)
) figure(2)retcolor(x/ld,y/ld,real(Ez)),shading
interp,axis equal caxis(caxis/10) figure(3) st
10ldst11/10ldxlinspace(st1,st,101)y0z0
e,e_oc,e_resretoc(nh,nb,ns,S,0,0,0,x,y,z,k
0,struct('z0_varie',0)) Ez1squeeze(e_oc(1,,,3)
)Ez2squeeze(e_res(1,,,3)) plot(x/ld,real(Ez1)
,'r','linewidth',3),hold on plot(x/ld,real(Ez2),'g
--','linewidth',3)
E (normal)
y/l
gold _at_ l800 nm