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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) - PowerPoint PPT Presentation
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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- 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- surfaces -definition of scattering coefficients for the quasi-CW-dual wave picture microscopic model
=0°
0.95 1 1.05 1.1 1.15
0
0.1
0.2
a=0.68 µm
a=0.94 µm
a=2.92 µm
Tra
nsm
itta
nce
/a
RCWASPP model
What is due to SPP in the EOT
Dual wave picture
x-m exp(ik0x)
=exp(ikSPx)
= 940 nm
P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).
Dual wave picture
x-m exp(ik0x)
=exp(ikSPx)
= 940 nm
P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).
Dual wave picture
=exp(ikSPx)
= 940 nm
P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).
quasi-
Why calling it a quasi-CW?
Why calling it a quasi-CW?
Young’s slit experiment
laser
F L1 L2PBS S CCDλ/2
titanium
glass
Au
N. Kuzmin et al., Opt. Lett. 32, 445 (2007).
Young’s slit experiment
0 10 20 30 40
0 10 20 30 40
(°) in air
=850 nm
N. Kuzmin et al., Opt. Lett. 32, 445 (2007).S. Ravets et al., JOSA B 26, B28 (2009).
(°)
=0.6 µm
=1 µm
=3 µm
=10 µm
PC
S. Ravets et al., JOSA B 26, B28 (2009).
Computational resultsF
ar-
fiel
d in
ten
sity
(a.
u.)
SPP mainly
Quasi-CW only
Quasi-CW & SPP
Quasi-CW mainly
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- 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- surfaces -definition of scattering coefficients for the quasi-CW-dual wave picture microscopic model
Hinc
Es, Hs= scattered fieldE actual fieldEs = jµ0Hs Hs = -j(r)Es –jE
1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=pxx+pyy. When is it reliable?
Polarizability tensorpx=(xxEx,inc + xzEz,inc)x
pz=(zxEx,inc+zzEz,inc)z
Hinc
Es, Hs= scattered fieldE actual fieldEs = jµ0Hs Hs = -j(r)Es –jE
1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=pxx+pyy.
2/The effective dipoles px and py are unknown. They probably depend on many parameters especially for sub- indentation that are not much smaller than (such as resonant grooves)
Hinc
Es, Hs= scattered fieldE actual fieldEs = jµ0Hs Hs = -j(r)Es –jE
1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=pxx+pyy.
2/The effective dipoles px and py are unknown.
3/ We solve Maxwell's equation for both dipole sourceE = jµ0HH = -j(r)E +(Es,xx+ Es,yy) (x,y)
The bad scenario
=800 nm, gold
x/ x/
H (
a.u
.)The actual scenario
The two dipole sources approximately generate the same field
Hinc
Es, Hs= scattered fieldE actual fieldEs = jµ0Hs Hs = -j(r)Es –jE
1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=pxx+pyy.
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 :
(x,y)= [SPEinc] SP(x,y) + [CWEinc] CW(x,y)
Cauchy theorem
kSP /k0 kSP /k0
m2
md2
d
0
i
d)xikexp()0z,x(H
H = HSP + HCW
)xikexp(k
kSP
dm
md20
2SP
HSP =
HCW= Integral over a single real variable
A single pole singularity : SPP contributionTwo Branch-cut singularities : m and d
Dominated by the branch-point singularity
Analytical expression for the quasi-CW
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, 846866.R.W.P. King and M.F. Brown, Lateral electromagnetic waves along plane boundaries: a summarizing approach, Proc. IEEE (1984) 72, 595611.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, 6479.
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, 90739086. 4/Y. Ravel, Y.L. Sheng, "Rigorous formalism for the transient surface Plasmon polariton launched by subwavelength slit scattering", Opt. Express (2008) 16, 2190321913.5/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- metallic surfaces", Surf. Sci. Rep. (review article under proof corrections, see ArXiv too)
Analytical expression for the quasi-CW
Maxwell's equationsE = jµ0HH = -j(r)E +(Es,xx+ Es,yy) (x,y)
Analytical solution Hz,CW
[m/mEs,x + nd/SEs,y] Ex,CW
Ey,CW
S is either d or m, whether the Dirac source is located in the dielectric material or in the metallic mediumUnder the Hypothesis that
•|m| >> d
•z < •x > /2
Hy
Ex
Ez
x/λ0 10 20 30
z
x
quasi-CW for gold at =940 nm
dominant
dominant
Intrinsic properties of quasi-CW
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 < •x > /2
z
x
Hy,CW
PL et al., Surf. Sci. Rep. (review article under production, 2009)
•linear z-dependence for z < •Main fields are almost null for z ( /2 m
•nearly an exp(ik0x) x-dependence for x >>
Hy,CW Hy,0
Ex,CW = F(x) Ex,0
Ez,CW Ez,0
Grazing plane-wave field
z
x
Hy,CW
PL et al., Surf. Sci. Rep. (review article under production, 2009)
102100
10-2
100
10-4
10-6
|F(x
)| (
a.u
.)
x/
1/x3
1/x
silver @ =1 µm
Closed-form expression for F(x)
PL et al., Surf. Sci. Rep. (review article under production, 2009)
Closed-form expression for F(x)
Highly accurate form for any xF(x) = exp(ik0x) W[2nSP-ndx/] (x/)3/2 with W(t) an Erflike function
Highly accurate for x < 10
F(x) = exp(ik0x) (x/)-m
m varies from 0.9 in the visible to 0.5 in the far IR
m=0.83 for silver @ 852 nm
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- 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- surfaces -definition of scattering coefficients for the quasi-CW-dual wave picture microscopic model
Scaling law
(result for silver)
10-1
10-2
100
10-3
101 102100 101 102100
=0.633 µm =1 µm
=9 µm=3 µm
x/ x/
|H| (
a.u.
)|H
| (a.
u.)
|H| (
a.u.
)|H
| (a.
u.)
x/ x/
10-1
10-2
100
10-3
HSP
HCW
(x/)-1/2 (PC)
PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)
Scaling law
(result for silver)
10-1
10-2
100
10-3
101 102100 101 102100
=0.633 µm =1 µm
=9 µm=3 µm
x/ x/
|H| (
a.u.
)|H
| (a.
u.)
|H| (
a.u.
)|H
| (a.
u.)
x/ x/
10-1
10-2
100
10-3
HSP
HCW
(x/)-1/2 (PC)
PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)
m2nd
3
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- 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- surfaces -definition of scattering coefficients for the quasi-CW-dual wave picture microscopic model
d
|S|2
|S0|2
|S/S
0|2
d (µm)
Slit-Groove experiment
G. Gay et al. Nature Phys. 2, 262 (2006)
d Fall off for d < 5
frequency = 1.05 k0
kSP=k0 [1-1/(2Ag)] 1.01k0
silver
=852 nm
promote an other model than SPP & quasi-CW (CDEW model)
L. AIGOUYESPCI, Paris
2 µm
2 µm
Near field validation
TM
slit slit
G. JULIE, V. MATHETIEF, Orsay
gold experiment
computation
=975 nm
E//E
Field recorded on the surface
=974 nm
gold
Ez = ASP sin(kSPx) + Ac [ik0-m/(x+d)] - Ac [ik0+m/(x-d)]
m0
d)(xx)exp(ik
m0
d)(xx)exp(-ik
standing SPP right-travelingcylindrical wave
left-travelingcylindrical wave
fitted parameterASP (real)Ac (complex)(m=0.5)
slit slit
L. Aigouy et al., PRL 98, 153902 (2007)
total field
cylindrical waveSPP
x
z
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?
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- 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- surfaces -definition of scattering coefficients for the quasi-CW-dual wave picture microscopic model
SPPCW
Defining scattering coefficients for CWs
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.
?
The SPP is a normal mode
CW-to-SPP cross-conversion
CW SPP
tc
CW
SPP
rc
Other scattering coefficients
CW SPP
CW = SP
CW = SP
CW
CW
SP
SP
X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)
Cross-conversion scattering coefficients
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)
Scaling law for rc and tc
-0.4
-0.2
0
0.2
0 5 10 5 100
-0.4
-0.2
0
0.2
100
10110
-4
10-2
10-1
(µm)
|m|1
Im(rc)
Re(rc)
Im(tc)
Re(tc)
tc
(µm) (µm)
|rc|2 & |tc|2
|tc|2
|rc|2
rc
Other scattering coefficients
CW SPP
CW = SP
CW = SP
CW
CW
SP
SP
X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)
Mixed SPP-CW model for the extraordinary optical
transmission
pure SPP model
mixed SPP-CW model
tA = t +2
(P-1+1) (+) rA =
2
(P-1+1) (+)
P = 1/(u-1 - 1) +n=1, HCW(na)
xk( )
SPP
CW
H. Liu et al. (submitted)
tA = t +2
u-1 (+)rA =
22
u-1 (+)
=0°
0.95 1 1.05 1.1 1.15
0
0.1
0.2
a=0.68 µm
a=0.94 µm
a=2.92 µm
Tra
nsm
itta
nce
/a
RCWASPP model
Pure SPP-model prediction of the EOT
H. Liu & P. Lalanne, Nature 452, 448 (2008).
=0°
0.95 1 1.05 1.1 1.15
0
0.1
0.2
0.3
a=0.68 µm
a=0.94 µm
a=2.92 µm
Tra
nsm
itta
nce
/a
RCWASPP modelSPP-CW model
Mixed SPP-CW model for the extraordinary optical
transmission
RCWA CW-SPP model
xk( )
SPP
CW
a=900 nm
SPP+quasi-CW coupled-mode equations
AnBn
An-1Bn-1
It is not necessary to be periodicIt is not necessary to deal with the same indentations
Conclusion
Two different microscopic waves, the SPP mode and the quasi-CW, are at the essence of the rich physics of metallic sub- surfaces
Their relative weights strongly vary with the metal permittivity
They echange their energy through a cross-conversion process, whose efficiency scales as |m|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
gold @ =800 nm
Quasi-SWCylindrical SPP
Quasi-Spherical Wave
Cylindrical SPPy/
E (
nor
mal
)
SPP, quasi-CW, C-SPP, quasi-SP