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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)

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1 the emblematic example of the eot extraordinary optical transmission eot

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


1 the emblematic example of the eot extraordinary optical transmission eot

What is due to SPP in the EOT

RCWA

SPP model

a=0.68 µm

q=0°

a=0.94 µm

0.2

Transmittance

a=2.92 µm

0.1

l/a

0

0.95

1

1.05

1.1

1.15


1 the emblematic example of the eot extraordinary optical transmission eot

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).


1 the emblematic example of the eot extraordinary optical transmission eot

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.

l = 940 nm

Dual wave picture

=exp(ikSPx)

 x-m exp(ik0x)

P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).


1 the emblematic example of the eot extraordinary optical transmission eot

l is cylindrical with (1/r) = 940 nm

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).


Young s slit experiment
Young’s slit experiment is cylindrical with (1/r)

L1

F

PBS

λ/2

L2

S

CCD

laser

Au

titanium

glass

N. Kuzmin et al., Opt. Lett. 32, 445 (2007).


Young s slit experiment1
Young’s slit experiment is cylindrical with (1/r)

q

0

10

20

30

40

l=850 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).


Computational results
Computational results is cylindrical with (1/r)

l=0.6 µm

SPP mainly

l=1 µm

Quasi-CW & SPP

l=3 µm

Far-field intensity (a.u.)

l=10 µm

Quasi-CW mainly

Quasi-CW only

PC

q (°)

S. Ravets et al., JOSA B 26, B28 (2009).


1 the emblematic example of the eot extraordinary optical transmission eot

1.The emblematic example of the EOT is cylindrical with (1/r)

-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


1 the emblematic example of the eot extraordinary optical transmission eot

E is cylindrical with (1/r)s, 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 p=pxx+pyy. When is it reliable?

Polarizability tensor

px=(axxEx,inc + axzEz,inc)x

pz=(azxEx,inc+azzEz,inc)z


1 the emblematic example of the eot extraordinary optical transmission eot

E is cylindrical with (1/r)s, 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 p=pxx+pyy.

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)


1 the emblematic example of the eot extraordinary optical transmission eot

E is cylindrical with (1/r)s, 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 p=pxx+pyy.

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)


1 the emblematic example of the eot extraordinary optical transmission eot

The bad scenario is cylindrical with (1/r)


1 the emblematic example of the eot extraordinary optical transmission eot

H (a.u.) is cylindrical with (1/r)

x/l

x/l

l=800 nm, gold

The actual scenario

The two dipole sources approximately generate the same field


1 the emblematic example of the eot extraordinary optical transmission eot

E is cylindrical with (1/r)s, 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 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 :

F(x,y)= [aSPEinc] FSP(x,y) + [aCWEinc] FCW(x,y)


1 the emblematic example of the eot extraordinary optical transmission eot

H is cylindrical with (1/r)SP =

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


1 the emblematic example of the eot extraordinary optical transmission eot

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


1 the emblematic example of the eot extraordinary optical transmission eot

0/ theory in wireless telegraphy with a Hertzian dipole radiating over the earthH. 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)


1 the emblematic example of the eot extraordinary optical transmission eot

Analytical expression for the quasi-CW theory in wireless telegraphy with a Hertzian dipole radiating over the earth

  • 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| >> ed

  • z < l

  • x > l/2p


1 the emblematic example of the eot extraordinary optical transmission eot

quasi-CW for gold at theory in wireless telegraphy with a Hertzian dipole radiating over the earthl=940 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/λ


1 the emblematic example of the eot extraordinary optical transmission eot

Intrinsic properties of quasi-CW theory in wireless telegraphy with a Hertzian dipole radiating over the earth

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 < l

  • x > l/2p

PL et al., Surf. Sci. Rep. (review article under production, 2009)


1 the emblematic example of the eot extraordinary optical transmission eot

Grazing plane-wave field theory in wireless telegraphy with a Hertzian dipole radiating over the earth

Hy,CW Hy,0

Ex,CW = F(x)Ex,0

Ez,CW Ez,0

Hy,CW

z

x

  • linear z-dependence for z < l

  • Main fields are almost null for z (l/2p)|em|1/2

  • nearly an exp(ik0x) x-dependence for x >> l

PL et al., Surf. Sci. Rep. (review article under production, 2009)


1 the emblematic example of the eot extraordinary optical transmission eot

theory in wireless telegraphy with a Hertzian dipole radiating over the earth1/x

1/x3

Closed-form expression for F(x)

100

10-2

|F(x)| (a.u.)

10-4

10-6

silver @ l=1 µm

100

102

x/l

PL et al., Surf. Sci. Rep. (review article under production, 2009)


1 the emblematic example of the eot extraordinary optical transmission eot

Closed-form expression for F(x) theory in wireless telegraphy with a Hertzian dipole radiating over the earth

Highly accurate form for any x

F(x) = exp(ik0x) W[2p(nSP-nd)x/l] (x/l)3/2with W(t) an Erf-like function

Highly accurate for x < 10l

F(x) = exp(ik0x) (x/l)-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

1.The emblematic example of the EOT theory in wireless telegraphy with a Hertzian dipole radiating over the earth

-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


1 the emblematic example of the eot extraordinary optical transmission eot

H theory in wireless telegraphy with a Hertzian dipole radiating over the earthSP

HCW

(x/l)-1/2 (PC)

100

10-1

|H| (a.u.)

|H| (a.u.)

10-2

l=1 µm

l=0.633 µm

10-3

x/l

x/l

100

10-1

|H| (a.u.)

|H| (a.u.)

10-2

l=3 µm

l=9 µm

10-3

100

101

102

100

101

102

x/l

x/l

Scaling law

(result for silver)

PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)


1 the emblematic example of the eot extraordinary optical transmission eot

H theory in wireless telegraphy with a Hertzian dipole radiating over the earthSP

HCW

(x/l)-1/2 (PC)

100

10-1

|H| (a.u.)

|H| (a.u.)

10-2

l=1 µm

l=0.633 µm

10-3

x/l

x/l

100

10-1

|H| (a.u.)

|H| (a.u.)

10-2

l=3 µm

l=9 µm

10-3

|em|

100

101

102

100

101

102

x/l

x/l

2pnd3

Scaling law

(result for silver)

PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)


1 the emblematic example of the eot extraordinary optical transmission eot

1.The emblematic example of the EOT theory in wireless telegraphy with a Hertzian dipole radiating over the earth

-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


1 the emblematic example of the eot extraordinary optical transmission eot

d theory in wireless telegraphy with a Hertzian dipole radiating over the earth

|S/S0|2

d (µm)

Slit-Groove experiment

Fall off for d < 5l

silver

l=852 nm

d

frequency = 1.05 k0

kSP=k0 [1-1/(2eAg)] 1.01k0

|S|2

|S0|2

promote an other model than SPP & quasi-CW (CDEW model)

G. Gay et al. Nature Phys. 2, 262 (2006)


1 the emblematic example of the eot extraordinary optical transmission eot

2 µm theory in wireless telegraphy with a Hertzian dipole radiating over the earth

2 µm

Near field validation

l=975 nm

gold

experiment

G. JULIE, V. MATHET

IEF, Orsay

computation

L. AIGOUY

ESPCI, Paris

slit

slit

TM


1 the emblematic example of the eot extraordinary optical transmission eot

Field recorded on the surface theory in wireless telegraphy with a Hertzian dipole radiating over the earth

E

E//

gold

l=974 nm


1 the emblematic example of the eot extraordinary optical transmission eot

slit theory in wireless telegraphy with a Hertzian dipole radiating over the earth

slit

Ez = ASP sin(kSPx) + Ac [ik0-m/(x+d)] - Ac [ik0+m/(x-d)]

standing SPP

right-traveling

cylindrical wave

left-traveling

cylindrical wave

fitted parameter

ASP (real)

Ac (complex)

(m=0.5)


1 the emblematic example of the eot extraordinary optical transmission eot

total field theory in wireless telegraphy with a Hertzian dipole radiating over the earth

SPP

cylindrical wave

z

x

L. Aigouy et al., PRL 98, 153902 (2007)


1 the emblematic example of the eot extraordinary optical transmission eot

A direct observation? theory in wireless telegraphy with a Hertzian dipole radiating over the earth

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

1.The emblematic example of the EOT theory in wireless telegraphy with a Hertzian dipole radiating over the earth

-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


1 the emblematic example of the eot extraordinary optical transmission eot

CW theory in wireless telegraphy with a Hertzian dipole radiating over the earth

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.

?


Cw to spp cross conversion
CW-to-SPP cross-conversion theory in wireless telegraphy with a Hertzian dipole radiating over the earth

tc

CW

SPP

SPP

rc

CW


Other scattering coefficients
Other scattering coefficients theory in wireless telegraphy with a Hertzian dipole radiating over the earth

bCW

bSP

CW

SPP

aCW

aSP

bCW = bSP

aCW = aSP

X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)


Cross conversion scattering coefficients
Cross-conversion scattering coefficients theory in wireless telegraphy with a Hertzian dipole radiating over the earth

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)


1 the emblematic example of the eot extraordinary optical transmission eot

0.2 theory in wireless telegraphy with a Hertzian dipole radiating over the earth

0.2

|tc|2

|rc|2

0

0

-0.2

-0.2

-0.4

-0.4

0

5

10

0

5

10

Scaling law for rc and tc

rc

tc

|rc|2 & |tc|2

-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)


Other scattering coefficients1
Other scattering coefficients theory in wireless telegraphy with a Hertzian dipole radiating over the earth

bCW

bSP

CW

SPP

aCW

aSP

bCW = bSP

aCW = aSP

X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)


1 the emblematic example of the eot extraordinary optical transmission eot

2 theory in wireless telegraphy with a Hertzian dipole radiating over the eartha2

2ab

rA=

tA= t +

u-1- (r+t)

u-1- (r+t)

Mixed SPP-CW model for the extraordinary optical transmission

pure SPP model

CW

mixed SPP-CW model

2a2

2ab

rA=

tA= t +

SPP

(P-1+1)- (r+t)

(P-1+1)- (r+t)

P = 1/(u-1 - 1) + Sn=1, HCW(na)

H. Liu et al. (submitted)


1 the emblematic example of the eot extraordinary optical transmission eot

Pure SPP-model prediction of the EOT theory in wireless telegraphy with a Hertzian dipole radiating over the earth

RCWA

SPP model

a=0.68 µm

q=0°

a=0.94 µm

0.2

Transmittance

a=2.92 µm

0.1

l/a

0

0.95

1

1.05

1.1

1.15

H. Liu & P. Lalanne, Nature 452, 448 (2008).


1 the emblematic example of the eot extraordinary optical transmission eot

RCWA theory in wireless telegraphy with a Hertzian dipole radiating over the earth

SPP model

SPP-CW model

Mixed SPP-CW model for the extraordinary optical transmission

0.3

a=0.68 µm

a=0.94 µm

0.2

Transmittance

q=0°

a=2.92 µm

0.1

0

l/a

0.95

1

1.05

1.1

1.15


1 the emblematic example of the eot extraordinary optical transmission eot

RCWA theory in wireless telegraphy with a Hertzian dipole radiating over the earth

CW-SPP model

a=900 nm

CW

SPP


1 the emblematic example of the eot extraordinary optical transmission eot

SPP+quasi-CW coupled-mode equations theory in wireless telegraphy with a Hertzian dipole radiating over the earth

It is not necessary to be periodic

It is not necessary to deal with the same indentations

An-1

An

Bn-1

Bn


Conclusion
Conclusion theory in wireless telegraphy with a Hertzian dipole radiating over the earth

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


1 the emblematic example of the eot extraordinary optical transmission eot

Quasi-Spherical Wave theory in wireless telegraphy with a Hertzian dipole radiating over the earth

Quasi-SW

Cylindrical SPP

Cylindrical SPP

SPP, quasi-CW, C-SPP, quasi-SP

ld=.8;k0=2*pi/ld;nh=1;nb=retindice(ld,2);ns=nb;S=[1,0,0,0,0,0];

st=3*ld;st1=1/10*ld;x=[linspace(-st,-st1,31) linspace(st1,st,31)];y=x;z=0;

[e,e_oc,e_res]=retoc([nh,nb,ns],S,[0,0,0],x,y,z,k0,struct('z0_varie',0));

Ez=squeeze(e_oc(1,:,:,3));

figure(1);retcolor(x/ld,y/ld,real(Ez)),shading interp, axis equal

% caxis(caxis/10);

Ez=squeeze(e_res(1,:,:,3));

figure(2);retcolor(x/ld,y/ld,real(Ez)),shading interp,axis equal

% caxis(caxis/10);

figure(3);

st=10*ld;st1=1/10*ld;x=linspace(st1,st,101);y=0;z=0;

[e,e_oc,e_res]=retoc([nh,nb,ns],S,[0,0,0],x,y,z,k0,struct('z0_varie',0));

Ez1=squeeze(e_oc(1,:,:,3));Ez2=squeeze(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 @ l=800 nm