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1.The emblematic example of the EOT -extraordinary optical transmission (EOT)

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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)

-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

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

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).

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).

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

L1

F

PBS

λ/2

L2

S

CCD

laser

Au

titanium

glass

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

Young’s slit experiment

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

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)

-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

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

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)

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)

x/l

x/l

l=800 nm, gold

The actual scenario

The two dipole sources approximately generate the same 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)

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

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

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-l metallic surfaces", Surf. Sci. Rep. (review article under proof corrections, see ArXiv too)

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| >> ed
- z < l
- x > l/2p

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

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 < l
- x > l/2p

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

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/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)

Closed-form expression for F(x)

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)

-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

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)

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)

-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

|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)

2 µm

Near field validation

l=975 nm

gold

experiment

G. JULIE, V. MATHET

IEF, Orsay

computation

L. AIGOUY

ESPCI, Paris

slit

slit

TM

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)

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-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

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.

?

Other scattering coefficients

bCW

bSP

CW

SPP

aCW

aSP

bCW = bSP

aCW = aSP

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

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)

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 coefficients

bCW

bSP

CW

SPP

aCW

aSP

bCW = bSP

aCW = aSP

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

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)

Pure SPP-model prediction of 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

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

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

SPP+quasi-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

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

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

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