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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)

-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

-definition & properties

-scaling law with the wavelength

4.Multiple Scattering of SPPs & quasi-CWs

-definition of scattering coefficients for the quasi-CW


l

How to know how much SPP is generated?


x

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 backward-propagating 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.

  • 1) make use of the completeness theorem for the normal modes of waveguides

  • Hy= 

  • Ez= 

  • 2) Use orthogonality of normal modes

PL, J.P. Hugonin and J.C. Rodier, PRL 95, 263902 (2005)


x

General theoretical formalism

exp[-Im(kspx)]



General theoretical formalism applied to grooves ensembles
General theoretical formalism applied to grooves ensembles

  • 11-groove optimized SPP coupler with 70% efficiency.

  • The incident gaussian beam has been removed.


General theoretical formalism
General theoretical formalism

0.4

0.3

0.2

0.1

0

  • 11-groove optimized SPP coupler with 70% efficiency.

  • The incident gaussian beam has been removed.


General theoretical formalism1
General theoretical formalism

0.4

0.3

0.2

0.1

0

  • 11-groove optimized SPP coupler with 70% efficiency.

  • The incident gaussian beam has been removed.


1.Why a microscopic analysis?

-extraodinary optical transmission (EOT)

-limitation of classical "macroscopic" grating theories

-a microscopic pure-SPP model of the EOT

2.SPP generation by sub-l indentation

-rigorous calculation (orthogonality relationship)

-the important slit example

-scaling law with the wavelength

3.The quasi-cylindrical wave

-definition & properties

-scaling law with the wavelength

4.Multiple Scattering of SPPs & quasi-CWs

-definition of scattering coefficients for the quasi-CW

Important case for plasmonics

analytical expressions



SPP generation

n1

a-(x)

a+(x)

n2


SPP generation

n1

b -(x)

b+(x)

n2


Analytical model

1) assumption : the near-field 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).


|a|2

|b|2

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 near-field 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, |a|2 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 slit-width 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).


Analytical model

1) assumption : the near-field 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

  • Immersing the sample in a dielectric enhances the SP excitation (n2/n1)

  • The SP excitation probability |a+|2scales as |e(l)|-1/2

PL, J.P. Hugonin and J.C. Rodier, JOSAA23, 1608 (2006).


a-(x)

a+(x)

b-(x)

b+(x)


Grooves

S

S

S = b+ [t0a exp(2ik0neffh)]/ [1-r0exp(2ik0neffh)]

PL, J.P. Hugonin and J.C. Rodier, JOSAA23, 1608 (2006).


groove

S

S

S = b+ [t0a exp(2ik0neffh)]/ [1-r0exp(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!


Grooves

S

S

Mention the alpha squared dependence. Mode-to-mode 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)]/ [1-r0exp(2ik0neffh)]

r

r = r +a2exp(2ik0neffh)]/ [1-r0exp(2ik0neffh)]

PL, J.P. Hugonin and J.C. Rodier, JOSAA23, 1608 (2006).


1.Why a microscopic analysis?

-extraodinary optical transmission (EOT)

-limitation of classical "macroscopic" grating theories

-a microscopic pure-SPP model of the EOT

2.SPP generation by sub-l indentation

-rigorous calculation (orthogonality relationship)

-the important slit example

-scaling law with the wavelength

3.The quasi-cylindrical wave

-definition & properties

-scaling law with the wavelength

4.Multiple Scattering of SPPs & quasi-CWs

-definition of scattering coefficients for the quasi-CW


- 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)


A perturbative analysis fails

(E' H')

Unperturbed system

 E' = jwµ0H'

H' = -jwe(r)E'

Perturbed system

E = jwµ0H

H = -jwe(r)E -jwDeE

Es=E-E' & Hs=H-H'

Es = jwµ0Hs

Hs = -jwe(r)Es–jwDeE

(E H)

De

Indentation acts like a volume current source : J= -jwDeE


A perturbative analysis fails

wDeEd(r-r0)

c1

c1

One may find how c1 or c1 scale.


"Easy" task with the reciprocity

Jd(r-r0)

c1

c1

Source-mode reciprocity

c1= ESP(-1)(r0)J

c1= ESP(1)(r0)J

provided that the SPP pseudo-Poynting flux is normalized to 1

½ dz [ESP(1) HSP(1))]x = 1

ESP = N1/2 exp(ikSPx)exp(igSPz), with N|e|1/2/(4we0)


A perturbative analysis fails

wDeEd(r-r0)

c1

c1

What is wrong with that. Simply that the E-field 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 c1 scale.

Then, one may apply the first-order 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


Numerical verification

e-1/2

|HSP|

l (µm)

(results obtained for gold)

H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)


Numerical verification

e-1/2

|HSP|

l (µm)

(results obtained for gold)

H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)


Numerical verification

e-1/2

|HSP|

l (µm)

(results obtained for gold)

H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)


Numerical verification

|e|-1/2

|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

-

-

-

-

-

Local field corrections: small sphere (R<<l)

De = eh-eb

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  |e|-1/2


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